Instabilities of Highly Anisotropic Spinning Disks

C IPRIAN D. C OMAN

Department of Mathematics, University of Glasgow, University Gardens, Glasgow G12 8QW, Scotland, UK (Received 25 August 20091 accepted 31 August 2009)

Abstract: This work investigates the asymptotic structure of a boundary-value problem proposed recently in connection with in-plane instabilities of spinning disks. Assuming an orthotropic elastic material with cylindrical symmetry we consider a perturbation with respect to the constitutive behavior. The material is assumed to be very stiff in the azimuthal direction, a situation which is commonly encountered in the case of composite flywheels based on hoop-wound carbon fibers in a flexible polyurethane resin. The accuracy of the asymptotic strategy is confirmed by a number of direct computer simulations of the original problem.

Key Words: Boundary layers1 vibrations1 polar orthotropy1 flywheels

1. INTRODUCTION A large class of elastic stability problems involving imperfect mechanical structures can be cast as

1[1] 1 12 [1] 2 f2

(1)

where 1 and 2 are differential operators and the equation is typically solved subject to homogeneous boundary conditions. Here 1 plays the role of a loading parameter, while the arbitrary function f characterizes the degree of imperfection1 the unknown 1 and the right-hand side f are possibly multi-dimensional vector fields. From a purely mathematical point of view, the key to understanding (1) is the generalized eigenproblem that results by setting f 3 0. If 2 2 3 (the identity operator) and the eigenproblem has only a discrete (simple) spectrum consisting of an infinite sequence of numbers 41 j 5 j60 , it is a standard fact that 12

7 1 j20

Mathematics and Mechanics of Solids 16: 3â€“17, 2011 1 The Author(s), 2011. Reprints and permissions: 8 http://www.sagepub.co.uk/journalsPermissions.nav

cj 1j2 1 1 1j

(2)

DOI: 10.1177/1081286509349964

4 C. D. COMAN where 41 j 5 j60 denotes the corresponding set of eigenfunctions and c j 9 1 are uniquely determined by f. In deriving the above expansion we have assumed that the eigenfunctions form a complete set, an issue that is far from trivial when the eigenproblem is not self-adjoint. It is obvious that as 1 1j0 then 1 7, irrespective of how small the right-hand side f is â€“ a clear hallmark of structural instability1 . Books like [1] (see Chapter 1) discuss at length the imperfection approach to elastic stability, but without making recourse to any eigenvalue problem or the spectral representation (2). Rather, the strategy is the other way around: starting from first principles the critical instability thresholds are identified as those values of 1 for which some displacement (or some other mechanical field of interest) becomes unbounded. If 2 2 I the situation becomes more complicated because the eigenproblem might have a non-empty essential spectrum (cf. [2]) and the generalized resolvent 31 1 12 411 is then not amenable to a convenient analytic representation, as was the case in (2). This situation is encountered in stability problems that lack certain compactness properties like, for example, that of a spinning visco/hyperelastic cylinder [3, 4]1 it is also found in certain buckling models derived from three-dimensional elasticity [5] and in the classical problem of finding the natural frequencies for certain thin shell models (e.g., [6]). The present work is concerned with a particular stability situation that falls within the framework of (1). It involves the steady rotation of a flat disk, a problem that is described in many standard texts (e.g., [7, 8]). In these classical treatments the expression of the centrifugal forces acting on the disk ignores the radial displacement, and as a consequence instability is ruled out. Starting with Brunelle [9] it was found that, when the expression of this centrifugal force is modified to account for radial displacements, certain rotational speeds lead to unbounded values for the displacements and stresses in isotropic disks. Relatively recently, Tutuncu [10] revisited this problem within the framework of polarly orthotropic materials and provided numerical estimates for the critical speeds. Portnov et al. [11] studied a variation on the same theme: a polar orthotropic disk with an axis of anisotropy coinciding with the main central axis of inertia but displaced relative to the axis of rotation by a small amount. The investigation was carried out by assuming that the material of the disk is infinitely stiff (i.e. rigid) in the azimuthal direction. A further study by Belov and Portnov [12] relaxed this assumption and presented a range of numerical results that were found to correlate well with the earlier findings of [11]. Our main objective is to clarify the asymptotic structure that was left open in the works of Portnov and his associates [11, 12]. The paper starts with a brief recapitulation of the main equations, which are shown to be reducible to two coupled second-order differential equations with variable coefficients. The outer solution, in the language of matched asymptotics, is discussed in detail in Section 3, where we show that for mode numbers different from unity the perturbation that results from assuming infinite stiffness in the circumferential direction is in fact singular. Prompted by this observation, we then pursue the inner (or boundary-layer) solution in Section 4 and provide some numerical results in Section 5. The paper concludes with a discussion on akin asymptotic phenomena observed in different contexts.

INSTABILITIES OF HIGHLY ANISOTROPIC SPINNING DISKS 5

Figure 1. On the left: top view of an anisotropic disk spinning with angular speed 5 1 the shaded central area represents the cross-section of the rigid shaft perfectly bonded to the disk. On the right: side view of the same configuration.

2. OVERVIEW AND PERTINENT EQUATIONS The stability problem that governs the in-plane rotation of an anisotropic disk is derived from a number of arguments that are well rehearsed in the literature (e.g., [7, 8, 13]). Nevertheless, in order to make the presentation reasonably self-contained, we outline below the equations given in [12]. A sketch of some geometrical features appears in Figure 1: a cylindrical rigid shaft whose cross-section has radius R1 passes through the center of an anisotropic disk of radius R2 and thickness h. The shaft is perfectly bonded to the disk and is rotated around its longitudinal axis with a constant angular velocity 51 possible out-of-plane bending of this configuration is neglected. Letting R1 r R2 and 0 6 7 28 be the polar coordinates in the domain occupied by the disk, the balance of linear momentum for the Cauchy stresses 9 i j (i2 j 9 4r2 65) takes the well-known form

9 rr 1 9 r6 1 39 rr 1 9 66 4 Pr 2 02

r r 6 r

9 r6 1 9 66 2 9 r6 P6 2 0

r r 6 r Pr and P6 represent the radial and azimuthal components, respectively, of the volumetric inertial force whose expressions will be discussed shortly. The strains i j (i2 j 9 4r2 65) are linked to the in-plane displacements u 3 u3r2 64 and 3 3r2 64 through

6 C. D. COMAN

u 2 rr 2

r

66

1 2 r

2

3

u 2

6

r6

1 2 2

2

u

1

6

r r

3

We also recall that the cylindrical orthotropy of the disk material amounts to the generalized Hookeâ€™s law 4 7 6 1 4 7 0 84 7 5 Er 1 E 6 rr 5 8 9 rr 5 8 5 85 8 5 8 5 r 85 8 1 566 8 2 5 1 5 8 0 8 5 8 5 E 8 59 66 8 2 E r 6 5 8 6 9 6 9 5 8 6 9 1 r6 9 r6 0 0 2G r6 with Er , E 6 being the Youngâ€™s moduli in the radial and hoop directions, respectively1 r and 6 are the corresponding Poissonâ€™s ratios, while G r6 denotes the shear modulus characterizing changes of angle between the r - and 6 -directions. These material constants are not independent since the identity Er E 6 2 r 6 must be satisfied as well. Simple manipulations of all these equations (see [12] for full details) result in a fourdimensional system of partial differential equations for the unknown fields 9 rr , 9 r6 , u, and . First, we choose to rescale the governing equations in order to render them non-dimensional. Thus, :2 9 66

r 2 R2

u :2

9 66 :2 2 Er

9 r6

u 2 R2

:2

9 r6 :2 2 Er

2 R2

9 rr :2 2

P r :2

R2 Er

9 rr 2 Er 2

3 Pr 2

P 6 :2

R2 Er

3 P6 2

and after dropping the bars for notational convenience we find 2 2 3 3

9 rr 1 9 r6 11 9 rr 1 2 u Pr 2 02

6

6 2 3 2 3

9 r6 9 rr 2 u 2

9 r6 2 2 P6 2 02

6

6

6 2 3 2 3 2

u 2 02 1 11 9 rr u

6 2 3 1

u

1 1 1 9 r6 2 02

6

(3a)

(3b)

(3c)

(3d)

where :2

E6 2 Er

:2

Er 2 G r6

(4)

INSTABILITIES OF HIGHLY ANISOTROPIC SPINNING DISKS 7 and :2 6 . This system of equations is solved subject to the boundary constraints u32 64 2 32 64 2 0 for 32 64 9 45 [02 2842

(5a)

9 rr 32 64 2 9 r6 32 64 2 0 for 32 64 9 415 [02 2842

(5b)

derived by assuming that the central shaft is rigidly fastened to the disk and the outer rim is traction-free. Following [12], the density of the disk material 3 32 64 is assumed to be uniform across the thickness but is allowed to be variable in the plane of the disk, deviating slightly from its average value m . Next, we expand :2 1 m into a Fourier series 7 1

c 1 n 34 cos3n64 sn 34 sin3n64 2 32 64 2 0 34 2 n21

and look for solutions of (3)â€“(5) in a similar form:

9 rr 32 64

u32 64

9 r6 32 64

32 64

c 7 9 n 34 1

2

cos3n64

s 7 9 n 34 1

sin3n642 u sn 34

c 7 n 34 1 sin3n64 cos3n64 s c n21 n21

n 34

n 34

u cn 34

s 7 n 34 1 n21

2

n21

This device eliminates essentially the dependence on the azimuthal coordinate and thus leaves us with finding the amplitudes 9 in , in , u in , and ni (i 9 4c, s5). The cosine (â€˜câ€™) and sine (â€˜sâ€™) coefficients turn out to satisfy identical systems of equations and thus we can restrict our attention to just one set of functions. Recalling the discussion of (1) in Section 1, we can also leave out any inhomogeneous contributions from Pr or P6 , and this is precisely what we are going to do. It can be shown that (see [12]) Pr 2

7 1

h cn 34

1u cn 34

cos3n64

n21

7 1

h sn 34 1u sn 34 sin3n642

n21

with

c h n s hn

2 :2

5 Er 2

R22

3 cn s n

2 and 1 :2

5 2 R22 Er

3 m

furthermore, the component P6 is not immediately needed in the subsequent analysis. After dropping all sub/superscripts, it becomes clear that the stability of the spinning disk amounts to studying the following system of first-order ordinary differential equations:

8 C. D. COMAN 2

9

11

3

3 2 n n 9 1 1 2 u 1 2 2 02

2 n n 1 2 9 1 2 3u n 4 2 02 2 3 2 u 1 1 1 9 3u n 4 2 02

1

(6a) (6b)

(6c)

1 3nu 4 1 2 0

(6d)

for 7 7 1, and where the dash indicates differentiation with respect to the radial variable. The constraints recorded in (5) become u34 2 34 2 9 314 2 314 2 0

(7)

Our main objective will be to investigate the spectral problem defined by (6)â€“(7) in the asymptotic limit 12

2 43142

n 2 4314

The motivation comes from recent advances in the manufacturing of composite flywheels. Having comparable strengths to their isotropic counterparts, they are considerably lighter and allow much higher speeds of rotation. According to [12], for composites based on carbon fibers in a flexible polyurethane resin, 1 7 103 . The interested reader may want to consult [14] and the references therein for more information regarding the challenges posed by the design and production of such composite flywheels. Rather than work with the system (6) directly, we first reduce it to a coupled set of second-order differential equations with variable coefficients. This is easily achieved by using (6c) and (6d) to express 9 and , respectively, in terms of the corresponding displacements. With these results in hand, (6a) and (6b) are transformed into 11 u 51 34u 61 34 71 34u 81 34 2 02

(8a)

11 52 34u 62 34 72 34u 82 34 2 02

(8b)

where 1 2 2 3 1 n 61 34 3 2 2 2 3 n2 1 71 34 3 1 12 2

51 34 3

52 34 3 1 61 342 62 34 3

1 2

72 34 3 81 342

INSTABILITIES OF HIGHLY ANISOTROPIC SPINNING DISKS 9 2

81 34 3 1

3 2 1 n 2 2

2

82 34 3 1 n

2

3 2 1 1 2 2

and :2 1 1

2

The new boundary conditions are u 2 2 0 for 2 2

(9a)

u 3u n 4 2 02

(9b)

1 3nu 4 2 0 for 2 1

3. OUTER EXPANSIONS Direct numerical simulations of (8)â€“(9) show that, in the limit 1, both u34 and 34 exhibit sharp changes within a small region adjacent to the outer boundary of the disk, 2 1. We expect the behavior of the spinning disk away from that region to be somewhat simpler than the predictions of the fourth-order differential equations satisfied by the two displacement fields. This expectation is confirmed by adopting an ansatz of the form

u

out

2

u 0

0

u 1

1

112

u 2

2

11 2

(10a)

1 2 10 11 112 12 11 2

(10b)

where the functions u j 3 u j 34 and j 3 j 34, as well as the numbers 1 j 9 1 are to be found systematically as explained below. In the interest of clarity, we start by introducing some more notation, largely motivated by the observation that the coefficients that feature in (8) admit expansions in suitable powers of ,

5i 6i 7i 8i

2

0 0 Ci Di

Ai j B 7 i j 1 j20 Ci j Di j

1 j 32 1 i4

0 0 7 1 j20 1j 0

1 j2

3i 2 12 242

10 C. D. COMAN

where 2

3

1 A10 3 2

A20 3 1

B10 3 1A20 2

1 2 3 2 n 1 2 C20 3 1 2 2 2 3 1 1 2 2 D20 3 1 2 n 2

2 C10 3 1

1

n 2

B20 3

n2 2

3

1 2 2

D10 3 C20 2 1 C1 3 1 2

C2 3 nC1 2

D1 3 nC1 2

D2 3 n D1

There is no difficulty in writing down further terms but it turns out that these expressions will suffice for our immediate purposes. On substituting the ansatz (10) in the Equations (8), at leading order we find an algebraic system C1

D1

C2

D2

u0

0

2

0 0

2

(11)

and hence u 0 n 0 2 0

(12)

At 4312 4 the unknowns 3u 1 2 1 4 satisfy a system identical to (11), that is u 1 n 1 2 0 It is the 4314 equations that provide the information needed for the determination of 3u 0 2 0 41 more precisely, taking into account the fact that 2 0 1 11 ( 0 2 1, etc), we discover that

11 4 7 C1 D1 u 2 0 u 0 A10 u 0 B10 0 C10 u 0 D10 0 6 9 21 (13) 11 C2 D2

2 0 A20 u 0 B20 0 C20 u 0 D20 0 Since the coefficient matrix is singular, the existence of 3u 2 2 2 4 demands that the usual solvability condition is met. In plain language, this amounts to the fact that the standard scalar product between the right-hand side of (13) and 3u 0 2 0 4 is null. Together with (12), routine calculations establish that u 0 3 u 0 34 must satisfy

INSTABILITIES OF HIGHLY ANISOTROPIC SPINNING DISKS 11 3 2 1 9out [u 0 ] 3 1 2 u 0 n

3 2 1 1 2 u 0 n 2 3 1 1 2 n 2 1 2 1 10 u 0 2 0 2 n

1

1

(14)

Note that for n 2 1 this collapses into Equation (26) found in [11], u 0

1 u 1POB 0 u 0 2 02 0

10

2 3 1 11 :2 10 1

where POB

denotes the eigenvalue used by Portnov et al. in the reference just mentioned. The general solution of (14) can be expressed in terms of Bessel functions. Indeed, after introducing the auxiliary notation :2 1

1 n 2

and 2 :2

10

312 2

:2

1

3 12 2 1 2 n2 2 1 2 n

(15)

it is straightforward to check that u 0 34 2 K 1 J 34 K 2 Y 342

3K 1 2 K 2 9 14

(16)

The main dilemma in relation to the outer approximation arises when one wants to determine 10 because Equation (14) must be supplemented by appropriate boundary conditions. At the left-end point, 2 , no ambiguity arises â€“ owing to the compatibility between u 0 and

0 expressed by (12). However, if we naively try to use (9b), the following constraints transpire: u 0 314 2 0 and u 0 314 3n 2 1 14u 0 314 2 0

(17)

As u 0 is the solution of a second-order ordinary differential equation which must vanish at 2 , one of the two conditions in (17) must be dropped or they must both be replaced by a new expression involving u 0 314 and/or u 0 314. It is the latter option that applies in our circumstances and, to find out what this new boundary condition is, we need to consider the structure of u 3 u34 and 3 34 in a boundary layer near 2 11 this will be the object of our discussion next. It should be clear that if n 2 1 there is no boundary-layer effect in

12 C. D. COMAN

the original problem because the original boundary conditions unambiguously provide the constraints for u 0 34. Thus, the perturbation with respect to 1 is regular in this case and the ansatz (10) captures the entire perturbation structure of the problem. However, for n 2 1 one has to deal with a singular perturbation. For further reference, let us record here the equation that will supply the 43112 4 correction term in the expansion of the eigenvalue (10b),

9out [u 1 ] 2 111 u 0

(18)

this is found by repeating exactly the same arguments that established (14).

4. THE BOUNDARY LAYER NEAR 2 1 The study of the boundary-layer structure near the outer rim of the disk is facilitated by introducing the stretched variable X 2 4314 defined by 2 1 1 112 X. Expanding the two displacement functions according to

u

inn

2

U0

V0

U1

V1

112

U2

V2

11 2

(19)

we find a sequence of order equations for U j 3 U j 3X4 and V j 3 V j 3X41 generically, they may be cast as d 2U j 1 U j 1 nV j d X2

2 131j4 2

1 d2Vj 1 nU j 1 V j d X2

2 132j4 2

(20a) j 2 02 12 2

(20b)

for some right-hand side 131j4 and 132j4 whose exact expressions are found by performing the appropriate substitutions. Also, with the help of the ansatz (19) the original boundary conditions (9b) yield

dU j 2 U j11 nV j11 2 dX

d Vj 2 1 nU j11 V j11 dX

for

X 2 02

(21)

where U11 2 V11 3 0. Instead of solving the coupled equations (20), we can reduce them to

9inn [U j ] 3

d 4U j d 2U j d 2 11 3 j4 3 j4 2 2 1 3n 14 2 n 1 1 n 1 2 1 d X4 d X2 d X2

and then V j is immediately available from (20a).

3 j4

(22)

INSTABILITIES OF HIGHLY ANISOTROPIC SPINNING DISKS 13

Expressing (10a) in terms of the inner variable X , it follows that U0 3X4 u 0 3142

(23a)

U1 3X4 1u 0 314X u 1 3142

(23b)

1 U2 3X4 u 0 314X 2 1 u 1 314X u 2 3142 2

(23c)

as X 7. We shall use these matching constraints in conjunction with the explicit solutions of (22) to work out the desired boundary condition for the outer problems. Since 9inn [U0 ] 2 0 and dU0 d 3 U0 2 0 for 2 dX d X3

X 2 02

we find U0 3X 4 3 C and V0 3X4 3 1Cn, for some C 9 1 that will be fixed later. At next order, 9inn [U1 ] 2 0 but the boundary conditions are amended according to dU1 2 0 and dX

d V1 1 2 3n 2 1 14C dX n

for

X 2 0

Letting 0 be defined by 2 :2 n 2 1, the functions compatible with these requirements are given by

U1 3X4 2 K 3 exp31 X 4 X K 4 2 V1 3X4 2

1

K 3 3 2 1 14 exp31 X 4 1 K 3 X 1 K 4 2 n

where K 3 2 3n 2 1 14C 3 . Matching the first of these expressions with (23a) supplies C 2 u 0 314, and further use of (23b) eventually leads to u 0 314

n2 1 1 u 0 314 2 02 n 2 1

which represents the missing boundary condition for Equation (14). To gain insight into the structure of U2 , we first use (14) and (24) to establish u 0 314 2

n2 2 n 1 1 1 10 C2 2

and then routine algebraic manipulations indicate that

9inn [U2 ] 2 C3n 2 1 14 32 X 1 54 exp31 X4 1 2 u 0 314 The other function, V2 3X4, can be worked out from

(24)

14 C. D. COMAN 1 d 2 U2 324 V2 3X4 2 1 U2 1 11 2 n d X2 where 1324 1 2 30 1 X4 exp31 X4 2 and 0 :2 1

n2 1 1 2 2 n C2 1 n 2

23n 2 1 14 C2 3 2 2 3 n2 1 1 1 1 2 :2 1 1 11 u 0 314 2 n 2 1 :2

The functions U2 and V2 are somewhat unwieldy and we have found it helpful to employ the symbolic manipulation package 234561 in the interest of brevity we do not record their expressions here. Remembering that these two solutions must satisfy (21), i.e. dU2 2 3n 2 1 14C dX

and

d V2 n2 1 1 21 dX n

2

C K3 K4

3 for

X 2 02

as well as the matching relations (23b) and (23c), we finally get u 1 314

n2 1 1 3n 2 1 142 314 2 u 1 n 2 1 3

2 3 1 1 1 2 C

(25)

This completes all the information we need to fix both 10 and 11 in (10b). Let us point out briefly that matching with the X 2 -term on the right-hand side of (23c) is automatically satisfied (and hence confirms the accuracy of our calculations), while the last term u 2 314 will require an explicit expression for U3 3X4 (available without difficulty at higher orders).

5. NUMERICAL RESULTS The leading-order term in the expansion of 1 as given by (10b) can be determined either by integrating numerically (14) subject to u 0 34 2 0 and the condition (24), or simply by taking advantage of the built-in Bessel functions in software packages like 23456 or 237538. For this latter option the associated determinantal equation must be first expressed in a convenient form1 using standard properties of Bessel functions we have, for example, J 34 Y 34 2 02

J 11 34 1 J 1 34 J 34 Y 11 34 1 Y 1 34 Y 3 4

(26)

INSTABILITIES OF HIGHLY ANISOTROPIC SPINNING DISKS 15 where :2 23n 2 1 143n 2 14. There is no difficulty in finding the lowest positive root of (26), say , whence 2 10 2

2

1 1 n 2

3

(27)

It should be noticed that, after multiplying (18) by , the differential operator that appears on the left-hand side is self-adjoint. Thus, to derive the Fredholm solvability condition that must be enforced for the existence of u 1 , we multiply the aforementioned equation by u 0 34 and integrate the result over [2 1]. Repeated use of the integration by parts formula in conjunction with (24) and (25) gives

1

u 20 34 d

3n 2 1 142 11 2 1 3

2 3 1 1 1 2 u 20 314

(28)

All that remains to be done is to evaluate numerically the integral on the right-hand side since the rest of the quantities that enter in (28) are available1 this is best done using the closedform expression recorded in (16). To summarize, our analysis has yielded the following estimate: 1 10 11 112 4311 4

(29)

To get an idea of the typical agreement between asymptotics and direct numerical simulations of the original eigenproblem, we have included in Figure 2 a sample of comparisons. Although somewhat unfortunately our asymptotic results provide only upper bounds for all n 6 2, the relative errors involved are quite low (less than 0 1%). We do not explore numerically the dependence of 1 on and because results in this direction have already appeared in [12]. Although not immediately obvious from the data included in Figure 2, for 1 1000, 1 10 provides a reasonably accurate approximation on its own. It must be recalled, however, that the outer approximation (14) is of little value without the correct boundary condition at 2 11 this information is only available by having recourse to the matching with the inner solutions. We also wish to remark in passing that both the numerical and the asymptotic results are insensitive to variations in the Poisson ratio 0 7 7 0 5.

6. CONCLUSION A boundary-layer analysis has been presented for a spectral problem that arises in connection with the rotational instability of an anisotropic composite disk for which E6 1 and Er

Er 2 4314 G r6

16 C. D. COMAN

Figure 2. A sample of comparisons between the asymptotic predictions (29) (dotted curves) and the direct numerical simulations of (8)â€“(9) (continuous curves). Here 2 0 1, 2 1 5, and 2 0 3.

As explained in [11, 14], this regime is highly relevant to the recent technological developments regarding hoop-wound composite material disks having elastomeric resin and carbon fibers. Our analysis complements the numerical work reported in [12] and provides a theoretical framework for understanding other boundary-value problems that crop up in similar problems (e.g., [10]). Although our calculations have been limited to capturing the main asymptotic structure, the matching strategy lends itself to an immediate extension to higher orders, should that be required or desired. Stress concentration and singular perturbation problems for strongly anisotropic solids were investigated in the past within the framework of linear elasticity by a number of authors (e.g., [15â€“17]). At least formally, the boundary layer reported in the present investigation has a similar origin. One of the main differences, however, is that in the papers just cited the partial differential equations are posed over rectangular domains and no use was made of separable variables solutions. The connection with those earlier works will be discussed in more detail in a forthcoming study.

INSTABILITIES OF HIGHLY ANISOTROPIC SPINNING DISKS 17

NOTE 1. The usual Fredholm solvability condition is not invoked here because we have in mind arbitrary righthand sides in (1)1 while it may be true that in some instances c j0 2 0 and hence remains bounded as 1 1 j0 , this is an exceptional situation that we choose to ignore in the present context.

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

Timoshenko, S. P. and Gere, J. M. Theory of Elastic Stability, McGraw-Hill, New York, 1961. Sanchez-Hubert, J. and Sanchez-Palencia, E. Vibration and Coupling of Continuous Systems, Springer-Verlag, New York, 1989. Oden, J. T. and Lin, T. T. On the general rolling contact problem for finite deformations of a viscoelastic cylinder. Computer Methods in Applied Mechanics and Engineering, 57, 297–367 (1986). Rabier, P. J. and Oden, J. T. Bifurcation in Rotating Bodies, Mason/Springer-Verlag, Paris, 1989. Szabo, B. and Kiralyfalvi, G. Linear models of buckling and stress-stiffening. Computer Methods in Applied Mechanics and Engineering, 171, 43–59 (1999). Gol’denveizer, A. L. Qualitative analysis of free vibrations of an elastic thin shell. Journal of Applied Mathematics and Mechanics, 30, 94–108 (1965). Lekhnitskii, S. G. Anisotropic Plates, Gordon & Breach Science Publishers, New York, 1968. Soedel, W. Vibrations of Shells and Plates, Marcel Dekker, New York, 1981. Brunelle, E. I. Stress redistribution and instability of rotating beams and disks. AIAA Journal, 9, 758–759 (1971). Tutuncu, N. Effect of anisotropy on inertio-elastic instability of rotating disks. International Journal of Solids & Structures, 37, 7609–7616 (2000). Portnov, G, G., Ochan, M. Y. and Bakis, C. C. Critical state of imbalanced rotating anisotropic disks with small radial and shear moduli. International Journal of Solids & Structures, 40, 5219–5227 (2003). Belov, M. and Portnov, G, G. Rotation stability of anisotropic disks. Mechanics of Composite Materials, 39, 245– 254 (2003). Roseau, M. Vibrations in Mechanical Systems, Springer-Verlag, Berlin, 1987. Gabrys, C. W. and Bakis, C. E. Design and manufacturing of filament wound elastomeric matrix composite flywheels. Journal of Reinforced Plastics and Composites, 16, 488–502 (1997). Everstine, G. C. and Pipkin, A. C. Boundary layers in fibre-reinforced materials. ASME Journal of Applied Mechanics, 40, 518–522 (1973). Morland, L. W. A plane theory of inextensible transversely isotropic elastic composites. International Journal of Solids and Structures, 9, 1501–1518 (1973). Spencer, A. J. M. Boundary layers in highly anisotropic plane elasticity. International Journal of Solids and Structures, 10, 1103–1123 (1974).

Erratum

The publishers would like to apologize for the errors that occurred in the paper ‘Non-Linear Elastic Bodies Exhibiting Limiting Small Strain’ by K R Rajagopal Mathematics and Mechanics of Solids first published on February 3, 2010 as doi:10.1177/1081286509357272 Page 4: Lines 20–22 should read “We note that when T 1 01 1 1 02 For the purpose of illustration, we shall set 3 1 01 and only consider cases wherein trT is non-negative.” instead of “For the sake of ease of illustration we shall set 3 to be unity. We note that when T 1 0 1 1 02 For the purpose of illustration, as we shall only consider cases wherein is non-negative, we shall set to be zero.”

Mathematics and Mechanics of Solids 16: 140, 2011 1 The Author(s), 2011. Reprints and permissions: 2 http://www.sagepub.co.uk/journalsPermissions.nav

DOI: 10.1177/1081286510365555

A Note on Existence Result for Viscoplastic Models with Nonlinear Hardening

S ERGIY N ESENENKO

Department of Mathematics, Darmstadt University of Technology, Schlossgartenstrasse 7, D-64289 Darmstadt, Germany (Received 2 September 20081 accepted 2 February 2009)

Abstract: In the recent work of H.-D. Alber and K. Chelminski (Mathematical Models and Methods in the Applied Sciences, 17, 189â€“213, 2007) the existence of the solutions to a model of inelastic (viscoplastic) behavior of materials at small strain is derived. In this work we show that the conditions of the existence theorem of Alber and Chelminski can be relaxed and the same result can be proved under less restrictive assumptions. The relaxation of the conditions of the existence theorem of Alber and Chelminski (2007) makes it possible to answer the question raised by them concerning the solvability of the model of nonlinear kinematic hardening without assuming a higher exponent in the constitutive law for one of the internal variables than the exponent in the constitutive law for the other one.

Key Words: Existence, plasticity, viscoplasticity, maximal monotone operator, general duality principle, degenerate equation

1. SETTING OF THE PROBLEM Alber and Chelminski [1] studied the existence of solutions to the initial boundary value problem modeling the behaviour of viscoplastic materials at small strains. The problem is formulated as follows. Let 1 1 13 denote the set of material points of the body. 1 3 denotes the space of symmetric 3 2 3-matrices. One searches the displacement u2x3 t4 3 13 , the Cauchy stress tensor T 2x3 t4 3 1 3 and the vector of internal variables z2x3 t4 4 25 p 2x3 t43 z5 2x3 t44 3 1 3 2 1 N 66 of the following model equations: 6divx T 2x3 t4 4 b2x3 t43 2 1 T 2x3 t4 4 2 527 x u2x3 t44 6 5 p 2x3 t4 3

Mathematics and Mechanics of Solids 16: 141â€“154, 2011 1 The Author(s), 2011. Reprints and permissions: 8 http://www.sagepub.co.uk/journalsPermissions.nav

(1) (2)

DOI: 10.1177/1081286509103818

142 S. NESENENKO 3 4 6t 5 p 2x3 t4 4 g1 T 2x3 t43 65z 2x3 t4 3

(3)

3 4 6t z5 2x3 t4 4 g2 T 2x3 t43 65z 2x3 t4 3

(4)

with the initial condition 5 p 2x3 04 4 5204 p 2x43

z5 2x3 04 4 z5 204 2x4

(5)

and with the Dirichlet boundary condition u2x3 t4 4 7 2x3 t43 2x3 t4 3 61 2 [03 948

(6)

The term 527 x u2x3 t44 in these equations denotes the symmetric 3 2 3-matrix 527 x u2x3 t44 4

4 13 7 x u2x3 t4 27 x u2x3 t44T 3 1 3 3 2

the strain tensor. We denote by 2 : 1 3 1 3 a linear, symmetric, positive definite mapping, the elasticity tensor. The functions b : 1 2 [03 94 13 is the volume force and 7 : 61 2 [03 94 IR3 is the boundary data. The functions g1 : 1 3 2 1 N 66 1 3 and g1 : 1 3 2 1 N 66 1 N 66 are given such that 2T3 y4 2g1 2T3 y43 g2 2T3 y44 : 1 N 1 N is a monotone mapping. Functional spaces. Let 1 be an open bounded set with C 1 -boundary 61. Te denotes a positive number (time of existence) and for 0 9 t 9 Te 1 t 4 1 2 203 t43

Q 4 1 2 203 Te 48

We denote the Banach space of Lebesgue integrable with the power p together with their weak derivatives up to the order m functions by W m3 p 21 3 1 N 4. The norm in W m3 p 21 3 1 N 4 is denoted by m3 p31 . We choose the numbers p3 q satisfying 1 9 p3 q 9 9 and 1 p 1 q 4 18 For such p and q one can define the bilinear form on the product space L p 21 3 1 N 42 L q 21 3 1 N 4 by 5 2 3 41 4

1

2x4 2x4dx8

A NOTE ON EXISTENCE RESULT 143 We define another bilinear form on L p 21 3 1 N 42L q 21 3 1 N 4 by [ 3 ]1 4 22 3 41 8 Now we recall the definition of an evolution triple. Definition 1.1. By an â€œevolution tripleâ€? (known as â€œGelfand tripleâ€? or â€œspaces in normal positionâ€?), we mean a triple of spaces X 1 H 1 X such that (a) H is a separable Hilbert space identified with its dual1 (b) X is a separable reflexive Banach space1 (c) the embedding i : X H is continuous and i2X 4 is dense in H (note that i 2H 4 is also dense in X ). If 2X3 H3 X 4 is an evolution triple and 1 9 p3 q 9 9, 1 p 1 q 4 1, then 6 7 W p3q 203 Te 3 X 4 4 u 3 L p 203 Te 3 X 4 u 3 L q 203 Te 3 X 4 is a separable reflexive Banach space furnished with the norm

u 2W p3q 4 u 2L p 203Te 3X4 u

2L q 203Te 3X 4 3 where the time derivative of u2 4 is understood in the sense of vector-valued distributions. We recall that the embedding W p3q 203 Te 3 X 4 1 C2[03 Te ]3 H 4 is continuous ([2], p. 4, for instance). Main result. Following [1] we define the operator 3 : F2Q3 1 3 4 F2Q3 1 3 4, where F2Q3 1 3 4 denotes the set of all function from Q to 1 3 , by the following rule. Let 2h3 z5 4 be a solution of the problem 3 4 h2x3 t4 4 g1 T 2x3 t43 65z 2x3 t4 3

(7)

3 4 6t z5 2x3 t4 4 g2 T 2x3 t43 65z 2x3 t4 3

(8)

z5 2x3 04 4 z5 204 2x43 for given z5 204 and T and 2x3 t4 3 Q. Then the operator 3 on F2Q3 1 3 4 is given by

32T 4 4 h8 In terms of the operator 3 the problem (1)â€“(6) can be written as follows:

(9)

144 S. NESENENKO 6divx T 2x3 t4 4 b2x3 t43 2 1 T 2x3 t4 4 2 527 x u2x3 t44 6 5 p 2x3 t4 3

(10) (11)

6t 5 p 2x3 t4 4 32T 43

(12)

5 p 2x3 04 4 5204 p 2x43

(13)

u2x3 t4 4 7 2x3 t43

2x3 t4 3 61 2 [03 948

(14)

Now we state the existence result for the problem (10)â€“(14). Theorem 1.1. Let 2 p 9 9 and 1 9 q 2 be numbers with 1 p 1 q 4 1. Assume that 3 : L p 2Q3 1 3 4 L q 2Q3 1 3 4 satisfies (a) 3 is demicontinuous and monotone1 (b) 3 enjoys the growth condition

32T 4 q3Q C21 T p3Q 4 p q

(c) the inverse 361 is strongly coercive, i.e. 3 9 as 93 3 361 2 48

2 3 Suppose that b 3 L p 2Q3 13 4, 7 3 L p 203 Te 3 W 13 p 21 3 13 44 and 5204 p 3 L 21 3 1 4. Then there exists a solution

u 3 L p 203 Te W 13 p 21 3 13 443 T 3 L p 203 Te L p 21 3 1 3 443 5 p 3 W p3q 203 Te 3 L p 21 3 1 3 44 of the problem (10)â€“(14). We note that monotonicity of 3 is implied by monotonicity of the mapping 2T3 y4 2g1 2T3 y43 g2 2T3 y44 (see Lemma 4.1 of [1]). In [1] Theorem 1.1 is proved under the additional assumption that the operator 3 is coercive. This assumption caused difficulties in the derivation of the existence of the solutions to the model of nonlinear kinematic hardening. Since to show that the operator 3 defined by the constitutive relations (specific choice of the functions g1 and g2 ) used for modeling of nonlinear kinematic hardening is coercive the authors of [1] had to impose the restriction on the exponents in the constitutive relations for the different internal variables (see Section 5). Our approach is actually based on the constructions from [1] and repeats main steps of that work with only one difference that we use the general duality principle for the sum of two operators from [3] to derive the existence of the solutions to the problem (10)â€“(14). The application of this duality principle gives us the possibility to avoid the coercivity assumption on 3.

A NOTE ON EXISTENCE RESULT 145

2. THE HELMHOLTZ PROJECTION ON TENSOR FIELDS We borrow the material for this section from [1]. Therefore we state only the main results presented there without going into detail and for further reading we refer the reader to that work. In this work we need projection operators to spaces of tensor fields, which are symmetric gradients and to spaces of tensor fields with vanishing divergence. We recall [4] that a Dirichlet boundary value problem from the linear elasticity theory formed by equations 6divx T 2x4 4 b2x43

x 3 13

T 2x4 4 22527 x u2x44 6 5 p 2x443 u2x4 4 7 2x43

x 3 613

(15) x 3 13

(16) (17)

to given b 3 W 613q 21 3 13 4, 5 p 3 L p 21 3 1 3 4 and 7 3 W 13 p 21 3 13 4 has a unique weak solution 2u3 T 4 3 W 13 p 21 3 13 4 2 L p 21 3 1 3 4 with 1 9 p 9 9 and 1 p 1 q 4 1. For b 4 7 4 0 the solution of (15)â€“(17) satisfies the inequality

527 x u4 p31 C 5 p p31 with some positive constant C. Definition 2.1. For every 5 p 3 L p 21 3 1 3 4 we define a linear operator Pp : L p 21 3 1 3 4 L p 21 3 1 3 4 by Pp 5 p 4 527 x u43 13 p

where u 3 W0 21 3 IR3 4 is a unique weak solution of (15)â€“(17) to the given function 5 p and b 4 7 4 0. A subset 4 p of L p 21 3 1 3 4 is defined by

4 p 4 527 x u4 u 3 W013 p 21 3 IR3 48 The following lemma gives the main properties of Pp . Lemma 2.1. For every 1 9 p 9 9 the operator Pp is a bounded projector onto the subset 4 p of L p 21 3 1 3 4. The projector 2Pp 4 adjoint with respect to the bilinear form [ 3 ]1 on L p 21 3 1 3 4 2 L q 21 3 1 3 4 satisfy 2Pp 4 4 Pq 3

where

1 1

4 18 p q

146 S. NESENENKO p with This implies ker2Pp 4 4 Hsol p 4 3 L p 21 3 1 3 4 [ 3 ]1 4 0 for all 3 4 q 8 Hsol

Since 2 is symmetric, the relation [ 3 ]1 4 0 holds for all 3 4 q if and only if 22 3 7 x 41 4 22 3 527 x 441 4 [ 3 527 x 4]1 4 0 13q

for all 3 W0 21 3 IR3 4. Consequently Hsol 4 3 L p 21 3 1 3 4 div22 4 4 08 p

Therefore the projection operator Q p 4 2I 6 Pp 4 : L p 21 3 1 3 4 L p 21 3 1 3 4 p is a generalization of the classical Helmholtz projection. with Q p 2L p 21 3 1 3 44 4 Hsol

Corollary 2.1. Let 22 Pp 4T be the operator adjoint to

2 Pp : L p 21 3 IR N 4 L p 21 3 IR N 4 with respect to the bilinear form 2 3 41 on the product space L p 21 3 IR N 42L q 21 3 IR N 4. Then 22 Pp 4T 4 2 Pq : L q 21 3 IR N 4 L q 21 3 IR N 48 Moreover, the operator 2 Q 2 is non-negative and self-adjoint. The last result in this corollary is proved in [5].

3. MAXIMAL MONOTONIC OPERATORS In this section we present our necessary tools for the construction of the existence theory for the problem (10)â€“(14), which will be used in the next section. Let V be a reflexive Banach space with the norm , V be its dual space with the norm . The brackets 3 denotes the dual pairing between V and V . Under V we shall always mean a reflexive Banach space throughout this section. For a multivalued mapping A : V 2V the sets D2A4 4 3 V A 4

A NOTE ON EXISTENCE RESULT 147

and Gr A 4 [ 3 ] 3 V 2 V 3 D2A43

3 A

are called the effective domain and the graph of A, respectively.

Definition 3.1. The mapping A : V 2V is called monotone if the inequality 6 u 3 6 u 0 holds for all [u3 u ]3 [ 3 ] 3 Gr A. The mapping A : V 2V is called maximally monotone iff the inequality 6 u 3 6 u 0 24 [u3 u ] 3 Gr A implies [ 3 ] 3 Gr A. The mapping A : V 2V is called strongly coercive iff either D2A4 is bounded or D2A4 is unbounded and the condition 3 6 9 as 93

[ 3 ] 3 Gr A3

is satisfied for each 3 D2A4. It is well known ([6, p. 105]) that if A is a maximal monotonic operator, then, for any 3 D2A4, the image A is closed convex subset of V and the graph Gr A is demiclosed. (A set A 3 V 2 V is demiclosed if n converges strongly to 0 in V and n converges weakly to 0 in V (or n converges weakly to 0 in V and n converges strongly to 0 in V ) and [ n 3 n ] 3 Gr A, then [ 3 ] 3 Gr A). Remark 3.1. We recall that the subdifferential of a lower semi-continuous and convex function is maximal monotone (Theorem 2.25 [7, p. 27]). The next theorem gives the sufficient conditions for a monotone operator to be maximal.

Theorem 3.1. Let A : V 2V be a monotone mapping such that for each 3 V the image is a non-empty closed convex subset of V . If A is continuous from the line segment in V to V in the weak topology, then A is maximally monotone. Proof. See Theorem 2.3 [6, p. 106] or Theorem 1.33 [8, p. 309].

1

We also need the following result on the maximality of the sum of two maximal monotonic operators.

148 S. NESENENKO

Theorem 3.2. Let V be a reflexive Banach space, and let A and B be maximally monotone. Suppose that the condition intD2A4 D2B4 4 is fulfilled. The the sum A B is a maximal monotone operator.

1

Proof. See Theorem III.3.6 in [6] or Theorem II.1.7 in [9].

For deeper results on the maximality of the sum of two maximal monotone operators we refer the reader to [10], see also [11]. The next surjectivity result on maximal monotone operators is the key tool in the proof of our main existence result.

Theorem 3.3. If V is a (strictly convex) reflexive Banach space and A : V 2V is maximally monotone and coercive, then A is surjective.

1

Proof. See Theorem III.2.10 in [6].

For further reading on maximal monotone operators we refer the reader to [6â€“9, 12] or [13].

4. PROOF OF THE EXISTENCE RESULT This section is devoted to the study of the existence of the solutions for the initial boundary value problem (10)â€“(14). To simplify the proof of Theorem 1.1 and avoid technicalities we assume that the initial value of 5 p is equal to zero, i.e. 5204 p 4 0. We suppose from now on that the numbers p and q satisfy the relations 1 9 q 2 p 9 93

1 1

4 18 p q

Proof of Theorem 1.1. Let 2u3 T3 5 p 4 be a solution of the initial-boundary value problem (10)â€“(14). The equations (10)â€“(11), (14) form a boundary value problem for the components 2u2t43 T 2t44 of the solution, the problem of linear elasticity theory. Due to linearity of this problem we can write these components of the solution in the form 2u2t43 T 2t44 4 2u2t43 5 T5 2t44 2 2t43 2t443 with the solution 2 2t43 2t44 of the Dirichlet boundary value problem (15)â€“(17) to the data b 4 b2t4, 7 4 7 2t4, 5 p 4 0, and with the solution 2u2t43 5 T5 2t44 of the problem (15)â€“(17) to the data b 4 7 4 0, 5 p 4 Bz2t4. We thus obtain

A NOTE ON EXISTENCE RESULT 149 5227 x u44 6 5 p 4 2Pp 6 I 45 p 5227 x 448 We insert this equation into (11) and (12) can be rewritten in the following form: 2 1 55 p 3 3 6 2 Q p 5 p 3

(18)

where the linear operator 5 : 6 6 is defined by

5z 4 6t z

with

D254 4 z 3 W p3q z204 4 08

The function can be determined from boundary value problem (15)â€“(17) to the given data b3 7 and therefore can be considered as known. Let us denote X

4 L p 21 3 1 N 43

6 4 L p 203 Te 3 X 43

H

4 L 2 21 3 1 N 43

3 4 L 2 203 Te 3 H 43

and M p 4 2 Q p : X X3

M2 4 2 Q 2 : H H8

26 3 33 6 4 is an evolution triple. Then, by the general duality principle [3], the inclusion (18) in 6 is equivalent to the following inclusion in 6 : M p 561 361 3

3 6 8

(19)

Indeed, (18) holds iff there exists 3 5z 326M p 5 p 4. Taking the inverse of the operators 5 and 3 gives (19). Thus if we solve (19), by the equivalence we obtain that the problem (18) has a solution as well. The next lemma shows that the operator M p 561 is maximally monotone. Lemma 4.1. The operator M p 561 : 6 6 is linear bounded and maximally monotone. Moreover, D2M p 561 4 4 6 . * Proof of Lemma 4.1. By definition, M p 561 is linear bounded and D2M p 561 4 4 6 . If we show that M p 561 is positive, then the lemma is proved. Let i : 6 33

and i : 3 6

be embedding maps. Recall that these embeddings are dense. To proceed further we note that

150 S. NESENENKO i M p 4 M2 i3

M p 561 4 561 Mq 8

Now we choose n 3 i 234. Then there exists n 3 3 such that n 4 i 2 n 4. By the density of the embeddings we obtain 8

M p 561 3

9 1 31

4 4 4 4 4

8 9 lim M p 561 n 3 n 1 31

n9

9 8 lim M p 561 i 2 n 43 i 2 n 4 1 31

n9

1 2 lim i2M p 561 4i 2 n 43 n 2

n9

3 lim

n9

1

1

M22 2i 561 i 4M22 n 3 n

4 2

3 4 1 1 lim 2i 561 i 4M22 n 3 M22 n 08

n9

2

Since the operator M2 is non-negative and self-adjoint, its square root is well-defined. Since M p 561 is monotone and continuous, it is maximally monotone due to Corollary 2.7 [14, p. 156] or Corollary 1.35 [8, p. 309] (see Theorem 3.1 as well). The proof of Lemma 4.1 is complete. 1 The operator 361 is maximally monotone as the inverse of a maximal monotone operator. Due to Lemma 4.1 the operators M p 561 and 361 satisfy the condition intD2M p 561 4 D2361 4 4 8 Therefore, by Theorem 3.2 the sum M p 561 361 is maximal monotone. Moreover, the coercivity of 361 implies that 8

9 M p 561 3 3 9 as 93

for 3 361 2 4. Thus in virtue of Theorem 3.3 the maximal monotone and coercive operator M p 561 361 is surjective. Therefore, Equation (19) is solvable and as consequence by the equivalence we obtain that the problem (18) has a solution as well. Substituting the solution of (18) into the boundary value problem formed by equations (10)â€“(11), (14) yields the existence of 2u3 T 4 by the existence theory for linear elliptic problems. This completes the proof of Theorem 1.1. 1

5. NONLINEAR KINEMATIC HARDENING Now we apply Theorem 1.1 to the model of of nonlinear hardening. The model of nonlinear kinematic hardening consists of the equations ([1, 15])

A NOTE ON EXISTENCE RESULT 151 6divx T T

4 b3

(20)

1 2 4 2 527 x u4 6 5 p 3

(21)

6t 5 p

4 c1 T 6 k25 p 6 5n 4r

6t 5n

4 c2 k25 p 6 5n 4m

5n 204 4 50n 3 u 4 73

T 6 k25 p 6 5n 4 3 T 6 k25 p 6 5n 4

k25 p 6 5n 4 3 k25 p 6 5n 4

5 p 204 4 50p 3 x 3 61 8

(22)

(23) (24) (25)

Equations (20)â€“(24) can be written in the general form (1)â€“(6) with g 4 2g1 3 g2 4 : 1 3 21 3 1 3 2 1 3 defined by g1 2T3 z5 4 4 c1 T k 1 2 z5 r

T k 1 2 z5 3 T k 1 2 z5

g2 2T3 z5 4 4 c1 k 1 2 T k 1 2 z5 r

T k 1 2 z5 z5

c2 k 1 2 k 1 2 z5 m 3 T k 1 2 z5 5z

(26)

(27)

where z5 4 k 1 2 25 p 6 5n 4. Monotonicity of the mapping 2T3 z5 4 2g1 2T3 z5 43 g2 2T3 z5 44 follows from the fact that g 4 2g1 3 g2 4 is the gradient of the convex function 2T3 z5 4 4

c1 c2 T k 1 2 z5 r 1

k 1 2 z5 m 1 8 r 1 m 1

Now we can prove the main result of this section. Theorem 5.1. Let c1 , c2 , k be positive constants and let r and 7 satisfy r3 m 1. Let us define p 4 1 r, q 4 1 1 r , p 4 maxp3 1 m and q 4 minq3 1 1 m. 2 3 204 Suppose that b 3 L p 2Q3 13 4, 7 3 L p 203 Te 3 W 13 p 21 3 13 44, 5204 p 3 L 21 3 1 4 and 5 n 3 2 3 L 21 3 1 4. Then there exists a solution u 3 L p 203 Te W 13 p 21 3 13 443 5p

3 W p3q 203 Te 3 L p 21 3 1 3 443

T 3 L p 203 Te L p 21 3 1 3 443 5n 3 W 13q 203 Te 3 L q 21 3 1 3 44

of the problem (20)â€“(24). Moreover, 5 p 6 5n 3 W p3 q 203 Te 3 L p 21 3 1 3 44.

152 S. NESENENKO

Proof of Theorem 5.1. In [1] Theorem 5.1 has already been proved under the restriction that m and r satisfy the inequality m r. This condition the authors of [1] used there in order to show that the operator 3 defined by Equations (22)â€“(24) according to the rule given in the first section is coercive. The demicontinuity and the verification of the growth condition are performed without using this additional assumption. Therefore, we refer the reader for the prove of the conditions (a)â€“(b) in Theorem 1.1 to that work and show only that the operator 3 defined by (22)â€“(24) has a coercive multivalued inverse 361 . We note first that 5 1 21 1 r c1 T 6 k25 p 6 5n 4r

32T 4 qq3Q 4 d2x3 t4 4 c1q T 6 k25 p 6 5n 4 pp3Q 8 Q

Equations (22)â€“(23) yield

32T 4 T

k T 6 k25 p 6 5n 4 4 6t 5 p 6 5n 2 2T 6 k25 p 6 5n 44 c1 T 6 k25 p 6 5n 4r 2 T 6 k25 p 6 5n 4

k25 p 6 5n 4 c2 k25 p 6 5n 4m

k25 p 6 5n 4 k25 p 6 5n 4

k 4 6t 5 p 6 5n 2 c1 T 6 k25 p 6 5n 4 p c2 k25 p 6 5n 4m 1 8 2 Then the integration and previous computations show that 232T 43 T 4 Q

4

k k

25 p 6 5n 42Te 4 21 6 25 p 6 5n 4204 21 2 2

c1 T 6 k25 p 6 5n 4 pp3Q c2 k25 p 6 5n 4 m 1 m 13Q k p c1 T 6 k25 p 6 5n 4 p3Q 6 25 p 6 5n 4204 21 2 k 4 c116q 32T 4 qq3Q 6 25 p 6 5n 4204 21 8 2 The last inequality implies the coercivity of the inverse 361 . Therefore Theorem 1.1 can be applied to obtain the existence and the required regularity of u3 T and 5 p . To show the existence of 5n we proceed as in [1]: The definition of 3 (see Section 1) implies that the function y 4 k 61 2 z5 4 5 p 6 5n solves the problem yt

4 c1 T 6 kyr

y204 4 y 204 3 204 where y 204 4 5204 p 6 5n .

T 6 ky y 6 c2 kym 3 T 6 ky y

(28) (29)

A NOTE ON EXISTENCE RESULT 153

The solvability of (28)â€“(29) can be obtained, for example, in the same manner as for Equation (19). Namely, the direct computations together with Youngâ€™s inequality with imply that the operator B : L p 2Q3 1 3 4 L q 2Q3 1 3 4 defined by B2T3 y4 4 6c1 T 6 kyr

T 6 ky y

c2 kym T 6 ky y

enjoys the inequality p p 2B2T3 y43 y4 Q C1 y m 1 m 1 C 2 T 6 ky p 6 C 3 T p

or p p 2B2T3 y43 y4 Q C1 y m 1 m 1 C 2 k y p 6 21 C 3 4 T p 3

where C1 3 C2 and C3 are some constants. The last inequality yields the coercivity of B in L p 2Q3 1 3 4. Repeating the arguments of the proof of the existence for (19) we get the solvability of (28)â€“(29). Uniqueness of the solution is obvious. From the existence of 5 p and y follows the existence and the required regularity of 5n . The last computations also show that the operator 3 defined by Equations (22)â€“(24) is well defined. The proof of Theorem 5.1 is complete. 1 For the time independent function b and 7 similar results for the model of nonlinear kinematic hardening were obtained by Pompe [16]. REFERENCES [1]

Alber, H.-D. and Chelminski, K. Quasistatic problems in viscoplasticity theory. II. Models with nonlinear hardening. Mathematical Models and Methods in the Applied Sciences, 17, 189â€“213 (2007). [2] Hu, Sh. and Papageorgiou, N. S. Handbook of Multivalued Analysis. Volume II: Applications, Mathematics and its Applications, Kluwer, Dordrecht, 2000. [3] Attouch, H. and Thera, M. A general duality principle for the sum of two operators. Journal of Convex Analysis, 3, 1â€“24 (1996). [4] Valent, T. Boundary Value Problems of Finite Elasticity. Local Theorems on Existence, Uniqueness, and Analytic Dependence on Data, Springer Tracts in Natural Philosophy, Vol. 31, Springer, New York, 1988. [5] Alber, H.-D. and Chelminski, K. Quasistatic Problems in Viscoplasticity Theory. I. Models with Linear Hardening, Operator Theoretical Methods and Applications to Mathematical Physics, Operator Theory Advances and Applications, Vol. 147, BirkhĂ¤user, Basel, pp. 105â€“129, 2004. [6] Pascali, D. and Sburlan, S. Nonlinear Mappings of Monotone Type, Editura Academiei, Bucharest1 Sijthoff and Noordhoff, Alphen aan den Rijn, 1978. [7] Phelps, R. R. Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Mathematics, Vol. 1364, Springer, Berlin, 1993. [8] Hu, Sh. and Papageorgiou, N. S. Handbook of Multivalued Analysis. Volume I: Theory, Mathematics and its Applications, Kluwer, Dordrecht, 1997. [9] Barbu, V. Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei, Bucharest1 Noordhoff, Leyden, 1976. [10] Simons, S. Minimax and Monotonicity, Lecture Notes in Mathematics, Vol. 1693, Springer, Berlin, 1998. [11] Brezis, H. and Nirenberg, L. Characterizations of ranges of some nonlinear operators and applications to boundary value problems. Annali della Scuola Normale Superiore di Pisa, 15, 225â€“325 (1978).

154 S. NESENENKO

[12] Browder, F. E. Nonlinear operators and nonlinear equations of evolution in Banach spaces, Nonlinear functional analysis. Proceedings of the Symposia of Pure Mathematics, Vol. XVIII, Part 2 (Chicago, IL, 1968), pp. 1–308, American Mathematical Society, Providence, RI, 1976. [13] Zeidler, E. Nonlinear Functional Analysis and its Applications, Vol. II/B, Springer, New York, 1990. [14] Cioranescu, I. Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Mathematics and its Applications, Kluwer, Dordrecht, 1990. [15] Alber, H.-D. Materials with Memory—Initial-Boundary Value Problems for Constitutive Equations with Internal Variables, Lecture Notes in Mathematics, Vol. 1682, Springer, Berlin, 1998. [16] Pompe, W. Quasistatic and dynamic problems in viscoplasticity theory-non-linearities with power growth. Mathematical Methods in Applied Sciences, 27, 1347–1365 (2004).

Corrigendum

Corrigendum to ‘A Note on Existence Result for Viscoplastic Models with Nonlinear Hardening’ by Sergiy Nesenenko. Mathematics and Mechanics of Solids first published on March 20, 2009 as doi:10.1177/1081286509103818 Theorem 1.1 should appear as Theorem 1.1. Let 2 1 p 1 2 and 1 1 q 1 2 be numbers with 12 p 3 12q 4 1. Assume that 1 : L p 3Q4 2 3 5 5 L q 3Q4 2 3 5 satisfies (a) 1 is demicontinuous and monotone1 (b) 1 enjoys the growth condition p 6171 3656 p4Q 1 C31 3 666q2 p4Q 58

(c) the inverse 171 is strongly coercive, i.e. 96 4 6 5 32 as 666

666 5 24

6 171 3657

Suppose that b L p 304 Te 8 W 714 p 38 4 13 55 and 9 L p 304 Te 8 W 14 p 38 4 13 55. Then there exists a solution u L q 304 Te 8 W 14q 38 4 13 554

T L p 304 Te 8 L p 38 4 2 3 554

p W 14q 304 Te 4 L q 38 4 2 3 55 of the problem (10)–(14). Section 4 should appear as 4. PROOF OF THE EXISTENCE RESULT This section is devoted to the study of the existence of the solutions for the initial boundary value problem (10)–(14). Proof of Theorem 1.1. Before we start the proof of Theorem 1.1, let us introduce the following notations

Mathematics and Mechanics of Solids 16: 155–157, 2011 1 The Author(s), 2011. Reprints and permissions: http://www.sagepub.co.uk/journalsPermissions.nav

DOI: 10.1177/1081286510364582

156 CORRIGENDUM

X 4 L p 38 4 2 3 54

3 4 L p 304 Te 8 W 54

M p 4 4 Q p : X 5 X7

We note that the operator M2 is non-negative by Corollary 2.0.1. Next, we define a linear maximal monotone operator 5 : 3 5 3 by

5z 4 t z

with

D355 4 z W p4q 304 Te 8 3 5 z305 4 07

The idea of the proof of Theorem 1.1 is to show the solvability of the abstract equation (18) in a reflexive Banach space 3 applying Theorem 3.3 and then, based on this result, to construct solutions for the initial boundary value problem (1)â€“(5). We note that the idea of the proof is strongly connected to the general duality principle for the sum of two operators obtained in [4]. Let us consider now the following inclusion in 3

571 Mq 3 171 4

3 4

(18)

where 363t54 3t55 is a solution of the Dirichlet boundary value problem (15)â€“(17) to the data b 4 b3t5, 9 4 9 3t5, p 4 0. The next lemma proves that the operator 571 Mq in (18) is maximal monotone. Lemma 4.1. The operator 571 Mq : D3571 Mq 5 3 5 3 is linear maximal monotone. Proof of Lemma 4.1. According to Theorem 3.1, the operator 571 Mq with D3571 Mq 5 4 6 3 Mq 6 D3571 5 is maximal monotone, if it is a densely defined closed monotone operator such that its adjoint 3571 Mq 5 is monotone. We note that the operator 571 Mq is the closure in 3 3 of the operator 50 given by

50 6 :4 571 Mq 64

6 D350 5 4 6 3 571 6 3 7

The last operator is monotone, what can be shown using the generalized integration by parts formula and the following identity

571 Mq 6 4 M p 571 64

6 D350 57

(19)

The identity (19) is proved in the end of this work. Therefore, the operator 571 Mq is monotone as the closure in 3 3 of the monotone operator 50 . Since the operator 3571 Mq 5 is the closure of 50 , thier adjoint operators coincide. The adjoint of 50 is easy to compute and is equal to 3571 5 Mq , by a well-known result from the functional analysis. Therefore, by arguing in the same way as above, we obtain that the adjoint 3571 Mq 5 (we recall that 3571 Mq 5 4 3571 5 Mq ) is monotone. Thus, since 571 Mq verifies all assumptions of Theorem 3.1, it is a maximal monotone operator. The proof of Lemma 4.1 is complete. 1

CORRIGENDUM 157 In order to apply Theorem 3.3 we note the operator 171 is maximal monotone as the inverse of a maximal monotone operator. By the assumption 3b5 of Theorem 1.1 the operators 571 Mq and 171 satisfy the condition D3571 Mq 5 int D3171 5 4 7 Therefore, by Theorem 3.2, the sum 571 Mq 3 171 is maximal monotone. Moreover, the coecivity of 171 implies that 2 1 71 5 Mq 6 3 6 4 6 96 4 6 5 32 as 666 666

666 5 24

for 6 171 365. Thus, in virtue of Theorem 3.3, the maximal monotone and coercive operator 571 Mq 3 171 is surjective. Therefore, the equition (18) has a solution. Denoting by 4 571 Mq we obtain from (18) that solves the problem

5 4 Mq G37 3 54

L p 38 Te 4 1 N 57

(20)

Using the last result, the construction of the solution of the problem (10)â€“(14) can be now performed as in [3]: Let 3 be the unique solution of (20). With the function let p W 14q 304 Te 4 L q 38 4 2 3 55 be the solution of 3 4 t p 3t5 4 1 7 3t5 3 3t5 4

for a.e. t 304 Te 5

p 305 4 07

(21) (22)

Moreover, by the linear elliptic theory, there is a unique solution 3u3t54 T 3t55 of problem (15)â€“(17) to the data b 4 9 4 0, p 4 p 3t5. The solution of (10)â€“(14) is now given as follows 3u4 T4 p 5 4 3u 3 64 T 3 4 p 5

L q 304 Te 8 W 14q 38 4 13 55 L p 38 Te 2 3 5 W 14q 304 Te 4 L q 38 4 2 3 557 To see that 3u4 T4 p 5 satisfies (12), we apply the operator Q q to (21)â€“(22) from the left and obtain that 3 4 t 3Q q p 5 4 Q q 1 7 3t5 3 3t5 4 t 4

Q q p 305 4 305 4 07

The last line implies that Q q p 4 . Thus T 4 T 3 4 7Q q p 3 4 7 3 L p 38 Te 2 3 57 The last observation completes the proof of Theorem 1.1.

1

The Local Conditions of Uniqueness and Plastic Strain Localization Part I: Determination of the Critical Hardening Moduli for Some Elastoplastic Materials

Z. S´ LODERBACH J. PAJAK 1

Department of Applications of Chemistry and Mechanics, Opole University of Technology, Opole, Poland

Abstract: The classical, most frequently cited constitutive equations of elastic–plastic materials with nonassociated and associated plastic flow laws are specified. Local sufficient uniqueness conditions for the solution of the boundary-value problem and local necessary conditions of initiation of deformation localization in the form of a Rice–Rudnicki localization plane are derived. We show that sufficient local uniqueness conditions for these constitutive equations with sign (1) on the right-hand side become local necessary conditions of non-uniqueness of the solution of the boundary-value problem.

Key Words: constitutive equations, local uniqueness condition, localization state, hardening function, potential and function of plasticity

1. INTRODUCTION Global and local necessary and sufficient uniqueness conditions for the solution of the incremental boundary-value problem of generalized coupled thermoplasticity (exactly thermoelastoplasticity) for small and large deformations and their velocities were derived in [1, 2]. The derived global and local uniqueness conditions are necessary and sufficient conditions, respectively, excluding the possibility of bifurcation state during isothermal and nonisothermal loads. Apart from pure mathematical meaning, the conditions have important practical value. They can serve as a tool for the estimation of critical load, beyond which the appearance of bifurcation is possible. It can be assumed that micro-concentrations of strains and stresses can occur in these points where conditions of uniqueness or bifurcation criteria were exceeded. The negative influence of such concentrations can be important during creep and variable mechanical and thermo- mechanical loads, that is, in the analysis of fracturing and fatigue resistance of materials (see, e.g., [3–5]). In the general case, constitutive equations of coupled thermo-elastoplasticity have the character of non-associated plastic flow laws and even in the case of the Gyarmati postulate

Mathematics and Mechanics of Solids 16: 18–35, 2011 1 The Author(s), 2011. Reprints and permissions: 2 http://www.sagepub.co.uk/journalsPermissions.nav

DOI: 10.1177/1081286510362407

PART I: DETERMINATION OF THE CRITICAL HARDENING MODULI 19

(see, e.g., [6]), they contain the effects of thermo-mechanical couplings and take into consideration the elastic–plastic coupling effects. This means that they can be applied not only to the description of metallic materials, but also porous materials, sintered powders, rocks and soils, concrete and less plastic (semi-brittle) metals and their alloys. In the present paper, some cases are formulated of local necessary and sufficient uniqueness conditions for small deformations of some porous materials, sintered powders, rocks and soils, concrete and less plastic (semi-brittle) metals and their alloys, for example exploited for a long time under heavy thermo-mechanical load. Plastic deformation instability (for example, as a localization plane, formation of a groove, neck, shear bands, etc.) for these materials may occur already for small deformations. These materials are described with ellipsoidal constitutive equations, Drucker–Prager equations, Janike–Shield equations, Rice–Rudnicki (R-R) equations and others. In the case of less general material models, constitutive functions occurring in local uniqueness conditions will be simpler, e.g. [1, 7–11]. Formal necessary local conditions for the appearance of plastic deformation localization as a R-R localization plane [12, 13] are also derived for these materials. Constitutive equations for some chosen materials presented in this paper are not exhaustive for all kinds. These are the equations most frequently used and cited in the literature. Recently, new papers referring to constitutive equations of non-associated plastic flow laws were published [14–19], but their practical applications to the description of real materials seems to require further studies and experimental verification. It should be pointed out that the local sufficient condition of the uniqueness of the solution for the incremental boundary-value problem of the generalized coupled thermoplasticity with an equals sign on the right-hand side becomes the local necessary condition of the nonuniqueness and of the possible appearance of the bifurcation state in the shape of a groove, neck, shear band or other state of deformation instability of elastoplastic materials, e.g. [8, 10–14, 20–33]. We demonstrate in Part II that in the case of a constitutive equation of material analyzed by Rudnicki and Rice [12] the local necessary condition of non-uniqueness of the solution for the incremental boundary-value problem of the generalized coupled elastoplasticity imposes higher values for the isothermal hardening function h than the R-R local necessary localization condition of plastic deformation in the shape of a localization plane. This means that the local necessary non-uniqueness condition is the upper estimation of hardening moduli values imposed by the R-R localization condition. Physically it is equivalent to the statement that the local necessary condition of non-uniqueness of the solution for the incremental boundary-value problem of the generalized coupled elastoplasticity permits the lower (more safe) values of critical loads than the R-R condition of plastic deformation localization. This happens because the higher value of the hardening modulus h means that lower loads are permissible. This implies that the local necessary non-uniqueness condition is a lower (safer) estimation of outer loads than the R-R local necessary localization condition. It can be assumed that similar results and conclusions will be valid for other constitutive equations of analyzed materials. This happens because the local uniqueness condition excludes instability of the plastic deformation, but the R-R plastic strain localization is a form of deformation instability.

Â´ 20 Z. SLODERBACH and J. PAJAK 1

2. CONSTITUTIVE EQUATIONS FOR THE DESCRIPTION OF SELECTED MATERIALS 2.1. Equations of Non-associated Laws of Plastic Flow 2.1.1. Ellipsoidal Equations

Ellipsoidal equations of non-associated laws of plastic flow are used to describe plastically compressible porous materials, sintered powders, rocks and soils, concrete and less plastic (semi-brittle) metals. They can be written (see, e.g., [9]) as 1

3 2I1 4 a1 32 5 b1 IIId 4 d 3 04

5 3 2I1 4 a2 32 5 b2 IIId 4

condition 2function3 of plasticity

plastic potential

(2.1) (2.2)

where I1 is the first invariant of stress tensor, I1 3 6 ii , IIId is the second invariant of stress deviator 6 i j , dev 6 i j 3 6 6i j , 1 IIId 3 77 2 3

1 6 6 6 6 2 ij ij

2

1 and 6 i6 j 3 6 i j 4 6 kk 8 i j 3

(2.3)

8 i j designates the Kronecker delta (unit tensor) for i4 j4 k 3 14 24 3, and a1 4 a2 4 b1 4 b2 and d are appropriate material parameters, dependent on the deformation history. If, for example, only parameter d is dependent on the deformation history and the other parameters are not dependent, then only the isotropic hardening is described. From expressions (2.1) and (2.2) we obtain that 3 91 4 3 b1 6 i6 j 5 2 26 kk 4 a1 3 8 i j 2f 3 3 4 4 5 6 ij 96 i j 4 95 4 4 6 gi j 3 3 b2 6 i6 j 5 2 26 kk 4 a2 3 8 i j 96 i j

(2.4)

Hence, 3 87 17

p

8

4 4 3 b2 6 i6 j 5 2 26 kk 4 a2 3 8i j b1 6 i6 j 6 kl 5 2 26 kk 4 a1 3 6 kk 4 5 ij h

6

4 4 6 4 6 e 3 1 i j 5 1 6 kk 8 i j ij 2 G 9 K

and

(2.5)

PART I: DETERMINATION OF THE CRITICAL HARDENING MODULI 21 3

6 7 6ij 4 b2

6

8

4 4 2 3 5 6 i6 j b1 6 i6 j 6 kl 5 2 26 kk 4 a1 3 6 kk 5 ij G h

4 4 6 6 26 kk 4 a2 3 7

8

4 6 kk 3 kk 5 b1 6 6i j 6 kl 5 2 26 kk 4 a1 3 6 kk 3K h

p

(2.6)

e

where i j is the plastic deformation velocity tensor, i j is the elastic deformation velocity

6

tensor, i j is the plastic deformation velocity deviator, and kk is the spherical part of the deformation velocity tensor. From expressions (2.5) and (2.6) after transformations we obtain

i j

6

3 5

7 1 6ij b2

8 5 6 i6 j b1 6 i6 j 6 kl 5 2 26 kk 4 a1 3 6 kk 2 G 2h

9

1 6 kk 6 26 kk 4 a2 3 7

8 b1 6 i6 j 6 kl 5 2 26 kk 4 a1 3 6 kk 8 i j 5 3 3K h

(2.7)

where i j is the total deformation velocity tensor, h is the hardening function (modulus), G is the shear modulus or shape elastic strain modulus (G 8 ), and K is the bulk or dilatational elastic strain modulus. As a results of expressions (2.1)â€“(2.7), the ellipsoidal equations describe elasticâ€“plastic compressible materials with non-associated laws of plastic flow and they consider neither elasticâ€“plastic nor thermalâ€“mechanical couplings effects, hence they describe materials on the basis of isothermal theory of elastoplasticity. The local sufficient condition of uniqueness of the solution for the incremental boundaryvalue problem of elastoplasticity for small deformations is the following: h 1

2

1 1 2 2 15 15 2 2 2 2 22b1 77 3 5 24 26 kk 4 a1 3 22b2 77 3 5 24 26 kk 4 a2 3 1 4 2 1 4 2

1 2

15 2 4 4b1 b2 77 5 24 26 kk 4 a1 3 26 kk 4 a2 3 1 4 2

(2.8)

where 3 5 2

2 21 5 3 3

1 4 2

and 3

E 2 21 5 3

or

K 3

3 5 2

3

and are Lame elastic constants, is Poissonâ€™s ratio, and E is Youngâ€™s modulus. Inequality (2.8) with an equals sign on the right-hand side formally becomes the local necessary condition of non-uniqueness of the solution of the incremental boundary-value problem of the generalized coupled elastoplasticity or the possible appearance of a bifurcation state. Formally, the expression for the critical value of the hardening function h cr describing the possibility of the appearance of a plastic deformation localization as a R-R

localization plane [12, 13] for ellipsoidal equations, obtained for the case when 69 6 (for

Â´ 22 Z. SLODERBACH and J. PAJAK 1

Figure 1. Scheme of plane of localization in the coordinate system 6 I 4 6 II 4 6 III , (adapted from Rudnicki, J. W. and Rice, J. R. Condition for the localisation of deformation in pressure-sensitive dilatant materials. Journal of the Mechanics and Physics of Solids, 23, 371â€“394 (1975). Reprinted with permission form Elsevier).

small deformations and if the effects of moleculesâ€™ co-rotations are neglected) and also when the expressions of O277 G3 order and higher orders are neglected, is the following: h cr 15 4 21 5 3 3 [N 5 2 226 kk 4 a1 4 a2 3]2 2a1 4 a2 32 4

14 2

(2.9)

Variable parameter N in Equation (2.9) depend on stress state and (see, e.g., [12, 13]) is equal to 6 6II 3 N 77 , where 6 6II is the second main component of the stress deviator in the coordinate system shown in Figure 1. It is also assumed after [12, 13] that the normal to the plane of the plastic localization n is perpendicular to the direction 6 II and 6 I 6 II 6 III , where 6 I 4 6 II 4 6 III denote the main stresses. Rudnicki and Rice [13] designated the angle as the angle-arranging plane of plastic localization in the main stress directions system ( is the angle between the normal n and the 6 III direction). The angle 0 in Figure 1 corresponds to the maximum value of the hardening modulus h cr max . For materials described with ellipsoidal equations, the conditions (2.8) and (2.9) are dependent not only on material parameters 4 4 a1 4 a2 4 b1 and b2 , but are also dependent on the spherical part of the stress tensor 6 kk , and on the stress intensity 77 through the parameter N. The maximum critical hardening modulus h cr max , based on Equation (2.9), is reached for N 3 42 226 kk 4 a1 4 a2 3. Then from (2.9), we obtain that h cr 4 21 5 3 max 3 2a1 4 a2 32

14

(2.10)

As a result of Equation (2.10), the critical maximum value of the hardening modulus h cr max is not dependent on stress, but depends only on the material parameters 4 4 a1 and a2 .

PART I: DETERMINATION OF THE CRITICAL HARDENING MODULI 23

2.1.2. Druckerâ€“Prager Equations

The Druckerâ€“Prager constitutive equations of non-associated laws of plastic flow can also be used to describe plastically compressible porous materials, sintered powders, rocks and soils, concrete and less plastic (semi-brittle) metals, for example exploited for a long time under thermo-mechanical loads. They can be written, after Drucker and Prager [34â€“36], in the following form: 1

1

3 I1 5 2IIId 3 2 4 k 3 04 1

5 3 I1 5 2IIId 3 2 4

condition 2function3 of plasticity

plastic potential

(2.11) (2.12)

There are the following relations among material parameters: 3 5 3 2 3 sin2 3 5 sin2 41 41 6 2 2 k 3 3c 5 cos2 3 5 sin2

(2.13)

where k is the parameter describing the yield point, is the dimensionless material parameter 2 13, is the internal friction angle, and c is the cohesion coefficient. The meaning of the other physical quantities and material parameters is the same as for ellipsoidal equations. From (2.11) and (2.12) we obtain 3 6 i6 j 91 4 4 f 3 3 5 8 i j 2 3 4 5 6 ij 96 i j 277 4 6 i6 j 95 4 4 6 gi j 3 3 5 8 i j 96 i j 277

(2.14)

Hence, from the theory of plastic flow and theory of elasticity we obtain 6 3 6ij

1 6 i6 j

p

4 4 6 kl 5 6 kk 4 5 i j 3 h 277 5 8 i j 277

6

4 4 e 6 4 6 3 1 i j 5 1 6 kk 8 i j ij 2 G 9 K 3

6 6ij 1 6 i6 j 6 i6 j

6

4 4 4 5 2i j 3 G 5 h 77 277 6 kl 5 6 kk

4 4

6 3 6 i6 j

4 6 kk 3 kk 5 6 5 6 kk 3K h 277 kl From expressions (2.15) and (2.16) after transformations we obtain

(2.15)

(2.16)

Â´ 24 Z. SLODERBACH and J. PAJAK 1

6

i j 3

1 6ij 1 6 i6 j 5 2 G 2h 77

1 2 6 6kl

3 6 6kl

1 6 kk

6 kl 5 6 kk 5 6 kl 5 6 kk 5 8i j 277 3 3K h 277

(2.17)

The meaning of the other physical quantities and material parameters in Equations (2.14)â€“(2.17) is the same as for the constitutive ellipsoidal equations. The local sufficient condition of uniqueness of the solution for the incremental boundaryvalue problem for small deformations is as follows: h 1

2

1 1 1 2 2 2 1 5 1 5 1 5 1 5 6 2 1 5 6 2 32 4 1 5 6 2 (2.18) 1 4 2 1 4 2 1 4 2

The equations expressing critical hardening modulus h cr and maximum critical hardening modulus h cr max , which describe the possibility of the appearance of the localization state of plastic deformations in the form of a plastic plane R-R localization, are as follows: h cr

3

15 15 [ 21 4 3]2 4 [N 5 21 5 3]2 2 21 4 3

(2.19)

h cr max

3

15 [ 21 4 3]2 21 4 3

(2.20)

Equation (2.20) for h cr max was obtained for the case when N 3 421 5 3. From (2.19) it results that this condition is dependent on the stress state through the variable parameter N , but expressions (2.18) and (2.20) depend only on material parameters (4 4 and ). 2.1.3. Jenikeâ€“Shield Equations

These equations are similar to Druckerâ€“Prager equations for the case when the dimensionless material parameter 3 0. The Jenikeâ€“Shield constitutive equations are also non-associated laws of plastic flow [37, 38]. However, they are used to describe elastically compressible and plastically non-compressible materials of non-associated laws of plastic flow. They are expressed as 1

1

3 I1 5 2IIId 3 2 4 k 3 04 1

5 3 2IIId 3 2 4

condition 2function3 of plasticity

plastic potential

(2.21) (2.22)

The meanings of the physical quantities and material parameters are the same as for the Druckerâ€“Prager equations for the case when 30 From (2.21) and (2.22) after transformations we obtain

(2.23)

PART I: DETERMINATION OF THE CRITICAL HARDENING MODULI 25 3 6 i6 j 91 4 4 f 3 3 5 8 i j 2 3 6 4 i j 5 96 i j 277

(2.24)

4 6 i6 j 95 4 4 6 gi j 3 3 96 i j 277 Hence, from the theory of plastic flow and theory of elasticity we obtain 2 1 3 1 6 i6 j 6 6kl

p

4 4 3 6 5 6 4 kl kk 5 ij h 277 277

6

4 4 6 4 6 e 3 1 i j 5 1 6 kk 8i j ij 2 G 9 K

(2.25)

3

6 6ij 1 6 i6 j 6 6kl

4

6

4 4 5 2i j 3 G 5 h 77 277 6 kl 5 6 kk

4 4

4 6 kk 3 6 kk 3K

(2.26)

then

As a result of the expressions above, the Jenikeâ€“Shield constitutive equations of nonassociated laws of plastic flow can be used to describe also less plastic metallic (semi-brittle) materials (for example metallics exploited under thermo-mechanical load) which are plastically non-compressible. From (2.25) and (2.26) after transformations we obtain

6

1 6ij 1 6 6i j i j 3 5 2 G 2h 77

1 6 kk 6 6kl

6 kl 5 6 kk 5 8i j 277 9 K

(2.27)

The meanings of the other physical quantities and material parameters in Equations (2.24)â€“(2.27) are the same as for the constitutive ellipsoidal and Druckerâ€“Prager equations. A local sufficient condition of uniqueness of the solution for the incremental boundaryvalue problem for small deformations, that is equivalent to Janike-Shield constitutive equations, is as follows: h 1

2

1 1 5 6 2

15 1 4 2

2

41

(2.28)

Inequality (2.28) with an equals sign on the right-hand side formally becomes the local necessary condition of non-uniqueness or of the possible appearance of bifurcation. The expressions for the critical hardening modulus h cr and maximum critical hardening modulus h cr max , which describe the possibility of the local appearance of plastic state localization as a R-R localization plane, are as follows:

Â´ 26 Z. SLODERBACH and J. PAJAK 1 h cr

3

15 2 15 4 2N 5 32 2 21 4 3

(2.29)

h cr max

3

15 2 21 4 3

(2.30)

Equation (2.30) for h cr max was received for the case where N 3 4. From expression (2.29), as previously, it follows that the condition depends on the stress state through parameter N . Equations (2.28) and (2.30) depend only on the respective material parameters (4 and ). 2.1.4. Riceâ€“Rudnicki Equations of Material Description

The R-R constitutive equations are a special kind of Druckerâ€“Prager equations for the case when 3 3 7 and 3 3. These equations can also be used to describe porous materials, sintered powders, rocks and soils, concrete or less plastic (semi-brittle) metals and other similar materials. They are expressed as 1

3

5 3

7 1 I1 5 2IIId 3 2 4 k 3 04 3 1 I1 5 2IIId 3 2 4 3

condition 2function3 of plasticity

plastic potential

(2.31) (2.32)

where

7 is the internal friction coefficient,

7 0.0â€“0.9, and is the dilatancy factor, 0.0â€“0.6. The meaning of the other physical quantities and material parameters is the same as in previous cases. The analytical dependence between the hardening modulus h and the graphically determined tangential hardening modulus h tan (see Figure 2), given by Rudnicki and Rice [12], is the following: h tan 3

h h 15

or

h tan 3

h 15

h E

(2.33)

The first equation in (2.33) is valid for pure shear and the second equation is for simple extension. Having determined a tangential hardening modulus we can determine the admissible stress and deformation state from the corresponding hardening curve [12]. From (2.31) and (2.32) we obtain 2 f 6 3i j 3

6 6i j 91

7 3 5 8i j 96 i j 277 3

(2.34)

PART I: DETERMINATION OF THE CRITICAL HARDENING MODULI 27

Figure 2. Scheme of curve of shear stress 7 versus shear strain showing the geometric interpretation of hardening modulus h , the tangent modulus h tan and the elastic shear modulus G 2G 8 3, where 26 3 4136 kk 3, (adapted from Rudnicki, J. W. and Rice, J. R. Condition for the localisation of deformation in pressure-sensitive dilatant materials. Journal of the Mechanics and Physics of Solids, 23, 371â€“394 (1975). Reprinted with permission form Elsevier).

gi j 3

6 i6 j 95 3 5 8i j 96 i j 277 3

(2.35)

Hence, from the theory of plastic flow and theory of elasticity we obtain 6 3 6 kl

1 6 i6 j

7

p 4 4 3 6 5 6 5 8 4 ij kl kk 5 ij h 277 3 277 3

6

4 4 6ij 1 1 6 kk

e 4 6 3 5 8i j ij 2 G 9 K

(2.36)

3

6 6ij

7

1 6 6i j 6 6kl

6 4 4 3 6 5 6 5 2 4 kl kk 5 ij G h 77 277 3

4 4 6

7

6 6kl

4 6 kk 3 kk 5 6 kl 5 6 kk 3K h 277 3

(2.37)

then

From Equations (2.36) and (2.37), after transformations, we obtain that

6

1 6 6i j 1 6ij 5 i j 3 2 G 2h 77

2

1 6 6kl

7

7

1 6 kk 6 4kl

5 6 kl 5 6 kk 5 6 kl 5 6 kk 8i j 277 3 3 3K h 77 3

(2.38)

The meanings of the other physical quantities and material parameters in Equations (2.34)â€“(2.38) are the same as for previous constitutive equations. The local sufficient condition of uniqueness of the solution for the incremental boundaryvalue problem for small deformations, which is equivalent to the R-R constitutive equation, is as follows:

Â´ 28 Z. SLODERBACH and J. PAJAK 1 3

72

7 2 15 (2.39)

5 h 22 5 33 5 22 5 33 4 5 22 5 33 26 3 3 3 or, after including (2.10), in dimensionless form: h 1

2

1 1 2 2 1 2 2 2 15 2 2 15 2 15 15 15

7 4 1 5

7 (2.40) 3 1 4 2 3 1 4 2 3 1 4 2

Inequalities (2.39) and (2.40) with equals signs on the right-hand side become, as previously, local necessary conditions of non-uniqueness or of the possible appearance of the bifurcation state for the coincidence of small deformations and their velocities. It also follows from (2.39) and (2.40) that these conditions are independent of the stress state. The more general form of the critical local necessary condition of the possible appearance of bifurcation state as a R-R plastic deformation localization plane [12,13] for the case

when 69 6 (small deformations and co-rotational effects are neglected), describing complex (non-linear) dependence on the stress state, is as follows: h cr

3 5

1 2 15 15 5

2 2 4 N5 2 4 3 9 21 4 3 2 3 1 2 1 22 4 4 3N 2 21 5 3 77 77 2 4 3 sin 2 0 5O 2

24 21 4 3

(2.41)

The less general critical value of the hardening function h cr describing the possibility of the appearance of the R-R plastic deformation localization plane, obtained for the case

when 69 6 (small deformations and co-rotational effects are neglected) and if neglected are expressions of the order O277 G3 and higher, depends on the square power of the deviator intensity 77 or on the second deviator component 6 6II (see, e.g., [12, 13]). It has the following formula: 15 15 h cr 3 24 2 4 3

9 21 4 3 2

1

5

N5 3

22 (2.42)

The changes in the critical modulus h cr on N for axially symmetric extension (6 I 6 II 3 6 III , then N 3 41 3), pure shear (6I 3 46 III , then N 3 0) and axially symmetric compression (6 I 3 6 II 6 III , then N 3 1 3) were presented by Rudnicki and Rice [12]. The maximum h cr 3. Then max solved from expression (2.42) is for N 3 4 2 5 3 15 h cr max 3 2 2 4 3

9 21 4 3

(2.43)

It is visible that the maximum critical modulus h cr max is not dependent on the stress state but depends only on the material parameters (4 4

and ), see, e.g., [12, 13].

PART I: DETERMINATION OF THE CRITICAL HARDENING MODULI 29 In order to determine Equations (2.42) and (2.43) in the variation range of ( and ) and so that the normal n to the plane of localization is not perpendicular to the direction of 6 III , the following condition must be fulfilled (see, e.g., [12, 13]): 5

1

3

(2.44)

2.2. Associated Plastic Flow Laws

Here we present selected examples of the plasticity conditions, which can also be used to describe plastically compressible porous materials, sintered powders, rocks and soils, concrete and less plastic (semi-brittle) metals. However, the Misesâ€“Hill plasticity condition considered in Section 2.2.2 is basically correct for plastically non-compressible metallic materials (metals and their alloys), see, e.g., [7, 39, 40]. 2.2.1. Misesâ€“Schleicher Plasticity Condition

The analytical form of the Misesâ€“Schleicher condition is as follows [8, 41]: 1

1 3 I1 5 2IIId 3 2 n 4 k 3 04

condition 2function3 of plasticity

(2.45)

where , k and n are positive material parameters (1 1 n 1 2), and IIId is the second invariant of the stress deviator. The local necessary and sufficient condition of uniqueness of the solution for the incremental boundary-value problem of elastoplasticity for small deformations, that is equivalent to the Misesâ€“Schleicher constitutive equations, is as follows: h 0

(2.46)

(a) A Case for n 3 1 Then from (2.45) we obtain that 2 f 6 3i j 3 gi j 3

6 6i j 91 3 5 8 i j 96 i j 277

(2.47)

From the theory of plasticity and elasticity it follows that: 6 3 1 6 6i j 6 kl

p

4 4 6 kl 5 6 kk 4 5 i j 3 h 277 5 8i j 277

6

4 4 6 4 6 e 3 1 i j 5 1 6 kk 8 i j ij 2 G 9 K and

(2.48)

Â´ 30 Z. SLODERBACH and J. PAJAK 1 3

6 6ij 1 6 i6 j 6 6kl

4

4 4 2 3 6 5 6 5 4 kl kk 5 ij G h 77 277

4 4 3 6 6kl

6

4 6 kk 3 kk 5 6 kl 5 6 kk 3K h 277

i j

6

1 6ij 1 6 i6 j 3 5 2 G 2h 77

(2.49)

2 1 6 kk 6 6kl

6 6kl

8 i j (2.50) 6 kl 5 6 kk 5 6 kl 5 6 kk 5 277 9K h 77

The expressions for the critical hardening modulus h cr and maximum critical hardening modulus h cr max for the case when n 3 1, which describes the possibility of the appearance of plastic state localization as the R-R localization plane, are as follows: h cr

3 4

15 2N 5 32 2

h cr max 3 0

(2.51) (2.52)

As a result of (2.52), the expressions are obtained from (2.51), when N 3 4. (b) A Case for n 3 2 Then from (2.45) we obtain that 2 f 6 3i j 3 gi j 3

91 3 6 6i j 5 8i j 96 i j

(2.53)

Hence, from the theory of plastic flow and theory of elasticity we obtain 3 87 6

17 6

p

8 4 4 4 5 i j 3 h 6 i j 5 8 i j 6 kl 6 kl 5 6 kk

6

4 4 1 6ij 1 6 kk

e 4 6 i j 3 5 8i j 2 G 9 K 3

6 7

4 6ij 1

6

8 4 4 5 6 i6 j 6 6kl 6 kl 5 6 kk 5 2i j 3 G h

4 4 3 7 6

6

8 4 6 kk 3 kk 5 6 kl 6 kl 5 6 kk 3K h

(2.54)

(2.55)

and

6

6 kk 1 6ij 1 6 6

3 6

i j 3 5 6 i j 6 kl 6 kl 5 6 kk 5 5 6 kl 6 kl 5 6 kk 8 i j 2 G 2h 3K h

(2.56)

PART I: DETERMINATION OF THE CRITICAL HARDENING MODULI 31

The expressions for the critical hardening modulus h cr and maximum critical hardening modulus h cr max for the case when n 3 2, which describes the possibility of the appearance of plastic state localization as the R-R localization plane, are as follows: h cr

3 4

2 15 6 26 II 5 2

h cr max 3 0

(2.57) (2.58)

As a result of (2.57), the expressions are obtained from (2.58) when 26 6II 3 4. 2.2.2. Misesâ€“Hill Plasticity Condition

The analytical form of this condition (the H-M-H condition) generalized for materials exhibiting hardening, is as follows [7, 40]: 1

1 3 2IIId 3 2 4 k 3 04

condition 2function3 of plasticity

(2.59)

The local necessary and sufficient condition of uniqueness of the solution for the incremental elastoplasticity boundary-value problem for small deformations, which is equivalent to Misesâ€“Hill constitutive equations, is as previously h 0

(2.60)

From (2.55) we obtain 2 f 6 3i j 3 gi j 3

6 i6 j 91 3 96 i j 277

(2.61)

From the theory of plastic flow and theory of elasticity, we have 3

6 2 1 6ij 1 6 i6 j 6 6kl

4

6 4 4 6 kl 5 2i j 3 G 5 h 77 277

4 4 4 6 kk 3 6 kk 3K 1 2 3 1 6 6i j 6 6kl

p 4 4 6 kl 4 5 i j 3 2h 77 277

6

4 4 4 6 e 3 1 6 i j 5 1 6 kk 8 i j ij 2 G 9 K

6 2 1

1 6ij 6 kk 1 6 6i j 6 6kl

i j 3 6 kl 5 5 8i j 2 G 2h 77 277 9K

(2.62)

(2.63)

(2.64)

Â´ 32 Z. SLODERBACH and J. PAJAK 1 The expressions for the critical hardening modulus h cr and maximum critical hardening modulus h cr max describing the possibility of the appearance of the plastic state localization as the R-R localization plane in this case are as follows: h cr

3 4

1 5 6 2 26 II 2

h cr max 3 0

(2.65) (2.66)

As a result of (2.66), the expressions are obtained from (2.65) when 6 6II 3 0. The meanings of the other physical quantities and material parameters in these equations are the same as for the previous constitutive equations analyzed previously in Section 2.1. As a result of Equations (2.51), (2.57), and (2.65), the negative value of the hardening modulus h cr is permissible, and, as previously for the R-R equations, they depend on the stress state. The stress state is represented here by means of a variable parameter N or a second deviatoric component 6 6II . The expressions (2.46), (2.52), (2.58), (2.60), and (2.66) have their simplest form and they are greater than or equal to zero and they do not depend on material parameters. 2.2.3. Drucker Plasticity Condition

This condition is expressed in dependence on the main stresses 6 1 , 6 2 , and 6 3 . It has the following [33] analytical form: 26 1 4 6 3 3 5 I1 4 k 3 0

(2.67)

where (6 1 6 2 6 3 ) are the maximum, mean, and minimum main stresses, respectively. 2.2.4. Coulombâ€“Mohr Plasticity Condition

This condition is expressed as [37, 38, 42] 26 1 4 6 3 3 3 26 1 5 6 3 4 2H 3 sin

(2.68)

where H is the tensile isotropic strength of the material. For conditions (2.67) and (2.68) the local necessary and sufficient condition of uniqueness or excluding bifurcation has the simplest form h 0

(2.69)

However, inequalities (2.46), (2.60), and (2.69) with equals signs on the right-hand side become local necessary conditions of the non-uniqueness solution of the incremental boundary-value problem elastoplasticity or possible appearance of the bifurcation state for the plasticity functions discussed in this section.

PART I: DETERMINATION OF THE CRITICAL HARDENING MODULI 33

Then h30

(2.70)

3. REMARKS AND CONCLUSIONS 1. In this work, selected cases have been formulated of constitutive equations, local uniqueness conditions for the solution of the incremental boundary-value problem of elastoplasticity, and local condition for the appearance of plastic deformation localization in the form of the R-R localization plane for some porous materials, sintered powders, rocks and soils, concrete and less plastic (semi-brittle) metals and their alloys, for example exploited for a long time under thermo-mechanical load. Such materials are described by means of constitutive equations of non-associated and associated laws of plastic flow valid for small deformations and their velocities. 2. For ellipsoidal constitutive equations, the local uniqueness condition of the solution for the incremental boundary-value (e.g. the condition excluding the appearance of bifurcation state) depends on the stress state. For other constitutive equations local conditions of uniqueness of solution for the incremental boundary-value are independent of the stress state, in a similar manner as in the case of maximum values of the critical hardening cr modulus h cr max (R-R). However, the critical hardening moduli h (R-R) for all equations considered in this paper are dependent on the stress state through the variable parameter N introduced by Rudnicki and Rice [12,13]. 3. During determination of the hardening modulus, it is convenient and practical to use the tangent hardening modulus h tan if the material strengthening curve is known. Graphical interpretation of the tangent hardening modulus h tan in the (7 4 ) coordinate system is shown in Figure 2. For a curve in the coordinate system (6 â€“ ) in the formula for the h tan , Youngâ€™s modulus E should be inserted in place of the G modulus, see Equations (2.33). After determining the tangent hardening modulus h tan , the equivalent stress and deformation state could be determined from the respective strengthening curve. 4. Rudnicki and Rice [12, 13] have presented an analysis of changes to the critical hardening moduli h cr on the stress state through the variable parameter N . Axial tension, pure shear, and axial compression cases, for which negative values of h cr can also result, were considered. 5. The formulated local sufficient uniqueness conditions, excluding the possibility of bifurcation state and local conditions describing the possibility of the appearance of the localization of plastic deformations, have important practical value in addition to pure mathematical and cognitive meaning. They can serve as a tool for the estimation of the critical load, beyond which the appearance of plastic deformation instability or a bifurcation state are possible [1, 11, 20, 22â€“24, 27, 28, 39, 43, 44]. 6. The problems of plastic deformation instability, localization of plastic deformation, and bifurcation state play important roles in reliability, safe engineering design, experiments and technology of production and repair problems of equipment, machines, and construc-

´ 34 Z. SLODERBACH and J. PAJAK 1

tion elements. Further studies and improvements in measuring and the calculation methods used to estimate the moment of appearance of these phenomena are necessary. 7. It can be assumed that micro-concentrations of strains and stresses can occur in these points of the material where the uniqueness conditions or bifurcation criteria were violated. Micro-cracks and micro-localizations of deformation can occur, growing and leading consequently to fracture and material destruction. The influence of such micro- and macro-concentrations can be important during creep and variable mechanical and thermomechanical loads, that is, in the analysis of fracturing and fatigue resistance of materials (see, e.g., [3–5]). 8. It seems worth pointing out that sufficient local conditions for the localization of plastic deformations as the R-R localization plane have not been derived previously. 9. Finally, we would like to point out that, as analyzed in this paper, the local sufficient conditions of uniqueness of solution for incremental boundary-value problem as inequality with sign 213 on the right-hand side formally becomes local necessary condition of nonuniqueness of solution for incremental boundary-value problem and possible appearance of bifurcation. REFERENCES [1] [2]

[3] [4] [5] [6] [7] [8] [9]

[10] [11] [12] [13] [14] [15]

´ Sloderbach, Z. Generalized coupled thermoplasticity. Part II. On the uniqueness and bifurcations criteria. Archives of Mechanics, 35, 351–367 (1983). ´ Sloderbach, Z. and Pajak J. Generalized coupled thermoplasticity taking into account large strains: part I. Conditions of uniqueness of the solution of boundary-value problem and bifurcation criteria. Mathematics and Mechanics of Solids (2009). Macha, E. Simulation investigations of the position of fatigue fracture plane in materials with biaxial loads. Materialwissenschaft und Werkstofftechnik, 20(I, 4/89), 132–136 and (II, 5/89), 153–163 (1989). Macha, E. A review of energy-based multiaxial fatigue failure criteria. Archive of Mechanical Engineering, 48, 71–101 (2001). Macha, E. and Sonsino, C. M. Energy criteria of multiaxial fatigue failure. Fatigue of Engineering Materials and Structures, 22, 1053–1070 (1999). ´ Sloderbach, Z. Generalized coupled thermoplasticity. Part I. Fundamental equations and identities. Archives of Mechanics, 35, 337–349 (1983). Hill, R. On the classical constitutive relations for elastic/plastic solids, in ed. B. Broberg et al., Recent Progress in Applied Mechanics – The Folke Odqvist Volume, Almquist and Wiksell, Stockholm, 1967, pp. 241–249. Hueckel, T. and Maier, G. Incremental boundary value problems in the presence of coupling of elastic and plastic deformations. A rock mechanics oriented theory. International Journal of Solids and Structures, 13, 1–15 (1977). Hueckel, T. and Maier, G. Non-associated and coupled flow rules of elastoplasticity for geotechnical media. Proceeding 9-th International Conference of Soil Mechanics Foundation of Engineering (JCSFE), Tokyo, Speciality session 7, Constitutive relations for soils, 1977, pp. 129–142. Ottosen, N. S. and Runesson, K. Properties of discontinuous bifurcation solutions in elasto-plasticity. International Journal of Solids and Structures, 27, 401–421 (1991). ´ Sloderbach Z. and Sawicki T. Determination of the critical adiabatical twisting moment in the case of thick and thin-walled metal tubes. Engineering Transactions, 31, 447–457 (1983). Rudnicki, J. W. and Rice, J. R. Condition for the localisation of deformation in pressure-sensitive dilatant materials. Journal of the Mechanics and Physics of Solids, 23, 371–394 (1975). Rice, J. R. The localisation of plastic deformation, in ed. W. T. Koiter, Theoretical and Applied Mechanics, NorthHolland, Amsterdam, 1976, pp. 207–220. Bigoni, D. Yield criteria for quasibrittle and frictional materials. International Journal of Solids and Structures, 41, 2855–2878 (2004). Lubarda, V. A., Mastilovic, S. and Knap, J. Some comments on plasticity postulates and non-associative flow rules. International Journal of Mechanical Science, 38, 247–258 (1996).

PART I: DETERMINATION OF THE CRITICAL HARDENING MODULI 35

[16] McDovell, G. R. A simple non-associated flow model for sand. Granular Materials, 4, 65–69 (2002). [17] Nova, R. The role of non-normality in soil mechanics and some of its mathematical consequences. Computer and Geotechnics, 31, 185–191 (2004). [18] Thomas, B. A non-associated flow rule for sheet metal forming. International Journal of Plasticity, 18, 687–714 (2002). [19] Thomas, B. and Yoon, J-W. A pressure-sensitive yield criterion under a non-associated flow rule for sheet metal forming. International Journal of Plasticity, 20, 705–731 (2004). [20] Hill, R. Bifurcation and uniqueness in non-linear mechanics of continua, in Problem of Continuum Mechanics (N.I. Muskhelishwili Anniversary Volume), SIAM, Philadelphia, PA, 1961, pp. 155–164. [21] Mróz, Z. On forms of constitutive laws for elastic-plastic solids. Archives of Mechanics, 18, 3–35 (1966). [22] Perzyna, P. Constitutive modelling for brittle dynamic fracture in dissipative solids. Archives of Mechanics, 38, 725–738 (1986). [23] Perzyna, P. Instability phenomena and adiabatic shear band localization in thermoplastic flow processes. Acta Mechanica, 106, 173–205 (1986). [24] Perzyna, P. and Duszek, M. K. The localisation of plastic deformation in thermoplastic solids. International Journal of Solids and Structures, 27, 1419–1443 (1991). [25] Petryk, H. On constitutive inequalities and bifurcation in elastic-plastic solids with a yield-surface vertex. Journal of the Mechanics and Physics of Solids, 37, 265–291 (1989). [26] P1echerski, R. B. Finite deformation plasticity with strain induced anisotropy and shear banding. Journal of Materials Processing Technology, 60, 35–44 (1996). [27] Raniecki, B. Uniqueness criteria in solids with non-associated plastic flow laws at finite deformations. Bulletin de l’Academie Polonaise des Sciences, Serie des Sciences Techniques, 27, 391–399 (1979). [28] Raniecki, B. and Bruhns, O. T. Bounds to bifurcation stress in solids with non-associated plastic flow at finite strain. Journal of the Mechanics and Physics of Solids, 29, 153–172 (1981). [29] Chau, K. T. and Rudnicki, J. W. Bifurcations of compressible pressure-sensitive materials in plane strain tension and compression. Journal of the Mechanics and Physics of Solids, 38, 875–898 (1990). [30] Bigoni, D. and Hueckel, T. Uniqueness and localization I. Associative and non-associative elastoplasticity. International Journal of Solids and Structures, 28, 197–213 (1991). [31] Benallal, A. and Bigoni, D. Effects of temperature and thermo-mechanical couplings on material instabilities and strain localization of inelastic materials. Journal of the Mechanics and Physics of Solids, 52, 725–753 (2004). [32] Bigoni, D. On smooth bifurcations in non-associative elastoplasticity. Journal of the Mechanics and Physics of Solids, 44, 1337–1351 (1996). [33] Nguyen, Q. S. Bifurcation and stability in dissipative media (plasticity, friction, fracture). Applied Mechanics Review, 47, 1–31 (1994). [34] Drucker, D. C. A more fundamental approach to plastic stress–strain relations. Proceedings of the 1st U.S. National Congress Applied Mechanics, 1952, pp. 481–487. [35] Drucker, D. C. Coulomb friction, plasticity and limit loads. Journal of Applied Mechanics, 21, 71–74 (1954). [36] Drucker, D. C. and Prager, W. Soil mechanics and plastic analysis of limit design. Quarterly Applied Mathematics, 10, 157–165 (1952). [37] Jenike, A. W. and Shield, R. T. On the plastic flow of Coulomb solids beyond original failure. Journal of Applied Mechanics—Transactions of the ASME, 26, 599–602 (1959). [38] Shield, R. T. On Coulombs law of failure in soil. Journal of the Mechanics and Physics of Solids, 4, 10–16 (1955). [39] Hill, R. On constitutive inequalities for simple materials. Part II. Journal of the Mechanics and Physics of Solids, 16, 315–322 (1968). [40] Hill, R. Mathematical Theory of Plasticity, Clarendon Press, Oxford, 1986. [41] Izbicki, J. R. and Mróz, Z. Carrying Capacity Limit in Soil and Rock Mechanics, IFTR – Polish Academy of Science, PWN, Warsaw-Pozna´n, 1976 (in Polish). [42] Collins, I. F. The upper bound theorem for rigid plastic solids generalized to include Coulomb friction. Journal of the Mechanics and Physics of Solids, 17, 323–338 (1969). [43] Hill, R. A general theory of uniqueness and stability in elasto-plastic solids. Journal of the Mechanics and Physics of Solids, 6, 236–249 (1958). [44] Hill, R. Eigenmodal deformations in elasto-plastic continua. Journal of the Mechanics and Physics of Solids, 15, 255–262 (1967).

Some Remarks on Applying Homogenization Theory to Modeling the Constitutive Response of Carbon Nanotubes

H AMID B ELLOUT F REDERICK B LOOM

Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115, USA (Received 11 March 20091 accepted 30 July 2009)

Abstract: An approach is described for constructing (using molecular dynamics simulations at the atomic scale) a mathematical model for the constitutive behavior of a carbon nanotube. The method is based on applying homogenization theory to a hexagonal array of carbon atoms with a specific chirality vector. The molecular dynamics simulations generate a set of periodic, rapidly varying, elastic constants for the nanotube. An example is presented to illustrate the technique for a specific array on a nanotube surface.

Key Words: Carbon nanotube, homogenization, chirality, hexagonal array, elastic constants

1. INTRODUCTION Considerable interest has been generated in nano-structured non-metallic materials because of their potential for providing significant improvements in both mechanical and physical properties with respect to traditional structural materials. Recent investigations [1] point to the fact that carbon nanotubes possess an extremely high elastic modulus (in the 1–5 TPa range) and can sustain an elastic strain of the order of 5% and a fracture strain up to 20%1 their strength is, therefore, ‘unmatched by any other known material’ [2]. Moreover, as clearly indicated in Harris [3, Chapter 6] and, for example, [4], because of the fact that small fiber composites are easier to process, the mechanical properties of carbon nanotubes, coupled with their small dimensions and large aspect ratios, make them candidates for the fabrication of the optimal carbon fiber reinforced materials. Further work, however, is needed in connection with the continuum modeling of the constitutive response of both single-walled carbon nanotube (SWNTs) and multi-walled carbon nanotubes (MWNTs) in order that reliable constitutive relations can be developed and used to predict the bulk mechanical properties of nanotube–polymer composites in terms of the molecular structure of the polymer, the nanotubes, and the polymer/nanotube interfaces. Indeed as Salvetat et al. [5] have pointed out, ‘in order to implement nanotube–composite design intelligently, it is vital to understand the mechanical behavior of nanotubes’ and as the authors of the review

Mathematics and Mechanics of Solids 16: 36–57, 2011 1 The Author(s), 2011. Reprints and permissions: 1 http://www.sagepub.co.uk/journalsPermissions.nav

DOI: 10.1177/1081286509349094

SOME REMARKS ON APPLYING HOMOGENIZATION THEORY TO MODELING 37

[6] clearly indicate, with respect to effecting such an understanding ‘even classical molecular dynamics computations are still limited to simulating on the order of 106 –108 atoms for a few nanoseconds. The simulation of larger systems or longer times must currently be left to continuum methods’. The effective continuum modeling of both carbon nanotubes and the graphene layers of which they are the ‘cylindrical’ images, and the delineation of the mapping which takes one from the graphene layer to the nanotube surface, as well as the dependence of that mapping on the orientation and magnitude of the chirality vector, form one of the central foci of the approach outlined in this paper. Almost all modeling of the response of carbon nanotubes to date, especially that modeling which has been used in the study of the buckling and bending of either SWNTs or MWNTs [1, 7–25], and irrespective of whether the authors treat the nanotubes as shells (with some ‘effective’ thickness) or as beams (with some effective ‘cross-section’), assumes that the ‘surface’ of the ‘cylindrical’ nanotube exhibits an isotropic mechanical response to deformations1 this assumption is based, in turn, on two hypotheses, namely, that: (i) the response of a graphene layer is a consequence of the isotropic behavior of graphite [26, Chapter 3] in the basal plane (or, alternatively, as argued in [9], that the isotropic response of a graphene sheet as an elastic medium is a consequence of the intrinsic hexagonal symmetry of the array of carbon atoms comprising the sheet) and (ii) that the nanotube can be viewed, as claimed in, for example, [13] as a ‘conformal mapping of a two dimensional honeycomb lattice to the surface of a cylinder that is subject to periodic (boundary conditions)’. As we indicate in Section 2, neither of the hypotheses (i) or (ii) is in accord with the recent literature [6] on molecular dynamics (MD) simulations of the response of carbon nanotubes, nor with experimental observations. In addition, maintaining the isotropy hypothesis for the response of the nanotube in its basal plane has led to various inconsistencies among the results for the bending stiffness of SWNTs, the representative thickness of such nanotubes, and experimental results on in-plane stiffness1 these inconsistencies are also highlighted in Section 2. Moreover, inferring the behavior of a graphene sheet from the basal plane response of bulk graphite is equivalent to invoking the continuum hypothesis, the validity of which would require a large number of atomic layers. As noted in [6], ‘the fundamental assumption of the continuum approximation, that quantities vary slowly over lengths on the order of the atomic scales, breaks down in 1 1 1 nano mechanics’. Indeed, the mathematical tool of homogenization (a form of multi-scale analysis) [27,28], which is designed to deal with problems where quantities vary rapidly on a small (microscopic) scale, can be used to establish the fact that the response of an equivalent (homogenized) continuum can be anisotropic even if the behavior at the microstructure level is isotropic. Homogenization theory is the main tool used in the approach proposed in this paper to establish effective continuum models for the graphitic sheet and its nanotube image and to determine the mappings between them1 in this regard we note the recent work in, for example, [29,30] which is directed at passing, in a consistent fashion, from atomic and molecular models to equivalent continuum models. The fundamental objective of the work initiated in this paper is, therefore, to construct a mathematical model for the constitutive behavior of a graphitic sheet by applying homogenization theory to a hexagonal lattice of carbon atoms with a specific chirality vector (e.g., Figure 1) and to then extend the analysis

38 H. BELLOUT and F. BLOOM

Figure 1. Unit cell for a standard (6,3) nanotube.

to specific arrays of carbon atoms on the surface of a SWNT of the same chirality so as to determine the mapping which takes the graphitic sheet, modeled as a continuum, onto the surface of the nanotube modeled as a continuum. This modeling effort will employ MD simulations, i.e. empirically determined potentials to generate periodic, rapidly varying elastic constants for the graphitic sheet and will incorporate the torsional component of strain which is induced by the deformation of the sheet onto the nanotube surface1 we note the novelty associated with applying homogenization theory at the structure level of the graphitic sheet, a task which has not previously been attempted.

2. PROBLEMS WITH THE CONTINUUM MODELING OF CARBON NANOTUBES Since their discovery in the early 1990s there has been a steadily increasing interest in the new forms of carbon known as nanotubes1 part of this interest has been due to their novel structure and properties. Even more important is the wealth of potentially important appli-

SOME REMARKS ON APPLYING HOMOGENIZATION THEORY TO MODELING 39

cations in which nanotubes could figure prominently. As indicated, for example, in [24], beyond being remarkably stiff and strong, SWNTs are outstanding conductors of electricity and are ‘projected to conduct heat even better than diamond’. Already under development are SWNT nanodevices and nanocircuits exhibiting superior logic and amplification functions. Nanotubes are being studied for use as storage devices for alkali ions (e.g., nanoscale power sources). Because nanotubes are hollow, so that matter can be stored inside them as well as transported through them, they can be used in ultra sensitive sensor devices as well as in a variety of biomedical applications, i.e. for drug delivery through the distribution of very small quantities of gas or liquid into living cells or onto their surfaces. However, the major potential application of carbon nanotubes is in the area of new structural materials because of their exceptional mechanical properties. Composite materials based on carbon nanotubes have the potential of providing strengthto-weight ratios which exceed those of any materials currently available. As nanotubes have diameters as small as 1 nm, and lengths up to several micrometers, aspect ratios for nanotube fibers of well over 1,000 are realizable. As indicated in, for example, [31], ‘in contrast to standard carbon fibers nanotubes remain curved and interwoven in the composite suggesting extreme flexibility’. Effective utilization of carbon nanotubes in composite applications depends, as Qian et al. [32] point out, ‘on the ability to disperse the nanotubes homogeneously throughout the matrix without destroying the integrity of the nanotube’. Additionally, a prerequisite for improving the mechanical properties of polymer composites is that there be good interfacial bonding between the nanotubes and the polymer in order to achieve load transfer across the nanotube–matrix interface1 the vehicle for such load transfer is the interfacial shear stress. Provided the load can be effectively transferred to the nanotube, the modulus of the composite is conjectured to be that of randomly oriented short fiber composites containing fibers of extremely high modulus and strength1 such composites can be modeled by standard techniques in the theory of heterogeneous materials (see, e.g., [32] or [33]), or would themselves become a suitable object to which to apply the homogenization techniques described in Section 3, and an obvious prerequisite for either approach is a firm understanding of the mechanical behavior (constitutive theory, effective thickness) of an individual nanotube at the same continuum level as the matrix material. The same considerations apply when one considers MWNTs as potential reinforcing agents1 in the this case, however, the strength of the composite is also affected by the ease with which individual nanotubes slide with respect to each other. In a MWNT the outer nanotube is loaded but, because of the relatively weak bonding between nanotube layers, the load in not transferred to the inner tubes if they slip with respect to the outertubes1 the interactions between SWNTs in a bundle are similar to the interlayer interactions of MWNTs and, as pointed out in [34], slippage within bundles can easily occur. Thus [34], ‘isolation of SWNTs is highly desirable for composite fabrication’. Clearly, however, an effective continuum model for the mechanical response of MWNTs depends on first having such a model for SWNTs1 the appropriate starting point for any such model is the graphene sheet of which the nanotube is a ‘cylindrical’ image, i.e. the nanotube is the result of ‘rolling’ a graphene sheet by specifying both the direction of rolling and the circumference of the cross-section1 such ‘rolling’ does not currently include deformation of the hexagonal lattice. A graphene sheet is a honeycomb lattice of carbon atoms in a hexagonal array as depicted in Figure 1. In terms of the base vectors a1 2 a2 of a hexagon, the nanotube chirality

40 H. BELLOUT and F. BLOOM vector r 2 na1 3 ma2 , where n2 m are both integers with 0 4 5m5 3 n. The chiral angle 4 is the angle formed by r and a1 . An 5n2 m6 nanotube is constructed by using r as a ‘roll-up’ vector, i.e. the nanotube is constructed by rolling up the honeycomb sheet so that, with respect to the crystallographically equivalent sites A2 A6 2 B, and B 6 (t 2 O7B is the translational vector of the nanotube – the direction of the nanotube axis), A and A6 coincide as do points B and B 6 . The rectangle A A6 B 6 B defines the unit cell for the nanotube. The two basic SWNT structures conform to chiral vectors of type 5n2 06 (zig-zag nanotubes) and type 5n2 n6 (armchair nanotubes). The unit cell is capped by one-half of a C60 molecule. As pointed out in [3], it is believed that most nanotubes do not exhibit the highly symmetric forms associated with zig-zag and armchair nanotubes but, rather, have structures in which the ‘hexagons’ are arranged helically around the tube axis (chiral nanotubes). We make the important observation that the ‘hexagons’ lying on the idealized nanotube surface may be deformed versions of those which form the graphitic sheet, for example, as is clearly indicated in [12], with reference to the C–C bonds on the nanotube surface: ‘...the angles between these bonds depend on the radius of the cylinder as well as on their orientations 1 1 1 all angles approach 1208 (the value in a perfect graphitic plane) with increasing cylindrical radius. For tubules with smaller radii, for example, the bond angles deviate up to 9% from their planar value.’ Moreover, ‘in a fully optimized graphitic plane, the nearest neighbor distances of carbon atoms are all equal. Upon forming curved surfaces for tubules, the nearest neighbor C–C distances change depending on their relative orientations with respect to the cylindrical axis direction’. Thus, statements such as those in, for example, [13] which point to the nanotube surface as being ‘a conformal mapping of a two-dimensional honeycomb lattice’ (as depicted in Figure 1) do not accurately depict the deformation which leads from a unit cell of the graphitic sheet, as specified by a given chirality vector, to the surface of the (idealized) cylindrical nanotube. In earlier work, however, some authors had suggested using the elastic moduli of graphite for SWNTs by neglecting the change in molecular structure that occurs when a graphene sheet is rolled into a nanotube. In recent work [6] it has been emphasized that ‘the deformation is, in fact, not homogeneous as the graphene sheet is rolled into (a carbon nanotube). Correspondingly, the energy of the (carbon nanotube) not only depends on the deformation gradient but also depends on higher-order derivatives of (the deformation gradient). In such case, a set of high(er)-order elastic constants that belongs to the framework of multipolar theory needs to be determined’. Examples of multipolar effects in continua may be found, for example, in [35–38]. Thus, even if one could justify modeling the graphitic sheet as an isotropic elastic continuum (and we have already indicated that, contrary to current consensus opinion on this point, the response is probably not isotropic) one can not carry such isotropic response over to the nanotube surface as a consequence of the relevant mapping somehow being conformal1 it is not. In fact, recent MD simulations, based on Brenner’s potential [39], have been reported by the authors of [6] which lead to a prediction of anisotropic behavior in the basal plane of a nanotube surface, thus further supporting the hypotheses indicated above. Two other lines of argument support the hypothesis that the response of the nanotube surface, within a continuum formulation, is not that of an elastic isotropic material. If the graphitic sheet can be considered to be an isotropic elastic continuum whose conformal image is the nanotube surface, then the response of the nanotube should 9 be independent of chirality (we note that an 5n2 m6 nanotube has diameter dt 2 01246 n 2 3 nm 3 m 2 78 and

SOME REMARKS ON APPLYING HOMOGENIZATION THEORY TO MODELING 41 9 9 an associated chiral angle 4 2 sin 1 5 3m72 n 2 3 nm 3 m 2 )). In particular, as we roll up a unit cell on the graphitic sheet the response of the resulting structure (SWNT) should be independent of the orientation of the edge of the cell (along r) with respect to the lattice vector a1 1 this is not the case, for example, if one ‘rolls up’ an arbitrarily oriented rectangular subset of a sheet of rectilinearly orthotropic material, such as paper, into a cylinder (as has been demonstrated in [40]). Chirality independence had been claimed by several authors1 for example, the ‘early’ work [9] asserted that because of the hexagonal symmetry of a graphite sheet its elastic properties are two-dimensionally isotropic, and so the helicity of the tube plays no significant role. Lu [8] also announced results for carbon nanotubes which are almost entirely independent of both tube size and chirality. However, as noted in [19], ‘the apparent insensibility of the Young’s modulus on the tube size and chirality observed by Lu is due to the fact that an empirical pair potential was used in his calculations, and such a model will not reflect the effects that the curvature will have on the bonding properties of the system. In the limit of large tube diameters, we would expect that the elastic properties would correspond to those of a plane defect-free graphitic sheet’. The observations made in [19] have appeared in many other recent references1 for example, Yao and Lordi [18] observe that ‘for a nanotube of small radius, Young’s modulus depends on the radius and helicity of the tube owing to changes in bonding that effect the torsional strain’. Such changes in bonding are directly related to the non-homogeneity of the map from the graphene sheet to the surface of the nanotube as indicated in [1]. The failure to include torsional strain contributions in MD simulations of nanotube behavior explains why, as noted in [9], earlier calculations ‘did not reveal significant effects of helicity on elastic properties’. The authors of the recent work [18] note that the ‘exact magnitude of physical and electronic properties depends on the diameter and chirality of the nanotube’. In [41] the observation is made that for the nanotube, ‘the strain energy under large strain conditions shows significant sensitivity to helicity’, while Zhao et al. [2] remark that ‘the behavior of nanotubes under large tensile strain depends strongly on their symmetry and diameter 1 1 1 different orientations of the carbon bonds with respect to the strain axis in tubes of different symmetry lead to completely different scenarios’. Finally, Zhao et al. [2] observed ‘the dependence of the elastic limit (the onset of plastic yielding) on the chirality of the (nano) tube: for the same radius an 5n2 06, thus zig zag, tube has nearly twice the limit of an 5n2 n6, thus armchair, tube’. If an ‘isotropic’ graphitic sheet were, indeed, mapped conformally onto the surface of an ‘isotropic’ (in the basal plane) SWNT such chirality dependence would not exist! One additional set of results in the literature mitigate strongly against modeling a SWNT as an elastic shell which is elastically isotropic in its basal plane, namely, those results which relate Young’s modulus (for whose computation one needs an ‘effective’ nanotube thickness), bending stiffness, and in-plane stiffness. Most authors use the equilibrium interlayer spacing between adjacent nanotubes h 2 0134 nm as the effective thickness of SWNTs. However, Yakobson et al. [9] report an in-plane stiffness for a SWNT of C 2 59 eV/atom 2 360 J/m2 and a flexural rigidity of D 2 0185 eV, as well as a Poisson ration of 9 2 0119 (computed by measuring the reduction in diameter of a tube stretched in simulations)1 if one uses these measured values in conjunction with the standard relations associated with an ‘isotropic’ shell, i.e. D 2 Y h 3 71251 9 2 6 and C 2 Y h then it is determined that h 2 01066 nm while the Young’s modulus Y 2 515 TPa. The value of h 2 01066 nm obtained is actually smaller than the typical C–C bond length in the graphitic sheet while the value of

42 H. BELLOUT and F. BLOOM Y 2 515 TPa is several times higher that the usually quoted range of 1â€“2 TPa. However, if (as Ru [24] points out) one begins with a value of Y 2 111 TPa (which is consistent with h 2 0134 nm and C 2 360 J/m2 ) and substitutes these values into the above expression for D, a computed D is obtained which is 25 times larger than the measured value of D 2 0185 eV. Our contention is that the inconsistencies noted above represent yet one more consequence of assuming, for ab initio, that the nanotube, modeled as an elastic shell, is isotropic in its basal plane, thus implying the expression indicated above for D. For a cylindrical shell which is, for example, the image of a rectilinearly orthotropic sheet one would obtain (see, e.g., Bellout and Bloom [40], or Bloom and Coffin [42]) for the in-plane principal bending stiffnesses D1 2 Y h 3 71251 597 62 6 and D2 2 2 Y h 3 71251 5 nu7 62 6 which are equal if and only if the orthotropic ratio 2 c22 7c11 2 1, where c11 and c22 are, respectively, the (in-plane) elastic constants in the axial and radial directions. Thus, the construction of a continuum (elastic shell) model for the response of a nanotube, which does not depend on the a priori assumption of basal plane isotropic response, could very well (i) lead to the resolution of the inconsistencies noted above with respect to the measured data for Y and C and the values of effective shell thickness h, and (ii) provide a basis for the formulation of more realistic sets of constitutive relations for the response of nanotube/polymer composites.

3. IMPLEMENTATION OF A MATHEMATICAL HOMOGENIZATION METHODOLOGY FOR THE GRAPHITIC SHEET We offer here a brief synopsis of the ideas underlying homogenization theory and then proceed to a discussion of the manner in which we believe such ideas can be applied at the level of a single-layer graphitic sheet. Homogenization theory relates to structures that have two naturally associated length scales, a microscale l and macroscale L1 the relevant physical properties of the medium vary rapidly on the scale l and more slowly on the larger scale L. The ratio 2 l7L plays a key role in the homogenization process wherein it is assumed that every property of the medium is of the form F5x2 y6 with x the position vector, in Cartesian coordinates, of a point in the medium and y 2 x7 the vector of stretched coordinates. It is usually the case that the microstructure is periodic (which is reflected in the periodicity of F5x2 y6 with respect to y)1 this is certainly the case for the graphitic sheet by virtual of the hexagonal symmetry displayed in Figure 1. For the graphitic sheet x 2 5x 1 2 x 2 62 y 2 5x1 72 x2 76 but in its (flat) undeformed state relevant physical properties will depend only on y, i.e. F 2 F5y6. In homogenization theory, each unknown field quantity u 5x6 2 u5x2 x76 is conceived of as having a double-scale asymptotic expansion of the form u 5x6 2 uo 5x2 y6 3 u1 5x2 y6 3 2 u2 5x2 y6 3

(3.1)

where each of the terms ui 5x2 y6 is periodic in y with the same period as the structure in question1 the ansatz (3.1) yields, in many cases, a rigorous procedure for deducing macroscopic equations, in x, for the overall behavior of the medium.

SOME REMARKS ON APPLYING HOMOGENIZATION THEORY TO MODELING 43

In the situation we are interested in, the global macroscopic behavior will be reflected in a set of homogenized elastic constants aihj which mediate the elastic response of a graphitic sheet at the continuum level. At the microscopic level the mechanical response of the graphitic sheet will be governed by a conservation law of the form

xi

1 2

u aij 5x6 2 f

x j

(3.2)

x where the aij 5x6 2 ai j 5x2 6 are periodic in y with period Y, i.e. ai j 5x2 y6 2 ai j 5x2 y 3 Y6 and satisfy aij 2 a ji 2 aik i k j j 2 0. The period Y will serve to define a unit cell Y for the structure and will be defined below for the graphitic sheet1 and Y for our specific problem will be chosen so as to reflect a distinct dependence on the magnitude of the chirality vector as well as on the chiral angle. The (microscale) forcing function f is also assumed to be Y periodic in y. The basic idea in the homogenization scheme is to substitute the expansion (3.1) into (3.2) and then identify powers of 1 using the fact that

5x2 x76 2

xj

1

1 3

x j yj

2 (3.3)

this procedure yields, at order O5 2 6,

yi

1 2

uo ai j 2 02

yj

(3.4)

which is an equation for the Y periodic function uo 5x2 y6 whose only solution is uo 5x2 y6 uo 5x6. At order O5 1 6, assuming uo 5x6 is known, we obtain for u1 the following elliptic equation in a unit cell Y whose solution must satisfy Y periodicity:

yi

1 3 1 42

u

uo ai j 2 01 3

yj

xj

(3.5)

It may be shown (see [27,28]) that u1 can be expressed in the form u1 5x2 y6 2

uo k 3 u 1 5x6

xk

(3.6)

where u 1 5x6 is an arbitrary function of x representing the mean value of u6 5x2 y6 on Y and 5 k WY 2 H 1 5Y 65 is Y

1 periodic and 2 5Y 5

is the unique solution of the variational equation

6 Y

7 5y6dy 2 0

44 H. BELLOUT and F. BLOOM 6 ai j Y

k dy 2

y j yi

6 aik Y

dy2 WY 1

yi

(3.7)

If we now write (3.2) in divergence form (in terms of the Cauchy stress tensor t) as

div t 2 f

with tik 2 aij

u k 2

xj

(3.8)

expand t as t 5x6 2 to 5x2 y6 3 t1 5x2 y6 3 , substitute into the expression for t the expansion (3.1) for u 5x6, and then set 2 0, we obtain 1 tiko 5x2 y6

2 ai j

u 1

u ok 3 k

xj

yj

2 1

(3.9)

The integral form of the conservation 8 law in (3.8), 8 in a macroscopic domain D composed of whole periods, is, of course, D t nd S 2 D f dV , which, to first order in , can be shown to yield [27,28] 6

D

to nd Sx 2

6

fdx1

(3.10)

D

From (3.10) we obtain the macroscopic (elasticity) equations 0

div t 2 f with tioj 2 aikh

u oj 2

xk

(3.11)

where the homogenized coefficients (elastic constants) aikh are completely determined by the microstructure and are given by aikh 5x6

1 2 5Y 5

2 6 1

k aik 3 ai j dy1

yi Y

(3.12)

If the original coefficients ai j 2 ai j 5y6 are independent of x then the aihj are constants1 this will be the case for homogenization of the graphitic sheet as depicted in Figure 1 because of the strict periodicity of the â€˜microâ€™ (atomistic) structure. Before we describe how we intend to apply the framework described above to the homogenization of a graphitic sheet, it is essential to make the following observation: the homogenized medium may be anisotropic even if its behavior at the microstructure level is isotropic. Even if ai j 2 a i j , so that a j j 2 a and ai j 2 02 i 2 j, it follows from (3.12) that aikh 2

1 5Y 5

6 a Y

k dy2

yi

(3.13)

in which case it does not necessarily follow that aikh 2 0 for i 2 k. This observation is of central importance in terms of what it may imply for the (homogenized) macroscopic behavior of a graphitic sheet.

SOME REMARKS ON APPLYING HOMOGENIZATION THEORY TO MODELING 45

Figure 2. (a) A hexagonal array of carbon atoms. (b) Hexagonal array rotated 4 radians counterclockwise.

In order to apply homogenization theory to a ‘nanotube unit cell’, in an undeformed graphitic sheet (as depicted in Figure 1) we must identify both the microscale unit cell Y and the elasticity coefficients ai j 5x2 y6 which appear in the defining relationship (3.12) for the homogenized coefficients aihj 1 to delineate our ideas in this direction we refer to Figures 2(a) and 2(b). In Figure 2(a) we depict one ‘hexagonal’ array of carbon atoms in the unit cell A A6 B 6 B shown in Figure 1. With out loss of generality, we have located the vertex at which the bravis lattice vectors a1 2 a2 originate as the point with coordinates 502 06 in the x2 y plane. The chiral angle for the array from which this ‘hexagon’ has been selected is 4 and the chirality vector r has components 5n2 m6 with respect to the basis vectors a1 2 a2 1 the ‘hexagon’ is shown as having a ‘banded’ structure with domain Y2 being an idealized rectilinear domain of width . In Y2 non-zero elastic constants are generated by the forces associated with C–C bonding1 these forces will be modeled by MD simulations as indicated below. In the ‘interior’ of the ‘hexagon’ the elastic constants on the microscale are taken to be zero. Three important points must be made: (i) the assumption that the arrays of carbon atoms in the graphitic lattice are perfect ‘hexagons’ is a mathematical idealization only1 (ii) the band width is not well defined and may be chosen to be a parameter in the model1 and (iii) the curves bounding the domain Y2 need not be rectilinear as shown but will be idealized to be so in a first model – a restriction that may be easily removed later. In order to make clear the dependence of the homogenized coefficients aihj on both the 9 chiral angle 4 and the magnitude m 2 3 n 2 of the chirality vector r, we now proceed as follows: (i) we rotate the banded hexagon in Figure 2(a) 4 radians counterclockwise so that the rotated lattice vector a1 2 i.e. a61 in Figure 2(b), is in the direction of the chirality vector r1

46 H. BELLOUT and F. BLOOM (ii) we identify the vector of periodicity as being a62 , select for the microscale measure the length l of the line9segment O P in Figure 2(a)1 (iii) we choose 2 l7 m 2 3 n 2 1 and (iv) we define the domain of integration in (3.13) to be Y Y4 2 Y14 Y24 where the Yi4 2 i 2 12 2, as shown in Figure 2(b) are the rotated images of the planar domains Yi 2 i 2 12 2 of Figure 2(a). In the present situation (underformed lattice) the molecular level ai j 2 ai j 5y6 with

ai j 5y6 2

9 4

a i j 5y62 y Y1 2

02

(3.14) y Y24 1

We remark that, while the ai j 5y6 may appear to be discontinuous along the boundary between Y14 and Y24 , one may always introduce standard mathematical â€˜cut-offâ€™ functions so as to effect a smooth transition from Y14 to Y24 . The remaining issue is the specification of the a i j 5y6 in (3.15)1 these are obtained from MD simulations (see [43], [44] or [45]) of the energy associated with the Câ€“C bonding in graphitic sheets. The total energy associated with a typical Câ€“C bond has the form E 2 E R 3 E 4 3 E 3 E 3 E dW 3 E el 2

(3.15)

where E R is associated with bond stretching, E 4 with bond angle bending, E with torsion, E with inversions (out-of plane bending), E dW with Van der Waals interactions, and E el with electrostatic interactions. Functional forms for each energy term in (3.15) are given, explicitly, for example, in Yao and Lordi [34] and are based on MD simulations1 alternative forms for the components of E may be found in many places in the literature, for example in [46] which compares the consequences of using either standard â€˜Brennerâ€™ or â€˜modified Morseâ€™ potentials. Employing the empirical relation for the classical Tersoffâ€“Brenner model the authors of the recent review [6] have shown how to generate the elasticity tensor associated with any particular MD model by treating the energy E as the potential function for a hyperelastic material1 such a procedure, if properly implemented, will serve to generate the a i j 5y6 in (3.15). As the graphene sheet is â€˜rolled upâ€™ into a nanotube, the hexagonal array in the unit cell A A6 B 6 B deforms (see Figure 3 which depicts the different surface arrays for two types of SWNTs)1 the deformation gradient associated with this mapping (the mapping f : A B of Figure 4) is determined, as pointed out in, for example, [6] by â€˜the direction and length of the chirality vector and the subsequent mechanical relaxation after rollingâ€™. As is apparent from, for example, Figure 3, the failure mechanism of a SWNT is highly dependent on the chirality of the nanotube. Using the approach indicated in [6, Section 2.4], one may, for various configurations of deformed â€˜hexagonal arraysâ€™ on nanotube â€˜surfacesâ€™, conformally map those surfaces onto the plane (the mapping g : B C in Figure 4), to apply mathematical homogenization to the planar configuration C, and, subsequently, to determine the composition of the mappings f and g, i.e. go f : A B. The deformation gradient associated with the mapping go f will serve to carry the homogenized coefficients aihj , computed for

SOME REMARKS ON APPLYING HOMOGENIZATION THEORY TO MODELING 47

Figure 3. Standard armchair and zigzag nanotubes.

Figure 4. Mappings of the graphitic sheet.

the graphitic sheet, onto a set of associated homogenized coefficients a ihj for the nanotube! While the homogenized coefficients aihj are expected to be constants, because (as noted explicitly in [6]) â€˜the deformation is in fact not homogeneous as the graphene sheet is rolled into a carbon nanotubeâ€™, we fully expect that the homogenized elasticity coefficients a ihj will not only not represent an isotropic material but we will have a ihj 2 a ihj 5x6. The result of the

48 H. BELLOUT and F. BLOOM

process delineated above may even lead, as anticipated in [6], to a set of higher-order elastic constants and, thus, the need for a multipolar theory. We remark that on a purely continuum level it should be obvious that different orientations of the ‘hexagons’ in a graphitic sheet should effect the sheet’s elastic response1 for example, see the recent work of Milton et al. [47] in which changing the orientations of elliptical voids in an elastic plate drastically affects the overall elastic response of the plate.

4. APPLICATION TO A PARTICULAR HEXAGONAL ARRAY As indicated in (3.15), the total energy associated with a typical C–C bond has the form E 2 E R 3 E 4 3 E 3 E 3 E dW 3 E el 2 where E R is associated with bond stretching, E 4 with bond angle bending, E with torsion, E with inversions (out-of plane bending), E dW with Van der Waal’s interactions, and E el with electrostatic interactions. Functional forms for each energy term in (3.15) are based on MD simulations1 forms for the components of E may be found in the literature which employ either standard ‘Brenner’ or ‘modified Morse’ potentials. In the example which follows, however, we employ an energy of the type which has appeared in the series of NASA papers and reports [48–51], namely, E2

K irj 5ri j rioj 62 3

i2 j

K i4j 54 i jk 4 iojk 62 2

(4.1)

i2 j

where ri j is the length of the bond joining the ith and jth atom in the deformed hexagonal array, rioj refers to the undeformed hexagonal array, 4 i jk is the angle formed by the vectors directed along the bonds from the ith atom to the jth and kth atoms, respectively, in the deformed array, and 4 i0jk refers to the same quantity in the undeformed array (see Figure 5). The symbols K irj 2 K i4j refer to the MD force constants associated with stretching and angle variance. In Figure 6 we show a schematic of the deformation from a portion of the undeformed nanotube N to its deformed image N 6 1 bonds joining atoms P2 Q2 R in N , and making an angle 4 are mapped onto the bonds joining the points P 6 2 Q 6 2 R 6 (occupied by the same atoms after the deformation) in N 6 and making an angle 4 6 . Locating points in the undeformed configuration by coordinates X and denoting the deformation by x i 2 x i 5X 6, we note the following: if P X 2 Q X 3 d X , then P 6 x i 2 Q 6 x i 3 dx i and 5P Q52 2 d X d X 2

5P 6 Q 6 52 2 dx i dx i 2

where we sum on repeated indices. However, dx i 2 Fi d X 2

Fi

xi 1

X

SOME REMARKS ON APPLYING HOMOGENIZATION THEORY TO MODELING 49

Figure 5. Bond deformation in the hexagonal array.

Figure 6. Deformation of infinitesimal small elements.

50 H. BELLOUT and F. BLOOM Similarly, if R X 3 d X 2 R 6 2 x i 3 d x i , then d x i 2 Fi d x 1 It follows that 5P 6 Q 6 52 2 5P Q52 3 e d X d X 2

(4.2)

where FFT I. Also

P 6 Q 6 P 6 R 6 2 5P 6 Q 6 55Q 6 R 6 5 cos 4 6 2 e d X d X 3 5P Q55P R5 cos 42 so 1 e d X d X 3 5P Q55P R5 cos 4 1 5P 6 Q 6 55P 6 R 6 5

cos 4 6 2

(4.3)

In terms of the undeformed and deformed portions of the hexagonal structure depicted in Figure 5 where, for example, ri j is the vector directed along the bond from atom i to atom

j, 5 r i j 5 2 ri j , etc., and V i j 2 ri j 75ri j 5, we have, as a consequence of (4.2) and (4.3), ri j

1 3 V ij e V ij2

2

rioj

2

cos 4 oijk 3 V i j e V ik 1 1 3 V i j e V i j 1 3 V ik e V ik

(4.4a)

cos 4 i jk

(4.4b)

We now apply these concepts to the undeformed hexagonal array depicted in 9 Figure 7, o o o o o o where l 0 is arbitrary. In this case r12 2 r45 2 l2 r23 2 r34 2 r56 2 r61 2 l7 22 4 o342 2

4 o615 2 8 72, while 4 o226 2 4 o231 2 4 o453 2 4 o564 2 3874. Also V 12 2 512 062 V 23 2 91 2 91 91 2 91 91 2 91 91 2 91 V 2

V 2 5 12 062 V 2

V 2

2 2 2 34 45 56 61 2 2 2 2 2 2 2 2

and V i j 2 V ji . With the notation e2

e11 e12

e12 2 e21 2

e21 e22

we have, for example,

o516

T

o o V ij V ij e V ij 2 o526

V ij

e11 e12 e21 e22

o516

V ij o526 2

V ij

SOME REMARKS ON APPLYING HOMOGENIZATION THEORY TO MODELING 51

Figure 7. Undeformed hexagonal structure.

with ‘T ’ indicating the transpose of the column vector. For small local deformations, (4.4a) yields 1 ri j rioj rioj 5V i j e V i j 62 2

(4.5)

so, for example, o r12 r12

2 2

o 1 o r12 5V 12 e V 12 6 2 1 e11 e12 1 1 l 2 0 0 e21 e22

1 le11 1 2

(4.6a)

In similar fashion we compute that o 2 r23 r23

l 9 5e11 3 2e12 3 e22 62 4 2

(4.6b)

o r34 r34 2

l 9 5e11 2e12 3 e22 62 4 2

(4.6c)

o 2 r45 r45

l le11 2 2

(4.6d)

o r56 r56 2

l 9 5e11 3 2e12 3 e22 62 4 2

(4.6e)

52 H. BELLOUT and F. BLOOM l o 2 9 5e11 3 2e12 3 e22 61 r61 r61 4 2

(4.6f)

Using (4.4b) we have the approximation 4 i jk cos 1

cos 4 iojk 3 1

V i j e V ik 1 51 3 V i j e V i j 651 3 V ik e V ik 6 2 2

(4.7)

so, for example, 4 126 cos 1

cos 4 o126 3 1

V 12 eV 16 1 1 51 3 V 12 e V 12 651 3 V 16 e V 16 2 2

However, 1 V 12 e V 12 2 1 3 e11 2 2

V 16 e V 16 2

1 5e11 2e12 3 e22 6 2

and 1 V 12 e V 16 2 9 5e11 e12 62 which leads to 2 3 4 1 4 126 cos 1 1 5ei j 6 2

(4.8a)

where 1 5ei j 6 2

1 3 e11 e12 1 1 1 51 3 e11 651 3 5e11 2e12 3 e22 66 2 4

(4.8b)

From (4.4a) and (4.4b) we obtain 3 4 126

4 o126

1

cos

3 4 4 1 1

1

9 1 5ei j 6 cos

9 1 2 2

However, cos 1 5x6 cos 1 5a6 so with

d 1 5x a62 cos 1 x5x2a 5x a6 2 9 dx 1 a2

(4.9)

SOME REMARKS ON APPLYING HOMOGENIZATION THEORY TO MODELING 53 1 1 x 2 9 1 5ei j 62 a 2 9 2 2 2 equation (4.9) yields 4 126 4 o126

1 3 e11 e12

1 1 1 51 3 e11 651 3 5e11 2e12 3 e22 66 2 4 1 3 e11 e12

1 1 1 3 53e11 2e12 3 e22 6 4

if we neglect terms in the denominator which are quadratic in the ei j . Finally, for strains ei j such that 5ei j 5 33 1, 4 126 4 o126 e11 e12 2

(4.10a)

and in a similar fashion the linear approximations for the remaining 4 i jk 4 iojk are 4 231 4 o231 e11 e12 2

(4.10b)

1 4 342 4 o342 5e11 e22 62 2

(4.10c)

4 453 4 o453 e11 e12 2

(4.10d)

4 564 4 o564 e11 3 e12 2

(4.10e)

4 615 4 o615

1 5e11 e22 61 2

(4.10f)

Employing the results in (4.6a)–(4.6f) and (4.10a)–(4.10f) in (4.1) leads to 5 E5ei j 6 2 K r

1 2 2 l2 l e11 3 5e11 3 2e12 3 e22 62 4 32

3

l2 1 2 5e11 2e12 3 e22 62 3 l 2 e11 32 4

3

l2 5e11 2e12 3 e22 62 32

7 l2 2 3 5e11 2e12 3 e22 6 32 5 7 1 3 2K 4 5e11 e12 62 3 5e11 3 e12 62 3 5e11 e22 62 4

(4.11)

54 H. BELLOUT and F. BLOOM

which simplifies to 2 2 2 3 12 e12 3 22 e22 3 12 e11 e22 E5ei j 6 2 11 e11

(4.12a)

with 9

11 2

12 2

5 9 Kr l 2 3 K 4 2 8 2 1 K r l 2 3 4K 4 2 2

22 2

12 2

(4.12b)

1 1 Kr l 2 3 K 4 2 8 2 1 Kr l 2 3 K 4 1 4

In (4.11) and (4.12a) and (4.12b), K r 2 K irj 2 K 4 2 K i4j for i2 j 2 12 22 1 1 1 2 61 typical values for carbon nanotubes are 9 k cal

K r 2 462 900 mole nm2 2

K4

2 63

k cal 1 mole rad2

In general, for an elastic material with a quadratic potential in a state of plane strain, 1 E5ei j 6 2 ai j ei e j 2

(4.13a)

E 2 ai j e j 2

ei

(4.13b)

and ti 2

where e1 2 e11 2 e2 2 e22 2 e6 2 2e12 and t1 2 t11 2 t2 2 t22 2 t6 2 t12 , the ti j being the components of the Cauchy stress tensor (in Cartesian coordinates). In expanded form, (4.13a) reads E5ei j 6 2

1 1 2 2 2 3 a22 e22 3 2a66 e12 a11 e11 2 2

3 a12 e11 e22 3 2a16 e11 e12 3 2a26 e22 e12 1 Comparing (4.14) with (4.12a) and (4.12b)) yields

(4.14)

SOME REMARKS ON APPLYING HOMOGENIZATION THEORY TO MODELING 55 9

a11 2

a22 2

a66 2

a12 2

5 K r l 2 3 9K 4 4 1 Kr l 2 3 K4 4

(4.15)

1 K r l 2 3 2K 4 4 1 Kr l 2 K4 4

with a16 2 a26 2 0. Using (4.13b) in conjunction with (4.14) yields

t11

A11

t12 2 A21 t22

A31

A12

A13

e11

A22

A23 e12

A32

A33

(4.16)

e22

where 9 A11 2 a11 2 A12 2 2a16 2 02 A13 2 a12 2

A21 2 a16 2 02 A22 2 2a66 2 A23 2 a26 2 02

A31 2 a12 2 A32 2 2a26 2 02 A33 2 a22 1 Inserting the values of the Ai j in (4.16), we obtain

5 2 4 K r l 3 9K 4 A2 0 1 Kr l 2 K4 4

0 1 K r l 2 3 4K 4 2 0

1 2 Kr l K 4 4 0 1 2 Kr l 3 K 4 4

(4.17)

i.e. the resulting linearized constitutive relation (for the nanotube whose intrinsic hexagonal array conforms to that depicted in Figure 7) is that of a rectilinear orthotropic elastic material. For the example considered above, therefore, we are in a position to define, explicitly the aij 5x6 which appear in (3.2) (aij 5x6 2 ai j 5x2 x76 where the ai j 5x2 y6 are given by (4.17) for x Y1 (Figure 9 2(a)), vanish for x Y2 (Figure 2(a)) and are periodic in y with period Y where 5Y5 2 572l). The final task in this program, which is the subject of future work, would be to attempt to solve (numerically) the variational equation (3.7) so as to be able to compute the homogenized coefficients in (3.12). Acknowledgement. This research was supported, in part, by NSF Grant CMS-0402900.

56 H. BELLOUT and F. BLOOM

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SOME REMARKS ON APPLYING HOMOGENIZATION THEORY TO MODELING 57

[29] Friesecke, G. and James, R. D. A scheme for the passage from atomic to continuum theory for thin films, nanotubes, and nanorods. Journal of the Mechanics and Physics of Solids, 48, 1519–1540 (2002). [30] Blane, X., Le Bris, C. and Lions, P. C. From molecular models to continuum mechanics. Archive for Rational Mechanics and Analysis, 164, 341–381 (2002). [31] Schadler, L. S., Giannaris, S. C. and Ajayan, P. M. Load transfer in carbon nanotube epoxy composites. Applied Physics Letters, 73, 3842–2870 (1998). [32] Qian, D., Dickey, E. C., Andrews, R. and Rantell, T. Load transfer and deformation mechanisms in carbon nanotubepolystyrene composites. Applied Physics Letters, 76, 2868–2870 (2000). [33] Geng, H., Rosen, R., Zheng, B., Shimoda, H., Heming, L., Liu, J. and Zhou, O. Fabrication and properties of composites of poly(ethylene oxide) and functionalized carbon nanotubes. Advanced Materials, 14, 1387–1390 (2003). [34] Lordi, V. and Yao, N. Molecular mechanics of binding in carbon-nanotube polymer composites. Journal of Materials Research, 15, 2770–2779 (2002). [35] Bellout, H., Bloom, F. and Ne2cas, J. Phenomenological behavior of multipolar viscous fluids. Quarterly of Applied Mathematics, L, 559–583 (1992). [36] Bellout, H. and Bloom, F. On the uniqueness of plane Poiseulle solutions of the equations of incompressible bipolar fluids. International Journal of Engineering Sciences, 31, 1535–1549 (1993). [37] Bellout, H. and Bloom, F. Steady plane Poiseulle flows of incompressible multipolar fluids. International Journal of Nonlinear Mechanics, 23, 503–518 (1993). [38] Bloom, F. and Hao, W. Steady flows of nonlinear bipolar viscous fluids between rotating cylinders. Quarterly of Applied Mathematics, LIII, 143–171 (1995). [39] Tersoff, J. and Ruoff, R. S. Structural properties of a carbon-nanotube crystal. Physics Review Letters, 73, 676–679 (1994). [40] Bellout, H. and Bloom, F. Modeling the buckling of rectilinearly orthotropic truncated conical shells. Mathematical and Computer Modelling, 34, 195–227 (2001). [41] Ozaki, T., Iwasa, Y. and Mitani, T. Stiffness of single-walled carbon nanotubes under large strain. Physics Review Letters, 84, 1712–1715 (2000). [42] Bloom, F. and Coffin, D. W. Handbook of Thin Plate Buckling and Postbuckling, CRC Press, Boca Raton, FL, 2000. [43] Wise, K. and Hinkley, J. Molecular dynamics simulations of nanotube-polymer composites. APS Society Spring Meeting, Seattle, WA, April 12–16, 2001. [44] Rapaport, D. C. The Art of Molecular Dynamics Simulation, Cambridge University Press, Cambridge, 1995. [45] Dresselhaus, M. S., Dresselhaus, G. and Eklund, P. C. Science of Fullerene and Carbon Nanotubes, Academic Press, San Diego, CA, 1995. [46] Belytschko, T., Xiao, S., Schetz, G. and Ruoff, R. S. Atomistic simulations of nanotube fracture. Physics Review B, 65, 235430-1–235430-8 [47] Milton, G. W., Serkov, S. K. and Movehan, A. B. Realizable (average stress, average strain) pairs in a plate with holes. SIAM Journal of Applied Mathematics, 63, 987–1028 (2003). [48] Gates, T. S. and Hinkley, J. A. Computational materials: modeling and simulation of nanostructured materials and systems. Technical Memo NASA/TM-2003-21263, March 2003. [49] Odegard, G. M., Gates, T. S., Nicholson, L. M. and Wise, K. E. Equivalent-continuum modeling of nano-structured materials. Technical Memo NASA/TM-2001-210863, May 2001. [50] Odegard, G. M., Gates, T. S. and Wise, K. E. Constitutive modeling of nanotube-reinforced polymer composites. A1AA-2002-1427. [51] Odegard, G. M. and Gates, T. S. Constitutive modeling of nanotube/polymer composites with various nanotube orientations. Proceedings of the 2002 SEM Annual Conference and Exposition on Experimental and Applied Mechanics, Milwaukee, WI, June 10–12, 2002.

Eshelby’s Problem for a Bonded Piezo-elastic Bimaterial

L. J. S UDAK

Department of Mechanical and Manufacturing Engineering, University of Calgary, Calgary, Alberta, Canada T2N 1N4 (Received 16 June 20091 accepted 28 July 2009)

Abstract: A general method is presented for the analytic solution of Eshelby’s problem concerned with a smooth inclusion of arbitrary shape embedded within the elastic half-plane of a bonded piezoelectric–elastic bimaterial. Since the well-known Stroh formulation, on which almost all existing works on Eshelby’s problem in piezoelectric media has been based, breaks down for isotropic elastic materials, the solutions available in the literature cannot be applied directly. Consequently, the method of solution for the current problem involves a combination of Stroh’s formulation for piezoelectric materials and the well-known Muskhelishvili method for isotropic elastic bodies. The general solution is derived in terms of an auxiliary function which can be constructed using conformal mappings that map the exterior of the inclusion onto the exterior of the unit circle. To illustrate the method, the case of a dilatational inclusion is examined and detailed results are shown for the mean stress within a thermal elliptic inclusion.

Key Words: Eshelby’s problem, piezoelectric, elasticity, inclusion, complex variable

1. INTRODUCTION Stress analysis of an infinite elastic body which contains a subdomain undergoing uniform stress-free strains, commonly referred to as the Eshelby problem, is a classic topic. Among the numerous physical phenomena that lead to Eshelby’s problem, of particular significance are the stresses caused by thermal mismatch between dissimilar materials. Practical examples of current interest include passivated interconnect lines, isolation trenches in large-scale integrated circuits and strained semiconductor laser devices (see, for example, [1–4]). In the majority of these cases, since the difference between material constants of buried active components and surrounding materials is usually small, a common simplification adopted by many researchers is that the elastic constants of all constituents are identical (see, for example, [5, 6]). Motivated by the aforementioned practical problems, a significant amount of effort has been devoted to the study of Eshelby’s problem in a plane or half-plane (see, for example, [7–11]). In all of these cases the subdomain is always embedded either within one of two dissimilar bonded elastic half-planes or within one of two dissimilar bonded piezoelectric half-planes. However, with the recent advancement of so-called ‘smart com-

Mathematics and Mechanics of Solids 16: 58–76, 2011 1 The Author(s), 2011. Reprints and permissions: 1 http://www.sagepub.co.uk/journalsPermissions.nav

DOI: 10.1177/1081286509352836

ESHELBY’S PROBLEM FOR A BONDED PIEZO-ELASTIC BIMATERIAL 59

posites’ there has been no similar work to investigate Eshelby’s problem in these types of materials. Smart composites are materials where piezoelectric ceramics are often bonded to nonpiezoelectric materials such as polymers (see, for example, [12–14]). In these kinds of situations, polymers are widely used as electrically inactive materials in electromechanical devices. For these materials, polymers have piezoelectric and dielectric constants several orders of magnitude lower than electroceramics, consequently, they can be treated as non-polarizable, insulating, isotropic elastic materials [15]. Although a non-piezoelectric isotropic elastic material can be treated as a special case of piezoelectric materials with vanishing piezoelectric constants, the well-known Stroh formulation, on which almost all existing works on Eshelby’s problem in piezoelectric media have been based, becomes complicated in the degenerate case of isotropic elastic materials due to the appearance of repeated eigenvalues [16]. Consequently, existing solutions based on the Stroh formulation cannot be applied directly to isotropic elastic materials. It is worth mentioning that Chen and Lai [17] developed an exact correspondence between the plane piezoelectric equations and those of generalized plane strain elasticity. In particular, they show that by setting a linkage between the two sets of material constants any problem of a plane deformation in piezoelectricity may be solved as a generalized plane strain in elasticity and vice versa. In [18], Ou and Chen propose a new type of general solution for the potential function in the Lekhnitskii’s theory for plane strain problems in anisotropic materials. In fact, they show that in the case of isotropic elasticity their solution degenerates to the well-known Muskhelishvili solution thus unifying both theories. Recently, in [19], Ru investigated the effects of a piezoelectric/elastic interfacial crack by combining both the Stroh and Muskhelishvili formalisms (additional works in this area can be found in [20–23]). However, the ability to unify simultaneously the theories of Muskhelishvili, Stroh and Lekhnitskii’s still remains an open question. The present work considers the Eshelby problem of an arbitrary shaped smooth inclusion embedded within an isotropic elastic half-plane that is bonded to a piezoelectric half-plane. In order to facilitate a method of solution for the current problem, the well-known Muskhelishvili’s formulation for isotropic elastic materials is combined with the well-established Stroh formulation for piezoelectric materials. The exact solution is derived in terms of an auxiliary function which can be constructed by the conformal mapping from the exterior of the inclusion to the exterior of the unit circle and the material properties of the two halfplanes. To illustrate the method, detailed results are shown for the mean stress within a thermal elliptic inclusion.

2. A PIEZO-ELASTIC BIMATERIAL WITH A SINGLE ARBITRARY INCLUSION 2.1. Preliminaries

In the absence of body forces and free charges, the basic equations for a linear piezoelectric material, in a fixed rectangular coordinate system xi , i 2 11 21 3, are

60 L. J. SUDAK 2 01

2 i j1 j

Di1i 2 01

1 4u i1 j 3 u j1i 51 2

3i j

2

2ij

2 Ci jkl 3 kl 4 eki j E k 1

E i 2 4u 41i 1 Dk 2 eki j 3 i j 3 6kl El 1

i1 j1 k1 l 2 11 21 31

(1)

where u i 1 i 2 11 21 3, and u 4 denote the elastic displacement and electrical potential, 2 i j , 3 i j , E i and Di denote the Cauchy stress, the strain, the electric field and the electric displacement and Ci jkl , ei jk and 6i j denote the elastic, piezoelectric and dielectric material parameters, respectively. For two-dimensional problems in which all quantities depend on x1 and x2 only, one can seek the solution in the form [24] 1 2T u 2 u 1 u 2 u 3 u 4 2 a f 4x 1 3 px2 51

(2)

where 4u 1 1 u 2 5 are the displacements in the 4x1 1 x2 5 plane and u 3 is the anti-plane displacement perpendicular to the 4x1 1 x2 5 plane, a is a 4 5 1 constant column vector, p is a complex number and f 465 is an arbitrary analytic function. Thus, it can be shown that all 22 equations of 415 are satisfied for f 465 if the eigenrelation 1 2 Q 3 p4R 3 RT 5 3 p 2 T a 2 01

(3)

where the 3 5 3 matrix R and the two 3 5 3 symmetric matrices Q and T have elements defined by Q ik 2 Ci1k1 1

Rik 2 Ci1k2 1

Tik 2 Ci2k2 7

(4)

For a stable material with positive-definite energy density, Equation 435 provides eight eigenvalues and eight associated eigenvectors. Hence, there are four distinct conjugate pairs with positive imaginary part. Let pi 1 4i 2 11 21 31 45 be the four distinct eigenvalues with positive imaginary parts and let ai 4i 2 11 21 31 45 be the four associated normalized eigenvectors. Then the general solution of 415 can be expressed as u 2

1 2T u 1 u 2 u 3 u 4 2 Af4z5 3 Af4z51

1 2

1 2T 8 1 8 2 8 3 8 4 2 Bf4z5 3 Bf4z51

f4z5 2 zj

1

f 1 4z 1 5

f 2 4z 2 5

2 x1 3 p j x2 1

f 3 4z 3 5

2T f 4 4z 4 5 1

p j 2 9 j 3 i j 1

j 01

j 2 11 21 31 41

(5)

where 1 denotes the generalized stress potential and is defined in terms of the electrical displacements and the stresses as follows,

ESHELBYâ€™S PROBLEM FOR A BONDED PIEZO-ELASTIC BIMATERIAL 61 D1 2 48 412 1

D2 2 8 411 1

2 j1 2 48 j12 1

2 j2 2 8 j11 1

E 1 2 4u 4 11 1

j 2 11 21 31

E 2 2 4u 4 12 1

(6)

and the constant matrices A, B are defined by 2 1 A 2 a1 a2 a3 a4 1

1 2 B 2 b1 b2 b3 b4 1

(7)

where bi 2 4RT 3 pi T5ai 2

41 4Q 3 pi R5ai 1 pi

i 2 11 21 31 47

(8)

Remark 1. We only consider transversely isotropic poled ferroelectrics that have their poling direction in either the x1 or x2 direction. In this case, with the poling axis parallel to the 4x 1 1 x 2 5 plane the in-plane deformation and the electric potential in the 4x1 1 x2 5 plane are decoupled from the anti-plane displacement u 3 . Thus, with zero displacement in the x3 direction the anti-plane component can be deleted. This means that the third row and the third column from all matrices in the above Stroh formulation will be removed and all matrices will subsequently be reduced from four to three dimensions. For plane isotropic elasticity the elastic stresses and displacements, in the absence of body forces, are given in terms of two complex potentials 4z5 and 4z5 by [25] 2 4u 1 3 iu 2 5 2 4z5 4 z 7 4z5 4 4z51 3 4 2 11 3 2 22 2 2 7 4z5 3 7 4z5 1 2 11 4 i2 12 2 7 4z5 3 7 4z5 4 z 77 4z5 4 7 4z5 1 4 3 [F1 3 i F2 ] 2 4i 4z5 3 z 7 4z5 3 4z5 1

(9)

AB

where z 2 x1 3 i x2 is the complex coordinate, 2 43 4 45 for plane strain and 2 43 4 541 3 5 for plane stress and 1 are the shear modulus and Poissonâ€™s ratio, respectively, 4F1 1 F2 5 denotes the resultant force and [6] AB denotes the increase undergone by the bracketed expression as the point z passes along the arc from A to B. 2.2. Formulation

Let us consider a piezoelectric half-plane 4x2 1 05 that is bonded to an isotropic dielectric elastic half-plane 4x2 2 05 along the whole real axis. Furthermore, let us assume that the lower half-plane contains a subdomain of arbitrary shape which has the same5material con6 stants as the lower half-plane and undergoes uniform stress-free eigenstrains 6611 1 6612 1 6622 .

62 L. J. SUDAK

Let S0 and S I denote the subdomain and its supplement to the lower half-plane, respectively, denotes the interface separating S0 and SI and SI I denotes the upper half-plane. Unless otherwise stated, all physical quantities in S0 , S I and SI I will be identified with the subscripts 01 I1 I I , respectively. Note that due to the compatibility conditions at the bonded interface the respective uniform remote loadings given in the two half-planes must be compatible in order to give rise to uniform stress and electric fields. The basic boundary value problem for the Eshelby problem considered in this work is given as 4u 1 3 iu 2 5 I 2 4u 1 3 iu 2 50 3 u 0 4z51 42 22 3 i2 12 50 2 42 22 3 i2 12 5 I 1 4 23 21 2 2 21 1

4 23 22 2 2 22 1

z 8 1

4 u3 1 2 u1 1

4 u3 2 2 u2 1

D23 2 01

x2 2 01

z 9 4x2 2 051

I 4z5 9 01

I 4z5 9 01

f i 4z5 9 01

z 9 4x2 1 051

4i 2 11 21 451

(10)

6 5 where u 0 4z5 is the additional displacement within S0 due to the eigenstrains 6611 1 6612 1 6622 and is given by 5 6 u 0 4z5 2 601 z 3 602 3 i603 z1 601 2

6611 3 6622 1 2

602 2

6611 4 6622 1 2

603 2 6612 7

(11)

The conditions at the interface, , represent the continuity of tractions and displacements. Along the real axis the superscripts ‘3’ and ‘4’ indicate the limit values taken from the upper piezoelectric half-plane and the lower isotropic elastic half-plane, respectively, and the final conditions represent that all physical quantities must vanish at infinity. Thus, the problem is to determine three analytic functions f k 465 4k 2 11 21 45 in the upper piezoelectric half-plane and four analytic functions k 4z51 k 4z5 4k 2 01 I 5 in the lower elastic half-plane. To accommodate an inclusion of arbitrary shape, conformal mapping is employed to map the exterior of the inclusion onto the exterior of the unit circle. It is well established that the exterior of the boundary curve in the z-plane can be mapped onto the exterior of the unit circle in the image plane by a conformal mapping function of the form [25] z 2 m4 5 2 3

7

ck 4k 1

(12)

k40

where is a real number and ck 4k 2 01 11 21 7 7 75 are some complex coefficients. Numerous methods for the accurate or approximate determination of the mapping function have been described in the literature (see, for example, [11] and [26] for details). For simple smooth boundary curves, the mapping function 4125 includes only a finite number of terms. However, in many other cases where the boundary curve is not so simple, the infinite series in

ESHELBYâ€™S PROBLEM FOR A BONDED PIEZO-ELASTIC BIMATERIAL 63 4125 can be truncated to a finite number of terms (say N terms) which would offer a good approximation to 4125. Hence, let us assume that such a polynomial mapping function, m4 5, exists for the curve . It can be shown that there exists a function D4z5 which satisfies the condition [9] z 2 D4z51

z 8 1

(13)

which is analytic in the exterior of except at infinity where it has a pole of finite degree determined by the asymptotic condition D4z5 9 P4z5 3 o4151

z 9 1

(14)

where P4z5 is a polynomial in z of finite degree. In the context of this presentation, let us assume then that the function D4z5 exists. On using 495 and 4115, the interface conditions 4105 can be written along in the equivalent form as follows I I 4z5 4 z 7I 4z5 4 I 4z5 2 I 0 4z5 4 z 70 4z5 4 0 4z5 1 6 2 5 3 2 I 601 z 3 602 3 i603 z 1

I 4z5 3 z 7I 4z5 3 I 4z5 2 0 4z5 3 z 70 4z5 3 0 4z51

z 8 7

(15)

Using 4135 and by adding the above two equations yields

0 4z5 2 I 4z5 4

6 2 5 2 I 1 0 61 z 3 602 3 i603 D4z5 1 1 3 I

z 8 7

(16)

Since the left-hand side and the right-hand side are analytic in the lower half-plane inside and outside the curve , respectively, any one of the functions can be analytically extended to the other across the curve . Hence, their derivatives must also be continuous across . Using this fact together with 4165 the second equation from 4155 is written as follows 0 4z5 2 I 4z5 3

6 6 2 5 5 2 I 1 0 261 D4z5 3 602 3 i603 D4z5D 7 4z5 3 602 4 i603 z 1 1 3 I

z 8 7

(17)

Note that the right-hand sides of 4155 and 4165 are analytic outside where they have poles of finite degree whereas the left-hand sides are analytic inside . Thus, let us define two new functions X 4z5 and Y 4z5 as follows

X 4z5 2

8 9

0 4z51

z 8 S0 1

6 2 5 2 I 1 0 9 I 4z5 4 61 z 3 602 3 i603 D4z5 1 1 3 I

z 8 SI 1

64 L. J. SUDAK 8 0 4z51 z 8 S0 1 9 9 9 9

6 6 2 5 5 2 I 1 0 Y 4z5 2 I 4z5 3 261 D4z5 3 602 3 i603 D4z5D 7 4z5 3 602 4 i603 z 1 9 1 3 I 9 9 9 z 8 SI 1

(18)

which are analytic in S0 and S I , respectively. Clearly, X 4z5 and Y 4z5 are continuous across and therefore analytic in the whole lower half-plane except at infinity where they have a pole of finite degree given by X 4z5 9 4 Y 4z5 9

6 2 5 2 I 1 0 61 z 3 602 3 i603 P4z5 1 1 3 I

z 9 1

5 6 5 6 2 2 I 1 0 261 P4z5 3 602 3 i603 Q4z5 3 602 4 i603 z 1 1 3 I

z 9 1 (19)

where Q4z5 is a polynomial in z and is defined as follows Q4z5 2 D4z5D 7 4z5 3 o4151

z 9 7

(20)

It immediately follows that

I 4z5 2

6 2 5 2 I 1 0 61 z 3 602 3 i603 D4z5 1 1 3 I

X 4z5 3

I 4z5 2 Y 4z5 4

z 8 SI 1

6 6 2 5 5 2 I 1 0 261 D4z5 3 602 3 i603 D4z5D 7 4z5 3 602 4 i603 z 1 1 3 I

z 8 SI 7

(21)

It now remains to examine the remaining interface conditions along the real axis (x2 2 0). In view of the fact that the in-plane displacements and electrical displacements are decoupled from the antiplane displacement for poled ferroelectrics, then replacing the complex variable z of the ith component of each Stroh function by the complex variable z i , the displacements, the associated stresses and the electrical displacements are obtained from 455 and are given by

f 17 4z 1 5 u 211 2 2 Re A f 7 4z 2 5 1 2 7 f 4 4z 4 5 4E 1 7

f 1 4z 1 5 2 21 2 22 2 2 Re B f 7 4z 2 5 1 2 7 f 4 4z 4 5 D2 u 111

p1 f 17 4z 1 5 u 212 2 2 Re A p2 f 7 4z 2 5 1 2 7 p4 f 4 4z 4 5 4E 2

2 11 p1 f 17 4z 1 5 2 12 2 42 Re B p2 f 7 4z 2 5 7 2 7 p4 f 4 4z 4 5 D1

u 112

(22)

ESHELBYâ€™S PROBLEM FOR A BONDED PIEZO-ELASTIC BIMATERIAL 65 Note that for a given poled ferroelectric material the eigenvalue problem 435 must be solved to obtain the eigenvalues pi 4i 2 11 21 45. Once these and the Stroh functions f k 465 4k 2 11 21 45 are determined, then all physical quantities in the upper half-plane can be calculated. For convenience, let us define a new set of analytic functions in the upper half-plane in terms of the Stroh function f k 465 4k 2 11 21 45 as follows

7 f 1 4z 1 5 h 1 4z5 h 2 4z5 2 B f 7 4z 2 5 7 2

(23)

f 47 4z 4 5

h 4 4z5

Since the 3 5 3 matrix B is invertible, then once the three functions h k 4z5 4k 2 11 21 45 are found the three Stroh functions, f k 465 4k 2 11 21 45, can be uniquely determined and all physical quantities are then known via 4225. In view of 4235 and the fact that z k z 4k 2 11 21 45 on the real axis (see [27]), it follows from 4225 that

2 21 h 1 4z5 2 22 2 2 Re h 2 4z5 1 D2

x 2 2 03 7

(24)

h 4 4z5

Thus, with the electrical boundary condition along the real axis and the asymptotic condition, D2 2 0, at infinity gives h 4 4z5 3 h 4 4z5 2 01 h 4 4z5 2 01

x2 2 03 1

z 9 1

(25)

which yields h 4 4z5 2 01

x2 1 07

(26)

It should be noted that due to the electromechanical coupling effect of piezoelectric materials, the in-plane electric field is coupled with the in-plane deformation in the upper half-plane even with h 4 4z5 2 0. In fact, it is seen from 4235 that even though h 4 4z5 2 0 does not necessarily imply that f 4 465 2 0. There are still three respective complex variables z k 4k 2 11 21 45 in the calculation of the electro-elastic fields in the upper half-plane (see 4225). Now, in view of the interface conditions 4105 and the relations 4951 4215 and 4245 the continuity of mechanical tractions along the whole real axis can be equivalently expressed as 1 2 h 2 4z5 3 h 2 4z5 3 i h 1 4z5 3 h 1 4z5 2

X 7 4z5 3

2 6 2 I 1 0 5 0 61 3 62 3 i603 D 7 4z5 3 X 7 4z5 1 3 I

66 L. J. SUDAK

3

6 6 2 1 2 2 I 5 0 2 I 1 0 5 0 61 3 62 4 i603 D 7 4z5 3 z X 77 4z5 3 Y 7 4z5 3 62 3 i603 z D 77 4z5 1 3 I 1 3 I

4

5 2 5 61 62 2 I 1 0 7 261 D 4z5 3 602 3 i603 D 7 4z5D 7 4z5 3 D4z5D 77 4z5 3 602 4 i603 1 1 3 I x2 2 07

(27)

Since D4z5 and its derivative are analytic in the upper half-plane including the real axis, the functions D4z5 and D 7 4z5 are analytic in the lower half-plane including the real axis. Thus, 4275 can be rewritten in the form h 2 4z5 3 i h 1 4z5 4

2 6 6 2 I 1 0 5 0 2 I 5 0 61 3 62 3 i603 D 7 4z5 4 62 3 i603 z D 77 4z5 1 3 I 1 3 I

3

61 62 5 2 5 2 I 1 0 7 261 D 4z5 3 602 3 i603 D 7 4z5D 7 4z5 3 D4z5D 77 4z5 3 602 4 i603 4 X 7 4z5 1 3 I

2

X 7 4z5 3

2 1 2 6 2 I 1 0 5 0 61 3 62 4 i603 D 7 4z5 3 z X 77 4z5 3 Y 7 4z5 4 h 2 4z5 4 i h 1 4z51 1 3 I

x 2 2 07

(28)

Since the right- and left-hand sides of 4285 are analytic in the lower and upper half-planes, respectively, and approach the same polynomial 6 6 6 2 I 5 0 2 I 5 0 2 I 5 0 62 4 i603 P 7 4z5 4 62 3 i603 P 7 4z5 4 62 3 i603 z P 77 4z5 1 3 I 1 3 I 1 3 I 3

5 5 6 62 2 I 1 0 7 261 P 4z5 3 602 3 i603 Q 7 4z5 3 602 4 i603 1 1 3 I

z 9 1

(29)

it is concluded that the right- and left-hand sides are equal to the above polynomial in the upper and lower half-planes, respectively. Hence, it follows from the continuity condition 4285 that h 2 4z5 3 i h 1 4z5 4 X 7 4z5 4

6 6 2 2 I 1 0 5 0 2 I 5 0 61 3 62 3 i603 D 7 4z5 4 62 3 i603 z D 77 4z5 1 3 I 1 3 I

3

5 2 5 61 62 2 I 1 0 7 261 D 4z5 3 602 3 i603 D 7 4z5D 7 4z5 3 D4z5D 77 4z5 3 602 4 i603 1 3 I

2

6 6 6 2 I 5 0 2 I 5 0 2 I 5 0 62 4 i603 P 7 4z5 4 62 3 i603 P 7 4z5 4 62 3 i603 z P 77 4z5 1 3 I 1 3 I 1 3 I

3

5 5 6 62 2 I 1 0 7 261 P 4z5 3 602 3 i603 Q 7 4z5 3 602 4 i603 1 1 3 I

x2 1 01

(30)

ESHELBYâ€™S PROBLEM FOR A BONDED PIEZO-ELASTIC BIMATERIAL 67

and 1 2 2 6 2 I 1 0 5 0 X 7 4z5 3 z X 77 4z5 3 Y 7 4z5 4 h 2 4z5 4 ih 1 4z5 3 61 3 62 4 i603 D 7 4z5 1 3 I 2

6 6 6 2 I 5 0 2 I 5 0 2 I 5 0 62 4 i603 P 7 4z5 4 62 3 i603 P 7 4z5 4 62 3 i603 z P 77 4z5 1 3 I 1 3 I 1 3 I

3

5 5 6 62 2 I 1 0 7 261 P 4z5 3 602 3 i603 Q 7 4z5 3 602 4 i603 1 1 3 I

x2 07

(31)

Substituting 4305 into 4315 and eliminating h 2 4z5 yields 1

2 2 61 2 I 5 0 z X 77 4z5 3 Y 7 4z5 4 62 4 i603 D 77 4z5 4 P 77 4z5 z 1 3 I

3

61 6 5 22 2 I 1 0 5 7 261 D 4z5 4 P 7 4z5 3 602 4 i603 D 7 4z5D 7 4z5 3 D4z5D 77 4z5 4 Q 7 4z5 1 3 I

3

5 5 6 6 62 2 I 5 0 2 I 1 0 7 62 3 i603 z P 77 4z5 4 261 P 4z5 3 602 3 i603 Q 7 4z5 3 602 4 i603 1 3 I 1 3 I

2 2i h 1 4z51

x2 07

(32)

In order to consider the remaining boundary condition along the real axis we note, from 4225, that f 17 4z 1 5 u 211 2 2 Im YB f 27 4z 2 5 1 f 47 4z 4 5 4E 1

u 111

(33)

where the constant matrix Y is defined in terms of the matrices A1 B as Y 2 iAB41 . Thus, in view of 4235 and 4265, Equation 4335 can be rewritten as h 1 4z5 u 4z5 h 2 2 Im Y 211 2 1 0 4E 1

u 111

x2 2 07

(34)

along the real axis. Since the displacements across the real axis must be continuous, it immediately follows via 495 and 4345 that the continuity of displacements across the real axis can be equivalently expressed as

68 L. J. SUDAK 42 I 4Y21 3 iY11 5h 1 4z5 3 2 I 4Y21 3 iY11 5h 1 4z5 2 I 4Y22 3 i Y12 5h 2 4z5 3 2 I 4Y22 3 iY12 5h 2 4z5 2 I 7I 4z5 4 7I 4z5 4 z 77I 4z5 4 7I 4z51

x2 2 07

(35)

Using 4305 to eliminate h 2 4z5 from 4355 yields the following 1 2 4 2 I [4Y21 3 Y12 5 3 i 4Y11 4 Y22 5] h 1 4z5 4 2 I 4Y22 3 iY12 5 3 I X 7 4z5 3 2 I

15 6 5 62 2 I 5 0 6 Y22 3 Y22 3 i Y12 3 Y12 62 3 i603 P 7 4z5 1 3 I

4 2 I 4Y22 3 iY12 5 5

6 5 22 61 2 I 1 0 5 7 261 D 4z5 4 P 7 4z5 3 602 3 i603 D 7 4z5D 7 4z5 3 D4z5D 77 4z5 4 Q 7 4z5 1 3 I

4 2 I 4Y22 3 iY12 5 3

61 2 2 I 5 0 62 3 i603 D 77 4z5 4 P 77 4z5 z 3 2 I 4Y22 3 iY12 5 1 3 I

6 2 2 I 1 0 5 0 61 3 62 3 i603 D 7 4z5 1 3 I

2 6 6 2 I 1 0 5 0 2 I 5 0 61 3 62 3 i603 D 7 4z5 3 62 3 i603 z D 77 4z5 1 3 I 1 3 I

61 62 5 2 5 2 I 1 0 7 261 D 4z5 3 602 3 i603 D 7 4z5D 7 4z5 3 D4z5D 77 4z5 3 602 4 i603 1 3 I 15 6 5 62 5 6 2 1 2 42 I Y21 4 Y12 3 i Y11 3 Y22 h 1 4z5 4 2 I Y22 3 i Y12 3 1 X 7 4z5

4

3 2 I

15 6 5 62 2 I 5 0 6 Y22 3 Y22 3 i Y12 3 Y12 62 4 i603 P 7 4z5 1 3 I

5 6 2 I 5 0 61 5 6 2 62 4 i603 D 77 4z5 4 P 77 4z5 z 3 2 I Y22 3 iY12 4 2 I Y22 3 iY12 1 3 I 5

6 5 22 61 2 I 1 0 5 7 261 D 4z5 4 P 7 4z5 3 602 4 i603 D 7 4z5D 7 4z5 3 D4z5D 77 4z5 4 Q 7 4z5 1 3 I

5 6 2 I 1 0 5 0 6 2 4 2 I Y22 3 iY12 61 3 62 4 i603 D 7 4z5 1 3 I 3

2 1 2 6 2 I I 1 0 5 0 61 3 62 4 i603 D 7 4z5 4 z X 77 4z5 3 Y 7 4z5 1 1 3 I

x2 2 07

(36)

As before, D4z5 and its derivative are analytic in the upper half-plane including the real axis, then the functions D4z5 and D 7 4z5 are analytic in the lower half-plane including the real axis. Since the right and left of 4365 are analytic in the lower and upper half-planes, respectively, it follows that the right- and left-hand side of 4365 approach the same polynomial

ESHELBYâ€™S PROBLEM FOR A BONDED PIEZO-ELASTIC BIMATERIAL 69 5 6 65 6 2 2 I 15 2 I Y22 3 iY12 3 1 602 3 i603 P 7 4z5 3 601 3 2 I 4Y22 3 iY12 5 1 3 I 5

2 6 6 2 I I 1 0 5 0 2 I 5 0 62 4 i603 P 7 4z5 3 61 3 62 4 i603 P 7 4z5 1 3 I 1 3 I

3

6 6 62 5 5 2 I 1 0 7 2 I 5 0 62 3 i603 z P 77 4z5 4 261 P 4z5 3 602 3 i603 Q 7 4z5 3 602 4 i603 1 1 3 I 1 3 I

z 9 7

(37)

This implies that the right- and left-hand side of 4365 are equal to the polynomial 4375 in the lower and upper half-planes, that is, 1

5 6 2 2 1 z X 77 4z5 3 Y 7 4z5 3 2 I Y22 3 iY12 3 1 X 7 4z5 15 6 5 62 42 I Y21 4 Y12 3 i Y11 3 Y22 h 1 4z5

5 6 2 I 1 0 5 6 2 2 I Y22 3 iY12 261 D 7 4z5 4 P 7 4z5 1 3 I 5 0 6 1 22 3 62 4 i603 D 7 4z5D 7 4z5 3 D4z5D 77 4z5 4 Q 7 4z5 3

6 62 5 5 2 I 1 0 7 261 P 4z5 3 602 3 i603 Q 7 4z5 3 602 4 i603 1 3 I

4

5 6 25 65 6 2 I 1 2 I Y22 3 iY12 4 I 602 4 i603 D 7 4z5 4 P 7 4z5 1 3 I

5 6 2 I 5 0 61 2 62 4 i603 D 77 4z5 4 P 77 4z5 z 4 2 I Y22 3 iY12 1 3 I 6 5 6 2 2 I 5 0 2 I 1 62 3 i603 z P 77 4z5 4 2 I Y22 3 iY12 3 1 1 3 I 1 3 I 1 0 5 0 2 6 5 61 3 62 3 i603 P 7 4z5 1 x2 01

4

(38)

and 1 2 2 I [4Y21 3 Y12 5 3 i 4Y11 4 Y22 5] h 1 4z5 3 2 I 4Y22 3 iY12 5 3 I X 7 4z5 5 6 2 2 I 1 2 I 4Y22 3 iY12 5 4 1 2601 D 7 4z5 4 P 7 4z5 2 1 3 I 5 0 65 61 6 5 2 4 62 3 i603 D 7 4z5 4 P 7 4z5 4 602 3 i603 D 77 4z5 4 P 77 4z5 z 5 61 2 3 602 3 i603 D 7 4z5D 7 4z5 3 D4z5D 77 4z5 4 Q 7 4z5 4

5 2 21 6 2 I 1 2 I 4Y22 3 iY12 5 3 I 601 3 602 4 i603 P 7 4z5 1 1 3 I

x2 1 07

(39)

70 L. J. SUDAK

Thus, the exact solution for the Eshelby problem of an inclusion of arbitrary shape embedded within an isotropic elastic half-plane and bonded to a piezoelectric half-plane is given explicitly by 43051 43251 4385 and 4395 for the unknown functions h 1 4z51 h 2 4z51 X 4z5 and Y 4z5 in terms of the auxiliary function D4z5 and the associated polynomials P4z5 and Q4z5. Note that D4z5 is given explicitly in terms of the inverse of the conformal mapping function for the boundary curve of the inclusion. In fact, the solution is exact provided the conformal mapping function includes only a finite number of terms. Alternatively, if the mapping function includes an infinite number of terms, then a suitable truncated conformal mapping function should be used to give an approximate solution. Remark 2. If we consider the matrix Y for an elastic isotropic, non-piezoelectric dielectric in plane strain [28] 1 4 II 1 4 II 0 i I I I I 1 4 1 4 I I I I Y2 0 4i 1 I I I I 1 0 0 4 6

then by letting I I 9 it can be shown that the expressions for Y 4z5 and X 4z5 given by 4385 and 4395 are identical to those in [11] in the lower elastic half-plane. The stresses and the electric field in the entire upper half-plane can be calculated via 42251 however, one first needs to find out the three functions f k 465 4k 2 11 21 45 via 4235 in terms of the known h k 4z5 4k 2 11 21 45 and to substitute the respective complex variables z k 4k 2 11 21 45. For the isotropic elastic material in the lower half-plane the stress field in the entire lower half-plane and its subdomain of arbitrary shape, not restricted to the limit values on the real axis, can be easily calculated from 495 once the original complex potentials I 4z51 I 4z5 have been determined via 4215 and X 4z5 2 0 4z51

z 8 S0 1

Y 4z5 2 0 4z51

z 8 S0 1

(40)

respectively. In view of their practical significance let us discuss in the following section, as an example, the thermal inclusion.

3. THERMAL INCLUSION In the case of a thermal inclusion, the stress-free eigenstrains are caused by a uniform change in temperature throughout the whole plane and thermal mismatch between the inclusion and the surrounding materials. For simplicity, we ignore the thermal mismatch between the upper

ESHELBYâ€™S PROBLEM FOR A BONDED PIEZO-ELASTIC BIMATERIAL 71

and lower half-planes and consider only thermal stresses caused by the thermal mismatch between the inclusion and the surrounding material. Thus, 601 2

90 4 9 I T1 2

602 2 603 2 01

(41)

where 9 denotes the thermal expansion coefficient and T is the uniform chance in temperature. Hence, upon simplification, the expressions (30), (32), (38) and 4395 reduce to the following 1 2 2 I [4Y21 3 Y12 5 3 i 4Y11 4 Y22 5] h 1 4z5 3 2 I 4Y22 3 iY12 5 3 I X 7 4z5 5 62 21 2 I 1 2 I 4Y22 3 i Y12 5 4 1 2601 D 7 4z5 4 P 7 4z5 1 3 I 4

2 2 I 601 1 2 I 4Y22 3 i Y12 5 3 I 1 x 2 1 01 1 3 I 5 6 2 15 6 5 62 2 1 1 2 I Y22 3 iY12 3 1 X 7 4z5 3 2 I Y21 4 Y12 3 i Y11 3 Y22 3 2i h 1 4z5 5 6 21 5 6 2 2 I 601 1 2 I Y22 3 i Y12 3 1 2 D 7 4z5 4 P 7 4z5 4 1 1 1 3 I

x 2 07

(42)

Clearly, 4425 represents a set of linear algebraic equations for the unknown function h 1 4z5 and X 7 4z5. Solving for X 7 4z5 yields 41 3 I 5 X 7 4z5 2 4 I 601

6 5 7 D 4z5 4 P 7 4z5 4 2 I 601 1

x2 01

(43)

where 1 5 6 2 15 6 5 62 2 2 I 2 I Y22 3 iY12 3 1 Y21 3 Y12 4 i Y11 4 Y22 1 5 6 21 15 6 5 62 2 4 2 I Y22 4 iY12 4 1 2 I Y21 4 Y12 3 i Y11 3 Y22 3 2i 1 15 6 5 62 1 5 6 2 2 2 I Y21 3 Y12 4 i Y11 4 Y22 2 I Y22 3 i Y12 3 1 1 15 6 5 62 21 5 6 2 4 2 I Y21 4 Y12 3 i Y11 3 Y22 3 2i 2 I Y22 4 i Y12 3 I 7

(44)

Thus, the general solution of 4435 is given explicitly by z 41 3 I 5 X 4z5 2 z0

4 I 601

5

D 7 4z5

4

6

P 7 4z5

4

2 I 601

dz 3 C0 1

x2 01

(45)

where z 0 is an arbitrarily chosen point in the lower half-plane with the integration taken along any arbitrary chosen curve in the lower half-plane and C0 is an arbitrary constant of integration determined by z 0 and the asymptotic behavior at infinity. In particular, one can select z 0 2 4i and then C0 should be determined so that

72 L. J. SUDAK z 41 3 I 5 X 4z5 2

4 I 601

4i

5

D 7 4z5

4

6

P 7 4z5

4

2 I 601

dz1

x 2 01

(46)

from which the exact solution for X 4z5 is given by X 4z5 2

6 2 I 601 4 I 601 5 D4z5 4 P4z5 4 z1 1 3 I 1 3 I

x2 07

(47)

It follows that the explicit expressions for h 1 4z5 can be obtained from 4425 and the remaining complex potentials that being h 2 4z5 and Y 4z5 can be obtained as follows h 2 4z5 2

X 7 4z5 4 i h 1 4z5 4 4

6 2 I 601 I 601 5 7 D 4z5 4 P 7 4z5 3 1 1 3 I 1 3 I

x2 1 01 Y 7 4z5 2 2i h 1 4z5 4 z X 77 4z5 3 4

6 I 601 7 60 5 P 4z5 4 4 I 1 D 7 4z5 4 P 7 4z5 1 1 3 I 1 3 I

x2 01

(48)

and then the exact piezo-elastic fields can be determined throughout the whole plane. The expressions 4425, 4475 and 4485 give the exact solution for the Eshelby problem of an arbitrary shaped thermal inclusion embedded within an elastic half-plane and subsequently bonded to a piezoelectric half-plane in terms of the auxiliary function D4z5 and the associated polynomial P4z5. It is worth mentioning that because of the presence of D4z5 and P4z5, the elastic stresses within the thermal inclusion cannot be uniform in general. For many practical problems of thermal stresses, the internal stress field in the inclusion is of significant interest. For the sake of illustration let us consider, as an example, the elliptical thermal inclusion. 3.1. Elliptical Thermal Inclusion

Consider the ellipse with the semi-major axis a and the foci 2c 42c a5. Also assume that the center of the ellipse is located at x1 2 0 and x 2 2 y0 4y0 05 and its principal axes are parallel to the Cartesian axes. Thus, the required conformal mapping and its inverse are given by [29] 1 z 2 m4 5 2 c R 3 3 i y0 1 R 2 z 4 i y 2c 0 1 1 3 1 4 2 m 41 4z5 2 2cR z 4 i y0

(49)

ESHELBYâ€™S PROBLEM FOR A BONDED PIEZO-ELASTIC BIMATERIAL 73

where R2

a3

a 2 4 4c2 17 2c

(50)

It can be shown that the function D4z5 and the polynomial P4z5 are of the form D4z5 2 P4z5 2

c 41 4 R 4 5m 41 4z5 4 i y0 1 R 1 3 R2 z 7 4 i y0 R2 R2

R 2 4z 4 i y0 5 3

(51)

For the sake of illustrations, let us assume that the depth of the inclusion is much larger than its size, then for z 8 S0 we have z 4 iy0 3O m 4z5 2 cR 41

c2

y0 2

1

y0 cR7

(52)

Since the piezo-elastic fields are related to the derivative of X 4z5, then with aid of 4515, Equation 4435 takes the simple form X 7 4z5 2 4

2 I 601 1 1 3 I

x 2 07

Then using 4425 and 4485 the remaining functions h 1 4z51 h 2 4z5 and Y 7 4z5 yield the following h 1 4z5 2 01

x2 1 01

h 2 4z5 2 01

x2 1 01

Y 7 4z5 2 4

I 601 1 R 2 41 3 I 5

x2 07

(53)

Thus, it is clear that the stresses and associated electrical displacements in the upper halfplane are zero. However, since thermally induced void formation is controlled mainly by the mean stress, the average mean stress within the inclusion is of major concern. In view of 495 and 4405 the average mean within an inclusion is determined by the real part of 70 4z5 as follows 1 2 1 2 42 11 3 2 22 5avg 2 4 Re 70 4z5 2 4 Re X 7 4z5 1

z 8 S0 7

(54)

Thus, it is readily seen that when depth of the inclusion is much larger than its size the mean stress inside the elliptic inclusion is uniform and there is no dependence on the piezoelectric material constants. This result is identical to that for an elliptic inclusion in a half-plane (see [9]). On the other hand, if we consider the general case of an elliptic thermal inclusion embedded in the elastic layer, then using 4545 it is found that the internal mean stress in the elliptic inclusion is

74 L. J. SUDAK 8 9 9 9 9 1 4 R4 8 I 601

13 Re 42 11 3 2 22 5 2 4 1 3 I 9 R2 9 9 9

1 1 4 2c 14 z 3 i y0

9 9 9 9 ! 1 2 9 9 9 9 "

z 8 S0 7

(55)

The above result shows that the internal mean stress is non-uniform and the effect of the piezoelectric layer on the mean stress decreases with the depth of the inclusion. Another interesting aspect that is worth discussing is when we consider the upper half-plane to be a transversely isotropic poled ferroelectric with the poling axis in either the x 1 or x2 direction. An example of practical interest is when the poling direction is perpendicular to the interface. In the case, when the poling axis is in the positive x2 -direction the matrix Y takes the form [30] 1 CL Y 2 4i9 4i

i9

i

1 1 e 1 4 6

1 CT 1 e

4C L 1 C T 1 e1 61 95 01

4 05 7

(56)

Alternatively, if the poling axis is along the negative x2 -direction, the matrix Y takes the form 1 i9 CL 1 Y 2 4i9 CT 1 i 4 e

4i

1 4 1 e 1 4 6

4C L 1 C T 1 e1 61 95 01

4 05 7

(57)

In both cases the ratio yields a purely real number given by 1 1 1 9 9 2 I 2 I 39 31 4 ! CT CL CT 9 9 1 1 1 9 3 2 I " 32 9 4 9 4 1 2 I 29 3 3 CT CT CT 7 28 1 1 1 9 9 9 9 2 2 3 9 3 1 4 I I !

CT CL CT 9 9 1 1 1 9 " 3 2 I 4 9 3 I 2 I 29 3 3 32 9 CT CT CT 8 9 9

(58)

ESHELBYâ€™S PROBLEM FOR A BONDED PIEZO-ELASTIC BIMATERIAL 75

Thus, the mean stress within the elliptic inclusion is then given by 8 9 9 9 9 9 9 9 9

! 0 4 14 R 1 8 I 61 13 Re 1 4 42 11 3 2 22 5 2 4 2 9 1 1 3 I 9 R2 9 9 2c 9 9 9 9 14 " z 3 i y0 z 8 S0 7

(59)

In particular, along the minor principal axis (with x1 2 0), the internal mean stress is 8 9 9 9 9 0

1 4 R4 8 I 61 13 42 11 3 2 22 5 2 4 1 3 I 9 R2 9 9 9 z 8 S0 7

1 1 4 2c 13 y 3 i y0

9 9 9 9 ! 1 2 9 9 9 9 " (60)

Clearly, these results illustrate that the piezoelectric upper-half plane has a significant effect on the internal stresses especially when the inclusion depth is small.

4. CONCLUSION A concise analytical solution for the Eshelby problem of a smooth inclusion of arbitrary shape that is embedded within the isotropic elastic layer of a piezo-elastic bimaterial has been derived. The method of solution involves combining the well-known Muskhelishviliâ€™s formulation, for isotropic elastic materials, with the well-established Stroh formulation for piezoelectric materials. The solution is constructed in terms of an auxiliary function, D4z5, which can be determined from a conformal mapping function associated with the boundary curve of the inclusion. One remarkable feature of the method is that relatively elementary expressions can be obtained for the elastic fields and the mean stress inside and outside the arbitrary shaped subdomain. In the case of a dilatational inclusion, the solution admits a very simple form. In particular, when the thermal inclusion is elliptic in shape and the depth of the inclusion is much larger than its size the mean stress inside the elliptic inclusion is uniform and there is no dependence on the piezoelectric material constants. Alternatively, when the inclusion depth is small the internal mean stress is non-uniform and the piezoelectric upperhalf plane has a significant effect on the internal stresses. Acknowledgement. The financial support of the Natural Sciences and Engineering Research Council of Canada through grant NSERC No. 249516 is appreciated.

76 L. J. SUDAK

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Hu, M. Stress from isolation trenches in silicone substrates. Journal of Applied Physics, 67, 1092–1101 (1990). Hu, S. M. Stress-related problems in silicone industry. Journal of Applied Physics, 70, R53–R80 (1991). Niwa, H., Yagi, H., Tsuchikava, H. and Kato, M. Stress distribution in an aluminum interconnect of very large scale integration. Journal of Applied Physics, 68, 328–333 (1990). Downes, J. R., Faux, D. A. and O’Reilly, E. P. A simple method for calculating strain distribution in quantum wire dot structures. Journal of Applied Physics, 81, 6700–6702 (1997). Gosling, T. J. and Willis, J. R. Mechanical stability and electronic properties of buried strained quantum wire arrays. Journal of Applied Physics, 77, 5601–5610 (1995). Wang, X., Sudak, L. J. and Ru, C. Q. Elastic fields in two imperfectly bonded half-planes with a thermal inclusion of arbitrary shape. Zeitschrift für angewandte Mathematik und Physik, 58, 488–509 (2007). Seo, K. and Mura, T. The elastic fields in a half-space due to an ellipsoid inclusion with uniform dilatational eigenstrains. ASME Journal of Applied Mechanics, 46, 568–572 (1979). Yu, H. Y. and Sanday, S. C. Elastic field in jointed semi-infinite solids with an inclusion. Proceedings of the Royal Society London A, 434, 521–530 (1991). Ru, C. Q. Analytic solution for Eshelby’s problem of an inclusion of arbitrary shape in a plane or half-plane. ASME Journal of Applied Mechanics, 66, 315–322 (1999). Ru, C. Q. A two dimensional Eshebly problem for two bonded piezoelectric half-planes. Proceedings of the Royal Society London A, 457, 865–883 (2001). Ru, C. Q., Schiavone, P. and Mioduchowski, A. Elastic fields in two jointed half-planes with an inclusion of arbitrary shape. Zeitschrift für angewandte Mathematik und Physik, 52, 18–32 (2001). Janas, V. F. and Safari, A. Overview of fine-scale piezoelectric ceramic/polymer composite processing. Journal of the American Ceramics Society, 78, 2945–2955 (1995). Choi, J. S., Ashida, F. and Noda, N. Control of thermally induced elastic displacement of an isotropic structural plate bonded to a piezoelectric ceramic plate. Acta Mechanica, 122, 49–63 (1997). Li, Q. and Chen, Y. H. Solution for a semi-permeable interface crack in elastic dielectric/piezoelectric bimaterials. ASME Journal of Applied Mechanics, 75, 011010 (2008). Ou, Z. C. and Chen, Y. H. Interface crack problem in elastic dielectric/piezoelectric bimaterials. International Journal of Fracture, 130, 427–454 (2004). Ting, T. C. T. Anisotropic Elasticity. Theory and Applications. Oxford University Press, Oxford, 1996. Chen, T. and Lai, D. An exact correspondence between plane piezoelectricity and generalized plane strain in elasticity. Proceedings of the Royal Society London A, 453, 2689–2713 (1997). Ou, Z. C. and Chen, Y. H. General solution of the stress potential function in Lekhnitskii’s elastic theory for anisotropic and piezoelectric materials. Advanced Studies in Theoretical Physics, 1, 357–366 (2007). Ru, C. Q. A hybrid complex variable solution for piezoelectric/isotropic elastic interfacial cracks. International Journal of Fracture, 152, 169–178 (2008). Chen, Y. H. and Lu, T. J. Cracks and fracture in piezoelectric materials. Advances in Applied Mechanics, 39, 127–221 (2003). Chen, Y. H. and Lu, T. J. Recent developments and applications in invariant integrals. ASME Applied Mechanics Review, 56, 515–552 (2003). Li, Q. and Chen, Y. H. Solution of the semi-permeable interface crack in dissimilar piezoelectric materials. ASME Journal of Applied Mechanics, 74, 833–844 (2007). Chen, Y. H. Advances in conservation laws and energy release rates. Kluwer Academic, Dordrecht, 2002. Stroh, A. N. Dislocation and cracks in anisotropic elasticity. Philosophical Magazine, 7, 625–646 (1958). Muskhelishvili, N. I. Some Basic Problems of the Mathematical Theory of Elasticity. P. Noordhoff, Reinhold, 1963. Kantorovich, L. V. and Krylov, V. I. Approximate Method of Higher Analysis. Interscience, New York, 1958. Clements, D. L. A crack between dissimilar anisotropic media. International Journal of Engineering Science, 9, 257–265 (1971). Suo, Z., Kuo, C. M., Barnett, D. M. and Willis, J. R. Fracture mechanics for piezoelectric ceramics. Journal of the Mechanics and Physics of Solids, 40, 739–765 (1992). Ru, C. Q. and Schiavone, P. On the elliptical inclusion in anti-plane shear. Mathematics and Mechanics of Solids, 1, 327–333 (1996). Ru, C. Q. Exact solution for finite electrode layers embedded at the interface of two piezoelectric half-planes. Journal of the Mechanics and Physics of Solids, 48, 693–708 (2000).

Derivation of a Hierarchy of Nonlinear Two-dimensional Models For Heterogeneous Plates

E. P RUCHNICKI

Ecole Polytechnique Universitaire de Lille, Cité Scientifique, Avenue Paul Langevin, F 59655 Villeneuve d’Ascq Cedex, France (Received 10 July 20091 accepted 27 July 2009)

Abstract: This paper is concerned with the asymptotic analysis of plates with periodically rapidly varying heterogeneities. The formal asymptotic procedure is performed when both the periods of changes of the material properties and the thickness of the plate are of the same orders of magnitude. Our approach is based on a sequence of recursive minimization problems. We consider a plate made of Ciarlet–Geymonat type materials. Depending on the order of magnitude of the applied loads, we obtain a nonlinear membrane model and a nonlinear membrane inextensional bending model as announced by Pruchnicki.

Key words: Asymptotic analysis, plates, nonlinear elasticity, minimization problem

1. INTRODUCTION Plates are three-dimensional bodies characterized by a small transverse dimension, called the thickness, that are very small in comparison with the two other dimensions of the middle surface. This feature suggests the use of the method of asymptotic expansion with the thickness regarded as a small parameter, as a means of deriving two-dimensional mathematical models of plates from the three-dimensional theory. A precise mathematical formalization of the asymptotic approach in the context of linear and nonlinear elasticity is given by Ciarlet [1]. For heterogeneous plates, the size of the heterogeneities is a second small parameter. The length scale of this microstructure can also be larger than, comparable to or smaller than the thickness. The second case is the more important because the two others can be deduced and this is confirmed by the rigorous results of 1 convergence (Lewi´nski and Telega [2]). For plates made of nonlinear heterogeneous Saint Venant–Kirchoff materials, Pruchnicki [3– 5] showed by using the asymptotic expansion method that the model of the highest possible order is of membrane type. Nevertheless the Saint Venant–Kirchoff material is not realistic because it does not exclude the possibility of squashing a volume into a point. For this work, we consider a nonlinear material proposed by Ciarlet and Geymonat [6] which is in agree-

Mathematics and Mechanics of Solids 16: 77–108, 2011 1 The Author(s), 2011. Reprints and permissions: 1 http://www.sagepub.co.uk/journalsPermissions.nav

DOI: 10.1177/1081286509349965

78 E. PRUCHNICKI

ment with this fundamental physical requirement. Moreover, the stored energy function is polyconvex with respect to the invariants of the deformation gradient which ensures powerful results for the minimization problem of the potential energy (Ball [7], Ciarlet [8]). Moreover we consider composite plates for which the stiffness of each material is of the same order of magnitude. When this assumption is not satisfied, the asymptotic analysis depends on another small parameter which is the ratio of the mechanical characteristics of the matrix to that of the reinforcement. This leads to non-classical models of homogenization (Panasenko [9]). The outline of this paper reads as follows. We adapt the method developed previously by Pantz [10, 11] to solve a sequence of minimization problems for heterogeneous plates. This method is also used by Meunier [12] and Trabelsi [13] for the cases of homogeneous rods and plates, respectively. We obtain a hierarchy of two-dimensional models depending on the order of magnitude of the applied loads. For loading of first order with respect to the small parameter considered in Section 6, we obtain a two-dimensional energy minimization problem modeling the nonlinear membrane behavior of the plate and whose solution is the leading term in the expansion of the displacement field. In Section 9, the loading is of order two in powers of the small parameter so we obtain a model without internal constraint. In Section 10, we are thus led to define a generalization of the space of nonlinear nonextensional displacements for heterogeneous plates. When this space does not reduce to {0} and for loading of order three with respect to the small parameter, the displacement field can be identified as a solution of a two-dimensional nonlinear membrane inextensional bending model. This model generalizes the ones proposed by Giroud [14], Caillerie and SanchezPalencia [15] and Trabelsi [16].

2. THE NONLINEAR BOUNDARY-VALUE PROBLEM OF PLATES In this section, we provide the main notation and describe in detail the problem we will consider. 2.1. Notation

Throughout this paper, the summation convention is adopted. The Latin indices take values 1, 2 and 3. The Greek indices (except for 2) run over 1 and 2. Boldface letters represent vector-valued functions, tensors or spaces. 3 is an open subset of the Euclidean space 1n (n 2 2). The boundary and the closure of 3 are denoted by 43 and 3, respectively. Let 5e1 6 e2 6 e3 7 denote the basis of the space 13 8 We shall write x 3 5x1 6 x2 6 x3 7 for vectors in 13 , and x4 3 5x1 6 x2 7 for vectors in 12 8 The Euclidean scalar product, the exterior product and the tensor product of vectors a, b 2 13 are denoted by a 5 b, a 6 b and a 7 b, the Euclidean norm on 1n 8 m is denoted by 959. I denotes1the2unit matrix. 9 is the Kronecker symbol. det 5A7 is the determinant of the matrix A 3 ai j and Cof5A7 is the cofactor matrix of the matrix A. The notation 3 4

3 (3 4 compactly contained in 3) means that 3 4 is compact and contained in 3. 3 1 3 2 is the complement of the subset 3 2 contained in 3 1 . A domain 3 in 1n is a bounded, open, connected subset of 1n with a Lipschitz-continuous boundary 43

TWO-DIMENSIONAL MODELS FOR HETEROGENEOUS PLATES 79 in the sense of Ne2cas [17]. The measure of a Lebesgue measurable set M in 1n is denoted by meas 5M7. We define L p 53 6 1m 7 to be the collection of all m-tuples 5 f 1 6 f 2 6 8 8 8 6 f m 7 of real functions in L p 537. Similarly, we say that u 2 W16 p 53 6 1m 7 if u belongs to L p 53 6 1m 7 together with its partial distributional derivatives 4x j u i 1 we adopt this notation to represent the partial 4u i derivative 4x for 1 i m, 1 j n. j n m C 53 6 1 7 (n 2 2 ) is the space of n times continuously differentiable mappings from a domain 3 of 1 p to 1m . v belongs to Cn 536 1m 7 (n 2 2 ) if it is in Cn 53 6 1m 7 and for each multi-index with 9 9 3 1 5 5 5 n n, there exists a mapping i in C0 536 17 9 9 i such that the restriction to 3 of the partial derivative 4 i 3 4 x 41 5554x

n 5i 3 16 8 8 8 6 m7 1

n

2 is equal to i . S3 represents the set of all symmetric 3 8 3 matrices. C per 5 8 Y 7 is 3 4 162 the set of functions of C 5 8 12 8 22 6 22 76 Y 4 periodic in y4 . Wper 5 8 Y 2 7 is the 2 162 completion of the space Cper 5 8Y 7 with respect to the norm of W 5 8Y 2 7 and W162 5 8 1 162 2 3 per2 2 4 2 3 2 3 2 2 Y 6 1 7 3 Wper 5 8 Y 7 8 In the same way, Cper 5Y 7 is the subset of C 51 8 2 6 2 7 2 2 of Y 4 periodic in y4 and W162 per 5Y 7 is the completion of the space Cper 5Y 7 with respect to the 1 2 3 162 162 5Y 2 6 13 7 3 Wper 5Y 2 7 8 norm of W162 5Y 2 7. Finally, Wper

2.2. Description of the Plate Geometry

Let be a domain4 in 12 .3 Let us consider a plate of thickness 2 and mid-surface 8 The 2 2 1 2 set 3 3 3 8 22 6 22 6 2 2 1 is called the reference configuration of this plate. The boundary of the plate 15 is6 divided 4 upper 5 62into two parts: the first is composed of the lower and 3 boundaries 8 22 22 1 the second is the lateral boundary 1 2 3 4 8 22 6 22 8 This plate is heterogeneous and the size of the heterogeneities, which is assumed to be of the same order of magnitude as the thickness, is very small with regard to the global length-scale between both the local length-scale of the shell x4 3 5x 1 6 x 2 7. Thus we can define the ratio 4 y4 3 5y1 6 y2 7 and the global one 2 3 xy 4 . We note that the microstructure of the plate is

periodic with respect to global coordinates and it is4sufficient define the distribution of 3 to 2 2 2 4 4 the constituents on the smallest period Y 3 Y 8 2 6 2 5Y 3 506 17 8 506 177, which is also called the unit cell. A current point in Y 2 is defined by y2 3 5y1 6 y2 6 x32 7. Since the thickness of the plate is of the order of magnitude 2, we introduce y3 such that x32 3 2 y3 . As the coordinates of a point of the domain and those y2 of a point of the domain Y 2 are independent, we can define the coordinates of a point of the domain 8 Y 2 by the following element of 15 : 5x 4 6 y 4 6 x 32 7. We suppose that the plate is composed of both a matrix and an inclusion, which occupy two domains Y12 and Y22 , such that Y 2 3 Y12 Y22 and Y12 Y22 3 3. 2 The common boundary of both the matrix and the inclusion is denoted I . 4 The boundary 3 1 2 2 2 2 4 4Y of the unit cell is divided into the lateral boundary , and into both 4Y 3 4Y 8 6 2 2 15 6 L 5 62 the lower and the upper boundaries 4Y2 3 Y 4 8 22 22 . 2.3. Loads and Boundary Conditions

The plate is submitted to dead loading which is periodic with respect to the local lengthscale y4 , and it is sufficient to define this dead loading only over one unit cell. The density

80 E. PRUCHNICKI of body force is the vector denoted f i2 ei 1 we assume that the vector f2 3 5 f i2 7 belongs to L2 5 8 Y 2 6 13 7. Let force5 per6unit area h i2 ei acting on the upper face 1 62 3 5 2 6us consider the surface

8 2 and lower face 1 62 3 8 22 of the plate1 then their components 1are2functions 15 6 5 62 2 2 4 h i2 :

18 We suppose that the vector h2 3 h i2 belongs 1 8 Y 82 3 22 2 2 to L 8 4Y 6 1 . The plate is clamped on its lateral boundary 1 2 . When subjected to the given loading, the plate field u i2 5x4 6 y2 7 ei 8 Next we denote 1 2 2undergoes the displacement 2 3 the vector field u 3 u i : 8 Y 2 1 . The deformation gradient is defined by F2 5u2 7 3 I grad u2 and grad u2 is the gradient of the displacement field u2 , 1 5grad u2 7i 3 4x u 2i 4 y u 2i 6 2

5grad u2 7i3 3 4x32 u i2 8

2.4. Constitutive Equations of each Constituent

The plate is made of a two-phase hyperelastic of which the stored energy functions defined by (Ciarlet and Geymonat [6], see also Ciarlet [8]) 1 1 22 W 5F2 7 3 a 9F2 92 b 9CofF2 92 1 det F2T F2 e6

(1)

with a 06 b 06 1 597 3 c 9 d ln5976 c 06 d 06 e 2 1 are polyconvex and satisfy W 5F2 7 3

1 2 5trE2 72 trE22 O E23 8 2

(2)

1 1 22 In the above, E2 3 12 F2T F2 I stands for the Greenâ€“Saint Venant strain tensor and 0, 0 are two given LamĂŠ constants. Relation 527 is the beginning of the expansion of the stored energy function of isotropic material near a natural state. The Saint Venantâ€“Kirchhoff material is the simplest one that agrees with this expansion. The constants a, b6 c, d and e are given in terms of the LamĂŠ constants 9

7 8 1 a 3 s 6 b 3 s 6 c3s 6 2 2 2 9

9

1 6 e 3 2s 6 d 3 2 2 4 and as a6 b6 c and d are real positive functions, we infer that necessarily 9

9 1 6 6 8 s 2 min 2 2 2 2

TWO-DIMENSIONAL MODELS FOR HETEROGENEOUS PLATES 81

Moreover, we have

lim

det5F2 70

W 5F2 7 3 , which means that an infinite amount of energy

is necessary to annihilate a volume. From expression 517 of the function W 5F2 7, we can establish the following coerciveness inequality: 2 1 W 5F2 7 A 9F2 92 9CofF2 92 5det F2 72 B6

A 08

If the plate is composed of two homogeneous materials then both and are piece-wise constant functions defined as 1 2 2 1 x4 6 y4 6 x 32 3 , x4 6 y4 6 x32 3 ,

1 2 s x4 6 y4 6 x32 3 s ,

1 2 in which x4 6 y4 6 x32 2 8 Y 2 8 Hence , s and represent the LamĂŠ constants of each material constituting the plate ( 0, 0, s 0). To carry out the computations involved in the asymptotic analysis, we can express the stored energy function 517 as W 5F2 7 3 a Cii2 5F2 7 b 5Cof C2 5F2 77ii 1 5det 5C2 5F2 777 e2 6

(3)

in which C2 5F2 7 3 F2T F2 denotes the right Cauchyâ€“Green strain tensor.

3. THE TWO-SCALE THREE-DIMENSIONAL MINIMIZATION PROBLEM In this section, we give the double-scale three-dimensional minimization problem posed on the family of plates parameterized by 2, 2 0. Then we suppress the difficulty originating from the dependence on the small parameter of the plate by applying a change of variables. 3.1. The Three-dimensional Double Scale Boundary Value Problem

P5 8 Y 2 7

1 2 We formulate the family of double-scale problems posed on 8 Y 2 . Let V2 8 Y 2 6 13 be the set of admissible displacement field 1 2 2 3 2 2 V2 8 Y 2 6 13 3 u2 2 W162 per 5 8 Y 6 1 7 : det 5F 5u 77 0 and u2 3 0 on 4 8 Y 2

8

Then the equilibrium state of the plate satisfies the minimization problem P 2 5 8 Y 2 7 1 2 Find v2 2 V2 8 Y 2 6 13 such that J 2 5v2 7 3

inf

u2 2V2 5 8Y 2 6 13 7

where the total energy functional J 2 takes the form J 2 5u2 7 3 I 2 5u2 7 l 2 5u2 7 8

J 2 5u2 7 6

82 E. PRUCHNICKI Here the linear form l 2 is the work of the external forces 2 2 2 2 4 2 l 5u 7 3 f i u i dx dy

h i2 u 2i da8

84Y2

8Y 2

2 1 In the previous equation, da denotes the area element along the boundary 4 8 Y2 , while I 2 measures the internal energy of the plate I 2 5u2 7 3

W 2 5F2 7 dx4 dy2 8

8Y 2

3.2. Transformation into a Problem

P 52 8 Y 7 Posed over a Domain Independent of 2

In this section, we transform problems P5 8Y 2 7 into problems P52 8Y 7 posed over a set that does not depend on 2 by using the following change of variables (Ciarlet [1], Caillerie [18], LewiÂ´nski [19], Ciarlet and Lods [20], Miara [21], Collard and Miara [22], Mardare [23]): 1 4 4 22 x 6 y 6 x3 2 8 Y 4 8 [26 2] 5x4 6 y7 2 8 Y 4 8 [16 1]. Thus the domains Y 2 and Y 2 become the domains Y 3 Y 4 8516 17 and Y , respectively. The boundaries 4Y 2 , 4Y2 , 4Y L2 , 1 2I become 4Y , 4Y 3 Y 4 851 17 6 4Y L 3 4Y 4 8[16 1] 6 1 I , respectively. For any function g 2 defined on 8 Y 4 8 [26 2], we associate a function g527 defined on 8 Y 4 8 [16 1] in the following manner: 1 2 g 2 x4 6 y4 6 x 32 3 g5275x4 6 y78 For the sake of simplicity, we omit 2 in brackets for the material constants. This change of variable implies the following relations concerning the partial derivatives 4x g 2 3 4x g527, 4 y g 2 3 4 y g527,

4x32 g 2 3

1 4 y g527. 2 3

Consequently, the space V2 5 8 Y 2 6 13 7 becomes 1 2 3 V 527 8 Y6 13 3 u 2 W162 per 5 8 Y6 1 7 : det 5F 527 5u77 0

and u 3 0 on 4 8 Y 6 in which the components of the deformation gradient are defined by

TWO-DIMENSIONAL MODELS FOR HETEROGENEOUS PLATES 83 1 1 Fi 527 5u 5277 3 9 i 4x u i 527 4 y u i 527 , Fi3 527 5u 5277 3 9 i3 4 y3 u i 527 8 2 2 The determination of a displacement field belonging to the space V2 5 8 Y 2 6 13 7 and satisfying the nonlinear minimization problem P 2 5 8 Y 2 7 is equivalent to finding in the space a displacement field satisfying the following problem P52 8 Y 7: 1 2 Find v 527 2 V 527 8 Y6 13 such that J 527 5v 5277 3

inf

u5272V5275 8Y6 13 7

J 527 5u 5277 6

where the total energy functional J 527 takes the form J 527 5u 5277 3 I 527 5u 5277 l 527 5u 5277 6 with I 527 5u 5277 3 2

W 527 5F 5277 dx4 dy6

8Y

4

f i 527 u i 527 dx dy

l 527 5u 5277 3 2

8Y

h i 527 u i 527 da8

84Y

4. SETTING OF THE ASYMPTOTIC PROCEDURE 4.1. Assumption of the Asymptotic Expansions

The formal asymptotic two-scale method is based on the hypothesis that the unknown vector field u 527, the data f 527 and h 527 admit asymptotic expansion as a power of the small parameter 28 When the Greenâ€“Lagrange strain tensor is small enough, the general stored energy function 537 becomes of Saint Venantâ€“Kirchhoff type and in this case Pruchnicki [3, 5] and Miara [21] show that the first power of 2 cannot be negative, so the asymptotic expansions of both the displacement fields and external loading are necessary as follows: 1 2 u2 x4 6 y2 3 u5275x4 6 y7 3 u0 5x4 6 y7 2 u1 5x4 6 y7 22 u2 5x4 6 y7 5 5 5 6 1 2 f2 x4 6 y2 3 f5275x4 6 y7 3 f0 5x4 6 y7 2 f1 5x4 6 y7 22 f2 5x4 6 y7 5 5 5 6 1 2 3 h5275x4 6 y7 3 2 h1 5x4 6 y7 22 h2 5x4 6 y7 5 5 5 8 h2 x4 6 y2 The coefficients of the asymptotic expansions of both the body load f p1 (for p 0) 2 and p 2 3 the1surface load h (for p 0)2 are supposed to belong to the spaces L 8 Y6 1 and L2 8 Y 4 8 51 17 6 13 , respectively. To each functional G 527, we associate the functional defined by

84 E. PRUCHNICKI 2 2 1 1 527 u0 6 u1 6 u2 6 8 8 8 3 G 527 u0 2 u1 22 u2 5 5 5 8 G 1 2 1 2 From now on, u and v will stand for the sequences ui i22 and vi i22 , respectively. 4.2. Asymptotic Expansion of the Energy Terms

The interest of the following section is to recast the sequence of problems P 52 8Y 7 into a sequence of simpler variational problems. We begin by computing the asymptotic expansion of F 527 5u 5277, 527 5u7 3 F 527 5u 5277 3 F

Fn 5u7 2n 6

(4)

n3 1

in which the components of F1 5u7, F0 5u7 and Fl 5u7 (for an integer l 0) are defined by Fi1 j 5u7 3

2 1 grady u0 i j 3 4 y j u i0 6

1 1 2 2 Fi0j 5u7 3 9 i j gradx4 u0 i j grady u1 i j 6 Filj 5u7 3

1 2 2 1 gradx4 ul i j grady ul1 i j

5l 07 8

(5)

527 admits the formal asymptotic expansion Lemma 1. The tensor C 527 5u7 3 C 527 5u 5277 3 C

Cn 5u7 2n 6

(6)

n32

where the components of the tensor Cn 5u7 have the following expressions: Cn 5u7 3

n1

F pT 5u7 Fn p 5u7 8

(7)

p31

Proof. From expansion 547, we get 527 5u7 3 C

F pT 5u7 Fk 5u7 2 pk 3

p31k31

3

n1

F pT 5u7 Fn p 5u7 2n

n32 p31

Cn 5u7 2n 8

n32

1

TWO-DIMENSIONAL MODELS FOR HETEROGENEOUS PLATES 85 7 8 527 Lemma 2. The asymptotic expansion of CofC is ii

7 8 Cof C 527 5u7 3 Cofiin 5u7 2n 6 ii

n34

where Cofiin 5u7 has the following expression: Cofiin 5u7 3

n2

p

n p

p

n p

p

n p

C11 5u7 C22 5u7 C11 5u7 C33 5u7 C22 5u7 C33 5u7

p32

n2

p n p p n p C12 5u7 C12 5u7 C13 5u7 C13 5u7

p32

n2

p n p C23 5u7 C23 5u7 8

(8)

p32

Proof. The proof is a direct consequence of the equality 9

7 8 7 8 1 7 8 2 C 527 5u7 C 527 5u7 C 527 5u7 kk jk jk 2 7 8 7 8 7 8 7 8 527 5u7 C 527 5u7 527 5u7 C 527 5u7 C 3 C

7 8 527 5u7 3 Cof C ii

11

22

11

33

7 7 8 8 527 5u7 527 5u7 C C 22

33

97 7 7 82 82 82 C 527 5u7 C 527 5u7 C 527 5u7 6 12

13

23

1

and we use 567 to conclude. Let us define 7 8 527 5u7 3 det C 527 5u7 8 D 527, Lemma 3. We have the asymptotic expansion for the function D 527 5u7 3 D

n36

D n 5u7 2n 6

86 E. PRUCHNICKI where D n 5u7 satisfies the following expression:

D n 5u7 3

1 2 p C11 5u7 Ck22 5u7 Cl33 5u7 Ck23 5u7 Cl23 5u7

pkl3n

p C12 5u7

1 k 2 2C13 5u7 Cl23 5u7 Ck12 5u7 Cl33 5u7

pkl3n

p C13 5u7 Ck13 5u7 Cl22 5u7 8

(9)

pkl3n

Proof. We note that 97

8 7 7 82 8 7 8 527 5u7 C 527 5u7 527 5u7 527 5u7 C det C 3 C 11

22

11

7 7 7 8 8 8 527 5u7 C 527 5u7 527 5u7 C 2 C 12

13

23

7 7 7 8 7 8 82 82 527 5u7 C 527 5u7 8 527 5u7 C 527 5u7 C C 33

12

22

13

527 5u7 567 6 the result follows. Then by considering the asymptotic expansion of the tensor C 1 Let us introduce the following notation: 527 5u7 3 ln D 527 5u7 8 L 527 admits the asymptotic expansion Lemma 4. The function L 527 5u7 3 L

L n 5u7 2n 6

n30

where L 0 5u7 3 ln

0

D n 5u7 2n

6

n36

n 1 L n 5u7 3 p p31

p

p 1 D ki 5u7 0 q i31 p D 5u7 2q q36

ki 3n

i31

n 2 18

TWO-DIMENSIONAL MODELS FOR HETEROGENEOUS PLATES 87

Proof. We have

527 5u7 3 ln D 527 5u7 3 ln L

D n 5u7 2n

n36

3 ln

0

D q 5u7 2q 2

q36

D n1 5u7 2n

n30

n

517n1 3 L 5u7 0 n n31 0

2 Dq

5u7 2

q

n D

k1

5u7 2

k

8

k30

q36

Consequently, p n ! p 1 1 0 ki L 527 5u7 3 L 5u7 D 5u7 2n 8 0 p q i31 p q n31 p31 D 2 5u7 " k 3n i q36 i31

1 Lemma 5. The functional I 527, which represents the elastic energy, admits the expression I 527 5u7 3

I n 5u7 2n meas 5 7

n35

e dy6 Y

with I 5u7 3 n

2 1 a Ciin1 5u7 b Cofiin1 5u7 dx4 dy

8Y

1 2 c D n1 5u7 d L n1 5u7 dx4 dy6

8Y

7 8 7 8 527 , Cof C 527 and L 527 6 in which the coefficients of the asymptotic expansions of C ii ii which are not defined in the above lemmas, are equal to zero. Moreover, the linear functional l 527 6 which represents the external loading contribution, can be written as l 527 5u7 3

l n 5u7 2n 6 n30

88 E. PRUCHNICKI

where l n 5u7 3

n1

f i p u in1 p dx4 dy

8Y p30

n h ip u in p da8

84Y p30

Finally, the total energy functional J 527 admits the following asymptotic expansion: J 527 5u7 3

J 5u7 2 meas 5 7 n

n

n35

e dy6

J n 5u7 3 I n 5u7 l n 5u7 8

Y

Proof. Again, this is obtained by replacing the different terms in the asymptotic expansions of I 527 5u7, l 527 5u7 and J 527 5u7 by their respective asymptotic expansions. 1 7 8 527 5u7 0 should be for 2 small enough so that this Finally, the constraint det F condition is equivalent to the positivity of the leading term in the 7 asymptotic 8 expansions in 527 5u7 as follows: powers of 2. Now we can write the asymptotic expansion of det F 7 8 527 5u7 3 Fndet 5u7 2n 8 det F n33 3 Here we necessarily have F3 5u7 0, so we do not need to express Fndet 5u7 det 5u7 3 det F in terms of Fq 5u7 6 q 2 18

4.3. The Asymptotic Procedure

The next proposition shows that the solution v 527 of problem P52 8 Y 7 can be obtained by solving a sequence of partial variational problems. This idea was introduced by Pantz [11] in the case of homogeneous plates. Proposition 6. The solution of problem P52 8 Y 7 is such that 1 2 # v 3 v0 6 v1 6 v2 6 8 8 8 2 Vn 5 8 Y6 13 76 n35

where $ v 2 Vn 5 8 Y6 13 7 : J n 5v7 3 Vn1 5 8 Y6 1 7 3 3

V5 5 8 Y6 13 7 3

162 v 2 Wper 51 8 Y6 13 72 :

inf

u2Vn 5 8Y 7

% J 5u7 6 n

vn 2n 2 V526 8 Y 7 8

n31

TWO-DIMENSIONAL MODELS FOR HETEROGENEOUS PLATES 89

Proof. The proof results from a simple adaptation to this heterogeneous case of the reason1 ings of Pantz [10] and Meunier [12]. Let Pn 5 8 Y 7 be the following boundary-value problem: Finding v 2 Vn 5 8 Y6 13 7 such that J n 5v7 3

inf

u2Vn 5 8Y6 13 7

J n 5u7 8

We are going to solve the first variational problems Pn 5 8 Y 78

5. SOLVING PROBLEMS Pn 5 8 Y 7 for n 3 0 The purpose of the first part of this section is to identify the set V0 5 8 Y6 13 7 by solving problems 5Pn 5 8 Y 77n 0 8 1These problems 2 are independent of the external loading. We show that any solutions v 3 v0 6 v1 6 v2 6 8 8 8 of P1 5 8 Y 7 are such that the leading-order term v0 does not depend on the microscopic scale of the plate. Next we show that problem P0 5 8Y 7 is trivial. Before studying this problems, we observe that the displacement field vi (i 0) cannot vanish on the clamped boundary 4 8 Y . Then a boundary layer term is considered by Lions [24, p. 49] for elastic heterogeneous three-dimensional structures, Dauge and Gruais [25, 26] for elastic homogeneous plates and Pruchnicki [27, 28] for hyperelastic heterogeneous three-dimensional structures and plates. Thus to avoid the boundary-layer region around the neighborhoods of the clamped part of the plate, we replace this boundary condition by its average on the unit cell. 5.1. Identification of the Set V0 5 8 Y6 13 7

Lemma 7. The set V0 5 8 Y6 13 7 is defined by 3 2 : det F0 5v7 0, V0 5 8 Y6 13 7 3 v 2 W162 per 5 8 Y6 1 7 1 1 2 2 v x4 6 y 3 v0 x4 , v0 3 0 and

v dy 3 0 on 4

0

! "

8

Y

Proof. We consider the first problem Pn 5 8 Y 7 55 3 n 3 17 which consists of finding v in 2n 5 8 Y 7 such that J n 5v7 3

inf

u2Vn 5 8Y6 13 7

J n 5u7 6

By Lemmas 4 and 5 of Section 4.2, we have

J n 5u7 3 I n 5u7 8

90 E. PRUCHNICKI I n 5u7 3

2 1 a Ciin1 5u7 b Cofiin1 5u7 c D n1 5u7 dx4 dy8

8Y

8 7 8 7 527 5u7 requires the positivity of the 527 5u7 and D 527 5u7 6 Cof C The positivity of C ii ii first-order term of their asymptotic expansions. Then, for the lowest order problem P5 5 8 Y 76 we necessarily have inf

u2V5 5 8Y6 13 7

J 5 5u7 3 06

1 2 since J 5 507 3 08 Then the element v in V4 8 Y6 13 satisfies I 5 5v7 3 0 and, by 22 1 recalling that D 6 5v7 3 det F3 5v7 3 0, we get V4 5 8 Y6 13 7 3

162 5 8 Y6 13 72 : det F 5v73 3 06 v 2 Wper

! 7 8 v dy 3 0 on 4 8 det F 527 5v7 0 and " Y

In a similar way, we solve recursively problems Pn 5 8 Y 7 for n 3 16 8 8 8 6 4. Thus we get D q 5v7 3 Cofiiq 5v7 3 Ciiq 5v7 3 0 for q 3 26 8 8 8 6 58 In particular, C2 ii 5v7 7 8 3 1 1 1 Fik 5v7 Fik 5v7 3 0 which implies that Fik 5v7 3 08 Then the condition det F 527 5v7 0 becomes 7 8 527 5v7 3 det F0 5v7 2 Fn 5u7 2n1 6 det F det n31

for 2 small enough. From now on, we require det F0 5v7 08

1 5.2. Solving Problem P0 5 8 Y 7

By using the above results, we see the following corollary. Corollary 8. For all v 2 V0 5 8 Y6 13 7, we have F1 5v7 3 C2 5v7 3 C1 5v7 3 0, Cofiip 5v7 3 0 for 4 3 p 3 1 and D q 5u7 3 0 for 6 3 q 3 18 The function 527 admits the asymptotic expansion L

TWO-DIMENSIONAL MODELS FOR HETEROGENEOUS PLATES 91

527 5v7 3 L

L n 5v7 2n 6

n30

where 2 1 L 0 5v7 3 ln D 0 5v7 6

p p 9 n ki 1 1 L n 5v7 3 D 5v7 p D 0 5v7 i31 p p31

n 2 18

(10)

ki 3n

i31

Proof. Formula 557 shows that F1 5v7 3 0 so then Lemmas 1â€“3 trivially imply that C2 3 C1 3 0, Cofiip 3 0 for 4 3 p 3 1 and D q 5u7 3 0 for 6 3 q 3 18 The result of the second part is deduced from Lemma 4. 1 The above corollary implies that J 0 5u7 3 I 0 5u7 l 0 5u7 =0 so then the problem P0 5 8 Y 7 becomes trivial and V0 5 8 Y6 13 7 3 V1 5 8 Y6 13 78

6. A NONLINEAR MEMBRANE PLATE THEORY In this section, we show that nonlinear membrane plate theory arises as problem P1 5 8 Y 78 2 1 Theorem 9. If v 3 v0 6 v1 6 8 8 8 is a solution of the problem P1 5 8 Y 7 then the first term v0 in the asymptotic expansion of the displacement field solves the global minimization problem 1 2 Jm v0 3

inf

u0 2U0 5 6 13 7

1 2 Jm u0 6

(11)

in which U0 5 6 13 7 is defined by 5 6 U0 5 6 13 7 3 u0 2 W162 5 6 13 7 u0 3 0 on 4 6 and the membrane energy is defined by 1 2 Jm u0 3

7 1 28 &0 u0 Wm F h i1 dy4 u i0 dx4 6 dx4 f i0 dy

Y

4Y

&0 is a 3 8 2 matrix of which entries are defined by F&0 3 9 i 4x u 0 and Wm is the in which F

i i

membrane elastic energy obtained by computing the following local minimization problem:

92 E. PRUCHNICKI 7 1 28 Wm F&0 u0 3

inf u1 2U1 5 8Y6 13 7

1 2 W 0 u0 6 u1 dy8

(12)

Y

1 2 The set U1 8 Y6 13 is ! 1 2 1 2 162 51 8 Y6 13 76 det F0 u0 6 u1 0, u1 dy 3 0 6 U1 8 Y6 13 3 u1 2 Wper " Y

and finally 1 2 1 2 1 2 W 0 u0 6 u1 3 a Cii0 u0 6 u1 b Cof0ii u0 6 u1 1 2 1 2 c D 0 u0 6 u1 d L 0 u0 6 u1 8

(13)

Proof. Let us consider problem P1 5 8 Y 7 J 1 5v7 3

inf

u2V1 5 8Y6 13 7

J 1 5u7 8

1 2 ' W 0 u0 6 u1 dx4 dy are well defined for all u0 2 U0 5 6 13 7 and The integrals I 1 5u7 3 1 2 8Y u1 2 U1 8 Y6 13 , we can apply Fubiniâ€™s theorem and the contribution of the stored energy is I 5u7 3 1

1 2 W 0 u0 6 u1 dy dx4 6

Y

while the contribution of the external forces to the energy is

1 2 l 1 u0 3

f i0 dy h i1 dy4 u i0 dx4 8 4Y

Y

Then problem P1 5 8 Y 7 becomes J 1 5v7 3

inf

u0 2U0 5 6 13 7

inf

u1 2U1 5 8Y6 13 7

1 2 1 2 W 0 u0 6 u1 dy dx4 l 1 u0 8

Y

As a consequence, we get the result 1 2 Jm v0 3 J 1 5v7 3

inf

u0 2U0 5 6

13 7

1 2 Jm u0 8

1

TWO-DIMENSIONAL MODELS FOR HETEROGENEOUS PLATES 93 ' In the proof of the previous theorem, we suppose implicitly that the infinimum of W 0 Y 1 0 2 u 6 5 dy is attained. For the computation of this infinimum, the global scale x4 can be considered as fixed and it plays the role of a parameter so it is not necessary to consider the regularity of the displacement field of first order with respect to the global scale. To apply the results of Ball [7], more regularity on u1 is necessary. Thus, in the' next 1theorem, 2 we define a new space Us1 5 8 Y6 13 7 and we show that the minimum of W 0 u0 6 8 dy is Y

attained on this space. Theorem 10. We define the set 1 2 Us1 8 Y6 13 3

1 2 3 For all x4 2 6 u1 x4 6 5 2 W162 per 5Y6 1 76

1 1 1 2 1 222 Cof F0 u0 x4 6 u1 x4 6 5 2 L2 5Y6 13 76 1 1 1 2 1 222 2 L2 5Y6 13 76 det F0 u0 x4 6 u1 x4 6 5 1 222 1 1 1 2 06 det F0 u0 x4 6 u1 x4 6 5

! u1 556 y7 dy 3 0 8 "

Y

' 01 0 2 W u 557 6 u1 556 y7 dy and the set Moreover, we assume that inf u1 2Us1 5 8Y6 13 7 Y 1 2 s 3 U1 1 8 Y6 1 2 is non-empty8 Then there exists at least one function v1 such that v1 2 Us1 8 Y6 13 and

1 2 W u0 6 v1 dy 3

0

Y

u1 2Us1

inf 5 8Y6 13 7

1 2 W 0 u0 6 u1 dy8

Y

0 which is defined by Proof. By the coerciveness of the functional W 1 1 22 1 2 0 F0 u0 6 u1 3 W 0 u0 6 u1 6 W we get

1 2 0 F0 dy inf 5A 7 W

316 2

Y

7 ( ( 0 (2 ( 28 1 (F ( (CofF0 (2 det F0 2 dy inf 5B 7 8

316 2

Y

By taking into account the PoincarĂŠâ€“Wirtinguer inequality and the condition

' Y

we see that there exists real constants f 0 and g, independent of u1 , such that

u1 dy 3 0,

94 E. PRUCHNICKI

1 2 0 F0 dy W

7 ( 1 (2 ( 1 2( 8 (u ( (grad y u1 (2 dy f

Y

Y

7 8 ( ( ( ( 2 2 (det F0 ( (CofF0 ( dy g8

(14)

Y

Let u1k be an infimizing sequence for the functional 1 2 satisfies u1k 2 Us1 8 Y6 13 for all k, and

1 2 W 0 u0 6 u1k dy 3

lim

k Y

'

1 2 W 0 u0 6 5 dy, i.e. a sequence that

Y

inf u1 2Us1 5 8Y6 13 7

1 2 W 0 u0 6 u1 dy.

Y

' 0 1 0 12 inf W u 6 u dy is bounded and by the coerciveness inequal5 8Y6 13 7 Y ity (14), the sequence

By assumption,

u1 2Us1

1 1k 1 4 2 1 1 2 1 22 1 1 2 1 222 u x 6 5 6 Cof F0 u0 x4 6 u1k x4 6 5 6 det F0 u0 x4 6 u1k x4 6 5 1 4 32 1 2 is bounded for all x4 2 in the reflexive Banach space W12 8 L2 Y 4 6 13 8 per Y 6 1 1 2 L2 Y 4 6 13 . Hence there exists a subsequence 1 1 2 1 22 1 1 2 1 222 1 1l 1 4 2 u x 6 5 6 Cof F0 u0 x4 6 u1l x4 6 5 6 det F0 u0 x4 6 u1l x4 6 5 1 1 4 32 2 that converges weakly to an element v1 5x4 6 57 6 H 5x4 6 57 6 3 5x4 6 57 in the space W12 Y 61 per 1 4 32 1 4 32 2 2 8 L Y 6 1 8 L Y 6 1 and we get 1 1 2 1 22 1 2 1 1 2 1 22 1 2 H x4 6 5 3 CofF0 u0 x4 6 v1 x4 6 5 , 3 x4 6 5 3 det F0 u0 x4 6 v1 x4 6 5 (Ciarlet [8, Theorem 7.6.1]). Now, we must check that

'

v1 556 y7 dy 3 0, but this follows

Y

from the implication that 1 2 1 2 1 4 32 u1l x4 6 5 converges weakly to v1 x4 6 5 in W12 per Y 6 1 3

u 556 y7 dy

v1 556 y7 dy8

1l

Y

Y

As in 1the proof of Theorem 7.7.1 of Ciarlet [8], it can be shown that for all x 4 2 6 2 det F0 u0 5x 4 7 '6 v1 5x14 6 57 0,2 so v1 5x 4 6 57 is a solution of the minimization problem of the functional W 0 u0 5x 4 7 6 5 dy. 1 Y

TWO-DIMENSIONAL MODELS FOR HETEROGENEOUS PLATES 95

By assuming the existence of solutions of the local problem of minimization (12), the solutions v of the problem P1 5 8 Y 7 are the elements of V2 5 8 Y6 13 7 defined by $ 1 2 1 2 V2 5 8 Y6 1 7 3 v 3 v0 6 v1 6 8 8 8 6 v0 solves Jm v0 3 3

inf

u0 2U0 5 6

13 7

1 2 Jm u0 6

v1 6 & v1 2 U0 5 6 13 7 and v1 solves v1 3 v1 & 1 2 1 2 inf W 0 v0 6 v1 dy 3 W 0 v0 6 u1 dy u1 2U1 5 8Y6 13 7 Y

Y

& v 3 0 and

v p dy 3 06 p 2 on 4

1

! "

8

Y

7. THE VARIATIONAL FORMULATIONS AND THE EULERâ€“LAGRANGE BOUNDARY-VALUE PROBLEMS In this section, we give both the variational and the Eulerâ€“Lagrange equations satisfied by the nonlinear membrane plate described by the total energy given in the previous section. We assume that the displacements fields are smooth enough to carry out all necessary computations. For a similar development in homogenization theory, we can refer to Bakhvalov and Panasenko [29, Chapter 6]. 1 2 Theorem 11. For all v 3 v0 6 v1 6 8 8 8 2 V2 5 8 Y6 13 7, the leading term v0 of the asymptotic expansion of the displacement field satisfies the two-dimensional nonlinear variational membrane problem 1 M 5 7: find 1 2 v0 2 U0 6 13 , such that

0 i

1 2 h i1 dy4 u i0 dx4 3 06 gradx4 v0 4x u i0 f i0 dy Y

4Y

1 2 for all u0 2 U0 6 13 . The Eulerâ€“Lagrange boundary value problem associated to the previous global variational problem is: find 1 2 v0 2 U0 6 13 , such that

96 E. PRUCHNICKI 1 2 0 4x i

gradx4 v0

f i0 dy

Y

h i1 dy4 3 08

4Y

1 2 ' The macroscopic tensor 0 ( 0 3 Y 0 v1 dy) is an implicit function of the covariant derivatives of the displacement field of order zero and the geometrical characteristics of the mid-surface of the shell, through the solution v1 of the three-dimensional nonlinear local variational problem 1l 5Y 7: find 1 2 v1 2 U1 8 Y6 13 6 such that

1 2 0i j gradx4 v0 6 grady v1 4 y j u i1 dy 3 06

1 2 u1 2 U1 8 Y6 13 6

Y

where the first Piolaâ€“Kirchhoff tensor of zero order is defined by i0j

1 1 22 0 F0 v0 6 v1 1 2 4W 0 1 gradx4 v 6 grady v 3 8 4 Fi0j

The local Eulerâ€“Lagrange boundary value problem associated to the variational problem 1l 5Y 7 consists of finding 1 2 v1 2 U1 8 Y6 13 , such that 1 2 4 y j i0j gradx4 v0 6 grady v1 3 08 Proof. Let us consider the local minimization problem 51271 obviously, we see that a solution of the displacement field v1 verifies 1 2 IY1 v0 6 v1 t u1 3

1 2 W 0 v0 6 v1 t u1 dy 2

Y

1 2 W 0 v0 6 v1 dy6

Y

1 2 for all t 2 1 and u1 2 U1 8 Y6 13 8 Moreover, assume that IY1 is differentiable at v1 , so the previous inequality implies the nullity of the directional derivative of IY1 at v1 in the direction u1 , D

IY1

1 0 12 1 v6 v u 3

Y

1 1 22 0 F0 v0 6 v1 4W 4 y j u 1i dy 3 06 4 Fi0j

TWO-DIMENSIONAL MODELS FOR HETEROGENEOUS PLATES 97 where D denotes the differential with respect to v1 . Therefore, the displacement field v1 is a solution of the three-dimensional nonlinear local variational problem Pl 5Y 7 : find v1 2 U1 5 8 Y6 13 7 such that 1 2 1 2 i0j gradx4 v0 6 grady v1 4 y j u i1 dy 3 0 u1 2 U1 8 Y6 13 6 Y

in which i0j denotes the first Piolaâ€“Kirchoff tensor of order zero, i0j

1 1 22 0 F0 v0 6 v1 1 2 4W 0 1 gradx4 v 6 grady v 3 8 4 Fi0j

We observe that the microscopic displacement field of first order u1 depends on the gradient of the global displacement field of order zero gradx4 u0 8 Now integrating by parts the local variational problem Pl 5Y 7 gives the following equality: 4 y j i0j u 1i dy 3 08 Y 4 Since u i1 5x 4 6 57 belongs to W162 per 5Y 7 for almost all x in 6 the above integration by parts is explicit when, for instance, the displacement field u i1 5x 4 6 57 is such that i0j 5x 4 6 57 belongs to W162 per 5Y 78 Then the boundary-value local problem follows. From the global minimization problem 5117 and by noting that a variation of v0 implies one of v1 6 we necessarily have

1 2 1 2 1 2 Jm v0 tu0 2 Jm v0 for all t 2 16 and u0 2 U0 6 13 8 Next we assume the differentiability of the functional Jm with respect to v0 1 therefore, by considering the previous inequality, we cancel the directional derivative of Jm at v0 in the direction u0 , 1 02 0 1 2 i0j gradx4 v0 6 grady v1 dy 4x j u i0 dx4 l 1 5u7 D Jm v u 3

Y

3 08 the2 Then the leading term v0 of the asymptotic expansion of the displacement field satisfies 1 0 0 two-dimensional nonlinear variational membrane problem PM 5 7: find v 2 U 6 13 such that 1 2 1 2 1 2 0i gradx4 v0 4x u i0 dx4 3 l 1 u0 for all v0 2 U0 6 13 6

98 E. PRUCHNICKI

in which the macroscopic tensor i0j

1 2 gradx4 v0 3

1 1 22 i0j gradx4 v0 6 grady v1 gradx4 v0 dy6

Y

since v1 depends on gradx4 v0 through the three-dimensional nonlinear local variational problem Pl 5Y 7. When u0 and 0 are sufficiently regular, an integration by parts of the problem PM 5 7 gives its Eulerâ€“Lagrange associated boundary value problem. 1

8. THE NONLINEAR MEMBRANE PLATE MODEL 8.1. Constitutive Properties of Wm

The constitutive law of each elastic material is physically realistic as it satisfies the additional property W 5F7

as det 5grad 5F77 0 8

As a consequence, the orientation-preserving condition is naturally imposed and the twodimensional model precludes singular folds of the mid-surface. We claim the following. Theorem 12. The membrane elastic energy satisfies the following condition: 7 8 &0 Wm F

(2 7 8 ( &0 3 ((F &0 (( 0 8 &0 T F &0 6 F as det F 1 2

7 8 1 2 &0k F &0k 0 &0 u0k be such that lim F &0 and det F &0k T F &0k 3 F Proof. Let the sequence F ( (k 7 8 &0 T F &0 3 0. As a consequence, ((F &0k (( is bounded. We argue by contradicwith det F 7 8 &0k 3 c 8 c is finite since the memtion. Suppose to the contrary that lim inf Wm F k brane 7energy 88 is bounded below8 Without loss 7 of8generality, we suppose that the sequence 7 0k & &0k is bounded. Theorem 10 yields the exisWm F is convergent and then Wm F k22 1 2 tence of a displacement field u1k 2 Us1 8 Y6 13 such that

7 8 1 2 &0k . W 0 u0k 6 u1k dy 3 Wm F

Y

1 2 From Equation 5137 which defines W 0 u0k 6 u1k , it is easy to establish that there exists K 1 2 1 such that

TWO-DIMENSIONAL MODELS FOR HETEROGENEOUS PLATES 99 7 8 ( ( &0k inf 5a 7 (F0k (2 dy K 1 6 Wm F

316 2

Y

( ( 7 8 &0k grad u1k 8 Since ((F &0k are bounded, we deduce that &0k (( and Wm F in which F0k 3 F y ( '( (grady u1k ( dy is bounded. We set 9 k 3 det F0k and again by using formula 5137, we can Y

write 7 8 1 2 &0k c 9 k 2 d ln59 k 2 7 dy8 Wm F Y

Let us suppose that 0 d1 d2 , so the previous inequality can be rewritten as

1 2 W 0 u0k 6 u1k dy d1

Y

ln59 k 2 7 dy c1

Y

9 k 2 dy Y1

2 1 c2 9 k 2 5d2 d1 7 ln59 k 2 7 dy8

Y2

It is easy to see that c1

'

9 k 2 dy

Y1

'1 c2 9 k

2

2 5d2 d1 7 ln59 k 2 7 dy is bounded below and

Y2

we deduce that there exists K 2 2 1 such that 1 0k 1k 2 0 dy 2 d1 ln59 k 7 dy K 2 8 W u 6u Y

Y

By using the Jensen inequality, we obtain

1 2 W 0 u0k 6 u1k dy 2 d1 ln 9 k dy K 2 8

Y

(15)

Y

Now 9 k can be expressed as ( ( 7 8 &0k 4 y u1k (( I3 4 y u1k 6 &0k 4 y u1k ((F 9 k 3 det F 1 2 3 1 2 where I3 denotes the third column of the 3 8 3 unit matrix. We want to show that 8 7 ( ( 1 2 k &0k (( I3 4 y u1k dy for all u1k x 4 6 5 2 W162 5Y6 13 78 &0k ((F 9 dy 3 det F 3 1 2 per Y

Y

(16)

100 E. PRUCHNICKI

We can obviously split

9 dy 3 k

Y

7 ( ( 8 &0k (( I3 4 y u1k dy &0k ((F det F 3 1 2

Y

( ( 8 7 (&0k 1k ( 1k dy det 4 y1 u1k (F 4 u 4 u I ( y 3 y 2 3 2

Y

7 8 ( ( &0k 4 y u1k (4 y u1k ( I3 4 y u1k dy8 det F 1 2 3 1

Y 1k 1k 3 Firstly, we assume that u1k 2 C 5i 3 16 37 per 5Y6 1 7, so the Fourier series of u and 4 yi u are

1 2 S p6q u1k 5y7 3

p q

Cn 1 n2 5y3 7 e2i5n1 y1 n 2 y2 7 6

n 1 3 p n 2 3q p q 2 1 S p6q 4 y u1k 5y7 3 2i n Cn1 n 2 5y3 7 e2i5n1 y1 n2 y2 7 6 n 1 3 p n 2 3q

1 2 S p6q 4 y3 u1k 5y7 3

p q

1 2 4 y3 Cn1 n 2 5y3 7 e2i5n 1 y1 n 2 y2 7 8

n 1 3 p n 2 3q

The result follows by observing that these Fourier series converge uniformly and thus their product converges uniformly too and can be integrated term by term. Due to the Y 4 -periodicity, the integrals of all non-constant harmonics are zero and formula 5167 is true for all u1k 3 3 162 3 2 C per 5Y6 1 7. By using the density of the space Cper 5Y6 1 7 in the space Wper 5Y6 1 76 formula 5167 holds.' ( ( Recalling that (grady u1k ( dy is bounded, we infer that there exists K 3 0 such that Y

( ( ( ( ( (&0k &0k ( (&0k &0k ( (( 1k ( F 9 k dy (F 6 F 4 u 6 F I dy K ( ( 3 y3 3 1 2 1 2 (8

Y

Y

From inequality 5157, we deduce that ( ( 7 8 1 2 &0k ((7 K 2 8 &0k 6 F &0k 3 W 0 u0k 6 u1k dy 2 d1 ln 5K 3 ((F Wm F 1 2 Y

7 8 &0k cannot be Consequently, we have shown the following contradiction: lim inf Wm F k

finite.

1

TWO-DIMENSIONAL MODELS FOR HETEROGENEOUS PLATES 101

Moreover, this membrane energy inherits the fact that the material frame indifference property is satisfied, which means that the energy is independent of the Cartesian frame in &0 6 F &0 3 06 we have &0 such that F which it is computed, namely for all 3 8 2 matrices F 1 2 7 8 7 8 &0 3 Wm F &0 for all R2 SO5378 Wm R F Similarly, we see that Wm is isotropic meaning that 7 Wm

8 7 8 &0 R 3 Wm F &0 for all R2 SO5278 F

Remark 13. By considering the rigorous 1-convergence argument of Muller [30], it can be shown that membrane energy is obtained by minimizing the local energy on k basic cells (Y becomes kY in formula (12)) and then on k. Nevertheless, as shown by Geymonat et al. [31] and Michel et al. [32], for a specific deformation gradient F0 , the one cell homogenized energy is the correct one. For homogeneous material the analytical expression of function Wm is continuous (Trabelsi [33]). However, for heterogeneous material, no mathematical results would allow us to show the continuity of Wm (Geymonat et al. [31, Section 5.2]).

9. CONTINUATION OF THE ASYMPTOTIC EXPANSION 9.1. The nonlinear membrane model revisited

We expect to find bending in the higher-order expansion. We want to exclude membrane ' ' effects and therefore we assume that g0 3 f0 dy h1 dy4 3 0. Then problem P1 5 8 Y 7 Y 4Y 1 1 22 1 2 0 F0 v0 6 v1 3 0 so we deduce that C0 v0 6 v1 3 I8 Now let us consider implies that W the components of the displacement field U, 1 2 Ui 3 gradx4 v0 i y v1i for i 3 16 38

(17)

Then we can write that 5I grady U7T 5I grady U7 3 C0 3 I8 If the displacement field v1 5x4 6 57 is regular enough, for example v1 5x4 6 57 2 C1 5Y1 6 13 7 C1 5Y2 6 13 7, it is easy to see that y U is a rigid deformation and then we have necessarily that 1 2 1 2 yi Ui 5x4 6 y7 3 i x4 Q ik x4 yk 6

(18)

where w is a vector field and Q is an orthogonal matrix. It follows from the expressions (17 ) and (18) that 1 1 1 2 1 1 2 1 22 2 1 2 2 i1 5x4 6 y7 3 i x4 Q i x4 9 i gradx4 v0 x4 i y Q i3 x4 9 i3 y3 8 (19)

102 E. PRUCHNICKI ' Since v1 is Y 4 periodic with respect to the variable y4 and v1 dy 3 0, formula (19) shows Y 1 2 that Q i 3 9 i gradx4 v0 i and w 3 08 Since Q is an orthogonal matrix, we immediately 1 2 1 2 &0 v0 6 F &0 v0 is the get that i1 5x4 6 y7 3 &i1 5x4 7 n i 5x4 7 y3 y3 9 i3 in which n 3 F 1 2 unit normal to the deformed mid-surface of the plate. Thus the leading-order term of the displacement field u0 inevitably belongs to the space of the inextensional displacement field 3 Uiso 0 5 6 1 7 3

5 v 2 W262 516 13 76 v 3 0 on 4 , E 5v7 3

% 8 2 1 2 1 71 I gradx4 5v7 k I gradx4 5v7 k 9 3 0 6 2

which is assumed to be different from 08 More precisely, the solutions v of the problem 3 P1 5 8 Y 7 are the elements of Viso 2 5 8 Y6 1 7, where 3 Viso 2 5 8 Y6 1 7 3

5 1 2 3 v 3 v0 6 v1 6 8 8 8 6 such that v0 2 Uiso 0 5 6 1 76 2 1 i1 x4 6 y 3 &i1 5x4 7 n i 5x4 7 y3 y3 9 i3 with & v1 2 U0 5 6 13 7 1 2 1 2 1 &0 v0 6 F &0 v0 6 & n3F v 3 0, 1 2

v p dy 3 06 p 2 on 4

! "

8

Y

9.2. Model Without Internal Energy

In this section we solve problem P2 5 8 Y 78 Theorem 14. For external loading satisfying g0 3

'

f0 dy

'

h1 dy4 3 06 if v 3 5v0 6 v1 6 8 8 87

4Y

Y

is a solution of problem P2 5 8 Y 7, then the first term v0 in the asymptotic expansion of the displacement field solves the global minimization problem 1 2 J 2 v0 3

inf

3 u0 2Uiso 0 5 6 1 7

1 2 J 2 u0 6

where 1 2 2 ' 1 0 1 02 J 2 u0 3 pi n i u gi1 u i0 dx4 with

p0 3

1 f0 y3 dy h1 dy4 h1 dy4 6 2 4Y

Y

g1 3

f1 dy

Y

4Y

h2 dy4 8

4Y

TWO-DIMENSIONAL MODELS FOR HETEROGENEOUS PLATES 103 Proof. We consider problem P2 5 8 Y 76 finding a displacement field v such that J 2 5v7 3

inf

3 u2Viso 2 5 8Y661 7

J 2 5u7 8

3 Let u 2 Viso 2 5 8 Y6 1 78 The contribution of the internal energy is given by

6 5 a C1ii 5u7 b Cofii1 5u7 c D 1 5u7 d L 1 5u7 dx4 dy8

I 5u7 3 2

8Y

By using formulae 577â€“5107, we get C1ii 5u7 3 2 F&i 0 Fi 1 2n i Fi31 6 Cof1ii 5u7 3 4 F&i 0 Fi 1 4n i Fi31 6 L 1 5u7 3

D 1 5u7 3 2 F&i 0 Fi 1 2n i Fi31 8

Recalling that a 2b 3 d c, we see that the internal energy vanishes. By computing the contribution of the external energy, we complete the proof. 1 We observe that problem P2 5 8 Y 7 is without internal energy and becomes trivial if we choose 1 h1 dy4 3 0 and g1 3 f1 dy h2 dy4 3 08 p0 3 f0 y3 dy h1 dy4 2 Y

4Y

4Y

Y

4Y

This leads us to choose p0 3 g1 3 0. Then problem P2 5 8 Y 7 reduces to a trivial one 3 iso 3 and therefore Viso 3 5 8 Y6 1 7 3 V2 5 8 Y6 1 78

10. A NONLINEAR MEMBRANE INEXTENSIONAL BENDING PLATE THEORY In this section, we assume that external loading satisfies f0 3 0, h1 3 0 and g1 3 0 then we justify a membrane inextensional bending plate model by solving P3 5 8 Y 78 1 2 Theorem 15. If v 3 v0 6 v1 6 8 8 8 is a solution of the problem P3 5 8 Y 7 then both the v1 in the asymptotic expansion of the leading term v0 and the macroscopic first-order term & displacement field solve the global minimization problem 1 2 Jbm v0 , & v1 3

u0 2Uiso 0 5 6

inf

13 76

& u1 2U0 5 6613 7

1 2 &1 6 Jbm u0 6 u

104 E. PRUCHNICKI

where the membrane non-extensional bending energy is defined by 1 2 &1 3 Jbm u0 6 u

1 1 2 1 1 22 4 Wbm gradx4 u0 6 gradx4 & u dx

f i 2 dy h i3 dy4 u i0 dx4 Y

4Y

f i1 y3 dy 1 h 2i dy4 h i2 dy4 n i dx4 8 2

4Y

Y

4Y

Wbm is the membrane inextensional bending elastic energy obtained by computing the local minimization problem 1 1 2 1 1 22 u 3 Wbm gradx4 u0 6 gradx4 &

inf u2 2U2 5 8Y6 13 7

1 2 u1 6 u2 dy6 W 2 u0 6 &

Y

in which we set 7 82 1 2 W 2 u0 6 & u1 6 u2 3 5a b7 F&i10 Fi21 F&i20 Fi11 7 87 8 0 1 1 5a b7 F&i 0 Fi31 n i Fi 1 F j3 n j F j

F&j

7 8 0 1 1 4 5a b7 F&i10 Fi11 F&j2 F j2 F&i 0 Fi 1 n j F j3 82 7 1 2 d F&i 0 Fi 1 n j F j3 6 1 2 where Fi 1 3 4x u&i1 y3 4x n i 4 y u 2i 6 Fi31 3 4 y3 u i2 and the set U2 8 Y6 13 is defined by ! 1 2 3 2 162 3 u2 dy 3 0 8 U2 8 Y6 1 3 u 2 Wper 51 8 Y6 1 76 " Y

Proof. We consider problem P3 5 8 Y 76 finding a displacement field v such that J 3 5v7 3

inf

u2Viso 3 5 8Y 7

J 3 5u7 8

(20)

1 2 2 1 2 3 3 Let u 3 u0 6 u1 6 8 8 8 2 Viso 3 5 8 Y6 1 7 and u 2 U2 8 Y6 1 . The contribution of the internal energy is given by

TWO-DIMENSIONAL MODELS FOR HETEROGENEOUS PLATES 105

6 5 a C2ii 5u7 b Cofii2 5u7 c D 2 5u7 d L 2 5u7 dx4 dy6

I 3 5u7 3

8Y

By using formulae 577â€“5107, we get Cii2 5u7 3 2 F&i 0 Fi 2 Fi 1 Fi 1 Fi31 Fi31 2 n i Fi32 6 Cofii2 5u7 3 4 F&i 0 Fi 2 2 Fi 1 Fi 1 2 Fi31 Fi31 4 n i Fi32 82 7 0 1 1 4 F&i10 Fi11 F&j2 F j2 4 F&i 0 Fi 1 n j F j3 F&i10 Fi21 F&i20 Fi11

7

F&i 0 4 y3 u 2i n i Fi 1

8 7

8 0 1 6 4 y3 u 2j n j F j

F&j

0 1 D 2 5u7 3 2 F&i 0 Fi 2 Fi 1 Fi 1 Fi31 Fi31 2 n i Fi32 4 F&i10 Fi11 F&j2 F j2 7 87 8 4 F&0 F 1 n j F 1 F&0 F 1 F 1 n i F&0 F 1 F 1 n j i

7

i

j3

F&i10 Fi21 F&i20 Fi11

i

82

i3

i

j

j3

j

6

22 D 1 5u7 8 D 5u7 2 1

L 5u7 3 2

2

in which Fi 1 3 4x u&i1 y3 4x n i 4 y u i2 6 Fi31 3 4 y3 u i2 and Fi32 3 4 y3 u i3 . Since a 2b 3 d c6 the internal energy can be written as I 3 5u7 3

1 2 W 2 u0 6 & u1 6 u2 dy dx4 , Y

with 82 7 1 2 u1 6 u2 3 5a b7 F&i10 Fi21 F&i20 Fi11 W 2 u0 6 & 7 87 8 0 1 1 5a b7 F&i 0 Fi31 n i Fi 1 F j3 n j F j

F&j

8 7 0 1 1 F j2 F&i 0 Fi 1 n j F j3 4 5a b7 F&i10 Fi11 F&j2 82 7 1 2 d F&i 0 Fi 1 n j F j3 8 The contribution of external forces to the total energy J 3 5u7 is represented by

(21)

106 E. PRUCHNICKI l 3 5u7 3

f i 2 dy h i3 dy4 u i0 dx4 Y

4Y

1 f i1 y3 dy h 2i dy4 h i2 dy4 n i dx4 8 2 Y

4Y

(22)

4Y

The proof of the theorem arises by combining formulae 5207 6 5217 and (2278

1

11. CONCLUSION In this work, we applied to inhomogeneous nonlinear plates a general systematic and objective method based on the asymptotic expansion analysis in order to determine a hierarchy of two-dimensional models. These results are in agreement with the models obtained by Gamma convergence for homogeneous plates by Friesecke et al. [34]. These methods present the drawback of separating the scales associated with membrane and bending behavior to a greater extent than desired. To overcome this difficulty, Ciarlet and Lods [35] and Ciarlet and Roquefort [36] proposed, in the spirit of the works of Koiter [37], a twodimensional model independent of the order of magnitude of the loads which generalize a hierarchy of two-dimensional models obtained by Ciarlet and Lods [20], Miara [21], Ciarlet et al. [38] and Lods and Miara [39]. By using a thickness-wise expansion of the displacement field, Steigmann [40] independently obtained a two-dimensional modeling of thin domains depending on the thickness and combining the effects of various levels of scales. This approach can be used to obtain a two-dimensional modeling of thin heterogeneous domains to which the method of homogenization will be applied. REFERENCES [1] [2] [3] [4] [5]

[6] [7] [8]

Ciarlet, P. G. Mathematical Elasticity, Vol. II. Theory of Plates, North-Holland, Amsterdam, 1997. LewiÂ´nski, T. and Telega, J. J. Plates, Laminates and Shells: Asymptotic Analysis and Homogenization, World Scientific, Singapore, 2000. Pruchnicki, E. Non linearly elastic membrane model for heterogeneous shells by using a new double scale variational formulation: A formal asymptotic approach. J. Elasticity, 84, 245â€“280 (2006). Pruchnicki, E. Two-dimensional nonlinear models for heterogeneous plates. Comptes Rendus de lâ€™AcadĂŠmie des sciences, Paris, 337(5), 297â€“302 (2009). Pruchnicki, E. Nonlinearly elastic membrane model for heterogeneous plates : a formal asymptotic approach by using a new double scale variationnal formulation. International Journal of Engineering Science, 40, 2183â€“2202 (2002). Ciarlet, P. G. and Geymonat, G. Sur les lois de comportement en ĂŠlasticitĂŠ non linĂŠaire compressible. Comptes Rendus de lâ€™AcadĂŠmie des sciences, Paris, Series II, 295, 423â€“426 (1982). Ball, J. M. Convexity condition and existence theorems in nonlinear elasticity. Archive for Rational Mechanics and Analysis, 63(4), 337â€“403 (1976). Ciarlet, P. G. Mathematical Elasticity, Vol. I: Three-Dimensional Elasticity. North-Holland, Amsterdam, 1988.

TWO-DIMENSIONAL MODELS FOR HETEROGENEOUS PLATES 107

[9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]

[32] [33] [34] [35] [36]

Panasenko, G. Muticomponent homogenization of the vibration problem for incompressible media with heavy and rigid inclusions. Comptes Rendus de l’Académie des sciences, Paris, Series I, 321, 1109–1114 (1995). Pantz, O. Quelques problèmes de modélisation en élasticité non linéaire, PhD Thesis, University of Paris VI, 2001. Pantz, O. Dérivation des modèles de plaques membranaires non linéaires à partir de l’élasticité tri-dimensionnelle. Comptes Rendus de l’Académie des sciences, Paris, Series I, 331, 171–174 (2000). Meunier, N. Recursive derivation of one-dimensional models from three-dimensional nonlinear elasticity. Mathematics and Mechanics of Solids, 13(2), 172–194 (2008). Trabelsi, K. Nonlinear thin plate models for a family of Ogden materials. Comptes Rendus de l’Académie des sciences, Paris, Series I, 337, 819–824 (2003). Giroud, P. Analyse asymptotique de coques inhomogènes en élasticité linéarisé anisotrope. Comptes Rendus de l’Académie des sciences, Paris, Series I, 327, 1011–1014 (1998). Caillerie, D. and Sanchez-Palencia, E. Elastic thin shells: Asymptotic theory in the anisotropic and heterogeneous cases. Mathematical Models and Methods in Applied Science, 8(4), 473–496 (1995). Trabelsi, K. Nonlinearly elastic thin plate models for a class of Ogden materials: II. The bending model. Analysis and Applications (Singapore), 3, 271–283 (2005). Ne2cas, J. Les méthodes directes en théorie des équations elliptiques, Masson. Paris, 1967. Caillerie, D. Thin elastic and periodic plates. Mathematical Models and Methods in Applied Science, 6, 159–191 (1984). Lewi´nski, T. Effective models of composite periodic plates I. Asymptotic solution. International Journal of Solids and Structures, 27, 1173–1174 (1991). Ciarlet, P. G. and Lods, V. Asymptotic analysis of linearly elastic shells I. Justification of membrane shell equations. Archive for Rational Mechanics and Analysis, 136, 119–161 (1996). Miara, B. Nonlinearly elastic shell models: a formal asymptotic approach I. The membrane model. Archive for Rational Mechanics and Analysis, 142, 331–353 (1998). Collard, C. and Miara, B. Asymptotic analysis of the stresses in thin elastic shells. Archive for Rational Mechanics and Analysis, 148, 233–264 (1999). Mardare, C. Asymptotic analysis of linearly elastic shells: error estimates in the membrane case. Asymptotic Analysis, 17, 31–51 (1998). Lions, J. L. Some Methods in the Mathematical Analysis of Systems and their Control, Gordon and Breach Science Publishers, New York, 1981. Dauge, M. and Gruais, I. Développement asymptotique d’ordre arbitraire pour une élastique mince encastrée. Comptes Rendus de l’Académie des sciences, Paris, Series I, 321, 375–380 (1995). Dauge, M. and Gruais, I. Asymptotics of arbitrary order for a thin elastic clamped plate. I: Optimal error estimates. Asymptotic Analysis, 13, 167–197 (1996). Pruchnicki, E. Hyperelastic homogenized law for reinforced elastomer at finite strain with edge effects. Acta Mechanica, 129, 139–162 (1998). Pruchnicki, E. Overall properties of thin hyperelastic plate at finite strain with edge effects using asymptotic method. International Journal of Engineering Science, 36, 973–1000 (1998). Bakhvalov, N. S. and Panasenko G. Homogenisation: Averaging Processes in Periodic Media (Mathematics and its Applications Soviet Series), Kluwer Academic, Dordrecht (1989). Muller, S. Homogenization of nonconvex cellular elastic materials. Archive for Rational Mechanics and Analysis, 99, 189–212 (1987). Geymonat, G., Muller, S. and Triantafyllidis, N. Homogenization of non linearly elastic materials, microscopic bifurcation and macroscopic loss of rank-one convexity. Archive for Rational Mechanics and Analysis, 122, 231– 290 (1993). Michel, J. C., Lopez-Pamies, O., Ponte-Castañeda, P. and Triantafyllidis, N. Microscopic and macroscopic instabilities in finitely strained porous elastomers. Journal of the Mechanics and Physics of Solids, 55, 900–938 (2007). Trabelsi, K. Nonlinearly elastic thin plate models for a class of Ogden materials: I The membrane model. Analysis and Applications (Singapore), 3, 195–221 (2005). Friesecke, G., James, R. D. and Muller, S. A herarchy of plate models derived from nonlinear elasticity by gammaconvergence. Archive for Rational Mechanics and Analysis, 180, 183–236 (2006). Ciarlet, P. G. and Lods, V. Asymptotic analysis of linearly elastic shells. III. Justification of Koiter’s shell equations. Archive for Rational Mechanics and Analysis, 136, 191–200 (1996). Ciarlet P. G. and Roquefort, A. Justification of a two-dimensional nonlineat shell model of Koiter’s type. Chinese Annals of Mathematics, 22B, 129–144 (2001).

108 E. PRUCHNICKI

[37] Koiter, W. T. On the nonlinear theory of thin elastic shells. Proceedings of the Academic Congress, Netherlands, Wetensch, B69, 1–54 (1966). [38] Ciarlet, P. G., Lods, V. and Miara, B. Asymptotic analysis of linearly elastic shells. II. Justification of flexural shell equations. Archive for Rational Mechanics and Analysis, 136, 163–190 (1996). [39] Lods, V. and Miara, B. Nonlinearly elastic shell models: A formal asymptotic approach II. The flexural model. Archive for Rational Mechanics and Analysis, 142, 355–374 (1998). [40] Steigmann, D. J. Two-dimensional models for the combined stretching of plates and shells on three-dimensional linear elasticity. International Journal of Engineering Science, 46, 654–676 (2008).

Derivation of a Hierarchy of Nonlinear Two-dimensional Models For Heterogeneous Plates

E. P RUCHNICKI

Ecole Polytechnique Universitaire de Lille, Cité Scientifique, Avenue Paul Langevin, F 59655 Villeneuve d’Ascq Cedex, France (Received 10 July 20091 accepted 27 July 2009)

Abstract: This paper is concerned with the asymptotic analysis of plates with periodically rapidly varying heterogeneities. The formal asymptotic procedure is performed when both the periods of changes of the material properties and the thickness of the plate are of the same orders of magnitude. Our approach is based on a sequence of recursive minimization problems. We consider a plate made of Ciarlet–Geymonat type materials. Depending on the order of magnitude of the applied loads, we obtain a nonlinear membrane model and a nonlinear membrane inextensional bending model as announced by Pruchnicki.

Key words: Asymptotic analysis, plates, nonlinear elasticity, minimization problem

1. INTRODUCTION Plates are three-dimensional bodies characterized by a small transverse dimension, called the thickness, that are very small in comparison with the two other dimensions of the middle surface. This feature suggests the use of the method of asymptotic expansion with the thickness regarded as a small parameter, as a means of deriving two-dimensional mathematical models of plates from the three-dimensional theory. A precise mathematical formalization of the asymptotic approach in the context of linear and nonlinear elasticity is given by Ciarlet [1]. For heterogeneous plates, the size of the heterogeneities is a second small parameter. The length scale of this microstructure can also be larger than, comparable to or smaller than the thickness. The second case is the more important because the two others can be deduced and this is confirmed by the rigorous results of 1 convergence (Lewi´nski and Telega [2]). For plates made of nonlinear heterogeneous Saint Venant–Kirchoff materials, Pruchnicki [3– 5] showed by using the asymptotic expansion method that the model of the highest possible order is of membrane type. Nevertheless the Saint Venant–Kirchoff material is not realistic because it does not exclude the possibility of squashing a volume into a point. For this work, we consider a nonlinear material proposed by Ciarlet and Geymonat [6] which is in agree-

Mathematics and Mechanics of Solids 16: 77–108, 2011 1 The Author(s), 2011. Reprints and permissions: 1 http://www.sagepub.co.uk/journalsPermissions.nav

DOI: 10.1177/1081286509349965

78 E. PRUCHNICKI

ment with this fundamental physical requirement. Moreover, the stored energy function is polyconvex with respect to the invariants of the deformation gradient which ensures powerful results for the minimization problem of the potential energy (Ball [7], Ciarlet [8]). Moreover we consider composite plates for which the stiffness of each material is of the same order of magnitude. When this assumption is not satisfied, the asymptotic analysis depends on another small parameter which is the ratio of the mechanical characteristics of the matrix to that of the reinforcement. This leads to non-classical models of homogenization (Panasenko [9]). The outline of this paper reads as follows. We adapt the method developed previously by Pantz [10, 11] to solve a sequence of minimization problems for heterogeneous plates. This method is also used by Meunier [12] and Trabelsi [13] for the cases of homogeneous rods and plates, respectively. We obtain a hierarchy of two-dimensional models depending on the order of magnitude of the applied loads. For loading of first order with respect to the small parameter considered in Section 6, we obtain a two-dimensional energy minimization problem modeling the nonlinear membrane behavior of the plate and whose solution is the leading term in the expansion of the displacement field. In Section 9, the loading is of order two in powers of the small parameter so we obtain a model without internal constraint. In Section 10, we are thus led to define a generalization of the space of nonlinear nonextensional displacements for heterogeneous plates. When this space does not reduce to {0} and for loading of order three with respect to the small parameter, the displacement field can be identified as a solution of a two-dimensional nonlinear membrane inextensional bending model. This model generalizes the ones proposed by Giroud [14], Caillerie and SanchezPalencia [15] and Trabelsi [16].

2. THE NONLINEAR BOUNDARY-VALUE PROBLEM OF PLATES In this section, we provide the main notation and describe in detail the problem we will consider. 2.1. Notation

Throughout this paper, the summation convention is adopted. The Latin indices take values 1, 2 and 3. The Greek indices (except for 2) run over 1 and 2. Boldface letters represent vector-valued functions, tensors or spaces. 3 is an open subset of the Euclidean space 1n (n 2 2). The boundary and the closure of 3 are denoted by 43 and 3, respectively. Let 5e1 6 e2 6 e3 7 denote the basis of the space 13 8 We shall write x 3 5x1 6 x2 6 x3 7 for vectors in 13 , and x4 3 5x1 6 x2 7 for vectors in 12 8 The Euclidean scalar product, the exterior product and the tensor product of vectors a, b 2 13 are denoted by a 5 b, a 6 b and a 7 b, the Euclidean norm on 1n 8 m is denoted by 959. I denotes1the2unit matrix. 9 is the Kronecker symbol. det 5A7 is the determinant of the matrix A 3 ai j and Cof5A7 is the cofactor matrix of the matrix A. The notation 3 4

3 (3 4 compactly contained in 3) means that 3 4 is compact and contained in 3. 3 1 3 2 is the complement of the subset 3 2 contained in 3 1 . A domain 3 in 1n is a bounded, open, connected subset of 1n with a Lipschitz-continuous boundary 43

TWO-DIMENSIONAL MODELS FOR HETEROGENEOUS PLATES 79 in the sense of Ne2cas [17]. The measure of a Lebesgue measurable set M in 1n is denoted by meas 5M7. We define L p 53 6 1m 7 to be the collection of all m-tuples 5 f 1 6 f 2 6 8 8 8 6 f m 7 of real functions in L p 537. Similarly, we say that u 2 W16 p 53 6 1m 7 if u belongs to L p 53 6 1m 7 together with its partial distributional derivatives 4x j u i 1 we adopt this notation to represent the partial 4u i derivative 4x for 1 i m, 1 j n. j n m C 53 6 1 7 (n 2 2 ) is the space of n times continuously differentiable mappings from a domain 3 of 1 p to 1m . v belongs to Cn 536 1m 7 (n 2 2 ) if it is in Cn 53 6 1m 7 and for each multi-index with 9 9 3 1 5 5 5 n n, there exists a mapping i in C0 536 17 9 9 i such that the restriction to 3 of the partial derivative 4 i 3 4 x 41 5554x

n 5i 3 16 8 8 8 6 m7 1

n

2 is equal to i . S3 represents the set of all symmetric 3 8 3 matrices. C per 5 8 Y 7 is 3 4 162 the set of functions of C 5 8 12 8 22 6 22 76 Y 4 periodic in y4 . Wper 5 8 Y 2 7 is the 2 162 completion of the space Cper 5 8Y 7 with respect to the norm of W 5 8Y 2 7 and W162 5 8 1 162 2 3 per2 2 4 2 3 2 3 2 2 Y 6 1 7 3 Wper 5 8 Y 7 8 In the same way, Cper 5Y 7 is the subset of C 51 8 2 6 2 7 2 2 of Y 4 periodic in y4 and W162 per 5Y 7 is the completion of the space Cper 5Y 7 with respect to the 1 2 3 162 162 5Y 2 6 13 7 3 Wper 5Y 2 7 8 norm of W162 5Y 2 7. Finally, Wper

2.2. Description of the Plate Geometry

Let be a domain4 in 12 .3 Let us consider a plate of thickness 2 and mid-surface 8 The 2 2 1 2 set 3 3 3 8 22 6 22 6 2 2 1 is called the reference configuration of this plate. The boundary of the plate 15 is6 divided 4 upper 5 62into two parts: the first is composed of the lower and 3 boundaries 8 22 22 1 the second is the lateral boundary 1 2 3 4 8 22 6 22 8 This plate is heterogeneous and the size of the heterogeneities, which is assumed to be of the same order of magnitude as the thickness, is very small with regard to the global length-scale between both the local length-scale of the shell x4 3 5x 1 6 x 2 7. Thus we can define the ratio 4 y4 3 5y1 6 y2 7 and the global one 2 3 xy 4 . We note that the microstructure of the plate is

periodic with respect to global coordinates and it is4sufficient define the distribution of 3 to 2 2 2 4 4 the constituents on the smallest period Y 3 Y 8 2 6 2 5Y 3 506 17 8 506 177, which is also called the unit cell. A current point in Y 2 is defined by y2 3 5y1 6 y2 6 x32 7. Since the thickness of the plate is of the order of magnitude 2, we introduce y3 such that x32 3 2 y3 . As the coordinates of a point of the domain and those y2 of a point of the domain Y 2 are independent, we can define the coordinates of a point of the domain 8 Y 2 by the following element of 15 : 5x 4 6 y 4 6 x 32 7. We suppose that the plate is composed of both a matrix and an inclusion, which occupy two domains Y12 and Y22 , such that Y 2 3 Y12 Y22 and Y12 Y22 3 3. 2 The common boundary of both the matrix and the inclusion is denoted I . 4 The boundary 3 1 2 2 2 2 4 4Y of the unit cell is divided into the lateral boundary , and into both 4Y 3 4Y 8 6 2 2 15 6 L 5 62 the lower and the upper boundaries 4Y2 3 Y 4 8 22 22 . 2.3. Loads and Boundary Conditions

The plate is submitted to dead loading which is periodic with respect to the local lengthscale y4 , and it is sufficient to define this dead loading only over one unit cell. The density

80 E. PRUCHNICKI of body force is the vector denoted f i2 ei 1 we assume that the vector f2 3 5 f i2 7 belongs to L2 5 8 Y 2 6 13 7. Let force5 per6unit area h i2 ei acting on the upper face 1 62 3 5 2 6us consider the surface

8 2 and lower face 1 62 3 8 22 of the plate1 then their components 1are2functions 15 6 5 62 2 2 4 h i2 :

18 We suppose that the vector h2 3 h i2 belongs 1 8 Y 82 3 22 2 2 to L 8 4Y 6 1 . The plate is clamped on its lateral boundary 1 2 . When subjected to the given loading, the plate field u i2 5x4 6 y2 7 ei 8 Next we denote 1 2 2undergoes the displacement 2 3 the vector field u 3 u i : 8 Y 2 1 . The deformation gradient is defined by F2 5u2 7 3 I grad u2 and grad u2 is the gradient of the displacement field u2 , 1 5grad u2 7i 3 4x u 2i 4 y u 2i 6 2

5grad u2 7i3 3 4x32 u i2 8

2.4. Constitutive Equations of each Constituent

The plate is made of a two-phase hyperelastic of which the stored energy functions defined by (Ciarlet and Geymonat [6], see also Ciarlet [8]) 1 1 22 W 5F2 7 3 a 9F2 92 b 9CofF2 92 1 det F2T F2 e6

(1)

with a 06 b 06 1 597 3 c 9 d ln5976 c 06 d 06 e 2 1 are polyconvex and satisfy W 5F2 7 3

1 2 5trE2 72 trE22 O E23 8 2

(2)

1 1 22 In the above, E2 3 12 F2T F2 I stands for the Greenâ€“Saint Venant strain tensor and 0, 0 are two given LamĂŠ constants. Relation 527 is the beginning of the expansion of the stored energy function of isotropic material near a natural state. The Saint Venantâ€“Kirchhoff material is the simplest one that agrees with this expansion. The constants a, b6 c, d and e are given in terms of the LamĂŠ constants 9

7 8 1 a 3 s 6 b 3 s 6 c3s 6 2 2 2 9

9

1 6 e 3 2s 6 d 3 2 2 4 and as a6 b6 c and d are real positive functions, we infer that necessarily 9

9 1 6 6 8 s 2 min 2 2 2 2

TWO-DIMENSIONAL MODELS FOR HETEROGENEOUS PLATES 81

Moreover, we have

lim

det5F2 70

W 5F2 7 3 , which means that an infinite amount of energy

is necessary to annihilate a volume. From expression 517 of the function W 5F2 7, we can establish the following coerciveness inequality: 2 1 W 5F2 7 A 9F2 92 9CofF2 92 5det F2 72 B6

A 08

If the plate is composed of two homogeneous materials then both and are piece-wise constant functions defined as 1 2 2 1 x4 6 y4 6 x 32 3 , x4 6 y4 6 x32 3 ,

1 2 s x4 6 y4 6 x32 3 s ,

1 2 in which x4 6 y4 6 x32 2 8 Y 2 8 Hence , s and represent the LamĂŠ constants of each material constituting the plate ( 0, 0, s 0). To carry out the computations involved in the asymptotic analysis, we can express the stored energy function 517 as W 5F2 7 3 a Cii2 5F2 7 b 5Cof C2 5F2 77ii 1 5det 5C2 5F2 777 e2 6

(3)

in which C2 5F2 7 3 F2T F2 denotes the right Cauchyâ€“Green strain tensor.

3. THE TWO-SCALE THREE-DIMENSIONAL MINIMIZATION PROBLEM In this section, we give the double-scale three-dimensional minimization problem posed on the family of plates parameterized by 2, 2 0. Then we suppress the difficulty originating from the dependence on the small parameter of the plate by applying a change of variables. 3.1. The Three-dimensional Double Scale Boundary Value Problem

P5 8 Y 2 7

1 2 We formulate the family of double-scale problems posed on 8 Y 2 . Let V2 8 Y 2 6 13 be the set of admissible displacement field 1 2 2 3 2 2 V2 8 Y 2 6 13 3 u2 2 W162 per 5 8 Y 6 1 7 : det 5F 5u 77 0 and u2 3 0 on 4 8 Y 2

8

Then the equilibrium state of the plate satisfies the minimization problem P 2 5 8 Y 2 7 1 2 Find v2 2 V2 8 Y 2 6 13 such that J 2 5v2 7 3

inf

u2 2V2 5 8Y 2 6 13 7

where the total energy functional J 2 takes the form J 2 5u2 7 3 I 2 5u2 7 l 2 5u2 7 8

J 2 5u2 7 6

82 E. PRUCHNICKI Here the linear form l 2 is the work of the external forces 2 2 2 2 4 2 l 5u 7 3 f i u i dx dy

h i2 u 2i da8

84Y2

8Y 2

2 1 In the previous equation, da denotes the area element along the boundary 4 8 Y2 , while I 2 measures the internal energy of the plate I 2 5u2 7 3

W 2 5F2 7 dx4 dy2 8

8Y 2

3.2. Transformation into a Problem

P 52 8 Y 7 Posed over a Domain Independent of 2

In this section, we transform problems P5 8Y 2 7 into problems P52 8Y 7 posed over a set that does not depend on 2 by using the following change of variables (Ciarlet [1], Caillerie [18], LewiÂ´nski [19], Ciarlet and Lods [20], Miara [21], Collard and Miara [22], Mardare [23]): 1 4 4 22 x 6 y 6 x3 2 8 Y 4 8 [26 2] 5x4 6 y7 2 8 Y 4 8 [16 1]. Thus the domains Y 2 and Y 2 become the domains Y 3 Y 4 8516 17 and Y , respectively. The boundaries 4Y 2 , 4Y2 , 4Y L2 , 1 2I become 4Y , 4Y 3 Y 4 851 17 6 4Y L 3 4Y 4 8[16 1] 6 1 I , respectively. For any function g 2 defined on 8 Y 4 8 [26 2], we associate a function g527 defined on 8 Y 4 8 [16 1] in the following manner: 1 2 g 2 x4 6 y4 6 x 32 3 g5275x4 6 y78 For the sake of simplicity, we omit 2 in brackets for the material constants. This change of variable implies the following relations concerning the partial derivatives 4x g 2 3 4x g527, 4 y g 2 3 4 y g527,

4x32 g 2 3

1 4 y g527. 2 3

Consequently, the space V2 5 8 Y 2 6 13 7 becomes 1 2 3 V 527 8 Y6 13 3 u 2 W162 per 5 8 Y6 1 7 : det 5F 527 5u77 0

and u 3 0 on 4 8 Y 6 in which the components of the deformation gradient are defined by

TWO-DIMENSIONAL MODELS FOR HETEROGENEOUS PLATES 83 1 1 Fi 527 5u 5277 3 9 i 4x u i 527 4 y u i 527 , Fi3 527 5u 5277 3 9 i3 4 y3 u i 527 8 2 2 The determination of a displacement field belonging to the space V2 5 8 Y 2 6 13 7 and satisfying the nonlinear minimization problem P 2 5 8 Y 2 7 is equivalent to finding in the space a displacement field satisfying the following problem P52 8 Y 7: 1 2 Find v 527 2 V 527 8 Y6 13 such that J 527 5v 5277 3

inf

u5272V5275 8Y6 13 7

J 527 5u 5277 6

where the total energy functional J 527 takes the form J 527 5u 5277 3 I 527 5u 5277 l 527 5u 5277 6 with I 527 5u 5277 3 2

W 527 5F 5277 dx4 dy6

8Y

4

f i 527 u i 527 dx dy

l 527 5u 5277 3 2

8Y

h i 527 u i 527 da8

84Y

4. SETTING OF THE ASYMPTOTIC PROCEDURE 4.1. Assumption of the Asymptotic Expansions

The formal asymptotic two-scale method is based on the hypothesis that the unknown vector field u 527, the data f 527 and h 527 admit asymptotic expansion as a power of the small parameter 28 When the Greenâ€“Lagrange strain tensor is small enough, the general stored energy function 537 becomes of Saint Venantâ€“Kirchhoff type and in this case Pruchnicki [3, 5] and Miara [21] show that the first power of 2 cannot be negative, so the asymptotic expansions of both the displacement fields and external loading are necessary as follows: 1 2 u2 x4 6 y2 3 u5275x4 6 y7 3 u0 5x4 6 y7 2 u1 5x4 6 y7 22 u2 5x4 6 y7 5 5 5 6 1 2 f2 x4 6 y2 3 f5275x4 6 y7 3 f0 5x4 6 y7 2 f1 5x4 6 y7 22 f2 5x4 6 y7 5 5 5 6 1 2 3 h5275x4 6 y7 3 2 h1 5x4 6 y7 22 h2 5x4 6 y7 5 5 5 8 h2 x4 6 y2 The coefficients of the asymptotic expansions of both the body load f p1 (for p 0) 2 and p 2 3 the1surface load h (for p 0)2 are supposed to belong to the spaces L 8 Y6 1 and L2 8 Y 4 8 51 17 6 13 , respectively. To each functional G 527, we associate the functional defined by

84 E. PRUCHNICKI 2 2 1 1 527 u0 6 u1 6 u2 6 8 8 8 3 G 527 u0 2 u1 22 u2 5 5 5 8 G 1 2 1 2 From now on, u and v will stand for the sequences ui i22 and vi i22 , respectively. 4.2. Asymptotic Expansion of the Energy Terms

The interest of the following section is to recast the sequence of problems P 52 8Y 7 into a sequence of simpler variational problems. We begin by computing the asymptotic expansion of F 527 5u 5277, 527 5u7 3 F 527 5u 5277 3 F

Fn 5u7 2n 6

(4)

n3 1

in which the components of F1 5u7, F0 5u7 and Fl 5u7 (for an integer l 0) are defined by Fi1 j 5u7 3

2 1 grady u0 i j 3 4 y j u i0 6

1 1 2 2 Fi0j 5u7 3 9 i j gradx4 u0 i j grady u1 i j 6 Filj 5u7 3

1 2 2 1 gradx4 ul i j grady ul1 i j

5l 07 8

(5)

527 admits the formal asymptotic expansion Lemma 1. The tensor C 527 5u7 3 C 527 5u 5277 3 C

Cn 5u7 2n 6

(6)

n32

where the components of the tensor Cn 5u7 have the following expressions: Cn 5u7 3

n1

F pT 5u7 Fn p 5u7 8

(7)

p31

Proof. From expansion 547, we get 527 5u7 3 C

F pT 5u7 Fk 5u7 2 pk 3

p31k31

3

n1

F pT 5u7 Fn p 5u7 2n

n32 p31

Cn 5u7 2n 8

n32

1

TWO-DIMENSIONAL MODELS FOR HETEROGENEOUS PLATES 85 7 8 527 Lemma 2. The asymptotic expansion of CofC is ii

7 8 Cof C 527 5u7 3 Cofiin 5u7 2n 6 ii

n34

where Cofiin 5u7 has the following expression: Cofiin 5u7 3

n2

p

n p

p

n p

p

n p

C11 5u7 C22 5u7 C11 5u7 C33 5u7 C22 5u7 C33 5u7

p32

n2

p n p p n p C12 5u7 C12 5u7 C13 5u7 C13 5u7

p32

n2

p n p C23 5u7 C23 5u7 8

(8)

p32

Proof. The proof is a direct consequence of the equality 9

7 8 7 8 1 7 8 2 C 527 5u7 C 527 5u7 C 527 5u7 kk jk jk 2 7 8 7 8 7 8 7 8 527 5u7 C 527 5u7 527 5u7 C 527 5u7 C 3 C

7 8 527 5u7 3 Cof C ii

11

22

11

33

7 7 8 8 527 5u7 527 5u7 C C 22

33

97 7 7 82 82 82 C 527 5u7 C 527 5u7 C 527 5u7 6 12

13

23

1

and we use 567 to conclude. Let us define 7 8 527 5u7 3 det C 527 5u7 8 D 527, Lemma 3. We have the asymptotic expansion for the function D 527 5u7 3 D

n36

D n 5u7 2n 6

86 E. PRUCHNICKI where D n 5u7 satisfies the following expression:

D n 5u7 3

1 2 p C11 5u7 Ck22 5u7 Cl33 5u7 Ck23 5u7 Cl23 5u7

pkl3n

p C12 5u7

1 k 2 2C13 5u7 Cl23 5u7 Ck12 5u7 Cl33 5u7

pkl3n

p C13 5u7 Ck13 5u7 Cl22 5u7 8

(9)

pkl3n

Proof. We note that 97

8 7 7 82 8 7 8 527 5u7 C 527 5u7 527 5u7 527 5u7 C det C 3 C 11

22

11

7 7 7 8 8 8 527 5u7 C 527 5u7 527 5u7 C 2 C 12

13

23

7 7 7 8 7 8 82 82 527 5u7 C 527 5u7 8 527 5u7 C 527 5u7 C C 33

12

22

13

527 5u7 567 6 the result follows. Then by considering the asymptotic expansion of the tensor C 1 Let us introduce the following notation: 527 5u7 3 ln D 527 5u7 8 L 527 admits the asymptotic expansion Lemma 4. The function L 527 5u7 3 L

L n 5u7 2n 6

n30

where L 0 5u7 3 ln

0

D n 5u7 2n

6

n36

n 1 L n 5u7 3 p p31

p

p 1 D ki 5u7 0 q i31 p D 5u7 2q q36

ki 3n

i31

n 2 18

TWO-DIMENSIONAL MODELS FOR HETEROGENEOUS PLATES 87

Proof. We have

527 5u7 3 ln D 527 5u7 3 ln L

D n 5u7 2n

n36

3 ln

0

D q 5u7 2q 2

q36

D n1 5u7 2n

n30

n

517n1 3 L 5u7 0 n n31 0

2 Dq

5u7 2

q

n D

k1

5u7 2

k

8

k30

q36

Consequently, p n ! p 1 1 0 ki L 527 5u7 3 L 5u7 D 5u7 2n 8 0 p q i31 p q n31 p31 D 2 5u7 " k 3n i q36 i31

1 Lemma 5. The functional I 527, which represents the elastic energy, admits the expression I 527 5u7 3

I n 5u7 2n meas 5 7

n35

e dy6 Y

with I 5u7 3 n

2 1 a Ciin1 5u7 b Cofiin1 5u7 dx4 dy

8Y

1 2 c D n1 5u7 d L n1 5u7 dx4 dy6

8Y

7 8 7 8 527 , Cof C 527 and L 527 6 in which the coefficients of the asymptotic expansions of C ii ii which are not defined in the above lemmas, are equal to zero. Moreover, the linear functional l 527 6 which represents the external loading contribution, can be written as l 527 5u7 3

l n 5u7 2n 6 n30

88 E. PRUCHNICKI

where l n 5u7 3

n1

f i p u in1 p dx4 dy

8Y p30

n h ip u in p da8

84Y p30

Finally, the total energy functional J 527 admits the following asymptotic expansion: J 527 5u7 3

J 5u7 2 meas 5 7 n

n

n35

e dy6

J n 5u7 3 I n 5u7 l n 5u7 8

Y

Proof. Again, this is obtained by replacing the different terms in the asymptotic expansions of I 527 5u7, l 527 5u7 and J 527 5u7 by their respective asymptotic expansions. 1 7 8 527 5u7 0 should be for 2 small enough so that this Finally, the constraint det F condition is equivalent to the positivity of the leading term in the 7 asymptotic 8 expansions in 527 5u7 as follows: powers of 2. Now we can write the asymptotic expansion of det F 7 8 527 5u7 3 Fndet 5u7 2n 8 det F n33 3 Here we necessarily have F3 5u7 0, so we do not need to express Fndet 5u7 det 5u7 3 det F in terms of Fq 5u7 6 q 2 18

4.3. The Asymptotic Procedure

The next proposition shows that the solution v 527 of problem P52 8 Y 7 can be obtained by solving a sequence of partial variational problems. This idea was introduced by Pantz [11] in the case of homogeneous plates. Proposition 6. The solution of problem P52 8 Y 7 is such that 1 2 # v 3 v0 6 v1 6 v2 6 8 8 8 2 Vn 5 8 Y6 13 76 n35

where $ v 2 Vn 5 8 Y6 13 7 : J n 5v7 3 Vn1 5 8 Y6 1 7 3 3

V5 5 8 Y6 13 7 3

162 v 2 Wper 51 8 Y6 13 72 :

inf

u2Vn 5 8Y 7

% J 5u7 6 n

vn 2n 2 V526 8 Y 7 8

n31

TWO-DIMENSIONAL MODELS FOR HETEROGENEOUS PLATES 89

Proof. The proof results from a simple adaptation to this heterogeneous case of the reason1 ings of Pantz [10] and Meunier [12]. Let Pn 5 8 Y 7 be the following boundary-value problem: Finding v 2 Vn 5 8 Y6 13 7 such that J n 5v7 3

inf

u2Vn 5 8Y6 13 7

J n 5u7 8

We are going to solve the first variational problems Pn 5 8 Y 78

5. SOLVING PROBLEMS Pn 5 8 Y 7 for n 3 0 The purpose of the first part of this section is to identify the set V0 5 8 Y6 13 7 by solving problems 5Pn 5 8 Y 77n 0 8 1These problems 2 are independent of the external loading. We show that any solutions v 3 v0 6 v1 6 v2 6 8 8 8 of P1 5 8 Y 7 are such that the leading-order term v0 does not depend on the microscopic scale of the plate. Next we show that problem P0 5 8Y 7 is trivial. Before studying this problems, we observe that the displacement field vi (i 0) cannot vanish on the clamped boundary 4 8 Y . Then a boundary layer term is considered by Lions [24, p. 49] for elastic heterogeneous three-dimensional structures, Dauge and Gruais [25, 26] for elastic homogeneous plates and Pruchnicki [27, 28] for hyperelastic heterogeneous three-dimensional structures and plates. Thus to avoid the boundary-layer region around the neighborhoods of the clamped part of the plate, we replace this boundary condition by its average on the unit cell. 5.1. Identification of the Set V0 5 8 Y6 13 7

Lemma 7. The set V0 5 8 Y6 13 7 is defined by 3 2 : det F0 5v7 0, V0 5 8 Y6 13 7 3 v 2 W162 per 5 8 Y6 1 7 1 1 2 2 v x4 6 y 3 v0 x4 , v0 3 0 and

v dy 3 0 on 4

0

! "

8

Y

Proof. We consider the first problem Pn 5 8 Y 7 55 3 n 3 17 which consists of finding v in 2n 5 8 Y 7 such that J n 5v7 3

inf

u2Vn 5 8Y6 13 7

J n 5u7 6

By Lemmas 4 and 5 of Section 4.2, we have

J n 5u7 3 I n 5u7 8

90 E. PRUCHNICKI I n 5u7 3

2 1 a Ciin1 5u7 b Cofiin1 5u7 c D n1 5u7 dx4 dy8

8Y

8 7 8 7 527 5u7 requires the positivity of the 527 5u7 and D 527 5u7 6 Cof C The positivity of C ii ii first-order term of their asymptotic expansions. Then, for the lowest order problem P5 5 8 Y 76 we necessarily have inf

u2V5 5 8Y6 13 7

J 5 5u7 3 06

1 2 since J 5 507 3 08 Then the element v in V4 8 Y6 13 satisfies I 5 5v7 3 0 and, by 22 1 recalling that D 6 5v7 3 det F3 5v7 3 0, we get V4 5 8 Y6 13 7 3

162 5 8 Y6 13 72 : det F 5v73 3 06 v 2 Wper

! 7 8 v dy 3 0 on 4 8 det F 527 5v7 0 and " Y

In a similar way, we solve recursively problems Pn 5 8 Y 7 for n 3 16 8 8 8 6 4. Thus we get D q 5v7 3 Cofiiq 5v7 3 Ciiq 5v7 3 0 for q 3 26 8 8 8 6 58 In particular, C2 ii 5v7 7 8 3 1 1 1 Fik 5v7 Fik 5v7 3 0 which implies that Fik 5v7 3 08 Then the condition det F 527 5v7 0 becomes 7 8 527 5v7 3 det F0 5v7 2 Fn 5u7 2n1 6 det F det n31

for 2 small enough. From now on, we require det F0 5v7 08

1 5.2. Solving Problem P0 5 8 Y 7

By using the above results, we see the following corollary. Corollary 8. For all v 2 V0 5 8 Y6 13 7, we have F1 5v7 3 C2 5v7 3 C1 5v7 3 0, Cofiip 5v7 3 0 for 4 3 p 3 1 and D q 5u7 3 0 for 6 3 q 3 18 The function 527 admits the asymptotic expansion L

TWO-DIMENSIONAL MODELS FOR HETEROGENEOUS PLATES 91

527 5v7 3 L

L n 5v7 2n 6

n30

where 2 1 L 0 5v7 3 ln D 0 5v7 6

p p 9 n ki 1 1 L n 5v7 3 D 5v7 p D 0 5v7 i31 p p31

n 2 18

(10)

ki 3n

i31

Proof. Formula 557 shows that F1 5v7 3 0 so then Lemmas 1â€“3 trivially imply that C2 3 C1 3 0, Cofiip 3 0 for 4 3 p 3 1 and D q 5u7 3 0 for 6 3 q 3 18 The result of the second part is deduced from Lemma 4. 1 The above corollary implies that J 0 5u7 3 I 0 5u7 l 0 5u7 =0 so then the problem P0 5 8 Y 7 becomes trivial and V0 5 8 Y6 13 7 3 V1 5 8 Y6 13 78

6. A NONLINEAR MEMBRANE PLATE THEORY In this section, we show that nonlinear membrane plate theory arises as problem P1 5 8 Y 78 2 1 Theorem 9. If v 3 v0 6 v1 6 8 8 8 is a solution of the problem P1 5 8 Y 7 then the first term v0 in the asymptotic expansion of the displacement field solves the global minimization problem 1 2 Jm v0 3

inf

u0 2U0 5 6 13 7

1 2 Jm u0 6

(11)

in which U0 5 6 13 7 is defined by 5 6 U0 5 6 13 7 3 u0 2 W162 5 6 13 7 u0 3 0 on 4 6 and the membrane energy is defined by 1 2 Jm u0 3

7 1 28 &0 u0 Wm F h i1 dy4 u i0 dx4 6 dx4 f i0 dy

Y

4Y

&0 is a 3 8 2 matrix of which entries are defined by F&0 3 9 i 4x u 0 and Wm is the in which F

i i

membrane elastic energy obtained by computing the following local minimization problem:

92 E. PRUCHNICKI 7 1 28 Wm F&0 u0 3

inf u1 2U1 5 8Y6 13 7

1 2 W 0 u0 6 u1 dy8

(12)

Y

1 2 The set U1 8 Y6 13 is ! 1 2 1 2 162 51 8 Y6 13 76 det F0 u0 6 u1 0, u1 dy 3 0 6 U1 8 Y6 13 3 u1 2 Wper " Y

and finally 1 2 1 2 1 2 W 0 u0 6 u1 3 a Cii0 u0 6 u1 b Cof0ii u0 6 u1 1 2 1 2 c D 0 u0 6 u1 d L 0 u0 6 u1 8

(13)

Proof. Let us consider problem P1 5 8 Y 7 J 1 5v7 3

inf

u2V1 5 8Y6 13 7

J 1 5u7 8

1 2 ' W 0 u0 6 u1 dx4 dy are well defined for all u0 2 U0 5 6 13 7 and The integrals I 1 5u7 3 1 2 8Y u1 2 U1 8 Y6 13 , we can apply Fubiniâ€™s theorem and the contribution of the stored energy is I 5u7 3 1

1 2 W 0 u0 6 u1 dy dx4 6

Y

while the contribution of the external forces to the energy is

1 2 l 1 u0 3

f i0 dy h i1 dy4 u i0 dx4 8 4Y

Y

Then problem P1 5 8 Y 7 becomes J 1 5v7 3

inf

u0 2U0 5 6 13 7

inf

u1 2U1 5 8Y6 13 7

1 2 1 2 W 0 u0 6 u1 dy dx4 l 1 u0 8

Y

As a consequence, we get the result 1 2 Jm v0 3 J 1 5v7 3

inf

u0 2U0 5 6

13 7

1 2 Jm u0 8

1

TWO-DIMENSIONAL MODELS FOR HETEROGENEOUS PLATES 93 ' In the proof of the previous theorem, we suppose implicitly that the infinimum of W 0 Y 1 0 2 u 6 5 dy is attained. For the computation of this infinimum, the global scale x4 can be considered as fixed and it plays the role of a parameter so it is not necessary to consider the regularity of the displacement field of first order with respect to the global scale. To apply the results of Ball [7], more regularity on u1 is necessary. Thus, in the' next 1theorem, 2 we define a new space Us1 5 8 Y6 13 7 and we show that the minimum of W 0 u0 6 8 dy is Y

attained on this space. Theorem 10. We define the set 1 2 Us1 8 Y6 13 3

1 2 3 For all x4 2 6 u1 x4 6 5 2 W162 per 5Y6 1 76

1 1 1 2 1 222 Cof F0 u0 x4 6 u1 x4 6 5 2 L2 5Y6 13 76 1 1 1 2 1 222 2 L2 5Y6 13 76 det F0 u0 x4 6 u1 x4 6 5 1 222 1 1 1 2 06 det F0 u0 x4 6 u1 x4 6 5

! u1 556 y7 dy 3 0 8 "

Y

' 01 0 2 W u 557 6 u1 556 y7 dy and the set Moreover, we assume that inf u1 2Us1 5 8Y6 13 7 Y 1 2 s 3 U1 1 8 Y6 1 2 is non-empty8 Then there exists at least one function v1 such that v1 2 Us1 8 Y6 13 and

1 2 W u0 6 v1 dy 3

0

Y

u1 2Us1

inf 5 8Y6 13 7

1 2 W 0 u0 6 u1 dy8

Y

0 which is defined by Proof. By the coerciveness of the functional W 1 1 22 1 2 0 F0 u0 6 u1 3 W 0 u0 6 u1 6 W we get

1 2 0 F0 dy inf 5A 7 W

316 2

Y

7 ( ( 0 (2 ( 28 1 (F ( (CofF0 (2 det F0 2 dy inf 5B 7 8

316 2

Y

By taking into account the PoincarĂŠâ€“Wirtinguer inequality and the condition

' Y

we see that there exists real constants f 0 and g, independent of u1 , such that

u1 dy 3 0,

94 E. PRUCHNICKI

1 2 0 F0 dy W

7 ( 1 (2 ( 1 2( 8 (u ( (grad y u1 (2 dy f

Y

Y

7 8 ( ( ( ( 2 2 (det F0 ( (CofF0 ( dy g8

(14)

Y

Let u1k be an infimizing sequence for the functional 1 2 satisfies u1k 2 Us1 8 Y6 13 for all k, and

1 2 W 0 u0 6 u1k dy 3

lim

k Y

'

1 2 W 0 u0 6 5 dy, i.e. a sequence that

Y

inf u1 2Us1 5 8Y6 13 7

1 2 W 0 u0 6 u1 dy.

Y

' 0 1 0 12 inf W u 6 u dy is bounded and by the coerciveness inequal5 8Y6 13 7 Y ity (14), the sequence

By assumption,

u1 2Us1

1 1k 1 4 2 1 1 2 1 22 1 1 2 1 222 u x 6 5 6 Cof F0 u0 x4 6 u1k x4 6 5 6 det F0 u0 x4 6 u1k x4 6 5 1 4 32 1 2 is bounded for all x4 2 in the reflexive Banach space W12 8 L2 Y 4 6 13 8 per Y 6 1 1 2 L2 Y 4 6 13 . Hence there exists a subsequence 1 1 2 1 22 1 1 2 1 222 1 1l 1 4 2 u x 6 5 6 Cof F0 u0 x4 6 u1l x4 6 5 6 det F0 u0 x4 6 u1l x4 6 5 1 1 4 32 2 that converges weakly to an element v1 5x4 6 57 6 H 5x4 6 57 6 3 5x4 6 57 in the space W12 Y 61 per 1 4 32 1 4 32 2 2 8 L Y 6 1 8 L Y 6 1 and we get 1 1 2 1 22 1 2 1 1 2 1 22 1 2 H x4 6 5 3 CofF0 u0 x4 6 v1 x4 6 5 , 3 x4 6 5 3 det F0 u0 x4 6 v1 x4 6 5 (Ciarlet [8, Theorem 7.6.1]). Now, we must check that

'

v1 556 y7 dy 3 0, but this follows

Y

from the implication that 1 2 1 2 1 4 32 u1l x4 6 5 converges weakly to v1 x4 6 5 in W12 per Y 6 1 3

u 556 y7 dy

v1 556 y7 dy8

1l

Y

Y

As in 1the proof of Theorem 7.7.1 of Ciarlet [8], it can be shown that for all x 4 2 6 2 det F0 u0 5x 4 7 '6 v1 5x14 6 57 0,2 so v1 5x 4 6 57 is a solution of the minimization problem of the functional W 0 u0 5x 4 7 6 5 dy. 1 Y

TWO-DIMENSIONAL MODELS FOR HETEROGENEOUS PLATES 95

By assuming the existence of solutions of the local problem of minimization (12), the solutions v of the problem P1 5 8 Y 7 are the elements of V2 5 8 Y6 13 7 defined by $ 1 2 1 2 V2 5 8 Y6 1 7 3 v 3 v0 6 v1 6 8 8 8 6 v0 solves Jm v0 3 3

inf

u0 2U0 5 6

13 7

1 2 Jm u0 6

v1 6 & v1 2 U0 5 6 13 7 and v1 solves v1 3 v1 & 1 2 1 2 inf W 0 v0 6 v1 dy 3 W 0 v0 6 u1 dy u1 2U1 5 8Y6 13 7 Y

Y

& v 3 0 and

v p dy 3 06 p 2 on 4

1

! "

8

Y

7. THE VARIATIONAL FORMULATIONS AND THE EULERâ€“LAGRANGE BOUNDARY-VALUE PROBLEMS In this section, we give both the variational and the Eulerâ€“Lagrange equations satisfied by the nonlinear membrane plate described by the total energy given in the previous section. We assume that the displacements fields are smooth enough to carry out all necessary computations. For a similar development in homogenization theory, we can refer to Bakhvalov and Panasenko [29, Chapter 6]. 1 2 Theorem 11. For all v 3 v0 6 v1 6 8 8 8 2 V2 5 8 Y6 13 7, the leading term v0 of the asymptotic expansion of the displacement field satisfies the two-dimensional nonlinear variational membrane problem 1 M 5 7: find 1 2 v0 2 U0 6 13 , such that

0 i

1 2 h i1 dy4 u i0 dx4 3 06 gradx4 v0 4x u i0 f i0 dy Y

4Y

1 2 for all u0 2 U0 6 13 . The Eulerâ€“Lagrange boundary value problem associated to the previous global variational problem is: find 1 2 v0 2 U0 6 13 , such that

96 E. PRUCHNICKI 1 2 0 4x i

gradx4 v0

f i0 dy

Y

h i1 dy4 3 08

4Y

1 2 ' The macroscopic tensor 0 ( 0 3 Y 0 v1 dy) is an implicit function of the covariant derivatives of the displacement field of order zero and the geometrical characteristics of the mid-surface of the shell, through the solution v1 of the three-dimensional nonlinear local variational problem 1l 5Y 7: find 1 2 v1 2 U1 8 Y6 13 6 such that

1 2 0i j gradx4 v0 6 grady v1 4 y j u i1 dy 3 06

1 2 u1 2 U1 8 Y6 13 6

Y

where the first Piolaâ€“Kirchhoff tensor of zero order is defined by i0j

1 1 22 0 F0 v0 6 v1 1 2 4W 0 1 gradx4 v 6 grady v 3 8 4 Fi0j

The local Eulerâ€“Lagrange boundary value problem associated to the variational problem 1l 5Y 7 consists of finding 1 2 v1 2 U1 8 Y6 13 , such that 1 2 4 y j i0j gradx4 v0 6 grady v1 3 08 Proof. Let us consider the local minimization problem 51271 obviously, we see that a solution of the displacement field v1 verifies 1 2 IY1 v0 6 v1 t u1 3

1 2 W 0 v0 6 v1 t u1 dy 2

Y

1 2 W 0 v0 6 v1 dy6

Y

1 2 for all t 2 1 and u1 2 U1 8 Y6 13 8 Moreover, assume that IY1 is differentiable at v1 , so the previous inequality implies the nullity of the directional derivative of IY1 at v1 in the direction u1 , D

IY1

1 0 12 1 v6 v u 3

Y

1 1 22 0 F0 v0 6 v1 4W 4 y j u 1i dy 3 06 4 Fi0j

TWO-DIMENSIONAL MODELS FOR HETEROGENEOUS PLATES 97 where D denotes the differential with respect to v1 . Therefore, the displacement field v1 is a solution of the three-dimensional nonlinear local variational problem Pl 5Y 7 : find v1 2 U1 5 8 Y6 13 7 such that 1 2 1 2 i0j gradx4 v0 6 grady v1 4 y j u i1 dy 3 0 u1 2 U1 8 Y6 13 6 Y

in which i0j denotes the first Piolaâ€“Kirchoff tensor of order zero, i0j

1 1 22 0 F0 v0 6 v1 1 2 4W 0 1 gradx4 v 6 grady v 3 8 4 Fi0j

We observe that the microscopic displacement field of first order u1 depends on the gradient of the global displacement field of order zero gradx4 u0 8 Now integrating by parts the local variational problem Pl 5Y 7 gives the following equality: 4 y j i0j u 1i dy 3 08 Y 4 Since u i1 5x 4 6 57 belongs to W162 per 5Y 7 for almost all x in 6 the above integration by parts is explicit when, for instance, the displacement field u i1 5x 4 6 57 is such that i0j 5x 4 6 57 belongs to W162 per 5Y 78 Then the boundary-value local problem follows. From the global minimization problem 5117 and by noting that a variation of v0 implies one of v1 6 we necessarily have

1 2 1 2 1 2 Jm v0 tu0 2 Jm v0 for all t 2 16 and u0 2 U0 6 13 8 Next we assume the differentiability of the functional Jm with respect to v0 1 therefore, by considering the previous inequality, we cancel the directional derivative of Jm at v0 in the direction u0 , 1 02 0 1 2 i0j gradx4 v0 6 grady v1 dy 4x j u i0 dx4 l 1 5u7 D Jm v u 3

Y

3 08 the2 Then the leading term v0 of the asymptotic expansion of the displacement field satisfies 1 0 0 two-dimensional nonlinear variational membrane problem PM 5 7: find v 2 U 6 13 such that 1 2 1 2 1 2 0i gradx4 v0 4x u i0 dx4 3 l 1 u0 for all v0 2 U0 6 13 6

98 E. PRUCHNICKI

in which the macroscopic tensor i0j

1 2 gradx4 v0 3

1 1 22 i0j gradx4 v0 6 grady v1 gradx4 v0 dy6

Y

since v1 depends on gradx4 v0 through the three-dimensional nonlinear local variational problem Pl 5Y 7. When u0 and 0 are sufficiently regular, an integration by parts of the problem PM 5 7 gives its Eulerâ€“Lagrange associated boundary value problem. 1

8. THE NONLINEAR MEMBRANE PLATE MODEL 8.1. Constitutive Properties of Wm

The constitutive law of each elastic material is physically realistic as it satisfies the additional property W 5F7

as det 5grad 5F77 0 8

As a consequence, the orientation-preserving condition is naturally imposed and the twodimensional model precludes singular folds of the mid-surface. We claim the following. Theorem 12. The membrane elastic energy satisfies the following condition: 7 8 &0 Wm F

(2 7 8 ( &0 3 ((F &0 (( 0 8 &0 T F &0 6 F as det F 1 2

7 8 1 2 &0k F &0k 0 &0 u0k be such that lim F &0 and det F &0k T F &0k 3 F Proof. Let the sequence F ( (k 7 8 &0 T F &0 3 0. As a consequence, ((F &0k (( is bounded. We argue by contradicwith det F 7 8 &0k 3 c 8 c is finite since the memtion. Suppose to the contrary that lim inf Wm F k brane 7energy 88 is bounded below8 Without loss 7 of8generality, we suppose that the sequence 7 0k & &0k is bounded. Theorem 10 yields the exisWm F is convergent and then Wm F k22 1 2 tence of a displacement field u1k 2 Us1 8 Y6 13 such that

7 8 1 2 &0k . W 0 u0k 6 u1k dy 3 Wm F

Y

1 2 From Equation 5137 which defines W 0 u0k 6 u1k , it is easy to establish that there exists K 1 2 1 such that

TWO-DIMENSIONAL MODELS FOR HETEROGENEOUS PLATES 99 7 8 ( ( &0k inf 5a 7 (F0k (2 dy K 1 6 Wm F

316 2

Y

( ( 7 8 &0k grad u1k 8 Since ((F &0k are bounded, we deduce that &0k (( and Wm F in which F0k 3 F y ( '( (grady u1k ( dy is bounded. We set 9 k 3 det F0k and again by using formula 5137, we can Y

write 7 8 1 2 &0k c 9 k 2 d ln59 k 2 7 dy8 Wm F Y

Let us suppose that 0 d1 d2 , so the previous inequality can be rewritten as

1 2 W 0 u0k 6 u1k dy d1

Y

ln59 k 2 7 dy c1

Y

9 k 2 dy Y1

2 1 c2 9 k 2 5d2 d1 7 ln59 k 2 7 dy8

Y2

It is easy to see that c1

'

9 k 2 dy

Y1

'1 c2 9 k

2

2 5d2 d1 7 ln59 k 2 7 dy is bounded below and

Y2

we deduce that there exists K 2 2 1 such that 1 0k 1k 2 0 dy 2 d1 ln59 k 7 dy K 2 8 W u 6u Y

Y

By using the Jensen inequality, we obtain

1 2 W 0 u0k 6 u1k dy 2 d1 ln 9 k dy K 2 8

Y

(15)

Y

Now 9 k can be expressed as ( ( 7 8 &0k 4 y u1k (( I3 4 y u1k 6 &0k 4 y u1k ((F 9 k 3 det F 1 2 3 1 2 where I3 denotes the third column of the 3 8 3 unit matrix. We want to show that 8 7 ( ( 1 2 k &0k (( I3 4 y u1k dy for all u1k x 4 6 5 2 W162 5Y6 13 78 &0k ((F 9 dy 3 det F 3 1 2 per Y

Y

(16)

100 E. PRUCHNICKI

We can obviously split

9 dy 3 k

Y

7 ( ( 8 &0k (( I3 4 y u1k dy &0k ((F det F 3 1 2

Y

( ( 8 7 (&0k 1k ( 1k dy det 4 y1 u1k (F 4 u 4 u I ( y 3 y 2 3 2

Y

7 8 ( ( &0k 4 y u1k (4 y u1k ( I3 4 y u1k dy8 det F 1 2 3 1

Y 1k 1k 3 Firstly, we assume that u1k 2 C 5i 3 16 37 per 5Y6 1 7, so the Fourier series of u and 4 yi u are

1 2 S p6q u1k 5y7 3

p q

Cn 1 n2 5y3 7 e2i5n1 y1 n 2 y2 7 6

n 1 3 p n 2 3q p q 2 1 S p6q 4 y u1k 5y7 3 2i n Cn1 n 2 5y3 7 e2i5n1 y1 n2 y2 7 6 n 1 3 p n 2 3q

1 2 S p6q 4 y3 u1k 5y7 3

p q

1 2 4 y3 Cn1 n 2 5y3 7 e2i5n 1 y1 n 2 y2 7 8

n 1 3 p n 2 3q

The result follows by observing that these Fourier series converge uniformly and thus their product converges uniformly too and can be integrated term by term. Due to the Y 4 -periodicity, the integrals of all non-constant harmonics are zero and formula 5167 is true for all u1k 3 3 162 3 2 C per 5Y6 1 7. By using the density of the space Cper 5Y6 1 7 in the space Wper 5Y6 1 76 formula 5167 holds.' ( ( Recalling that (grady u1k ( dy is bounded, we infer that there exists K 3 0 such that Y

( ( ( ( ( (&0k &0k ( (&0k &0k ( (( 1k ( F 9 k dy (F 6 F 4 u 6 F I dy K ( ( 3 y3 3 1 2 1 2 (8

Y

Y

From inequality 5157, we deduce that ( ( 7 8 1 2 &0k ((7 K 2 8 &0k 6 F &0k 3 W 0 u0k 6 u1k dy 2 d1 ln 5K 3 ((F Wm F 1 2 Y

7 8 &0k cannot be Consequently, we have shown the following contradiction: lim inf Wm F k

finite.

1

TWO-DIMENSIONAL MODELS FOR HETEROGENEOUS PLATES 101

Moreover, this membrane energy inherits the fact that the material frame indifference property is satisfied, which means that the energy is independent of the Cartesian frame in &0 6 F &0 3 06 we have &0 such that F which it is computed, namely for all 3 8 2 matrices F 1 2 7 8 7 8 &0 3 Wm F &0 for all R2 SO5378 Wm R F Similarly, we see that Wm is isotropic meaning that 7 Wm

8 7 8 &0 R 3 Wm F &0 for all R2 SO5278 F

Remark 13. By considering the rigorous 1-convergence argument of Muller [30], it can be shown that membrane energy is obtained by minimizing the local energy on k basic cells (Y becomes kY in formula (12)) and then on k. Nevertheless, as shown by Geymonat et al. [31] and Michel et al. [32], for a specific deformation gradient F0 , the one cell homogenized energy is the correct one. For homogeneous material the analytical expression of function Wm is continuous (Trabelsi [33]). However, for heterogeneous material, no mathematical results would allow us to show the continuity of Wm (Geymonat et al. [31, Section 5.2]).

9. CONTINUATION OF THE ASYMPTOTIC EXPANSION 9.1. The nonlinear membrane model revisited

We expect to find bending in the higher-order expansion. We want to exclude membrane ' ' effects and therefore we assume that g0 3 f0 dy h1 dy4 3 0. Then problem P1 5 8 Y 7 Y 4Y 1 1 22 1 2 0 F0 v0 6 v1 3 0 so we deduce that C0 v0 6 v1 3 I8 Now let us consider implies that W the components of the displacement field U, 1 2 Ui 3 gradx4 v0 i y v1i for i 3 16 38

(17)

Then we can write that 5I grady U7T 5I grady U7 3 C0 3 I8 If the displacement field v1 5x4 6 57 is regular enough, for example v1 5x4 6 57 2 C1 5Y1 6 13 7 C1 5Y2 6 13 7, it is easy to see that y U is a rigid deformation and then we have necessarily that 1 2 1 2 yi Ui 5x4 6 y7 3 i x4 Q ik x4 yk 6

(18)

where w is a vector field and Q is an orthogonal matrix. It follows from the expressions (17 ) and (18) that 1 1 1 2 1 1 2 1 22 2 1 2 2 i1 5x4 6 y7 3 i x4 Q i x4 9 i gradx4 v0 x4 i y Q i3 x4 9 i3 y3 8 (19)

102 E. PRUCHNICKI ' Since v1 is Y 4 periodic with respect to the variable y4 and v1 dy 3 0, formula (19) shows Y 1 2 that Q i 3 9 i gradx4 v0 i and w 3 08 Since Q is an orthogonal matrix, we immediately 1 2 1 2 &0 v0 6 F &0 v0 is the get that i1 5x4 6 y7 3 &i1 5x4 7 n i 5x4 7 y3 y3 9 i3 in which n 3 F 1 2 unit normal to the deformed mid-surface of the plate. Thus the leading-order term of the displacement field u0 inevitably belongs to the space of the inextensional displacement field 3 Uiso 0 5 6 1 7 3

5 v 2 W262 516 13 76 v 3 0 on 4 , E 5v7 3

% 8 2 1 2 1 71 I gradx4 5v7 k I gradx4 5v7 k 9 3 0 6 2

which is assumed to be different from 08 More precisely, the solutions v of the problem 3 P1 5 8 Y 7 are the elements of Viso 2 5 8 Y6 1 7, where 3 Viso 2 5 8 Y6 1 7 3

5 1 2 3 v 3 v0 6 v1 6 8 8 8 6 such that v0 2 Uiso 0 5 6 1 76 2 1 i1 x4 6 y 3 &i1 5x4 7 n i 5x4 7 y3 y3 9 i3 with & v1 2 U0 5 6 13 7 1 2 1 2 1 &0 v0 6 F &0 v0 6 & n3F v 3 0, 1 2

v p dy 3 06 p 2 on 4

! "

8

Y

9.2. Model Without Internal Energy

In this section we solve problem P2 5 8 Y 78 Theorem 14. For external loading satisfying g0 3

'

f0 dy

'

h1 dy4 3 06 if v 3 5v0 6 v1 6 8 8 87

4Y

Y

is a solution of problem P2 5 8 Y 7, then the first term v0 in the asymptotic expansion of the displacement field solves the global minimization problem 1 2 J 2 v0 3

inf

3 u0 2Uiso 0 5 6 1 7

1 2 J 2 u0 6

where 1 2 2 ' 1 0 1 02 J 2 u0 3 pi n i u gi1 u i0 dx4 with

p0 3

1 f0 y3 dy h1 dy4 h1 dy4 6 2 4Y

Y

g1 3

f1 dy

Y

4Y

h2 dy4 8

4Y

TWO-DIMENSIONAL MODELS FOR HETEROGENEOUS PLATES 103 Proof. We consider problem P2 5 8 Y 76 finding a displacement field v such that J 2 5v7 3

inf

3 u2Viso 2 5 8Y661 7

J 2 5u7 8

3 Let u 2 Viso 2 5 8 Y6 1 78 The contribution of the internal energy is given by

6 5 a C1ii 5u7 b Cofii1 5u7 c D 1 5u7 d L 1 5u7 dx4 dy8

I 5u7 3 2

8Y

By using formulae 577â€“5107, we get C1ii 5u7 3 2 F&i 0 Fi 1 2n i Fi31 6 Cof1ii 5u7 3 4 F&i 0 Fi 1 4n i Fi31 6 L 1 5u7 3

D 1 5u7 3 2 F&i 0 Fi 1 2n i Fi31 8

Recalling that a 2b 3 d c, we see that the internal energy vanishes. By computing the contribution of the external energy, we complete the proof. 1 We observe that problem P2 5 8 Y 7 is without internal energy and becomes trivial if we choose 1 h1 dy4 3 0 and g1 3 f1 dy h2 dy4 3 08 p0 3 f0 y3 dy h1 dy4 2 Y

4Y

4Y

Y

4Y

This leads us to choose p0 3 g1 3 0. Then problem P2 5 8 Y 7 reduces to a trivial one 3 iso 3 and therefore Viso 3 5 8 Y6 1 7 3 V2 5 8 Y6 1 78

10. A NONLINEAR MEMBRANE INEXTENSIONAL BENDING PLATE THEORY In this section, we assume that external loading satisfies f0 3 0, h1 3 0 and g1 3 0 then we justify a membrane inextensional bending plate model by solving P3 5 8 Y 78 1 2 Theorem 15. If v 3 v0 6 v1 6 8 8 8 is a solution of the problem P3 5 8 Y 7 then both the v1 in the asymptotic expansion of the leading term v0 and the macroscopic first-order term & displacement field solve the global minimization problem 1 2 Jbm v0 , & v1 3

u0 2Uiso 0 5 6

inf

13 76

& u1 2U0 5 6613 7

1 2 &1 6 Jbm u0 6 u

104 E. PRUCHNICKI

where the membrane non-extensional bending energy is defined by 1 2 &1 3 Jbm u0 6 u

1 1 2 1 1 22 4 Wbm gradx4 u0 6 gradx4 & u dx

f i 2 dy h i3 dy4 u i0 dx4 Y

4Y

f i1 y3 dy 1 h 2i dy4 h i2 dy4 n i dx4 8 2

4Y

Y

4Y

Wbm is the membrane inextensional bending elastic energy obtained by computing the local minimization problem 1 1 2 1 1 22 u 3 Wbm gradx4 u0 6 gradx4 &

inf u2 2U2 5 8Y6 13 7

1 2 u1 6 u2 dy6 W 2 u0 6 &

Y

in which we set 7 82 1 2 W 2 u0 6 & u1 6 u2 3 5a b7 F&i10 Fi21 F&i20 Fi11 7 87 8 0 1 1 5a b7 F&i 0 Fi31 n i Fi 1 F j3 n j F j

F&j

7 8 0 1 1 4 5a b7 F&i10 Fi11 F&j2 F j2 F&i 0 Fi 1 n j F j3 82 7 1 2 d F&i 0 Fi 1 n j F j3 6 1 2 where Fi 1 3 4x u&i1 y3 4x n i 4 y u 2i 6 Fi31 3 4 y3 u i2 and the set U2 8 Y6 13 is defined by ! 1 2 3 2 162 3 u2 dy 3 0 8 U2 8 Y6 1 3 u 2 Wper 51 8 Y6 1 76 " Y

Proof. We consider problem P3 5 8 Y 76 finding a displacement field v such that J 3 5v7 3

inf

u2Viso 3 5 8Y 7

J 3 5u7 8

(20)

1 2 2 1 2 3 3 Let u 3 u0 6 u1 6 8 8 8 2 Viso 3 5 8 Y6 1 7 and u 2 U2 8 Y6 1 . The contribution of the internal energy is given by

TWO-DIMENSIONAL MODELS FOR HETEROGENEOUS PLATES 105

6 5 a C2ii 5u7 b Cofii2 5u7 c D 2 5u7 d L 2 5u7 dx4 dy6

I 3 5u7 3

8Y

By using formulae 577â€“5107, we get Cii2 5u7 3 2 F&i 0 Fi 2 Fi 1 Fi 1 Fi31 Fi31 2 n i Fi32 6 Cofii2 5u7 3 4 F&i 0 Fi 2 2 Fi 1 Fi 1 2 Fi31 Fi31 4 n i Fi32 82 7 0 1 1 4 F&i10 Fi11 F&j2 F j2 4 F&i 0 Fi 1 n j F j3 F&i10 Fi21 F&i20 Fi11

7

F&i 0 4 y3 u 2i n i Fi 1

8 7

8 0 1 6 4 y3 u 2j n j F j

F&j

0 1 D 2 5u7 3 2 F&i 0 Fi 2 Fi 1 Fi 1 Fi31 Fi31 2 n i Fi32 4 F&i10 Fi11 F&j2 F j2 7 87 8 4 F&0 F 1 n j F 1 F&0 F 1 F 1 n i F&0 F 1 F 1 n j i

7

i

j3

F&i10 Fi21 F&i20 Fi11

i

82

i3

i

j

j3

j

6

22 D 1 5u7 8 D 5u7 2 1

L 5u7 3 2

2

in which Fi 1 3 4x u&i1 y3 4x n i 4 y u i2 6 Fi31 3 4 y3 u i2 and Fi32 3 4 y3 u i3 . Since a 2b 3 d c6 the internal energy can be written as I 3 5u7 3

1 2 W 2 u0 6 & u1 6 u2 dy dx4 , Y

with 82 7 1 2 u1 6 u2 3 5a b7 F&i10 Fi21 F&i20 Fi11 W 2 u0 6 & 7 87 8 0 1 1 5a b7 F&i 0 Fi31 n i Fi 1 F j3 n j F j

F&j

8 7 0 1 1 F j2 F&i 0 Fi 1 n j F j3 4 5a b7 F&i10 Fi11 F&j2 82 7 1 2 d F&i 0 Fi 1 n j F j3 8 The contribution of external forces to the total energy J 3 5u7 is represented by

(21)

106 E. PRUCHNICKI l 3 5u7 3

f i 2 dy h i3 dy4 u i0 dx4 Y

4Y

1 f i1 y3 dy h 2i dy4 h i2 dy4 n i dx4 8 2 Y

4Y

(22)

4Y

The proof of the theorem arises by combining formulae 5207 6 5217 and (2278

1

11. CONCLUSION In this work, we applied to inhomogeneous nonlinear plates a general systematic and objective method based on the asymptotic expansion analysis in order to determine a hierarchy of two-dimensional models. These results are in agreement with the models obtained by Gamma convergence for homogeneous plates by Friesecke et al. [34]. These methods present the drawback of separating the scales associated with membrane and bending behavior to a greater extent than desired. To overcome this difficulty, Ciarlet and Lods [35] and Ciarlet and Roquefort [36] proposed, in the spirit of the works of Koiter [37], a twodimensional model independent of the order of magnitude of the loads which generalize a hierarchy of two-dimensional models obtained by Ciarlet and Lods [20], Miara [21], Ciarlet et al. [38] and Lods and Miara [39]. By using a thickness-wise expansion of the displacement field, Steigmann [40] independently obtained a two-dimensional modeling of thin domains depending on the thickness and combining the effects of various levels of scales. This approach can be used to obtain a two-dimensional modeling of thin heterogeneous domains to which the method of homogenization will be applied. REFERENCES [1] [2] [3] [4] [5]

[6] [7] [8]

Ciarlet, P. G. Mathematical Elasticity, Vol. II. Theory of Plates, North-Holland, Amsterdam, 1997. LewiÂ´nski, T. and Telega, J. J. Plates, Laminates and Shells: Asymptotic Analysis and Homogenization, World Scientific, Singapore, 2000. Pruchnicki, E. Non linearly elastic membrane model for heterogeneous shells by using a new double scale variational formulation: A formal asymptotic approach. J. Elasticity, 84, 245â€“280 (2006). Pruchnicki, E. Two-dimensional nonlinear models for heterogeneous plates. Comptes Rendus de lâ€™AcadĂŠmie des sciences, Paris, 337(5), 297â€“302 (2009). Pruchnicki, E. Nonlinearly elastic membrane model for heterogeneous plates : a formal asymptotic approach by using a new double scale variationnal formulation. International Journal of Engineering Science, 40, 2183â€“2202 (2002). Ciarlet, P. G. and Geymonat, G. Sur les lois de comportement en ĂŠlasticitĂŠ non linĂŠaire compressible. Comptes Rendus de lâ€™AcadĂŠmie des sciences, Paris, Series II, 295, 423â€“426 (1982). Ball, J. M. Convexity condition and existence theorems in nonlinear elasticity. Archive for Rational Mechanics and Analysis, 63(4), 337â€“403 (1976). Ciarlet, P. G. Mathematical Elasticity, Vol. I: Three-Dimensional Elasticity. North-Holland, Amsterdam, 1988.

TWO-DIMENSIONAL MODELS FOR HETEROGENEOUS PLATES 107

[9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]

[32] [33] [34] [35] [36]

Panasenko, G. Muticomponent homogenization of the vibration problem for incompressible media with heavy and rigid inclusions. Comptes Rendus de l’Académie des sciences, Paris, Series I, 321, 1109–1114 (1995). Pantz, O. Quelques problèmes de modélisation en élasticité non linéaire, PhD Thesis, University of Paris VI, 2001. Pantz, O. Dérivation des modèles de plaques membranaires non linéaires à partir de l’élasticité tri-dimensionnelle. Comptes Rendus de l’Académie des sciences, Paris, Series I, 331, 171–174 (2000). Meunier, N. Recursive derivation of one-dimensional models from three-dimensional nonlinear elasticity. Mathematics and Mechanics of Solids, 13(2), 172–194 (2008). Trabelsi, K. Nonlinear thin plate models for a family of Ogden materials. Comptes Rendus de l’Académie des sciences, Paris, Series I, 337, 819–824 (2003). Giroud, P. Analyse asymptotique de coques inhomogènes en élasticité linéarisé anisotrope. Comptes Rendus de l’Académie des sciences, Paris, Series I, 327, 1011–1014 (1998). Caillerie, D. and Sanchez-Palencia, E. Elastic thin shells: Asymptotic theory in the anisotropic and heterogeneous cases. Mathematical Models and Methods in Applied Science, 8(4), 473–496 (1995). Trabelsi, K. Nonlinearly elastic thin plate models for a class of Ogden materials: II. The bending model. Analysis and Applications (Singapore), 3, 271–283 (2005). Ne2cas, J. Les méthodes directes en théorie des équations elliptiques, Masson. Paris, 1967. Caillerie, D. Thin elastic and periodic plates. Mathematical Models and Methods in Applied Science, 6, 159–191 (1984). Lewi´nski, T. Effective models of composite periodic plates I. Asymptotic solution. International Journal of Solids and Structures, 27, 1173–1174 (1991). Ciarlet, P. G. and Lods, V. Asymptotic analysis of linearly elastic shells I. Justification of membrane shell equations. Archive for Rational Mechanics and Analysis, 136, 119–161 (1996). Miara, B. Nonlinearly elastic shell models: a formal asymptotic approach I. The membrane model. Archive for Rational Mechanics and Analysis, 142, 331–353 (1998). Collard, C. and Miara, B. Asymptotic analysis of the stresses in thin elastic shells. Archive for Rational Mechanics and Analysis, 148, 233–264 (1999). Mardare, C. Asymptotic analysis of linearly elastic shells: error estimates in the membrane case. Asymptotic Analysis, 17, 31–51 (1998). Lions, J. L. Some Methods in the Mathematical Analysis of Systems and their Control, Gordon and Breach Science Publishers, New York, 1981. Dauge, M. and Gruais, I. Développement asymptotique d’ordre arbitraire pour une élastique mince encastrée. Comptes Rendus de l’Académie des sciences, Paris, Series I, 321, 375–380 (1995). Dauge, M. and Gruais, I. Asymptotics of arbitrary order for a thin elastic clamped plate. I: Optimal error estimates. Asymptotic Analysis, 13, 167–197 (1996). Pruchnicki, E. Hyperelastic homogenized law for reinforced elastomer at finite strain with edge effects. Acta Mechanica, 129, 139–162 (1998). Pruchnicki, E. Overall properties of thin hyperelastic plate at finite strain with edge effects using asymptotic method. International Journal of Engineering Science, 36, 973–1000 (1998). Bakhvalov, N. S. and Panasenko G. Homogenisation: Averaging Processes in Periodic Media (Mathematics and its Applications Soviet Series), Kluwer Academic, Dordrecht (1989). Muller, S. Homogenization of nonconvex cellular elastic materials. Archive for Rational Mechanics and Analysis, 99, 189–212 (1987). Geymonat, G., Muller, S. and Triantafyllidis, N. Homogenization of non linearly elastic materials, microscopic bifurcation and macroscopic loss of rank-one convexity. Archive for Rational Mechanics and Analysis, 122, 231– 290 (1993). Michel, J. C., Lopez-Pamies, O., Ponte-Castañeda, P. and Triantafyllidis, N. Microscopic and macroscopic instabilities in finitely strained porous elastomers. Journal of the Mechanics and Physics of Solids, 55, 900–938 (2007). Trabelsi, K. Nonlinearly elastic thin plate models for a class of Ogden materials: I The membrane model. Analysis and Applications (Singapore), 3, 195–221 (2005). Friesecke, G., James, R. D. and Muller, S. A herarchy of plate models derived from nonlinear elasticity by gammaconvergence. Archive for Rational Mechanics and Analysis, 180, 183–236 (2006). Ciarlet, P. G. and Lods, V. Asymptotic analysis of linearly elastic shells. III. Justification of Koiter’s shell equations. Archive for Rational Mechanics and Analysis, 136, 191–200 (1996). Ciarlet P. G. and Roquefort, A. Justification of a two-dimensional nonlineat shell model of Koiter’s type. Chinese Annals of Mathematics, 22B, 129–144 (2001).

108 E. PRUCHNICKI

[37] Koiter, W. T. On the nonlinear theory of thin elastic shells. Proceedings of the Academic Congress, Netherlands, Wetensch, B69, 1–54 (1966). [38] Ciarlet, P. G., Lods, V. and Miara, B. Asymptotic analysis of linearly elastic shells. II. Justification of flexural shell equations. Archive for Rational Mechanics and Analysis, 136, 163–190 (1996). [39] Lods, V. and Miara, B. Nonlinearly elastic shell models: A formal asymptotic approach II. The flexural model. Archive for Rational Mechanics and Analysis, 142, 355–374 (1998). [40] Steigmann, D. J. Two-dimensional models for the combined stretching of plates and shells on three-dimensional linear elasticity. International Journal of Engineering Science, 46, 654–676 (2008).

The Local Conditions of Uniqueness and Plastic Strain Localization Part II: Comparison of the Local Uniqueness Condition with the Rice–Rudnicki Local Plastic Strain Localization

Z S´ LODERBACH J PAJAK 1

Department of Applications of Chemistry and Mechanics, Opole University of Technology, Opole, Poland (Received 4 August 20092 accepted 30 December 2009)

Abstract: Based on previous work of the first author, the local sufficient condition of uniqueness of the solution of the incremental boundary-value problem in generalized elastoplasticity which excludes the possibility of bifurcation state is compared with the local necessary condition for the localization of plastic deformations as a Rice–Rudnicki localization plane. The analytical results of limitations imposed on the isothermal hardening functions (moduli) by the Rice–Rudnicki condition of plastic strain localization are compared with those resulting from the local uniqueness condition. It is shown that the local sufficient condition of uniqueness of the solution of the incremental boundary-value problem with an equals sign becomes a local necessary condition of non-uniqueness or local necessary condition of the possible appearance of a bifurcation state.

Key Words: local uniqueness condition, localization of plastic deformation, plane of localization, potential and function of plasticity, constitutive equations

1. INTRODUCTION The uniqueness conditions and criteria (the conditions and criteria excluding the possibility of a bifurcation state) for the solution of the general incremental boundary-value problem, derived by the present authors [1–3], take into account the thermal–mechanical and elastic– plastic coupling effects and contain the cases of non-associated plastic flaw laws. These conditions can be applied to the description of metallic materials, less plastic (semi-brittle) metals and their alloys, exploited under thermo-mechanical loads, porous materials, sintered powders, rocks and soils. In this work we present the results of calculations based on a sufficient local uniqueness condition (more specifically, using a local necessary non-uniqueness condition) and compare these results with the results based on a necessary condition for R-R plastic state localization. In other words, we compare the results of limitations imposed on the isothermal hardening function by a local necessary non-uniqueness condition and by a necessary condition for R-R

Mathematics and Mechanics of Solids 16: 109–121, 2011 1 The Author(s), 2011. Reprints and permissions: 1 http://www.sagepub.co.uk/journalsPermissions.nav

DOI: 10.1177/1081286510362408

Â´ 110 Z SLODERBACH and J PAJAK 1

plastic strain localization. An analysis of the relations between these conditions is performed. The results of calculations are presented in the form of figures and diagrams. It should be pointed out that the local sufficient condition of uniqueness of the solution of the incremental boundary-value problem of the generalized coupled thermoplasticity (exactly thermo-elastoplasticity) with sign (2) on the right-hand side becomes a local necessary condition of non-uniqueness and of the possible appearance of a bifurcation state in the shape of a groove, neck, shear band or other states of deformation instability of elastoplastic materials, see e.g. [4â€“22]. The compared conditions, in addition to theoretical meaning, have important practical value. They can serve as a tool to estimate critical values of load, beyond which the appearance of bifurcation state or localization of the plastic deformation state are possible. It can be also be assumed that micro-concentrations of strains and stresses can occur in these points where conditions of uniqueness or bifurcation criteria were exceeded.

2. ESTIMATION OF THE STATE OF PLASTIC DEFORMATION LOCALIZATION 2.1. Short Description of Riceâ€“Rudnicki Localization Conditions

Rice and Rudnicki [20, 21] derived conditions for the possible appearance of a plastic deformation localization plane for compressed elastoplastic materials obeying non-associated flow laws. Constitutive equations for such materials and for a small deformation 13 4 15 (where 13 denotes Jaumannâ€™s objective derivative of Cauchyâ€™s stress tensor 1 ) have the following form (see, e.g., [20, 21]): 252i6 j

7

25 kk

7

2 15 kl 1 6kl

15 kl 8 3 249 3 1 2 15 kk 15 kl 6 1 6kl

8 15 kl 8 3 23 8 35 h 249 3 15 i j 1 1 i6 j 8 3 h 49

1

(2.1)

where 25 6 7 dev 25 7 1 6 7 dev 1 , h is the isothermal hardening function, 6 is the dilatancy factor, 6 0 086, 3

is the internal friction coefficient, 3

080 089, 3 and 5 are Lame elastic constants, and 3 G is the shear modulus or shape elastic strain modulus. The symbol 49 designates the second invariant of stress deviator 1 6 1 49 7

1 6 6 1 1 2 ij ij

2 12 (2.2)

In Figure 1(a), a portion of yield surface is presented showing the geometric interpretation of the coefficient of internal friction 3

and the dilatancy factor 6. In Figure 1(b), a curve of shear stress 4 versus shear strain 9 showing the geometric interpretation of the harden-

PART II: COMPARISON OF THE LOCAL UNIQUENESS CONDITION 111

Figure 1. Scheme of yield surface (adapted from Rudnicki, J. W. and Rice, J. R. Condition for the localisation of deformation in pressure-sensitive dilatant materials. Journal of the Mechanics and Physics of Solids, 23, 371â€“394 (1975). Reprinted with permission form Elsevier).

3ing modulus 4h, the tangent modulus h tan and the elastic shear modulus (G 3 , where 1 7 13 1 kk , are presented. In general, if the internal friction coefficient 3

is not equal to the dilatancy factor 6 (3

7 6) and if both coefficients are different from zero 3

7 6 7 0, then constitutive equations (2.1) obey non-associated plastic flow laws and they include compressible elasticâ€“plastic effects. Flow laws presented in the form of (2.1) do not take into account elasticâ€“plastic coupling effects. When 3

7 6 7 0, then the constitutive equations (2.1) are associated. When 3

7 6 7 0, Equations (2.1) became Prandtlâ€“Reuss constitutive equations, which do

Â´ 112 Z SLODERBACH and J PAJAK 1

Figure 2. Scheme of the plane of localization in the coordinate system 1 I 7 1 II 7 1 III , (adapted from Rudnicki, J. W. and Rice, J. R. Condition for the localisation of deformation in pressure-sensitive dilatant materials. Journal of the Mechanics and Physics of Solids, 23, 371â€“394 (1975). Reprinted with permission form Elsevier).

not take into consideration elasticâ€“plastic coupling effects and material compressibility, see e.g. [1, 20, 21]. The critical value of hardening function h cr , describing the possible appearance of a R-R plastic deformation localization plane [20, 21], obtained for the case when 13 4 15 (small deformations and co-rotational effects are neglected) and when expressions of order 0 (49 G) and higher are neglected, is as follows: h cr 18

18

2 7 6 3

3 9 1

2

1

683

N8 3

22 (2.3)

where is Poissonâ€™s ratio. The variable parameter N in Equation (2.3) depends on the stress state and is equal to 1 6II 7 N 49 , where 1 6II is the second main component of the stress deviator in a coordinate system as in Figure 2. It is also assumed after Rudnicki and Rice [20] that the normal to the plane of plastic localization n is perpendicular to the direction 1 II and 1 I 1 II 1 III , where 1 I 7 1 II 7 1 III denote main stresses. Rudnicki and Rice designated the angle as the angle-arranging plane of plastic localization in the main stress direction system ( is the angle between the normal n and the 1 III direction). The angle 0 in Figure 2 corresponds to the maximum value of the hardening modulus h cr max . The changes in critical modulus h cr with N for axially symmetric extension (1 I 1 II 7 1 III , then N 7 1 3), pure shear (1 I 7 1 III , then N 7 0), and axially symmetric compression (1 I 7 1 II 1 III , then N 7 1 3) have been presented by Rudnicki and Rice [20]. The maximum h cr

3. Then max derived from expression (2.3) is for N 7 6 8 3

18

h cr max 7

2 6 3

3 9 1

(2.4)

PART II: COMPARISON OF THE LOCAL UNIQUENESS CONDITION 113

It is visible that the maximum critical modulus h cr max is not dependent on the stress state but depends only on the material parameters (67 37 3

and ) [20, 21]. In order to determine Equations (2.3) and (2.4) in the variation range of 6 and 3

and in order that the normal n to the plane of localization is not perpendicular to the direction of 1 III , the following condition must be fulfilled (e.g. [20, 21]): 3

86 2

3

(2.5)

2.2. Sufficient Local Uniqueness Conditions for the Case of Riceâ€“Rudnicki Constitutive Equations

The R-R constitutive equations can be used to describe porous materials, sintered powders, rocks and soils, concrete or less plastic (semi-brittle) metals, and other similar materials. They are expressed as 5 3

1 6 6 7 7 3 I1 8 IIId 2 k 7 07 condition (function) of plasticity 6 6 8 7 6 I 8 I 12 7 plastic potential 1 IId 3

(2.6)

From (2.6) after differentiations we obtain 5 1 i6 j 3 9 6 6 6 f 7 7 8 i j

1 ij 6 7 1 i j 249 3

(2.7)

6 1 i6 j 6 6 6 6 7 8 i j 8 gi j 7 1 i j 249 3 For an elastic isotropic body (e.g. [1, 2, 18, 19]) we have Mi jkl 7 3 ik jl 8 jk il 8 5 i j kl 7

23 8 35 2 1 8

7 3 1 2

and 23 7

E 7 (2.8) 18

where Mi jkl is the tensor of isothermal moduli of elasticity, i j is Kroneckerâ€™s delta, and E is Youngâ€™s modulus. Substituting expression (2.7) into local sufficient condition of uniqueness of the solution of the incremental boundary-value problem (see, e.g., [2, 3]) and taking into expressions (2.8), we obtain after some transformations, that h 3

9

1 2

7

h 8 3

2 1 8 62 3

1

18

1 2

2

2 2 18 3

3

1

18

1 2

2

2 1 8 63

3

1

18

1 2

2

(2.9)

Â´ 114 Z SLODERBACH and J PAJAK 1

This is a requested form of the local sufficient condition of uniqueness solution. When right side of the expression (2.9) is equal to h3 , then it becomes formally a local necessary condition of non-uniqueness of the solution or a local necessary condition for the possible appearance of a bifurcation state. 2.3. Comparison of the Local Necessary Non-uniqueness Condition with the Maximum Condition of Riceâ€“Rudnicki Plastic Plane Localization

The curves presenting the analytical results obtained from expression (2.4) on maximal value of hardening function h cr max and from (2.9) expression on h , according to [2, 3], obtained from local necessary condition of non-uniqueness of solution, are shown in Figure 3. Curve 1 is obtained from condition (2.4) on the maximum value of the hardening function h cr

6. This introduced max , which is dependent on the introduced parameter z 7 3 parameter z does not have any physical meaning but is a useful parameter for making appropriate calculations. Curves 2, 3, 4, 5, and 6 are obtained from the local necessary condition of non-uniqueness of the solution h3 , i.e. (2.9), which is dependent on the introduced parameter z. Non uniqueness of condition h3 is dependent on z and 6 and represents a two-parameter family of local necessary conditions, because of the parameters z and 6. When z is treated as an independent variable, then expression h3 becomes a one-parameter family of local necessary non-uniqueness conditions dependent on parameter 6. The results of analytical calculations, presented in Figure 3, were obtained with assumption (after [20]) that Poissonâ€™s ratio 7 083, the internal friction coefficient 3

0.0â€“0.9 and the dilatancy factor 6 0.0â€“0.6. From the curves presented in Figure 3 it follows that the expression on h3 , i.e. (2.9), which describes the local necessary condition of non-uniqueness of the solution for the incremental boundary-value problem of the generalized coupled elastoplasticity imposes higher values on isothermal hardening function h than R-R h cr max maximal critical local necessary localization condition of plastic deformation in the shape of a localization plane (2.4) and also on h cr (2.3). This means that the local necessary condition of non-uniqueness of the solution is the higher estimation of the hardening modulus values than the estimation imposed by the R-R localization condition. Physically, it is equivalent to the statement that the local necessary condition of non-uniqueness of the solution of the incremental boundary-value problem of the generalized coupled elastoplasticity permits the lower (more safe) values of critical loads than the R-R condition of plastic deformation localization. This happens because the higher value of the hardening modulus h means that the lower loads are admissible. This implies that the local necessary condition of non-uniqueness of the solution is the lower (safer) estimation of outer loads than the R-R local necessary localization condition. Then we can write the following inequalities: h cr 2 h cr max 2 h

(2.10)

The expression h 2 h is the local necessary condition of non-uniqueness of the solution and the respective local sufficient condition of non-uniqueness of the solution is h 2 0. In the range 0 h 2 h , non-uniqueness of the solution (meaning bifurcation of the equilibrium

PART II: COMPARISON OF THE LOCAL UNIQUENESS CONDITION 115

Figure 3. The results obtained from Equation (2.4) on the maximal value of the hardening function h cr max and from expression (2.8) on h .

Â´ 116 Z SLODERBACH and J PAJAK 1 state) can occur. However, in the range h cr max h 2 h other different types (different modes) of material non-stability, of plastic deformations and the appearance of a bifurcation state in the form of a groove, neck, shear band, etc., can occur. The curves in Figure3 indicate that h h cr B on curves 6 and max . Points A and 5 inFigure 3are the border points for which z 7 08532 z 7 3 182 and z 7 08832 z 7 3 089 , and for a suitable greater value

of the parameter z than z 08532 for curve 6 and z 08832 for curve 5, the inequality (2.5) is not satisfied. A point at which the conditions (2.4) and (2.9) are equal is the beginning of the coordinate system. At this point z 7 0 and h 7 0 for the case when 3

7 6. For 3

7 6 the constitutive equations (2.1) of plastic flow laws have the associated form. Then the local necessary condition of non-uniqueness of the solution (2.9) also becomes the sufficient condition and is equal to the R-R local necessary condition (2.4) for plastic strain localization plane and they have the form h cr max 7 h 7 0. At this point the critical values of the hardening cr function h (see Equations (2.3)) can even be negative. If we compare Equations (2.4) and (2.9) and make some transformations, we can obtain an expression for the existence of mutual points dependent on material constants (67 3

and ) and when the condition 6 3

2 0875 is fulfilled in the following form: 1 2

3

2 7 26 3

6 3

3 1

2

(2.11)

For the case of non-compressible elastic material 7 085, so from expression (2.11) we obtain that 6 3

7 34 . The plot for ( 7 085) or for (6 7 3)

forms the curve 1 presented in Figure 4. Curve 2 is for ( 7 083) and curve 3 for ( 7 080). On the basis of research on the validity of expressions (2.3) and (2.4) it can be concluded (see [20, 21]) that for the case of inequality 3 6 83

2

(2.12)

the normal n to plastic strain localization plane is orthogonal to the 1 II direction and the conditions (2.3) and (2.4) are independent of the stress state. When we compare (2.5) with (2.12) we can obtain that 3 26 83

2 3 2

(2.13)

and then we can see that the normal n to the plastic strain localization plane is not orthogonal to the 1 II and 1 III direction [20, 21]. In the case when 3

8 6 3, the normal n to the plane of plastic strain localization is orthogonal to 1 III , see [20, 21], but this case is not analyzed in the present paper. A graphical interpretation of conditions (2.5), (2.11), (2.12), and (2.13) for various material parameters is presented in Figure 4. At the point S, 6 7 3

7 0, plastic flow laws have an associated form. The estimation coefficient which was introduced here has the following form:

PART II: COMPARISON OF THE LOCAL UNIQUENESS CONDITION 117

Figure 4. Geometrical interpretation of condition (2.11), imposed by the inequalities (2.12) and (2.13), resulting from the R-R condition of plastic strain localization.

7

h tan

3

h cr max8tan

h h cr max 3 3 2 1 2 71 h h cr max 18 18 3 3

(2.14)

Â´ 118 Z SLODERBACH and J PAJAK 1

Figure 5. Percentage change of the estimation coefficient versus dilatancy factor 6 for z 7 084.

Tangential hardening moduli are used in Equation (2.14). A geometrical interpretation of hardening modulus (hardening function) h and tangent hardening modulus h tan is presented in Figure 1(b). In Figure 5, the percentage change of the coefficient versus the dilatancy factor 6 for the assumed value of the parameter z 7 084 is presented. cr The values of h3 and h max for the assumed z dependent on 6 were taken from curves in 3 Figure 3. From the curve in Figure 5 we see that this estimation is quite precise. For 6 7 086, 7 2% and for 6 7 080, 7 12%. On the basis of the results presented in Figure 5 we can draw a more general conclusion. Namely, the estimation is more precise for more plastic compressible materials (the dilatancy factor 6 is greater and tends to the value 0.6) and less precise for less plastic compressible materials (the dilatancy factor 6 is lower and tends to the value 0.0).

3. GENERAL REMARKS ON THE LOCAL UNIQUENESS CONDITION AND THE R-R LOCAL PLASTIC STRAIN LOCALIZATION The local condition h 2 h cr for the appearance of the R-R plastic plane strain localization is a mathematically necessary condition. Derived in [2, 3], the local condition h h of uniqueness of the solution of the incremental boundary-value elastoplasticity is a mathematically sufficient condition. We can tell that the mathematically local necessary conditions for non-uniqueness of the solution of incremental boundary-value problem is h 2 h and the mathematically sufficient condition is h 2 0. This is illustrated in Figure 6.

PART II: COMPARISON OF THE LOCAL UNIQUENESS CONDITION 119

Figure 6. The scheme of relations between local uniqueness conditions and the R-R local plane plastic strain localization.

In Figure 6 at point A h 7 h cr max and at point B h 7 h . On the left from point A when 683

cr N 7 3 then h 2 h max and the plastic deformation as the R-R localization plane can occur or not, so expression (2.4) is a mathematically necessary condition. Also on the left

from point A when N 7 683 3

then h 2 h cr and the R-R plane of plastic strain localization can occur. There is also a possibility of the appearance of various instabilities of plastic deformations, different to the R-R localization plane, for example in the form of a shear band (see, e.g., [1, 6, 12, 14, 17, 22â€“24]). On the left from point 0 (h 2 0) the bifurcation of the equilibrium state must appear and mathematically this is the sufficient condition for non-uniqueness of the solution of the incremental boundary-value problem of elastoplasticity. In the range 0 h 2 B, the bifurcation of the equilibrium state (non-uniqueness state) can occur (or not) since mathematically it is the necessary condition. However, in the range between the points A and B (A h 2 B), the other types of instability of elasticâ€“plastic deformations and the other types of non-uniqueness of the solution or bifurcation equilibrium state can appear, different to those occurring for the h 2 0 range. These instabilities can appear in the form of a groove, neck, shear band, or other states of deformation instability of elastoplastic materials.

4. REMARKS AND CONCLUSIONS 1. It has been demonstrated that the local necessary condition of non-uniqueness of the solution of the incremental boundary-value problem of the elastoplasticity imposes higher values for the isothermal hardening function h than the R-R local necessary localization condition of plastic deformation in the shape of a localization plane. This means that the local necessary non-uniqueness condition is the upper estimation of the hardening moduli values imposed by the R-R localization condition. Physically it is equivalent to the statement that the local necessary condition of non-uniqueness of the solution of the incremental boundary-value problem of the generalized coupled elastoplasticity permits lower (more safe) estimations of values of critical loads than the R-R condition of the plastic

´ 120 Z SLODERBACH and J PAJAK 1

2.

3.

4.

5.

deformation localization plane. This happens because the higher value of the hardening modulus h means that lower loads are permissible. This implies that the local necessary non-uniqueness condition is the lower (safer) estimation of outer loads than the R-R local necessary localization condition. Intuitively it was possible to predict this, because the local uniqueness condition excludes all forms of instability of plastic deformations, but the R-R plastic strain localization is a form of plastic instability. cr It was proven that h h cr max and hence h h . The results are presented on appropriate curves in Figures 3, 4, and 5. It seems worth pointing out that a sufficient local condition for localization in the form of plastic deformations as the R-R localization plane has not yet been derived in the range h 2 h cr max , that is on the left from point A in Figure 6 and this problem is still open. At point 0 (starting point of the coordinate system) in Figures 3 and 4 both conditions are equal (h 7 h cr 7 0). The formulated local sufficient condition of uniqueness of the solutions, excluding the possibility of a bifurcation state and local conditions describing the possibility of appearance of localization of plastic deformations have important practical value in addition to the pure mathematical and cognitive meaning. They can serve as a tool for the estimation of the critical load, beyond which the appearance of plastic deformation instability or a bifurcation state are possible (see, e.g., [2–22]). The problems of plastic deformation instability, localization of plastic deformation and bifurcation state play an important role in reliability, safe engineering design, experiments and technology of production and repair problems of equipment, machines, and construction elements. Micro-concentrations of strains and stresses may occur in these points of material where uniqueness conditions or bifurcation criteria are violated. Micro-cracks and micro-localizations of deformation can occur, then grow and lead to fracture and material destruction. Such micro- and then macro-concentrations might be important during creep and variable mechanical and thermo-mechanical loads (e.g. Macha [25, 26]). Further studies and improvements in measuring and calculating methods to estimate the moment of appearance of these phenomena seem necessary. It will also be interesting to derive an explicit form of local sufficient condition of uniqueness condition (or local necessary non-uniqueness condition) for the case of large deformation in the frame of non-linear mechanics of continuum solids2 see, for example, [4–6, 17–19, 24, 27, 28].

REFERENCES [1] [2] [3]

[4] [5]

´ Sloderbach, Z. Generalized coupled thermoplasticity. Part I. Fundamental equations and identities. Archives of Mechanics, 35, 337–349 (1983). ´ Sloderbach, Z. Generalized coupled thermoplasticity. Part II. On the uniqueness and bifurcations criteria. Archives of Mechanics, 35, 351–367 (1983). ´ Sloderbach, Z. and Pajak, J. Generalized coupled thermoplasticity taking into account large strains: part I. Conditions of uniqueness of the solution of boundary-value problem and bifurcation criteria. Mathematics and Mechanics of Solids 2010, in press. Hill, R. A general theory of uniqueness and stability in elasto-plastic solids. Journal of the Mechanics and Physics of Solids, 6, 236–249 (1958). Hill, R. Bifurcation and uniqueness in non-linear mechanics of continua, Problem of Continuum Mechanics (N.I. Muskhelishwili Anniversary Volume), SIAM, Philadelphia, PA, 1961, pp. 155–164.

PART II: COMPARISON OF THE LOCAL UNIQUENESS CONDITION 121

[6] [7] [8] [9] [10]

[11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

Hill, R. Mathematical Theory of Plasticity, Clarendon Press, Oxford, 1986. Hueckel, T. Coupling of elastic and plastic deformations of bulk solids. Acta Mechanica, 11, 227–235 (1976). Hueckel, T. and Konig, J.A. Some problems in elastoplasticity. Academia Pollaca Della Scienze, Conference 74, Ossolineum, Warszawa, 1979. Hueckel, T. and Maier, G. Incremental boundary value problems in the presence of coupling of elestic and plastic deformations. A rock mechanics oriented theory. International Journal of Solids and Structures, 13, 1–15 (1977). Hueckel, T. and Maier, G. Non-associated and coupled flow rules of elastoplasticity for geotechnical media. Proceeding 9th International Conference of Soil Mechanics Foundation of Engineering (JCSFE), Speciality session 7, Constitutive relations for soils, Tokyo, 1977, pp. 129–142. Maier, G. A minimum principle for incremental elastoplasticity, with non-associated plastic flow laws. Journal of the Mechanics and Physics of Solids, 18, 319–330 (1970). Marciniak, Z. Limit deformations in sheet metal stamping. WNT, Warszawa, 1971. Mróz, Z. On forms of constitutive laws for elastic–plastic solids. Archives of Applied Mechanics, 18, 3–35 (1966). Perzyna, P. Instability phenomena and adiabatic shear band localisation in thermoplastic flow processes. Acta Mechanica, 106, 173–205 (1986). Perzyna, P. and Duszek, M.K. The localisation of plastic deformation in thermoplastic solids, International Journal of Solids and Structures, 27, 1419–1443 (1991). Petryk, H. On constitutive inequalities and bifurcation in elastic-plastic solids with a yield-surface vertex. Journal of the Mechanics and Physics of Solids, 37, 265–291 (1989). P1echerski, R.B. Finite deformation plasticity with strain induced anisotropy and shear banding. Journal of Materials Processing Technology, 60, 35–44 (1996). Raniecki, B. Uniqueness criteria in solids with non-associated plastic flow laws at finite deformations. Bulletin De l’Academie Polonaise des Sciences, Serie des Sciences Techniques, 27(8–9), 391–399 (1979). Raniecki, B. and Bruhns, O.T. Bounds to bifurcation stress in solids with non-associated plastic flow at finite strain. Journal of the Mechanics and Physics of Solids, 29, 153–172 (1981). Rudnicki, J.W. and Rice J.R., Condition for the localisation of deformation in pressure-sensitive dilatant materials. Journal of the Mechanics and Physics of Solids, 23, 371–394 (1975). Rice, J.R. The localisation of plastic deformation, in ed. W.T. Koiter, Theoretical and Applied Mechanics, NorthHolland, Amsterdam, 1976, pp. 207–220. ´ Sloderbach, Z. and Sawicki, T. Determination of the critical adiabatical twisting moment in the case of thick and thin-walled metal tubes. Engineering Transactions, 31, 447–457 (1983). Izbicki, J.R. and Mróz, Z. Carrying capacity limit in soil and rock mechanics [in Polish], IFTR – PAS (Institute of Fundamental Technological Research – Polish Academy of Sciences), PWN, Warszawa-Pozna´n, 1976. Lubliner, J. Plasticity Theory, Macmillan, New York, 1990. Macha, E. Simulation investigations of the position of fatigue fracture plane in materials with biaxial loads. Materialwissenschaft und Werkstofftechnik 20(I, 4/89), 132–136, (II, 5/89), 153–163 (1989). Macha, E. A review of energy-based multiaxial fatigue failure criteria. The Archive of Mechanical Engineering, 48, 71–101 (2001). Lubliner, J., On the thermodynamic foundations of non-linear solid mechanics. International Journal of NonLinear Mechanics, 7, 237–254 (1972). 3 Zyczkowski, M. and Szuwalski, K. On the termination of the process of finite plastic deformations. Journal de Mécanique Theorique et Appliqee, 1, 175–186 (1982).

Non-Linear Elastic Bodies Exhibiting Limiting Small Strain

K.R. R AJAGOPAL

Department of Mechanical Engineering, College Station, TX 77843-3123, USA (Received 28 July 20091 accepted 30 September 2009)

Abstract: In this paper we develop a constitutive model for describing the elastic response of solids that does not stem from either classical Cauchy or Green elasticity. In contrast to the classical theory, we show that it is possible to obtain a constitutive model wherein the linearized strain is related to the stress in a non-linear manner. The specific choice that we make allows for the stress to be arbitrarily large while the strain remains small (consistent with the assumption used in the linearization of the non-linear strain) or below some limiting value. Such models are worth investigating in detail as they have relevance to problems involving cracks as well as other problems wherein one finds strain singularities within the classical theory of linearized elasticity, and to models that exhibit limited stretch.

Key Words: Cauchy elasticity, Green elasticity, linearized strain

1. INTRODUCTION In a paper entitled â€˜Elasticity of elasticityâ€™ Rajagopal [1] showed that Cauchy elastic and Green elastic solids (see Truesdell and Noll [2] for a definition of such bodies) form a small sub-class of solids that are incapable of dissipation. In addition to implicit relations between the stress and the stretch, Rajagopal also discussed the possibility of implicit rate-type models to describe the elastic response. Related ideas that form the basis for the models discussed in [1] were investigated in an earlier paper by Rajagopal [3] on implicit constitutive relations for both fluids and solids developed to describe the general response of bodies. Later, Rajagopal and Srinivasa [4, 5] built upon the earlier work by Rajagopal to provide a thermodynamic basis for the non-dissipative response of solids. In [5] a methodology was also presented to describe non-dissipative solids from a purely Eulerian point of view without having the need to define a reference configuration or the notion of a deformation gradient. This is in marked departure from the usual way in which elastic materials are described in terms of the deformation gradient F which is a Lagrangian measure. It is worth noting that the class of bodies developed by Rajagopal and Srinivasa [5] are more general than the class of hypoelastic bodies. In this paper we consider an elastic body which is defined by requiring the deformation gradient to be a function of the stress. Explicit expressions for the stretch in terms of the stress form a special sub-class of models of the implicit type presented in [1].

Mathematics and Mechanics of Solids 16: 122â€“139, 2011 1 The Author(s), 2011. Reprints and permissions: 1 http://www.sagepub.co.uk/journalsPermissions.nav

DOI: 10.1177/1081286509357272

NON-LINEAR ELASTIC BODIES EXHIBITING LIMITING SMALL STRAIN 123

Using that as the starting point one can obtain a model wherein the linearized strain bears a non-linear relation to the stress. An interesting consequence of the generalization of elastic bodies introduced by Rajagopal [1] is that one can provide a rational justification for developing a model wherein one has a non-linear relationship between the linearized strain and the stress. When the general implicit model is linearized under the usual assumption that the displacement gradient should be small, we obtain a non-linear relationship between the linearized strain and the stress. Such models are not possible within the purview of either Cauchy elasticity or Green elasticity, wherein linearization leads to the classical linearized elastic solid model. Recently, Bustamante and Rajagopal [6] have shown that for models wherein the linearized strain depends non-linearly on the stress, when one considers plane problems one can once again introduce the classical Airyâ€™s stress function to simplify the governing equations. However, instead of a biharmonic equation that governs the classical plane problems of linearized elasticity, one obtains a highly non-linear partial differential equation for the model considered by Bustamante and Rajagopal [6]. Bustamante [7] has provided a weak formulation for this class of problems in order to solve boundary value problems for such bodies within a numerical framework. In this paper, we consider a specific model that belongs to the general class introduced by Rajagopal [1] with a view towards illustrating the usefulness of such models in describing problems which within the classical linearized elastic model leads to singularities in the strain. The model can also describe the behavior of certain polymeric solids that exhibit limited stretch. A discussion of elastic models that can describe the phenomenon of limited chain extensibility can be found in Horgan and Saccomandi [8]. The worm-like chain model that has been proposed to describe limited chain extensibility has been provided with a three-dimensional framework by Ogden, Saccomandi and Sgura [9]. The models studied by Saccomandi and his co-workers fall under the umbrella of classical non-linear models that have been developed for describing the response of elastic bodies. Recently, Rajagopal and Saccomandi [10] have studied implicit constitutive models for compressible elastic solids wherein the material moduli depend on the mean normal stress. They also discuss models that can describe the phenomenon of limited chain extensibility. The class of models that is presented below is not to be construed as â€˜theâ€™ only class of models to describe the limited stretch that has been observed in certain elastic bodies1 it is merely meant to bring attention to a new arsenal that has become available. Models belonging to this class have far-reaching technological implications as they allow for the possibility of finite strains even though the stress can become unbounded.

2. PRELIMINARIES Let x denote the current position of a particle which is at X in the stress-free reference configuration. Let x 2 1 1X2 t3 denote the motion of a particle and let us denote by u and F the displacement and deformation gradient through

124 K.R. RAJAGOPAL

u

:2

x 3 X2

(2.1)

F

:2

41 5 4X

(2.2)

We shall assume that 1 is sufficiently differentiable to make all of the operations meaningful. We define the stretch tensors B and C through B :2 FFT 2

C :2 FT F2

(2.3)

and Greenâ€“St. Venant strain E and the Almansiâ€“Hamel strain e through E :2

1 1C 3 13 2 2

e :2

2 11 1 3 B31 5 2

(2.4)

Under the assumption max 41u4 2 0 163 2 X 6 K R 1B3 2

6 5 12

t 6 12

(2.5)

where 474 stands for the usual trace norm, we find 1 2 E 2 2 8 0 62 2

1 2 e 2 2 8 0 62 2

(2.6)

where 22

4 13 1u 8 11u3T 5 2

(2.7)

Let us consider the class of elastic models given by f 1T2 B3 2 05

(2.8)

If f is an isotropic function, then it follows that (see Spencer [11]): 7 0 1 8 7 1 T 8 7 2 B 8 7 3 T2 8 7 4 B2 8 7 5 1TB 8 BT3 1 2 1 8 7 6 T2 B 8 BT2 8 7 7 B2 T 8 TB2 8 7 8 T2 B2 8 B2 T2 3 2 02

(2.9)

where the material moduli 7 i 2 i 2 02 5558 depend on 1 2 1 2 1 2 82 trT2 trB2 trT2 2 trB2 2 trT3 2 trB3 2 tr 1TB3 2 tr T2 B 2 tr B2 T 2 tr T2 B2 5

(2.10)

Models defined through (2.9) and (2.10) are not amenable to use as it is impossible to outline an experimental program wherein the material functions can be determined.

NON-LINEAR ELASTIC BODIES EXHIBITING LIMITING SMALL STRAIN 125

A much simpler, but yet far too general, sub-class is given by B 2 79 0 1 8 79 1 T 8 79 2 T2 2

(2.11)

where 79 i 2 i 2 12 22 3 depend on 82 trT2 trT2 2 and trT3 . Equation (2.11) will form the starting point for an approximation which leads to a different â€˜small displacement gradientâ€™ theory that allows us to have a non-linear relationship between the linearized strain and the stress. We shall use exactly the same small displacement gradient approximation that leads to the classical linearized theory of elasticity. We note that under the approximation (2.5), the model (2.11) reduces to 2 2 99 0 1 8 99 1 T 8 99 2 T2 2

(2.12)

1 2 B 2 1 8 22 8 0 6 2 5

(2.13)

since

Since the strain is dimensionless, the material moduli 9 1 and 9 2 need to have dimensions that are the inverse of the stress and the square of the stress, respectively. With regard to the relation (2.12), while 2 is required to be small there are no such demands on the stress and it can be arbitrarily large. We consider a special simple model that belongs to the class defined by (2.12), namely the model 227

5 68 7

9

13e

391trT3 [1861trT2 3]

18

T 2 1 21 1 8 trT2 2

(2.14)

which is not an implicit model, but provides an explicit relationship for the linearized strain in terms of the stress. In the above equation 7, 9, and 6 are constant with appropriate units so that the right hand side is dimensionless. For the sake of ease of illustration, we shall set 6 to be unity. We note that when T 2 02 2 2 0. For the purpose of illustration, as we shall only consider cases wherein is non-negative, we shall set to be zero. Otherwise, we need to consider the full model. Also, if we require linearity in T, the above model reduces to 2 2 79 1trT3 1 8 7 T2

(2.15)

and thus 79 2

313 2 E

7 2

11 8 3 2 E

(2.16)

leads to the classical linearized elastic solid. We reiterate that the model (2.14) is merely a small displacement gradient approximation and is not to be viewed as a model in its own right in that it holds for large strains. Another possible model that exhibits small strains even when the stresses are large is the following:

126 K.R. RAJAGOPAL

27 13

1 18

trT [186trT2 ]

1 1 22 189 18 1 1 8 trT2

n T2

(2.17)

where 72 92 , and n are constants. One can construct infinity of such models wherein the strains are bounded, the bound being as small or large as one wishes, as the stress tends to infinity. We do not consider model (2.17) in this paper, but merely restrict ourselves to a study of the response of model (2.14). In general, problems involving the specification of the linearized strain in terms of a nonlinear function of the stress, or in the more general case the stretch B as a non-linear function of the stress, are more complicated as we have to solve the constitutive relation (2.12) (or (2.11)) and the balance of linear momentum 8

dv 2 divT 8 8b2 dt

(2.18)

where 8 is the density, v is the velocity, and b is the body force, simultaneously. The usual methodology of providing a constitutive equation for the stress T in terms of the strain allows one to substitute that expression for the stress into Equation (2.18) to arrive at a single partial differential equation for the displacement field. Instead of a differential equation for the displacement (three scalar equations), in the current situation we have to solve nine scalar equations simultaneously (six constitutive relations and three from the balance of linear momentum). Of course, in both cases we need to also solve the balance of mass. We adopt a semi-inverse approach to find solutions to problems where we can make assumptions concerning the form of both the displacement and stress fields. We seek semiinverse solutions to the problem of extension, shear, torsion, circumferential shear, telescopic shear, and the combination of circumferential and telescopic shear. As the constitutive relation that we consider is non-linear, the combination of two of the problems cannot be obtained as the superposition of the two individual solutions.

3. SPECIAL PROBLEMS 3.1. Extension

Let us consider the problem of extension wherein we assume that the state of stress T takes the form T 2 T 1ex ex 3 2 where T is a constant and ex denotes the unit vector in the x coordinate direction. Thus, 2 1

39T 227 13e T 2 18 [1 8 T ]

(3.1)

(3.2)

NON-LINEAR ELASTIC BODIES EXHIBITING LIMITING SMALL STRAIN 127

and thus

x x

2 7 1 3 e39T 8

T 11 8 T 3

2

(3.3)

and

yy 2 zz 2 7 1 3 e39T 5

(3.4)

Within the context of linearized elasticity, when one makes an assumption for the stress one needs to ensure that the strains meet the compatibility equations. In this specific example the compatibility conditions are met trivially as all of the strains are constant. The displacements u x 2 u y 2 u z in the x2 y2 and z direction, respectively, can be obtained trivially by integration and are given by ux uy uz

2 7 1 3 e39T 8

T 11 8 T 3

x 8 C1 2

2 7 1 3 e39T y 8 C2 2 2 7 1 3 e39T z 8 C3 5

(3.5) (3.6) (3.7)

3 Note that as T 2 x x 2 7 1 8 2 yy 2 zz 7, and thus 4 4 7 2 13 8 2

41 8 2 3 2 , in marked contrast to the linearized model wherein the strain increases as the stress increases thereby violating the assumption that the displacement gradient and hence the strain are sufficiently small. Since stress has dimensions, it would be better to introduce a dimensionless stress T 2 9T and allow the dimensionless stress to tend to infinity1 however, the outcome of such a process remains the same. 3.2. Simple Shear

Consider the deformation x

2

X 8 f 1Y 3 2

(3.8a)

y 2 Y2

(3.8b)

z 2

(3.8c)

Z2

in a Cartesian coordinate system and assume that the stress tensor takes the form 2 1 T 2 T ex e y 8 e y ex 2

(3.9)

where ex and e y are unit vectors in the x and y coordinate directions, respectively, and T is a constant. We are assuming both the form for the stress field as well as the deformation. It is possible that these two assumptions might not be compatible1 however, whether such an

128 K.R. RAJAGOPAL

assumption is consistent or otherwise will be borne out by the analysis. In the case of this example we find that the assumptions (3.8) and (3.9) are consistent and as we shall see the strains that are a consequence of (3.9) and (2.14) meet the compatibility for the strain. We are looking for semi-inverse solutions and whether such solutions are possible or not can only be ascertained by carrying out the calculations. In virtue of assumption (3.9), in the absence of body forces, the equations of equilibrium are met automatically, and since trT 2 0, the constitutive relation (2.14) is given by 7

22

2 1 T2 1 1 8 trT2 2

(3.10)

and thus

x y 2

7 T 5 1 8 2T

(3.11)

Both the assumed form for the deformation (3.8), and the constitutive relation (2.14) together with (3.9), imply that all of the other components of the strain are zero, and thus our assumptions (3.8) and (3.9) are consistent. Also, since the shear straingiven by (3.11) is a constant, it follows that f 1Y 3 2 Y . When T 2 x y 7 2 2and, thus, the shear strain reaches a critical value as the shear stress tends to infinity. In the linearized theory x y

as T contradicting the original assumption of the theory that the displacement gradient and hence the strain is very small. We see that there is no such contradiction in the case of the model (2.14). 3.3. Torsion

Consider the deformation 1R2 2 Z3 1r2 2 z3 such that r 2 R2

2 8 Z2

z 2 Z5

(3.12)

Let us further suppose that the stress has the form T 2 T 1e ez 8 ez e 3 2

(3.13)

where e and ez are unit vectors along the and z directions and T is a constant. Once again, the equations of equilibrium are met automatically. A trivial calculation leads to 7 T

z 2

5 1 8 2T

(3.14)

In virtue of the constitutive relation, all of the other components of the strain are zero and this is consistent with the assumption (3.12). Since, we can find the shear strain from (3.12), equating that to (3.14) will lead to an expression for z .

NON-LINEAR ELASTIC BODIES EXHIBITING LIMITING SMALL STRAIN 129 We find that z 7 2 as T , i.e. we once again have a limiting strain. 3.4. Circumferential Shear

We consider the circumferential shearing of an annular cylinder of inner radius Ri and outer radius R0 . Consider the deformation 1R2 2 Z3 1r2 2 z3 such that r 2 R2

2 8 g 1R3 2

z 2 Z2

(3.15)

in a cylindrical polar coordinate system. Let us suppose that the stress has the form T 2 T 1r3 [er e 8 e er ] 5

(3.16)

We find that both the constitutive relation (2.14) and (3.16), as well as (3.15) speak with one voice and the only non-zero component of the strain is r . It follows from the equations of equilibrium that dT 2T 8 2 02 dr r

(3.17)

and thus T 1r3 2 T 1R3 2

C 5 R2

(3.18)

Next, 1 dg R 2 2 dR

(3.19)

1 dg 7 T 1R3 5 R 2 2 dr 1 8 2T 1R3

(3.20)

r 2 and by (2.14)

r 2

We first note that the maximum value that r can take is 7 2 . It also follows from (3.20) that dg 27 C 2 5 dR R R 2 8 2C

(3.22)

The solution for g 1R3 is obtained by integrating (3.22) and then enforcing the boundary conditions

130 K.R. RAJAGOPAL

Figure 1. Variation of angular displacement g with the radius R when displacements at the boundary are specified and Ri 2 055R0 5

g 1Ri 3 2 02

(3.24)

g 1R0 3 2 2

(3.25)

where Ri and R0 are the inner and outer radius of the annular region. A straightforward calculation leads to 37

C 2 g 1R3 2 ln 1 8 8 D2 (3.26) 2 R2 where

8

3 7 2

13e 2 C2 8 2 2 3 7

2 2 R0 e 3 Ri Ri2 R02

(3.27)

and D2

7

2

C 2 ln 1 8 5 Ri2

(3.28)

The function g, the angular displacement at the radial coordinate R, is portrayed in Figure 1 when the inner radius is half the outer radius and in Figure 2 when the inner radius is 0.99

NON-LINEAR ELASTIC BODIES EXHIBITING LIMITING SMALL STRAIN 131

Figure 2. Variation of angular displacement g with radius R when displacements at the boundary are specified and Ri 2 0599R0 5

times the outer radius (i.e. when the cylinder has a thin wall). It is very interesting to note that the structure of the displacement is different, being concave in one instance and convex in the other which immediately implies that the structure of the strain is different. The corresponding ratios of [1 r 317 3] for the two cases are plotted as a function of the radius R corresponding to these two cases in Figures 3 and 4, respectively. Since the product 17 3 can be chosen to be very small, the strain predicted in these two cases by the model is as small as we wish it to be. Moreover, we once again see that the manner in which the strain varies in the case of the thick-walled and thin-walled cylinder are both qualitatively and quantitatively different. In the first case it decreases, close to linearly, while in the second case it increases non-linearly. In fact, it might be more reasonable to prescribe the shear stress that is applied at the outer radius R0 . Thus, in addition to (3.24) we could prescribe 9 27

T dg 1R0 3 2 2 dR R0 1 8 2T9

(3.29)

where T9 is the shear stress at R 2 R0 . A simple calculation leads to C 2 T9 R02 2 with D once again given by (3.28). Thus, the angular displacement g 1R3 is given by

(3.30)

132 K.R. RAJAGOPAL

Figure 3. Variation of the strain r with the radius R when displacements at the boundary are specified and Ri 2 055R0 5

Figure 4. Variation of the strain r with the radius R when displacements at the boundary are specified and Ri 2 0599R0 5

NON-LINEAR ELASTIC BODIES EXHIBITING LIMITING SMALL STRAIN 133

Figure 5. Variation of the angular displacement with the radius R when displacement is specified at Ri and traction is specified at R0 and Ri 2 055R0 . (Note that the curves corresponding to the cases when T 2 102 000 and T 2 100 essentially lie on top of each other.)5

7

g 1R3 2 2

!

ln 1 8

" T9 R02 2 T9 R02 2 3 ln 1 8 5 Ri2 R2

(3.31)

Figures 5 and 6 show how the angular displacement g varies with the radius R in the case of a thick-walled and thin-walled cylinder, respectively. While the displacement is essentially linear in the thin-walled case, we note that in the case of a thick-walled cylinder the displacement can vary non-linearly with radius. 3.5. Telescopic Shearing

Let us next consider the deformation from 1R2 2 Z 3 1r2 2 z3, in a cylindrical polar coordinate system so that r 2 R2

2 2

z 2 Z 8 f 1R3 5

(3.32)

A trivial calculation shows that 123 2

1# # 2

0

0

f 1R3

0

0

0

f 1R3 0

0

$ $5

(3.33)

134 K.R. RAJAGOPAL

Figure 6. Variation of the angular displacement with the radius R when displacement is specified at Ri and traction is specified at R0 and Ri 2 0599R0 . (Note that the curves corresponding to the cases when T 2 102 000 and T 2 100 essentially lie on top of each other.)5

Let us suppose that the stress tensor is given by T 2 T 1R3 1er ez 8 ez er 3 5

(3.34)

It immediately follows from (2.14) and (3.18) that

r z 2

1 7 T 1R3 f 1R3 2 2 1 8 2T 1R3

(3.35)

and all other components the strain are zero consistent with (3.27). As before, the maxi of

mum strain possible is 7 2 . The equations of equilibrium reduce to dTr z Trr 8 2 05 dr r

(3.36)

Thus, Trr 2

C 2 r

(3.37)

NON-LINEAR ELASTIC BODIES EXHIBITING LIMITING SMALL STRAIN 135

and it follows from (3.31) and (3.33) that 27 C df 2 2 dR R 8 2C

(3.38)

which can be integrated to yield f 1R3 2 27 C ln R 8 2C 8 D2

(3.39)

where D is a constant. The boundary conditions f 1Ri 3 2 U2

(3.40)

f 1R0 3 2 02

(3.41)

and the solution for f 1R3 can be obtained by solving (3.35), (3.36) and (3.37) where U is the displacement at the inner radius. It follows that C is obtained as a solution to Ri 8 2C U C ln 2 2 (3.42) 27

R0 8 2C and D 2 327 C ln R0 8 2C 5

(3.43)

The solution f 1R3 is plotted in Figures 7 and 8. Interestingly, unlike the previous case of circumferential shearing, the axial displacements have the same qualitative features for both the thick- and thin-walled cylinders. 3.6. Circumferential and telescopic shear

In this case the deformation from 1R2 2 Z3 1r2 2 z3 takes the form r 2 R2

2 8 g 1R3 2

z 2 Z 8 f 1R3 2

(3.44)

and the strain is given by 123 2

0

1# # Rg 1R3 2 f 1R3

Rg 1R3

f 1R3

0

0

0

0

$ $5

(3.45)

136 K.R. RAJAGOPAL

Figure 7. Variation of axial displacement with the radius R when displacements at the boundary are specified and Ri 2 055R0 5

Figure 8. Variation of axial displacement with the radius R when displacements at the boundary are specified and Ri 2 0599R0 5

NON-LINEAR ELASTIC BODIES EXHIBITING LIMITING SMALL STRAIN 137

Let us suppose that the stress takes the form T 2 T1 1R3 1er e 8 e er 3 8 T2 1R3 1er ez 8 ez er 3 5

(3.46)

It follows from (2.14) that the assumption (3.38) is consistent with the expression (3.37) for the strain stemming from (3.36). Thus,

r

2

1 7 T1 1R3 Rg 2 1 21 2 2 1 8 2 T12 8 T22 2

(3.47)

r z

2

1 7 T2 1R3 f 2 21 2 1 2 1 8 2 T12 8 T22 2

(3.48)

and thus 5 2 1 2 6 T1 8 T22 4 4 2 7 2

5 7 1 8 2 1T 2 8 T 2 2 12 1 2

(3.49)

Thus 4 4 has a limit 7 . We do not find the solutions to T1 1R3 2 T2 1R3 2 g 1R3 2 or f 1R3 as our main interest is in showing that for the model being considered, the strain cannot exceed a critical value and will remain small being consistent with the spirit of the approximation that was used to linearize the non-linear strain, unlike the case of the classical linearized theory wherein for all of the problems considered the strain becomes unbounded as the stresses become unbounded. In all of the examples considered in this section, we note that the strain never exceeds a critical value that could be chosen to be arbitrarily small and can be decided by the modeler a priori.

4. CONCLUDING REMARKS The model (2.14) that we have considered exhibits a limiting strain in all of the examples that were considered, while the linearized strains in the corresponding problems within the context of the theory of linearized elasticity would blow up. It would be interesting to study the problems of an elliptic hole in a plate subject to say biaxial loading and then to consider the limit as the ratio of the minor to the major axes of the ellipse tends to zero, thereby modeling a crack. It is conjectured that the strain corresponding to the model (2.14) would 1 not exhibit the classical singularity at the crack tip. Of course, problems involving r crack propagation and fracturing of solids leads to dissipation in a region adjacent to the crack tip, which implies that the response in that region cannot be described by an elastic response. This has led to the use of a â€˜plastic zoneâ€™ adjacent to the crack tip. This dissipation

138 K.R. RAJAGOPAL

notwithstanding, the fact that the strains remain small makes the model a highly desirable model to study stresses near cracks and the model can be modified near the crack tip to allow for dissipation. It is always possible to include small strain inelastic response in addition to the small strain that stems from the elastic response with models that have limiting strain. One need not restrict oneself to the models that relate the linearized strain to the stress. It is possible to have limited stretch or limited strain, with the stretch and strain not being restricted to being small. For instance, an interesting counterpart to (2.14) and (2.17) would be the constitutive model 5 61 3 4

B 22 7 T 2 (4.1) 1 8 e391trT3 1 8

21 1 72 1 8 6 trT2 2 where 7, 9, and 6 are constants and have units so that the right hand side is dimensionless. Also, as before, we set delta to be unity. Another possible model is n 8 7 1 1 1 22 T2 B2 18 189 18 1 2 1 8 trT 1 8 trT2

(4.2)

respectively. In the case of the uniaxial extension problem, given a stress state of the form (3.1) we find that 2 11

39T B27 T 5 (4.3) 18 18e [1 8 T ] 2 Once again, we see that as T 2 B remains bounded. It is not our intent here to consider the models (4.1) and (4.2) in any detail, rather it is merely to point out that such models allow for limited stretch as the stress becomes arbitrarily large. Acknowledgement. K. R. Rajagopal is grateful to the National Science Foundation for supporting this research.

REFERENCES [1] [2] [3] [4] [5] [6] [7]

Rajagopal, K. R. Elasticity of Elasticity, ZAMP, 58, 309â€“317 (2007). Truesdell, C. and Noll, W. The Non-Linear Field Theories of Mechanics, Antman, S. S. (ed.), Springer, Berlin, 2004. Rajagopal, K. R. On implicit constitutive theories. Appl. Math., 28, 279â€“319 (2003). Rajagopal, K. R. and Srinivasa, A. R. On the response of non-dissipative solids. Proc. R. Soc. A, 463, 357â€“367 (2007). Rajagopal, K. R. and Srinivasa, A. R. On a class of non-dissipative materials that are not hyperelastic. Proc. R. Soc. A, 465, 493â€“500 (2009). Bustamante, R. and Rajagopal, K. R. A note on plane strain and plane stress problems for a new class of elastic bodies. Math. Mech. Solids (2009), in press. Bustamante, R. Some topics on a new class of elastic bodies. Proc. R. Soc. A, 465, 1377â€“1392 (2009).

NON-LINEAR ELASTIC BODIES EXHIBITING LIMITING SMALL STRAIN 139

[8]

Horgan, C. O. and Saccomandi, G. Phenomenological hyperelastic strain-stiffening constitutive models for rubber. Rubber Chem. Technol., 71, 152–169 (2006). [9] Ogden, R. W., Saccomandi, G. and Sgura, I. On worm-like chain models within three dimensional continuum mechanics framework. Proc. R. Soc. A, 462, 749–768 (2006). [10] Rajagopal, K. R. and Saccomandi, G. The mechanics and mathematics of the effect of pressure on the shear modulus of elastomers, Proceedings of the Royal Society, London, 465, 3859–3874 (2009). [11] Spencer, A. J. M. Theory of Invariants in Continuum Physics, Vol. 2, Eringen, A. C. (ed.), Academic Press, New York, 1975.

Non-Linear Elastic Bodies Exhibiting Limiting Small Strain

K.R. R AJAGOPAL

Department of Mechanical Engineering, College Station, TX 77843-3123, USA (Received 28 July 20091 accepted 30 September 2009)

Abstract: In this paper we develop a constitutive model for describing the elastic response of solids that does not stem from either classical Cauchy or Green elasticity. In contrast to the classical theory, we show that it is possible to obtain a constitutive model wherein the linearized strain is related to the stress in a non-linear manner. The specific choice that we make allows for the stress to be arbitrarily large while the strain remains small (consistent with the assumption used in the linearization of the non-linear strain) or below some limiting value. Such models are worth investigating in detail as they have relevance to problems involving cracks as well as other problems wherein one finds strain singularities within the classical theory of linearized elasticity, and to models that exhibit limited stretch.

Key Words: Cauchy elasticity, Green elasticity, linearized strain

1. INTRODUCTION In a paper entitled â€˜Elasticity of elasticityâ€™ Rajagopal [1] showed that Cauchy elastic and Green elastic solids (see Truesdell and Noll [2] for a definition of such bodies) form a small sub-class of solids that are incapable of dissipation. In addition to implicit relations between the stress and the stretch, Rajagopal also discussed the possibility of implicit rate-type models to describe the elastic response. Related ideas that form the basis for the models discussed in [1] were investigated in an earlier paper by Rajagopal [3] on implicit constitutive relations for both fluids and solids developed to describe the general response of bodies. Later, Rajagopal and Srinivasa [4, 5] built upon the earlier work by Rajagopal to provide a thermodynamic basis for the non-dissipative response of solids. In [5] a methodology was also presented to describe non-dissipative solids from a purely Eulerian point of view without having the need to define a reference configuration or the notion of a deformation gradient. This is in marked departure from the usual way in which elastic materials are described in terms of the deformation gradient F which is a Lagrangian measure. It is worth noting that the class of bodies developed by Rajagopal and Srinivasa [5] are more general than the class of hypoelastic bodies. In this paper we consider an elastic body which is defined by requiring the deformation gradient to be a function of the stress. Explicit expressions for the stretch in terms of the stress form a special sub-class of models of the implicit type presented in [1].

Mathematics and Mechanics of Solids 16: 122â€“139, 2011 1 The Author(s), 2011. Reprints and permissions: 1 http://www.sagepub.co.uk/journalsPermissions.nav

DOI: 10.1177/1081286509357272

NON-LINEAR ELASTIC BODIES EXHIBITING LIMITING SMALL STRAIN 123

Using that as the starting point one can obtain a model wherein the linearized strain bears a non-linear relation to the stress. An interesting consequence of the generalization of elastic bodies introduced by Rajagopal [1] is that one can provide a rational justification for developing a model wherein one has a non-linear relationship between the linearized strain and the stress. When the general implicit model is linearized under the usual assumption that the displacement gradient should be small, we obtain a non-linear relationship between the linearized strain and the stress. Such models are not possible within the purview of either Cauchy elasticity or Green elasticity, wherein linearization leads to the classical linearized elastic solid model. Recently, Bustamante and Rajagopal [6] have shown that for models wherein the linearized strain depends non-linearly on the stress, when one considers plane problems one can once again introduce the classical Airyâ€™s stress function to simplify the governing equations. However, instead of a biharmonic equation that governs the classical plane problems of linearized elasticity, one obtains a highly non-linear partial differential equation for the model considered by Bustamante and Rajagopal [6]. Bustamante [7] has provided a weak formulation for this class of problems in order to solve boundary value problems for such bodies within a numerical framework. In this paper, we consider a specific model that belongs to the general class introduced by Rajagopal [1] with a view towards illustrating the usefulness of such models in describing problems which within the classical linearized elastic model leads to singularities in the strain. The model can also describe the behavior of certain polymeric solids that exhibit limited stretch. A discussion of elastic models that can describe the phenomenon of limited chain extensibility can be found in Horgan and Saccomandi [8]. The worm-like chain model that has been proposed to describe limited chain extensibility has been provided with a three-dimensional framework by Ogden, Saccomandi and Sgura [9]. The models studied by Saccomandi and his co-workers fall under the umbrella of classical non-linear models that have been developed for describing the response of elastic bodies. Recently, Rajagopal and Saccomandi [10] have studied implicit constitutive models for compressible elastic solids wherein the material moduli depend on the mean normal stress. They also discuss models that can describe the phenomenon of limited chain extensibility. The class of models that is presented below is not to be construed as â€˜theâ€™ only class of models to describe the limited stretch that has been observed in certain elastic bodies1 it is merely meant to bring attention to a new arsenal that has become available. Models belonging to this class have far-reaching technological implications as they allow for the possibility of finite strains even though the stress can become unbounded.

2. PRELIMINARIES Let x denote the current position of a particle which is at X in the stress-free reference configuration. Let x 2 1 1X2 t3 denote the motion of a particle and let us denote by u and F the displacement and deformation gradient through

124 K.R. RAJAGOPAL

u

:2

x 3 X2

(2.1)

F

:2

41 5 4X

(2.2)

We shall assume that 1 is sufficiently differentiable to make all of the operations meaningful. We define the stretch tensors B and C through B :2 FFT 2

C :2 FT F2

(2.3)

and Greenâ€“St. Venant strain E and the Almansiâ€“Hamel strain e through E :2

1 1C 3 13 2 2

e :2

2 11 1 3 B31 5 2

(2.4)

Under the assumption max 41u4 2 0 163 2 X 6 K R 1B3 2

6 5 12

t 6 12

(2.5)

where 474 stands for the usual trace norm, we find 1 2 E 2 2 8 0 62 2

1 2 e 2 2 8 0 62 2

(2.6)

where 22

4 13 1u 8 11u3T 5 2

(2.7)

Let us consider the class of elastic models given by f 1T2 B3 2 05

(2.8)

If f is an isotropic function, then it follows that (see Spencer [11]): 7 0 1 8 7 1 T 8 7 2 B 8 7 3 T2 8 7 4 B2 8 7 5 1TB 8 BT3 1 2 1 8 7 6 T2 B 8 BT2 8 7 7 B2 T 8 TB2 8 7 8 T2 B2 8 B2 T2 3 2 02

(2.9)

where the material moduli 7 i 2 i 2 02 5558 depend on 1 2 1 2 1 2 82 trT2 trB2 trT2 2 trB2 2 trT3 2 trB3 2 tr 1TB3 2 tr T2 B 2 tr B2 T 2 tr T2 B2 5

(2.10)

Models defined through (2.9) and (2.10) are not amenable to use as it is impossible to outline an experimental program wherein the material functions can be determined.

NON-LINEAR ELASTIC BODIES EXHIBITING LIMITING SMALL STRAIN 125

A much simpler, but yet far too general, sub-class is given by B 2 79 0 1 8 79 1 T 8 79 2 T2 2

(2.11)

where 79 i 2 i 2 12 22 3 depend on 82 trT2 trT2 2 and trT3 . Equation (2.11) will form the starting point for an approximation which leads to a different â€˜small displacement gradientâ€™ theory that allows us to have a non-linear relationship between the linearized strain and the stress. We shall use exactly the same small displacement gradient approximation that leads to the classical linearized theory of elasticity. We note that under the approximation (2.5), the model (2.11) reduces to 2 2 99 0 1 8 99 1 T 8 99 2 T2 2

(2.12)

1 2 B 2 1 8 22 8 0 6 2 5

(2.13)

since

Since the strain is dimensionless, the material moduli 9 1 and 9 2 need to have dimensions that are the inverse of the stress and the square of the stress, respectively. With regard to the relation (2.12), while 2 is required to be small there are no such demands on the stress and it can be arbitrarily large. We consider a special simple model that belongs to the class defined by (2.12), namely the model 227

5 68 7

9

13e

391trT3 [1861trT2 3]

18

T 2 1 21 1 8 trT2 2

(2.14)

which is not an implicit model, but provides an explicit relationship for the linearized strain in terms of the stress. In the above equation 7, 9, and 6 are constant with appropriate units so that the right hand side is dimensionless. For the sake of ease of illustration, we shall set 6 to be unity. We note that when T 2 02 2 2 0. For the purpose of illustration, as we shall only consider cases wherein is non-negative, we shall set to be zero. Otherwise, we need to consider the full model. Also, if we require linearity in T, the above model reduces to 2 2 79 1trT3 1 8 7 T2

(2.15)

and thus 79 2

313 2 E

7 2

11 8 3 2 E

(2.16)

leads to the classical linearized elastic solid. We reiterate that the model (2.14) is merely a small displacement gradient approximation and is not to be viewed as a model in its own right in that it holds for large strains. Another possible model that exhibits small strains even when the stresses are large is the following:

126 K.R. RAJAGOPAL

27 13

1 18

trT [186trT2 ]

1 1 22 189 18 1 1 8 trT2

n T2

(2.17)

where 72 92 , and n are constants. One can construct infinity of such models wherein the strains are bounded, the bound being as small or large as one wishes, as the stress tends to infinity. We do not consider model (2.17) in this paper, but merely restrict ourselves to a study of the response of model (2.14). In general, problems involving the specification of the linearized strain in terms of a nonlinear function of the stress, or in the more general case the stretch B as a non-linear function of the stress, are more complicated as we have to solve the constitutive relation (2.12) (or (2.11)) and the balance of linear momentum 8

dv 2 divT 8 8b2 dt

(2.18)

where 8 is the density, v is the velocity, and b is the body force, simultaneously. The usual methodology of providing a constitutive equation for the stress T in terms of the strain allows one to substitute that expression for the stress into Equation (2.18) to arrive at a single partial differential equation for the displacement field. Instead of a differential equation for the displacement (three scalar equations), in the current situation we have to solve nine scalar equations simultaneously (six constitutive relations and three from the balance of linear momentum). Of course, in both cases we need to also solve the balance of mass. We adopt a semi-inverse approach to find solutions to problems where we can make assumptions concerning the form of both the displacement and stress fields. We seek semiinverse solutions to the problem of extension, shear, torsion, circumferential shear, telescopic shear, and the combination of circumferential and telescopic shear. As the constitutive relation that we consider is non-linear, the combination of two of the problems cannot be obtained as the superposition of the two individual solutions.

3. SPECIAL PROBLEMS 3.1. Extension

Let us consider the problem of extension wherein we assume that the state of stress T takes the form T 2 T 1ex ex 3 2 where T is a constant and ex denotes the unit vector in the x coordinate direction. Thus, 2 1

39T 227 13e T 2 18 [1 8 T ]

(3.1)

(3.2)

NON-LINEAR ELASTIC BODIES EXHIBITING LIMITING SMALL STRAIN 127

and thus

x x

2 7 1 3 e39T 8

T 11 8 T 3

2

(3.3)

and

yy 2 zz 2 7 1 3 e39T 5

(3.4)

Within the context of linearized elasticity, when one makes an assumption for the stress one needs to ensure that the strains meet the compatibility equations. In this specific example the compatibility conditions are met trivially as all of the strains are constant. The displacements u x 2 u y 2 u z in the x2 y2 and z direction, respectively, can be obtained trivially by integration and are given by ux uy uz

2 7 1 3 e39T 8

T 11 8 T 3

x 8 C1 2

2 7 1 3 e39T y 8 C2 2 2 7 1 3 e39T z 8 C3 5

(3.5) (3.6) (3.7)

3 Note that as T 2 x x 2 7 1 8 2 yy 2 zz 7, and thus 4 4 7 2 13 8 2

41 8 2 3 2 , in marked contrast to the linearized model wherein the strain increases as the stress increases thereby violating the assumption that the displacement gradient and hence the strain are sufficiently small. Since stress has dimensions, it would be better to introduce a dimensionless stress T 2 9T and allow the dimensionless stress to tend to infinity1 however, the outcome of such a process remains the same. 3.2. Simple Shear

Consider the deformation x

2

X 8 f 1Y 3 2

(3.8a)

y 2 Y2

(3.8b)

z 2

(3.8c)

Z2

in a Cartesian coordinate system and assume that the stress tensor takes the form 2 1 T 2 T ex e y 8 e y ex 2

(3.9)

where ex and e y are unit vectors in the x and y coordinate directions, respectively, and T is a constant. We are assuming both the form for the stress field as well as the deformation. It is possible that these two assumptions might not be compatible1 however, whether such an

128 K.R. RAJAGOPAL

assumption is consistent or otherwise will be borne out by the analysis. In the case of this example we find that the assumptions (3.8) and (3.9) are consistent and as we shall see the strains that are a consequence of (3.9) and (2.14) meet the compatibility for the strain. We are looking for semi-inverse solutions and whether such solutions are possible or not can only be ascertained by carrying out the calculations. In virtue of assumption (3.9), in the absence of body forces, the equations of equilibrium are met automatically, and since trT 2 0, the constitutive relation (2.14) is given by 7

22

2 1 T2 1 1 8 trT2 2

(3.10)

and thus

x y 2

7 T 5 1 8 2T

(3.11)

Both the assumed form for the deformation (3.8), and the constitutive relation (2.14) together with (3.9), imply that all of the other components of the strain are zero, and thus our assumptions (3.8) and (3.9) are consistent. Also, since the shear straingiven by (3.11) is a constant, it follows that f 1Y 3 2 Y . When T 2 x y 7 2 2and, thus, the shear strain reaches a critical value as the shear stress tends to infinity. In the linearized theory x y

as T contradicting the original assumption of the theory that the displacement gradient and hence the strain is very small. We see that there is no such contradiction in the case of the model (2.14). 3.3. Torsion

Consider the deformation 1R2 2 Z3 1r2 2 z3 such that r 2 R2

2 8 Z2

z 2 Z5

(3.12)

Let us further suppose that the stress has the form T 2 T 1e ez 8 ez e 3 2

(3.13)

where e and ez are unit vectors along the and z directions and T is a constant. Once again, the equations of equilibrium are met automatically. A trivial calculation leads to 7 T

z 2

5 1 8 2T

(3.14)

In virtue of the constitutive relation, all of the other components of the strain are zero and this is consistent with the assumption (3.12). Since, we can find the shear strain from (3.12), equating that to (3.14) will lead to an expression for z .

NON-LINEAR ELASTIC BODIES EXHIBITING LIMITING SMALL STRAIN 129 We find that z 7 2 as T , i.e. we once again have a limiting strain. 3.4. Circumferential Shear

We consider the circumferential shearing of an annular cylinder of inner radius Ri and outer radius R0 . Consider the deformation 1R2 2 Z3 1r2 2 z3 such that r 2 R2

2 8 g 1R3 2

z 2 Z2

(3.15)

in a cylindrical polar coordinate system. Let us suppose that the stress has the form T 2 T 1r3 [er e 8 e er ] 5

(3.16)

We find that both the constitutive relation (2.14) and (3.16), as well as (3.15) speak with one voice and the only non-zero component of the strain is r . It follows from the equations of equilibrium that dT 2T 8 2 02 dr r

(3.17)

and thus T 1r3 2 T 1R3 2

C 5 R2

(3.18)

Next, 1 dg R 2 2 dR

(3.19)

1 dg 7 T 1R3 5 R 2 2 dr 1 8 2T 1R3

(3.20)

r 2 and by (2.14)

r 2

We first note that the maximum value that r can take is 7 2 . It also follows from (3.20) that dg 27 C 2 5 dR R R 2 8 2C

(3.22)

The solution for g 1R3 is obtained by integrating (3.22) and then enforcing the boundary conditions

130 K.R. RAJAGOPAL

Figure 1. Variation of angular displacement g with the radius R when displacements at the boundary are specified and Ri 2 055R0 5

g 1Ri 3 2 02

(3.24)

g 1R0 3 2 2

(3.25)

where Ri and R0 are the inner and outer radius of the annular region. A straightforward calculation leads to 37

C 2 g 1R3 2 ln 1 8 8 D2 (3.26) 2 R2 where

8

3 7 2

13e 2 C2 8 2 2 3 7

2 2 R0 e 3 Ri Ri2 R02

(3.27)

and D2

7

2

C 2 ln 1 8 5 Ri2

(3.28)

The function g, the angular displacement at the radial coordinate R, is portrayed in Figure 1 when the inner radius is half the outer radius and in Figure 2 when the inner radius is 0.99

NON-LINEAR ELASTIC BODIES EXHIBITING LIMITING SMALL STRAIN 131

Figure 2. Variation of angular displacement g with radius R when displacements at the boundary are specified and Ri 2 0599R0 5

times the outer radius (i.e. when the cylinder has a thin wall). It is very interesting to note that the structure of the displacement is different, being concave in one instance and convex in the other which immediately implies that the structure of the strain is different. The corresponding ratios of [1 r 317 3] for the two cases are plotted as a function of the radius R corresponding to these two cases in Figures 3 and 4, respectively. Since the product 17 3 can be chosen to be very small, the strain predicted in these two cases by the model is as small as we wish it to be. Moreover, we once again see that the manner in which the strain varies in the case of the thick-walled and thin-walled cylinder are both qualitatively and quantitatively different. In the first case it decreases, close to linearly, while in the second case it increases non-linearly. In fact, it might be more reasonable to prescribe the shear stress that is applied at the outer radius R0 . Thus, in addition to (3.24) we could prescribe 9 27

T dg 1R0 3 2 2 dR R0 1 8 2T9

(3.29)

where T9 is the shear stress at R 2 R0 . A simple calculation leads to C 2 T9 R02 2 with D once again given by (3.28). Thus, the angular displacement g 1R3 is given by

(3.30)

132 K.R. RAJAGOPAL

Figure 3. Variation of the strain r with the radius R when displacements at the boundary are specified and Ri 2 055R0 5

Figure 4. Variation of the strain r with the radius R when displacements at the boundary are specified and Ri 2 0599R0 5

NON-LINEAR ELASTIC BODIES EXHIBITING LIMITING SMALL STRAIN 133

Figure 5. Variation of the angular displacement with the radius R when displacement is specified at Ri and traction is specified at R0 and Ri 2 055R0 . (Note that the curves corresponding to the cases when T 2 102 000 and T 2 100 essentially lie on top of each other.)5

7

g 1R3 2 2

!

ln 1 8

" T9 R02 2 T9 R02 2 3 ln 1 8 5 Ri2 R2

(3.31)

Figures 5 and 6 show how the angular displacement g varies with the radius R in the case of a thick-walled and thin-walled cylinder, respectively. While the displacement is essentially linear in the thin-walled case, we note that in the case of a thick-walled cylinder the displacement can vary non-linearly with radius. 3.5. Telescopic Shearing

Let us next consider the deformation from 1R2 2 Z 3 1r2 2 z3, in a cylindrical polar coordinate system so that r 2 R2

2 2

z 2 Z 8 f 1R3 5

(3.32)

A trivial calculation shows that 123 2

1# # 2

0

0

f 1R3

0

0

0

f 1R3 0

0

$ $5

(3.33)

134 K.R. RAJAGOPAL

Figure 6. Variation of the angular displacement with the radius R when displacement is specified at Ri and traction is specified at R0 and Ri 2 0599R0 . (Note that the curves corresponding to the cases when T 2 102 000 and T 2 100 essentially lie on top of each other.)5

Let us suppose that the stress tensor is given by T 2 T 1R3 1er ez 8 ez er 3 5

(3.34)

It immediately follows from (2.14) and (3.18) that

r z 2

1 7 T 1R3 f 1R3 2 2 1 8 2T 1R3

(3.35)

and all other components the strain are zero consistent with (3.27). As before, the maxi of

mum strain possible is 7 2 . The equations of equilibrium reduce to dTr z Trr 8 2 05 dr r

(3.36)

Thus, Trr 2

C 2 r

(3.37)

NON-LINEAR ELASTIC BODIES EXHIBITING LIMITING SMALL STRAIN 135

and it follows from (3.31) and (3.33) that 27 C df 2 2 dR R 8 2C

(3.38)

which can be integrated to yield f 1R3 2 27 C ln R 8 2C 8 D2

(3.39)

where D is a constant. The boundary conditions f 1Ri 3 2 U2

(3.40)

f 1R0 3 2 02

(3.41)

and the solution for f 1R3 can be obtained by solving (3.35), (3.36) and (3.37) where U is the displacement at the inner radius. It follows that C is obtained as a solution to Ri 8 2C U C ln 2 2 (3.42) 27

R0 8 2C and D 2 327 C ln R0 8 2C 5

(3.43)

The solution f 1R3 is plotted in Figures 7 and 8. Interestingly, unlike the previous case of circumferential shearing, the axial displacements have the same qualitative features for both the thick- and thin-walled cylinders. 3.6. Circumferential and telescopic shear

In this case the deformation from 1R2 2 Z3 1r2 2 z3 takes the form r 2 R2

2 8 g 1R3 2

z 2 Z 8 f 1R3 2

(3.44)

and the strain is given by 123 2

0

1# # Rg 1R3 2 f 1R3

Rg 1R3

f 1R3

0

0

0

0

$ $5

(3.45)

136 K.R. RAJAGOPAL

Figure 7. Variation of axial displacement with the radius R when displacements at the boundary are specified and Ri 2 055R0 5

Figure 8. Variation of axial displacement with the radius R when displacements at the boundary are specified and Ri 2 0599R0 5

NON-LINEAR ELASTIC BODIES EXHIBITING LIMITING SMALL STRAIN 137

Let us suppose that the stress takes the form T 2 T1 1R3 1er e 8 e er 3 8 T2 1R3 1er ez 8 ez er 3 5

(3.46)

It follows from (2.14) that the assumption (3.38) is consistent with the expression (3.37) for the strain stemming from (3.36). Thus,

r

2

1 7 T1 1R3 Rg 2 1 21 2 2 1 8 2 T12 8 T22 2

(3.47)

r z

2

1 7 T2 1R3 f 2 21 2 1 2 1 8 2 T12 8 T22 2

(3.48)

and thus 5 2 1 2 6 T1 8 T22 4 4 2 7 2

5 7 1 8 2 1T 2 8 T 2 2 12 1 2

(3.49)

Thus 4 4 has a limit 7 . We do not find the solutions to T1 1R3 2 T2 1R3 2 g 1R3 2 or f 1R3 as our main interest is in showing that for the model being considered, the strain cannot exceed a critical value and will remain small being consistent with the spirit of the approximation that was used to linearize the non-linear strain, unlike the case of the classical linearized theory wherein for all of the problems considered the strain becomes unbounded as the stresses become unbounded. In all of the examples considered in this section, we note that the strain never exceeds a critical value that could be chosen to be arbitrarily small and can be decided by the modeler a priori.

4. CONCLUDING REMARKS The model (2.14) that we have considered exhibits a limiting strain in all of the examples that were considered, while the linearized strains in the corresponding problems within the context of the theory of linearized elasticity would blow up. It would be interesting to study the problems of an elliptic hole in a plate subject to say biaxial loading and then to consider the limit as the ratio of the minor to the major axes of the ellipse tends to zero, thereby modeling a crack. It is conjectured that the strain corresponding to the model (2.14) would 1 not exhibit the classical singularity at the crack tip. Of course, problems involving r crack propagation and fracturing of solids leads to dissipation in a region adjacent to the crack tip, which implies that the response in that region cannot be described by an elastic response. This has led to the use of a â€˜plastic zoneâ€™ adjacent to the crack tip. This dissipation

138 K.R. RAJAGOPAL

notwithstanding, the fact that the strains remain small makes the model a highly desirable model to study stresses near cracks and the model can be modified near the crack tip to allow for dissipation. It is always possible to include small strain inelastic response in addition to the small strain that stems from the elastic response with models that have limiting strain. One need not restrict oneself to the models that relate the linearized strain to the stress. It is possible to have limited stretch or limited strain, with the stretch and strain not being restricted to being small. For instance, an interesting counterpart to (2.14) and (2.17) would be the constitutive model 5 61 3 4

B 22 7 T 2 (4.1) 1 8 e391trT3 1 8

21 1 72 1 8 6 trT2 2 where 7, 9, and 6 are constants and have units so that the right hand side is dimensionless. Also, as before, we set delta to be unity. Another possible model is n 8 7 1 1 1 22 T2 B2 18 189 18 1 2 1 8 trT 1 8 trT2

(4.2)

respectively. In the case of the uniaxial extension problem, given a stress state of the form (3.1) we find that 2 11

39T B27 T 5 (4.3) 18 18e [1 8 T ] 2 Once again, we see that as T 2 B remains bounded. It is not our intent here to consider the models (4.1) and (4.2) in any detail, rather it is merely to point out that such models allow for limited stretch as the stress becomes arbitrarily large. Acknowledgement. K. R. Rajagopal is grateful to the National Science Foundation for supporting this research.

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Rajagopal, K. R. Elasticity of Elasticity, ZAMP, 58, 309â€“317 (2007). Truesdell, C. and Noll, W. The Non-Linear Field Theories of Mechanics, Antman, S. S. (ed.), Springer, Berlin, 2004. Rajagopal, K. R. On implicit constitutive theories. Appl. Math., 28, 279â€“319 (2003). Rajagopal, K. R. and Srinivasa, A. R. On the response of non-dissipative solids. Proc. R. Soc. A, 463, 357â€“367 (2007). Rajagopal, K. R. and Srinivasa, A. R. On a class of non-dissipative materials that are not hyperelastic. Proc. R. Soc. A, 465, 493â€“500 (2009). Bustamante, R. and Rajagopal, K. R. A note on plane strain and plane stress problems for a new class of elastic bodies. Math. Mech. Solids (2009), in press. Bustamante, R. Some topics on a new class of elastic bodies. Proc. R. Soc. A, 465, 1377â€“1392 (2009).

NON-LINEAR ELASTIC BODIES EXHIBITING LIMITING SMALL STRAIN 139

[8]

Horgan, C. O. and Saccomandi, G. Phenomenological hyperelastic strain-stiffening constitutive models for rubber. Rubber Chem. Technol., 71, 152–169 (2006). [9] Ogden, R. W., Saccomandi, G. and Sgura, I. On worm-like chain models within three dimensional continuum mechanics framework. Proc. R. Soc. A, 462, 749–768 (2006). [10] Rajagopal, K. R. and Saccomandi, G. The mechanics and mathematics of the effect of pressure on the shear modulus of elastomers, Proceedings of the Royal Society, London, 465, 3859–3874 (2009). [11] Spencer, A. J. M. Theory of Invariants in Continuum Physics, Vol. 2, Eringen, A. C. (ed.), Academic Press, New York, 1975.