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

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Unified theory of variation principles in non-linear theory of elasticity

The purpose of this paper is to introduce and to discuss several main variation principles in nonlinear theory of elasticity——namely the classic potential energy principle, complementary energyprinciple, and other two complementary energy principles (Levinson principle and Fraeijs de Veu-beke principle) which are widely discussed in recent literatures. At the same time, the generalized variational principles are given also for all these principles. In this paper, systematic derivation and rigorous proof are given to these variational principles on the unified bases of principle of virtual work, and the intrinsic relations between these principles are also indicated. It is shown that, these principles have unified bases, and their differences are solely due to the adoption of different variables and Legendre tarnsformation. Thus, various variational principles constitute an organized totality in an unified frame. For those variational principles not discussed in this paper, the same frame can also be used, a diagram is given to illustrate the interrelationships between these principles.     

Further study of the equivalent theorem of Hellinger-Reissner and Hu-Washizu variational principles

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Generalized variational principles on nonlinear theory of elasticity with finite displacements

Two generalized variational principles on nonlinear theory of elasticity with finitedisplacements in which the σ_(ij),e_(i j)and u_i are all three kinds of independent functionsare suggested in this paper.It isproved that these two generalized variational principles areequivalent to each other if the stress-strain relation is satisfied as constraint.Some specialcases,i.e.generalized variational principles on nonlinear theory of elasticity with smalldeformation,on linear theory with finite deformation and on linear theory with smalldeformation together with the corresponding equivalent theorems are also obtained.All ofthem are related to the three kinds of independent variables.     

Variational principles in elasticity with nonlinear stress-strain relation

Since 1979, a series of papers have been published concerning the variational principles and generalized variational principles in elasticity such as [1] (1979), [6] (1980), [2,3] (1983) and [4,5] (1984). All these papers deal with the elastic body with linear stress-strain relations. In 1985, a book was published on generalized variational principles dealing with some nonlinear elastic body, but never going into detailed discussion. This paper discusses particularly variational principles and generalized variational principles for elastic body with nonlinear stress-strain relations. In these discussions, we find many interesting problems worth while to pay some attention. At the same time, these discussions are also instructive for linear elastic problems. When the strain is small, the high order terms may be neglected, the results of this paper may be simplified to the well-known principles in ordinary elasticity problems.     

Variational principles and generalized variational principles for nonlinear elasticity with finite displacement

In a previous paper (1979)^[1], the minimum potential energy principle and stationary complementary energy principle for nonlinear elasticity with finite displacement, together with various complete and incomplete generalized principles were studied. However, the statements and proofs of these principles were not so clearly stated about their constraint conditions and their Euler equations. In somecases, the Euler equations have been mistaken as constraint conditions. For example, the stress displacement relation should be considered as Euler equation in complementary energy principle but have been mistaken as constraint conditions in variation. That is to say, in the above mentioned paper, the number of constraint conditions exceeds the necessary requirement. Furthermore, in all these variational principles, the stress-strain relation never participate in the variation process as constraints, i.e., they may act as a constraint in the sense that, after the set of Euler equations is solved, the stress-strain relation may be used to derive the stresses from known strains, or to derive the strains from known stresses. This point was not clearly mentioned in the previous paper (1979)^[1]. In this paper, the high order Lagrange multiplier method (1983)^[2] is used to construct the corresponding generalized variational principle in more general form. Throughout this paper, V/.V. Novozhilov's results (1958)^[3] for nonlinear elasticity are used.     

Method of integro-differential relations in linear elasticity

Boundary-value problems in linear elasticity can be solved by a method based on introducing integral relations between the components of the stress and strain tensors. The original problem is reduced to the minimization problem for a nonnegative functional of the unknown displacement and stress functions under some differential constraints. We state and justify a variational principle that implies the minimum principles for the potential and additional energy under certain boundary conditions and obtain two-sided energy estimates for the exact solutions. We use the proposed approach to develop a numerical analytic algorithm for determining piecewise polynomial approximations to the functions under study. For the problems on the extension of a free plate made of two different materials and bending of a clamped rectangular plate on an elastic support, we carry out numerical simulation and analyze the results obtained by the method of integro-differential relations.     

some general theorems and generalized and piecewise generalized variational principles for linear elastodynamics

From the concept of four-dimensional space and under the four kinds of time limit conditions, some general theorems for elastodynamics are developed, such as the principle of possible work action, the virtual displacement principle, the virtual stress-momentum principle, the reciprocal theorems and the related theorems of time terminal conditions derived from it. The variational principles of potential energy action and complementary energy action, the H-W principles, the H-R principles and the constitutive variational principles for elastodynamics are obtained. Hamilton's principle, Toupin's work and the formulations of Ref. [5], [17]-[24] may be regarded as some special cases of the general principles given in the paper. By considering three cases: piecewise space-time domain, piecewise space domain, piecewise time domain, the piecewise variational principles including the potential, the complementary and the mixed energy action fashions are given. Finally, the general formulation of piecewise variati     

二维Nonlocal线弹性理论的变分原理与对偶方程

基于Eringen提出的Nonlocal线弹性理论的微分形式本构关系,导出了相应的能量密度表达式,进而得到二维Nonlocal线弹性理论的变分原理.利用变分原理导出了对偶平衡方程和相应的边界条件.进而给出了非局部动力问题的Lagrange函数,并引入对偶变量和Hamilton函数,得到了对偶体系下的变分方程.在Hamilton体系下,通过变分得到了二维Nonlocal线弹性理论的对偶平衡方程和相应的边界条件.     

The combined energy variational principles in linear elasticity

In this paper, a new kind of mixed energy variational principles in linear elasticity—the combined energy variational principles is presented. First, we define the conjugate body of an elastic body, which is obtained by changing the boundary conditions of the elastic body. Next, we decompose the conjugate body into two component-states, construct functionals of potential energy and complementary energy, respectively, for the component-states and define the additional hybrid energy between the component-states. Thus the functionals of combined energy can be constructed. Three typical combined energy variational principles are demonstrated and the other forms of combined energy variational principles are given. The application of the proposed principles to the calculation of thin plates with complicated boundaries is shown.     

On the unilateral contact problem of structures with a non quadratic strain energy density

The analysis of structures with “unilateral contact” boundary conditions is considered. The stress-strain relations are nonlinear and they are derived from a non quadratic strain energy density by “subdifferentiation”. It is proved that for the inequality constrained boundary value problem the “principles” of virtual and of complementary virtual work hold in an inequality form constituting a variational inequality. The theorems of minimum potential and complementary energy are proved to be valid to account for this type of boundary conditions. These theorems are used to formulate the analysis as a nonlinear programming problem. A numerical example of a structure having the “unilateral contact” boundary condition illustrates the theory.     

Large deformation symmetrical elasticity problems solved by the variational method

In this paper,based on the mathematical theory of classical mechanics and Chen‘s theorem,the variational method is used in the study of large deformation symmetrical elasticity problems.The generalized variational principles of potential energy and complementary energy,based on the instantaneous configuration are obtained,and the equivalence between the two principles is proved.Besides,the generalized variational principles of dynamical problems based on the instantaneous configuration are also given.     

Method of high-order lagrange multiplier and generalized variational principles of elasticity with more general forms of functionals

It is known[1]that the minimum principles of potential energy andcomplementary energy are the conditional variation principles underrespective conditions of constraints.By means of the method of La-grange multipliers,we are able to reduce the functionals of condi-tional variation principles to new functionals of non-conditionalvariation principles.This method can be described as follows:Mul-tiply undetermined Lagrange multipliers by various constraints,andadd these products to the original functionals.Considering these un-determined Lagrange multipliers and the original variables in thesenew functionals as independent variables of variation,we can see thatthe stationary conditions of these functionals give these undeter-mined Lagrange multipliers in terms of original variables.The sub-stitutions of these results for Lagrange multipliers into the abovefunctionals lead to the functionals of these non-conditional varia-tion principles.However,in certain cases,some of the undetermined Lagrangemultipliers ma     

A non-convex lower bound on the effective energy of polycrystalline shape memory alloys

This paper addresses the estimation of the effective free energy of polycrystalline shape memory alloys, in the framework of nonlinear elasticity and infinitesimal strains. The translation method is combined with a Hashin-Shtrikman type variational formulation to provide rigorous lower bounds on the effective free energy. Those bounds incorporate both intra-grain compatibility conditions (resulting in a non-convex bound) and inter-grain constraints (by taking one- and two-point statistics into account). Some examples are given to compare the results obtained with other bounds from the literature.     

Variational principles for Cosserat body

Two variational principles are derived for the mixed boundary value problem of Cosserat solid. These principles are a generalization of the stationary principle of potential energy and the stationary principle of complementary energy from non-linear theory of elasticity.     

On various variational principles in nonlinear theory of elasticity

In the present paper functionals for the various possible main variational principles in the nonlinear theory of e-lasticity are derived from the "full energy principle" and several of them are not found yet in the literatures available. Through the derivation of this paper we suggest a conjecture on the nonexistence of the eleventh and the sixth classes for the variational principles in Table 6.1 of H.C. Hu’s monograph [1].     

STUDY OF THE EQUIVALENT THEOREM OF GENERALIZED VARIATIONAL PRINCIPLES IN ELASTICITY

The relations of all generalized variational principles in elasticity are studied by employing the invariance theorem of field theory. The infinitesimal scale transformation in field theory was employed to investigate the equivalent theorem. Among the results found particularly interesting are those related to that all generalized variational principles in elasticity are equal to each other. Also studied result is that only two variables are independent in the functional and the stress-strain relation is the variational constraint condition for all generalized variational principles in elasticity. This work has proven again the conclusion of Prof. Chien Wei-zang.     

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．     

Analysis of micro-sized beams for various boundary conditions based on the strain gradient elasticity theory

Bending analysis of micro-sized beams based on the Bernoulli-Euler beam theory is presented within the modified strain gradient elasticity and modified couple stress theories. The governing equations and the related boundary conditions are derived from the variational principles. These equations are solved analytically for deflection, bending, and rotation responses of micro-sized beams. Propped cantilever, both ends clamped, both ends simply supported, and cantilever cases are taken into consideration as boundary conditions. The influence of size effect and additional material parameters on the static response of micro-sized beams in bending is examined. The effect of Poisson’s ratio is also investigated in detail. It is concluded from the results that the bending values obtained by these higher-order elasticity theories have a significant difference with those calculated by the classical elasticity theory.