International Journal of Metals

Volume 2015, Article ID 416824, 16 pages

http://dx.doi.org/10.1155/2015/416824

## Response of Functionally Graded Material Plate under Thermomechanical Load Subjected to Various Boundary Conditions

^{1}Jodhpur Institute of Engineering and Technology, Jodhpur, Rajasthan, India^{2}Jai Narain Vyas University, Jodhpur, Rajasthan, India

Received 8 October 2014; Revised 27 December 2014; Accepted 7 January 2015

Academic Editor: Massimo Pellizzari

Copyright © 2015 Manish Bhandari and Kamlesh Purohit. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Functionally graded materials (FGMs) are one of the advanced materials capable of withstanding the high temperature environments. The FGMs consist of the continuously varying composition of two different materials. One is an engineering ceramic to resist the thermal loading from the high-temperature environment, and the other is a light metal to maintain the structural rigidity. In the present study, the properties of the FGM plate are assumed to vary along the thickness direction according to the power law distribution, sigmoid distribution, and exponential distribution. The fundamental equations are obtained using the first order shear deformation theory and the finite element formulation is done using minimum potential energy approach. The numerical results are obtained for different distributions of FGM, volume fractions, and boundary conditions. The FGM plate is subjected to thermal environment and transverse UDL under thermal environment and the response is analysed. Numerical results are provided in nondimensional form.

#### 1. Introduction

Composite materials are widely in use due to their intrinsic mechanical property such as high strength, modulus of elasticity, and lower specific gravity. Further, as a result of intensive studies into metallurgical aspects of material and better understanding of structural property, it has become possible to develop new composite materials with improved physical and mechanical properties. The functionally graded material (FGM) is one such material whose property can be useful to accomplish the specific demands in various engineering applications to achieve the advantage of the properties of individual material. This is possible due to the material composition of the FGM which changes according to a law in a preferred direction. The thermomechanical analysis of FGM structures is one dimension which has attracted the attention of many researchers in the past few years. The applications of FGMs include design of aerospace structures, heat engine components, and nuclear power plants. A large number of research papers have been published to evaluate the behaviour of FGM using both experimental and numerical techniques which include both linearity and nonlinearity in various areas. A few of published literatures highlight the importance of the present work. The FGM can be produced by gradually and continuously varying the constituents of multiphase materials in a predetermined profile. Most researchers use the power law function (P-FGM), sigmoid function (S-FGM), or exponential function (E-FGM) to describe the effective material properties. Delale and Erdogan [1] indicated that the effect of Poisson’s ratio on the deformation is much less as compared to that of Young’s modulus. Praveen and Reddy [2] examined the thermoelastostatic response of simply supported square FG plates subjected to pressure loading and thickness varying temperature fields. They used the first order shear deformation plate theory (FSDT) to develop the governing equations. They reported that the basic response of the plates which corresponds to properties intermediate to those of the metal and the ceramic does not necessarily lie in between those of the ceramic and metal. Reddy [3] found that the nondimensional deflection reached a minimum value at a particular volume fraction index. They used the power law function to calculate the material gradient. Cheng and Batra [4] computed deformations due to thermal and mechanical loads applied to the top and bottom surfaces of the rigidly clamped elliptic FG plate separately. It was found that the through-thickness distributions of the in-plane displacements and transverse shear stresses in a functionally graded plate do not agree with those assumed in classical and shear deformation plate theories. Reddy and Zhen [5] solved the problem using a higher order shear and normal deformable plate theory (HONSDPT) since, in the HOSNDPT, the transverse normal and shear stresses are computed from equation of the plate theory rather than by integrating the balance of linear momentum with respect to the thickness coordinate. It was reported that the assumption of constant deflection is not true for thermal load but it was found to be true for mechanical load. Qian and Batra [6] found that the centroidal deflection for a clamped plate is nearly one-third of that for simply supported plate, and the maximum magnitude of the axial stress induced at the centroid of the top surface is nearly 40% larger than that for a simply supported plate. Dai et al. [7] analyzed the plate under the mechanical loading as well as thermal gradient and found that the relations between the deflection and the volume fraction exponent are quite different under the two loadings. Ferreira et al. [8] used collocation method third order shear order deformation theory and presented the effect of aspect ratio of the plate and the volume fraction of the constituents on the transverse deflection. Chi and Chung [9, 10] studied the effect of loading conditions on the mechanical behaviour of a simply supported rectangular FGM plate. They assumed that Young’s moduli vary continuously throughout the thickness direction according to the volume fraction of constituents defined by sigmoid function. The maximum tensile stress of the FGM plate was found to be at the bottom surface of the plate. Wang and Qin [11] concluded that the appropriate graded parameter can lead to low stress concentration and little change in the distribution of stress fields. They assumed that the thermoelastic constants and the temperature vary exponentially through the thickness. Mahdavian [12] obtained the equilibrium and stability equations based on the classical plate theory (CPT) and Fourier series expansion. They found that the critical buckling coefficients for FGM plates are considerably higher than isotropic plates. Ashraf and Daoud [13] derived the equilibrium and stability equations using sinusoidal shear deformation plate theory (SPT). It was concluded that the critical buckling temperature differences of functionally graded plates are generally lower than the corresponding ones for homogeneous ceramic plates. Alieldin et al. [14] proposed three approaches to determine the property details of an FG plate equivalent to the original laminated composite plate. They developed the equations of motion based on the combination of the first order plate theory and the Von Karman strains. Kyung-Su and Ji-Hwan [15] compared numerical results for three types of materials. It was found that the minimum compressive stress ratio is observed for the fully FGM plate with largest volume fraction index. Suresh et al. [16] studied the effect of shear deformation and nonlinearity response of functionally graded material plate and concluded that the effect of nonlinearity in functionally graded composite plates is more predominant in decreasing the deflections in thin plates for side to thickness ratio of 10. Mohammad and Singh [17] obtained the numerical results for different thickness ratios, aspect ratios, volume fraction index, and temperature rise with different loading and boundary conditions. They employed the boundary conditions; for example, all edges are simply supported (SSSS), all edges are clamped (CCCC), two edges are simply supported and two are clamped (SCSC), two edges are clamped and two are free (CFCF), all edges are hinged (HHHH), and two edges are clamped and two are hinged (CHCH). It was noticed that the maximum centre deflection was found for simply supported boundary conditions and least central deflection was found for clamped (CCCC) boundary conditions. Nguyen-Xuan et al. [18] applied the method for static, free vibration and mechanical/thermal buckling problems of functionally graded material (FGM) plates. They analysed the behaviour of FGM plates under mechanical and thermal loads numerically in detail through a list of benchmark problems. Alshorbagy et al. [19] concluded that FG plates provide a high ability to withstand thermal stresses, which reflects its ability to operate at elevated temperatures. Bhandari and Purohit [20] presented the response of FG plates under mechanical load for various boundary conditions, for example, SSSS, CCCC, SCSC, CFCF, CCSS, SSFF, SSSC, SSSF, and SSCF. The power law, sigmoid law, and exponential law were used for the calculations of the properties through the thickness. They also compared the behaviour of isotropic plates (ceramic and metal) with that of the FGM plates. It was concluded that the isotropic ceramic plate has the lowest tensile stress for all the boundary conditions. It was also reported that the maximum tensile stress occurs for CCFF boundary condition and the minimum tensile stress was observed for SCSC boundary condition. Bhandari and Purohit [21] presented the response of FG plates under mechanical load for varying aspect ratios. The power law, sigmoid law, and exponential law were used for the calculations of the properties through the thickness. They also compared the behaviour of isotropic plates (ceramic and metal) with that of the FGM plates.

With the increasing applications of functionally graded materials, it is vital to understand the behaviour of thermomechanical response of FG plates under various boundary conditions. It is also important to study the response of the FG materials following various material gradient laws, for example, power law function, sigmoid function, and exponential function for properties. In both power law and exponential functions, the stress concentrations appear in one of the interfaces in which the material is continuous but is rapidly changing. In sigmoid FGM, which is composed of two power law functions, there is a gradual change in volume fraction as compared to power law and exponential function. Power law function has been applied to many of the FGMs’ but application of sigmoid function is sparse in literature. Power law function, sigmoid function, and exponential function have been used in various research works separately but the comparisons of the three FGM laws for various volume fraction exponents, ceramic and metal, have been sparse. In most of the research works, FGM plates with edges simply supported and clamped have been considered. Plate subjected to other boundary conditions, for example, clamped, free, hinged, and combined, is also useful but they have been rarely reported. Thermomechanical loaded FGM structures have been researched but thermally loaded plates have been rarely reported.

Keeping this in consideration, the objective of the present work is to examine the thermomechanical behaviour for various boundary conditions, various material gradient laws, and various volume fraction exponents. The results are presented in the form of nondimensional parameters. The comparison of isotropic ceramic, metal, and FGM plates is also presented.

#### 2. Material Gradient of FGM Plate

The material properties in the thickness direction of the FGM plates vary with power law functions (P-FGM), exponential functions (E-FGM), or sigmoid functions (S-FGM). A mixture of the two materials composes the through-thickness characteristics. The FGM plate of thickness “” is modeled usually with one side of the material being ceramic and the other side being metal as shown in Figure 1.