压电热弹性体的变分原理及正则方程和齐次方程

结合对偶变量理论，为压电热弹性体混合层合板问题推导了齐次的控制方程和Hamilton等参元列式．首先根据广义的Hamilton变分原理推导了压电热弹性体非齐次的Hamilton正则方程．然后进一步考虑了热平衡方程与导热方程中变量的对偶关系，通过增加正则方程的维数，成功地将非齐次的正则方程转化为能独立求解压电热弹性体耦合问题的齐次控制方程．为了推导四节点Hamilton等参元列式的方便，可将温度梯度关系类比成本构关系并构建新的变分原理．齐次方程大大简化了人们在分析压电热弹性体耦合问题时，通常要求解非齐次方程和关于平衡方程和导热方程的二阶微分方程的繁琐方法，同时也减少了数值计算工作量．

Variation Principle of Piezothermoelastic Bodies, Canonical Equation and Homogeneous Equation

Combining the symplectic variations theory,the homogeneous control equation and isoparametric element homogeneous formulations for piezothermoelastic hybrid laminates problems were deduced.Firstly,based on the generalized Hamilton variation principle,the non-homogeneous Hamilton canonical equation for piezothermoelastic bodies was derived.Then the symplectic relationship of variations in the thermal equilibrium formulations and gradient equations was considered. The non-homogeneous canonical equation was transformed to homogeneous control equation for solving independently the coupling problem of piezothermoelastic bodies by the incensement of dimensions of the canonical equation.For the convenience of deriving Hamilton isoparametric element formulations with four nodes,one can consider the temperature gradient equation as constitutive relation and reconstruct new variation principle.The homogeneous equation simplifies greatly the solution programs which are often performed to solve non-homogeneous equation and second order differential equation on the thermal equilibrium and gradient relationship.