ABSTRACT
The external loads applied to a plate invariably change with time. If the variations in time are small and occur over an extended interval, the inertial effects may be neglected and the behavior of the plate can be approximately determined from considerations of equilibrium and material properties. However, in some modern engineering constructions, like aircraft, missiles, and launch vehicles, rapid time variations of loadings occur that must be considered in formulating structural design. In such cases, inertial effects must be taken into account and the dynamic behavior of the plate must be treated as a function of time. Many studies have been reported on the free and forced vibration of FGM
plates, for example, Yang and Shen (2001, 2002), Cheng and Reddy (2003), Vel and Batra (2004), Kim (2005), Elishakoff et al. (2005), Prakash and Ganapathi (2006), Efraim and Eisenberger (2007), and Li et al. (2008). According to mixture rules, Abrate (2006) found that the natural frequencies of FGM plates are proportional to those of the corresponding homogeneous plate and concluded that only one case is needed to determine the proportionality constant, and direct analysis is unnecessary. In most conditions of severe environments, when the plate deflection-to-
thickness ratio is greater than 0.4, the nonlinearity is very important and the nonlinear dynamic equations of plates are required to perform the analysis. Praveen and Reddy (1998) analyzed the nonlinear static and dynamic response of functionally graded ceramic-metal plates in a steady temperature field and subjected to dynamic transverse loads by finite element method (FEM). Reddy (2000) developed both theoretical and finite element formulations for thick FGM plates according to the HSDPT, and studied the nonlinear dynamic response of FGM plates subjected to suddenly applied uniform pressure. Yang et al. (2003) presented a large amplitude vibration analysis of an initially stressed FGM plate with surface-bonded piezoelectric layers by using a semianalytical method based on 1D differential quadrature and Galerkin technique. Chen (2005), Chen et al. (2006), Fung and Chen (2006), and Chen and Tan (2007) studied the large amplitude vibration of
Materials: of and Shells
an plate with or without geometric imperfections. In their studies, the initial stress was taken to be a combination of pure bending stress and an extensional stress in the plane of the plate, and the formulations were based on the FSDPT and classical plate theory (CPT), respectively. Also, Allahverdizadeh et al. (2008a-c) studied the nonlinear free and forced vibration of FGM circular thin plates based on the CPT. However, in the references cited above (Praveen and Reddy 1998, Reddy 2000, Yang et al. 2003, Chen 2005, Chen et al. 2006, Chen and Tan 2007, Allahverdizadeh et al. 2008a-c), the material properties were either assumed to be independent of temperature or considered in a constant temperature environment (T¼ 300 K). Since FGMs always serve in the high-temperature environments, the materials properties of FGM plates must be temperatureand position-dependent. Kitipornchai et al. (2004) and Yang and Huang (2007) studied nonlinear free and forced vibration of imperfect FGM laminated plates with various boundary conditions and with temperature-dependent material properties, respectively. On the other hand, ceramics and the metals used in FGM do store different amounts of heat, and therefore the heat conduction usually occurs. This leads to a nonuniform distribution of temperature through the plate thickness, but it is not accounted for in the above study. Also recently, Huang and Shen (2004, 2006) provided nonlinear free and forced vibration analysis of shear deformable FGM plates without or with surface-bonded piezoelectric layers in thermal environments. Xia and Shen (2008) provided small-and large-amplitude vibration analysis of compressively and thermally postbuckled sandwich plates with FGM face sheets in thermal environments. In these studies, heat conduction and temperaturedependent material properties were both taken into account. Sundararajan et al. (2005) calculated frequencies for nonlinear free flexural vibration of functionally graded rectangular and skew plates under thermal environments. In their studies, the material properties were based on the MoriTanaka scheme, and a remarkable synergism between the Mori-Tanaka scheme and the rule of mixture was found.
