Thermoelastic Closed-form Solutions of FGM Plates Subjected to Temperature Change in Axial and Thickness Directions

An analytical solution of simply supported FGM plates under thermal loads is developed based on medium-thick plate assumption. Further to assume constant Poisson’s ratio and thermal expansion coefficient, the closed-form solutions of the FGM plates with through-the-thickness Young’s modulus under temperature change in x - and z -directions are evaluated, expressed in terms of the thermal axial force and thermal bending moment. The closed-form solutions confirmed by finite element analysis give a complete insight into the thermal-mechanical behavior of FGM plates. Hence, the deflection, strain, stress, axial force, and bending moment of the FGM plate under thermal loads in axial and thickness directions are discussed. Results show that the use of FGM makes the maximum stress from the top or bottom surface move to the inner portion of the FGM plate, and significantly reduces the maximum stress of the plates. Moreover, although the FGM plate is subjected to thermal load in the thickness direction, the deflection of the FGM plate can be zero by properly choosing the steep material gradation, directly derived from the obtained closed-form solution.


Introduction
Composite media have been widely used because of the high performance demands of engineering devices.However, in the interface of two different materials there exists stress concentration occurred by the mismatch of material properties, especially in the environment of high-temperature change.Therefore, the concept of Functional Graded Material (FGM) was introduced to simultaneously reduce thermal expansion mismatch [1], increase interface bonding strength [2,3], and enhance coating toughness [4].
Literatures corresponding to FGM plates subjected to thermal loads have been rapidly increased recently.Lee and Erodgan [5] studied the exponentially metal-rich, ceramic-rich, and linear FGMs under uniform thermal loading and showed that the metal-rich has the lowest stress singularity.Chi and Chung [6] evaluated the stress intensity factors of cracked coating-substrate composite media by the finite element method, and indicated if the material strength of the coating is weaker than that of the substrate, although the use of the S-type functionally graded material can eliminate the stress singularity on the interfaces, the S-type functionally graded material that behaves like a bridge connecting the material difference of the coating and substrate will help the crack further extends into the substrate.Based on the classical plate theory and Fourier series expansion, Chung and Chang [7] obtained the series solutions for power-law FGM, sigmoid FGM, and exponential FGM plates with the coefficient of thermal expansion varying continuously throughout the thickness direction subjected to linear temperature change in the z direction.The mechanical behaviors of imply supported beams made of functionally graded materials (FGM) under an in-plane loading was investigated by Ma and Lee [8], indicating that the response of load-frequency for the beams is quite different from what was observed in the analysis for beams made of pure materials when effects of both the transverse shear deformation and the temperature dependent material properties are simultaneously taken into account.Chareonsuk [9] used a high-order control volume finite element method to explore thermal stress analysis for FGM structural components subjected to steady-state thermal and mechanical loads at steady state with the unstructured mesh capability for arbitrary-shaped domain.Ghannadpour [10] applied a finite strip method to analyze the buckling behavior of rectangular functionally graded plates (FGPs) under thermal loadings, and discussed the effects of geometrical parameters and material properties on the FGPs' buckling temperature difference.Thermal effect on buckling and free vibration behavior of functionally graded (FG) microbeams based on classical and first order shear deformation beam theories to count for the effect of shear deformations is presented by Nateghi [11], indicating that higher temperature changes signify size dependency of FG microbeam.Tahvilian [12] examined the thermal residual stress distribution in a functionally graded cemented tungsten carbide (FG WC-Co) hollow cylinder with an emphasis on the effects of key variables, such as gradient profile and gradient thickness on the magnitude and distribution of the stress field, and pointed that the effect of gradient thickness.By the investigation of two-dimensional thermoelastic sliding frictional contact of functionally graded material (FGM) coated half-plane under the plane strain deformation, Liu [13] showed that the distribution of the contact stress can be altered and therefore the thermoelastic contact damage can be modified by adjusting the gradient index, Peclet number and friction coefficient.To gain better understanding of the thermo-mechanical behavior of layered structures, Liu [14] investigated the problem of a finite line bond between two orthotropic functionally graded strips under thermal loading, by using Fourier transforms technique.Zhang et al. [15] applied to study the mechanical and thermal buckling behaviors of ceramic-metal functionally grade plates and to investigate the influences of volume fraction exponent, boundary condition, length-to-thickness ratio and loading type on the buckling behaviors of functionally grade plates.Taking into account the effects of transverse shear strains as well as the transverse normal strain, Zenkour [16] refined plate theory as well as different plate theories to study the thermoelastic response of multilayered cross-ply laminates and angle-ply sandwich plates resting on Pasternak's or Winkler's elastic foundation.Assuming that the material properties depended on the temperature vary in the thickness direction by a simple power law distribution, Parandvar and Farid [17] studied large amplitude vibration of functionally graded material (FGM) plates subjected to combined random pressure and thermal load using finite element modal reduction method.Kulikov and Plotnikova [18] developed the method of sampling surfaces and its implementation for the three-dimensional steady-state problem of thermoelasticity for laminated functionally graded plates subjected to thermomechanical loading and indicated that sampling surfaces method can be applied efficiently to the 3D stress analysis for thermoelastic laminated FG plates with a specified accuracy utilizing the sufficient number of sampling surfaces.Trabelsi, et al. [19] investigated geometrically nonlinear post-buckling responses of Functionally Graded Material shell structures exposed to uniform, linear and nonlinear temperature distributions through the thickness direction based on a modified first order shear deformation theory.And the effect the geometrical parameters, the volume fraction index and boundary conditions on nonlinear responses are performed.A hybrid genetic algorithm with the complex method is developed by Ding and Wu [20] for the optimization of the material composition of a multi-layered functionally graded material plate with temperature-dependent material properties in order to minimize the thermal stresses induced in the plate when it is subjected to steady-state thermal loads.Sator et al. [21] presented the development of completely 2D formulation for bending of functionally graded plates subjected to stationary thermal loading.Zhang et al. [22] investigated the dynamic thermal buckling and postbuckling of imperfect functionally graded material (FGM) annular plates based on the nonlinear plate theory.And the effects of the loads, the material gradient and the initial geometric imperfections on the dynamic responses and the buckling critical temperatures of the FGM annular plates are analyzed in detail.
FGMs may be utilized to plate structures in engineering applications as a thermal barrier.Hence understanding the mechanical behavior of the thermal barrier is important in assessing the safety of FGM plate structures.It is well known that the closed-form solutions can provide a much better understanding of the thermo-mechanical behavior of FGM plates.In this study, based on the Fourier series expansion, the closed-form solution to the problem of simply-supported rectangular FGM plates subjected to temperature distribution change in the x-or z-direction is developed and proved by finite element calculation.To the authors' best knowledge, the closed-form solution to the problem concerned that is not found in the literature.

Governing equations of FGM plates under thermal Loading
Consider a simply-supported rectangular FGM plate with uniform thickness exposed to a temperature change ( , , ) T x y z .Further assume that the through-thickness functionality of material properties is graded.For the non-homogeneous elastic FGM plate, the stress-strain relation under thermal loading ( , , ) T x y z based on the assumptions of small deformation is [7]: By the definitions of the in-plane axial forces , the in-plane axial forces and the bending moments expressed in matrix forms are: where ( ) ( ) ( , , ) 1 ( )  ( , , ) By introducing the stress function   the equilibrium equation of FGM plates under thermal loading is expressed in terms of the deflection w and the stress function   And the compatibility equation of an FGM plates, where the definitions of quantities ij Q , ij S and ij P can be found in Chi and Chung [23].The equilibrium equations, Eq. ( 8), and the compatibility equation, Eq. ( 9), provide the simultaneous equations to solve for the stress function   , xy  and the deflection w for an FGM plate subjected to thermal loads.

Series Solution of
, sin( )sin( ) Then, substituting Eq. ( 11) into Eqs.( 4) and ( 5) yields the temperature dependent quantities sin( ) sin( ) * 12 sin( ) sin( ) The quantities * N , * M represent axial force and bending moment caused by temperature change, called thermal axial force and thermal bending moment in this study.To satisfy the loading condition in Eq. ( 11) and the boundary conditions in Eqs.(10) Consequently, the strains at neutral surface are:


And the strain and stress fields of the FGM plate under thermal loads are found as:


With the aid of Eq. ( 7), the in-plane axial forces and the bending moments of the FGM plate subjected to thermal loads are also obtained: sin( )sin( ) sin( )sin( )

Solution of the FGM plate with uniform Poisson's ratio
Delale and Erdogan [24] indicated that the influence of the Poisson's ratio on the deformation of the FGM plates will be much less than that of Young's modulus.The same conclusion also obtained by Chi and Chung [25].Therefore, this paragraph will derive the solutions for the FGM plates with Poisson's ratio and the coefficient of thermal expansion being uniform, but Young's modulus varying in the thickness direction.Consequently, the relations of the quantities SC  .

linear temperature change in the x-direction
Assume that the temperature change of the FGM plate varies only in the x direction, i.e., 12 ( , , ) ( , ) sin( ) sin( ) where mn T can be evaluated from Eq. (11b).For the assumption of , and () Fz=1, the thermal axial force * N and thermal bending moment * M defined in Eq. ( 13) are simplified as: * 11 (1 ) (1 ) 0 and the parameters  and  are: It is noted that the thermal bending moment * 0 M  , Eq. ( 20), coincides with the phenomenon of FGM plates subjected to thermal change in the x-direction.And the thermal axial force * N is independent of Young's modulus.By the use of Eqs. ( 14), ( 15), ( 18), ( 20) and ( 21), the deflection and the stress functions of the FGM plates with constant Poisson's ratio and constant thermal expansion coefficient subjected to temperature change in the x-direction are then found as: ( , ) 0 w x y  (22a) The closed-form solution in Eqs.(23) reveals if an FGM plate with constant and  is subjected to temperature change in the x-direction, the strains and axial forces are independent of Young's modulus.The bending moment 0 expected.Moreover, the stresses in Eq. ( 23) indicate that the stresses are a function of Ez, and hence the stresses along the thickness of the FGM plate have the same shapes as the corresponding variations of Young's moduli.

linear temperature change in the z-direction
Consider the situation that the temperature change of the FGM plate varies in the z-direction, i.e.

Material gradation
The Young's modulus of the considered FGM plates is assumed to vary continuously in the thickness direction (z-axis) based on power-law function (simply called P-FGM), sigmoid function (S-FGM), or exponential function (E-FGM).

P-FGM plates
The Young's modulus of P-FGM plates is defined by: where 1 E and 2 E are Young's moduli of the lowest ( ) Then the coefficients 11 A and 11 C of the P-FGM plates evaluated from Eq. ( 6) are: 12(1 ) is the stiffness of homogeneous plate with Young's modulus 2 E .Thus the characteristic of 11 C is the stiffness of FGM plates.

S-FGM plates
The Young's modulus of S-FGM plates can be calculated by: , and h of S-FGM plates can be evaluated as: Consequently, the coefficients 11 A and 11 C of S-FGM plates expressed in terms of material properties and the plate thickness are found as: )

E-FGM plates
The Young's modulus of E-FGM plates is specified as 2 () 2 ( ) , where ln( ) With the manner similar with P-FGM plates, the quantities 1 h , 2 h , 11 A and 11 C of the E-FGM plates are：

Numerical results
The dimensions of the FGM plate are taken as 100 ab  cm and 2 h  cm.The Poisson's ratio and the coefficient of thermal expansion are 0.3 v  and  3 The axial forces x N at points ( / 2, ) ay of P-FGM plates under linear temperature change in the x-direction for different aspect ratios / ab .

Linear temperature change in the z-direction
Assume that the temperature change varies linearly in the z-direction from Then the quantities * N and * M defined in Eqs. ( 13) are evaluated as: for S-FGM plates, and    EE ratios are illustrated in Fig. 4. Notably, it can be seen from Fig. 4 that the deflection 0 w  as 21 /5  EE  .This indicates that for FGM plates under temperature change in the thickness direction, one can choose the ratio of 21 / EE to a certain value such that no deflection occurs.This occurrence can be easily achieved by setting * 0 M  .Herein, with the aid of Eq. (39b), the material steep provides * 0 M  , and consequently gives zero deflection for S-FGM plates.Taking ( 3 gives zero deflection for P-FGM plates.The maximum deflections max w of the S-FGM plates versus the aspect ratio is displayed in Fig. 5 for different 21 / EE ratios and in Figs. 6 for different 12 / TT ratios, revealing that the maximum deflections max w as the aspect ratio, or material gradient, or temperature steep increases.

Conclusion
Based on the medium-thick plate assumption, the analytical solution of the simply-supported FGM plates under thermal loading by Fourier series expansion has been successfully developed and leads to the following conclusion: (1) Under the assumption of constant Poisson's ratio and constant thermal expansion coefficient, the closed-form solutions of the FGM plates with through-the-thickness Young's modulus under temperature change in the radial and thickness directions are evaluated, and give a complete insight into the thermal-mechanical behavior of FGM plates.

where 1 h and 2 h
are the distances of the neutral surface to the bottom and top surfaces of the FGM plate.The entries of [ ], [ ], [ ] A B C in Eqs.(2) and (3) are the integration of the material properties of the FGM plate: Simply-Supported FGM plates under thermal loadingConsider a simply-supported rectangular FGM plate with length a , width b , and uniform thickness h subjected to a temperature change( , , )  temperature change in the z-direction and x-y plane, respectively.For the simply-supported FGM plates, the transverse and tangential components of displacements are restricted to move but the axial components are allowed on all four edges.Therefore, the boundary conditions of the simply supported rectangular FGM plate are: noted from Eq. (27) that the strains and stresses along the thickness of the FGM plate with constant and  subjected to temperature change in the z-direction are functions of z and () zE z , respectively, and

2 zh 5 . 1
 ) of the FGM plate is fixed to 2 21 E GPa  and that at the bottom of the plate varies according to the ratio of 21 EE =1、2、5、10、 20 and 50, while Young's modulus inside the FGM plate is determined based on the material gradation defined in Section 4.3.The theoretical solutions are obtained according to the closed-form solutions and compared with the finite element analysis.plate under linear temperature change in the x-direction versus the aspect ratio / ab .linear temperature change in the x-direction Simply assume that the temperature change of the FGM plate linearly varies in the x-direction from 0 T at 0 x  to a T at xa  .The distribution of the temperature change ( , , ) T x y z is expressed as: 5, ..... mn  .Hence, substituting mn T into Eqs.(23) gives the solutions of strains, stresses, axial forces and the bending moments for the FGM plates with the material of () E E z  subjected to the temperature change in x-directions.Taking 0 T = 10 C and , 500 displayed in Fig. 1(b) shows that the strain x  vanishes for small as well as large temperature changes as the aspect ratio /3 ab  .The stresses x  at the central points of the P-FGM ( 2 p  ), S-FGM ( 2 p  ), and E-FGM plates for 21 / EE  1, 2, 5, 10, 20, 50 are illustrated in Figs. 2, showing the decrease of the compressive stress x  with the increase of 21 / EE .This phenomenon is attributed to that the increase 21 / EE for fixed 2 E means the decrease of the overall strength of the FGM plate, causing a decreases of the stresses.Moreover, it can be seen from Figs. 2 that the stress distribution along the thickness direction exhibits () Ezcurves for P-, S-, and E-FGM plates.The distribution of axial force x N along y-direction at /2 xa  are plotted in Fig. 3 for different aspect ratio / ab .Fig. 3 reveals that the axial force x Nappears peaks near the edges of 0 y  and yb  for small aspect ratio.This event is attributed to the simply-supported edges.Moreover, the axial forcex N increases as / ab increases.

1 (
Substitution Eqs.(38~40) to Eqs. (27), one can obtain the analytical solution of the FGM plates with the material property of () E E z  subjected to the temperature linearly change in the z-direction.

Fig. 4
Fig. 4 The deflections at the points ( , / 2) xb of S-FGM plate under linear temperature change in the

Fig. 5 Fig. 6 Fig. 7
Fig. 5 The maximum deflections of S-FGM plate ( 2 p  , 12 / 10 TT ) under linear temperature change in the z-direction versus the aspect ratio / ab for different 21 / EE ratios.(a) (b) , the deflection function w and the stress function   and 8he comparison of the different kind of FGM plates are illustrated in Figs.8