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International Journal of Material Scie nce Vol. 2 Iss. 4, Dece mbe r 2012

Modeling of Large Deformations of Hyperelastic Materials E. Boudaia#1 , L. Bousshine*2 Department of Mechanical Engineering, Faculty of Science and Technology,

#

BP 523, Mghrila, 23000 Béni Mellal, Morocco Laboratoire des Technologies de Constructions et des Systèmes Industriels, ENSEM,

*

BP 8118, Oasis, Casablanca, Morocco boudelhassan@yahoo.fr, 2lbousshine@ yahoo.fr

1

Abstract The e lastomer prope rties (large de formations, damping) make the ir use more and more common in industries such as aerospace, automotive , construction and civil e ngineering or e ven the ente rtainme nt industry... Mode ling the behavior of such materials is highly nonlinear, the nonlinearities are both geome tric (due to large de formations impose d) and be havioral (behavior laws use d are nonlinear). This paper prese nts a de taile d de scription of the nume rical imple mentation of incompressible isotropic hype re lastic be havior. In this study, the analysis of large de formation proble ms of Ogde n’s hypere lastic is base d on the finite e leme nt method (FEM) and mathe matical programming. To solve the proble m of balance , we suggest using the combine d Ne wton-Raphson/Arc-le ngth proce dure . A typical e xample is prese nte d to illustrate the pe rformance of this formulation.

medium and medium formulation based on isotropic hyperelastic tensor structure. In a second step, the most widely used hyperelastic potential is exposed. Description of Large Stra ins Elastomers, with hyperelastic behavior, usually work   in large strains. Note x and X positions vectors of a particle P of a deformable body in the current configuration (where the solid occupies the volume Ω ) and initial (where the solid occupies the volume Ω0 ).

  

We will use the same reference (o, e1 , e2 , e3 ) for the initial configuration C0 given at time t 0 and deformed configuration Ct at time t (Fig.1).

Keywords Elastomers; Hyperelasticity; Large Strain; FEM; Newton-Raphson Algorithm; Arc-Length Scheme

Introduction The theoretical studies of hyperelastic materials have mobilized many researchers for decades; see eg the work of Treloar [1], Gent [2] and Frakley [3]. However, the first important works were proposed by Mooney [4] and Rivlin [5]. Valani and al. [6] proposed to write the strain energy in a form separated following the main directions. This led to the widely used model of Ogden [7, 8]. Recently, different models have been developed for hyperelastic materials especially in biomechanics (Zulliger and al. [9]). The objective of this paper is to present the model most widely used hyperelastic behavior and in particular the isotropic model. To do this, the different results of basic mechanics of the major changes are outlined in the first place. Are recalled and the various tensor quantities used in the modeling of continuous

FIG. 1 INITIAL AND DEFORMED CONFIGURATIONS

For a Lagrangian description, the movement of the body can be defined relative to a reference configuration C0, by a vector function:

C0 → Ct Φ:     X  x = Φ ( X , t ).

(1)

 By introducing the displacement vector u and we write the equation (1) in the equivalent form:     (2) x= X + u ( X , t ). Order to describe the geometric transformations associated with these large strains, we introduce the deformation gradient tensor F defined by:

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International Journal of Material Scie nce Vol. 2 Iss. 4, Dece mbe r 2012

  ∂Φ i ( X , t ) ∂xi ∂u Fij ( X , t= ) = = δ ij + i , ∂X j ∂X j ∂X j

(3)

σ=

1 FSF T . det( F )

(9)

δ ij denotes the Kronecker symbol.

For the Ogden model [7] for incompressible materials, the energy density is expressed in terms of eigenvalues λ j (j = 1, 2, 3) of the Cauchy-Green tensor

Or equation (3) can be written in matrix form:

right by:

Where

F = I + ∇u ,

(4)

Where

W (λ1 , λ= 2 , λ3 )

m

µk

k

k

∑ α ( λα

k

1

)

+ λ2α k + λ3α k − 3 ,

(10)

I is the unit tensor,

where the constants are the material parameters and m is the total strain energy terms.

∇ is the gradient tensor of displacements.

We recall that the incompressibility of the material

Due to the large displacements and large rotations, the strain tensor of Green-Lagrange E was adopted to describe the nonlinear relationship between the deformation and displacement:

results = in det( F ) 1= ou λ1λ2 λ3 1. Hence, the energy

= E

1 (C − I ), 2

m

µk 

k

k

∑α

 1 α α  λ1 k + λ2 k + α k α k − 3 . λ1 λ2  

In the case of a hyperelastic law, there is a strain energy density W which depends on one of the strain tensor and whose derivative gives the second stress tensor of Piola-Kirchhoff S:

∂W ∂W = 2 . ∂Eij ∂Cij

(6)

In the particular case of isotropic hyperelasticity [10], equation (6) can be written as follows:

 ∂W  C, −  ∂I 2 

(7)

I i (i = 1, 2, 3) are the invariants of the Cauchy-Green tensor on the right, C, such as:

1 2  I1 − tr (C 2 )  and I 3 = I1 = tr (C ) ; I 2 = det(C ), 2

Solving the problem balance of incompressible hyperelastic is reduced to problem of minimizing the functional Ψ :

(

(8)

)

 Min =  Ψ (u , S ) S .E (u ) − f .u d Ω − ∫ t .ud Γ  ∫   Ω Γt     Conditions of incompressibility,  Subject to  (u u on Γu ). =  Boundary conditions 

Ω is an isotropic structure, subject to surface

forces t on the surface Γ t , f is the gravity forces, is displacements imposed on the surface Γ u .

The nonlinear equations (12) are solved numerically with the finite element method whose the approximation of the displacement field is defined by the relation:

u (x) = N (x)U with ε = ∇( N (x))U ,

Where

Where

tr(•) and det(•) indicate, respectively, the trace and the determinant of a tensor.

x= < x, y, z > are the nodal coordinates,

Note that S has no physical meaning but it is symmetric and purely Lagrangian, and from which one can determine the expression of the Cauchy tensor σ :

(12)

Where

u

Where

120

(11)

Variational Formulation

C = F T F is the strain tensor of Cauchy-Green right.

 ∂W −1  ∂W ∂W = + I1 S 2  I3 C + I I ∂ ∂ ∂I 2 3  1 

W (λ1 , λ= 2 , λ3 )

(5)

Where

= Sij

density can be expressed only by λ1 and λ2 :

(13)

N(x) is the shape functions matrix,

U is the vector of unknown nodal displacements. Therefore, the functional Ψ takes the discretized form as follows:


International Journal of Material Scie nce Vol. 2 Iss. 4, Dece mbe r 2012

= Ψ (U )

∫ ( S .E ( N .U ) − N Ω

T

)

. f .U d Ω − ∫ N T .t .Ud Γ. Γt

(14)

Thus, the discrete system (14) can be solved iteratively by using the combined Newton-Raphson/Arc-length procedure (see [11] for details on the implementation of this method in the context finite element). Numerical Example In this example, we consider a plate with a central circular hole in the stress plane and tension is sought under a constant displacement up to 100% elongation. Geometry (L × D = 25.4 × 14.24 cm²) with a unit thickness is shown in Fig. 2. For more details see reference [12]. Due to symmetry of geometry, only one quarter of the plate is modeled, as shown below.

FIG. 4 ISOVALUES OF STRESS ALONG THE X AXIS (TOP) AND ALONG THE Y AXIS (BELOW).

Conclusions FIG. 2 GEOMETRY AND LOADING (LEFT) AND BOUNDARY CONDITIONS WITH MES HES (528 ELEMENTS T3) (RIGHT)

A computational method by using the mathematical programming and the analysis approach hyperelastic is developed here to investigate the large deformation. A finite elem ent model in conjunction with the combined Newton - Raphson / Arc - length procedure was adopted allowing investigating interesting applications. The obtained results illustrate the effectiveness of our approach developed. This work can be extended in the future by taking into account other parameters as frictional contact with heat transfer. REFERENCES [1]

L. R. G. Tre loar, The Physics of Rubber Elasticity. 3rd e dition, Oxford unive rsity press, 1975.

[2]

N.

Ge nt, “Rubbe r and Rubber Elasticity, a Revie w.”

Journal of Polyme r Scie nce : Polymer Symposia, vol. 48, pp. 1–17, 1974. [3]

P. K. Frakle y and A. R. Payne, The ory and Practice of Enginne ring with Rubbe r. Applie d Scie nce Publishers, London, 1978.

FIG. 3 DEFORMATION ALONG THE X AXIS (TOP) AND ALONG THE Y AXIS (BELOW).

[4]

M. Mooney, “A Theory of Large Elastic De formations.”

Journal of Applie d Mechanics, vol. 11, pp. 582–592, 1940.

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International Journal of Material Scie nce Vol. 2 Iss. 4, Dece mbe r 2012

[5]

R. S. Rivlin, “Large e lastic De formations of Isotropic

Inde pe nde nt

theory, Philosophical Transactions of the Royal Socie ty

Orie ntation of the Spatial Basis Attache d to De forming

of London, Series A, vol. 24, pp. 1379–397, 1948. [6]

of

the

Function of a Hype re lastic Mate rial in Terms of the

Me thods for Plasticity: The ory and Applications. John Wiley & Sons Ltd, 2008. [12] E. A. Ne to De Souza. Perlc Djordje and D. R. J. Owe n,

R. W. Ogde n, “Large De formation Isotropic Elastic - on

“Finite Elasticity In Spatial De scription: Linearization

the

Aspe cts with 3-D Me mbrane Applications.” Inte r. J. For

Corre lation

of

The ory

and

Expe riment

for

Royal Socie ty A, vol. 326, 565–584, 1972. R.W. Ogde n, “Elastic De formation of Rubber-Like Solids.” The Ronde y Hill 60th Annive rsary Volume , Editor: H.G. Hopkins and M.J. Se we ll, Pe rgamon, Oxford, 1982. M. A.

Zullige r, P. Fride z, K.

Hayashi and N.

Ste rgiopulos, “A Strain Ene rgy Function for Arteries Accounting For Wall Composition and Structure.” Journal of Biomechanics, vol. 37, 989–1000, 2004.

122

Prese rving

Me dium.” Comput. Mech. Vol. 21, 1053-1060, 2007.

Incompressible Rubberlike Solids.” Procee dings of the

[9]

for

[11] E. S. Neto, D. Perié and D. R. J. Owe n, Computational

pp. 2997–3002, 1967.

[8]

Algorithm

K. C. Valanis and R. F. Lande l, “The Strain-Ene rgy Exte nsion Ratios.” Journal of Applie d Physics, vol. 38,

[7]

[10] F. Peyraut, Z. Q. Fe ng and N. Labe d, “A Material-

Mate rials IV.” Further de ve lopments of the gene ral

Num. Me th. In Eng., vol. 38, 3365-3381, 1995. El Hassan Boudaia is an associate Professor at the Mechanical Engineering De partme nt of Faculty of Scie nce and Te chnology of Be ni Me llal, Morocco. His Ph.D thesis in 2007 was on the subject of a meshless and finite e le ment me thods analysis for e lasto-plastic contact problems with friction. His curre nt research activities are multidisciplinary and focus mainly on design approaches and re liability analysis, especially in structural e ngineering. Using the e lasto-plastic analysis and the meshless or the finite e le ment me thod, he has publishe d with his coworke rs many works on the re liability assessme nt of structures, espe cially in the non-associate d e lasto-plasticity area.


Modeling of Large Deformations of Hyperelastic Materials