【作者】 刘艳红；

【导师】 张惠明；

【作者基本信息】 天津大学 ， 动力机械及工程， 2009， 博士

【摘要】 压电传感器等部件在工作中其内部常常存在着不均匀的温度场,研究压电材料等类似材料的温度场—电场—变形等的耦合问题对于工程应用有很大的理论意义。本文基于弹性体的本构关系、热平衡方程和导热方程,根据弹性体修正后的H-R变分原理建立了热弹性材料、压电热弹性材料及电磁热弹性材料在温度梯度下的广义H-R变分原理,并推导了相应的非齐次广义Hamilton正则方程。然后根据对偶变量理论,将热流量及温度变量类比为应力与位移的对偶关系,导出了增维的齐次向量方程。齐次向量方程的特点是:方程中的变量是偶数,它们可分为广义的平面外应力与广义位移两大类,而且这两类变量是完全对偶的;另一方面,齐次向量方程具有控制方程特性,即可以独立求解稳态的三维板壳问题。采用各类材料相应的层合板、壳实例验证了所得出的齐次向量方程的正确性。本论文工作的主要意义是将复杂问题简单化,这主要表现在:(1)考虑温度梯度关系,建立了热弹性体、压电热弹性体和电磁热弹性体修正后的广义H-R变分原理,并推导了相应的非齐次Hamilton正则方程。广义H-R变分原理的主要应用价值是:一方面,基于变分原理可简化Hamilton正则方程的推导过程;另一方面,基于广义H-R变分原理,为有关复杂工程结构问题的数值分析(如有限元法,渐近法等)的推导提供了必要的理论基础。(2)根据热平衡和导热方程中变量的对偶关系,将非齐次的Hamilton正则方程化为能独立求解稳态的三维问题的齐次向量方程。齐次向量方程不但可大大简化分析耦合问题时通常要求联立非齐次Hamilton正则方程及由热平衡方程和导热方程导出的二阶微分方程的繁琐方法,而且也避免了求解非齐次方程时的矩阵求逆或卷积运算,在提高了运算效率的同时保证了数值结果的稳定性。总之,本论文的工作为相关工程问题分析提供了一种简便、可靠的方法。更多还原

【Abstract】 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.更多还原