SEMI-INVERSE METHOD AND GENERALIZED VARIATIONAL PRINCIPLES WITH MULTI-VARIABLES IN ELASTICITY

Semi-inverse method, which is an integration and an extension of Hu's try-and-error method, Chien's veighted residual method and Liu's systematic method, is proposed to establish generalized variational principles with multi-variables without arty variational crisis phenomenon. The method is to construct an energy trial-functional with an unknown function F, which can be readily identified by making the trial-functional stationary and using known constraint equations. As a result generalized variational principles with two kinds of independent variables (such as well-known Hellinger-Reissner variational principle and Hu-Washizu principle) and generalized variational principles with three kinds of independent variables (such as Chien's generalized variational principles) in elasticity have been deduced without using Lagrange multiplier method. By semi-inverse method, the author has also proved that Hu-Washizu principle is actually a variational principle with only two kinds of independent variables, stress-strain relations are still its constraints.

FURTHER STUDY OF THE EQUIVALENT THEOREM OF HELLINGER-REISSNER AND HU-WASHIZU VARIATIONAL PRINCIPLES

In this paper, it has been pro ved that the well_known Hu_Washizu variational principle is a pseudo_generalized variational principle(pseudo_GVP). The stationary conditions of its functional may satisfy all its field equations and boundary conditions if all the variables in the functional are considered as independent variations, but there might exi st some kinds of constraints. Some new pseudo_GVPs are established to distinguis h them from genuine ones by the so_called inverse Lagrange multiplier method. Th e constrained Hu_Washizu principle, therefore, is proved to be equivalent with t he Hellinger_Reissner principle under the constraints of stress_strain relations .

GURTIN-TYPE VARIATIONAL PRINCIPLES FOR DYNAMICS OF A NON-LOCAL THERMAL EQUILIBRIUM SATURATED POROUS MEDIUM

Based on the porous media theory and by taking into account the efects of the pore fuid viscidity, energy exchanges due to the additional thermal conduction and convection between solid and fuid phases, a mathematical model for the dynamic-thermo-hydro-mechanical coupling of a non-local thermal equilibrium fuid-saturated porous medium, in which the two constituents are assumed to be incompressible and immiscible, is established under the assumption of small de- formation of the solid phase, small velocity of the fuid phase and small temperature changes of the two constituents. The mathematical model of a local thermal equilibrium fuid-saturated porous medium can be obtained directly from the above one. Several Gurtin-type variational principles, especially Hu-Washizu type variational principles, for the initial boundary value problems of dy- namic and quasi-static responses are presented. It should be pointed out that these variational principles can be degenerated easily into the case of isothermal incompressible fuid-saturated elastic porous media, which have been discussed previously.

Highly accurate symplectic element based on two variational principles

For the stability requirement of numerical resultants, the mathematical theory of classical mixed methods are relatively complex. However, generalized mixed methods are automatically stable, and their building process is simple and straightforward. In this paper, based on the seminal idea of the generalized mixed methods, a simple, stable, and highly accurate 8-node noncompatible symplectic element(NCSE8) was developed by the combination of the modified Hellinger-Reissner mixed variational principle and the minimum energy principle. To ensure the accuracy of in-plane stress results, a simultaneous equation approach was also suggested. Numerical experimentation shows that the accuracy of stress results of NCSE8 are nearly the same as that of displacement methods, and they are in good agreement with the exact solutions when the mesh is relatively fine. NCSE8 has advantages of the clearing concept, easy calculation by a finite element computer program, higher accuracy and wide applicability for various linear elasticity compressible and nearly incompressible material problems. It is possible that NCSE8 becomes even more advantageous for the fracture problems due to its better accuracy of stresses.

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 the initial 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.

THE VARIATIONAL PRINCIPLES FOR PYROELECTRIC MEDIA

The pyroelectric medium is an important material in the application of smart materials and structures. It is necessary to systematically discuss ail kinds of variational principles which play a very significant role in the fundamental theory of mechanics and numerical analysis method. This paper firstly gives the quasi-static and the dynamic variational principles,then the principles for eigen problems. As a simple example, the principle was finally applied to derive the fundamental, equations for an anisotropic piezoelectric plate.

Variational principles for buckling and vibration of MWCNTs modeled by strain gradient theory

Variational principles for the buckling and vibration of multi-walled carbon nanotubes （MWCNTs） are established with the aid of the semi-inverse method. They are used to derive the natural and geometric boundary conditions coupled by small scale parameters. Hamilton＇s principle and Rayleigh＇s quotient for the buckling and vibration of the MWCNTs are given. The Rayleigh-Ritz method is used to study the buckling and vibration of the single-walled carbon nanotubes （SWCNTs） and double-walled carbon nanotubes （DWCNTs） with three typical boundary conditions. The numerical results reveal that the small scale parameter, aspect ratio, and boundary conditions have a profound effect on the buckling and vibration of the SWCNTs and DWCNTs.

UNCONVENTIONAL HAMILTON-TYPE VARIATIONAL PRINCIPLES FOR DYNAMICS OF REISSNER SANDWICH PLATE

According to the basic idea of classical yin-yang complementarity and modern dual-complementarity, in a simple and unified way proposed by Luo（1987）, some uncon ventional Hamilton-type variational principles for dynamics of Reissner sandwich plate can be established systematically. The unconventional Hamilton-type variation principle can fully characterize the initial boundary value problem of this dynamics. In this paper, an important integral relation 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 in dynamics of Reissner sandwich plate, but also to derive systematically the complementary functionals for fivefield, two-field and one-field unconventional Hamilton-type variational principles by the generalized Legender transformations. Furthermore, with this approach, the intrinsic relationship among the various principles can be explained clearly.

Generalized variational principles for boundary value problem of electromagnetic field in electrodynamics

An expression of the generalized principle of virtual work for the boundary value problem of the linear and anisotropic electromagnetic field is given. Using Chien＇s method, a pair of generalized variational principles （GVPs） are established, which directly leads to all four Maxwell＇s equations, two intensity-potential equations, two constitutive equations, and eight boundary conditions. A family of constrained variational principles is derived sequentially. As additional verifications, two degenerated forms are obtained, equivalent to two known variational principles. Two modified GVPs are given to provide the hybrid finite element models for the present problem.

Investigation of power-type variational principles in liquid-filled system

Starting from the basic equations of hydrodynamics, the maximum power- type variational principle of the hydrodynamics of viscous fluids was established by Weizang CHIEN in 1984. Through long-term research, it is clarified that the maximum power-type variational principle coincides with the Jourdian principle, which is one of the common principles for analytical mechanics. In the paper, the power-type variational principle is extended to rigid-body dynamics, elasto-dynamics, and rigid-elastic：liquid coupling dynamics. The governing equations of the rigid-elastic-liquid coupling dynamics in the liquid-filled system are obtained by deriving the stationary value conditions. The results show that, with the power-type variational principles studied directly in the state space, some transformations in the time domain space may be omitted in the establishing process, and the rigid-elastic-liqUid coupling dynamics can be easily numerically modeled. Moreover, the analysis of the coupling dynamics in the liquid-filled system in this paper agrees well with the numerical analyses of the coupling dynamics in the liquid-filled system offered in the literatures.

VARIATIONAL PRINCIPLES OF ASYMMETRIC ELASTICITY THEORY OF FINITE DEFORMATION

In this paper, based on the finite deformation S-R decomposition theorem, the definition of the body moment is renewed as the stem of its internal and external. The expression of the increment rate of the deformation energy is derived and the physical meaning is clarified. The power variational principle and the complementary power variational principle for finite deformation mechanics are supplemented and perfected.

Hamiltonian Systems in Elasticity and Their Variational Principles

As an inverse problem of Hamiltonian mechanics, a new Hamiltonian system inelasticity and its variational principle are derived from the basic equations of elasticity.

THE VARIATIONAL PRINCIPLES OF COUPLED SYSTEMS IN PHOTOELASTICITY

This paper presents an energy principle, zero different principle of coupledsystems in photoelasticity, from which the potential energy, the complementary energy,generalized potential energy and generalized complementary energy variationalprinciples of the coupled systems in photoelasticity are derived What is called the coupled systems means that two deformational bodies, forwhich figures, sizes,loads and boundary conditions are the same and they are all inactual states but they are made of different materials.Prototype body and model body in photoelasticity are essentially the coupledsystems, therefore the above principles become the theoretical basis of defining theinflunce of Poissons ratio v on accuracy of the frozen-stress method.

VARIATIONAL PRINCIPLES AND GENERALIZED VARIATIONAL PRINCIPLES IN HYDRODYNAMICS OF VISCOUS FLUIDS

In this paper, the variational principles of hydrodynamic problems for the incompressible and compressible viscous fluids are established. These principles are principles of maximum power losses. Their generalized variational principles are also discussed on the basis of Lagrangian multiplier methods.

A Revised Pre-Order Principle and Set-Valued Ekeland Variational Principles with Generalized Distances

In my former paper ＂A pre-order principle and set-valued Ekeland variational principle＂ （see [J. Math. Anal. Applo, 419, 904 937 （2014）]）, we established a general pre-order principle. From the pre-order principle, we deduced most of the known set-valued Ekeland variational principles （denoted by EVPs） in set containing forms and their improvements. But the pre-order principle could not imply Khanh and Quy＇s EVP in [On generalized Ekeland＇s variational principle and equivalent formulations for set-valued mappings, J. Glob. Optim., 49, 381-396 （2011）], where the perturbation contains a weak T-function, a certain type of generalized distances. In this paper, we give a revised version of the pre-order principle. This revised version not only implies the original pre-order principle, but also can be applied to obtain the above Khanh and Quy＇s EVP. In particular, we give several new set-valued EVPs, where the perturbations contain convex subsets of the ordering cone and various types of generalized distances.

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 unconventional Hamilton-type variational principles for geometrically nonlinear elastodynamics of orthogonal cable-net structures are established systematically, which can fully characterize the initial-boundary-value problem of this kind of dynamics. An ifnportant integral relation is made, which can be considered as the generalized principle of virtual work for geometrically nonlinear dynamics of orthogonal cable-net structures in mechanics. 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 functionals 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 potential energy functional for one-field unconventional Hamilton-type variational principle for geometrically nonlinear elastodynamics of orthogonal cable-net structures by the generalized Legendre transformation given in this paper, Furthermore, the intrinsic relationship among various principles can be explained clearly with this approach.

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.

Variational Principles and Hamiltonian Formulation for Nonlinear Water Waves

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REGION-WISE VARIATIONAL PRINCIPLES AND GENERALIZED VARIATIONAL PRINCIPLES ON LARGE STRAIN FOR CONSOLIDATION THEORY

The difference of constitutive character and large deformation as to soil mass are basic questions to analyze deformational feature. According to the description method of limited deformation, the large deformation consolidation equations of soil mass were created and its variational principles were rigorously testified. The regionwise variational principles of consolidation theory were deduced using sub-structure continuous condition of region-wise. Quoting the method of Lagrangian multiplier operator, generalized variational principles of region-wise of large deformation consolidation in the nonconstrained condition were created and approved.

VARIATIONAL PRINCIPLES OF FLUID FULLFILLED ELASTIC SOLIDS

The generalized variational principles of isothermal quasi-static fluid full-filled elastic solids are established by using Variational Integral Method. Then by introducing constraints, several kinds of variational principles are worked out, including five-field variable, four-field variable, three-field variable and two-field variable formulations. Some new variational principles are presented besides the principles noted in the previous works. Based on variational principles, finite element models can be set up.