Generalized H-R Variation Principles Of Three Thermoelastical Solids And Homogeneous Vector Equations

In order to investigate the generalized Hamilton canonical equation theory of thermoelasticity, piezothermoelasticity and electromagnetothermoelasticity, based on the constitutive relationships of thermoelastical solids, the modified generaliezed H-R (Hellinger-Reissner) variation principles of thermoelasticity, piezothermoelasticity and electromagnetothermoelasticity were established in terms of the modified H-R variation formulation of elastical solid. Meantime, the relevant nonhomogeneous generalized Hamilton canonical equations were derived from these principles in this paper. And then, based upon the symplectic variable theory, the nonhomogeneous generalized Hamilton canonical equations were transform into the homogeneous vector equations with the dimension expanding according to the symplectic relationship of the heat dynsity in the heat equilibrium equations and the temperature variable in the heat conduction equation were analogied as the symplectic variables. Moreover, many numerical examples (laminated plate/shell) with the simply supported boundary condition were used to verify the correctness of these homogeneous vector equations. There exit several charactisitics in the homogeneous vector equations: the numbers of variables in these equatinos is even, and these variables, which are symplectic variables, can be classified into two groups, namely, the generalized outplane stress and the generalized displacement. The homogeneous vector equations is governing equations, that is to say, these equations can be used to solve independly the stable three dimensional plate/shell problems.Main purpose of the paper is to simply the complex problems, main works are listed as follows:(1) Consider the gradient relationship of temperature field, the modified generalized H-R variation formulations of the thermoelastictity, piezothermoelasticity and electromagnetothermoelasticity was established and the nonhomogeneous generalized Hamilton canonical equations of the above three materials were derived in detail. Several nonhomogeneous generalized Hamilton canonical equations were derived in detail. The main values of application of these variation principles can be marked as follows: On the one hand, the deriving approach of Hamilton canonical equation is simpled greatly based on these variation theorems. On the other hand, these variation principles provides a fundation for the derivation of complex engeneering problems of nnumerical method (eg. the finite element methods). (2) On the basis of the dual relationships of variables in the heat equilibrium equations and the heat conduction equation, the nonhomogeneous generalized Hamilton canonical equations were simplied to the homogeneous vector equations. And they can be used independly to analyze the stable three dimensional problems. The main advantages of the homogeneous vector equations simplifies greatly the solution programs which are often performed to combining nonhomogeneous Hamilton canonical equation and second order differential equation on the thermal equilibrium and the gradent relationship. Meanwhile, this algorithm avoids the inverse matrix calculations or the convolution operation of nonhomogeneous equations, improves the computing efficiency and ensures the stablitiness of numerical resultions.In a word, a simply and reliable method for relevant engeneering problems is provided in this paper.