IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-3008, p-ISSN:2319-7676. Volume 9, Issue 5 (Jan. 2014), PP 43-54 www.iosrjournals.org

Dynamic Behavior of Functionally Graded Solid Sphere Subjected To Thermal Load S.P. Pawar1, K.C. Deshmukh2, G.D. Kedar2 1

S. N. Mor and Smt. G. D. Saraf Science College Tumsar Dist- Bhandara (MS), India. 2 Department of Mathematics, R.T.M. Nagpur University, Nagpur (MS.) India.

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Dynamic Behavior Of Functionally Graded Solid Sphere Subjected To Hermal Load inside the sphere from the heat conduction equation, it is regarded as a known function and is introduced in the thermoelastic equations to determine the thermal stresses. The objective of this paper is to provide an effective mathematical model to analyze the transient response for FGM sphere subjected to spherically symmetric thermal load.

II.

Formulation of The Problem

Consider the FGM solid sphere of radius a . The material properties of the sphere are assumed to be function of radius. Assume the material properties of the FGM as a mixture of ceramic and metal vary exponentially in the radial direction as,

k R   k m e pR

(1)

 R    m e vR

(2)

c p R   c pme wR

(3) In which p , v and

p  ln

kc km

v  ln

c m

w

are the dimensionless nonhomogeneity parameters ,

(4)

w  ln

c pc c pm

The subscripts c and m means the properties of ceramic and metal respectively. According to (4) the centre of the sphere is metal rich and outer surface is ceramic rich. Following dimensionless parameters are introduced,

r a T  T  Tw  T t  t0 R

t0 

a 2  m c pm km

(5) Where a  Radius of the sphere

t 0  Reference time k  Thermal conductivity Tw  Surface temperature

T  Ambient temperature T  T r, t   Temperature t  Time   Density c p  Specific heat

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Dynamic Behavior Of Functionally Graded Solid Sphere Subjected To Hermal Load Heat Conduction Problem Consider the transient temperature distribution in the FGM sphere. Initially the sphere is assumed to be at uniform temperature and equal to the ambient temperature T . The surface of the sphere is maintained at the temperature Tw . The heat conduction equation without internal heat generation for one dimensional unsteady

T r , t  stated in spherical coordinates r ,  ,   is given as [26],  2T T  2 1 dk   r c p r  T     2 r  r k r  dr  k r  t r

state distribution of temperature

(6) Subjected to the initial and boundary condition, T r , t  T at t  0 (7) T r , t   Tw at r  a

 

(8)

T 0 r

at r  0

(9) Substituting the material expressions (1-3) and the dimensionless parameters (5), the problem (6-9) is transformed into dimensionless form as,

 2   2  v  w p R     p  e 2 R  R  R  (10) The dimensionless initial and boundary conditions are,  R,  0 at   0 (11)  R,  1 on R  1 (12)

 0 R

Setting p ,

at

v

R0

(13) and w  0 the equation reduces to the case of a homogeneous material.

Thermoelasticity problem For one dimensional problem in the spherical coordinate system

r, ,  ,which means spherically

symmetric problem, the displacement technique is extensively used. The properties in spherical coordinates and  direction are identical. The only nonzero components of the displacement vector denoted by u .The strain displacement relation are as [1],

u r r  which can be

u r u  r

 rr 

 

(14) The thermoelastic stress-strain relations are,

 rr  c11 rr  2c12   c11  2c12 T    c12 rr  c11  c12    c11  2c12 T

(15) Where c ij are the elastic constants,

 , the thermal expansion coefficient , T  T  T

The thermoelastic constants are radially dependent so that

c11  c11 r  , c12  c12 r  and

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

  r . When the material is isotropic then, E 1       2 1   1  2  E c12   1   1  2  E  21   

c11 

(16) Where E is the Young’s modulus and  is Poisson’s ratio assumed to be constant. The stress components in terms of the displacement component u , using (14), (15) and (16) may be obtained as [1],

E  1    u  2 u  1   T   1   1  2   r r  E  u u         1   T   1   1  2   r r 

 rr   

(17) Two of the three equations of stress equilibrium are identically satisfied; the third takes the form as [29, page 277],

 rr 2  2u   rr       2 r r t (18) By inserting equation (16) in (15) and then in (18) the dynamic equation of motion for displacement u is obtained as,

 2 u u  2 1 dE  u  2 1 dE 2  1     1 dE  1   1  2    2 u          T    T      1    E t 2 E dr  r 2 r  r E dr  r  1   E dr r  1    r

(19) The solution of u is known from (19), the stresses can be determined from (17). It is expected that the stresses at the centre r  0 are finite; therefore the displacement u must vanish when r  0 . The sphere is free of surface tractions. So the initial and boundary conditions becomes,

ur , t   0 u 0 r ur , t   0  rr r, t   0

t 0 t 0 r 0

r a

(20) Introduce the following dimensionless parameters for displacement and stresses as [15],

u a m Tw  T   rr Sr  c11m m Tw  T    S  S   c11m m Tw  T 

U

(21) www.iosrjournals.org

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Dynamic Behavior Of Functionally Graded Solid Sphere Subjected To Hermal Load The thermoelastic constants E and

E r   E m e

 are considered as,

sR

 r    m e qR (22)

Where

s  ln

Ec Em

q  ln

c m

(23) The special case of a homogeneous material is obtained by letting s, q  0 .The Poisson’s ratio υ is assumed to be constant. Now by using the parameters in(21) ,the constants in (22) and the parameters defined in (5),the equation (19) and (17) can be expressed as, 2 2 d 2U dU  2  U  2 2  1   qR  d  1   1  2  ma v  s R  U     s  s   e   q  s  e  1    Emt02 dR2 dR  R  R 1   R  1    dR  2 

(24)

2 U  1   s  q R  U S r  e sR   e    R 1   R  1   (25)

1 U  1    s  q R   U S  e sR   e   1   R 1   R  1   (26) The dimensionless initial and boundary conditions are, U R,  0 at   0

U R,   0 at   0  U R,   0 at R  0 U 1,   2 U 1,   1   e q R 1 1 (27)

III.

Analytic Solutions

Solution of the Temperature Field A fully implicit finite difference scheme is used to evaluate the dimensionless temperature govern by equations (10-13).A backward-difference representation is used for the time derivative and central difference formulas are employed for other derivatives. The implicit schemes are unconditionally stable for any size of time step, but accuracy of the solution is only first-order in time. Hence, small time steps are needed to ensure reasonable accuracy of results. The differential equation (10) is represented in a finite difference form as,

in11  2in 1  in11

R 2

in11  in11  2 2R

n 1 n  v  w p iR i  i  p  e    iR 

(28)

Rearranging the terms one gets,

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 1 1 p  n 1  2 e v  w p iR  n 1  1 1 p  n 1    i            i  1 2 2 2  R  iR R 2R   R   R  iR R 2R i 1         e v  w p iR n  i  i  2,3,4,5..........N (29) For a solid sphere the equation (10) easily seen singularity at R  0 , R and

i 1 ( R  0 ). Hence we have a

 becomes zero at the node R

0 ratio at the origin. By application of L’Hospital’s rule, it can shown that the ratio 0

has the following determinate form,

 1      R R  i 1

      R  R    2     2    R  R 0 R R

(30) Thus (10) becomes,

3

 2   p  2 R  R

(31)

Additional relationship is obtained by discretizing equation (31) at R  0 . In order to use a second order accurate central difference formula at R  0 , a node is needed to the left of the origin R  0 . This is achieved by considering a fictitious node “0” at fictitious temperature  0 located at the distance R to the left of node i  1 on R  axis. The resulting finite difference approximation of (31) at i  1 becomes,

3

in11  2in 1  in11

R 2

p

in11  in11 2R

 e v  w p iR

in 1  in 

(32) As the centre of the sphere is metal rich, therefore’ v  w  p  0 by (4) Putting i  1 in (32) one gets,

3

0n 1  21n 1  2n 1

R 2

1n 1  1n 

(33) Where the fictitious temperature

0n 1 is determined by utilizing the symmetry condition at the node i  1 as,

in11  in11 2n 1  0n 1     0    2R 2R  R  i 1 (34)

0n 1  2n 1

Therefore (33) becomes,

3

2n 1  21n 1  2n 1

R 2

1n 1  1n 

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 6 1  n 1 6 1 n  1   2n 1   1 2 2  R          R   (35)

for R  0

Putting i  N in (29) one gets

 1 1 p  n 1  2 e v  w p  NR     R 2  NR R  2R  N 1    R 2      v  w p  NR e   Nn  Since  Nn 11  1 boundary condition  1 p  n 1  1 2 e v  w p NR     R 2  NR R  2R  N 1    R 2     

 n 1  1 1 p  n 1  N      R 2   NR R  2R  N 1    

 n 1 p  e v  w p NR n  1 1  N     N     2      N  R  R 2  R     R  

(36) e v  w p  NR n  1 1 p    N     2    N  R  R 2  R    R  

DN  

(37) Thus the finite difference form of equation (10-13) in matrix form is obtained as, 0 0 0 ... 0  1n 1   D1   B1 C1   A 0 0 ... 0   2n 1   D2   2 B2 C 2 0 A3 B3 C 3 0 ... 0  3n 1   D3        

0 :

0 :

: :

: :

: :

: :

:

:

:

:

:

:

0

0

0

0

0

AN

    0  :    :  :  :   :      C N 1   Nn 11   D N 1  B N   Nn 1   D N 

(38)

Where,

B1   C1 

6

R 

2

1 

6

R 2

D1  

1 1n 

Ai 

1

Ci 

1

1

p

iR R

1

p 2R

R 2 iR R 2R 2 e v  w p iR Bi     R 2

R 

Di  

2

e v  w p iR n i 

DN  

i  2,3,4.....N

e v  w p  NR n  1 1 p    N     2 NR R 2R    R 

The above equations represent a system of N+1 equations with N+1 unknowns for each time step.The resulted matrix is tridigonal. www.iosrjournals.org

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Dynamic Behavior Of Functionally Graded Solid Sphere Subjected To Hermal Load Solution of the Thermal Stresses A fully implicit scheme is used to evaluate the dimensionless displacement govern by the equations (24) and (27). The backward-difference scheme is used for time derivative, while central difference formulas are employed for other derivatives. The implcit scheme is unconditionally stable for any time step size, but the accuracy of the solution is only first-order in time. Hence, small time steps are needed to ensure reasonable accuracy of results. The differential equation (24) is represented in a finite difference form as,

U in11  2U in 1  U in11

R 2

n 1 U in11  U in11  2 2   U i  2  s  s    2R iR   iR  iR 1  

 1   1  2   m a 2 v  s iR U in 1  2U in  U in 1  1   qiR in11  in11 e   in 1 q  s   e   1    E m t 02 1  2  2R   

Rearranging terms one gets,

 1 1   1  2  ma 2 ev  s iR U n1 1 s  n 1  2 2s 2   U      i   i 1  2 2 2 1  Emt02  2   R  iR R 2R   R  iR 1    iR 

 1 1   1  2  ma 2 ev  s iR U n1  2U n 1 s  n 1 1   q iR  in11  in11 n 1         U  e  q  s   i  i  2  i 1 1   1    Emt02  2 i  2R   R  iR R 2R 

i  2,3,4,.....N (39) Putting i  N  1 one gets,  1 1   1  2   m a 2 e v  s  U n 1  1 s  n 1  2 2s   U  2      N  2 1    E m t 02  2  N 1 R 2R  R 2 1     R   n 1 n 1  1 1   1  2   m a 2 e v  s U n 1  U n 1 s  n 1 1   q  N  2   N  1       U  e  q  s      N  2 2 1    E m t o2  2 N 1 N 1 R 2R  1  2R  1  R 

(40) The value of U Nn 12 and

 Nn 12 can be obtained from (26) and (28), the equation (40) written as,

 2 4  1  1   1  2   m a 2 e v  s  n 1 n 1 U  2   1     U N 1  1    E m t o2  2  R 2 N  R 2 1    R   1   1  2   m a 2 e v  s 1 q  2  1   2e q  1   Nn 1 e q  2   U Nn 11  2U Nn 1   1    1  R  1   R  2  2  p R  E m t 02  2 2

(41) From this equation one can find the values of AN 1 , B N 1 and Dn 1 The finite difference form of equation (24) and (27) in Matrix form is given as, 0 0 0 ... 0  U 2n 1   D2   B2 C 2   A 0 0 ... 0  U 3n 1   D3   3 B3 C 3 0 A4 B4 C 4 0 ... 0            :    ... 0         :                         C N  U Nn 1   D N    0 0 0 0 0 AN 1 B N 1  U Nn 11   D N 1  

(42)

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Ai 

1

R 

2

1 s  iR R 2R

 2 1   1  2   m a 2 e v  s iR  2 2s Bi        2 1    E m t 02  2  iR 2 iR 1     R 

Ci  Di 

1

R 

2

1 s  iR R 2R

 1   1  2   m a 2 e v  s iR 1   q iR  in11  in11 e  in 1 q  s   U in 1  2U in  2 2 1    1 2R E m t 0    

i  2,3,4.........N 2 AN 1  R 2

 2 4  1  1   1  2   m a 2 e v  s  B N 1   2   1    2 1    E m t 02  2  R  1    R    1   1  2  m a 2 ev  s 1 q 2  1   2eq  1  Nn 1   DN 1  e q  2  U Nn 11  U Nn 1  1    1   R  1   R  2  2  p R  Emt02  2

The components of stress can be calculated from equations (25-26) by using the temperature and displacement histories already obtained. The finite difference form of stress function components at any time for nonzero values of n can be written as, U n  U in1 2 U in  1    s  q iR n S r  e s iR   i 1  e i  2R 1   iR  1    (43)   U in1  U in1 U in  1    s  q iR n 1 S   e s iR    e i  2R 1   iR   1   1  

(44)

i  2,3,4.....N

The values of stresses at the centre and on the curve surface of sphere are obtained as follows, At the centre, R  0 , i  1 , by using the L’Hospital’s rule the finite difference forms can be obtained as,

S r R  0, 

j

   1    U

n 2

  1n  

   1    U

n 2

  1n   

 1    R

(45) S  R  0, 

j

 1    R

(46) At the surface, R  1 ,

S r R  1, j   0

(47) S R  1, 

i  N  1, the finite difference form of stress components becomes, (Traction free surface)

  e 1    2 U 2

s

j

(46)

1   

2

n N 1

IV.

1   1  2  e s  q 1   2

(48)

Results and Discussion

For the numerical calculations, consider the functionally graded solid sphere has the material properties of ceramic ZrO2 at the heated circular surface R  1 and alloy Ti-6A1-4V.These material composites vary exponentially in the radial direction. The thermo mechanical properties of this material at room temperature are[15],

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Dynamic Behavior Of Functionally Graded Solid Sphere Subjected To Hermal Load Material

Young’s modulus

Conductivity

k , W / mK Metal, ZrO2 Ceramic, Ti-6A1-4V

2.09

151

7.5

116.7

Density

10  10 6 9.5  10 6

1/3

5.5  10 3 4.43  10 3

 , 1/ K

E , GPa

Poisson’s Ratio

Coefficient of thermal expansion

Specific heat

 , kg / m

1/3

3

c p , J / kgK 418 560

Set the value of

a  1m, T  0, Tw  10K

The nonhomogeneity parameters for this FGM are p  1.278 , v  0.216 , w  0.292 , s  0.258 , q  0.051 1 0.9 0.8

Dimensionless Temperature

0.7 0.6 t=0.01 0.5

t=0.1 t=0.2

0.4

t=0.35 0.3

t=0.5 t=1.0

0.2

t=2.0 0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

R

Figure 1: Variation of Dimensionless Temperature along radius 1.5

t=0.01 t=0.1 t=0.2

Dimensionless Displacement, U

1

t=0.35 t=0.5 t=1.0 t=2.0 0.5

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

R

Figure 2: Variation of Dimensionless displacement along radius 1.4

1.2

1

0.8 t=2.0 0.6

t=1.0 t=0.5 t=0.35

0.4

t=0.2 t=0.1

0.2

t=0.01 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

R

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Dynamic Behavior Of Functionally Graded Solid Sphere Subjected To Hermal Load 1.4 1.2 1

Dimensionless Tangential stress

0.8 0.6 0.4 t=0.01 0.2

t=0.1 t=0.2

0

t=0.35 t=0.5

-0.2

t=1.0 -0.4

t=2.0

-0.6 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 4: Variation of Dimensionless tangential stress along radius The ceramic surface of FG sphere is suddenly heated from the initial temperature

T  0 to

Tw  10 K . Fig 1 shows the temperature distribution in the sphere for different values of the dimensionless Fourier time  under heating process. For time   0.01,0.1,0.2,0.35,0.5,1.0,2.0 the temperature distributions are shown. For lesser values of times the temperatures gradually increase from centre to surface of sphere. As time progresses the temperature value tends towards steady state   1 .For very short time   0.01 the variation of temperature is observed near the surface and gradually it shifts towards centre. Fig 2 shows the transient displacement variation along radius of the sphere. In the beginning of heating process the variation in displacement has significant picture and it becomes linear as the time progresses. Fig 3 represents the radial stress variation along radial direction. For time   0.01 , after sudden change in the surface temperature ,there is a significant variation in radial stress values. Stresses are tensile at the centre and compressive towards surface. As time progresses the stress values distribution approaches to the steady state and extreme stress gradients occur at the heated outer surface. The tangential stress function gives tensile nature throughout in the beginning of heating process. But the nature changes as time progresses. The stresses are tensile at the centre and compressive at the surface. And it approaches to steady state as time progresses further.

V.

Conclusions

In this work the transient temperature distribution, displacement and thermal stresses are obtained by using fully implicit finite difference scheme. The results are presented for the FGM consists of ZrO 2 and Ti6A1-4V.The stresses are computed numerically by using the known temperature obtained independently. The transient results shown are illustrated graphically for   0.01,0.1,0.2,0.35,0.5,1.0,2.0 .For lesser values of Fourier time  the temperature has significant gradient and it reaches to steady state as time progresses further. The displacement has the significant variation for smaller times and it varies linearly as time progresses. Its effect on stresses gives the result which agreements the displacement. For   0.01 the radial stress is tensile at the centre and compressive towards the surface, the stresses gives the significant variation for the smaller values of  ..As time progresses the radial stress function change to compression at the surface. And the stresses in region close to the outer surface suddenly drop to zero as per the induced condition on the boundary. The tangential stress function gives tensile nature throughout in the beginning of heating process. . But the nature changes as time progresses. The stresses are tensile at the centre and compressive at the surface. This mathematical model is very useful in engineering field to study the transient thermal stresses in FGM solid spherical tools.

Acknowledgement The authors are thankful to University Grants Commission, New Delhi to provide the partial financial assistance under major research project scheme.

References [1] [2] [3] [4]

Noda N., Hetnarski R.B.and Tanigawa Y., Thermal Stresses, Taylor and Francis, New York, 302, 2nd Ed: (2003). Obata Y. and Noda N., Steady State Thermal stresses in a hollow circular cylinder and a hollow sphere of a functionally graded material, J. Thermal Stresses,17(3), 471- 487, (1994). Ootao Y., Akai T. and Tanigawa Y., Three dimensional Transient thermal stresses Analysis of a non-homogeneous hollow circular cylinder due to moving heat source in the axial direction , J. Thermal stresses vol. 18, 497-512,(1995) Lutz M. P., Zimmerman R. W., Thermal stresses and effective thermal expansion coefficient of A Functionally Graded Sphere, J. Thermal Stresses,19,39-54, (1996)

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Dynamic Behavior Of Functionally Graded Solid Sphere Subjected To Hermal Load [5] [6] [7]

[8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]