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.     

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 non-linear 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 generalizedvariational principles are given also for all these principles.In this paper,systematic derivation andrigorous proof are given to these variational principles on the unified bases of principle of virtualwork,and the intrinsic relations between these principles are also indicated.It is shown that,theseprinciples have unified bases,and their differences are solely due to the adoption of different varia-bles and Legendre tarnsformation.Thus,various variational principles constitute an organized totalityin an unified frame.For those variational principles not discussed in this paper,the same frame canalso be used,a diagram is giv     

A METHOD FOR ESTABLISHING GENERALIZED VARIATIONAL PRINCIPLE

A method for establishing generalized variational principle is proposed in this paper Itis based on the analysis of mechanical meaning and it can be applied to problems in whichthe variational principles are needed but no corresponding variational principle is availableIn this paper,the Hu-Washizu’s generalized variational principle and the Hu’s generalizedprinciple of complementary energy are derived from the mechanical meaning instead offrom the generalization of the principle of minimum potential energy and the correct proofsof these two generalized variational principles are given.It is also proved that this is wrongif one beleives that (?),e_(?) and u_i are independent variables each other based on thereason that these three kinds of variables are all contained in these two generalizedvariational principles.The condition of using these two variational principles in a correctmanner is also explained.     

BOUNDARY VALUE PROBLEMS FOR SECOND ORDER DELAY DIFFERENTIAL EQUATIONS

In this paper,using a fixed point principle and existence principle given in[1],westudy the boundary value problems for second order differential equations.Some newexistence results are obtained.     

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.     

EXTENDED VARIATIONAL PRINCIPLE IN NONLINEAR THEORY OF ELASTICITY

From the extended energy functional, a unified variational principle and an extended variational principle are given and proved in this paper. It is about the non-conservative, abrupt and divided domains static (or kinetic) system and by using the undetermined energy functional, we can immediately deduce various variational principles.     

Initial value problem for a class of fourth-order nonlinear wave equations

In this paper, existence and uniqueness of the generalized global solution and the classical global solution to the initial value problem for a class of fourth-order nonlinear wave equations are studied in the fractional order Sobolev space using the contraction mapping principle and the extension theorem. The sufficient conditions for the blow up of the solution to the initial value problem are given.     

A, φ-Ω METHOD FOR 3-D EDDY CURRENT ANALYSIS

After the field equations and the snonumuoo conditions between the interfaces for 3D eddy current problems Under various gauges were discussed, it was pointed cut in this paper that using the magnetic vector potential A. the electric scalar potential φ and Coulomb gauge ∇·A = 0 in eddy current regions and using the magntetic scalar potential Ω in the non-conducting regions are more suitable. All field equations, the boundary conditions, the interface continuity conditions and the corresponding variational principle of this method are also given.     

T-STRESS IN PIEZOELECTRIC SOLID

The non-singular and bounded terms for stresses near the crack tip were investigated. The crack problem in a transversely isotropic piezoelectric solid for the plane problem was dealt with. The principle of superposition and the Plemelj formulation were introduced. The non-singular terms are given by solving Rieman-Hilbert problem. It is shown that the non-singular terms are influenced by the elastic and electric constants.     

BIFURCATION AND STABILITY ANALYSIS FOR LIQUID SURFACES

A more comprehensive discussion on the bifurcation problems for the shape of liquidsurfaces is made in this paper.The necessary conditions for bifurcation are given,and thebifurcating solutions near bifurcation points can be obtained by perturbation technique.Finally the stability of the bifurcating states is analyzed by means of the principle ofminimum potential energy.     

Extensions of Appell's and Tzènoff's equations to general nonholonomic systems

In this paper we present a new variational principle, from which Gauss principle can be derived as a corollary. By using the new principle, Appell's and Tzènoff's equations are extended to nonholonomic systems of any order. As an example, a second order nonholonomic system is given to illustrate the application of the equations obtained.     

On a variational principle of Hashin and Shtrikman

In this paper a simple derivation of a new version of the variational principle of Hashin and Shtrikman is given. It appears possible to dispense with some subsidiary conditions which in the original principle had to be imposed on the class of functions considered in the variational process. However, through the use of the new principle the possibility of obtaining a pair of two sided bounds is lost and this has to be compensated by invoking one of the complementary principles which are formulated in the sequel.     

Extended variational principle in nonlinear theory of elasticity

From the extended energy functional, a unified variational principle and an extended variational principle are given and proved in this paper. It is about the non-conservative, abrupt and divided domains static (or kinetic) system and by using the undetermined energy functional, we can immediately deduce various variational principles.     

Saint-Venant's Principle

The principle of the elastic equivalence of statically equivalent systems of load, or Saint-Venant's Principle, is given a precise mathematical formulation and proof. Counterexamples to traditional verbal statements of the principle are given, and the results are compared with previous mathematical work on the Saint-Venant principle.     

On some general variational principles for creep with applications to thin shells

A constitutive relation for a viscous material subject to small strains but finite rotations is postulated and associated variational theorems are formulated. These are similar to the principle of minimum of potential energy and the Hellinger-Reissner theorem of an elastic solid. The derivation of strain-displacement relations for thin shells subject to small strains but moderately large rotations are given. On this basis a mixed variational principle for thin viscous shells is developed. For the problem of creep collapse of long cylindrical shells under external pressure it is demonstrated that the mixed variational principle may be advantageous compared to other variational theorems. A comparison with the creep collapse theory of Hoff et al. is given.     

APPELL方程和TZNOFF方程在一般非完整系统上的推广

本文给出一个适用于任何阶非完整系统的变分原理。Gauss原理可作为推论由之导出。由此原理出发,将Appell方程和Tzènoff方程推广到一般非完整系统,以一个二阶非完整系统为例,说明所得方程的应用。     

用于弹性蠕变损伤问题的参变量变分原理

参变量变分原理是近年来发展的用于处理数学物理问题中边界待定边值问题的一种有效方法,本文建立起用于蠕变损伤问题结构分析的参变量变分原理,该原理将原问题化为求解带约束条件的泛函极值,其约束条件就是由蠕变损伤本构关系推导出的系统状态方程组;该原理物理意义明确、表达式简单并且规范,容易为计算机实现。本文给出原理的证明,并就2.25Cr-1Mo钢在550℃下的蠕变问题给出实例。     

On a theory of micropolar protoelastic material bodies and constitutive equations for nonlocal micropolar elastic continua

In this paper the definition of micropolar protoclastic material bodies is given and with the help of the principle of virtual power, the variational principle of those bodies is derived. In terms of that same idea and the definition of micropolar protopotential presented here, the constitutive equations for nonlocal micropolar elastic continua are naturally derived.