The firstorder shear deformation plate model, accounting for the exact neutral plane position, is exploited to investigate the uncoupled thermomechanical behavior of functionally graded (FG) plates. Functionally graded materials are mainly constructed to operate in high temperature environments. Also, FG plates are used in many applications (such as mechanical, electrical, and magnetic), where an amount of heat may be generated into the FG plate whenever other forms of energy (electrical, magnetic, etc.) are converted into thermal energy. Several simulations are performed to study the behavior of FG plates, subjected to thermomechanical loadings, and focus the attention on the effect of the heat source intensity. Most of the previous studies have considered the midplane neutral one, while the actual position of neutral plane for functionally graded plates is shifted and should be firstly determined. A comparative study is performed to illustrate the effect of considering the neutral plane position. The volume fraction of the two constituent materials of the FG plate is varied smoothly and continuously, as a continuous power function of the material position, along the thickness of the plate.
Functionally graded materials (FGMs) are microscopically inhomogeneous composite materials, in which the volume fraction of the two or more materials is varied smoothly and continuously as a continuous function of the material position along one or more dimension of the structure. These materials are mainly constructed to operate in high temperature environments.
In conventional laminated composite structures, homogeneous elastic laminae are bonded together to obtain enhanced mechanical and thermal properties. The main inconvenience of such an assembly is the creation of stress concentration sources along the interfaces and specifically when the structure is exposed to elevated temperatures. This can lead to many deficiencies such as delaminations, matrix cracks, and other damage mechanisms which may result from the abrupt change of the mechanical properties at the interface between the layers. One of the ways to overcome this problem is to use functionally graded materials (FGMs) with continuous material properties variations, which can lead to a continuity of the material properties. The concept of functionally graded material (FGM) was proposed in 1984 by the material scientists in Japan [
The FGM is suitable for various applications, such as thermal coatings of barrier for ceramic engines, gas turbines, nuclear fusions, optical thin layers, and biomaterial electronics. Cheng and Batra [
Functionally graded materials are used in many applications, owing to their stability in high thermal environments. To this aim, many approaches are developed to study the thermoelastic behavior of functionally graded materials. One of these approaches is the finite element analysis of such material type.
In this paper the boundary value problem of the uncoupled thermoelastic behavior of FG plate is formulated and solved. First, the temperature distribution is predicted to be used in the thermoelastic analysis of FG plate. Then, the firstorder shear deformation plate theory is proposed, accounting for the exact neutral plane position, for modeling the functionally graded plates. A parametric study is developed to investigate the effects of different distributions of the material properties on the response of the plates. Moreover, some numerical comparisons are performed to show the lack of accuracy while neglecting the effect of neutral plane position for FG plates.
In this section, a mathematical model is derived for the uncoupled thermoelasticity systems considering the exact neutral plane position. The temperature distribution and the effective material properties of the FGMs are determined firstly and used in the developed model as input data. The effect of neutral plane position is typically neglected in most previous studies, while the position of neutral plane for functionally graded plates must be predetermined. Modifications over the model formulated at [
Based on the FSDT assumptions, the transverse normals would not remain perpendicular to the midsurface but remain straight after deformation. Thus, the transverse shear strains and consequently the shear stresses are constant throughout the laminate thickness. In practice a convenient shear correction factor, equals to 5/6, is assumed for the analysis of the plates [
Coordinate system used for a typical FG plate.
The total strain is the variation of the continuum deformation with respect to its volume, so the linear GreenLagrange strains components for small deformations and moderate rotations (
The governing equations for the plate equilibrium are derived based on the principle of minimum total potential energy. So, the total potential energy takes the form
Based on the concept that the equivalent singlelayer theories are built up, that a heterogeneous plate is treated as a statically equivalent, single layer having a complex constitutive behavior, reducing the 3D continuum problem to 2D problem, the equivalent layer of the FG plate can be obtained. By integrating for the plate material properties through the plate thickness the equivalent singlelayer material matrix can be determined to be
Prior to the determination of the material stiffnesses, the location of the neutral plane must be given. Clearly, due to varying young’s modulus of the plate, the neutral plane is no longer at the midplane but shifted from the midplane unless for a plate with symmetrical young’s modulus [
The equivalent material stiffnesses of isotropic FG plate are
So, by minimizing the total potential energy (
Knowledge of the temperature distribution within a body is basically important in many engineering problems. This information will be highly required in computing the capacity of the heat flow in or out the body. Further, if a body is not free to expand in all the directions, some stresses may be developed inside the body. The magnitude of these thermal stresses will influence dramatically on the design of devices such as boilers, steam turbines, and jet engines. The first step in calculating the thermal stresses is to determine the temperature distribution within the body.
FGMs are primarily used in situations where large temperature gradients are encountered. Also, FG plates are used in many applications (such as mechanical, electrical, and magnetic), where an amount of heat may be generated into the FG plate whenever other forms of energy (electrical, magnetic, etc.) are converted into thermal energy. Within our analysis, constant temperatures are imposed at the ceramic and metal surfaces, but a scalar temperature field is assumed to vary continuously along the
The temperature distribution along the thickness can be obtained by solving the onedimensional steady state heat transfer equation with the presence of a heat source (
If the plate is in a steady state without any heat sources, (
It is convenient here to mention that, in general, there are many approaches for homogenization of FGMs. The choice of the approach should be based on the gradient of gradation relative to the size of a typical representative volume element. One of such approaches is the approximation approach. The linear rule of mixtures and the modified rule of mixtures by Tamura are convenient methods for estimating the equivalent material properties of the FGMs based on the approximation approach. Previous studies predicted that the linear rule of mixtures cannot reflect the detailed constituent geometry and the microstructure and provides a highly questionable accuracy compared to the modified rule of mixtures. On the other hand, the modified rule of mixtures by Tamura provides a convenient accuracy for a wide range of volume fractions and loading conditions. But, the modified rule of mixtures is restricted to the Young’s modulus, so any appropriate averaging method must be used to estimate the other thermomechanical properties. Usually the linear rule of mixtures is being conventionally employed [
For the analyses of the flexural behavior of a functionally graded plate, subjected only to transversal applied load, we have to determine the location of the neutral plane before solving the equilibrium equation of the plate. Clearly, due to the varying of Young’s modulus of the FG plate through the thickness, the neutral plane is no longer located at the midplane but shifted from it. To determine the position of the neutral plane, we construct a new coordinate system such that the new
In this case, similar to the usual treatment in the FSDT, the axial force for an infinitely wide FG plate subjected to transverse mechanical load is given by
Equation (
The displacements and normal rotations at any point into a finite element
Ninenode quadratic Lagrange rectangular element.
So, the total potential energy can be obtained, and (
In this section we present several numerical simulations in order to assess the behavior of functionally graded plates subjected to thermomechanical loads. A simple supported plate is considered for the investigation. The plate is made up of a ceramic material at the top and a metallic at the bottom. The simple power law with different values of
The analysis of FG plates is performed for a combination of materials of type ceramicmetal. The lower plate surface is assumed to be aluminum, while the top surface is assumed to be zirconia. Material properties parameter
Material properties.
Property  Aluminum  Zirconia 

Young’s modulus 


Poisson’s ratio 


Thermal conductivity 


Thermal expansion 


A simply supported FG plate subjected to a uniformly distributed mechanical load and thermal loading.
To investigate the thermomechanical behavior of the plate, the temperature distribution through the FG plate thickness should be firstly determined. Figure
Temperature distribution through the FG plate thickness for various values of grading parameter
To investigate the elastostatic behavior of FG plates, with aluminum and zirconia material constituents, several numerical simulations are performed, for different values of the grading parameter
central deflection
nondimensional load intensity
thickness
where
Figure
Nondimensional deflection of midpoint versus nondimensional load intensity (
Nondimensional central deflection ( 



Metal 



Ceramic  
Midplane  Neutral plane  Midplane  Neutral plane  Midplane  Neutral plane  
1  0.0452  0.0317  0.0328  0.0287  0.03  0.0267  0.0277  0.021 
2  0.0905 ( 
0.0633 ( 
0.0656  0.0573 ( 
0.06  0.0534 ( 
0.0554  0.042 ( 
3  0.1357  0.095  0.0985  0.086  0.09  0.0801  0.0831  0.0629 
4  0.181 ( 
0.1266 ( 
0.1313  0.1146 ( 
0.119  0.1067 ( 
0.1108  0.0839 ( 
5  0.2262  0.1583  0.1641  0.1433  0.149  0.1334  0.1385  0.1049 
6  0.2714 ( 
0.1899 ( 
0.1969  0.172 ( 
0.179  0.1601 ( 
0.1662  0.1258 ( 
7  0.3167  0.2216  0.2298  0.2006  0.209  0.1868  0.1939  0.1468 
8  0.3619 ( 
0.2532 ( 
0.2626  0.2293 ( 
0.239  0.2135 ( 
0.2216  0.1678 ( 
9  0.4072  0.2849  0.2954  0.2579  0.269  0.2402  0.2493  0.1888 
10  0.4524 ( 
0.3166 ( 
0.3282  0.2866 ( 
0.299  0.2669 ( 
0.277  0.2097 ( 
11  0.4979  0.3482  0.361  0.3153  0.329  0.2935  0.3047  0.2307 
12  0.5428 ( 
0.3799 ( 
0.3939  0.3439 ( 
0.359  0.3202 ( 
0.3324  0.2517 ( 
Nondimensional center deflection of PFGM plate versus nondimensional load intensity (mechanical load): (a) neglecting neutral plane position, (b) considering neutral plane position.
Figure
Nondimensional deflection of midpoint versus nondimensional load intensity (
( 



Metal 


Ceramic  
Midplane  Neutral plane  Midplane  Neutral plane  
1  −0.205 ( 
−0.0665 ( 
−0.0338  −0.0547 ( 
−0.0338  −0.0879 ( 
2  −0.1598 ( 
−0.0379 ( 
−0.0038  −0.028 ( 
−0.0061  −0.0669 ( 
3  −0.1146 ( 
−0.0092 ( 
0.0262  −0.0013 ( 
0.0216  −0.0459 ( 
4  −0.069 ( 
0.0195 ( 
0.0562  0.0254 ( 
0.0493  −0.025 ( 
5  −0.024 ( 
0.0481 ( 
0.0862  0.0521 ( 
0.077  −0.004 ( 
6  0.021  0.0768  0.1161  0.0788  0.1047  0.017 
7  0.066  0.1054  0.1461  0.1054  0.1324  0.038 
8  0.1115  0.1341  0.1761  0.1321  0.1601  0.0589 
9  0.1568  0.1628  0.2061  0.1588  0.1878  0.0799 
10  0.202  0.1914  0.2361  0.1855  0.2155  0.1009 
11  0.2473  0.2201  0.2661  0.2122  0.2432  0.1219 
12  0.2925  0.2487  0.296  0.2389  0.2709  0.1428 
Nondimensional center deflection of PFGM in steady state without heat source present versus nondimensional load intensity (
The variation of the axial stress
Nondimensional axial stress distribution through the thickness of FG plate subjected to mechanical loads: (a) neglecting neutral plane position, (b) considering neutral plane position (*nondimensional axial stress
Figure
Nondimensional axial stress distribution, through the plate thickness, for steady state thermomechanical loads: (a) neglecting actual neutral plane position. (b) Considering the location of the neutral plane position (*nondimensional axial stress
The plate considered at the previous section is used to simulate a FG plate in a steady state with heat source strength (rate of heat generated per unit volume)
Figure
Temperature distributions through the plate thickness for various values of grading parameter
Figure
Nondimensional deflection of midpoint versus nondimensional load intensity (

( 


Metal 


Ceramic  
1  −0.2051  −0.0387  −0.0471  −0.0879 
2  −0.1599  −0.0087  −0.0194  −0.0669 
3  −0.1146  0.0213  0.0083  −0.0459 
4  −0.0694  0.0513  0.036  −0.025 
5  −0.0241  0.0813  0.0637  −0.004 
6  0.0211  0.1112  0.0914  0.017 
7  0.0663  0.1412  0.1192  0.038 
8  0.1116  0.1712  0.1469  0.0589 
9  0.1568  0.2012  0.1746  0.0799 
10  0.2021  0.2312  0.2023  0.1009 
11  0.2473  0.2612  0.2300  0.1219 
12  0.2925  0.2911  0.2577  0.1428 
Nondimensional center deflection of PFG plate in steady state with heat source present versus nondimensional load intensity (
Figure
Nondimensional axial stress distributions through the thickness of a steady state plate, with heat source present, for thermomechanical loads.
Nondimensional axial stress distributions through the thickness of a steady state plate, with heat source present, for thermal loads.
Non dimensional deflection of the central point of a steady state plate versus the heat source strength (
Figures
Figures
In this study, a finite element model based on the firstorder shear deformation plate (FSDT) theory is developed for the investigation of thermomechanical behavior of functionally graded plates. Different numerical simulations have been developed to investigate the thermoelastic behavior of a simply supported FG plate, with different material distributions along the thickness. The numerical results lead to the following conclusions:
there is a difference in plate deflection while considering the effect of shifting the neutral plane position;
the neutral plane of the FG plate is shifted towards the surface with the higher young’s modulus. Also, the position of the neutral plane depends mainly on the ratio of the young’s modulus of the two plate constituents;
FG plates provide a high ability to withstand thermal stresses, which reflects its ability to operate at elevated temperatures;
the FGMs are more sensitive to the variation of the intensity of the heat flow, in or out of the structure, than that may be happened in the case of the isotropic material structures. The FGMs provide a highly stable response for the thermal loading comparing to that of the isotropic materials;
due to the continuity of the material properties distribution along the thickness of the plates, the strains and stresses are varied smoothly without any sort of singularities and on contrary to what may be happened in the conventional laminated plates.