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．     

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     

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.     

Application of Chien’s Theorem to General Variational Principle in Finite Elasticity with Body Couple

An important theorem proved by W. Z. Chien[1] states the equivalence of the functionals in general variational principles of potential energy and complementary energy. The stated theorem is applied now in formulation of general variational principle in finite elasticity with body couple (polar elasticity). Comoving coordinate system is being used in the derivation throughout (refer to[6], [8]).     

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.     

Energy principles in theory of elastic materials with voids

According to the basic idea of dual-complementarity, in a simple and unified way proposed by the author^[1], various energy principles in theory of elastic materials with voids can be established systematically. In this paper, an important integral relation is given, which can be considered essentially as the generalized pr. inciple of virtual work. Based on this relation, it is possible not only to obtain the principle of virtual work and the reciprocal theorem of work in theory of elastic materials with voids, but also to derive systematically the complementary functionals for the eight-field, six-field, four-field and two-field generalized variational principles, and the principle of minimum potential and complementary energies. Furthermore, with this appro ach, the intrinsic relationship among various principles can be explained clearly. The project supported by the National Natural Science Foundation of China     

Variational formulations and a consistent finite-element procedure for a class of nonlocal elastic continua

The structural boundary-value problem in the context of nonlocal (integral) elasticity and quasi-static loads is considered in a geometrically linear range. The nonlocal elastic behaviour is described by the so-called Eringen model in which the nonlocality lies in the constitutive relation. The diffusion processes of the nonlocality are governed by an integral relation containing a recently proposed symmetric spatial weight function expressed in terms of an attenuation function. A firm variational basis to the nonlocal model is given by providing the complete set of variational formulations, composed by ten functionals with different combinations of the state variables. In particular the nonlocal counterpart of the classical principles of the total potential energy, complementary energy and mixed Hu–Washizu principle and Hellinger–Reissner functional are recovered. Some examples concerning a piecewise bar in tension are provided by using the Fredholm integral equation and the proposed nonlocal FEM.     

Gurtin变分原理及其应用的时间有限元法

动力学Ｇｕｒｔｉｎ变分原理完整地表征了动力学的全部特征。由于在Ｇｕｒｔｉｎ变分原理的泛函中含有双重卷积，给时间域的离散带来很大困难。本文通过在Ｌａｐｌａｃｅ空间构造泛函，获得了几个具有单重卷积的Ｇｕｒｔｉｎ原理。由于卷积降阶，所给出的泛函更加便于应用。本文还通过在时间域采用适当的插值多项式逼近广义节点坐标，进一步讨论了时间有限元法实施的基本原理和步骤。     

On the connections between the functionals of equivalent variational principles

It is proved in this paper that the functionals of equivalent variational principles are different essentially only in terms of weighted residues. Consequently, the simplest method for constructing equivalent variational principles is to add weighted residues to known functionals as first used by Courant [4].     

Principles of minimum transformed energy and minimum principles for dynamics of plates

In the present paper, we first by Laplace transform present a derivation of principle of transformed virtual work, three principles of minimum transformed energy with influence of rotatory enertia for dynamics of anisotropic linear elastic plates with three generalized displacements. Moreover, the forms with the original in place-time domain corresponding these variational principles are presented.Then by the introduction of the set of admissible weight functions the three minimum principles for the original place-time domain are derived.In each of the preceding groups of the variational principles there are two which are the dynamic counterparts to the static principles of minimum potetial energy and minimum complementary energy; the other principles are formulated in terms of the internal force alone, but have no counterpart in elastostatics of plates.     

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].     

Energy and variational principles for generalized (gyroscopic) conservative problems

The study of the local stability of the equilibrium of an elastic body subjected to conservative and gyroscopic forces, which was initiated in [6] is extended. Two variational principles are established in actual and equivalent energy terms respectively. These principles, together with the stability criterion formulated in [6], form a generalized version of (and reduce to) the classical energy principles governing the stability of conservative non-gyroscopic systems, and are applied to the example of a loaded (or unloaded) elastic shaft under constant rotation. In addition, a necessary and sufficient condition for the applied forces to be gyroscopic or conservative is obtained explicitly as a function of the change in force components due to a small disturbance superimposed on the equilibrium state.     

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.     

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.     

等价变分原理泛函之间的联系

本文证明了等价变分原理的泛函,实质上都只相差某种加权残差项,这也就表明了,在已知的泛函后附加加权残差项,是建立等价变分原理最简便的方法.     

Mathematical study of boundary-value problems within the framework of Steigmann–Ogden model of surface elasticity

Mathematical questions pertaining to linear problems of equilibrium dynamics and vibrations of elastic bodies with surface stresses are studied. We extend our earlier results on existence of weak solutions within the Gurtin–Murdoch model to the Steigmann–Ogden model of surface elasticity using techniques from the theory of Sobolev’s spaces and methods of functional analysis. The Steigmann–Ogden model accounts for the bending stiffness of the surface film; it is a generalization of the Gurtin–Murdoch model. Weak setups of the problems, based on variational principles formulated, are employed. Some uniqueness-existence theorems for weak solutions of static and dynamic problems are proved in energy spaces via functional analytic methods. On the boundary surface, solutions to the problems under consideration are smoother than those for the corresponding problems of classical linear elasticity and those described by the Gurtin–Murdoch model. The weak setups of eigenvalue problems for elastic bodies with surface stresses are based on the Rayleigh and Courant variational principles. For the problems based on the Steigmann–Ogden model, certain spectral properties are established. In particular, bounds are placed on the eigenfrequencies of an elastic body with surface stresses; these demonstrate the increase in the body rigidity and the eigenfrequencies compared with the situation where the surface stresses are neglected.     

Involutory transformations and variational principles with multi-varoables in thin plate bending problems

In this paper, the generalizd variational principles of plate bending, froblems are established from their minimum potential energy principle and minimum complementary energy principle through the elimination of their constraints by means of the method of Lagrange multipliers. The involutory transformations are also introduced in order to reduce the order of differentiations for the variables in the variation. Funhermore, these involutory transformations become infacl the additional constraints in the varialion. and additional Lagrange multipliers may be used in order to remove these additional constraints. Thus, various multi-variable variational principles are obtained for the plate bending problems. However, it is observed that. nol all the constrainls ofva’iaticn can be removed simply by the ordinary method of linear Lagrange multipliers. In such cases, the method of high-order Lagrange multipliers are usedto remove iliose constrainls left over by ordinary linear multiplier method. And consequently. some funct ionals of more general forms are oblained for the generaleed variational principles of plate bending problems.     

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.     

On some basic principles in dynamic theory of elastic materials with voids

According to the basic idea of dual-complementarity, in a simple and unified way proposed by the author^[1], some basic principles in dynamic theory of elastic materials with voids can be established systematically. In this paper, an important integral relation in terms of convolutions is given, which can be considered as the generalized principle of virtual work in mechanics. Based on this relation, it is possible not only to obtain the principle of virtual work and the reciprocal theorem in dynamic theory of elastic materials with voids, but also to derive systematically the complementary functionals for the eight-field, six-field, four-field and two-field simplified Gurtin-type variational principles. Furthermore, with this approach, the intrinsic relationship among various principles can be explained clearly. The project supported by the Foundation of Zhongshan University Advanced Research Center     

Two‐objective optimization strategies using the adjoint method and game theory in inverse natural convection problems

This paper considers the problem of estimating the strengths of two time‐varying heat sources simultaneously, from measurements of the temperature inside the square domain in a porous medium, when prior knowledge of the source functions is not available. This problem is an inverse natural convection problem. In order to circumvent this problem, we define several optimization criteria (objective functionals) that measure discrepancies between model and measured data, where objective functionals depend on two heat sources and use multi‐criteria optimization to identify Nash equilibria, which are solutions to the non‐cooperative game according to game theory. Two non‐cooperative game strategies are considered: competitive (Nash) game and hierarchical (modified Stackelberg) game. The methodology that we employ relies on a combination of mixed finite element space approximations, finite difference time discretizations, adjoint equation and sensitivity equation techniques, and nonlinear conjugate gradient algorithms for the solutions of estimating two heat sources. Applying the Sobolev gradient for the noise removal is investigated. The performance of the present technique of inverse analysis is evaluated, by means of several numerical experiments, and is found to be very accurate as well as efficient. Copyright © 2012 John Wiley & Sons, Ltd.