Recent Advances on Herglotz’s Generalized Variational Principle of Nonconservative Dynamics

This paper summarized the recent development on Herglotz’s generalized variational principle and its symmetries and conserved quantities for nonconservative dynamical systems.Taking Lagrangian mechanics,Hamiltonian mechanics and Birkhoffian mechanics as three research frames,we introduce Herglotz’s generalized variational principle,dynamical equations of Herglotz type,Noether symmetry and conserved quantities,and their generalization to time-delay dynamics,fractional dynamics and time-scale dynamics,and put forward some problems as suggestions for future research.

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.

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.

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.

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.

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.

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.

THE MATHEMATICAL MODELS AND GENERALIZED VARIATIONAL PRINCIPLES OF NONLINEAR ANALYSIS FOR PERFORATED THIN PLATES

On foe basis of the Kirchoff-Karman hypothses for the nonlinear bending of thin plates, the three kinds of boundary value problems of nonlinear analysis for perforated fhin plates are presented under the differenr in-plane boundary conditions and the corresponding generalized varialional principles are established. One can see that all mathematical models presented in this paper are completely new ones and differ from the ordinary von Karman theory. These mathematical models can be applied to the nonlinear analysis and the Stability analysis of perforaled thin plates in arbitraryplane boundary conditions.

GENERALIZED VARIATIONAL PRINCIPLE ON NONLINEAR THEORY OF NATURALLY CURVED AND TWISTED CLOSED THIN-WALLED COMPOSITE BEAMS

Naturally curved and twisted closed thin-walled slender beams of composite material undergoing small strains, large displacements and rotations have been investigated, and an incomplete generalized variational function on theory of elasticity with finite displacement is established far these beams with complete constrained boundaries at two ends. The balance equations as well as all boundary conditions concerned have been deduced from functional stationary value condition. The above-mentioned method can also be extended to other various incomplete constrained boundaries conveniently. In addition, the fundamental equations and concerned formulas in the small displacement theory of the beams can be derived by using above results.

USING GENERALIZED VARIATIONAL PRINCIPLES TO RESOLVE THE ST.VENANT’S TORSIONAL BAR WITH A CRACK

According to generalized variational principles suitable for linear elastic incompatible displacement elements given by Professor Chien Wei-zang, using crack tip singular element and isoparametric surrounding element given by the author of this paper, we will study the St. Venant's torsional bar with a radial vertical crack and compare the present computed results with the results of reference [2], The present computed results show that, using the method provided in this paper, satisfactory convergent solution can be obtained under lower degree of freedom.

A RECTANGULAR ELEMENT OF THIN PLATES BASED UPON THE GENERALIZED VARIATIONAL PRINCIPLES

Based on generalized variational principles, an element called MR-12 was constructed for the static and dynamic analysis of thin plates with orthogonal anisotropy. Numerical results showed that this incompatible element converges very rapidly and has good accuracy. It was demonstrated that generalized varialional principles arc useful and effective in founding incompatible clement.Moreover, element MR-12 is easy for implementation since it does not differ very much from the common rectangular element R-12 of thin plate.

THE APPLICATIONS OF GENERALIZED VARIATIONAL PRINCIPLES IN NONLINEAR STRUCTURAL ANALYSIS

This paper discusses the generalized variational principles founded by the technique of Lagrangian multipliers in structural mechanics and analyzes the nonlinear statically indeterminate structures. It is assumed that the stress-strain relationship of the materials of structures has the form of namely, the physical equations of structures have the shape of exponential functions. Several examples are given to illustrate the statically indeterminate structures such as the trusses, beams, frames and torsional bars.

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.

ANALYSES ON NONLINEAR COUPLING OF MAGNETO-THERMO-ELASTICITY OF FERROMAGNETIC THIN SHELL—I:GENERALIZED VARIATIONAL THEORETICAL MODELING

Based on the generalized variational principle of magneto-thermo-elasticity of the ferromagnetic elastic medium, a nonlinear coupling theoretical modeling for a ferromagnetic thin shell is developed. All governing equations and boundary conditions for the ferromagnetic shell are obtained from the variational manipulations on the magnetic scalar potential, temperature and the elastic displacement related to the total energy functional. The multi-field couplings and geometrical nonlinearity of the ferromagnetic thin shell are taken into account in the modeling. The general modeling can be further deduced to existing models of the magneto-elasticity and the thermo-elasticity of a ferromagnetic shell and magneto-thermo-elasticity of a ferromagnetic plate, which are coincident with the ones in literature.

Generalized mixed finite element method for 3D elasticity problems

Without applying any stable element techniques in the mixed methods, two simple generalized mixed element(GME) formulations were derived by combining the minimum potential energy principle and Hellinger–Reissner(H–R) variational principle. The main features of the GME formulations are that the common C0-continuous polynomial shape functions for displacement methods are used to express both displacement and stress variables, and the coefficient matrix of these formulations is not only automatically symmetric but also invertible. Hence, the numerical results of the generalized mixed methods based on the GME formulations are stable. Displacement as well as stress results can be obtained directly from the algebraic system for finite element analysis after introducing stress and displacement boundary conditions simultaneously. Numerical examples show that displacement and stress results retain the same accuracy. The results of the noncompatible generalized mixed method proposed herein are more accurate than those of the standard noncompatible displacement method. The noncompatible generalized mixed element is less sensitive to element geometric distortions.

ROLE OF UNDERGROUND STRUCTURE DEFORMATION VELOCITY IN THE ANALYSIS OF BLAST-RESISTANT STRUCTURES

The structural deformation velocity plays a significant role in the dynamic calculation of underground blast-resistant structures. The motion differentiating equation of a structure system taking into account the role of deformation velocity of the structure will truthfully describe the actual situation of structural vibration. With the one-dimensional plane wave theory, the expression of load on the structural periphery is developed, and the generalized variation principle for the dynamic analysis of underground arched-bar structures is given. At the same time, the results of the numerical calculation are compared.

DYNAMIC ANALYSIS OF UNDERGROUND COMPOSITE STRUCTURES UNDER EXPLOSION LOADING

In selecting rational types of underground structures resisting explosion,in order to improve stress states of the structural section and make full use of material strength of each part of the section,the research method of composite structures is presented. Adopting the analysis method of micro-section free body,equilibrium equations,constraint equations and deformation coordination equations are given. Making use of the concept of generalized work and directly introducing Lagrange multiplier specific in physical meaning,the validity of the constructed generalized functional is proved by using variation method. The rational rigidity matching relationship of composite structure section is presented through example calculations.

ON THE LARGE DEFORMATION RUBBER-LIKE MATERIALS: CONSTITUTIVE LAWS AND FEM

This paper mainly discusses the constitutive laws of incompressible rubber-like materials and the associated finite element analysis method. By a multiplicative decomposition of the deformation gradient into distortional and dilatational parts, the YEOH mode type constitutive laws of rubber-like materials and their numerical implementation are presented. In order to deal with incompressible problems, a three-field variational principle is developed in which deformation, Jacobian and pressure field are treated independently. The connection between the three-field principle and the Hu-Wasizhu generalized variational principle is established. It is shown that the approach proposed can be degenerated to the B-bar method in the linear case. The derailed FE formulation is given in which deformation is ap proximated by isoparametric conforming element, and Jacobian and pressure by discontinuous approximation. Finally, two numerical examples are presented to show the effectiveness and reliability of the method proposed. The work in this paper provides a corner stone of FEA of this kind of problem. This paper features the combination of the multiplicative decomposition, the three-field principle and YEOH model of rubber-like materials, especially under Lagrangian description, giving an effective way for solving this kind of problems. The Lagrangian description is compatible with usually geometrically nonlinear FEM and the constitutive laws are expressed by the second Kirchhoff stress and the Green strain.

FUNDAMENTAL OF DYNAMICAL ANALYSIS OF THE BLAST-RESISTANT UNDERGROUND STRUCTURES

In this paper, the generalized variational principle of dynamic analysis for the blast-resistant underground structures is established, and the corresponding generalized functional of elastoplastic analysis for underground structures is derived, and the generalized variational principle of nonconservative system is given, thus the fundamental of dynamical analysis for underground structures to resist blast is proposed. Finally, for the underground cylindrical structure to resist blast, dynamical calculations are made, and compared with the test results.

Changes in the natural frequency of a ferromagnetic rod in a magnetic field due to magnetoelastic interaction

Based on the magnetoelastic generalized variational principle and Hamilton＇s principle, a dynamic theoretical model characterizing the magnetoelastic interaction of a soft ferromagnetic medium in an applied magnetic field is developed in this paper. From the variational manipulation of magnetic scale potential and elastic displacement, all the fundamental equations for the magnetic field and mechanical deformation, as well as the magnetic body force and magnetic traction for describing magnetoelastic interaction are derived. The theoretical model is applied to a ferromagnetic rod vibrating in an applied magnetic field using a perturbation technique and the Galerkin method. The results show that the magnetic field will change the natural frequencies of the ferromagnetic rod by causing a decrease with the bending motion along the applied magnetic field where the magnetoelastic buckling will take place, and by causing an increase when the bending motion of the rod is perpendicular to the field. The prediction by the mode presented in this paper qualitatively agrees with the natural frequency changes of the ferromagnetic rod observed in the experiment.