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Indian Journal of Materials Science

Volume 2013 (2013), Article ID 206239, 7 pages

http://dx.doi.org/10.1155/2013/206239

## Stress and Strain Analysis of Functionally Graded Rectangular Plate with Exponentially Varying Properties

^{1}Mechanical Engineering Department, University of Tehran, Tehran, Iran^{2}Department of Mechanical Engineering, Yasooj Branch, Islamic Azad University, Yasooj, Iran

Received 26 June 2013; Accepted 16 July 2013

Academic Editors: S. Banerjee and D. L. Sales

Copyright © 2013 Amin Hadi et al. 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

The bending of rectangular plate made of functionally graded material (FGM) is investigated by using three-dimensional elasticity theory. The governing equations obtained here are solved with static analysis considering the types of plates, which properties varying exponentially along direction. The value of Poisson’s ratio has been taken as a constant. The influence of different functionally graded variation on the stress and displacement fields was studied through a numerical example. The exact solution shows that the graded material properties have significant effects on the mechanical behavior of the plate.

#### 1. Introduction

Recently, a new category of composite materials known as functionally graded materials (FGMs) has attracted the interest of many researchers. The FGMs are heterogeneous composite materials in which the mechanical properties vary continuously in certain direction. FGMs are used in many engineering applications such as aviation, rocketry, missiles, chemical, aerospace, and mechanical industries. Therefore, composites that are made of FGMs were considerably attractive in recent years.

Several studies have been performed to analyze the behavior of functionally graded beam, plates, and shells. Hadi et al. [1, 2] studied an Euler-Bernoulli and Timoshenko beam made of functionally graded material subjected to a transverse loading at which Young’s modulus of the beam varies by specific function. Reddy [3] has analyzed the static behavior of functionally graded rectangular plates based on his third-order shear deformation plate theory. Cheng and Batra [4] have related the deflections of a simple supported functionally graded polygonal plate given by the first-order shear deformation theory and a third-order shear deformation theory to an equivalent homogeneous Kirchhoff plate. Cheng and Batra [5] also presented results for the buckling and steady state vibrations of a simple supported functionally graded polygonal plate based on Reddy’s plate theory. Loy et al. [6] studied the vibration of functionally graded cylindrical shells by using Love’s shell theory.

Analytical 3D solutions for plates are useful because provided benchmark results to assess the accuracy of various 2D plate theories and finite element formulations. Cheng and Batra [7] used the method of asymptotic expansion to study the 3D thermoelastic deformations of a functionally graded elliptic plate. Recently, Vel and Batra [8] have presented an exact 3D solution for the thermoelastic deformation of functionally graded simple supported plates of finite dimensions. Reiter et al. [9] performed detailed finite element studies of discrete models containing simulated particulate and skeletal microstructures and compared the results with those that are computed from homogenized models in which effective properties were derived by the Mori-Tanaka and the self-consistent methods.

Tanigawa [10] used a layerwise model to solve a one-dimensional transient heat conduction problem and the associated thermal stress problem of an inhomogeneous plate. He further formulated the optimization problem of the material composition to reduce the thermal stress distribution. Tanaka et al. [11, 12] have designed FGM property profiles by using sensitivity and optimization methods based on the reduction of thermal stresses. Jin and Noda [13] used the minimization of thermal stress intensity for a crack in a metal-ceramic functionally gradient material as a criterion for optimizing material property variation. In the same context, also were studied both the steady state [14] and the transient [15] heat conduction problems by them, but neglected the thermomechanical coupling (see also [16, 17]).

The response of functionally graded ceramic-metal plates has been investigated by Praveen and Reddy [18] with using a plate finite element that accounts for the transversal shear strains, rotatory inertia, and moderately large rotations in von Kármán sense. Reddy and Chin [19] have studied the dynamic thermoelastic response of functionally graded cylinders and plates. Najafizadeh and Eslami [20] presented the buckling analysis of radially loaded solid circular plate that is made of functionally graded material.

In this paper, an exact solution bending of rectangular plate made of functionally graded material (exponential form along direction) subjected to top and bottom pressures and , respectively, is investigated by using three-dimensional elasticity theory.

#### 2. Analysis

The equilibrium of a weightless homogeneous transversally isotropic elastic FGM plate is considered. The geometry of the elastic FGM plate in relation to the coordinate axes is shown in Figure 1.

The plate is assumed under the action of top and bottom pressures and , respectively, where , and are top and bottom pressures in , .

In order to account for the changing material properties along the direction, an exponential relationship is used as follows: Here is the module of elasticity at , and is the inhomogeneity constants determined empirically. Equilibrium equations in three dimensions are defined as follows: where , , and are normal stress components in , , and direction, respectively. , , and are shear stresses components.

The displacement in the , , and directions is denoted by , , and , respectively, six strain components can be expressed as where , , and are normal strain components in , , and directions, respectively. , , and are shear strain components.

The stress-strain relations are where is shear modulus, with the modulus of elasticity and Poisson’s ratio. The value of Poisson’s ratio has been taken as constant.

The boundary conditions of the problem are the following. Along the sides of the plate, we have the boundary conditions on the plate faces are as below: By the displacement field below, the boundary conditions (7) are satisfied: Substituting (9) into (5) and the resulting expressions for stress components into equilibrium equations (3), we see that the equilibrium equation is expressed as system of second-order ordinary differential equations: where Therefore The general solution of (10) is as follows: where are the roots of the equation below: where are arbitrary integration constants.

The resulting displacement field is defined as follows: Substituting (15) into (5), (6) stress component is calculated as follows:

The resultant moments on a unit of length are obtained by using relations (16a)–(16f):

Using (16a), (16b), (16c), (16d), (16e), and (16f) the transverse shearing forces on a unit of length are by definition

#### 3. Results and Discussion

In the following, the obtained solution will be employed to analyze the effect of material inhomogeneity on the elastic field in the rectangular plate. Consider a rectangular plate with length , width , and thickness , with material property , subjected to top and bottom pressures and , respectively. It is assumed the Poisson’s ratio has a constant value of 0*.*3. Dimensionless and normalized variables are used. For different values of , dimensionless modulus of elasticity along the direction is plotted in Figure 2. According to this figure, at the same position , dimensionless modulus of elasticity is increasing as the parameter * is *decreasing, while for , dimensionless modulus of elasticity is increasing as the parameter * is *increasing.

Figure 3 displays the nondimensional displacement of the plate in direction for different values of parameter . This plot displays that the magnitude of changing by is low, so for, assuming plane strain is true and reasonable, but, for , the amount of changing by is high.

In Figure 4 is shown the nondimensional displacement in the direction versus for , . In this plot displacement is decreasing as the parameter is increasing. In Figure 5 is shown the nondimensional displacement in the y direction versus for , .

In Figure 6 is shown the variation of nondimensional stress in the direction versus nondimensional thickness at the , . This plot shows that the boundary conditions at the up and down surfaces are satisfied. Also at the constant , by increasing the parameter , it is observed the stress is decreased.

Figure 7 shows the according to the for the variable amount of : this figure probes that the stress’s component can be deleted proportionally to other components.

#### 4. Conclusion

It is apparent that close form solutions are important to simplified kinds of real engineering problems. In this paper is studied the rectangular plate that is made of functionally graded material with the variable properties (exponential form) by using 3D elasticity theory. Then some exact solution packages for stresses, displacements are presented.

To show the effect of inhomogeneity on the stress distributions, different values were considered for material inhomogeneity parameter . The presented results show that the material inhomogeneity has a significant influence on the mechanical behaviors of the solid rectangular plate made of exponentially FG.

#### References

- A. R. Daneshmehr, A. Hadi, and S. M. N. Mehrian, “Investigation of elastoplastic functionally graded Euler-Bernoulli beam subjected to distribute transverse loading,”
*Journal of Basic and Applied Scientific Research*, vol. 2, no. 10, pp. 10628–10634, 2012. View at Google Scholar - A. Hadi, A. R. Daneshmehr, S. M. N. Mehrian, M. Hosseini, and F. Ehsani, “Elastic analysis of functionally graded timoshenko beam subjected to transverse loading,”
*Technical Journal of Engineering and Applied Sciences*, vol. 3, no. 13, pp. 1246–1254, 2013. View at Google Scholar - J. N. Reddy, “Analysis of functionally graded plates,”
*International Journal for Numerical Methods in Engineering*, vol. 47, no. 1–3, pp. 663–684, 2000. View at Google Scholar · View at Scopus - Z.-Q. Cheng and R. C. Batra, “Deflection relationships between the homogeneous Kirchhoff plate theory and different functionally graded plate theories,”
*Archives of Mechanics*, vol. 52, no. 1, pp. 143–158, 2000. View at Google Scholar · View at Scopus - Z.-Q. Cheng and R. C. Batra, “Exact correspondence between eigenvalues of membranes and functionally graded simply supported polygonal plates,”
*Journal of Sound and Vibration*, vol. 229, no. 4, pp. 879–895, 2000. View at Publisher · View at Google Scholar · View at Scopus - C. T. Loy, K. Y. Lam, and J. N. Reddy, “Vibration of functionally graded cylindrical shells,”
*International Journal of Mechanical Sciences*, vol. 41, no. 3, pp. 309–324, 1999. View at Google Scholar · View at Scopus - Z.-Q. Cheng and R. C. Batra, “Three-dimensional thermoelastic deformations of a functionally graded elliptic plate,”
*Composites Part B: Engineering*, vol. 31, no. 2, pp. 97–106, 2000. View at Google Scholar · View at Scopus - S. S. Vel and R. C. Batra, “Exact solution for thermoelastic deformations of functionally graded thick rectangular plates,”
*AIAA Journal*, vol. 40, no. 7, pp. 1421–1433, 2002. View at Google Scholar · View at Scopus - T. Reiter, G. J. Dvorak, and V. Tvergaard, “Micromechanical models for graded composite materials,”
*Journal of the Mechanics and Physics of Solids*, vol. 45, no. 8, pp. 1281–1302, 1997. View at Google Scholar · View at Scopus - Y. Tanigawa, “Theoretical approach of optimum design for a plate of functionally gradient materials under thermal loading,” in
*Thermal Shock and Thermal Fatigue Behavior of Advanced Ceramics*, vol. 241 of*Nato Science Series E*, pp. 171–180, 1992. View at Google Scholar - K. Tanaka, Y. Tanaka, K. Enomoto, V. F. Poterasu, and Y. Sugano, “Design of thermoelastic materials using direct sensitivity and optimization methods. Reduction of thermal stresses in functionally gradient materials,”
*Computer Methods in Applied Mechanics and Engineering*, vol. 106, no. 1-2, pp. 271–284, 1993. View at Google Scholar · View at Scopus - K. Tanaka, Y. Tanaka, H. Watanabe, V. F. Poterasu, and Y. Sugano, “An improved solution to thermoelastic material design in functionally gradient materials: Scheme to reduce thermal stresses,”
*Computer Methods in Applied Mechanics and Engineering*, vol. 109, no. 3-4, pp. 377–389, 1993. View at Google Scholar · View at Scopus - Z. H. Jin and N. Noda, “Minimization of thermal stress intensity factor for a crack in a metal ceramic mixture, Ceramic Trans,”
*Functionally Graded Material*, vol. 34, pp. 47–54, 1993. View at Google Scholar - N. Noda and Z. H. Jin, “Thermal stress intensity factors for a crack in a strip of a functionally gradient material,”
*International Journal of Solids and Structures*, vol. 30, no. 8, pp. 1039–1056, 1993. View at Google Scholar · View at Scopus - Z. H. Jin and N. Noda, “Transient thermal stress intensity factors for a crack in a semi-infinite plate of a functionally gradient material,”
*International Journal of Solids and Structures*, vol. 31, no. 2, pp. 203–218, 1994. View at Google Scholar · View at Scopus - Y. Obata and N. Noda, “Steady thermal stresses in a hollow circular cylinder and a hollow sphere of a functionally gradient material,”
*Journal of Thermal Stresses*, vol. 17, no. 3, pp. 471–487, 1994. View at Google Scholar · View at Scopus - Y. Obata, N. Noda, and T. Tsuji, “Steady thermal stresses in a functionally gradient material plate,”
*Transactions of the JSME*, vol. 58, pp. 1689–1695, 1992. View at Google Scholar - G. N. Praveen and J. N. Reddy, “Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates,”
*International Journal of Solids and Structures*, vol. 35, no. 33, pp. 4457–4476, 1998. View at Google Scholar · View at Scopus - J. N. Reddy and C. D. Chin, “Thermomechanical analysis of functionally graded cylinders and plates,”
*Journal of Thermal Stresses*, vol. 21, no. 6, pp. 593–626, 1998. View at Google Scholar · View at Scopus - M. M. Najafizadeh and M. R. Eslami, “Buckling analysis of circular plates of functionally graded materials under uniform radial compression,”
*International Journal of Mechanical Sciences*, vol. 44, no. 12, pp. 2479–2493, 2002. View at Publisher · View at Google Scholar · View at Scopus