ON A GENERALIZATION OF BERTRAND’S THEOREM

Bertrand’s theorem for the determination of the applied forces to a holonomicsystem from one of its first integrals,is extended to nonholonomic systems.Someinteresting applications of this new result are also given.     

THE CONSERVATION LAW OF NONHOLONOMIC SYSTEM OF SECOND-ORDER NON-CHETAVE''''S TYPE IN EVENT SPACE

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A COMMENT ON THE PROOF OF FERMAT'S LAST THEOREM

I.IntroductionItiswell-knownthatFermat'sLastTheoremll'2'31:"TheequationX" yn=Z"(l.1)hasnosolutioninpositiveintegersx,.1',:whicharealldifferentfrom0foranarbitrarynumbern>2"isahardmathematicalproblem,proposedbyFrenchmathematicianPierredeFermatinabout1637.Sincethenmorethan300yearspast,manyfamousmathematicianshavemadegreatcontributionsintheresearchesanddevelopmentsinthenumbertheoryofalgebraaswellasothermathematicalbranchestolookouttheproofoftheproblematthemoment,buttilefinalproofhasonlybeenobt…     

THE CONSERVATION LAW OF NONHOLONOMIC SYSTEM OF SECOND-ORDER NON-CHETAVE’S TYPE IN EVENT SPACE

The conservation law of nonholonomic system of second-order non-Chataev’s type in event space is studied. The Jourdain’s principle in event space is presented. The invariant condition of the Jourdain’s principle under infinitesimal transformation is given by introducing Jourdain’s generators in event space. Then the conservation law of the system in event space is obtained under certain conditions. Finally a calculating example is given.     

ON THE FREE ENERGY AND VARIATIONAL THEOREN OF ELASTIC PROBLEM IN THERMAL SHOCK

In some investigations on variational principle for cou-pled thermoelastic problems,the free energy φ(e_(ij),θ) ,Wherethe state variables are elastic strain e_(ij) and temperatureincrement θ,is expressed asφ(e_(ij),θ)=λ/2e_(kk)e(ij) μe_(ki)e_(kj)-γe_(kk)θ-c/2ρθ~2/T_o (0.1)This expression is employed only under the condition of|θ|≤T_o (absolute temperature of reference)But the value of temperature increment is great, evengreater than T_o in thermal shock. And the material pro-perties (λ,μ,γ,c,etc.) will not remain constant,theyvary with θ.The expression of free energy for this con-dition is derived in this paper. Equation (0.1) is itsspecial case.Euler’s equations Will be nonlinear While this expres-sion of free energy has been introduced into variationaltheorem. In order to linearise, the time interval of ther-mal shock is divided into a number of time elements △t_k(△t_k=t_k-t_(k-1),k=1,2,…,n),Which are so small that the tempera-ture increment θ_k within it is very small,too.Thus,t     

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.     

THE SIZE EFFECT ON THE DISPERSION OF A PARTICLE IN A HOMOGENEOUS ISOTROPIC TURBULENCE

The mechanism of the response motion of a suspended particle to turbulent motion ofits,surrounding fluid is different according to size of turbulent eddies.The particle isdragged by the viscous force of large eddies.and meanwhile driven randomly by smalleddies.Based on this understanding,the dispersion of a particle with finite size in ahomogeneous isotropic turbulence is calculated in this study.Results show that there aretwo competing effects:when enhanced by the inertia of a particle,the long-term particlediffusivity is reduced by the finite size of the particle.     

THE EFFECT OF THE VASOMOTION ON THE MASS TRANSFER IN MICROCIRCULATION

This paper deals with the mass transfer in the microcirculation with vasomotion. The effect of the vasomotion on the mass transfer is analysed. It is shown that the vasomotion is of great importance to accelerate the mass transfer.     

A CATASTROPHE ANALYSIS ON THE STABILITY OF THE CRACK GROWTH IN THREE-POINT BENDING SPECIMENS

This paper presents an attempt at the application of catastrophe theory to the stabilityanalysis of J-controlled crack growth in three-point bending specimens. By introducing the solutions ofJ-integral in the completely yielding state for the ideal plastic material, the critical condition of losingstability for the crack propagation in the specimen is obtained, based on the cusp catastrophe theory.The process of the crack growth from geometrical sense is described.     

THE FINITE ROTATION PARAMETER OF A DEFORMABLE BODY AND GYROMAGNETIC EFFECT

The derivation of synge[2]for the finite rotation formula interms of Euler parameter is put into tensor form.Hence a fur-ther clarification of the geometric meaning of the orthogonaltransformation obtained in the author’s paper[7]is made.Thetensor properties of rotation axis vector are also discussed.By means of the method of co-moving coordinate system establi-shed in[8],[9],we explain the gyromagnetic effect.and derivea simple formula for calculating bodg couple which is inducedby magnetization.     

A DISCUSSION ABOUT THE EFFECTIVENESS ON USING A VARIATE-TRANSFORMATION TO FIND OUT SOLUTIONS OF MODE Ⅲ CRACK PROBLEMS IN POWER HARDENING MEDIA

By the use of the transformations of physical plane to strain plane and physical planeto stress plane,an analytic expression of the asymptotic solution near a modeⅢcrack tipin a power hardening medium can be obtained.In this paper the effectiveness of thetransformation is discussed.Analytical results show that the transformation is effectiveexcept for a special limit case of power hardening media—the ideal plastic materials.     

UNIFORM CONVERGENCE FOR THE DIFFERENCE SCHEME IN THE CONSERVATION FORM OF ORDINARY DIFFERENTIAL EQUATION WITH A SMALL PARAMETER

In this paper,we consider a singular perturbation boundary problem for a self-adjointordinary differential equaiton.We construct a class of difference schemes with fittedfactors,and give the sufficient conditions under which the solution of difference schemeconverges uniformly to the solution of differential equation.From this we propose severalspecific schemes under weaker conditions,and give much higher order of uniformconvergence.     

VARIATIONAL PRINCIPLE OF HYBRID ENERGY AND THE FUNDAMENTALS OF 3-D LAMINATE THEORY—A NEW APPROACH FOR THE ANALYSIS OF INTERLAMINAR STRESSES IN COMPOSITE LAMINATES

This paper discusses the discuntinuity of stresses and strains at interlaminar surfaces of the eomposne laminate and presents a 3-D laminate theory for composite materials. This paper also presents a new type of clastic energy based on the globally continuous variables in laminates, different from the traditional potemial energy and complementary energy. Then. a variational principle corresponding to the 3-D laminate theory is developed. The theory and the principle could be a basis of verifying the 2-D lamimte theory and deterplining the interlaminar stresses mear the free edges.     

THE BOUNDARY INTEGRAL-VARIATIONAL THEOREMS OF AN ARBITRARY ELEMENT IN THE SOLID—COMPUTE THE ENERGY RELEASE RATE OF AN ARBITRARY CRACK EXTENSION

Based on the solid mechanics of the discrete form and its variational principles proposed by Niu, this paper puts forward four kinds of boundary integral-variational theorems of an arbitrarg element.In the course of the fracture analysis, they can be used to compute the energy release rate along the normal direction of the crack boundary. When there is a hole in the solid, and whether there are given surface forces on the hole boundary or not they can be used to compute the variation of the energy along the normal direction of the hole boundary. In the course of the discrete analysis, they can be used to establish the discrete equations, so that the values of the unknown functions are solved. At the same time, from this paper we know that the J-integral proposed by Rice represents an integral to be independent of a path imperfectly.     

THE BOUNDARY INTEGRAL-VARIATIONAL THEOREMS OF AN ARBITRARY ELEMENT IN THE SOLID—COMPUTE THE ENERGY RELEASE RATE OF AN ARBITRARY CRACK EXTENSION

Based on the solid mechanics of the discrete form and its variational principles proposed by Niu^[1–2], this paper puts forward four kinds of boundary integral-variational theorems of an arbitrary element. In the course of the fracture analysis, they can be used to compute the energy release rate along the normal direction of the crack boundary. When there is a hole in the solid, and whether there are given surface forces on the hole boundary or not they can be used to compute the variation of the energy along the normal direction of the hole boundary. In the course of the discrete analysis, they can be used to establish the discrete equations, so that the values of the unknown functions are solved. At the same time, from this paper we know that the J-integral proposed by Rice^[3] represents an integral to be independent of a path imperfectly.     

EFFECTS OF POISSON’S RATIO ON SCALING LAW IN HERTZIAN FRACTURE

In this paper the Auerbach＇s scaling law of Hertzian fracture induced by a spherical indenter pressing on a brittle solid is studied. In the analysis, the singular integral equation method is used to analyze the fracture behavior of the Hertzian contact problem. The results show that the Auerbach＇s constant sensitively depends on the Poisson＇s ratio, and the effective Auerbach＇s domain is also determined for a given value of the Poisson＇s ratio.     

ON THE UNIFICATION OF THE HAMILTON PRINCIPLES IN NONHOLONOMIC SYSTEM AND IN HOLONOMIC SYSTEM

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A THEOREM ON POTENTIAL OF ISOTROPIC SYMMETRIC SECOND-ORDER TENSOR FUNCTION

In this paper we propose and prove the following theorem:If the second-order tensor H is an isotropic function of a symme-tric second-order tensor T,and there exists a potential function for H,then there will certainly exist a potential function for T,too.     

ON THE POSITIVE DEFINITENESS OF A CLASS OF OPERATORS

In this paper,a proof of the positive definiteness for a class of operators is given.Theoperators considered are general enough to include those in two-and three-dimensionalelasticity,thin plates and shells as their special cases.     

ENERGY ELEMENT AND ITS APPLICATION IN THE DYNAMIC CALCULATION OF CONTINUOUS MEDIUM

The present paper which discusses the problem re-lated to the combination of the finite difference me-thod and the finite element method,describes an en-ergy element calculating mode including the specialfeatures of the above two methods.