电磁热弹性介质的一些基本理论

具有电磁热弹性耦合性质的介质是许多传感器及机敏结构或系统中的关键材料，本文第一部分动态变分原理已就具有这种性质的机敏材料动力学理论作了较为系统的分析。在此第二部分则系统地讨论了在数值计算中极具重要性的各类变分原理，包括准静态变分原理、动态变分原理和关于固有频率的变分原理。     

智能板结构振动特性的半解析法

为智能板结构振动特性的分析提出了一种半解析法.根据压电材料修正后的Hellinger-Re issner变分原理,推导了压电材料的Ham ilton ian等参元.智能板结构的基体板和压电块(压电传感器或驱动器)被看作独立的三维体,并用Ham ilton ian等参元分别离散板和压电块.考虑到板和压电块在连接界面上广义应力和广义位移的连续性,联立板和压电块的方程得到整个结构的方程组.数值实例的分析结果证明了方法的可靠性.     

含压电片层合壳的有限元分析与控制仿真

本文首先针对含压电材料的一般结构，推导了按位移和电位移求解时的混合变分原理，在此基础上，通过对面电荷取变分，直接得到了含按电压驱动压电片层合壳的有限元方程，最后给出了利用压电片进行静变形和振动控制的仿真算例，并进行了分析。结果表明，本文建立的有限元方程准确、可靠。     

压电层合梁静动力学分析的高精度梁单元

给出了一个对复合材料压电层合梁进行数值分析的高精度压电层合梁单元。基于Shi三阶剪切变形板理论的位移场和Layer-wise理论的电势场,用力-电耦合的变分原理及Hamilton原理推导了压电层合梁单元列式。采用拟协调元方法推导了一个可显式给出单元刚度矩阵的两节点压电层合梁单元,并应用于压电层合梁的力-电耦合弯曲和自由振动分析。计算结果表明,该梁单元给出的梁挠度和固有频率与解析解吻合良好,并优于其它梁单元的计算结果,说明了本文所给压电层合梁单元的可靠性和准确性。研究结果可为力-电耦合作用下压电层合梁的力学分析提供一个简单、精确且高效的压电层合梁单元。     

饱和多孔介质耦合系统的几类变分原理

采用变积方法，建立了一组等温准静态下饱和多孔介质的六类变量的广义变分原理，在此基础上，引入约束条件得到五类变量，四类变量，三类变量和二类变量的变分原理，为建立饱和多孔介质的有限元模型提供了基础。     

不可压流体饱和多孔弹性梁的变分原理及有限元方法

基于不可压饱和多孔弹性梁动力弯曲的数学模型,建立了以多孔弹性梁挠度和孔隙流体压力等效力偶为宗量的Gurtin型变分原理,并给出了特殊边界条件下解耦时的仅以挠度为宗量的变分原理.同时,作为动力响应的退化情形,讨论了拟静态情形下的相应变分原理.根据所建立的变分原理,导出了一个有限元离散公式.由于Gurtin型变分原理是关于时间的卷积型的泛函,空间的有限元离散导致一个关于时间的对称微分一积分方程组,此方程组可进一步转化为常微分方程组.利用隐式Euler法,给出了时间区域的计算格式.作为一个数值例子,分析了饱和多孔弹性悬臂梁在自由端简谐载荷作用下的动力响应,分析了流相与固相相互作用对饱和多孔弹性悬臂梁动力响应的影响.     

刚塑性广义变分不等原理及其在平面应变分析中的应用

首先利用Lagrangian乘子法，从势能角度出发构造了考虑摩擦效应这一能导致变分不等形式的广义能量泛函，把一般的有条件的变分原理化为无条件的变分原理唯一确定，得出了各Lagrangian乘子所代表的物理意义。建立了刚塑性理论中的Coulomb摩擦约束的广义变分不等原理。而后基于退化的摩擦约束广义变分等式原理，对长矩形板镦粗进行了塑性加工工步分析，所得结果与经典上限法结果相吻合。     

Reisner板的哈密尔顿体系及其辛正交解析法

基于Ｒｅｉｓｎｅｒ板理论，通过对混合能变分原理的修正，建立了更一般的哈密尔顿型广义变分原理，并给出了Ｒｅｉｓｎｅｒ板问题的哈密尔顿正则方程及其共轭辛正交解析法     

变水深环境下中厚度浮板耦振问题的一个变分解

本文从Reissner 板的理论出发,提出并论证了变水深环境下中厚度浮板耦振问题的一个局部变分原理,它是建立中厚度浮板各种近似分析方法的一个有力工具,借助这个变分原理,文中还导得了一个计算中厚度浮板固有频率的一个变分式,数值算例表明本文方法具有精度高和省机时的优点。     

Reissner板的哈密尔顿体系及其辛正交解析法

基于Ｒｅｉｓｓｎｅｒ板理论，通过对混合能变分的修正，建立了更一般的哈密尔顿型广义变分原理，并给出了Ｒｅｉｓｓｎｅｒ板问题的哈密尔顿正则方程及其共轭辛正交解析法。     

Generalized variational principles for linear elastic materials with voids

In this paper, the idea of variational principles of linear elastic theory is used to establish generalized variational principles for linear elastic materials with voids. The fundamental equations of linear elastic materials with voids used have already been established in Ref. [5].     

Generalized variational principles of the viscoelastic body with voids and their applications

From the Boltzmann‘ s constitutive law of viscoelastic materials and the linear theory of elastic materials with voids, a constitutive model of generalized force fields for viscoelastic solids with voids was given. By using the variational integral method, the convolution-type functional was given and the corresponding generalized variational principles and potential energy principle of viscoelastic solids with voids were presented. It can be shown that the variational principles correspond to the differential equations and theinitial and boundary conditions of viscoelastic body with voids. As an application, a generalized variational principle of viscoelastic Timoshenko beams with damage was obtained which corresponds to the differential equations of generalized motion and the initial and boundary conditions of beams. The variational principles provide a way for solving problems of viscoelastic solids with voids.     

Variational analysis of thermomechanically coupled steady-state rolling problem

A steady-state, rigid-plastic rolling problem for temperature and strain-rate dependent materials with nonlocal friction is considered. A variational formulation is derived, coupling a nonlinear variational inequality for the velocity and a nonlinear vari- ational equation for the temperature. The existence and uniqueness results are obtained by a proposed fixed point method.     

微孔压电弹性动力学的能量原理

根据古典阴阳互补和现代对偶互补的基本思想,通过作者早已提出一条简单而统一的新途径,系统地建立了微孔压电弹性动力学的能量原理,给出一个重要的以卷积表示的积分关系式,可以认为,在力学上它是广义虚功原理的表式,从该式出发,不仅能得到微孔压电弹性动力学的虚功原理和互等定理,而且通过作者所给出的一系列广义Legendre变换,能系统地导出成互补关系的11类变量、9类变量、6类变量和3类变量简化Gurtin型变分原理的泛函,同时,通过这条新途径,还能清楚地阐明这些原理之间的内在联系。     

参变量变分原理的提出、发展与应用

参变量变分原理及其参数二次规划算法是由钟万勰院士1985年针对弹性接触边界非线性问题首次提出来的,经过将近40年的不断发展,目前参变量变分原理已经成功应用于各个领域,其中包括弹塑性分析、接触问题、润滑力学、岩土力学、变刚度杆系结构、先进材料性能分析、材料的蠕变与损伤、柔性结构力学和LQ最优控制等各个工程领域。本文首先回顾了参变量变分原理的起源,介绍了参变量变分原理的基本概念,然后以弹塑性分析问题为例,阐明建立参变量变分原理的理论模型以及实现数值参数二次规划求解原理,最后详细回顾了参变量变分原理的基本理论与相应数值算法在各个领域的发展及其工程应用,展示了参变量变分原理在求解各类非线性问题的特色与优势。     

Plastic collapse in non-associated hardening materials with application to Cam-clay

This paper presents a variational formulation for the analysis of plastic collapse conditions for a class of hardening materials that accounts for some non-associated flow laws such as the modified Cam-clay model of soils. In this framework, classical statical and kinematical principles of limit analysis do not hold. The variational principle is formulated for the general class of materials whose flow equations are derived from a kind of generalized potentials named bipotentials by de Saxcé.The plastic collapse phenomenon for hardening materials is considered first and formulated as a system of equations. In particular, the case of the usual modified Cam-clay model is analyzed. The paper follows with the proposal of a minimization principle whose solution is then related to the solution of the plastic collapse problem. We demonstrate the use of this minimum principle in a simple example of triaxial compression of a modified Cam-clay material. Finally, we discuss the particular form of the proposed variational formulation for the case of associated plasticity.     

子空间变分原理的修正及其应用于确定梁的剪切系数

首先简化了空间变分原理的数学结构，据此表明子空间变分原理存在一定的奇异性，并提出消除奇异性的修正子空间变分原理。作为应用，计算了单材料多种截面的剪切系数并与Ｃｏｗｐｅｒ解做了比较，结果表明修正后的子空间变分原理是正确的。我们还进一步计算了一个夹层梁在Ｃｏｗｐｅｒ意义下的剪切系数，说明了子空间变分原理复杂截面的能力。     

各向异性弹性力学问题Hamilton正则方程的一般形式

本文从修正后的Ｈｅｌｌｉｎｇｅｒ－Ｒｅｉｓｓｎｅｒ变分原理出发，导出了由２１个弹性常数组成的各向异性材料的混合方程，井证明它们即是Ｈａｍｉｌｔｏｎ正则方程。由该统一形式还给出了角铺设材料和正交各向异性材料的Ｈａｎｉｌｔｏｎ正则形式。     

Stationary variational estimates for the effective response and field fluctuations in nonlinear composites

This paper presents a variational method for estimating the effective constitutive response of composite materials with nonlinear constitutive behavior. The method is based on a stationary variational principle for the macroscopic potential in terms of the corresponding potential of a linear comparison composite (LCC) whose properties are the trial fields in the variational principle. When used in combination with estimates for the LCC that are exact to second order in the heterogeneity contrast, the resulting estimates for the nonlinear composite are also guaranteed to be exact to second-order in the contrast. In addition, the new method allows full optimization with respect to the properties of the LCC, leading to estimates that are fully stationary and exhibit no duality gaps. As a result, the effective response and field statistics of the nonlinear composite can be estimated directly from the appropriately optimized linear comparison composite. By way of illustration, the method is applied to a porous, isotropic, power-law material, and the results are found to compare favorably with earlier bounds and estimates. However, the basic ideas of the method are expected to work for broad classes of composites materials, whose effective response can be given appropriate variational representations, including more general elasto-plastic and soft hyperelastic composites and polycrystals.     

The random variational principle and finite element method

In this paper, we introduced the random materials, geometrical shapes, force and displacement boundary condition directly into the functional variational formulations and developed a unified random variational principle and finite element method with the small parameter perturbation method. Numerical examples showed that the methods have the advantages of the simple and convenient program implementation, and are effective for the random mechanics problems.