A note on the expression for the rotation tensor in terms of Euler-angles

It is pointed out in this paper that the derivation of the expression for the rotation tensor in terms of Euler angles by N. E, Kochin^[1] is erroneous. To point out Kochin’s deficit on this problem is advantageous for perfecting the concept of the rotation tensor and its method of expression.     

THE IMPROVEMENT OF THE GENERAL EXPRESSION FOR THE STRESS FUNCTION φ OF THE TWO-DIMENSIONAL PROBLEM

In this paper,it is pointed that the general expression for the stress functionφ_0 of theplane problem in polar coordinates is incomplete.The problems of the curved bar with anarbitrary distributive load at the boundries can’t be solved by this stress function.For thisreason,we suggest two new stress functions and put them into the general expression.Then,the problems of the curved bar applied with an arbitrary distributive load at r=a,bboundaries can be solved.This is a new stress function including geometric boundaryconstants.     

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.     

THEOREMS ON THE LIMIT ANALYSIS DEALING WITH THE YIELD CONDITION EXPRESSED BY THE SUM OF THE HOMOGENEOUS LINEAR FORM OF STRESS TENSOR AND THE HOMCGENEOUS QUADRATIC FORM OF STRESS TENSOR

In this paper a generelized variational principle on the limitanalysis dealing With the yield condition expressed by thesum of the homogeneous linear form of stress tensor and thehomogeneous quadratic form of stress tensor is suggested.Thisvariational principle can be applied to the limit analgsis inrock mechanics and it takes the situation.in which the yieldcondition is expressed by the homogeneous linear form of stresstensor or the homogeneous quadrdtic form of stress tensor.asits special case.     

THE EXPRESSION OF FREE ENERGY FOR THERMOELASTIC MATERIAL AND ITS RELATION TO THE VARIABLE MATERIAL COEFFICIENTS

The expression of free energy is expanded in a power series, in which there aren't any terms of order higher than third in the temperature increments θ~* and second in the strains γ_(ij) in this paper. The regular patterns of the material coefficients changing with temperature increments can be derived from this expression, These regulations accord with the experimental graph in references but the constants in the expression of free energy must be determined by experimental data. It is pointed out that the variable modulus of elasticity E and shearing modulus of elasticity G are independent of each other, but the rest of the coefficients are related to them.     

A direct proof of uniqueness of square-root of a positive semi-definite tensor

Understanding of the basic properties of the positive semi-definite tensor is a prerequisite for its extensive applications in theoretical and practical fields, especially for its square-root. Uniqueness of the square-root of a positive semi-definite tensor is proven in this paper without resorting to the notion of eigenvalues, eigenvectors and the spectral decomposition of the second-order symmetric tensor.     

DISCUSSION ON THE SIF FOR POINTS ON BORDER OF ELLIPTICAL FLAT CRACK INSIDE INFINITE SOLID UNDER UNIFORM TENSION

Using the results of crack surface displacement field in Green-Sneddon’s solution and coordinate transformation,this paper hasderived an expression K_1(x_1,z_1,α)for SIF at any point and at anyorientation on the border of elliptical flat crack inside infinitesolid under uniform tension.As a complement of Irwin’s work,it is shown that for any pointed point on the elliptical border theSIF defined on normal plane takes the maximum value.And it shouldbe pointed out that in some works some idea concerning Irwin’s con-tents is open to question.An expression K_I in terms of polarangle which is more intuitional than centrifugal angle is proposedfor SIF at any point on the elliptical border.     

THE ORDER OF A DAMAGE TENSOR

In this paper,the stress deformation constitutive relations for continua are discussedand a stress deformation constitutive relation expressed by functional tensorial expressionis found When we study the anisotropic damage of anisotropic materials either from amacroscopic continuum mechanics model or from a micro-defect model,there exists a limitto the order of a damage tensor,and the condition under which the damage variable,may bedescribed by a tensor lower than those of the highest order is found.     

THE CONVERGENT CONDITION AND UNITED FORMULA OF STEP REDUCTION METHOD

The step reduction method was first suggested by Prof. Yeh Kai-yuan. This method has more advantages than other numerical methods. By this method, the analytic expression of solution can he obtained for solving nonuniform elastic mechanics. At the same time. its ealculuting time is very short and convergent speed very fast. In this paper, the convergent condition and nited formula of step reduction method are given by mathemutical method. It is proved that the solution of displacement and stress resultants obtained by this method can eonverge to exact solution uniformly, when the convergent condition is sutisfied. By united formula, the analytic solution solution can be expressed as matrix form, and therefore the former complicated expression can be avoMed. Two numerical examples are given at the end of this paper which indicate that. by the theory in this paper, a right model can be obtained for step reduction method.     

THE DISTRIBUTION OF THE SPECIFIC PRESSURE IN ROLLING STRIPS

The principle of the distribution of the specific pressure in rol-ling strips is used not only for calculating the total rolling pres-sure but also for providing the basis in calculating the widening andin designing the rational roll profile.Hitherto,the results of an-alysis on this subject in the references are expressed as a functionof one dimension,and they cannot reflect the variation of the speci-fic pressure along the width of the contact surface.This paper dealswith the two-dimensional expression of the principle.of the distribu-tion of the specific pressure with the help of the calculus of varia-tions.     

ON THE TORSION OF A CYLINDER WITH SEVERAL CRACKS

In this paper, based on paper, the analytic expression of the torsion function for a cylinder containing arbitrary oriented cracks is obtained. The probtem is reduced to solve a system of singular integral equations for the unknown dislocation density functions. Using the numerical method of the singular integral equations, the torsional rigidities and stress intensity factors are evaluated for several multicracked cylinders. Next, the cr(?)kcutting method is firstly extended to lve the torsion problem for a rectangular prism. The numerical results show that the method presented here is successful.     

THEORETICAL MODEL OF EFFECTIVE STRESS COEFFICIENT FOR ROCK/SOIL-LIKE POROUS MATERIALS

Physical mechanisms and influencing factors on the effective stress coefficient for rock/soil-like porous materials are investigated, based on which equivalent connectivity index is proposed. The equivalent connectivity index, relying on the meso-scale structure of porous material and the property of liquid, denotes the connectivity of pores in Representative Element Area （REA）. If the conductivity of the porous material is anisotropic, the equivalent connectivity index is a second order tensor. Based on the basic theories of continuous mechanics and tensor analysis, relationship between area porosity and volumetric porosity of porous materials is deduced. Then a generalized expression, describing the relation between effective stress coefficient tensor and equivalent connectivity tensor of pores, is proposed, and the expression can be applied to isotropic media and also to anisotropic materials. Furthermore, evolution of porosity and equivalent connectivity index of the pore are studied in the strain space, and the method to determine the corresponding functions in expressions above is proposed using genetic algorithm and genetic programming. Two applications show that the results obtained by the method in this paper perfectly agree with the test data. This paper provides an important theoretical support to the coupled hydro-mechanical research.     

Some discussions on finite deformation of continuous media

Three problems are discussed in this paper:1.The physical significance of the characteristic tensorθ~ii of finitedeformation is discussed as a complement to the literature.2.The four characterietic tensors of finite deformation,introduced inthe.literature,ard analyzed and discussed more thoroughly.3.The representation of the general finite deformation through simple.loading process is not always possible,the condition for its realization beingthat the given finite deformation satisfies the equations of compatibility.It is pointed out in this paper that for the finite deformation set forth byL.I.Sedov in an illustrative example in[9],the equations of compatibilitycannot be satisfied even for k=1,henceforth for whatever value of k,thefinite deformation set forth by Sedov can not be represented through simpleloading process.     

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.     

Suggestion of a new definition of angular strain

A new definition of angular strain is put forward in this paper. Analogous to the definition of linear strain, angular strain is also defined as a rate of change (of angular displacement). It is verified that the new definition is equivalent to the original besides being more convenient for geometrical interpretations. Using this new definition we have deduced the transformation formulas for rotation of coordinate axes, established the strain tensor and proved the Hooke’s law for shearing.     

SUGGESTION OF A NEW DEFINITION OF ANGULAR STRAIN

A new definition of angular strain is put forward in this paper. Analogous to the definition of linear strain, angular strain is also defined as a rate of change (of angular displacement). It is verified that the new definition is equivalent to the original besides being more convenient for geometrical interpretations. Using this new definition we have deduced the transformation formulas for rotation of coordinate axes, established the strain tensor and proved the Hooke's law for shearing.     

THEORY OF ELASTICITY OF AN ANISOTROPIC BODY FOR THE BENDING OF BEAMS

A theory of elasticity for the bending of orthogonal anisotropic beams has been developed by analogy with the special case, which can be obtained by applying the theory of elasticity for bending of transversely isotropic plates to the problems of two deminsions. In this paper, we present a method to solve the problems of bending of orthogonal anisotropic beams and a new theory of the deep-beam whose ratio of depth to length is larger. It is pointed out that Reissner's theory to account for the effect of transverse shear deformation is not very approximate in the components of stress,     

ON THE GENERAL EQUATION AND THE GENERAL SOLUTION IN PROBLEMS FOR PLASTODYNAMICS WITH RIGID-PLASTIC MATERIAL

This work is the continuation of the discussion of refs.[1-2].We discuss thedynamics problems of ideal rigid—plastic material in the flow theory of plasticity in thispaper.From introduction of the theory of functions of complex variable under Dirac-Paulirepresentation we can obtain a group of the so-called“general equations”(i.e.have twoscalar equations)expressed by the stream function and the theoretical ratio.In this paperwe also testify that the equation of evolution for time in plastodynamics problema is neitherdissipative nor disperive,and the eigen-equation in plastodynamics problems is a stationarySchr(?)dinger equation,in which we take partial tensor of stress-increment as eigenfunctionsand take theoretical ratio as eigenvalues.Thus,we turn nonlinear plastodynamics problemsinto the solution of linear stationary Schr(?)dinger equation,and from this we can obtain thegeneral solution of plastodynamics problems with rigid-plastic material.     

AN ANALYSIS ON ENTRANCE REGION EFFECT OF THE LAMINAR RADIAL FLOW BETWEEN TWO PARALLEL DISKS

In this paper, B. B. Golubef method is used for calculating the radial diffuse flow between two parallel disks for the first step. The momentum integral equation together with the energy integral equation is derived from the boundary layer momentum equation and the expression of secondary approximation explicit function in which the channel length of entrance region varies with the boundary layer thickness can be obtained by using Picard iteration in the solution of the energy integral equation. Therefore, this has made it possible to analyze directly and analytically the coefficients of the entrance region effect. In particular, when the outer diameter of disk is smaller than the entrance region length, the advantage of this method can be prominently manifest. Only because the energy integral equation is employed, the terms in the pressure loss coefficient can be independently derived theoretically. The computable value of the pressure loss coefficient presented in this paper is nearer to the testing v     

EFFECTIVE SPECIFIC HEATS OF MULTI-PHASE THERMOELASTIC COMPOSITES

This paper studies the effective properties of multi-phase thermoelastic composites. Based on the Helmholtz free energy and the Gibbs free energy of individual phases, the effective elastic tensor, thermal-expansion tensor, and specific heats of the multi-phase composites are derived by means of the volume average of free-energies of these phases. Particular emphasis is placed on the derivation of new analytical expressions of effective specific heats at constant-strain and constant-stress situations, in which a modified Eshelby’s micromechanics theory is developed and the interaction between inclusions is considered. As an illustrative example, the analytical expression of the effective specific heat for a three-phase thermoelastic composite is presented.