VARIATIONAL METHODS FOR THE PROBLEMS OF NONCONSERVATIVE FORCE FIELDS IN THE MICROPOLAR ELASTODYNAMICS

Based on the properties of the convolution and the convolute commutation, somequasi-variational principles for the problems of nonconservative force field in themicropolar elastodynamics are given and verified in this paper. The theorem given in thispaper can be applied to the theories of the nonlocal elastic mediums and the nonlocalmicropolar elastic mediums.     

An eigen theory of static electromagnetic field for anisotropic media

Static electromagnetic fields are studied based on standard spaces of the physical presentation, and the modal equations of static electromagnetic fields for anisotropic media are derived. By introducing a new set of first-order potential functions, several novel theoretical results are obtained. It is found that, for isotropic media, electric or magnetic potentials are scalar; while for anisotropic media, they are vectors. Magnitude and direction of the vector potentials are related to the anisotropic subspaces. Based on these results, we discuss the laws of static electromagnetic fields for anisotropic media.     

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.     

CONSTITUTIVE RELATION OF UNSATURATED SOIL BY USE OF THE MIXTURE THEORY (Ⅱ)-LINEAR CONSTITUTIVE EQUATIONS AND FIELD EQUATIONS

The linear constitutive equations and field equations of unsaturated soils were obtained through linearizing the nonlinear equations given in the first part of this work. The linear equations were expressed in the forms similar to Biot’s equations for saturated porous media. The Darcy’s laws of unsaturated soil were proved. It is shown that Biot’s equations of saturated porous media are the simplification of the theory. All these illustrate that constructing constitutive relation of unsaturated soil on the base of mixture theory is rational.     

NONLINEAR SATURATION OF BAROCLINIC INSTABILITY IN THE GENERALIZED PHILLIPS MODEL (Ⅰ) -THE UPPER BOUND ON THE EVOLUTION OF DISTURBANCE TO THE NONLINEARLY UNSTABLE BASIC FLOW

On the basis of the nonlinear stability theorem in the context of Arnol’s second theorem for the generalized Phillips model, nonlinear saturation of baroclinic instability in the generalized Phillips model is investigated. By choosing appropriate artificial stable basic flows, the upper bounds on the disturbance energy and potential enstrophy to the nonlinearly unstable basic flow in the generalized Phillips model are obtained, which are analytic completely and without the limitation of infinitesimal initial disturbance.     

APPLICATIONS 0F THE RECIPROCAL THEOREM TO SOLVING THE EQUATIONS OF THE DEFLECTION SURFACE OF THE RECTANGULAR PLATES WITH VARIOUS EDGE CONDITIONS

This paper points out that on certain condition the reciprocaltheorem is equivalent to the superposition principle of displacements.On the basis of[6],applications of the reciprocal theorem are fur-ther extended;a convenient.general and new method for solving theequations of deflection surface of the rectangular plates and thestraight beams with various edge conditions under various loads is pre-sented.     

THE EXISTENCE AND UNIQUENESS OF THE SCREW TENSOR FOR THE FINITE DISPLACEMENT OF A RIGIDBODY

The existence and uniqueness theorem of the screw tensor for the finite displacement of a rigidbody is proposed and then proved using the screw calculus. As a consequence, formulae are obtained for determining the screw tensor in terms of the finite displacement data of the rigidbody.     

An eigen theory of waves in piezoelectric solids

Based on the standard spaces of the physical presentation, both the quasi-static mechanical approximation and the quasi-static electromagnetic approximation of piezoelectric solids are studied here. The complete set of uncoupled elastic wave and electromagnetic wave equations are deduced. The results show that the number and propagation speed of elastic waves and electromagnetic waves in anisotropic piezoelectric solids are determined by both the subspaces of electromagnetically anisotropic media and ones of mechanically anisotropic media. Based on these laws, we discuss the propagation behaviour of elastic waves and electromagnetic waves in the piezoelectric material of class 6 mm.     

THE LINEAR APPROXIAIATION OF THE LINE CONTINUOUS DISTRIBUTION METHOD OF SINGULARITIES IN CREEPING MOTION

The linear approximation of the line continuous distri-bution method of singularities is proposed to treat the creeping motion of the arbitrary prolate axisymmetrical body. The analytic expressions in closed form for the flow field are obtained. The numerical results for the proiate spheroid and Cassini oval demonstrate that the convergence and the accuracy of the proposed method are better than the constant density approximation, furthermore, it can be applied to greater slender ratio. In this paper the example is yielded to show that the linear approximation of the singularities for the density on the partitioned segments can be utilized to consider the creeping motion of the arbitrary pointed prolaue axisymmetrical body.     

ON THE EMBEDDING AND COMPACT PROPERTIES OF FINITE ELEMENT SPACES

In this paper, the generalized Sobolev embedding theorem and the generalized Rellich-Kondrachov compact theorem for finite element spaces with multiple sets of functions are established. Specially, they are true for nonconforming, hybrid and quasi-conforming element spaces with certain conditions.     

THE SOLUTION OF LARGE DEFLECTION PROBLEM OF THIN CIRCULAR PLATE BY THE METHOD OF COMPOSITE EXPANSION

In this paper, the method of composite expansion in perturbation theory is used for the solution of large deflection problem of thin circular plate. In this method. the outer field solution and the inner boundary layer solution are combined together to satisfy all the boundary conditions. In this paper, Hencky's membrane solution is used for the first approximation in outer field solution, and then the second approximate solution is obtained. The inner boundary layer solution is found on the bases of boundary layer coondinate. In this paper, the reciprocal ratio of maximum deflection and thickness of the plate is used as the small parameter. The results of this paper improves quite a bit in comparison with the results obtained in 1948 by Chien Wei-zang.     

ON THE METHOD OF RECIPROCAL THEOREM TO FIND SOLUTIONS OF THE PLANE PROBLEMS OF ELASTICITY

In this paper the method of reciprocal theorem is extended to find solutions of planeproblems of elasticity of the rectangular plates with various edge conditions.First we give the basic solution of the plane problem of the rectangular plate with fouredges built-in as the basic system and then find displacement expressions of the actualsystem by using the reciprocal theorem between the basic system and actual system withvarious edge conditions.When only displacement edge conditions exist,obtaining displacement expressions bymeans of the method of reciprocal theorem is actual.But in other conditions,when staticforce edge conditions or mixed ones exist,the obtained displacements are admissible.Inorder to find actual displacement,the minimum potential energy theorem must be applied.Calculations show that the method of reciprocal theorem is a simple,convenient andgeneral one for the solution of plane problems of elasticity of the rectangular plates withvarious edge conditions.Evidently,it is a new method.     

AN UPPER BOUND SOLUTION FOR THE FORCE OF COMBINED BACKWARD-FORWARD EXTRUSION

A rigid-triangle velocity field for combined backward-forwardextrusion based on the experiments and the slip-line field isproposed in this paper.The flow separation point in the ri-gid-triangle velocity field is defined in accordance with theslip-line theory.A formula of minimum upper bound solutionfor the punch pressure of the combined extrusion is derived.The values from this formula are compared with those from theslip-line solution and with experimental results.The formulaof upper bound solution can be used in practice.     

THE THEOREM OF THE STABILITY OF LINEAR NONAUTONOMOUS SYSTEMS UNDER THE FREQUENTLY-ACTING PERTURBATION

In the paper a theorem on the stability of linear nonautonomous system under thefrequently-acting perturbation has been given and proved on the basis of Malkin＇sTheorem.     

DYNAMICS OF MULTI—DEFORMABLE BODIES

The Gibbs-Appell equations provide what is the powerful tool to formulate equations of motion not only for rigid-body, but also for deformable body. The application of the Gibbs-Appell equations in formulating equations of motion of multi-deformable boodies is developed and exploited in this paper. The advantages of using Gibbs-Appell equation are shown herein. Equations of motion of multi-deformable bodies with explicit form have been obtained and their physical meaning is apparent. In addition, recently developed new ideas are also employed. These ideas include the use of Euler parameters, quasi-coordinates, and relative coordinates.     

ANALYSIS OF SURFACE CRACK ON FORWARD EXTRUDING BAR DURING AXISYMMETRIC CUP-BAR COMBINED EXTRUSION PROCESS

In this paper,the kinematically admissible velocity field with surface crack on forwardextruding bar is put forward during the axisymmetric cup-bar combined extrusion process,in accordance with the results of model experiments.On the basis of velocity field,the necessary condition for surface crack formation onthe forward extruding bar is derived,with the help of upper bound theorem and theminimum energy principle.Meanwhile,the relationships between surface crack formationand combination of reduction in area for the part of forward and backward extursions(εb,εf),relative residual thickness of billet(T/R_0),frictional factor(m)or relativeland length of ram and chamber(l_b/R_0,l_f/R_0)are calculated during the extrusion process.Therefore,whether the surface crack on forward exturding bar occurs can be predictedbefore extruding the lower-plasticity metals for axisymmetric cup-bar combined extrusionprocess.The analytical results agree very well with experimental results of aluminium alloyLY12 (ASTM2024)an     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅹ) —MASTER BALANCE LAW

Through a comparison between the expressions of master balance laws and the conservation laws derived by Noether's theorem, a unified master balance law and six physically possible balance equations for micropolar continuum mechanics are naturally deduced. Among them, by extending the well-known conventional concept of energymomentum tensor, the rather general conservation laws and balance equations named after energy-momentum, energy-angular momentum and energy-energy are obtained. It is clear that the forms of the physical field quantities in the master balance law for the last three cases could not be assumed directly by perceiving through the intuition. Finally, some existing results are reduced immediately as special cases.     

BRIDGE-TOUGHENING ANALYSIS OF THE FIBER REINFORCED COMPOSITE CONTAINING INTERPHASE EFFECT

A detailed fracture mechanics analysis of bridge-toughening in a fiberreinforced composite is presented in this paper.The integral equation governing bridge-toughening as well as crack opening displacement (COD) for the composite withinterfacial layer is derived from the Castigliano's theorem and interface shear-lagmodel.A numerical result of the COD equation is obtained using the iteration solutionof the second Fredholm integral equation.In order to investigate the effect of variousparameters on the toughening,an approximate analytical solution of the equation ispresent and its error analysis is performed,which demonstrates the approximatesolution to be appropriate.A parametric study of the influence of the crack length,interracial shear modules,thickness of the interphase,fiber radius,fiber volumefraction and properties of materials on composite toughening is therefore carried out.The results are useful for experimental demonstration and toughening design includingthe fabrication process of the composite.     

CONSOLIDATION HEORY OF UNSATURATED SOIL BASED ON THE THEORY OF MIXTURE(Ⅰ)

Unsaturated soil is a three-phase media and is composed of soil grain, water and gas. In this paper, the consolidation problem of unsaturated soil is investigated based on the theory of mixture. A theoretical formula of effective stress on anisotropic porous media and unsaturated soil is derived. The principle of effective stress and the principle of Curie symmetry are taken as two fundamental constitutive principles of unsaturated soil. A mathematical model of consolidation of unsaturated soil is proposed, which consists of 25 partial differenfial equations with 25 unknowns. With the help of increament linearizing method, the model is reduced to 5 governing equations with 5 unknowns, i.e., the three displacement components of solid phase, the pore water pressure and the pore gas pressure. 7 material parameters are involved in the model and all of them can he measured using soil tests. It is convenient to use the model to engineering practice. The well known Biot’s theory is a special case of the model.     

Numerical simulation of motion and deformation of ring bubble along body surface

Numerical simulation for fluid flow over an attached rigid body with a deformable ring bubble is analyzed based on the velocity potential theory together with the boundary element method （BEM）. The analysis is focused on the axisymmetric case. The bubble surface is treated as a well defined air-liquid interface and is tracked by a mixed Eulerian-Lagrangian method. The points of intersection between the bubble and body are treated, specially in the numerical procedure. The auxiliary function method is adopted to calculate the pressure on the body surface and in the flow field. The convergence study is undertaken to assess the developed numerical method and the computation code. Some case studies are undertaken in which the interactions between the bubble/body and the incoming flow field are simulated. The effects of various physical parameters on the interactions are investigated.