ABSTRACT
When a flat plate is under the action of edge compression in its middle plane, the plate is deformed but remains completely flat when the edge forces are sufficiently small unless there is an initial geometric imperfection. By increasing the load, a state is reached when the plate bends slightly. The in-plane compressive load which is just sufficient to keep the plate in a slightly bent form is called the critical load or buckling load. Once the buckling load is exceeded, the load-deflection relationship exhibits a stable character due to membrane forces which come into play. Actually, the buckling mode will change in the postbuckling range. These changes occur when the energy stored in the plate is sufficient to carry the plate from one buckled form to the other. To obtain an accurate analysis of FGM plates in a wide postbuckling range, the changes in buckling mode must be taken into account. In the usual postbuckling analysis, the buckling mode of the plate is assumed to remain unchanged. This is reasonable assumption in the immediate postbuckling range, e.g., the postbuckling load less than about three times the buckling load. Many studies have been reported on the buckling and postbuckling analy-
sis of FGM plates subjected to mechanical or thermal loading. Among those, thermal and mechanical buckling of simply supported FGM rectangular plates was studied by Javaheri and Eslami (2002a-c) based on the classical and higher order shear deformation plate theories. Na andKim (2004, 2006a,b) used solid finite elements to calculate buckling temperature of FGM plates with fully clamped edges. Najafizadeh and Eslami (2002a,b) and Najafizadeh and Heydari (2004a,b, 2008) considered axisymmetric buckling of simply supported and clamped circular FGM plates under a uniform temperature rise or a radial compression based on the first order and higher order shear deformation theory, respectively. Ma and Wang (2003a,b) did the postbuckling analysis of simply supported and clamped FGM circular plates under a radial compression or nonlinear temperature change across the plate thickness based on the classical von Kármán plate theory. Subsequently, they gave the relationships between axisymmetric buckling solutions of FGM circular
Materials: of and Shells
plate theory and classical theory (Ma and Wang 2004). The effect of initial geometric imperfections on the postbuckling behavior of FGM circular plates subjected to mechanical edge loads and heat conduction was then studied by Li et al. (2007). Naei et al. (2007) calculated buckling loads of radially loaded FGM circular thin plates with variable thickness by using finite element method (FEM). Buckling of FGM plates without or with piezoelectric layers subjected to various nonuniform in-plane loads, along with heat and applied voltage, was considered by Chen and Liew (2004) and Chen et al. (2008) using the first-order shear deformation theory. Ganapathi et al. (2006) and Ganapathi and Prakash (2006) presented the buckling loads for simply supported FGM skew plates subjected to in-plane mechanical loads and heat conduction. In their analysis, the material properties were based on the Mori-Tanaka scheme and rule of mixture, respectively. This work was then extended to the case of thermal postbuckling of FGM skew plates by Prakash et al. (2008). Furthermore, Yang and Shen (2003) studied the postbuckling behavior of FGM thin plates under fully clamped boundary conditions. This work was then extended to the case of shear deformable FGM plates with various boundary conditions and various possible initial geometric imperfections by Yang et al. (2006). Woo et al. (2005) studied the postbuckling behavior of FGM plates and shallow shells under edge compressive loads and a temperature field based on the higher order shear deformation theory. Wu (2004) studied the thermal buckling behavior of simply supported FGM rectangular plates under uniform temperature rise and gradient through the thickness based on the first-order shear deformation plate theory. Shariat and Eslami (2005, 2006) performed the thermal buckling of imperfect FGM rectangular plates under three types of thermal loading as uniform temperature rise, nonlinear temperature rise through the thickness, and axial temperature rise, based on the first-order shear deformation plate theory and the classical thin plate theory, respectively. Wu et al. (2007) studied the postbuckling of FGM rectangular plates under various boundary conditions subjected to uniaxial compression or uniform temperature rise based on the first-order shear deformation plate theory. In the above studies, however, the materials properties were virtually assumed to be temperature-independent (T-ID). Park and Kim (2006) presented thermal postbuckling and vibration of simply supported FGM plates with temperature-dependent (T-D) materials properties by using FEM. Shukla et al. (2007) studied the postbuckling of clamped FGM rectangular plates subjected to thermomechanical loads. In their analysis, the temperature-dependent materials properties were considered and the analytical approach was based on fast-converging Chebyshev polynomials. It has been pointed out by Shen (2002) that the governing differential equations for an FGM plate are identical in form to those of unsymmetric cross-ply laminated plates, and applying in-plane compressive edge loads to such plates will cause bending curvature to appear. Consequently, the bifurcation buckling did not exist due to the stretching=bending coupling effect, as previously
FGM 47
Qatu and Leissa (1993), the solutions are physically incorrect for simply supported FGM rectangular plates subjected to in-plane compressive edge loads and=or temperature variation. Moreover, Liew et al. (2003, 2004) studied thermal postbuckling behavior
of FGM hybrid plates with different kinds of boundary conditions. In their analysis, the material properties were assumed to be temperatureindependent and temperature-dependent, respectively, but their results were only for the simple thermal loading case of uniform temperature rise. They confirmed that the FGM plates with all four edges simply supported (SSSS) have no bifurcation buckling temperature, even for the loading case of uniform temperature change. Obviously, when the FGM plate is geometrically midplane symmetric, as reported in Birman (1995) and Feldman and Aboudi (1997), a bifurcation buckling load under in-plane compressive edge loads and=or temperature variation does exist. Recently, Shen (2005) provided a postbuckling analysis for simply supported, midplane symmetric FGM plates with fully covered or embedded piezoelectric actuators subjected to the combined action of mechanical, thermal, and electronic loads. In his study, the material properties were considered to be temperature-dependent and the effect of temperature rise and applied voltage on the postbuckling response was reported. This work was then extended to the case of postbuckling analysis of sandwich plates with FGM face sheets subjected to mechanical and thermal loads (Shen and Li 2008). On the other hand, due to the temperature gradient the plate is subjected to additional moments along with the membrane forces and the problem cannot be posed as an eigenvalue problem, when the four edges of the plate are simply supported. Therefore, the bifurcation solutions for FGM plates subjected to transverse temperature variation, i.e., linear and=or nonlinear gradient through the thickness, may also be physically incorrect. More recently, Shen (2007b) provided a thermal postbuckling analysis for simply supported, midplane symmetric FGM plates under in-plane nonuniform parabolic temperature distribution and heat conduction, and concluded that for the case of heat conduction, the postbuckling path for geometrically perfect plates is no longer of the bifurcation type.
