Solutions for behavior of a functionally graded thick-walled tube subjected to mechanical and thermal loads

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Solutions for behavior of a functionally graded thick-walled tube subjected to mechanical and thermal loads Libiao Xin, Guansuo Dui, Shengyou Yang

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Received date: 29 January 2015 Revised date: 18 March 2015 Accepted date: 21 March 2015 Cite this article as: Libiao Xin, Guansuo Dui, Shengyou Yang, Solutions for behavior of a functionally graded thick-walled tube subjected to mechanical and thermal loads, International Journal of Mechanical Sciences, http://dx.doi.org/ 10.1016/j.ijmecsci.2015.03.016 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Solutions for behavior of a functionally graded thick-walled tube subjected to mechanical and thermal loads

Libiao Xin a, Guansuo Dui a , *, Shengyou Yang b. a

Institute of Mechanics, Beijing Jiaotong University, Beijing, 100044, China

b

Department of Mechanical Engineering, University of Houston, Houston, TX 77204, USA

Abstract For the problem of a functionally graded thick-walled tube subjected to internal pressure, we have already presented the solution of the elastic problem based on the Voigt method with the assumption of a uniform strain field within the representative volume element. This paper discusses the thermoelastic problem of the functionally graded thick-walled tube subjected to axisymmetric mechanical and thermal loads, and the solution is given in terms of volume fractions of constituents. We assume that the tube consists of two linear elastic constituents and the volume fraction of one phase is a power function varied in the radial direction. The theoretical solutions of the displacement and the stresses are presented under the assumption of a uniform strain field within the representative volume element. Comparisons of the theoretical solutions and the finite element analysis demonstrate the validity of the assumption. Based on the relation of the volume average stresses of constituents and the macroscopic stresses of the composite material in micromechanics, the present method can avoid the assumption of the distribution regularities of unknown overall material parameters appeared in existing papers. Further, the present method is valid for the materials with different Poisson’s ratios of constituents. The effects of the volume fraction, the ratio of two thermal expansion coefficients and the ratio of two thermal conduction *Corresponding author. Tel.: +86 01051688437; fax: +86 01051682094. E-mail address: [email protected]

1

coefficients on the displacement and stresses are systematically studied. Keywords: Functionally graded materials, Thermoelastic analysis, Thick-walled tube, The Voigt method. Nomenclature a

inner radius

b

outer radius

r

radial coordinate of the tube

c(r)

volume fraction of material A

c0, k, n

material parameters

pa, pb

internal and external pressures, respectively

Cij (i, j=1, 2)

constant thermal parameters

fj (j=1, 2)

constants on the inner and outer radii, respectively

k(r)

thermal conductivity of the tube

ki (i=1, 2)

thermal conduction coefficient of the components

T(r)

temperature variation

Ci (i=0, 1, 2)

constants in temperature function

αi (i=1, 2)

thermal expansion coefficient of the components

λi, µi (i=1, 2)

Lamé constants of the components

Ei, vi (i=1, 2)

Young's modulus and Poisson's ratio of the components

ε r( i ) , ε θ( i )

the radial and circumferential strains of the components, respectively

σr(i ) , σθ(i ) , σz(i )

the radial, circumferential and axial stresses of the components, respectively

σr , σθ , σz

the radial, circumferential and axial average stresses of the tube, respectively

2

u

the displacement in the radial direction

x

a new variable about r

F

hypergeometric function

α, β, δ

coefficients in hypergeometric function

Bi (i=0, 1, 2)

intergral constants

r

non-dimensional expression for radial coordinate

a, b

non-dimensional expressions for inner and outer radii, respectively

u

non-dimensional expression for the radial displacement

σ ij

non-dimensional expressions for the stresses

1. Introduction Functionally graded materials (FGMs) are composite materials formed of two or more constituent phases with a continuously variable composition. FGMs have a lot of advantages that make them attractive in potential applications, including a potential reduction of in-plane and transverse through-the-thickness stresses, an improved residual stress distribution, enhanced thermal properties, higher fracture toughness, and reduced stress intensity factors [1].

In the last two decades, FGMs have been widely used in engineering applications, particularly in high-temperature environment, power transmission equipment, etc. On the other hand, thick-walled tube is a kind of typical structures such as pressure vessels and cylinders that are utilized in reserving or transferring chemical gas, oil, etc. Owing to these reasons, the thermoelastic problems of thick-walled tubes made of special nonhomogeneous materials have been studied by many 3

researchers. Some researchers presented an exact solution for one-dimensional thermal stresses of FGM cylinders [2] and spheres [3] whose Young's modulus and thermal expansion coefficient vary linearly with the radius. However, the linear function assumption is not sufficient for describing more complex cases. To capture Young's modulus and thermal expansion coefficient of the FGM thick-walled tube more precisely, some researchers [4-18] proposed another assumption of Young’s modulus such as the form of E(r) = E0rm1 (E0 and m1 are material constants, r is the radial coordinate of the cylinder) and thermal expansion coefficient such as the form of α ( r ) = α r m2 ( α0 0

and m2 are material constants). For convenience, thermal conduction coefficient of the FGM long tube was mostly assumed as the form of k (r) = k r m3 (k0 and m3 are material constants). Further, 0

these functions of material properties with two variables are not sufficient for describing more complex cases. To capture material parameters of the FGM thick-walled tube more precisely, researchers [19] proposed the three variable controlled material properties such as the Young’s k modulus was assumed as the form E(r) = E0 1− nE (r / b) E  (E0, nE and kE are material constants, b is

the outer radius of the cylinder), thermal expansion coefficient was assumed as the form α (r) = α0 1 − nα (r / b) α  ( α0 , nα and kα are material constants) and thermal conduction coefficient was k

k assumed as the form λ (r) = λ0 1 − nλ (r / b) λ  ( λ0 , nλ and kλ are material constants).

In the above-mentioned works, varying material properties are usually treated as specific gradient variation such as linear form or power-law form. Nevertheless, it is difficult to satisfy the 4

engineering manufacture in practice. For this reason, authors [20-21] proposed a new method which assumed the material properties as some special form in terms of the volume fraction of one phase such as Young's modulus with the form E(r)=EcVc+EmVm (Ec and Em are material constants, Vc and Vm are volume fractions) and the thermal expansion coefficient with the form

α(r) = αcVc + αmVm ( αc and α m are material constants). Though some researchers [22-23] considered the variations of both Young's modulus and the thermal expansion coefficient accurately by using the Mori-Tanaka method, they did not obtain the theoretical solutions due to the complex calculation of the equilibrium equation presented in their work.

Recently, the Carrera Unified Formulation (CUF), which was developed by Carrera for multi-layered structures [24-28] is extended to also account for functionally graded shells under mechanical and thermal loadings. In [29], the Principle of Virtual Displacements (PVD) has been proposed and the extension to Reissner's Mixed Variational Theorem (RMVT) has been given in [30]. The thermo-mechanical bending problem of functionally graded plates has already been proposed in [31]. Authors studied the static response of functionally graded shells and the governing equations are derived from the Principle of Virtual Displacements, considering the temperature as an external load [32].

In most actual systems, the overall elastic modulus and the thermal expansion coefficient, the thermal conduction coefficient of the FGM tube can not be found directly, and they can be 5

obtained in terms of their properties of constituents and the volume fractions in a certain regulation.

In this paper, we first define a volume fraction varied radially rather than the assumption of Young's modulus, thermal expansion coefficient and thermal conduction coefficient of the FGM tube. Then by using the elastic results in [33], the thermoelastic response of the functionally graded thick-walled tube subjected to axisymmetric mechanical and thermal loads is investigated in this work. In Section 2 the basic equations of the FGM long tube and the analysis of thermoelastic mechanical behaviors of the tube are described first. Then Section 3 gives the FEM results and discusses the effect of parameter n, the ratio of two thermal expansion coefficients and the ratio of two thermal conduction coefficients. Finally some concluding remarks are given in Section 4. 2. Theoretical analysis A state of axial symmetry is considered in the problem of a FGM thick-walled tube subjected to axisymmetric mechanical and thermal loads on its inner and outer surfaces (Fig. 1). Cylindrical polar coordinates ( r,θ , z ) are used and the inner and outer radii of the thick-walled tube are designated as a and b, respectively.

The tube consists of two linear elastic materials A and B, and the volume fraction c(r) ∈[0,1] of

material A is given by

c(r) = c0 1 − k(r / b)n  where r is the radius, c0, k and n are the material parameters. 6

(1)

2.1. Heat transfer

To obtain desired thermal stresses in the FGM long tube, it is natural to first determine temperature distribution in the thick-wall tube. To this end, we need to first consider a steady-state heat conduction problem without internal heat source. Thus, the heat equation for steady-state heat conduction with no heat source reads 1 [ rk (r)T ′(r)]′ = 0 r

(2)

and thermal boundary conditions for the FGM tube at the inner and outer surfaces are C11T (a ) + C12T ′(a) = f1 C21T (b) + C22T ′(b) = f 2

(3)

where T(r) is the radial temperature variation between the tube and the ambient condition, k(r) is the thermal conductivity, Cij (i, j=1, 2) are the constant thermal parameters relative to the conduction and convection coefficients, fj (j=1, 2) are known constants on the inner and outer radii, respectively.

The thermal conductivity k(r) is assumed as [34] 1 c( r ) 1 − c( r ) = + k (r) k0 k1

(4)

where c(r) is given by Eq. (1), and k0 and k1 are the thermal conduction coefficients of material A and B, respectively.

Then substituting Eq. (4) into (2), one can easily find  c k (k − k )r n  T ( r ) = C1  n 0 0 1 + ln r  + C2 nb ( c k + k − c k ) 0 1 0 0 0   7

(5)

With thermal boundary conditions Eq. (3), we have

C1 =

f1C21 − f 2C11 C0

C2 =

 C  c k (k − k )   f 2 C1   c0k (k0 − k1 ) − + ln b + 22  0 0 1 + 1  C21  C21 C21   n(c0k1 + k0 − c0k0 )  b  c0k1 + k0 − c0k0  

(6)

where n n   a  C  c0 k ( k0 − k1 ) a n   c k ( k − k )( a − b ) C0 = C21 C11  0 n 0 1 + ln  + 12  + 1  n b  a  ( c0 k1 + k0 − c0k0 )b   nb ( c0 k1 + k0 − c0k0 )  

C C − 11 22 b

 c0k ( k0 − k1 )   c k + k − c k + 1  0 1 0 0 0 

(7)

In particular, two typically thermal boundary conditions are that heat flux is prescribed and temperature change is prescribed. The former corresponds to Cj1=0 and the latter corresponds to Cj2=0. In the next numerical analysis, the latter is used with C12=C22=0, C11=C21=1. And then we

have f1=T(a) and f2=T(b). Thus the constants in Eq. (6) can be simplified as C1 =

f1 − f 2 c0k (k0 − k1 )(a n − bn ) a + ln nbn (c0k1 + k0 − c0k0 ) b

(8)

 c0k (k0 − k1 )  + ln b  C2 = f 2 − C1   n(c0k1 + k0 − c0k0 ) 

2.2. Thermoelastic problem

The average stress tensor over a representative volume element V are defined as [35] σ =

1 V

∫

V

σ (i ) =

σˆ ( x )dx ,

1 Vi

∫

Vi

σˆ ( i ) ( x )dx.

(9)

where σˆ is the stress tensor at random over a representative volume element; σ is the overall volume average stress tensor of composite material; σˆ ( i ) is the stress tensor at random over the constituents of composite material; σ ( i ) is the overall volume average stress tensor in their subvolumes Vi. 8

For the FGM tube consists of two phases of material A and B, Eq. (9) reduces to σ =c ( r )σ (0) + [1 − c ( r )]σ (1)

(10)

where σ(0) and σ(1) are the average stresses of material A and B, respectively. By using the assumption of a uniform strain field within the representative volume element, the components of the strain tensor are

εθ(0) = εθ(1) = εθ , ε r(0) = ε r(1) = ε r

(11)

where i=0, 1 denotes material A and B, ε r( i ) and ε θ( i ) are the radial and circumferential strains, respectively. For the small deformation, the strain-displacement relation is εr =

du , dr

εθ =

u r

(12)

where u is the displacement in the radial direction. For isotropic materials, the stress and the strain are related through the generalized Hooke’s law σ r( i ) = λiεθ( i ) + ( λi + 2 µi ) ε r( i ) − ( 3λi + 2 µi ) αiT (r ) σ θ( i ) = ( λi + 2 µi ) εθ(i ) + λiε r( i ) − ( 3λi + 2 µi ) αiT (r )

(13)

σ z( i ) = λi (εθ(i ) + ε r(i ) ) − ( 3λi + 2µi )αiT ( r )

the subscript i=0, 1 denotes material A and B, λi and µi are Lamé constants and αi is the coefficient of thermal expansion; σr(i ) , σθ(i ) and σz(i ) are the radial, circumferential and axial stresses, respectively; ε r( i ) , ε θ( i ) and ε z( i ) are the radial, circumferential and axial strains, respectively. The radial temperature variation T(r) is given by Eq. (5). Substituting Eq. (13) into (10), the average stresses of the FGM tube are

9

u du + ( λ + 2 µ ) −  c( r )α 0 (3λ0 + 2 µ0 ) + (1 − c( r ) )α1 (3λ1 + 2 µ1 )  T ( r ) r dr u du −  c( r )α 0 (3λ0 + 2 µ0 ) + (1 − c( r ) ) α1 (3λ1 + 2 µ1 )  T ( r ) σ θ = (λ + 2µ ) + λ r dr   u du  σ z = λ  +  −  c ( r )α 0 (3λ0 + 2 µ0 ) + (1 − c ( r ) ) α1 (3λ1 + 2 µ1 )  T ( r )  r dr 

σr = λ

(14)

where λ = c ( r )λ0 + [1 − c ( r ) ] λ1

(15)

µ = c ( r ) µ 0 + [1 − c ( r ) ] µ1

The equilibrium equation is dσ r σ r − σ θ + =0 dr r

(16)

2.3. Thermal displacement and stresses

Substituting Eq. (14) into (16) and using Eq. (1) and (5), the governing differential equation for the radial displacement u is r  c0 (λ0 + 2µ0 ) + (1 − c0 )(λ1 + 2 µ1 ) − c0k ( r / b)n (λ0 + 2µ0 − λ1 − 2 µ1 )

d 2u dr 2

du dr u − {c0k ( r / b) n [ (n − 1)(λ0 − λ1 ) + 2( µ1 − µ0 )] + c0 (λ0 + 2 µ0 ) + (1 − c0 )(λ1 + 2 µ1 )} r dT ( r ) = nc0 k ( r / b)n T ( r ) [α1 (3λ1 + 2µ1 ) − α 0 (3λ0 + 2µ0 )] + r {α1 (3λ1 + 2µ1 ) dr +  c0 (λ0 + 2 µ0 ) + (1 − c0 )(λ1 + 2 µ1 ) − ( n + 1)c0k ( r / b) n (λ0 + 2 µ0 − λ1 − 2 µ1 ) 

(17)

}

+ c0 1 − k ( r / b)n  [α 0 (3λ0 + 2 µ0 ) − α1 (3λ1 + 2 µ1 )]

For notational convenience, we introduce a new variable x = χ ( r) =

c0k (λ0 + 2µ0 − λ1 − 2µ1 ) rn b [ c0 (λ0 + 2µ0 ) + (1 − c0 )(λ1 + 2µ1 )] n

(18)

and a transformation y(x)=u(r)/r. A simplified form of Eq. (17) is x (1 − x )

d2y dy + [δ − (α + β + 1) x ] − αβ y = x (1 − x ) f ( x ) 2 dx dx

where

10

(19)

f ( x) =

Ax ln x + B + Cx 2 + Dx x 2 (1 − x )

1 2

2 n

δ =1 + , α =  δ − δ 2 −

(20)

8 λ0 + µ 0 − λ1 − µ1  , β = δ − α n λ0 + 2 µ0 − λ1 − 2 µ1 

(21)

and A=

C1 [α1 (3λ1 + 2 µ1 ) − α 0 (3λ0 + 2 µ0 ) ] n 2 ( λ0 + 2 µ0 − λ1 − 2 µ1 )

B=

C1 [ c0α 0 (3λ0 + 2 µ0 ) + α1 (1 − c0 )(3λ1 + 2 µ1 ) ] n 2 [ c0 (λ0 + 2 µ0 ) + (1 − c0 )(λ1 + 2 µ1 ) ]

2( k0 − k1 ) A[ c0 (λ0 + 2 µ0 ) + (1 − c0 )( λ1 + 2 µ1 ) ] C= (λ0 + 2 µ0 − λ1 − 2 µ1 )( c0 k1 + k0 − c0k0 ) D=

(22)

CB ( nC2 + C1 ) A + 2A C1

Solution for Eq. (19) is the sum of the general solution and the particular solution, and the general solution have been given by [33]  B1F (α , β , δ ; x ) + B2 x1−δ F (α − δ + 1, β − δ + 1,2 − δ ; x ), x < 1  x ( −1)−α x1−δ x  yc ( x ) =  B1 (1 − x )−α F (α , δ − β , δ ; ) + B2 F (α − δ + 1,1 − β ,2 − δ ; ), x < −1 1−δ +α − 1 ( − 1) −1 x x x   B1 x −α F (α ,α − δ + 1, α − β + 1;1 x ) + B2 x − β F ( β , β − δ + 1,1 − α + β ;1 x ), x > 1 

(23)

where B1 and B2 are constants and F is the hypergeometric function defined in the range x < 1 by the power series (α )m (β )m xm (δ )m m! m=0 ∞

F (α , β ,δ ; x) = ∑

(24)

Here (q)m is the Pochhammer symbol, which is defined by 1 (q)m =  q(q + 1) ⋅⋅⋅ (q + m − 1)

m=0 m>0

Note that the solutions at the regular singular points x = 1 are not discussed here and the variable

x = χ(r) mainly lies in the interval x < 1 , Eq. (23)1 is mainly used throughout. For convenience, rewrite the general solution

11

yc ( x) = B1 y1( x) + B2 y2 ( x)

(25)

y1 ( x) = F (α , β ,δ ; x), y2 ( x) = x1−δ F (α − δ + 1, β − δ + 1,2 − δ ; x)

(26)

where

In the following, we will give out the particular solution. And the particular solution can be easily obtained by using the method of variation of parameters as y p ( x ) = − y1 ( x ) ∫

x xa

x y (t ) f (t ) y2 (t ) f (t ) dt + y 2 ( x ) ∫ 1 dt x a W (t ) W (t )

(27)

where xa = χ (a) from Eq. (18) and the Wronskian is

W( x) = y1( x) y2′ ( x) − y2 ( x) y1′( x)

(28)

Then the solution of the Eq. (19) can be obtained using Eqs. (25) and (27) as x y (t ) f (t ) x y (t ) f (t )     y( x) = y1 ( x)  B1 − ∫ 2 dt  + y2 ( x)  B2 + ∫ 1 dt  xa x a W (t ) W (t )    

(29)

And then the radial displacement can be reduced by using y(x)=u(r)/r as x y (t ) f (t ) x y (t ) f (t )      u ( r ) = r  y1 ( x )  B1 − ∫ 2 dt  + y2 ( x )  B2 + ∫ 1 dt   xa x a W (t ) W (t )     

(30)

Substituting Eq. (30) into (14), the stresses of the FGM tube can be obtained as σ r = 2 y ( x ) ( λ + µ ) + nxy ′( x ) ( λ + 2 µ ) −  c ( r )α 0 (3λ0 + 2 µ0 ) + (1 − c ( r ) ) α1 (3λ1 + 2 µ1 )  T ( r ) σ θ = 2 y ( x ) ( λ + µ ) + nxy ′( x )λ −  c ( r )α 0 (3λ0 + 2 µ 0 ) + (1 − c ( r ) ) α1 (3λ1 + 2 µ1 )  T ( r ) σ z = λ [ 2 y ( x ) + nxy ′( x ) ] −  c ( r )α 0 (3λ0 + 2 µ 0 ) + (1 − c ( r ) ) α1 (3λ1 + 2 µ1 )  T ( r )

With stress controlled boundary conditions, σ

r=a

12

= − pa , σ

r =b

= − p b , we have

(31)

{ (

)

B1 =  2 λ (b) + µ (b) y2 ( xb ) + nxb ( λ (b) + 2 µ (b) ) y2′ ( xb )   − pa + ( c( a )α 0 (3λ0 + 2 µ0 ) + (1 − c( a ))  

(

)

(

× (α1 (3λ1 + 2µ1 ) ) T (a )  −  2 λ ( a ) + µ ( a ) y2 ( xa ) + nxa ( λ (a ) + 2 µ ( a ) ) y2′ ( xa )   2 λ (b) + µ (b)  

)

xb y (t ) f ( t ) xb y ( t ) f (t ) xb y (t ) f ( t )    ×  y1 ( xb ) ∫ 2 dt − y2 ( xb ) ∫ 1 dt  + nxb ( λ (b) + 2µ (b) )  y1′( xb ) ∫ 2 dt x xa x a a W (t ) W (t ) W (t )    xb y ( t ) f (t )    − y2′ ( xb ) ∫ 1 dt  + ( c(b)α 0 (3λ0 + 2 µ0 ) + (1 − c(b))α1 (3λ1 + 2µ1 ) ) T (b) − pb   B0 xa W (t )   

(

(32)

)

B2 = {( c( a )α 0 (3λ0 + 2 µ0 ) + (1 − c( a ))α1 (3λ1 + 2 µ1 ) ) T ( a ) − pa − B1  2 λ ( a ) + µ ( a ) y1 ( xa ) 

} 2 (λ (a) + µ (a) ) y ( x ) + nx y′ ( x ) ( λ (a) + 2µ (a) )

+ nxa ( λ (a ) + 2 µ ( a ) ) y1′( xa ) 

2

a

a

2

a

where xa = χ (a) and xb = χ (b) from Eq. (18) and

(

)

(

B0 =  2 λ ( a ) + µ ( a ) y1 ( xa ) + nxa ( λ ( a ) + 2 µ ( a ) ) y1′ ( xa )   2 λ (b) + µ (b)  

(

)

)

× y2 ( xb ) + nxb ( λ (b) + 2 µ (b) ) y2′ ( xb )  −  2 λ ( a ) + µ ( a ) y2 ( xa ) + nxa y2′ ( xa ) 

(

(33)

)

× ( λ ( a ) + 2 µ ( a ) )   2 λ (b) + µ (b) y1 ( xb ) + nxb ( λ (b) + 2 µ (b) ) y1′ ( xb )   

The thermoelastic solutions of the displacement and the stresses of a long FGM tube subjected to axisymmetric mechanical and thermal loads can be obtained from Eqs. (30) and (31). 3. Numerical results and discussion In the simulation, the non-dimensional expressions for radial coordinate, inner radius, stress and radial displacement are defined as r =

σ ij u r a and u = α f b , respectively; , a = , σ ij = E0α0 f2 b b 0 2

where a and b are the inner and outer radii, respectively, α0 is the thermal expansion coefficient of material A, E0 is Young's modulus of material A and f2 is constant at outer radius. The inner radius is taken as a = 0.7 throughout, which is reasonable for a thick-walled tube. The temperature at inner and outer radii are T(a)=0

℃ and T(b)=100℃, respectively.

Since thermal stresses are only concerned in this work, in what follows we solely seek the stresses and radial displacement induced by temperature change at the surfaces. As a consequence, for the sake of simplicity the inner and outer surfaces are assumed traction-free, namely σ r 13

r=a

= 0 and

σr

r=b

= 0.

3.1. Comparisons of the theoretical and FEM results

Analysis is also carried out with the ANSYS finite element software for purposes of comparison. Due to the symmetry, only a quarter of the tube is taken into account. The tube is discretized using brick 8-noded elements (thermal solid) (Fig. 2). The cross-sectional contours of the radial displacement and the stresses via FEM (finite element method) are given in Fig. 3(a)-(d). The theoretical results of the radial displacement and the stresses can be obtained from Eqs. (30) and (31), respectively. Both the theoretical and FEM results of the radial displacement u , the radial stress σr , the circumferential stress σ θ , and the axial stress σz are plotted in Fig. 4. Furthermore, Tables 1 and 2 provide results for the radial displacement and stresses in non-dimensionalized form for different positions. It is clearly shown that the FEM and theoretical results agree well with each other, which in turn shows the validity of the theoretical solutions presented in this paper. 3.2. Effects of parameter n

In order to examine the influence of the volume fraction, numerical results are evaluated for different parameter n and shown in Figs. 5-8. Fig. 5 shows the plot of the radial displacement along the radius. It is obvious that the magnitude of the radial displacement is decreased as the parameter n increased. And the radial displacement reached the minimum for different values of n where the position closes to the inner radius. From Fig. 6, it is seen that σ r

r =a

= 0 and σ r

r =b

= 0 , as expected. In particular, the peak-value

position of σr in Fig. 6 is viewed to progressively shift towards the inner surface when n is raised. 14

The effects of n on the normalized circumferential and axial stresses are different from those seen in Fig. 6. Form Figs. 7 and 8, the circumferential and axial stresses decrease when r begins to increase from the inner surface. Meanwhile, the minimum value of the circumferential stress take place at the outer surface. In general, the influence of n on the axial stress is more pronounced for positions closer to the outer surface than those close to the inner surface. In addition, n does not change the axial stresses at the outer surface, but affects its values for positions closer to the outer surface. 3.3. Effects of thermal conduction coefficient

Figs. 9-12 show the influence of the radial displacement and thermal stresses corresponding to different ratio k1/k0 of two thermal conduction coefficients. In Fig. 9, the radial displacement monotonically increases with the increases of k1/k0. Additionally, the minimum of the radial displacement where occurs is at inner surface for different values of k1/k0. Compared to the effects on the radial displacement, however, the effects of k1/k0 on the thermal stresses are not obvious (see Figs. 10-12). In particular, some interesting phenomenon is that the radial stress σr has a stable value 0.05 for different ratio k1/k0 at r = 0.88 and the circumferential stress σ θ has two stable value for different ratio k1/k0 at r = 0.77 and r = 0.94 . Additionally, the axial stress σz is near same both at the inner and outer surface for different ratio k1/k0, which means that k1/k0 affects the axial stress σz at internal position. 3.4. Effects of thermal expansion coefficient

Fig. 13 shows the evolution of the radial displacement corresponding to different ratio α1/α0 of two thermal expansion coefficients. It is not difficult to see that the radial displacement monotonically increases with the increase of the ratio α1/α0. 15

The effects of the ratio α1/α0 on the radial stress σr , the circumferential stress σ θ and the axial stress σz are plotted in Figs. 14, 15 and 16, respectively. Clearly, the radial stress σr obeys the boundary condition σ r

r =a

= 0 and σ r

r =b

= 0 as shown in Fig. 14. Moreover, we can see that the

normalized radial stress σr in magnitude exhibits a maximum value at the internal position. This maximum value increases with an increasing ratio α1/α0. Although the maximum value of the radial stress happened at an internal position, these positions are slightly shifted with α1/α0 varying. From Figs. 15 and 16, the circumferential and axial stresses decrease when r begins to increase from the inner surface. In particular, there is a critical position at about r = 0.86 at which σ θ has almost the same value -0.07 for all values of α1/α0. Also, another interesting phenomenon is that the axial stress σz has a stable value -0.09 for different ratio α1/α0 at r = 0.75 , that is, the curves of σz intersect in this position.

3.5. Effects of Poisson's ratio

One of the contributions of this paper is that the method is available for the materials with different Poisson’s ratios for two phases. To illustrate the effects of Poisson’s ratio on the mechanical behavior, we choose three groups of Poisson’s ratio: v0= v1=0.3; v0=0.3, v1=0.2 and v0=0.3, v1=0.4.

Fig. 17 shows that the radial displacement u highly depends on the variation of Poisson’s ratio. From Figs. 18-20 the effects of Poisson’s ratio on the stresses are obvious, especially for the radial

16

and axial stresses. These results show that the assumption of different Poisson’s ratios is necessary if a precious analysis is needed in practice. Importantly, this paper builds the theoretical foundation for this thermoelastic problem.

4. Conclusions The thermoelastic analysis for axisymmetrical problems of a FGM thick-walled long tube is investigated in this paper. Different from the previous, the method presented in this paper has clear physical meaning and can avoid the assumption of the distribution regularities of unknown overall material parameters including Young's modulus, thermal expansion coefficient and thermal conduction coefficient. Furthermore, numerical results of the radial displacement and the thermal stresses are evaluated and presented graphically. Different parameter n is first discussed to elucidate the individual effect of volume fraction on the radial displacement and thermal stresses. For the ratio of two thermal expansion coefficients, it has great impact on the radial stress in the middle but negligible at the inner and outer radii, which is opposite to the effects on the circumferential and axial stresses. Interestingly, the ratio of two thermal conduction coefficients plays a significant role in the radial displacement of the FGM tube, however, its effects on the stresses are negligible. Importantly, this paper builds the thermoelastic theoretical foundation for different Poisson's ratio of constituents. In engineering design, this permits one to choose appropriate gradients to make the FGM tube have high reliability in structural integrity when subjected to high-temperature change of the inner or outer environments. Acknowledgments

17

The authors acknowledge the financial support of National Natural Science Foundation of China (Nos.11132003, 11172033, 11272044, 11272136) and National Basic Research Program of China (973 Program) (2010CB7321004).

References [1] Shariyat M. Nonlinear transient stress and wave propagation analyses of the FGM thick cylinders, employing a unified generalized thermoelasticity theory. Int J Mech Sci 2012;65:24-37. [2] Zimmerman RW, Lutz MP. Thermal stresses and thermal expansion in a uniformly heated functionally graded cylinder. J Therm Stresses 1999;22:177-88. [3] Lutz MP, Zimmerman RW. Thermal stresses and effective thermal expansion coefficient of a functionally gradient sphere. J Therm Stresses 1996;19:39-54. [4] Jabbari M, Sohrabpour S, Eslami MR. Mechanical and thermal stresses in a functionally graded hollow cylinderdue to radially symmetric loads. Int J Pres Ves Pip 2002;79:493-97. [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. J Appl Mech 2003;70:111-8. [6] Eslami MR, Babaei MH, Poultangari R. Thermal and mechanical stresses in a functionally graded thick sphere. Int J Pres Ves Pip 2005;82:522-27. [7] Rahimi GH, Nejad MZ. Exact solutions for thermal stresses in a rotating thick-walled cylinder of functionally graded materials. J Appl Sci 2008;8:3267-72. [8] Kordkheili SAH, Naghdabadi R. Thermoelastic analysis of functionally graded cylinders under axial loading. J Therm Stresses 2008;31:1-17. [9] Poultangari R, Jabbari M, Eslami MR. Functionally graded hollow spheres under 18

non-axisymmetric thermo-mechanical loads. Int J Pres Ves Pip 2008;85:295-305. [10] Hosseini SM, Akhlaghi M. Analytical solution in transient thermo-elasticity of functionally graded

thick

hollow

cylinders

(Pseudo-dynamic

analysis).

Math

Method

Appl

Sci

2009;32:2019-34. [11] Nie GJ, Batra RC. Stress analysis and material tailoring in isotropic linear thermoelastic incompressible functionally graded rotating disks of variable thickness. Compos Struct 2010;92:720-29. [12] Peng XL, Li XF. Thermoelastic analysis of a cylindrical vessel of functionally graded materials. Int J Pres Ves Pip 2010;87:203-10. [13] Peng XL, Li XF. Transient response of temperature and thermal stresses in a functionally graded hollow cylinder. J Therm Stresses 2010;33:485-500. [14] Sadeghian M, Toussi HE. Axisymmetric yielding of functionally graded spherical vessel under thermo-mechanical loading. Comput Mater Sci 2011;50:975-81. [15] Sadeghi H, Baghani M, Naghdabadi R. Strain gradient elasticity solution for functionally graded micro-cylinders. Int J Eng Sci 2012;50:22-30. [16] Tahvilian L, Fang ZZ. An investigation on thermal residual stresses in a cylindrical functionally graded WC-Co component. Mater Sci Eng A 2012;557:106-12. [17] Jabbari M, Bahtui A, Eslami MR. Axisymmetric mechanical and thermal stresses in thick short length FGM cylinders. Int J Pres Ves Pip 2009;86:296-306. [18] Nayebi A, Ansari Sadrabadi S. FGM elastoplastic analysis under thermomechanical loading. Int J Pres Ves Pip 2013;111-112:12-20. [19] Ozturk A, Gulgec M. Elastic-plastic stress analysis in a long functionally graded solid cylinder with fixed ends subjected to uniform heat generation. Int J Eng Sci 2011;49:1047-61. 19

[20] Shao ZS. Mechanical and thermal stresses of a functionally graded circular hollow cylinder with finite length. Int J Pres Ves Pip 2005;82:155-63. [21] Wu LH, Jiang ZQ, Liu J. Thermoelastic stability of functionally graded cylindrical shells. Compos Struct 2005;70:60-8. [22] Ching HK, Chen JK. Thermal stress analysis of functionally graded composites with temperature-dependent material properties. J Mech Mater Struct 2007;2:633-53. [23] Alavi F, Karimi D, Bagri A. An investigation on thermoelastic behaviour of functionally graded thick spherical vessels under combined thermal and mechanical loads. J Achi Mater Manuf Eng 2008;31:422-28. [24] Carrera E. A Class of Two Dimensional Theories for Multilayered Plates Analysis. Atti Accademia delle Scienze di Torino, Mem Sci Fis 1995;19-20:49-87. [25] Carrera E. Theories and finite elements for multilayered, anisotropic, composite plates and shells. Arch Comput Method E 2002;9:87-140. [26] Carrera E. Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking. Arch Comput Method E 2003;10: 215-96. [27] Carrera E, Brischetto S, Nali P. Plates and Shells for Smart Structures. John Wiley & Sons; 2011. [28] Carrera E, Cinefra M, Petrolo M, Zappino E. Finite Element Analysis of Structures through Unified Formulation. John Wiley & Sons; 2014. [29] Carrera E, Brischetto S, Robaldo A. A Variable Kinematic Model for the Analysis of Functionally Graded Materials Plates. AIAA J 2008;46:194-203. [30] Brischetto S, Carrera E. Mixed Theories for the Analysis of Functionally Graded Materials 20

Plates. presented on AIMETA 2007, Brescia (Italy), 14-17 September 2007. [31] Brischetto S, Leetsch R, Carrera E, Wallmersrerger T, Kröplin B. Thermo-Mechanical Bending of Functionally Graded Plates. J Therm Stresses 2008;31:286-308. [32] Cinefra M, Carrera E, Brischetto S, Belouettar S. Thermo-mechanical analysis of functionally graded shells. J Therm Stresses 2010;33:942-63. [33] Xin L, Dui G, Yang S, Zhang J. An elasticity solution for functionally graded thick-walled tube subjected to internal pressure. Int J Mech Sci 2014;89:344-49. [34] Wang J, Carson JK, North MF, Cleland DJ. A new approach to modelling the effective thermal conductivity of heterogeneous materials. Int J Heat Mass Tran 2006;49:3075-83. [35] Qu JM, Cherkaoui M. Fundamentals of Micromechanics of Solids. New Jersey: John Wiley &Sons, Inc.; 2006.

21

Table 1 Non-dimensional radial displacement u at top (t), middle (m) and bottom (b) of the FGM tube (n=1.5, k1/k0=2,

α1/α0=2, v0=0.3, v1=0.2). u (t) u (m) u (b)

ANSYS

This work

0.72022 0.80554 1.0746

0.71402 0.81637 1.0682

Table 2 Non-dimensional stresses σ ij at top (t), middle (m) and bottom (b) of the FGM tube (n=1.5, k1/k0=2, α1/α0=2,

v0=0.3, v1=0.2). σ r (t) σ r (m) σ r (b) σ θ (t) σ θ (m) σ θ (b) σ z (t) σ z (m) σ z (b)

ANSYS

This work

0 0.04775 0 0.65521 -0.05012 -0.47743 0.19468 -0.42508 -0.715

0 0.04732 0 0.64902 -0.04888 -0.48769 0.18948 -0.4311 -0.702

22

Figure captions

Fig. 1. The cross-sectional contour of a long FGM tube subjected to mechanical and thermal loads. Fig. 2. The cross-sectional contour of finite element modeling of the tube with brick 8-noded element. Fig. 3. The cross-sectional contours of the radial displacement and stresses contours based on the FEM: (a) The radial displacement, (b) The radial stress, (c) The circumferential stress, (d) The axial stress (n=1.5, k1/k0=2, α1/α0=2, v0=0.3, v1=0.2). Fig. 4. Comparisons of the theoretical and FEM results (n=1.5, k1/k0=2, α1/α0=2, v0=0.3, v1=0.2). Fig. 5. Evolution of the radial displacement u with different n (k1/k0=2, α1/α0=2, v0=0.3, v1=0.2).

Fig. 6. Evolution of the radial stress σr with different n (k1/k0=2, α1/α0=2, v0=0.3,

v1=0.2).

Fig. 7. Evolution of the circumferential stress σ θ with different n (k1/k0=2, α1/α0=2, v0=0.3, v1=0.2).

Fig. 8. Evolution of the axial stress σz with different n (k1/k0=2, α1/α0=2, v0=0.3, v1=0.2). Fig. 9. Evolution of the radial displacement u with different ratios k1/k0 (n=1.5, α1/α0=2, v0=0.3, v1=0.2).

Fig. 10. Evolution of the radial stress σr with different ratios k1/k0 (n=1.5, α1/α0=2, v0=0.3, v1=0.2).

Fig. 11. Evolution of the circumferential stress σ θ with different ratios k1/k0 (n=1.5, α1/α0=2, v0=0.3, v1=0.2).

Fig. 12. Evolution of the axial stress σz with different ratios k1/k0 (n=1.5, α1/α0=2, v0=0.3, v1=0.2). 23

Fig. 13. Evolution of the radial displacement u with different ratios α1/α0 (n=1.5, k1/k0=2, v0=0.3, v1=0.2).

Fig. 14. Evolution of the radial stress σr with different ratios α1/α0 (n=1.5, k1/k0=2, v0=0.3, v1=0.2).

Fig. 15. Evolution of the circumferential stress σ θ with different ratios α1/α0 (n=1.5, k1/k0=2, v0=0.3, v1=0.2).

Fig. 16. Evolution of the axial stress σz with different ratios α1/α0 (n=1.5, k1/k0=2, v0=0.3, v1=0.2).

Fig. 17. Evolution of the radial displacement u with different Poisson's ratios (n=1.5, k1/k0=2, α1/α0=2). Fig. 18. Evolution of the radial stress σr with different Poisson's ratios (n=1.5, k1/k0=2, α1/α0=2). Fig. 19. Evolution of the circumferential stress σ θ with different Poisson's ratios (n=1.5, k1/k0=2, α1/α0=2). Fig. 20. Evolution of the axial stress σz with different Poisson's ratios (n=1.5, k1/k0=2, α1/α0=2).

24

Tb

a

Ta pa

b

pb

Fig. 1. The cross-sectional contour of a long FGM tube subjected to mechanical and thermal loads.

Fig. 2. The cross-sectional contour of finite element modeling of the tube with brick 8-noded element.

25

(a) (b)

(c) (d) Fig. 3. The cross-sectional contours of the radial displacement and stresses contours based on the FEM: (a) The radial displacement, (b) The radial stress, (c) The circumferential stress, (d) The axial stress (n=1.5, k1/k0=2, α1/α0=2, v0=0.3, v1=0.2).

26

1.2 1.0

u

stresses and displacement

0.8 0.6

σθ

0.4 0.2

σr

0.0

-0.2

σz

-0.4 -0.6 -0.8 0.70

theoretical results FEM 0.75

0.80

0.85

radial coordinate

0.90

0.95

1.00

Fig. 4. Comparisons of the theoretical and FEM results (n=1.5, k1/k0=2, α1/α0=2, v0=0.3, v1=0.2).

1.0

0.8

u

0.6

0.4

n=1.5 n=3 n=5 n=10

0.2

0.0 0.70

0.75

0.80

0.85

0.90

0.95

1.00

r

Fig. 5. Evolution of the radial displacement u with different n (k1/k0=2, α1/α0=2, v0=0.3, v1=0.2).

0.07

n=1.5 n=3 n=5 n=10

0.06 0.05

σr

0.04 0.03 0.02 0.01 0.00 0.70

0.75

0.80

0.85

0.90

0.95

1.00

r

Fig. 6. Evolution of the radial stress σr with different n (k1/k0=2, α1/α0=2, v0=0.3, v1=0.2).

27

n=1.5 n=3 n=5 n=10

0.8 0.6 0.4

σθ

0.2 0.0 -0.2 -0.4 -0.6 0.70

0.75

0.80

0.85

0.90

0.95

1.00

r

Fig. 7. Evolution of the circumferential stress σ θ with different n (k1/k0=2, α1/α0=2, v0=0.3, v1=0.2). n=1.5 n=3 n=5 n=10

0.2

0.0

σz

-0.2

-0.4

-0.6

-0.8

-1.0 0.70

0.75

0.80

0.85

0.90

0.95

1.00

r

Fig. 8. Evolution of the axial stress σz with different n (k1/k0=2, α1/α0=2, v0=0.3, v1=0.2).

1.0

0.8

u

0.6

0.4

k1/k0=0.5 k1/k0=1

0.2

k1/k0=2 k1/k0=5 0.0 0.70

0.75

0.80

0.85

0.90

0.95

1.00

r

Fig. 9. Evolution of the radial displacement u with different ratios k1/k0 (n=1.5, α1/α0=2, v0=0.3, v1=0.2). 28

k1/k0=0.5

0.05

k1/k0=1 k1/k0=2

0.04

k1/k0=5

σr

0.03

0.02

0.01

0.00 0.70

0.75

0.80

0.85

0.90

0.95

1.00

r

Fig. 10. Evolution of the radial stress σr with different ratios k1/k0 (n=1.5, α1/α0=2, v0=0.3, v1=0.2). k1/k0=0.5 0.6

k1/k0=1 k1/k0=2

0.4

k1/k0=5

σθ

0.2 0.0 -0.2 -0.4 -0.6 0.70

0.75

0.80

0.85

0.90

0.95

1.00

r

Fig. 11. Evolution of the circumferential stress σ θ with different ratios k1/k0 (n=1.5, α1/α0=2, v0=0.3, v1=0.2). 0.2

k1/k0=0.5 0.1

k1/k0=1 k1/k0=2

0.0

k1/k0=5

σz

-0.1 -0.2 -0.3 -0.4 -0.5 -0.6 0.70

0.75

0.80

0.85

0.90

0.95

1.00

r

Fig. 12. Evolution of the axial stress σz with different ratios k1/k0 (n=1.5, α1/α0=2, v0=0.3, v1=0.2).

29

2.1 1.8 α1/α0=0.5 α1/α0=1

1.5

α1/α0=2 α1/α0=5

u

1.2 0.9 0.6 0.3 0.0 0.70

0.75

0.80

0.85

0.90

0.95

1.00

r

Fig. 13. Evolution of the radial displacement u with different ratios α1/α0 (n=1.5, k1/k0=2, v0=0.3, v1=0.2). 0.12 α1/α0=0.5 α1/α0=1

0.10

α1/α0=2 α1/α0=5

σr

0.08

0.06

0.04

0.02

0.00 0.70

0.75

0.80

0.85

0.90

0.95

1.00

r

Fig. 14. Evolution of the radial stress σr with different ratios α1/α0 (n=1.5, k1/k0=2, v0=0.3, v1=0.2). 1.5 α1/α0=0.5 α1/α0=1

1.0

α1/α0=2 α1/α0=5

σθ

0.5

0.0

-0.5

-1.0

-1.5 0.70

0.75

0.80

0.85

0.90

0.95

1.00

r

Fig. 15. Evolution of the circumferential stress σ θ with different ratios α1/α0 (n=1.5, k1/k0=2, v0=0.3, v1=0.2). 30

0.3

0.0

σz

-0.3

-0.6

-0.9

α1/α0=0.5 α1/α0=1

-1.2

α1/α0=2 α1/α0=5

0.70

0.75

0.80

0.85

0.90

0.95

1.00

r

Fig. 16. Evolution of the axial stress σz with different ratios α1/α0 (n=1.5, k1/k0=2, v0=0.3, v1=0.2). 1.2

1.0

u

0.8

0.6

0.4

v0=0.3,v1=0.2

0.2

0.0 0.70

v0=0.3,v1=0.3 v0=0.3,v1=0.4 0.75

0.80

0.85

0.90

0.95

1.00

r

Fig. 17. Evolution of the radial displacement u with different Poisson's ratios (n=1.5, k1/k0=2, α1/α0=2). 0.07

v0=0.3,v1=0.2 v0=0.3,v1=0.3

0.06

v0=0.3,v1=0.4 0.05

σr

0.04 0.03 0.02 0.01 0.00 0.70

0.75

0.80

0.85

0.90

0.95

1.00

r

Fig. 18. Evolution of the radial stress σr with different Poisson's ratios (n=1.5, k1/k0=2, α1/α0=2). 31

v0=0.3,v1=0.2

0.8

v0=0.3,v1=0.3 0.6

v0=0.3,v1=0.4

0.4

σθ

0.2 0.0 -0.2 -0.4 -0.6 0.70

0.75

0.80

0.85

0.90

0.95

1.00

r

Fig. 19. Evolution of the circumferential stress σ θ with different Poisson's ratios (n=1.5, k1/k0=2, α1/α0=2).

0.5

v0=0.3,v1=0.2 v0=0.3,v1=0.3

0.0

v0=0.3,v1=0.4 -0.5

σz

-1.0 -1.5 -2.0 -2.5 -3.0 0.70

0.75

0.80

0.85

0.90

0.95

1.00

r

Fig. 20. Evolution of the axial stress σz with different Poisson's ratios (n=1.5, k1/k0=2, α1/α0=2).

32

1.

A thermoelastic solution for a FGM thick-walled tube is given.

2.

The volume fraction is a power function varied in the radial direction.

3.

It avoids assuming the distribution regularities of unknown material parameters.

4.

It is available for the materials with different Poisson’s ratios.

33