Nonlinear bending and post-buckling of a functionally graded circular plate under mechanical and thermal loadings

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Abstract

Based on the classical nonlinear von Karman plate theory, axisymmetric large deflection bending of a functionally graded circular plate is investigated under mechanical, thermal and combined thermal–mechanical loadings, respectively, and axisymmetric thermal post-buckling behavior of a functionally graded circular plate is also investigated. The mechanical and thermal properties of functionally graded material (FGM) are assumed to vary continuously through the thickness of the plate, and obey a simple power law of the volume fraction of the constituents. Governing equations for the problem are derived, and then a shooting method is employed to numerically solve the equations. Effects of material constant n and boundary conditions on the temperature distribution, nonlinear bending, critical buckling temperature and thermal post-buckling behavior of the FGM plate are discussed in details.

Introduction

Extensive investigations on the thermal bending and post-buckling of isotropic and composite plates and shells were carried out by Tauchert and Huang (1987); Tauchert (1991); Meyers and Hyer (1991) and Leissa (1992), etc. However, there are few works on the stability, vibration, bending and buckling behavior of functionally graded structures, and these are still open problems. Loy et al. (1999) and Pradhan et al. (2000) examined the free vibration of functionally graded cylindrical shell by using the Rayleigh–Ritz method. An analytical solution of the dynamic response of simply supported functionally graded cylinder due to low-velocity impact was given by Gong et al. (1999). Based on the classical small deflection theory of plate, Yang and Shen (2001) investigated the dynamic response of a functionally graded rectangular thin plate with initial stress subjected to partially distributed impulsive lateral loads and without or resting on an foundation. Ng et al., 2000, Ng et al., 2001 studied the dynamic stability of functionally graded rectangular plate and cylindrical shell, respectively. A modified classical lamination theory to account for piezoelectric coupling terms under applied electric field was developed by Almajid et al. (2001), and then the theory was applied to predict the out-of-plane displacement and stress field of actuators and the functionally graded material (FGM) bimorph. Mian and Spencer (1998) established a large class of exact solutions of the three-dimensional elasticity equations for functionally graded and laminated elastic materials.

The response of functionally graded ceramic–metal plate accounting for the transverse shear strains, rotary inertia and moderately large rotations in the von Karman sense was studied by Praveen and Reddy (1998), in which finite element method was employed to investigate the static and dynamic responses of the functionally graded plate by varying the volume fraction of the ceramic and metallic constituents. Effect of imposed temperature field on the response of the functionally graded plate was also discussed. Reddy and Chin (1998) investigated the dynamic thermo-elastic response of functionally graded cylinders and plates. A thermo-elastic boundary value problem was derived by using the first-order shear deformation plate theory that account for coupling with a three-dimensional heat conduction equation for a functionally graded plate. Using the first-order shear deformation theory of Mindlin plate, axisymmetric bending of functionally graded annular and circular plates was studied by Reddy et al. (1999), in which the solutions were expressed in terms of the classical solutions based on the Kirchhoff plate theory. Based on the higher-order shear deformation theory of plate, Reddy (2000) developed both theoretical and finite element formulations for thick FGM plates, and the nonlinear dynamic responses of FGM plates subjected to suddenly applied uniform pressure were studied. Based on the von Karman theory, Woo and Meguid (2001) derived an analytical solution expressed in terms of Fourier series for the large deflection of functionally graded plates and shallow shells under transverse mechanical loading and a temperature field. Cheng and Batra (2000) studied three-dimensional thermo-mechanical deformations of an isotropic linear thermo-elastic functionally graded elliptic plate. A closed form solution was obtained which shows that the through-thickness distribution of the in-plane displacements and transverse shear stress in a functionally graded plate do not agree with those assumed in classical and shear deformation plate theories. Moreover, a new set of field equations in terms of displacement and stress potential functions for inhomogeneous plates had been presented and reformulated by Cheng (2001), and mixed Fourier series technique was employed to solve the equations. Using an asymptotic method, the three-dimensional thermo-mechanical deformations of functionally graded rectangular plate were investigated by Reddy and Cheng (2001) and the distributions of temperature, displacements and stresses in the plate were calculated for different volume fraction of ceramic constituent.

Assuming that the material properties throughout the structure are produced by a spatial distribution of the local reinforcement volume fraction vf=vf(x,y,z), Feldman and Aboudi (1997) studied the elastic bifurcation buckling of functionally graded plate under in-plane compressive loading. More recently, Javaheri and Eslami, 2002a, Javaheri and Eslami, 2002b studied the thermal buckling of functionally graded rectangular plate based on the classical and the higher-order shear deformation theories of plate, respectively, and obtained the closed form solutions under several types of thermal loads. Ma and Wang (in press) studied the axisymmetric post-buckling behavior of a functionally graded circular plate under uniformly distributed radial compression on the basis of classical nonlinear plate theory.

To the authors’ knowledge, only few works on the nonlinear bending of functionally graded plates are concerned, but the thermal post-buckling of FGM plates has not been carried out in the previous works. In the present paper, axisymmetric nonlinear bending and thermal post-buckling behavior of a functionally graded circular plate are studied under mechanical, thermal and combining thermal–mechanical loading in the framework of von Karman plate theory. Simply supported and clamped boundary conditions are considered. The material properties are assumed to vary continuously through the thickness of the plate. Effects of material properties and boundary conditions on the large deflection bending and thermal post-buckling behavior of the FGM plate are discussed in details.

Section snippets

Basic equations

A functionally graded circular plate with thickness h and radius b is considered here. It is assumed that the mechanical and thermal properties of FGM vary through the thickness of plate, and the material properties P can be expressed as (Reddy and Chin, 1998; Reddy et al., 1999)P(z)=(Pm−Pc)Vm+Pc,where the subscripts m and c denote the metallic and ceramic constituents, respectively, Vm denotes the volume fraction of metal and follows a simple power law asVm=h−2z2hn,where z is the thickness

Shooting method

In what follows, a shooting method (Li et al., 1996) is employed to numerically solve the problems. Here, the governing equations , and boundary conditions , , can be rewritten in the following formdYdx=H(x,Y),B0Y(0)=b0,B1Y(1)=b1,whereY=y1y2y3y4y5y6y7T=wdwdxd2wdx2d3wdx3ududxδT,H=y2y3y4ϕy6ψ0T.For bending problem, δ=Q. For buckling and post-buckling problems, δ=λ. Expressions of ϕ, ψ, B0, B1, b0 and b1 are as followsϕ=−2xd3wdx31x2d2wdx2+1x3dwdx+F2dudx+νxu+12dwdx2d2wdx2+F2νdudx+1xu+ν2dwdx21xdwdx

Numerical results and discussions

In what follows, the well-known Runge–Kutta method in conjunction with a Newton iterative formulation are employed to numerically solve Eqs. , , . If one obtains the solution of Eqs. , , for a sufficiently small value of parameter ξ, then the solutions of Eqs. , , , , can be obtained for large scale of ξ by using the so-called analytical continuation method in which the parameter ξ increases step by step,δ=d3=δ(ξ),ξ>0.For bending problem, Eq. (37) is the solution of deflection–load curves, Q=

Conclusions

Axisymmetric nonlinear bending and thermal post-buckling of a functionally graded circular plate are investigated under uniformly distributed transverse mechanical, thermal and combined mechanical–thermal loadings, respectively. Based on the classical nonlinear von Karman plate theory, governing equations for the problem are derived, and then a shooting method is employed to numerically solve the equations. Effects of material constant n and boundary conditions on the temperature distribution,

Acknowledgements

This work was supported by the National Natural Science Foundation of China (10125212) and the Fund from The Ministry of Education of China.

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