Simplified Gurtin-type generalized variational principles for fully dynamic magneto-electro-elasticity with geometrical nonlinearity

The fundamental equations, governing all the variables of the initial boundary value problem in fully dynamic magneto-electro-elasticity with geometrical nonlinearity, are expressed in covariant differential form. The generalized principle of virtual work is given in terms of convolutions for the present problem. Two simplified Gurtin-type generalized variational principles, directly leading to all the fundamental equations, are deduced by using He’s semi-inverse method instead of Laplace transforms. By enforcing some fundamental equations as constraint conditions, one of various constrained variational principles is given as an example. By simply dropping out selected field functions, several reduced variational principles are obtained as special forms for piezoelectricity, elastodynamics, and electromagnetics, respectively. This paper aims at providing a more complete theoretical foundation for the finite element applications for the discussed problem.     

A Universal Variational Formulation for Two Dimensional Fluid Mechanics

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电磁弹性动力学初边值问题12类变量广义变分原理

基于Yao建立的电磁弹性固体广义变分原理,运用关于非传统Hamilton型广义变分原理的方法,建立了电磁弹性动力学初边值问题的12类变量广义变分原理,可反映该问题的全部特征,其独立变分变量为该问题的全部变量,即位移、速度、动量、应变、应力、电位移、磁感应强度、电场强度、磁场强度、电标量势、磁标量势和磁矢量势.本文建立的...     

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.     

带孔隙损伤的弹性固体的广义变分原理

应用弹性微结构理论,建立了具广义力场带孔隙损伤线弹性固体的基本模型．应用变积方法,同时分别建立了带孔隙损伤弹性固体四类和两类变量的广义变分原理,这些变分原理对应着带孔隙损伤弹性固体微分方程和初值边值条件．应用弹性微结构理论,建立了带孔隙损伤的弹性Timoshenko 梁的基本方程,得到带孔隙损伤的弹性Timoshenko 梁两类变量的广义变分原理．这些广义变分原理为近似求解带孔隙损伤的弹性问题提供了有效途径．     

Semi-inverse method and generalized variational principles with multi-variables in elasticity

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GENERALIZED VARIATIONAL PRINCIPLESFOR VISCOELASTIC THIN AND THICK PLATES WITH DAMAGE

From the constitutive model with generalized force fields for a viscoelastic body with damage, the differential equations of motion for thin and thick plates with damage are derived under arbitrary boundary conditions. The convolution-type functionals for the bending of viscoelastic thin and thick plates with damage are presented, and the corresponding generalized variational principles are given. From these generalized principles, all the basic equations of the displacement and damage variables and initial and boundary conditions can be deduced. As an example, we compare the difference between the dynamical properties of plates with and without damage and consider the effect of damage on the dynamical properties of plates.     

相空间中非保守系统的Herglotz广义变分原理及其Noether定理

与经典变分原理相比,基于由微分方程定义的作用量的Herglotz广义变分原理给出了非保守动力学系统的一个变分描述,它不仅能够描述所有采用经典变分原理能够描述的动力学过程,而且能够应用于经典变分原理不能适用的非保守或耗散系统.将Herglotz广义变分原理拓展到相空间,研究相空间中非保守力学系统的Herglotz广义变分原理与Noether定理及其逆定理.首先,提出相空间中Herglotz广义变分原理,给出相空间中非保守系统的变分描述,导出相应的Hamilton正则方程;其次,基于非等时变分与等时变分之间的关系,导出相空间中Hamilton-Herglotz作用量变分的两个基本公式;再次,给出Noether对称变换的定义和判据,提出并证明相空间中非保守系统基于Herglotz变分问题的Noether定理及其逆定理,揭示了相空间中力学系统的Noether对称性与守恒量之间的内在联系.在经典条件下,Herglotz广义变分原理退化为经典变分原理,与之相应的相空间中的Noether定理退化为经典Hamilton系统的Noether定理.文末以著名的Emden方程和平方阻尼振子为例说明上述方法和结果的有效性.     

耦合热弹性动力学中各类Hamilton型拟变分原理

根据古典阴阳互补和现代对偶互补的基本思想,通过作者早已提出的一条简单而统一的途径,系统地建立了耦合热弹性动力学的各类Hamilton型拟变分原理。这种以单一泛函的变分式表示的Hamilton型拟变分原理,能精确反映耦合热弹力动力学初值－边值问题的全部特征。文中首先给出一个在力学上可以认为是广义拟虚功原理的表式。然后从该式出发,通过所给出的一系列广义Legendre变换,系统地推导出耦合热弹性动力学的8类变量、6类变量、4类变量和2类变量Hamilton型拟变分原理。同时,通过这条途径还能阐明这些原理的内在联系。     

分析力学初值问题的一种变分原理形式

明确了分析力学初值问题的控制方程，按照广义力和广义位移之间的对应关系，将 各控制方程卷乘上相应的虚量，代数相加，进而在 原空间中建立了分析力学初值问题的一种变分原理形式，即建立了分析力学初值问题的卷积 型变分原理和卷积型广义变分原理. 推导了分析力学初值问题卷积型变分原理和卷积型广义 变分原理的驻值条件. 在建立分析力学初值问题的一种变分原理形式的同时, 将变积方法推广为卷变积方法.     

A generalized variational formulation for convective heat transfer

The Scope of this paper is to develop the basic equations for a variational formulation which can be used to solve problems related to convection and/or diffusion dominated flows. The formulation is based on the introduction of a generalized quantity defined as the hear displacement. The governing equation is expressed in terms of this quantity and a variational formulation is developed which leads to a system of equations similar in form to Lagrange's equations of mechanics. These equations can be used for obtaining approximate solutions, though they are of particular interest for application of the finite element method. As an example of the formulation two finite element models are derived for solving convectiondiffusion boundary value problems. The performance of the two models is investigated and numerical results are given for different cases of convection and diffusion with two types of boundary conditions. The applications of the developed formulations are not limited to convection-diffusion problems but can also be applied to other types of problems such as mass transfer, hydrodynamics and wave propagation.     

Classification of variational principles in elasticity

In this paper, variational principels in elasticity are classified according to the differences in the constraints used in these principles. It is shown in a previous paper [4] that the stress-strain relations are the constraint conditions in all these variational principles, and cannot be removed by the method of linear Lagrange multiplier. The other possible constraints are four of them: (1) equations of equilibrium, (2) strain-displacement relations, (3) boundary conditions of given external forces and (4) boundary conditions of given boundary displacements. In variational principles of elasticity, some of them have only one kind of such constraints, some have two kinds or three kinds of constraints and at the most four kinds of constraints. Thus, we have altogether 15 kinds of possible variational principles. However, for every possible variational principle, either the strain energy density or the complementary energy density may be used. Hence, there are altogether 30 classes of functional of variational principles in elasticity. In this paper, all these functionals are tabulated in detail.     

调和函数边界积分方程的充要条件

本文以调和函数的边值问题为例,探讨了边界积分方程的充要条件.文中首次提出了超定问题的概念,并建立了超定问题有解的一个充要条件,它也就是直接变量边界积分方程的一个充要条件.文中首次阐明了边界积分方程与变分原理的内在的联系,还指出了间接变量与直接变量两类边界积分方程之间存在着一一对应的关系.文中的慨念、思路和论点不难用于其它有变分原理的问题的边界积分方程.     

Variational principles for hydrodynamic impact problems

We first establish the rigorous field equations of the two continuous stages before and after entering water. Then correspondently, we obtain the specific variational principles, bounded theorems, and boundary integral equations of the second stage problems. The existence of solutions are proved and the scheme of solving the solutions are provided. Finally, as a numerical example, the ship's wave resistence problem is used to demonstrate the specific application of the second stage problems and its accuracy. Then we provide a rigorous and sound theoretical basis of variational finite element method and boundary element method for calculating the accurately fundamental equations.     

On the method of Lagrange multiplier and others

The fundamentals for the correct use of the method of Lagrange multiplier are presented and illustrated by examples. Some misunderstandings of the method are clarified. Equivalent variational principles are defined. It is pointed out that for a given problem of mechanics, there may be many equivalent and unequivalent variational principles. The functional of the so called generalized variational principles of elasticity with more general forms^[16] are linear combinations of the well known functionals given by Reissner and Hu-Washizu.     

电磁弹性固体三维问题的广义变分原理

提出了以电磁弹性固体所有变量应力、应变、位移、电位移、电场强度、电势、磁感应强度、磁场强度和磁势为自变量的电磁弹性固体三维问题最一般形式的广义变分原理。它们涵盖了电磁弹性固体问题所有的基本方程和边界条件。在此基础上还可以进一步给出部分变量为自变量的其它形式广义变分原理。     

Necessary and sufficient condition for the correct formulation of boundary integral equations of harmonic functions

A necessary and sufficient condition for the correct formulation of boundary integral equations of harmonic functions is established in the present paper. A super-determined problem of harmonic functions is proposed for the first time. Then a necessary and sufficient condition for the existence of solution of the super-determined problem is proved. At the same time, it is a necessary and sufficient condition for the correct formulation of boundary integral equations with direct unknown quantities. A relation between boundary integral equations and variational principles is discovered for the first time. And a one-to-one correspondence between boundary integral equations with direct and indirect unknown quantities is indicated. The concept and route of the present paper can be applied to other boundary value problems possessing variational principles.     

非保守弹性动力学初值问题的简单Gurtin型拟变分原理

按照广义力和广义位移之间的对应关系,将弹性动力学基本方程卷积乘上相应的虚量,然后积分且代数相加,并利用体积力和面积力均为伴生力这一特征,建立了非保守系统初值问题的简单Gurtin型五类变量的完全拟变分原理.更进一步地还建立了非保守系统初值问题的简单Gurtin型不完全拟变分原理和有条件不完全拟变分原理.在建立非保守系统初值问题的各类简单Gurtin型拟变分原理的同时,还将变积方法推广为卷变积方法.最后,介绍了寻求伴生力的方法.     

大变位热弹性扁壳的一种边值模型

本文给出了正则形式基本方程及广义变分原理,并给出了关于应力函数、挠度、温度三种变量的一种边值模型。     

A NEW ELEMENT FOR THIN PLATE OF BENDING WITH CURVILINEAR BOUNDARY—CURVILINEAR BOUNDARY QUADRILATERAL ELEMENT

This paper presents a curvilinear boundary quadrilateral element for the problem of thin plate of bending with curvilinear boundary. A coordinate transformation of two dimensions is performed in the calculation of FEM. The introduction of an additional stiffness matrix based on the generalized variational principles results in high accuracy and less computation time. The numerical results agree with the analytical solution very well.