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.     

CONSTITUTIVE RELATION OF UNSATURATED SOIL BY USE OF THE MIXTURE THEORY(Ⅰ)—NONLINEAR CONSTITUTIVE EQUATIONS AND FIELD EQUATIONS

IntroductionSoilisthemostcommonlyusedconstructionmaterialincivilengineeringandhydraulicengineering .Thecharacteristicsofsoilhavebeeninvestigatingfornearlyonehundredyears.Butbecauseofitscomplexstructure,changeableenvironmentandbeingsensitivetotheoutsideconditions,thesoiloftenshowsvariedproperties[1,2 ].Themaindifficultytothedevelopmentofgeotechnicalmechanicsishowtosetupconstitutiveequationswhichcouldsatisfactorilyaccountforengineeringpropertiesofsoil[3].Manyconstitutivemodelshavebeenformedinth…     

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

The nonlinear constitutive equations and field equations of unsaturated soils were constructed on the basis of mixture theory. The soils were treated as the mixture composed of three constituents. First, from the researches of soil mechanics, some basic assumptions about the unsaturated soil mixture were made, and the entropy inequality of unsaturated soil mixture was derived. Then, with the common method usually used to deal with the constitutive problems in mixture theory, the nonlinear constitutive equations were obtained. Finally, putting the constitutive equations of constituents into the balance equations of momentum, the nonlinear field equations of constituents were set up. The balance equation of energy of unsaturated soil was also given, and thus the complete equations for solving the thermodynamic process of unsaturated soil was formed. Foundation items: the National Natural Science Foundation of China (59678003); Special Research Plan of the Education Department of Shaanxi Province (01JK178) Biographies: HUANG Yi (1936-) ZHANG Yin-ke (1964-)     

ENERGY THEOREMS FOR SOLVING EQUATIONS OF DEFLECTIONS

In this paper the unit-dummy-load method is ge-neralized on the basis of Castigliano’s Theorem.Onthese grounds the general equations of deflection sur-faces of the structures.such as a kind of beams,platesand shells.are directly derived by the force method.We derived the equations of the deflection sur-faces of the rectangular thin plates and thick platesconsidering the effect of transverse shearing defor-mations with the inhomogeneous displacement boundaryconditions.At the same time we give the equations ofdeflection axes of the corresponding straight beams.The applications of the reciprocal theorem arealso generalized.Three simple calculated examples are given.     

GIBBS-APPELL’S EQUATIONS OF VARIABLE MASS NONLINEAR NONHOLONOMIC MECHANICAL SYSTEMS

In this paper,the Gibbs-Appell’s equations of motion are extended to the most generalvariable mass nonholonomic mechanical systems.Then the Gibbs-Appell’s equations ofmotion in terms of generalized coordinates or quasi-coordinates and an integral variationalprinciple of variable mass nonlinear nonholonomic mechanical systems are obtained.Finally,an example is given.     

KANE’S EQUATIONS FOR PERCUSSION MOTION OF VARIABLE MASS NONHOLONOMIC MECHANICAL SYSTEMS

In this paper,the Kane’s equations for the Routh’s form of variable massnonholonomic systems are established.and the Kane’s equations for percussion motionof variable mass holonomic and nonholonomic systems are deduced from them. Secondly,the equivalence to Lagrange’s equations for percussion motion and Kane’sequations is obtained,and the application of the new equation is illustrated by anexample.     

AN APPLICATION OF SCHAUDER’S FIXED POINT THEOREM TO THEEXISTENCE OF SOLUTlONS OF IMPULSIVELYDIFFERENTIAL EQUATIONS

In this paper the existence of solutions of a boundary value problem forimpulsively differential equations that is difficult to solve by the upper and lowersolution method will be proved by means of Schauder’s fixed point theorem,whichimproves some existing results.     

DISCUSSION ON BARENBLATT’S EQUATIONS OF MEAN MOVEMENT OF THE INHOMOGENEOUS FLUID

In this paper,it is proved that Barenblatt’s equation of mean movement of the inhomogeneous fluid is not cor-rect,because his equations are based on the assumption that the characteristics of inhomogeneity of fluid are only af-fected with respect to gravitational terms.The correct form of the equation of mean movement of the inhomogeneous fluid is derived in this paper.     

STRAIN PATH THEORY OF PLASTIC CONSTITUTIVE RELATION

Based on Illiushin s two hypotheses,a new plastic constitutive equation of integral typeis proposed.The deviatoric stress induced by a strain path consists of two parts which varyaccording to different laws.Comparison of theoretical calculation with recent experimentsis satisfactory.The present theory is not an endochronic theory.     

ROUTH’S EQUATIONS FOR GENERAL NONHOLONOMIC MECHANICAL SYSTEMS OF VARIABLE MASS

In this paper,Routh’s equations for the mechanical systems of the variable masswith nonlinear nonholonomic constraints of arbitrary orders in a noninertial referencesystem have been deduced not from any variational principles,but from the dynamicalequations of Newtonian mechanics.And then again the other forms of equations fornonholonomic systems of variable mass are obtained from Routh’s equations.     

ON PERIODIC SOLUTIONS OF SEVERAL CLASSES OF RICCATI’S EQUATION AND SECOND-ORDER DIFFERENTIAL EQUATIONS

In this paper,the problem on periodic solutions of severalclasses of Riccati’s equation with periodic coefficients is dis-cussed,and the conditions,under which several classes of se-condorder equations with periodic coefficients have periodic so-lutions,are given.     

THE CONTINUAL DIFFERENTIABLE PEAK-UNIMODAL SOLUTIONS OF FEIGENBAUM’S FUNCTIONAL EQUATIONS

For the famous Feigenbaum’s equations,in this paper,we established its constructivetheorem of the peak-unimodal,then we found out other paths to explore the peak-unimodalsolutions.For example,we proceed on the direction to try the non-symmetrical continuouspeak-unimodal solutions and C~1 solutions.     

EXTRACTION OF CHARACTERISTIC ROOTS ON BESSEL-NEUMANN’S MIXED EQUATIONS

This paper discusses the problem of the extraction of characteristic roots{λ_(?)}(λ_0<λ_1<λ_2…)on Bessel-Neumann’s mixed equations.It gives the expressions and the evaluationof the minimum root.The advantage of the method has no use for the table of the multi-figure number Bessel function and it does not need computer but can calculate all thecharacteristic roots{λ_(?)}.The precision of these roots is still high.     

SOME SOLUTIONS OF MAXWELL EQUATIONS

This article gives a class of solution of Maxwellequations which implies the solution of arbitrary ho-lomorphic hypercomplex functional with brief discus-sion of their physical meaning.It exhibites somewider nature than classical electromagnetic theory:(l)An electromagnetic field may spread non-periodically and may have the holographic nature andfree shape.(2)The electromagnetic field may spread more slow-ly,equally or more quickly than light speed.(3)The electromagnetic field may spread in arbi-trary curved direction.(4)There is a branch electromagnetic field in in-tergrowth space.An intergrowth world may exist.This article also makes an inference and a descrip-tion about the example of the above-mentioned strangeelectromagnetic field.It relates to some hard-to-explain phenomena through different physical models.     

A CONSERVATIVE INTEGRAL INDEPENDENT OF THE CONSTITUTIVE RELATION

J-integral defined by Rice depends on the con-stitutive relation of mediums.In the present paper,a conservative integral independent of the constitu-tive relation of mediums is defined in order to ex-tend Rice＇s definition of the J-integral which de-pends on the constitutive relation.Its main pro-perties which are similar to those of the prevailingJ-integral are proved.In the case of the elasticmediums,the extensive conservative integral is thesame with Rice＇s J-integral.     

ASYMPTOTIC SOLUTION OF SINGULAR PERTURBATION PROBLEMS FOR THE FOURTH-ORDER ELLIPTIC DIFFERENTIAL EQUATIONS

In this paper we consider the singularly perturbed boundary value problem for thefourth-order elliptic differential equation,establish the energy estimates of the solutionandits derivatives and construct the formal asymptotic solution by Lyuternik-Vishik’s method.Finally,by means of the energy estimates we obtain the bound of the remainder of theasymptotic solution.     

DISCUSSION ON BARENBLATT’S EQUATIONS OF MEAM MOVEMENT THE INHOMOGENEOUS FLUID

In this paper, it is proved that Barenblatt’s equation of mean movement of the inhomogeneous fluid is not correct, because his equations are based on the assumption that the characteristics of inhomogeneity of fluid are only affected with respect to gravitational terms. The correct form of the equation of mean movement of the inhomogeneous fluid is derived in this paper.     

THE CONTINUAL DIFFER ENTI ABLE PEAK-UNIMODAL SOLUTIONS OF FEIGENBAUM’S FUNCTIONAL EQUATIONS

For the famous Feigenbaum’s equations, in this paper, we established its constructive theorem of the peak-unimodal, then we found out other paths to explore the peak-unimodal solutions. For example, we proceed on the direction to try the non-symmetrical continuous peak-unimodal solutions and C1 solutions.     

A NOTE ABOUT ODEN’S CONSTITUTIVE VARIATIONAL PRINCIPLE

A remark about some unreasoning things of Oden’s constitutive variational principledescribed in Ref.[1]is given in this paper.According to Oden’s idea,the other twoconstitutive variational principles,which are complementary to one another,are developed.An example is performed to demonstrate the application of the constitutive variationalprinciples.     

CRITERIA FOR THE EXISTENCE OF SOLUTIONS TO RANDOM INTEGRAL AND DIFFERENTIAL EQUATIONS

In[1],we proved a general random fixed point theorem and gave some applications.Inthis paper,we shall give further applications of the theorem.We first obtain a randomDarbo’s fixed point theorem,using the theorem,we give the criteria for the existence ofsolutions under compactness hypotheses to nonlinear random volterra integral equationsand the Cauchy problem of nonlinear random differential equations.Our theorems improveand generalize some main results of Lakshmikanthem,Vaughn as well as De Blasiand Myjak.