Unconventional Hamilton-type variational principles for nonlinear elastodynamics of orthogonal cable-net structures

According to the basic idea of classical yin-yang complementarity and modem dual-complementarity,in a simple and unified new way proposed by Luo,the unconven- tional Hamilton-type variational principles for geometrically nonlinear elastodynamics of orthogonal cable-net structures are established systematically,which can fully charac- terize the initial-boundary-value problem of this kind of dynamics.An important in- tegral relation is made,which can be considered as the generalized principle of virtual work for geometrically nonlinear dynamics of orthogonal cable-net structures in mechan- ics.Based on such relationship,it is possible not only to obtain the principle of virtual work for geometrically nonlinear dynamics of orthogonal cable-net structures,but also to derive systematically the complementary functionais for five-field,four-field,three-field and two-field unconventional Hamilton-type variational principles,and the functional for the unconventional Hamilton-type variational principle in phase space and the poten- tial energy functional for one-field unconventional Hamilton-type variational principle for geometrically nonlinear elastodynamics of orthogonal cable-net structures by the general- ized Legendre transformation given in this paper.Furthermore,the intrinsic relationship among various principles can be explained clearly with this approach.     

有限变形弹性动力学的非传统Gurtin型变分原理

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，系统地建立了有限变形弹性动力学的各类非传统Gurtin型变分原理，给出一个以卷积表示的重要关系式，可以认为，该式是有限变形动力学的广义虚功原理的表式。从该式出发，不仅能得到有限变形动力学的虚功原理，而且通过给出的一系列广义Legendre变换，还能系统地成对导出5类变量，3类变量，2类变量和1类变量非传统Gurtin型变分原理的互补泛函。通过这条途径还能清楚地阐明这些原理之间的内在联系。     

UNCONVENTIONAL GURTIN-TYPE VARIATIONAL PRINCIPLES FOR FINITE DEFORMATION ELASTODYNAMICS

According to the basic idea of classical Yin-Yang complementarity and modern dual-complementarity,in a simple and unified new way proposed by Luo,the unconventional Gurtin-type variational prinicples for finite deformation elastodynamics can be established systemati-cally.In this paper,an important integral relation in terms of convolution is given,which canbe considered as the expression of the generalized principle of virtual work for finite deformationdynamics.Based on this relation,it is possible not only to obtain the principle of virtual work forfinite deformation dynamics,but also to derive systematically the complementary functionals forfive-field,three-field,two-field and one-field unconventional Gurtin-type variational principles bythe generalized Legendre transformations given in this paper.Furthermore,with this approach,the intrinsic relationship among various principles can be clearly explained.     

有限变形弹性动力学的非传统简化Gurtin型变分原理--文献[1]的续篇

根据古典阴阳互补和现代对偶互补的基本思想,通过作者提出的一条简单而统一的新途径,建立了有限变形弹性动力学的另一种单卷积形式的变分原理一各类非传统简化Gurtin型变分原理．首先给出一个以卷积表示的关系式,在力学上它是有限变形动力学的广义虚功原理的另一种表式．然后从该式出发,不仅能得到有限变形动力学另一种形式的虚功原理,而且通过文中所给出的一系列广义Legendre变换,还能成对导出5类变量、3类变量、2类变量和1类变量非传统简化Gurtin型变分原理的互补泛函．同时,通过这条新途径还能阐明这些原理之间的内在联系。     

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.     

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.     

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.     

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.     

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 Formulation of the First Principle of Continuum Thermodynamics

The First Principle of Continuum Thermodynamics is formulated as a variational condition whose test fields are piecewise constant virtual temperatures. Lagrange multipliers theorem is applied to relax the constraint of piecewise constancy of test fields. This provides the existence of square summable vector fields of heat flow through the body fulfilling a virtual thermal work principle, analogous to the virtual work principle in Mechanics. The issue of compatibility of thermal gradients is dealt with and expressed by the complementary variational condition. Primal, complementary and mixed variational inequalities leading to computational methods in heat-conduction boundary-value problems are briefly discussed.     

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。     

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]).     

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 micromechanics-based nonlocal constitutive equation incorporating three-point statistics for random linear elastic composite materials

A variational formulation employing the minimum potential and complementary energy principles is used to derive a micromechanics-based nonlocal constitutive equation for random linear elastic composite materials, relating ensemble averages of stress and strain in the most general situation when mean fields vary spatially. All information contained in the energy principles is retained; we employ stress polarization trial fields utilizing one-point statistics so that the resulting nonlocal constitutive equation incorporates up through three-point statistics. The variational structure is developed first for arbitrary heterogeneous linear elastic materials, then for randomly inhomogeneous materials, then for general n-phase composite materials, and finally for two-phase composite materials, in which case explicit variational upper and lower bounds on the nonlocal effective modulus tensor operator are derived. For statistically uniform infinite-body composites, these bounds are determined even more explicitly in Fourier transform space. We evaluate these in detail in an example case: longitudinal shear of an aligned fiber or void composite. We determine the full permissible ranges of the terms involving two- and three-point statistics in these bounds, and thereby exhibit explicit results that encompass arbitrary isotropic in-plane phase distributions; we also develop a nonlocal “Milton parameter”, the variation of whose eigenvalues throughout the interval [0, 1] describes the full permissible range of the three-point term. Example plots of the new bounds show them to provide substantial improvement over the (two-point) Hashin–Shtrikman bounds on the nonlocal operator tensor, for all permissible values of the two- and three-point parameters. We next discuss further applications of the general nonlocal operator bounds: to any three-dimensional scalar transport problem e.g. conductivity, for which explicit results are given encompassing the full permissible ranges of the two- and three-point statistics terms for arbitrary three-dimensional isotropic phase distributions; and to general three-dimensional composites, where explicit results require future research. Finally, we show how the work just summarized, treating elastostatics, can be generalized to elastodynamics, first in general, then explicitly for the longitudinal shear example.     

A method for establishing generalized variational principle

A method for establishing generalized variational principle is proposed in this paper. It is based on the analysis of mechanical meaning and it can be applied to problems in which the variational principles are needed but no corresponding variational principle is available. In this paper, the Hu-Washizu ’s generalized variational principle and the Hu’s generalized principle of complementary energy are derived from the mechanical meaning instead of from the generalization of the principle of minimum potenlial energy and the correct proofs of these two generaleed variational principles are given. It is also proved that this is wrong if one beleives that σ_ij, e_ij and ui are independent variables each other based on the reason that these three kinds of variables are all contained in these two generalized variational principles. The condition of using these two variational principles in a correct manner is also explained.     

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.     

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.     

A variational principle for non-linear elastodynamics and its application to the hybrid stress model

A variational principle is derived for the mixed initial-boundary value problem of non-linear elastodynamics. This principle involves stress quantities only. It is an extension of a similar one derived by Gurtin[1] for linear elastodynamics. The principle is then specialized to the class of semilinear materials, and generalized for the use in the hybrid stress model of finite element analysis. An incremental procedure for the numerical solution is described.     

On the variational principles in linear elastodynamics

A new approach is proposed for the systematic derivation of varïous variational principles in linear elastodynamics. Based on an important integral relation in terms of convolutions given by the authors, the new approach can be used to derive the complementary functionals for the five-field, four-field, three-field, two-field and one-field variational principles more simply and directly. Furthermore, with this approach, it is possible not only to derive the variational principles given by Herrera and Bielak, Oden and Reddy, but also to develop new more general variational principles. And the intrinsic relationship among various principles can be explained clearly.     

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