- Email: [email protected]
Transient thermoelastic analysis for a multilayered thick strip with piecewise exponential nonhomogeneity
Composites: Part B 42 (2011) 973–981
Contents lists available at ScienceDirect
Composites: Part B journal homepage: www.elsevier.com/locate/compositesb
Transient thermoelastic analysis for a multilayered thick strip with piecewise exponential nonhomogeneity Yoshihiro Ootao Department of Mechanical Engineering, Graduate School of Engineering, Osaka Prefecture University, 1-1 Gakuen-cho, Nakaku, Sakai, Osaka 599-8531, Japan
a r t i c l e
i n f o
Article history: Received 19 August 2010 Received in revised form 30 November 2010 Accepted 10 December 2010 Available online 21 December 2010 Keywords: A. Layered structures B. Elasticity C. Analytical modeling
a b s t r a c t This paper is concerned with the theoretical treatment of transient thermoelastic problem involving a multilayered thick strip with piecewise exponential nonhomogeneity due to nonuniform heat supply in the width direction. The thermal and thermoelastic constants of each layer are assumed to vary exponentially in the thickness direction, and their values continue on the interfaces. We obtain the exact solution for the two-dimensional temperature change in a transient state, and thermoelastic response of a simple supported strip under the state of plane strain. Some numerical results for the temperature change, the displacement and the stress distributions are shown in figures. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Functionally graded materials (FGMs) was proposed as a new material which is adaptable for a super-high-temperature environment in Japan at first. FGMs are nonhomogeneous material systems that two or more different material ingredients change continuously and gradually. In recent years, the concept of FGMs has been applied in many industrial fields in addition to the aerospace field [1,2]. When FGMs are used under high-temperature conditions or are subjected to several thermal loading, it is necessary to analyze the thermal stress problems for FGMs. It is wellknown that thermal stress distributions in a transient state can show large values compared with the one in a steady state. Therefore, the transient thermoelastic problems for FGMs become important. The governing equations for the temperature field and the associate thermoelastic field of FGMs become of nonlinear form in generally, the analytical treatment is very difficult. As the analytical treatment of the thermoelastic problems of FGMs, there are two pieces of treatment mainly. One is introducing the theory of laminated composites, which have a number of homogeneous layers along the thickness direction. Using the theory of laminated composites, we analyzed theoretically the thermal stress problems of several analytical models for transient temperature field [3–8]. Sugano et al. reported an approximate three-dimensional analysis of thermal stresses in a nonhomogeneous plate with temperature change and nonhomogeneous properties only in the thickness direction [9] and an one-dimensional analysis of transient thermal
E-mail address: [email protected] 1359-8368/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2010.12.011
stress in a circular plate with arbitrary variation of heat-transfer coefficient [10]. On the other hand, the other analytical treatment is the exact analysis under the assumption that the material properties are given in the specific functions containing the variable of the thickness coordinate without using the laminated composite model. As the exact treatment without laminated composite models, Sugano analyzed exactly one-dimensional thermal stresses of nonhomogeneous plate where the thermal conductivity and Young’s modulus vary exponentially, whereas Poisson’s ratio and the coefficient of linear thermal expansion vary arbitrarily in the thickness direction [11]. Vel and Batra [12] analyzed the three-dimensional transient thermal stresses of the functionally graded rectangular plate. We analyzed the transient thermal stress problems of a functionally graded thick strip [13] and a functionally graded rectangular plate [14], where the thermal conductivity, the coefficient of linear thermal expansion and Young’s modulus vary exponentially in the thickness direction, due to nonuniform heat supply. The onedimensional solutions for transient thermal stresses of functionally graded hollow cylinders and hollow spheres whose material properties vary with the power product form of radial coordinate variable were obtained [15]. Zhao et al. analyzed the one-dimensional transient thermo-mechanical behavior of a functionally graded solid cylinder, whose thermoelastic constants vary exponentially through the thickness [16]. Shao et al. analyzed one-dimensional transient thermo-mechanical behavior of functionally graded hollow cylinders, whose thermoelastic constants are expressed as Taylor’s series [17]. In cases of actual engineering problems, we encounter case of nonuniform distributed heating. Then it is necessary to analyze the transient thermoelastic problems due to
974
Y. Ootao / Composites: Part B 42 (2011) 973–981
nonuniform heating under the convective boundary conditions. Shao et al. obtained the analytical solutions for transient thermomechanical response of functionally graded cylindrical panels [18] and functionally graded hollow cylinders [19]. We obtained the two-dimensional analytical solution for transient thermal stresses of a functionally graded cylindrical panel whose material properties vary with the power product form of radial coordinate variable [20]. Ohmichi et al. analyzed the transient thermal stress problem of the strip with boundaries oblique to the functionally graded direction theoretically [21]. However, these studies discuss the thermoelastic problems of one-layered FGM models, which have the big limitation of nonhomogeneity. On the other hand, the arbitrary nonhomogeneity can be expressed in the theory of laminated composites approximately, but the material properties are discontinuous on the interfaces. Guo and Noda proposed a piecewise-exponential model, for the crack problems in FGMs with arbitrary properties [22]. To the author’s knowledge, however, the exact analysis for transient thermoelastic problems of a multilayered thick strip with piecewise exponential nonhomogeneity has not been reported. From the viewpoint of above mentioned, we analyze the transient thermoelastic analysis for a multilayered thick strip with piecewise exponential nonhomogeneity under a plane strain condition.
heat conduction equations for the ith layer are taken in the following forms:
kti ðzi Þ
@2T i @ @T i @T i ¼ ci ðzi Þqi ðzi Þ þ k ðz Þ ti i @x2 @zi @zi @t
ð1Þ
The thermal conductivity kti and the heat capacity per unit volume ciqi in each layer are assumed to take the following forms:
kti ðzi Þ ¼ k0ti expðai zi =BÞ ci ðzi Þqi ðzi Þ ¼
c0i
ð2Þ
0 i
q expðki zi =BÞ
ð3Þ
where
ai ¼
ln k0t;iþ1 =k0ti h i
;
ki ¼
0iþ1 =c0i q 0i ln c0iþ1 q h
ð4Þ
i
Substituting the Eqs. (2) and (3) into the Eq. (1), the transient heat conduction equations in dimensionless form are
@2T i @T i @ 2 T i @T i þ ai þ 2 ¼ K2i eðki ai Þzi 2 @ x @zi @s @ zi
ð5Þ
where
c0 q 0 K2i ¼ i0 i kti
ð6Þ
The initial and thermal boundary conditions in dimensionless form are
2. Analysis Consider an infinite long, multilayered thick strip with piecewise exponential nonhomogeneity. The thermal and thermoelastic constants of each layer are assumed to vary exponentially in the thickness direction, and their values continue on the interfaces as show in Fig. 1. The length of the side in the width direction is denoted by Lx. The thickness of the laminated composite strip is represented by B. Let hi be the thickness ith, and coordinate axes x, z and zi are chosen as shown in Fig. 1. The thermal and thermoelastic constants of each layer are assumed to vary exponentially in the thickness direction, and their values continue on the interfaces.
s ¼ 0; T i ¼ 0; i ¼ 1; 2; . . . ; N
ð7Þ
@T 1 Ha T 1 ¼ Ha T a fa ðxÞ @ z1 ; z ¼ 0; T ¼ T ; i ¼ 1; 2; . . . ; N 1 zi ¼ h i iþ1 i iþ1 z1 ¼ 0;
; z ¼ 0; zi ¼ h i iþ1
@T i @T iþ1 ¼ ; @zi @ziþ1
i ¼ 1; 2; . . . ; N 1
@T N þ Hb T N ¼ Hb T b fb ðxÞ @zN x ¼ 0; Lx ; T i ¼ 0; i ¼ 1; 2; . . . ; N
N ; zN ¼ h
ð8Þ ð9Þ ð10Þ ð11Þ ð12Þ
In Eqs. (4)–(12), we have introduced the following dimensionless values:
2.1. Heat conduction problem We assume that the multilayered thick strip is initially at zero temperature and is suddenly heated from the lower and upper surfaces by surrounding media with relative heat-transfer coefficients ha and hb. We denote the temperatures of the surrounding media by the functions Tafa(x) and Tbfb(x) and assume its end surfaces (x = 0, Lx) are held zero temperature. Then the temperature distribution shows a two-dimensional distribution, and the transient
ðT i ; T a ; T b Þ k0 ; Lx Þ ¼ ðx; zi ; hi ; Lx Þ ; ; k0ti ¼ ti ; ðx; zi ; h i T0 B kt0 0 0 c q kt0 0i ¼ i i ; ðHa ; Hb Þ ¼ ðha ; hb ÞB j0 ¼ ; c0i q ð13Þ c 0 q0 c0 q0
ðT i ; T a ; T b Þ ¼
s¼
j0 t B2
;
where Ti is the temperature change; t is time; ji is thermal diffusivity; and T0, kt0, c0q0 and j0 are typical values of temperature, thermal conductivity, heat capacity per unit volume and thermal diffusivity,
Fig. 1. Analytical model and coordinate systems.
975
Y. Ootao / Composites: Part B 42 (2011) 973–981
respectively. Introducing the finite sine transformation with respect to the variable x and Laplace transformation with respect to the variable s, the solution of Eq. (5) can be obtained so as to satisfy the conditions (7)–(12). These solutions are shown as follows:
Ti ¼
1 X
T ik ðzi ; sÞ sin qk x;
i ¼ 1; 2; . . . ; N
ð14Þ
k¼1
e2i;2i1 ¼ e
where
ai 2 1 0 ai ð1ci0 Þzi T ik ðzi ; sÞ ¼ Ai e 2 þ B0i e 2 ð1þci0 Þzi Lx F " # 1 X ða1 k1 Þ2 l21j 2 ai s exp zi þ 2 4 l1j D0 ðl1j Þ j¼1
h ai ki zi þ Bi Y ci gi l1j Ai J ci ½gi l1j exp 2
ai ki zi if ai – ki exp 2
a 2i ð1ci0 Þh i
e2i;2iþ1 ¼ 1; e2iþ1;2i1 ¼
;
e2i;2i ¼ e
a 2i ð1þci0 Þh i
ð18Þ
;
e2i;2iþ2 ¼ 1; ai ai ð1 ci0 Þe 2 ð1ci0 Þhi ; 2
ai ai aiþ1 ð1 þ ci0 Þe 2 ð1þci0 Þhi ; e2iþ1;2iþ1 ¼ ð1 ciþ1;0 Þ; 2 2 aiþ1 ¼ ð1 þ ciþ1;0 Þ; i ¼ 1; 2; . . . ; N 1 2
e2iþ1;2i ¼ e2iþ1;2iþ2 ð15Þ
where Jci ½ and Y ci ½ are the Bessel functions of the first and second kind of order ci, respectively. And D and F are the determinants of 2N 2N matrix [akl] and [ekl], respectively; the coefficients Ai and Bi are defined as the determinant of the matrix similar to the coefficient matrix [akl], in which the (2i 1)th column or 2ith column is replaced by the constant vector {ck}, respectively. Similarly, the coefficients A0i and B0i are defined as the determinant of the matrix similar to the coefficient matrix [ekl], in which the (2i 1)th column or 2ith column is replaced by the constant vector {ck}, respectively. The nonzero elements of the coefficient matrices [akl] , [ekl] and the constant vector {ck} are given as
a1 a1 k1 a k a1;1 ¼ Ha þ þ c1 Jc1 ðK1 l1 Þ þ 1 1 K1 l1 Jc1 þ1 ðK1 l1 Þ; 2 2 2
a1 a1 k1 a1 k1 c1 Y c1 ðK1 l1 Þ þ K1 l1 Y c1 þ1 ðK1 l1 Þ; a1;2 ¼ Ha þ þ 2 2 2
aN aN kN aN aN kN a2N;2N1 ¼ e 2 hN Hb cN JcN KN lN e 2 hN 2 2 aN kN aN kN aN kN KN lN e 2 hN JcN þ1 KN lN e 2 hN ; þ 2
aN aN kN aN aN kN a2N;2N ¼ e 2 hN Hb cN Y cN KN lN e 2 hN 2 2 aN kN aN kN aN kN ð16Þ KN lN e 2 hN Y cN þ1 KN lN e 2 hN þ 2 ai ai ki ai ai ki a2i;2i1 ¼ e 2 hi J ci Ki li e 2 hi ; a2i;2i ¼ e 2 hi Y ci Ki li e 2 hi ; a2i;2iþ1 ¼ J ciþ1 ðKiþ1 liþ1 Þ; a2i;2iþ2 ¼ Y ciþ1 ðKiþ1 liþ1 Þ
ai ai ki ai ai ki ci Jci Ki li e 2 hi a2iþ1;2i1 ¼ e 2 hi 2 2 a k a k ai ki i 2 ih i 2 ih iJ i ; Ki l i e K l e þ i ci þ1 i 2
ai ai ki ai ai ki ci Y ci Ki li e 2 hi a2iþ1;2i ¼ e 2 hi 2 2 ai ki ai ki ai ki Ki li e 2 hi Y ci þ1 Ki li e 2 hi ; þ 2
aiþ1 aiþ1 kiþ1 ciþ1 Jciþ1 ðKiþ1 liþ1 Þ a2iþ1;2iþ1 ¼ 2 2 aiþ1 kiþ1 Kiþ1 liþ1 Jciþ1 þ1 ðKiþ1 liþ1 Þ ; þ 2
aiþ1 aiþ1 kiþ1 a2iþ1;2iþ2 ¼ ciþ1 Y ciþ1 ðKiþ1 liþ1 Þ 2 2 aiþ1 kiþ1 Kiþ1 liþ1 Y ciþ1 þ1 ðKiþ1 liþ1 Þ ; þ 2 i ¼ 1; 2; . . . ; N 1
a1 a1 ð1 c10 Þ Ha ; e1;2 ¼ ð1 þ c10 Þ Ha ; 2 2 h a i aN N 2 ð1cN0 Þh N e2N;2N1 ¼ ð1 cN0 Þ þ Hb e ; 2 h a i aN N e2N;2N ¼ ð1 þ cN0 Þ þ Hb e 2 ð1þcN0 ÞhN 2
e1;1 ¼
ð17Þ
C 1 ¼ Ha T a^f a ðqÞ;
C 2N ¼ Hb T b ^f b ðqÞ
ð19Þ ð20Þ
In Eq. (20), q represents the parameter of finite sine transformation with respect to the variable x and a symbol (^) represents the image 0 function. In Eqs. (14) and (15), qk, D (l1j), ci0, ci and gi are
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2i þ 4q2k kp dD ; D0 ðl1j Þ ¼ ; c ¼ ; i0 dl1 l1 ¼l1j a2i Lx sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a1 k1 a2i þ 4q2k Ki ci ¼ g ¼ ; i ai ki ðai ki Þ2 qk ¼
ð21Þ
and l1j represent the jth positive roots of the following transcendental equation
Dðl1 Þ ¼ 0
ð22Þ
Fig. 2. Temperature change (Case 1): (a) variation on the heated surface ðz ¼ 1Þ and (b) variation in the radial direction ð x ¼ 3Þ.
976
Y. Ootao / Composites: Part B 42 (2011) 973–981
In Eqs. (16) and (17), the relations between li, lN and l1 are
Ki li ¼ gi l1 ;
i ¼ 1; 2; . . . ; N
ð23Þ
The Young’s modulus of elasticity Ei , the coefficient of linear thermal expansion ai and Poisson’s ratio mi are assumed to take the following forms:
Ei ðzi Þ ¼ E0i expðli zi Þ;
2.2. Thermoelastic problem
a i ðzi Þ ¼ a 0i expðbizi Þ; mi ¼ const
ð27Þ
where Let analyze the transient thermoelasticity of a multilayered thick strip as a plane strain problem. The displacement–strain relations are expressed in dimensionless form as follows:
exxi ¼ u i;x ;
ezzi ¼ w i;z ;
ezxi ¼ ðu i;z þ w i;x Þ=2;
eyyi ¼ eyzi ¼ exyi ¼ 0
; i ¼ 1; 2; . . . ; N
ð24Þ
where a comma denotes partial differentiation with respect to the variable that follows. The stress–strain relation in dimensionless form is given by the following relation:
8 9 r xxi > > > > > > < r yyi = > r zzi > > > > > : ; r zxi
2 ¼
6 Ei 6 6 ð1 þ mi Þð1 2mi Þ 4
1 mi
mi mi
0 8 9 8 9 >1> > > > < exxi > = > <1> = a i Ei T i ezzi > >1> 1 2mi : > > > > ezxi ; > : ; 0
mi mi
0
1 mi
0
0
1 2mi
0
3 7 7 7 5
ð25Þ
ln E0iþ1 =E0i ; h
bi ¼
i
In Eqs. (24)–(28), introduced:
0 iþ1 =a 0i ln a h
ð28Þ
i
the following dimensionless values
are
0 ai ; a0i 0 Ei ; Ei e kli 0 i; a i ¼ r kli ¼ ; ekli ¼ ; a ; Ei ;Ei ¼ ; E0 a0 E0 T 0 a0 T 0 a0
rkli
iÞ ¼ i ; w ðu
ðui ;wi Þ a0 T 0 B
ð29Þ
where rkli are the stress components, ekli are the strain components, (ui,wi) are the displacement components, and a0 and E0 are the typical values of the coefficient of linear thermal expansion and Young’s modulus of elasticity, respectively. Substituting Eqs. (24), (25) and (27) into Eq. (26), the displacement equations of equilibrium are written as
i;zi þ li w i;x Þ þ w i;xx þ ð1 2mi Þðu i;zi z þ li u i;xzi 2ð1 mi Þu
The equilibrium equation is expressed in dimensionless form as follows:
r xxi;x þ r zxi;zi ¼ 0; r zxi;x þ r zzi;zi ¼ 0
li ¼
ð26Þ
Fig. 3. Variation of displacements on the heated surface ðz ¼ 1Þ (Case 1): (a) and (b) displacement w. displacement u
¼ 2ð1 þ mi Þebi zi T i;x
ð30Þ
i;xx þ 2ð1 mi Þðw i;zi zi þ li w i;x þ ð1 2mi Þw i;zi Þ i;xzi þ 2mi li u u ¼ 2ð1 þ mi Þebi zi ½ðli þ bi ÞT i þ T i;zi
ð31Þ
xx (Case 1): (a) variation on the heated surface ðz ¼ 1Þ and Fig. 4. Thermal stress r (b) variation in the thickness direction ð x ¼ 3Þ.
977
Y. Ootao / Composites: Part B 42 (2011) 973–981
If the lower and upper surfaces are traction free, and the interfaces of the each layer are perfectly bonded, then the boundary conditions of lower and upper surfaces and the conditions of continuity on the interfaces can be represented as follows:
r zz1 ¼ 0; r zx1 ¼ 0 ziþ1 ¼ 0; r zzi ¼ r zz;iþ1 ; r zxi ¼ r zx;iþ1 ;
z1 ¼ 0; ; z ¼ b i
i
i ¼ u iþ1 ; u N ; zN ¼ b
i ¼ w iþ1 ; w
r zzN ¼ 0; r zxN ¼ 0
i ¼ 1; 2; . . . ; N 1
ð32Þ ð33Þ
i ¼ 0 r xxi ¼ 0; w
½W cik ðzi Þ þ W pik ðzi Þ sin qk x
Case 1. Real Roots for si Given Ni real roots for si ; U cik ðzi Þ and W cik ðzi Þ are given by the following expressions:
ð35Þ U cik ðzi Þ ¼
Ni X
F kJi expðsiJ zi Þ;
J¼1
W cik ðzi Þ ¼
Ni X
M kJi ðsiJ ÞF kJi expðsiJ zi Þ
ð39Þ
J¼1
ð36Þ
k¼1
where
M qJi ðsiJ Þ ¼
In Eq. (36), the first term on the right side gives the homogeneous solution and the second term of right side gives the particular solution. We now consider the homogeneous solution. We express Ucik and Wcik as follows:
ðU cik ; W cik Þ ¼ U 0cik ; W 0cik expðsi zi Þ
ð38Þ
From Eq. (38), there might be four real roots, two real roots and one pair of conjugate complex roots, or two pairs of conjugate complex roots.
1 X i ¼ ½U cik ðzi Þ þ U pik ðzi Þ cos qk x; u
i ¼ w
exist leads to the following
mi 2 2 2 s4i þ 2li s3i þ li 2q2k s2i 2li q2k si þ q2k li þ qk ¼ 0 1 mi
We assume the solutions of Eqs. (30) and (31) in order to satisfy Eq. (35) in the following form.
k¼1 1 X
U 0cik ; W 0cik
ð34Þ
We now consider the case of a simply supported strip given by the following relations:
x ¼ 0; Lx ;
non-trivial solutions of equation.
ð37Þ
Substituting the first term on the right side of Eqs. (36) and (37) into the homogeneous equations of Eqs. (30) and (31), the condition that
zx (Case 1): (a) variation on the interface ðz ¼ 0:5Þ and (b) Fig. 5. Thermal stress r variation in the thickness direction ð x ¼ 2Þ.
ðsiJ þ 2mi li Þqk 2ð1 mi ÞsiJ ðsiJ þ li Þ ð1 2mi Þq2k
ð40Þ
In Eq. (39), FkJi are unknown constants. Case 2. Complex Roots for si If the complex roots for si is expressed by siJ = aiJ ± jbiJ, and given J 0i pairs of complex roots for si ; U cik ðzi Þ and W cik ðzi Þ are given by the following expressions:
zz (Case 1): (a) variation on the interface ðz ¼ 0:5Þ and (b) Fig. 6. Thermal stress r variation in the thickness direction ð x ¼ 3Þ.
978
Y. Ootao / Composites: Part B 42 (2011) 973–981 0
Ji X U cik ðzi Þ ¼ ½C 1Ji expðaiJ zi Þ cosðbiJ zj Þ þ C 2Ji expðaiJ zi Þ sinðbiJ zj Þ; J¼1 0
W cik ðzi Þ ¼
Ji X fC 1Ji expðaiJ zi Þ½CiJ cosðbiJ zj Þ XiJ sinðbiJ zj Þ J¼1
þ C 2Ji expðaiJ zi Þ½XiJ cosðbiJ zj Þ þ CiJ sinðbiJ zj Þg
ð41Þ
where
CiJ ¼ Re½MkJi jsiJ ¼aiJ þjbiJ ;
XiJ ¼ Im½MkJi jsiJ ¼aiJ þjbiJ
ð42Þ
pffiffiffiffiffiffiffi In Eq. (42), j, Re[ ] and Im[ ] are imaginary unit j ¼ 1, real part and imaginary part, respectively. Furthermore, in Eq. (41), C1Ji and C2Ji are unknown constants. In order to obtain the particular solution, we use the series expansions of the Bessel functions. Since the order ci of the Bessel functions in Eq. (15) is not integer in general, Eq. (15) can be written as the following expression.
h a i h a i i 0 i T ik ðzi ; sÞ ¼ a0i0 exp ð1 þ ci0 Þzi þ bi0 exp ð1 ci0 Þzi 2 2 1 X ðai ki Þ ai þ ain ðsÞ exp ð2n þ ci Þ zi 2 2 n¼0 ðai ki Þ ai ð2n ci Þ zi þ bin ðsÞ exp 2 2
ð43Þ
where
2 0 2 0 0 Bi ; bi0 ¼ Ai ; Lx F Lx F " # 1 ða1 k1 Þ2 l21j 2 X 2 exp s ain ðsÞ ¼ 0 4 Lx j¼1 l1j D ðl1j Þ
gi l1j 2nþci cos ci p ð1Þn ðAi þ Bi Þ ; sin ci p n!Cðci þ n þ 1Þ 2 " # 1 ðai ki Þ2 l21j 2 X 2 bin ðsÞ ¼ s exp 0 4 Lx j¼1 l1j D ðl1j Þ 2nci l1j Bi ð1Þn sin ci p n!Cðci þ n þ 1Þ 2
a0i0 ¼
ð44Þ
Fig. 7. Variation of temperature change in the thickness direction (Case 2, x ¼ 3).
U pik ðzi Þ and W pik ðzi Þ of the particular solution are obtained as the function system like Eq. (43). Then, the stress components can be evaluated by substituting Eq. (36) into Eq. (24), and later into Eq.
Fig. 8. Variation of displacements on the heated surface ðz ¼ 1Þ (Case 2) (a) and (b) displacement w. displacement u
xx (Case 2): (a) variation on the heated surface ðz ¼ 1Þ and Fig. 9. Thermal stress r (b) variation in the thickness direction ð x ¼ 3Þ.
Y. Ootao / Composites: Part B 42 (2011) 973–981
(25). The unknown constants in Eqs. (39) and (41) are determined so as to satisfy the boundary conditions (32)–(34). 3. Numerical results We consider the functionally graded materials composed of aluminum alloy and alumina. We assume that the multilayered thick strip is heated from the upper surface (alumina 100%) by surrounding media, the temperature of which is denoted by the symmetric function with respect to the center of strip. The material of the lower surface is aluminum alloy 100%. The material properties gi of the interface between ith layer and (i + 1)th layer are assumed as follows:
g i ¼ g a þ ðg b g a Þfi ;
0 6 fi 6 1; i ¼ 1; 2; . . . ; N 1
ð45Þ
where ga is the material property of the lower surface, and gb is the material property of the upper surface. The numerical parameters of heat conduction, shape and fi are presented as follows:
Ha ¼ Hb ¼ 5:0; T a ¼ 0; T b ¼ 1:0; Lx ¼ 6:0; fb ðxÞ ¼ 1 x02 =x20 Hðx0 jx0 jÞ; x0 ¼ x Lx =2; 1 ¼ h 2 ¼ 0:5; f1 ¼ 0:3; 0:5; 0:7 Case 1 : N ¼ 2; h
x0 ¼ 1:0
ð46Þ
ð47Þ 1 ¼ h 2 ¼ h 3 ¼ 1=3; Case 2 : N ¼ 3; h ðf1 ¼ 0:2; f2 ¼ 0:8Þ; ðf1 ¼ 0:3; f2 ¼ 0:7Þ; ðf1 ¼ 0:4; f2 ¼ 0:6Þ ð48Þ
Case 3 : N ¼ 1
ð49Þ
The multilayered thick strip is heated from the upper surface. The material constants for aluminum alloy are taken as,
zx (Case 2): (a) variation on the interface ðz ¼ 1=3Þ and (b) Fig. 10. Thermal stress r variation in the thickness direction ð x ¼ 2Þ.
979
j ¼ 90:6 106 m2 =s; c ¼ 905 J=ðkg KÞ; q ¼ 2688 kg=m3 ; kt ¼ 23:7 10 W=ðm KÞ; a ¼ 23:2 106 1=K; E ¼ 7:0 10 GPa; m ¼ 0:33 ð50Þ for alumina,
j ¼ 11:9 106 m2 =s; c ¼ 778:9 J=ðkg KÞ; q ¼ 3880 kg=m3 ; kt ¼ 3:6 10 W=ðm KÞ; a ¼ 8:0 106 1=K; E ¼ 34:3 10 GPa; m ¼ 0:33 ð51Þ The typical values of material properties such as j0, k0, a0 and E0 used to normalize the numerical data, are based on those of aluminum alloy. In order to examine the influence of the material property distribution for the two-layered FGM model, the numerical results for Case 1 are shown in Figs. 2–6. Fig. 2 shows the variations of temperature change. The variation on the heated surface ðz ¼ 1Þ is shown in Fig. 2a and the variation in the thickness direction at the midpoint ð x ¼ Lx =2Þ is shown in Fig. 2b, respectively. As shown in Fig. 2a, the temperature change rise can clearly be seen in the heated region. From Fig. 2, the temperature rises as time proceeds and is greatest in the steady state. It can be seen from Fig. 2 that the maximum value of the temperature change decreases when the parameter f1 decreases. Fig. 3 shows the variations of displacements on the heated surface ðz ¼ 1Þ. The variation of the displace is shown in Fig. 3a and the variation of the displacement w ment u is shown in Fig. 3b, respectively. From Fig. 3, the absolute values of and w in a steady state decrease when the the displacements u parameter f1 decreases, while those in a transient state (s = 0.1) increase when the parameter f1 decreases. Fig. 4 shows the variations
zz (Case 2): (a) variation on the interface ðz ¼ 2=3Þ and (b) Fig. 11. Thermal stress r variation in the thickness direction ð x ¼ 3Þ
980
Y. Ootao / Composites: Part B 42 (2011) 973–981
xx . The variation on the heated surface is shown of normal stress r in Fig. 4a and the variation in the thickness direction at the midpoint ð x ¼ Lx =2Þ of the strip is shown in Fig. 4b, respectively. From Fig. 4a, the large compressive stress occurs on the heated region in a transient state. From Fig. 4b, the maximum tensile stress occurs in the second layer in the transient state. It can be seen from Fig. 4 that the maximum values of the tensile and compressive stress decrease when the parameter f1 decreases. Fig. 5 shows the variations zx . The variation on the interface between the of shearing stress r first layer and the second layer ðz ¼ 0:5Þ is shown in Fig. 5a and the variation in the thickness direction at the edge ð x ¼ 2Þ of the heated region is shown in Fig. 5b, respectively. As shown in Fig. 5a, the maximum shearing stress occurs near x ¼ 2; 4 in a
transient state. From Fig. 5b, the maximum shearing stress occurs inside the second layer in a transient state. Fig. 6 shows the varia zz . The variation on the interface between tions of normal stress r the first layer and the second layer ðz ¼ 0:5Þ is shown in Fig. 6a and the variation in the thickness direction at the midpoint ð x ¼ Lx =2Þ of the strip is shown in Fig. 6b, respectively. As shown in Fig. 6a, the maximum tensile stress occurs in the heated region in a transient state. From Fig. 6b, the maximum tensile stress occurs inside the second layer in a transient state. In order to examine the influence of the material property distribution for the three-layered FGM model, the numerical results for Case 2 are shown in Figs. 7–11. Fig. 7 shows the variation of temperature change in the thickness direction at the midpoint ð x ¼ Lx =2Þ. Fig. 8 shows the variations of displacements on the is heated surface ðz ¼ 1Þ. The variation of the displacement u is shown shown in Fig. 8a and the variation of the displacement w in Fig. 8b, respectively. From Figs. 7 and 8, the maximum value of the temperature change decreases when the parameter f1 increases and w increases. and f2 decreases, while that of displacements u xx . The variation on Fig. 9 shows the variations of normal stress r the heated surface is shown in Fig. 9a and the variation in the thickness direction at the midpoint ð x ¼ Lx =2Þ of the strip is shown in Fig. 9b, respectively. Fig. 10 shows the variations of shearing zx . The variation on the interface between the first layer stress r and the second layer ðz ¼ 1=3Þ is shown in Fig. 10a and the variation in the thickness direction at the edge ð x ¼ 2Þ of the heated region is shown in Fig. 10b, respectively. Fig. 11 shows the variations zz . The variation on the interface between the of normal stress r second layer and the third layer ðz ¼ 2=3Þ is shown in Fig. 11a and the variation in the thickness direction at the midpoint ð x ¼ Lx =2Þ of the strip is shown in Fig. 11b, respectively. It can be seen from Figs. 9–11 that maximum values of the thermal stresses r xx ; r zx and r zz decrease when the parameter f1 increases and f2 decreases. In order to assess the influence of the functional grading, the numerical results for Case 3, i.e. one-layered FGM model, are shown in Fig. 12. Fig. 12a and c show the variations of normal xx and r zz in the thickness direction at the midpoint stresses r ð x ¼ Lx =2Þ of the strip, respectively. Fig. 12b shows the variation zx in the thickness direction at the edge of shearing stress r ð x ¼ 2Þ of the heated region. In comparison with the numerical results for Case 1 and Case 2, it is possible to decrease the maximum xx ; r zx and r zz using the multilayered values of thermal stresses r FGM model with piecewise exponential nonhomogeneity.
4. Conclusion
xx ð Fig. 12. Results for one-layered thick strip (Case 3): (a) normal stress r x ¼ 3Þ, (b) zx ð zz ð shearing stress r x ¼ 2Þ and (c) normal stress r x ¼ 3Þ.
In the present article, we analyzed the transient thermoelastic problem involving a multilayered thick strip with piecewise exponential nonhomogeneity due to nonuniform heat supply in the width direction. The thermal and thermoelastic constants of each layer are assumed to vary exponentially in the thickness direction, and their values continue on the interfaces. We obtained the exact solution for the transient two-dimensional temperature and transient thermoelastic response of a multilayered thick strip under the plane strain condition. As an illustration, we carried out numerical calculations for the two or three layered composite thick strips that the lower surface is pure aluminum alloy and the upper surface is pure alumina and examined the behaviors in the transient state for the temperature change, the displacement, and thermal stress displacements. Furthermore, the influence of the functionally grading on the temperature change, displacements, and thermal stresses is investigated. It is concluding that it is possible to decrease the maximum values of thermal stresses using the multilayered FGM model with
Y. Ootao / Composites: Part B 42 (2011) 973–981
piecewise exponential nonhomogeneity as compared with the onelayered FGM model. References [1] Suresh S, Mortensen A. Fundamentals of functionally graded materials: processing and thermomechanical behavior of graded metals and metal– ceramic composites. London: IOM Communications; 1998. [2] Miyamoto Y, Kaysser WA, Rabin BH, Kawasaki A, Ford RG. Functionally graded materials: design, processing and applications. Kluwer Academic Publishers.; 1999. [3] Ootao Y, Fukuda T, Tanigawa Y. Transient thermal stress analysis of a multilayered composite hollow cylinder due to asymmetric heating and its application to nonhomogeneous materials. Theor Appl Mech 1989;38:177–88. [4] Ootao Y, Tanigawa Y, Fukuda T. Axisymmetric transient thermal stress analysis of a multilayered composite hollow cylinder. J Therm Stresses 1991;14:201–13. [5] Tanigawa Y, Ootao Y, Kawamura R. Thermal bending of laminated composite rectangular plates and nonhomogeneous plates due to partial heating. J Therm Stresses 1991;14:285–308. [6] Ootao Y, Tanigawa Y. Three-dimensional transient thermal stress analysis of nonhomogeneous hollow sphere with respect to rotating heat source. Trans Jpn Soc Mech Eng 1994;60A:2273–9. [7] Ootao Y, Akai T, Tanigawa Y. Three-dimensional transient thermal stress analysis of nonhomogeneous hollow circular cylinder due to a moving heat source in the axial direction. J Therm Stresses 1995;18:497–512. [8] Ootao Y, Tanigawa Y. Three-dimensional transient thermal stresses of functionally graded rectangular plate due to partial heating. J Therm Stresses 1999;22:35–55. [9] Sugano Y, Sato K, Kimura N, Sumi N. Three-dimensional analysis of transient thermal stresses in a nonhomogeneous plate. Trans Jpn Soc Mech Eng 1996;62A:728–36.
981
[10] Sugano Y, Chiba R, Hirose K. Analytical solutions of transient temperature and thermal stress in a circular plate with arbitrary variation of heat-transfer coefficient. Trans Jpn Soc Mech Eng 2001;67A:542–8. [11] Sugano Y. An expression for transient thermal stress in a nonhomogeneous plate with temperature variation though thickness. Ing Arch 1987;57:147–56. [12] Vel SS, Batra RC. Three-dimensional analysis of transient thermal stresses in functionally graded plates. Int J Solids Struct 2003;40:7181–96. [13] Ootao Y, Tanigawa Y. Transient thermoelastic problem of functionally graded thick strip due to nonuniform heat supply. Comp Struc 2004;63:139–46. [14] Ootao Y, Tanigawa Y. Three-dimensional solution for transient thermal stresses of functionally graded rectangular plate due to nonuniform heat supply. Int J Mech Sci 2005;47:1769–88. [15] Ootao Y, Tanigawa Y. One-dimensional solutions for transient thermal stresses in functionally graded hollow cylinders and hollow spheres. AIP Conf Proc 2008;973:682–7. [16] Zhao J, Ai X, Li Y, Zhou Y. Thermal shock resistance of functionally gradient solid cylinders. Mater Sci Eng A 2006;418:99–110. [17] Shao ZS, Wang TJ, Ang KK. Transient thermomechanical analysis of graded hollow circular cylinders. J Therm Stresses 2007;30:81–104. [18] Shao ZS, Wang TJ, Ang KK. Transient thermomechanical stresses of functionally graded cylindrical panels. AIAA J 2007;45:2487–96. [19] Shao ZS, Ang KK, Reddy JN, Wang TJ. Nonaxisymmetric thermomechanical analysis of functionally graded hollow cylinders. J Therm Stresses 2008;31:515–36. [20] Ootao Y, Tanigawa Y. Transient thermoelastic problem of a functionally graded cylindrical panel due to nonuniform heat supply. J Therm Stresses 2007;30:441–57. [21] Ohmichi M, Noda N, Ishihara M. The effect of oblique functional gradation to transient thermal stresses in the functionally graded infinite strip. Acta Mech 2010;212:219–32. [22] Guo LC, Noda N. Modeling method for a crack problem of functionally graded materials with arbitrary properties – piecewise-exponential model. Int J Solids Struct 2007;44:6768–90.
Contents lists available at ScienceDirect
Composites: Part B journal homepage: www.elsevier.com/locate/compositesb
Transient thermoelastic analysis for a multilayered thick strip with piecewise exponential nonhomogeneity Yoshihiro Ootao Department of Mechanical Engineering, Graduate School of Engineering, Osaka Prefecture University, 1-1 Gakuen-cho, Nakaku, Sakai, Osaka 599-8531, Japan
a r t i c l e
i n f o
Article history: Received 19 August 2010 Received in revised form 30 November 2010 Accepted 10 December 2010 Available online 21 December 2010 Keywords: A. Layered structures B. Elasticity C. Analytical modeling
a b s t r a c t This paper is concerned with the theoretical treatment of transient thermoelastic problem involving a multilayered thick strip with piecewise exponential nonhomogeneity due to nonuniform heat supply in the width direction. The thermal and thermoelastic constants of each layer are assumed to vary exponentially in the thickness direction, and their values continue on the interfaces. We obtain the exact solution for the two-dimensional temperature change in a transient state, and thermoelastic response of a simple supported strip under the state of plane strain. Some numerical results for the temperature change, the displacement and the stress distributions are shown in figures. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Functionally graded materials (FGMs) was proposed as a new material which is adaptable for a super-high-temperature environment in Japan at first. FGMs are nonhomogeneous material systems that two or more different material ingredients change continuously and gradually. In recent years, the concept of FGMs has been applied in many industrial fields in addition to the aerospace field [1,2]. When FGMs are used under high-temperature conditions or are subjected to several thermal loading, it is necessary to analyze the thermal stress problems for FGMs. It is wellknown that thermal stress distributions in a transient state can show large values compared with the one in a steady state. Therefore, the transient thermoelastic problems for FGMs become important. The governing equations for the temperature field and the associate thermoelastic field of FGMs become of nonlinear form in generally, the analytical treatment is very difficult. As the analytical treatment of the thermoelastic problems of FGMs, there are two pieces of treatment mainly. One is introducing the theory of laminated composites, which have a number of homogeneous layers along the thickness direction. Using the theory of laminated composites, we analyzed theoretically the thermal stress problems of several analytical models for transient temperature field [3–8]. Sugano et al. reported an approximate three-dimensional analysis of thermal stresses in a nonhomogeneous plate with temperature change and nonhomogeneous properties only in the thickness direction [9] and an one-dimensional analysis of transient thermal
E-mail address: [email protected] 1359-8368/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2010.12.011
stress in a circular plate with arbitrary variation of heat-transfer coefficient [10]. On the other hand, the other analytical treatment is the exact analysis under the assumption that the material properties are given in the specific functions containing the variable of the thickness coordinate without using the laminated composite model. As the exact treatment without laminated composite models, Sugano analyzed exactly one-dimensional thermal stresses of nonhomogeneous plate where the thermal conductivity and Young’s modulus vary exponentially, whereas Poisson’s ratio and the coefficient of linear thermal expansion vary arbitrarily in the thickness direction [11]. Vel and Batra [12] analyzed the three-dimensional transient thermal stresses of the functionally graded rectangular plate. We analyzed the transient thermal stress problems of a functionally graded thick strip [13] and a functionally graded rectangular plate [14], where the thermal conductivity, the coefficient of linear thermal expansion and Young’s modulus vary exponentially in the thickness direction, due to nonuniform heat supply. The onedimensional solutions for transient thermal stresses of functionally graded hollow cylinders and hollow spheres whose material properties vary with the power product form of radial coordinate variable were obtained [15]. Zhao et al. analyzed the one-dimensional transient thermo-mechanical behavior of a functionally graded solid cylinder, whose thermoelastic constants vary exponentially through the thickness [16]. Shao et al. analyzed one-dimensional transient thermo-mechanical behavior of functionally graded hollow cylinders, whose thermoelastic constants are expressed as Taylor’s series [17]. In cases of actual engineering problems, we encounter case of nonuniform distributed heating. Then it is necessary to analyze the transient thermoelastic problems due to
974
Y. Ootao / Composites: Part B 42 (2011) 973–981
nonuniform heating under the convective boundary conditions. Shao et al. obtained the analytical solutions for transient thermomechanical response of functionally graded cylindrical panels [18] and functionally graded hollow cylinders [19]. We obtained the two-dimensional analytical solution for transient thermal stresses of a functionally graded cylindrical panel whose material properties vary with the power product form of radial coordinate variable [20]. Ohmichi et al. analyzed the transient thermal stress problem of the strip with boundaries oblique to the functionally graded direction theoretically [21]. However, these studies discuss the thermoelastic problems of one-layered FGM models, which have the big limitation of nonhomogeneity. On the other hand, the arbitrary nonhomogeneity can be expressed in the theory of laminated composites approximately, but the material properties are discontinuous on the interfaces. Guo and Noda proposed a piecewise-exponential model, for the crack problems in FGMs with arbitrary properties [22]. To the author’s knowledge, however, the exact analysis for transient thermoelastic problems of a multilayered thick strip with piecewise exponential nonhomogeneity has not been reported. From the viewpoint of above mentioned, we analyze the transient thermoelastic analysis for a multilayered thick strip with piecewise exponential nonhomogeneity under a plane strain condition.
heat conduction equations for the ith layer are taken in the following forms:
kti ðzi Þ
@2T i @ @T i @T i ¼ ci ðzi Þqi ðzi Þ þ k ðz Þ ti i @x2 @zi @zi @t
ð1Þ
The thermal conductivity kti and the heat capacity per unit volume ciqi in each layer are assumed to take the following forms:
kti ðzi Þ ¼ k0ti expðai zi =BÞ ci ðzi Þqi ðzi Þ ¼
c0i
ð2Þ
0 i
q expðki zi =BÞ
ð3Þ
where
ai ¼
ln k0t;iþ1 =k0ti h i
;
ki ¼
0iþ1 =c0i q 0i ln c0iþ1 q h
ð4Þ
i
Substituting the Eqs. (2) and (3) into the Eq. (1), the transient heat conduction equations in dimensionless form are
@2T i @T i @ 2 T i @T i þ ai þ 2 ¼ K2i eðki ai Þzi 2 @ x @zi @s @ zi
ð5Þ
where
c0 q 0 K2i ¼ i0 i kti
ð6Þ
The initial and thermal boundary conditions in dimensionless form are
2. Analysis Consider an infinite long, multilayered thick strip with piecewise exponential nonhomogeneity. The thermal and thermoelastic constants of each layer are assumed to vary exponentially in the thickness direction, and their values continue on the interfaces as show in Fig. 1. The length of the side in the width direction is denoted by Lx. The thickness of the laminated composite strip is represented by B. Let hi be the thickness ith, and coordinate axes x, z and zi are chosen as shown in Fig. 1. The thermal and thermoelastic constants of each layer are assumed to vary exponentially in the thickness direction, and their values continue on the interfaces.
s ¼ 0; T i ¼ 0; i ¼ 1; 2; . . . ; N
ð7Þ
@T 1 Ha T 1 ¼ Ha T a fa ðxÞ @ z1 ; z ¼ 0; T ¼ T ; i ¼ 1; 2; . . . ; N 1 zi ¼ h i iþ1 i iþ1 z1 ¼ 0;
; z ¼ 0; zi ¼ h i iþ1
@T i @T iþ1 ¼ ; @zi @ziþ1
i ¼ 1; 2; . . . ; N 1
@T N þ Hb T N ¼ Hb T b fb ðxÞ @zN x ¼ 0; Lx ; T i ¼ 0; i ¼ 1; 2; . . . ; N
N ; zN ¼ h
ð8Þ ð9Þ ð10Þ ð11Þ ð12Þ
In Eqs. (4)–(12), we have introduced the following dimensionless values:
2.1. Heat conduction problem We assume that the multilayered thick strip is initially at zero temperature and is suddenly heated from the lower and upper surfaces by surrounding media with relative heat-transfer coefficients ha and hb. We denote the temperatures of the surrounding media by the functions Tafa(x) and Tbfb(x) and assume its end surfaces (x = 0, Lx) are held zero temperature. Then the temperature distribution shows a two-dimensional distribution, and the transient
ðT i ; T a ; T b Þ k0 ; Lx Þ ¼ ðx; zi ; hi ; Lx Þ ; ; k0ti ¼ ti ; ðx; zi ; h i T0 B kt0 0 0 c q kt0 0i ¼ i i ; ðHa ; Hb Þ ¼ ðha ; hb ÞB j0 ¼ ; c0i q ð13Þ c 0 q0 c0 q0
ðT i ; T a ; T b Þ ¼
s¼
j0 t B2
;
where Ti is the temperature change; t is time; ji is thermal diffusivity; and T0, kt0, c0q0 and j0 are typical values of temperature, thermal conductivity, heat capacity per unit volume and thermal diffusivity,
Fig. 1. Analytical model and coordinate systems.
975
Y. Ootao / Composites: Part B 42 (2011) 973–981
respectively. Introducing the finite sine transformation with respect to the variable x and Laplace transformation with respect to the variable s, the solution of Eq. (5) can be obtained so as to satisfy the conditions (7)–(12). These solutions are shown as follows:
Ti ¼
1 X
T ik ðzi ; sÞ sin qk x;
i ¼ 1; 2; . . . ; N
ð14Þ
k¼1
e2i;2i1 ¼ e
where
ai 2 1 0 ai ð1ci0 Þzi T ik ðzi ; sÞ ¼ Ai e 2 þ B0i e 2 ð1þci0 Þzi Lx F " # 1 X ða1 k1 Þ2 l21j 2 ai s exp zi þ 2 4 l1j D0 ðl1j Þ j¼1
h ai ki zi þ Bi Y ci gi l1j Ai J ci ½gi l1j exp 2
ai ki zi if ai – ki exp 2
a 2i ð1ci0 Þh i
e2i;2iþ1 ¼ 1; e2iþ1;2i1 ¼
;
e2i;2i ¼ e
a 2i ð1þci0 Þh i
ð18Þ
;
e2i;2iþ2 ¼ 1; ai ai ð1 ci0 Þe 2 ð1ci0 Þhi ; 2
ai ai aiþ1 ð1 þ ci0 Þe 2 ð1þci0 Þhi ; e2iþ1;2iþ1 ¼ ð1 ciþ1;0 Þ; 2 2 aiþ1 ¼ ð1 þ ciþ1;0 Þ; i ¼ 1; 2; . . . ; N 1 2
e2iþ1;2i ¼ e2iþ1;2iþ2 ð15Þ
where Jci ½ and Y ci ½ are the Bessel functions of the first and second kind of order ci, respectively. And D and F are the determinants of 2N 2N matrix [akl] and [ekl], respectively; the coefficients Ai and Bi are defined as the determinant of the matrix similar to the coefficient matrix [akl], in which the (2i 1)th column or 2ith column is replaced by the constant vector {ck}, respectively. Similarly, the coefficients A0i and B0i are defined as the determinant of the matrix similar to the coefficient matrix [ekl], in which the (2i 1)th column or 2ith column is replaced by the constant vector {ck}, respectively. The nonzero elements of the coefficient matrices [akl] , [ekl] and the constant vector {ck} are given as
a1 a1 k1 a k a1;1 ¼ Ha þ þ c1 Jc1 ðK1 l1 Þ þ 1 1 K1 l1 Jc1 þ1 ðK1 l1 Þ; 2 2 2
a1 a1 k1 a1 k1 c1 Y c1 ðK1 l1 Þ þ K1 l1 Y c1 þ1 ðK1 l1 Þ; a1;2 ¼ Ha þ þ 2 2 2
aN aN kN aN aN kN a2N;2N1 ¼ e 2 hN Hb cN JcN KN lN e 2 hN 2 2 aN kN aN kN aN kN KN lN e 2 hN JcN þ1 KN lN e 2 hN ; þ 2
aN aN kN aN aN kN a2N;2N ¼ e 2 hN Hb cN Y cN KN lN e 2 hN 2 2 aN kN aN kN aN kN ð16Þ KN lN e 2 hN Y cN þ1 KN lN e 2 hN þ 2 ai ai ki ai ai ki a2i;2i1 ¼ e 2 hi J ci Ki li e 2 hi ; a2i;2i ¼ e 2 hi Y ci Ki li e 2 hi ; a2i;2iþ1 ¼ J ciþ1 ðKiþ1 liþ1 Þ; a2i;2iþ2 ¼ Y ciþ1 ðKiþ1 liþ1 Þ
ai ai ki ai ai ki ci Jci Ki li e 2 hi a2iþ1;2i1 ¼ e 2 hi 2 2 a k a k ai ki i 2 ih i 2 ih iJ i ; Ki l i e K l e þ i ci þ1 i 2
ai ai ki ai ai ki ci Y ci Ki li e 2 hi a2iþ1;2i ¼ e 2 hi 2 2 ai ki ai ki ai ki Ki li e 2 hi Y ci þ1 Ki li e 2 hi ; þ 2
aiþ1 aiþ1 kiþ1 ciþ1 Jciþ1 ðKiþ1 liþ1 Þ a2iþ1;2iþ1 ¼ 2 2 aiþ1 kiþ1 Kiþ1 liþ1 Jciþ1 þ1 ðKiþ1 liþ1 Þ ; þ 2
aiþ1 aiþ1 kiþ1 a2iþ1;2iþ2 ¼ ciþ1 Y ciþ1 ðKiþ1 liþ1 Þ 2 2 aiþ1 kiþ1 Kiþ1 liþ1 Y ciþ1 þ1 ðKiþ1 liþ1 Þ ; þ 2 i ¼ 1; 2; . . . ; N 1
a1 a1 ð1 c10 Þ Ha ; e1;2 ¼ ð1 þ c10 Þ Ha ; 2 2 h a i aN N 2 ð1cN0 Þh N e2N;2N1 ¼ ð1 cN0 Þ þ Hb e ; 2 h a i aN N e2N;2N ¼ ð1 þ cN0 Þ þ Hb e 2 ð1þcN0 ÞhN 2
e1;1 ¼
ð17Þ
C 1 ¼ Ha T a^f a ðqÞ;
C 2N ¼ Hb T b ^f b ðqÞ
ð19Þ ð20Þ
In Eq. (20), q represents the parameter of finite sine transformation with respect to the variable x and a symbol (^) represents the image 0 function. In Eqs. (14) and (15), qk, D (l1j), ci0, ci and gi are
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2i þ 4q2k kp dD ; D0 ðl1j Þ ¼ ; c ¼ ; i0 dl1 l1 ¼l1j a2i Lx sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a1 k1 a2i þ 4q2k Ki ci ¼ g ¼ ; i ai ki ðai ki Þ2 qk ¼
ð21Þ
and l1j represent the jth positive roots of the following transcendental equation
Dðl1 Þ ¼ 0
ð22Þ
Fig. 2. Temperature change (Case 1): (a) variation on the heated surface ðz ¼ 1Þ and (b) variation in the radial direction ð x ¼ 3Þ.
976
Y. Ootao / Composites: Part B 42 (2011) 973–981
In Eqs. (16) and (17), the relations between li, lN and l1 are
Ki li ¼ gi l1 ;
i ¼ 1; 2; . . . ; N
ð23Þ
The Young’s modulus of elasticity Ei , the coefficient of linear thermal expansion ai and Poisson’s ratio mi are assumed to take the following forms:
Ei ðzi Þ ¼ E0i expðli zi Þ;
2.2. Thermoelastic problem
a i ðzi Þ ¼ a 0i expðbizi Þ; mi ¼ const
ð27Þ
where Let analyze the transient thermoelasticity of a multilayered thick strip as a plane strain problem. The displacement–strain relations are expressed in dimensionless form as follows:
exxi ¼ u i;x ;
ezzi ¼ w i;z ;
ezxi ¼ ðu i;z þ w i;x Þ=2;
eyyi ¼ eyzi ¼ exyi ¼ 0
; i ¼ 1; 2; . . . ; N
ð24Þ
where a comma denotes partial differentiation with respect to the variable that follows. The stress–strain relation in dimensionless form is given by the following relation:
8 9 r xxi > > > > > > < r yyi = > r zzi > > > > > : ; r zxi
2 ¼
6 Ei 6 6 ð1 þ mi Þð1 2mi Þ 4
1 mi
mi mi
0 8 9 8 9 >1> > > > < exxi > = > <1> = a i Ei T i ezzi > >1> 1 2mi : > > > > ezxi ; > : ; 0
mi mi
0
1 mi
0
0
1 2mi
0
3 7 7 7 5
ð25Þ
ln E0iþ1 =E0i ; h
bi ¼
i
In Eqs. (24)–(28), introduced:
0 iþ1 =a 0i ln a h
ð28Þ
i
the following dimensionless values
are
0 ai ; a0i 0 Ei ; Ei e kli 0 i; a i ¼ r kli ¼ ; ekli ¼ ; a ; Ei ;Ei ¼ ; E0 a0 E0 T 0 a0 T 0 a0
rkli
iÞ ¼ i ; w ðu
ðui ;wi Þ a0 T 0 B
ð29Þ
where rkli are the stress components, ekli are the strain components, (ui,wi) are the displacement components, and a0 and E0 are the typical values of the coefficient of linear thermal expansion and Young’s modulus of elasticity, respectively. Substituting Eqs. (24), (25) and (27) into Eq. (26), the displacement equations of equilibrium are written as
i;zi þ li w i;x Þ þ w i;xx þ ð1 2mi Þðu i;zi z þ li u i;xzi 2ð1 mi Þu
The equilibrium equation is expressed in dimensionless form as follows:
r xxi;x þ r zxi;zi ¼ 0; r zxi;x þ r zzi;zi ¼ 0
li ¼
ð26Þ
Fig. 3. Variation of displacements on the heated surface ðz ¼ 1Þ (Case 1): (a) and (b) displacement w. displacement u
¼ 2ð1 þ mi Þebi zi T i;x
ð30Þ
i;xx þ 2ð1 mi Þðw i;zi zi þ li w i;x þ ð1 2mi Þw i;zi Þ i;xzi þ 2mi li u u ¼ 2ð1 þ mi Þebi zi ½ðli þ bi ÞT i þ T i;zi
ð31Þ
xx (Case 1): (a) variation on the heated surface ðz ¼ 1Þ and Fig. 4. Thermal stress r (b) variation in the thickness direction ð x ¼ 3Þ.
977
Y. Ootao / Composites: Part B 42 (2011) 973–981
If the lower and upper surfaces are traction free, and the interfaces of the each layer are perfectly bonded, then the boundary conditions of lower and upper surfaces and the conditions of continuity on the interfaces can be represented as follows:
r zz1 ¼ 0; r zx1 ¼ 0 ziþ1 ¼ 0; r zzi ¼ r zz;iþ1 ; r zxi ¼ r zx;iþ1 ;
z1 ¼ 0; ; z ¼ b i
i
i ¼ u iþ1 ; u N ; zN ¼ b
i ¼ w iþ1 ; w
r zzN ¼ 0; r zxN ¼ 0
i ¼ 1; 2; . . . ; N 1
ð32Þ ð33Þ
i ¼ 0 r xxi ¼ 0; w
½W cik ðzi Þ þ W pik ðzi Þ sin qk x
Case 1. Real Roots for si Given Ni real roots for si ; U cik ðzi Þ and W cik ðzi Þ are given by the following expressions:
ð35Þ U cik ðzi Þ ¼
Ni X
F kJi expðsiJ zi Þ;
J¼1
W cik ðzi Þ ¼
Ni X
M kJi ðsiJ ÞF kJi expðsiJ zi Þ
ð39Þ
J¼1
ð36Þ
k¼1
where
M qJi ðsiJ Þ ¼
In Eq. (36), the first term on the right side gives the homogeneous solution and the second term of right side gives the particular solution. We now consider the homogeneous solution. We express Ucik and Wcik as follows:
ðU cik ; W cik Þ ¼ U 0cik ; W 0cik expðsi zi Þ
ð38Þ
From Eq. (38), there might be four real roots, two real roots and one pair of conjugate complex roots, or two pairs of conjugate complex roots.
1 X i ¼ ½U cik ðzi Þ þ U pik ðzi Þ cos qk x; u
i ¼ w
exist leads to the following
mi 2 2 2 s4i þ 2li s3i þ li 2q2k s2i 2li q2k si þ q2k li þ qk ¼ 0 1 mi
We assume the solutions of Eqs. (30) and (31) in order to satisfy Eq. (35) in the following form.
k¼1 1 X
U 0cik ; W 0cik
ð34Þ
We now consider the case of a simply supported strip given by the following relations:
x ¼ 0; Lx ;
non-trivial solutions of equation.
ð37Þ
Substituting the first term on the right side of Eqs. (36) and (37) into the homogeneous equations of Eqs. (30) and (31), the condition that
zx (Case 1): (a) variation on the interface ðz ¼ 0:5Þ and (b) Fig. 5. Thermal stress r variation in the thickness direction ð x ¼ 2Þ.
ðsiJ þ 2mi li Þqk 2ð1 mi ÞsiJ ðsiJ þ li Þ ð1 2mi Þq2k
ð40Þ
In Eq. (39), FkJi are unknown constants. Case 2. Complex Roots for si If the complex roots for si is expressed by siJ = aiJ ± jbiJ, and given J 0i pairs of complex roots for si ; U cik ðzi Þ and W cik ðzi Þ are given by the following expressions:
zz (Case 1): (a) variation on the interface ðz ¼ 0:5Þ and (b) Fig. 6. Thermal stress r variation in the thickness direction ð x ¼ 3Þ.
978
Y. Ootao / Composites: Part B 42 (2011) 973–981 0
Ji X U cik ðzi Þ ¼ ½C 1Ji expðaiJ zi Þ cosðbiJ zj Þ þ C 2Ji expðaiJ zi Þ sinðbiJ zj Þ; J¼1 0
W cik ðzi Þ ¼
Ji X fC 1Ji expðaiJ zi Þ½CiJ cosðbiJ zj Þ XiJ sinðbiJ zj Þ J¼1
þ C 2Ji expðaiJ zi Þ½XiJ cosðbiJ zj Þ þ CiJ sinðbiJ zj Þg
ð41Þ
where
CiJ ¼ Re½MkJi jsiJ ¼aiJ þjbiJ ;
XiJ ¼ Im½MkJi jsiJ ¼aiJ þjbiJ
ð42Þ
pffiffiffiffiffiffiffi In Eq. (42), j, Re[ ] and Im[ ] are imaginary unit j ¼ 1, real part and imaginary part, respectively. Furthermore, in Eq. (41), C1Ji and C2Ji are unknown constants. In order to obtain the particular solution, we use the series expansions of the Bessel functions. Since the order ci of the Bessel functions in Eq. (15) is not integer in general, Eq. (15) can be written as the following expression.
h a i h a i i 0 i T ik ðzi ; sÞ ¼ a0i0 exp ð1 þ ci0 Þzi þ bi0 exp ð1 ci0 Þzi 2 2 1 X ðai ki Þ ai þ ain ðsÞ exp ð2n þ ci Þ zi 2 2 n¼0 ðai ki Þ ai ð2n ci Þ zi þ bin ðsÞ exp 2 2
ð43Þ
where
2 0 2 0 0 Bi ; bi0 ¼ Ai ; Lx F Lx F " # 1 ða1 k1 Þ2 l21j 2 X 2 exp s ain ðsÞ ¼ 0 4 Lx j¼1 l1j D ðl1j Þ
gi l1j 2nþci cos ci p ð1Þn ðAi þ Bi Þ ; sin ci p n!Cðci þ n þ 1Þ 2 " # 1 ðai ki Þ2 l21j 2 X 2 bin ðsÞ ¼ s exp 0 4 Lx j¼1 l1j D ðl1j Þ 2nci l1j Bi ð1Þn sin ci p n!Cðci þ n þ 1Þ 2
a0i0 ¼
ð44Þ
Fig. 7. Variation of temperature change in the thickness direction (Case 2, x ¼ 3).
U pik ðzi Þ and W pik ðzi Þ of the particular solution are obtained as the function system like Eq. (43). Then, the stress components can be evaluated by substituting Eq. (36) into Eq. (24), and later into Eq.
Fig. 8. Variation of displacements on the heated surface ðz ¼ 1Þ (Case 2) (a) and (b) displacement w. displacement u
xx (Case 2): (a) variation on the heated surface ðz ¼ 1Þ and Fig. 9. Thermal stress r (b) variation in the thickness direction ð x ¼ 3Þ.
Y. Ootao / Composites: Part B 42 (2011) 973–981
(25). The unknown constants in Eqs. (39) and (41) are determined so as to satisfy the boundary conditions (32)–(34). 3. Numerical results We consider the functionally graded materials composed of aluminum alloy and alumina. We assume that the multilayered thick strip is heated from the upper surface (alumina 100%) by surrounding media, the temperature of which is denoted by the symmetric function with respect to the center of strip. The material of the lower surface is aluminum alloy 100%. The material properties gi of the interface between ith layer and (i + 1)th layer are assumed as follows:
g i ¼ g a þ ðg b g a Þfi ;
0 6 fi 6 1; i ¼ 1; 2; . . . ; N 1
ð45Þ
where ga is the material property of the lower surface, and gb is the material property of the upper surface. The numerical parameters of heat conduction, shape and fi are presented as follows:
Ha ¼ Hb ¼ 5:0; T a ¼ 0; T b ¼ 1:0; Lx ¼ 6:0; fb ðxÞ ¼ 1 x02 =x20 Hðx0 jx0 jÞ; x0 ¼ x Lx =2; 1 ¼ h 2 ¼ 0:5; f1 ¼ 0:3; 0:5; 0:7 Case 1 : N ¼ 2; h
x0 ¼ 1:0
ð46Þ
ð47Þ 1 ¼ h 2 ¼ h 3 ¼ 1=3; Case 2 : N ¼ 3; h ðf1 ¼ 0:2; f2 ¼ 0:8Þ; ðf1 ¼ 0:3; f2 ¼ 0:7Þ; ðf1 ¼ 0:4; f2 ¼ 0:6Þ ð48Þ
Case 3 : N ¼ 1
ð49Þ
The multilayered thick strip is heated from the upper surface. The material constants for aluminum alloy are taken as,
zx (Case 2): (a) variation on the interface ðz ¼ 1=3Þ and (b) Fig. 10. Thermal stress r variation in the thickness direction ð x ¼ 2Þ.
979
j ¼ 90:6 106 m2 =s; c ¼ 905 J=ðkg KÞ; q ¼ 2688 kg=m3 ; kt ¼ 23:7 10 W=ðm KÞ; a ¼ 23:2 106 1=K; E ¼ 7:0 10 GPa; m ¼ 0:33 ð50Þ for alumina,
j ¼ 11:9 106 m2 =s; c ¼ 778:9 J=ðkg KÞ; q ¼ 3880 kg=m3 ; kt ¼ 3:6 10 W=ðm KÞ; a ¼ 8:0 106 1=K; E ¼ 34:3 10 GPa; m ¼ 0:33 ð51Þ The typical values of material properties such as j0, k0, a0 and E0 used to normalize the numerical data, are based on those of aluminum alloy. In order to examine the influence of the material property distribution for the two-layered FGM model, the numerical results for Case 1 are shown in Figs. 2–6. Fig. 2 shows the variations of temperature change. The variation on the heated surface ðz ¼ 1Þ is shown in Fig. 2a and the variation in the thickness direction at the midpoint ð x ¼ Lx =2Þ is shown in Fig. 2b, respectively. As shown in Fig. 2a, the temperature change rise can clearly be seen in the heated region. From Fig. 2, the temperature rises as time proceeds and is greatest in the steady state. It can be seen from Fig. 2 that the maximum value of the temperature change decreases when the parameter f1 decreases. Fig. 3 shows the variations of displacements on the heated surface ðz ¼ 1Þ. The variation of the displace is shown in Fig. 3a and the variation of the displacement w ment u is shown in Fig. 3b, respectively. From Fig. 3, the absolute values of and w in a steady state decrease when the the displacements u parameter f1 decreases, while those in a transient state (s = 0.1) increase when the parameter f1 decreases. Fig. 4 shows the variations
zz (Case 2): (a) variation on the interface ðz ¼ 2=3Þ and (b) Fig. 11. Thermal stress r variation in the thickness direction ð x ¼ 3Þ
980
Y. Ootao / Composites: Part B 42 (2011) 973–981
xx . The variation on the heated surface is shown of normal stress r in Fig. 4a and the variation in the thickness direction at the midpoint ð x ¼ Lx =2Þ of the strip is shown in Fig. 4b, respectively. From Fig. 4a, the large compressive stress occurs on the heated region in a transient state. From Fig. 4b, the maximum tensile stress occurs in the second layer in the transient state. It can be seen from Fig. 4 that the maximum values of the tensile and compressive stress decrease when the parameter f1 decreases. Fig. 5 shows the variations zx . The variation on the interface between the of shearing stress r first layer and the second layer ðz ¼ 0:5Þ is shown in Fig. 5a and the variation in the thickness direction at the edge ð x ¼ 2Þ of the heated region is shown in Fig. 5b, respectively. As shown in Fig. 5a, the maximum shearing stress occurs near x ¼ 2; 4 in a
transient state. From Fig. 5b, the maximum shearing stress occurs inside the second layer in a transient state. Fig. 6 shows the varia zz . The variation on the interface between tions of normal stress r the first layer and the second layer ðz ¼ 0:5Þ is shown in Fig. 6a and the variation in the thickness direction at the midpoint ð x ¼ Lx =2Þ of the strip is shown in Fig. 6b, respectively. As shown in Fig. 6a, the maximum tensile stress occurs in the heated region in a transient state. From Fig. 6b, the maximum tensile stress occurs inside the second layer in a transient state. In order to examine the influence of the material property distribution for the three-layered FGM model, the numerical results for Case 2 are shown in Figs. 7–11. Fig. 7 shows the variation of temperature change in the thickness direction at the midpoint ð x ¼ Lx =2Þ. Fig. 8 shows the variations of displacements on the is heated surface ðz ¼ 1Þ. The variation of the displacement u is shown shown in Fig. 8a and the variation of the displacement w in Fig. 8b, respectively. From Figs. 7 and 8, the maximum value of the temperature change decreases when the parameter f1 increases and w increases. and f2 decreases, while that of displacements u xx . The variation on Fig. 9 shows the variations of normal stress r the heated surface is shown in Fig. 9a and the variation in the thickness direction at the midpoint ð x ¼ Lx =2Þ of the strip is shown in Fig. 9b, respectively. Fig. 10 shows the variations of shearing zx . The variation on the interface between the first layer stress r and the second layer ðz ¼ 1=3Þ is shown in Fig. 10a and the variation in the thickness direction at the edge ð x ¼ 2Þ of the heated region is shown in Fig. 10b, respectively. Fig. 11 shows the variations zz . The variation on the interface between the of normal stress r second layer and the third layer ðz ¼ 2=3Þ is shown in Fig. 11a and the variation in the thickness direction at the midpoint ð x ¼ Lx =2Þ of the strip is shown in Fig. 11b, respectively. It can be seen from Figs. 9–11 that maximum values of the thermal stresses r xx ; r zx and r zz decrease when the parameter f1 increases and f2 decreases. In order to assess the influence of the functional grading, the numerical results for Case 3, i.e. one-layered FGM model, are shown in Fig. 12. Fig. 12a and c show the variations of normal xx and r zz in the thickness direction at the midpoint stresses r ð x ¼ Lx =2Þ of the strip, respectively. Fig. 12b shows the variation zx in the thickness direction at the edge of shearing stress r ð x ¼ 2Þ of the heated region. In comparison with the numerical results for Case 1 and Case 2, it is possible to decrease the maximum xx ; r zx and r zz using the multilayered values of thermal stresses r FGM model with piecewise exponential nonhomogeneity.
4. Conclusion
xx ð Fig. 12. Results for one-layered thick strip (Case 3): (a) normal stress r x ¼ 3Þ, (b) zx ð zz ð shearing stress r x ¼ 2Þ and (c) normal stress r x ¼ 3Þ.
In the present article, we analyzed the transient thermoelastic problem involving a multilayered thick strip with piecewise exponential nonhomogeneity due to nonuniform heat supply in the width direction. The thermal and thermoelastic constants of each layer are assumed to vary exponentially in the thickness direction, and their values continue on the interfaces. We obtained the exact solution for the transient two-dimensional temperature and transient thermoelastic response of a multilayered thick strip under the plane strain condition. As an illustration, we carried out numerical calculations for the two or three layered composite thick strips that the lower surface is pure aluminum alloy and the upper surface is pure alumina and examined the behaviors in the transient state for the temperature change, the displacement, and thermal stress displacements. Furthermore, the influence of the functionally grading on the temperature change, displacements, and thermal stresses is investigated. It is concluding that it is possible to decrease the maximum values of thermal stresses using the multilayered FGM model with
Y. Ootao / Composites: Part B 42 (2011) 973–981
piecewise exponential nonhomogeneity as compared with the onelayered FGM model. References [1] Suresh S, Mortensen A. Fundamentals of functionally graded materials: processing and thermomechanical behavior of graded metals and metal– ceramic composites. London: IOM Communications; 1998. [2] Miyamoto Y, Kaysser WA, Rabin BH, Kawasaki A, Ford RG. Functionally graded materials: design, processing and applications. Kluwer Academic Publishers.; 1999. [3] Ootao Y, Fukuda T, Tanigawa Y. Transient thermal stress analysis of a multilayered composite hollow cylinder due to asymmetric heating and its application to nonhomogeneous materials. Theor Appl Mech 1989;38:177–88. [4] Ootao Y, Tanigawa Y, Fukuda T. Axisymmetric transient thermal stress analysis of a multilayered composite hollow cylinder. J Therm Stresses 1991;14:201–13. [5] Tanigawa Y, Ootao Y, Kawamura R. Thermal bending of laminated composite rectangular plates and nonhomogeneous plates due to partial heating. J Therm Stresses 1991;14:285–308. [6] Ootao Y, Tanigawa Y. Three-dimensional transient thermal stress analysis of nonhomogeneous hollow sphere with respect to rotating heat source. Trans Jpn Soc Mech Eng 1994;60A:2273–9. [7] Ootao Y, Akai T, Tanigawa Y. Three-dimensional transient thermal stress analysis of nonhomogeneous hollow circular cylinder due to a moving heat source in the axial direction. J Therm Stresses 1995;18:497–512. [8] Ootao Y, Tanigawa Y. Three-dimensional transient thermal stresses of functionally graded rectangular plate due to partial heating. J Therm Stresses 1999;22:35–55. [9] Sugano Y, Sato K, Kimura N, Sumi N. Three-dimensional analysis of transient thermal stresses in a nonhomogeneous plate. Trans Jpn Soc Mech Eng 1996;62A:728–36.
981
[10] Sugano Y, Chiba R, Hirose K. Analytical solutions of transient temperature and thermal stress in a circular plate with arbitrary variation of heat-transfer coefficient. Trans Jpn Soc Mech Eng 2001;67A:542–8. [11] Sugano Y. An expression for transient thermal stress in a nonhomogeneous plate with temperature variation though thickness. Ing Arch 1987;57:147–56. [12] Vel SS, Batra RC. Three-dimensional analysis of transient thermal stresses in functionally graded plates. Int J Solids Struct 2003;40:7181–96. [13] Ootao Y, Tanigawa Y. Transient thermoelastic problem of functionally graded thick strip due to nonuniform heat supply. Comp Struc 2004;63:139–46. [14] Ootao Y, Tanigawa Y. Three-dimensional solution for transient thermal stresses of functionally graded rectangular plate due to nonuniform heat supply. Int J Mech Sci 2005;47:1769–88. [15] Ootao Y, Tanigawa Y. One-dimensional solutions for transient thermal stresses in functionally graded hollow cylinders and hollow spheres. AIP Conf Proc 2008;973:682–7. [16] Zhao J, Ai X, Li Y, Zhou Y. Thermal shock resistance of functionally gradient solid cylinders. Mater Sci Eng A 2006;418:99–110. [17] Shao ZS, Wang TJ, Ang KK. Transient thermomechanical analysis of graded hollow circular cylinders. J Therm Stresses 2007;30:81–104. [18] Shao ZS, Wang TJ, Ang KK. Transient thermomechanical stresses of functionally graded cylindrical panels. AIAA J 2007;45:2487–96. [19] Shao ZS, Ang KK, Reddy JN, Wang TJ. Nonaxisymmetric thermomechanical analysis of functionally graded hollow cylinders. J Therm Stresses 2008;31:515–36. [20] Ootao Y, Tanigawa Y. Transient thermoelastic problem of a functionally graded cylindrical panel due to nonuniform heat supply. J Therm Stresses 2007;30:441–57. [21] Ohmichi M, Noda N, Ishihara M. The effect of oblique functional gradation to transient thermal stresses in the functionally graded infinite strip. Acta Mech 2010;212:219–32. [22] Guo LC, Noda N. Modeling method for a crack problem of functionally graded materials with arbitrary properties – piecewise-exponential model. Int J Solids Struct 2007;44:6768–90.