Dimensional Two Steady State Thermal and Mechanical Stresses of a Poro-FGM Spherical Vessel

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Procedia - Social and Behavioral Sciences 46 (2012) 4880 – 4885

WCES 2012

Dimensional two steady state thermal and mechanical stresses of a Poro-FGM spherical vessel Sajad Karampour a * a

Department of mechanical engineering, Omidiyeh Branch, Islamic Azad University , Omidiyeh, Iran

Abstract In this study, an analyti tate a functionally graded porous material hollow sphere (FGpm). It is assumed that properties of poro, and FGM material is changed through thickness according to power law functions. heat conduction equation is obtained for obtaining temperature distribution and navier equations analytically using legendre polynomials and Euler differential equations system for investigating displacements changes and stress and potential functions for different indices power indices. © 2012 Published by Elsevier Ltd. Selection and/or peer review under responsibility of Prof. Dr. Hüseyin Uzunboylu Keywords: FGM; Poroelastic; Hollow Sphere; Thermoporoelasticity

1.Intruduction Functionally graded materials (FGMs) are made of a mixture with arbitrary composition of two different materials,and the volume fraction of each material changes continuously and gradually. The FGMs concept is applicable to many industrial fields such as chemical plants, electronics, biomaterials and so on[1]. Thick hollow sphere analysis made of FGM under mechanical and thermal loads and in asymmetric and twowas conducted investigating navier equations and using legendre polynomials [2]. Ootao and Tanigawa derived the threedimensional transient thermal stresses of a non-homogeneous hollow sphere with a rorating heat source [3]. Jabbari presented the analytical solution of one and two-dimensional steady state thermoelastic problems of the FGM cylinder [4].Two-dimensionalnon-axisymmetric transient mechanical and thermal stresses in a thick hollow cylinder is presented by Jabbari et al [5]. Porous spheres of nanometer to micrometer dimensions are being pursued with great interest because of several possible technical applications in catalysis, drug delivery systems, separation techniques, photonics, as well as piezoelectric and other dielectric devices [6-7]. The study of the thermomechanical response of fluid saturated porous materials is important for several branches of engineering [8-12]. Inspite of conducted studies on spherical and cylindrical vessel made of FGM to obtain mechanical displacements and mechanical and thermal stresses, it has not done any study on composition of poro and FGM materials. This study investigates the effect of Fluid trapped in the porous medium is located in undrain conditions. 2. Analysis Consider a thick spherical vessel of inside radius a and outside radius b made of poro FGM. It is assume that the mechanical and thermal loads and their associated boundary conditions are such that the stress field is a function of variables r and . For the assumed condition, the strain displacement relations are 1 1 u 1 u u,r , ( u, ), (1) cot , r rr , , ,r r r 2 r 2 r r * Corresponding author. Tel.: +98-916-737-3814; fax: +98-652-322-2533. E-mail address: [email protected]

1877-0428 © 2012 Published by Elsevier Ltd. Selection and/or peer review under responsibility of Prof. Dr. Hüseyin Uzunboylu doi:10.1016/j.sbspro.2012.06.353

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Sajad Karampour / Procedia - Social and Behavioral Sciences 46 (2012) 4880 – 4885

where u and v are the displacement components along the r- and graded porous material hollow sphere (FGpm) can be express C33 rr C13 C13 p rr z r T (r , ), rr C11

Where

ij

C13

and

ij

C12

rr

p

z T (r, ),

are the stress and strain tensors, T

- directions, respectively.the linear constitutive relations for a functionally

C11 2C44

r

C13

C12

rr

,

r

p

z T (r , ) (2)

0

r

is the temperature distribution determined from the heat conduction equation, z i and is

r,

the coefficients of thermal expansion in effective stress, Cij are the elastic constants, and M are Biots coefficient of effective stress and Biots Moduls. there for , z C11 (3) z zr 2m , z r C33 r C13 C13 r C12 (C11 C12 C13 ) C13 (C33 2C13 ) Pressure equation in porous environment (4) p M( ) For undrain condition M M ( rr ) (5) 0, p and are variation of fluid content and volumetric strain. The equilibrium equations, disregarding the body forces and the inertia Where terms, are 1 1 1 1 (6) (( ) cot 3 r ) 0 (2 rr 0, r ,r , rr , r r , r cot ) r 2 r r Substituting Eqs.(5) in to Eqs.(2) lead to * * * * * * C33 C13 C13 z r T (r , ), C13 C11 C12 z T (r , ) rr rr rr r * C13

rr

* C12

* C11

Where 2 * C11* C11 M , C22 With this hypothesis that

Cij*

z T (r , ),

2

C22

* M , C33

C33

2

M , C13*

(7)

r

2

C13

M , C12*

2

C12

M

(8)

Cij*r m

(9)

Employing a change of variable cos equations) are obtained as 1 2 C13* (m 1) u, rr (m 2) u, r ( r r2 C33* ( v, Sin

(v,rr

2C44

r

v cot ) ,r

1 (m 2)v,r r 2 1/ 2

(1

)

r

2

(m 2

1 C13* (m 1) ( r2 C33*

1 C11* (v, r 2 C44 * 11

C

* 12

C

C44

)u,

and using Eqs. (1)-(9), the equilibrium equations in terms of the displacement components (Navier C11* C12* )u C33*

1 C44 ((1 r 2 C33*

2

C44 (C11* C12* ) )( v, Sin C33*

v cot )

1 (( m 2) r2

fz r m 1 r T, (1 C44

C12* C44

)u,

2 u, )

v cot )

(

1 1 2

2 1/ 2

)

1)

1 C13* ( r C33*

C44 ) C33*

zr m 1 r (2(m 1) f )T rT,r ), (10) C33*

11 )v C44

2

(1 r

)1 / 2 (1

C13* )u, r C44

(11)

Where f

z zr

3. Temperature distribution The heat conduction equation and the thermal boundary conditions, respectively, are

(12)

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T, rr

Sajad Karampour / Procedia - Social and Behavioral Sciences 46 (2012) 4880 – 4885

(

k k

2 )T, r r

1 (1 r2

2

)T,

0,

a

r

b,

1

1,

(13)

where k is the coefficient of thermal conduction. It is assumed that the thermal conduction coefficient for the poro FGM sphere is (14) k (r ) km r m

k (a) where k (a) is the conduction coefficient of the inner radius of FG material at r = a and m is the material constant. am Substituting Eq. (14) into Eq. (13) yields the FGM heat conduction equation as

Here, k m

1 1 2 2 (15) (m 2)T, r (1 )T, T, 0 r r2 r2 Solution of the conduction equation may be assumed in the form of Legendre series as T, rr

T (r , )

(16)

Tn (r ) pn ( ) n 0

where Tn (r ) as the coefficient of Legendre series may be illustrated as 2n 1 1 2n 1 (17) Tn (r ) T (r , ) p n ( )d T (r , ) pn (cos ) sin( )d 1 2 2 0 Using Eq. (12), Eq. (11) may be written as 1 1 2 (18) {(Tn (r ) (m 2)Tn (r )) pn ( ) ((1 ) pn ( ) 2 pn ( )Tn (r )} 0 2 r r n 0 Employing the change of variable given in Appendix B, results into the separation of independent variables in Eq. (18), which may be written as 1 n2 n (19) Tn (r ) (m 2) Tn (r ) Tn (r ) 0 r r2 The above equation is the Euler equation. Thus, the solution may be written in the form Tn (r ) An r (20) Substituting Eq. (20) into Eq. (19) yields (m 1)

(m 1) 4n(n 1) 2 The general solution of Eq. (18) is Tn (r ) An1r n1 An 2 r n 2 Thus, the temperature distribution becomes

(21)

n1, 2

T (r , )

( An1r

n1

An 2 r

n2

(22) (23)

) pn ( )

n 0

Which, here An1 , An 2 constants are obtained from thermal boundary conditions for internal and external radius as below. (24) C11 T a, C12 T a, f1 , C21 T b, C22 T b, f2 . Where f1 and f 2 4. Stress distributions

are inside and outside radius temperature shown boundary condition.

The Navier equations (10) and (11) may be solved by the direct method of analysis employing the series solution introduced by Jabbari et al. [4, 5]. The solution of the Navier equations (10) and (11) is assumed in the form of Legendre series as u n ( r ) p n ( ) , v(r , )

u (r , ) n 0

vn (r ) pn ( )(1

2

)

(25)

n 0

where u n r and vn r are functions of r. Substituting Eqs. (25) into Eqs. (10) and (11) and then using form of Legendre series to separate the independent variables r and lead to

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Sajad Karampour / Procedia - Social and Behavioral Sciences 46 (2012) 4880 – 4885

1 C13* m C ( * n(n 1) 44* 2 r C33 C33

1 un (r ) (m 2) un (r ) r

2C13* 2(C11* C12* ) C13* C44 1 u ( r ) ( n ( n 1 )( ))v (r ) n r C33* C33*

z n(n 1) C13* (m 1) (C44 (C11* C12* )vn (r ) r* ((2((m 1) f ) ( r2 C33* C33* C33 1 C11* (( m 2 ) n ( n 1 ) r2 C44

1 (m 2)vn (r ) r

vn (r )

fz r ( An1 r m C 44*

n1

1

An 2 r m

n2

1

n1

) An1

m

n1

1

((2(m 1) f ) An 2 An 2

C13* 1 (1 )un (r ) r C44

C12* vn (r ) C44*

n2

))r m

n2

(26)

1

1 C11* C12* ( m 2 )un (r ) r2 C44

(27)

)

This is the system of Euler differential equations. Thus, the solution of homogeneous part of Eqs. (26) and (27) may be assumed in the form (28) ung (r ) Br , v ng (r ) Cr where B and C are constants to be found using the given boundary conditions. Substituting Eqs. (28) into the homogeneous parts of Eqs. (26) and (27) yields 2

(m 2)

(2

C13 ) C44

(1

C13* m C33*

n(n 1)

2(C13*

C44 C33*

C11* C12* ) B C44

(m 2

(C11* C12* )) ) B C33* 2

(m 2)

n(n 1)(

C13* C44 ) C33*

((m 2) n(n 1)

C11* C44

* * * n(n 1) ( C13 (m 1) (C44 (C11 C12 )) ) C 0 * C33

C12* C C44

(29)

0

To obtain the non-trivial solution of the above equation, the determinant of coefficients of constants B and C must be vanished. This leads to the evaluation of the eigenvector obtained as (

2

(m 2)

(2

2(C13* (C11* C12* )) )) (( C33*

C13* m C n(n 1) 44* C33* C33

2

(m 2)

((m 2) n(n 1)

C11* C44

C12* ) C44

C13* C44 C13* (m 1) (C44 (C11* C12* )) C13 C11* C12* (30) ) ( 1 )( )) ( ( 1 ) ( 2 ))) 0 n n m C44 C44 C33* C33* Thus, the general solution, utilizing the linearity lemma, is a linear combination of all values of eigenvalues and are obtained as ((n(n 1)(

4

ung (r )

Bnj r

,

nj

4

vng (r )

j 1

N nj Bnj r

(31)

nj

j 1

Where, using Eq. (29), N nj may be founded as N nj

* 2 * C33 (C33 (m 2) (n(n 1)(C13* C44 )

2C13* m n(n 1)C44 n(n 1)(C13* (m 1)

2(C13* (C11* C12* ))) j 1,2,3,4 , (C44 C11* C12* )))

(32)

The particular solution of Eqs. (26) and (27) are assumed As m

1

Dn3r m

, vnp (r ) Substituting Eqs. (33) into Eqs. (26)-(27) yields unp (r )

d1 Dn1r

d 7 Dn3 r

Dn1 r m

n1

m

m 1

n1

1

Dn 2 r

d 2 Dn 2 r 1

n2

d8 Dn 4 r

m 1

m

n2

d 3 Dn 3 r 1

m

n1

d 9 Dn1r

1

m

d 4 Dn 4 r 1

n1

m

1

n2

Dn 4 r m 1

d10 Dn 2 r

d5r m

n2

n1

1

1

m 1

(33) d6r

n2

m 1

(34)

m 1

m 1

(35)

d11r d12r where coefficients d1 through d 12 are presented in Appendix A. Equating the coefficients of the identical powers yields n1

n2

n1

d1 Dn1 d 3 Dn3

d5 ,

d 2 Dn 2 d 4 Dn 4

d6 ,

d 9 Dn1 d 7 Dn3

d11,

d10 Dn 2 d8 Dn 4

d12 ,

n2

n1

n2

(36)

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Sajad Karampour / Procedia - Social and Behavioral Sciences 46 (2012) 4880 – 4885

Here, Dni i 1,...,4 are obtained solving the two systems of algebraic equations. The complete solutions for displacements are the sum of Eqs. (31) and (33) and are 4

un (r )

Bnj r

Dn1r

n j

m 1

n1

Dn 2 r

m 1

n2

4

, v (r ) n

j 1

N nj Bnj r

Dn3 r m

nj

1

n1

Dn 4 r m

(37)

1

n2

j 1

For n

0 , the system of Navier equations (26)-(27) lead to the following single differential equation, as

u 0 (r )

zr 1 2 C * (m 1) C11* C12* m 2 u0 (r ) 2 ( 13 )u0 (r ) 2m 1 r r C33* C33*

f

01

) A01 r m

1

01

2m 1

f

02

) A02 r m

02

1

(38)

solving the homogeneous part of Eq. (38) provide the complete solution for u0g (r ) as 4

u 0g (r )

B0 j r

(39)

0 j

j 1

where B0 , and

0

are

constants to be found using the given boundary conditions. solving the non_homogeneous part of Eq. (38) provide the complete solution for u 0P (r ) as

u0p r

D01r

m 1

01

D02r

m 1

02

(40)

,

where D01 , and D02 are constants to be found using the given boundary conditions. Thus, the complete solution of radial and circumferential displacement components for all values of n, using Eqs. (37) and (40), is 2

u r,

B0 j r

D01r

0j

01

m 1

D02r

4

m 1

02

j 1

n 1

4

v r,

N nj Bnj r

Dn3 r

nj

n1

m 1

Dn 4 r

Bnj r Pn

m 1

n2

Dn1r

nj

m 1

n1

Dn 2 r

n2

m 1

(41)

Pn

j 1

(42)

2 1/ 2

(1

)

j 1

n 1

Using the strain displacement relations given by Eqs. (1) and relations (2), the radial, circumferential and shear stresses are obtained as strains are 2

C33* (

rr

(

2C13* ) B0 j r C33*

0j

j 1

(

4

( n 1

(

C13* D (2 n(n 1) N nj n3 )) r * Dn1 C33 2

C13* (

(

0j

n 1

r

C nj

j 1 4

n 1

(

* 11

N nj Bnj r

n1

2m

Dn 2 ((

* 12

C ) Bnj r C13*

nj

n2

m 1

Dn 3 r

nj

n1

m 1

2m

0j

0j

m 1

nj

Dn1 ((

m 1

Dnj ((

Dn 4 r

D0 j ((

4 n 1

j 1

Bnj ( N nj (

nj

1) 1)r

nj

m 1

n1

n2

Dn1 ((

0j

m 1) (

nj

2m

j 1

C44 (

2C13* )r C33*

m 1)

n1

0j

zr A01r C33*

2m

n2

2m

zr ( An1r C33*

C

* 12

C ))r nj C13* 1

2m

Dn3 1)r Dn1

n1

2m

2m

Dn 2 ((

(

n2

0j

An 2 r 2m

n2

2m

A02 r nj

02

2m

m 1

Dn1 ((

n1

2m

m 1)

n1

)) Pn ( )

z A01r C13*

zr ( An1r C13*

2m

C11 n(n 1) Pn ( ) C12* C13 C13*

m)

n1

C11* C12* )r m 1) C13* * 11

01

C13* (2 n(n 1) N nj ) Bnj r C33*

m 1)

C13* D (2 n(n 1) N nj n 4 ))r * Dn 2 C33

m 1)

C1* C12* ) B0 j r C13*

j 1

(

D0 j ((

C13* (2 n(n 1) N nj ) Bnj r C33*

nj

j 1

4

m 1

0j

(43)

01

2m

An 2 r

A02r n2

2m

02

Dn 4 1)r Dn 2

n2

2m

Pn ( )(1

1

(44)

2 1/ 2

) )

(45) Appendix A

1

)) Pn ( )

1 C12* C11* ) Pn ( ) * 1 C13

m)

2m

1 1

the

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Sajad Karampour / Procedia - Social and Behavioral Sciences 46 (2012) 4880 – 4885

d1

(m

1)(m

n1

(m 2)(m

n2

n1

1) (

n(n 1)(C13* (m 1) (C 44 C33* e 1)(m d 5 ( 22* )(m n1 C 33 n2

1) n(n 1)(

m

n1

1

d11

(1

C11* C12* )

n1

) (

e25 ), d 7 C 23*

((m 2) n(n 1) C13* )(m C 44

n1

1) (

2C13* m n(n 1) 2(C13* C33*

(

(m

) (m 2)(m

n1

),

d4

2C13* m n(n 1) 2(C13* (C11* C12* )) ), C33*

(C11* C12* ))

(

n(n 1)(C13* C33*

(m 2)e22 2e21 )(m C 33*

zr ( 2m 1 f ) C 33* C11* C 44

C12* ), d10 C 44

1) ((m 2)

n1

n1

m

, d3

C11* C12* ), d12 C 44

n(n 1)(C13* C33*

C 44 )

)(m

1) n(n 1)(

An 1 , d 8 n2

(

1 m (1

n2

e25 ), d 6 C 23*

zr ( 2m 1 f ) C 33* n2

C13* )(m C 44

m 2 m

n2

C 44 )

1) ( (

)(m

n2

n1

(m

1)(m

n2

n2

An 2 , d 9 1

1) ((m 2)

n2

)

1)

n(n 1)(C13* (m 1) (C 44 C33*

e22 )(m C 33* n2

d2

1)(m m

n2

n1

) (

C11* C12* )

(m 2)e22 2e21 ) C 33*

1 m

((m 2) n(n 1)

)

C11* C 44

n1

m 2

C12* ), C 44

C11* C12* ) C 44

References [1] [2]

Qin Q-H. Fracture mechanics of piezoelectric materials. Southampton:WIT; 2001. R. Poultangari, M. Jabbari, M.R. Eslami. Functionally graded hollow spheres under non-axisymmetric thermo-mechanical loads. Int J Pressure Vessels and Piping 2008;79:295 305 [3] Ootao Y, Tanigawa Y. Three-dimensional transient thermal stress analysis of a nonhomogeneous hollow sphere with respect to arotating heat source. Trans Jpn Soc Mech Eng A 1994;460:2273 9 [in Japanese]. [4] Jabbari M, Sohrabpour S, Eslami MR. Mechanical and thermal stresses in functionally graded hollow cylinder due to radially symmetric loads. Int J Pressure Vessel Piping 2002;79:493 7. [5] Jabbari M, Sohrabpour S, Eslami MR. General solution for mechanical and thermal stresses in a functionally graded hollow cylinder due to nonaxisymmetric steady-state loads. ASME J Appl Mech 2003;70:111 8. [6] U. Meyer, A. Larsson, H.P. Hentze, R.A. Caruso, Adv. Mater. 14 (2002) 1768 [7] A. Stein,Micropor.Mesopor.Mater. 44 (2001) 227 [8] Delage P, Sultan N, Cui YJ. On the thermal consolidation of Boomclay. Can Geotech J 2000;37(4):343 54. [9] Jing LR, Feng XT. Numerical modeling for coupled thermo-hydromechanical and chemical processes (THMC) of geological media international and Chinese experiences. Chin J Rock Mech En 2003;22(10):1704 15. [10] Kodres CA. Moisture induced pressures in concrete airfield pavements.J Mater Civil Eng 1996;8(1):41 50. [11] Tanaka N, Graham J, Crilly T. Stress strain behaviour of reconstituted illitic clay at different temperatures. Eng Geol 1997;47(7):339 50. [12] Romero E, Gens A, Lloret A. Suction effects on a compacted clay under non-isothermal conditions. Geotechnique 2003;53(1):65 81.