RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅸ)— THERMOMECHANICS

The existing fundamental laws of thermodynamics for micropolar continuum field theories are restudied and their incompleteness is pointed out. New first and second fundamental laws for thermostatics and thermodynamics for micropolar continua are postulated. From them all equilibrium equations and the entropy inequality of thermostatics as well as all balance equations and the entropy rate inequalities are naturally and simultaneously deduced. The comparisons between the new results presented here and the corresponding results demonstrated in existing monographs and textbooks concerning micropolar continuum mechanics are made at any time. It should be emphasized to note that, the problem of why the local balance equation of energy and the local entropy inequality could not be obtained from the existing fundamental laws of thermodynamics for micropolar continua, is believed to be clarified.     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅴ)--POLAR THERMOMECHANICAL CONTINUA

The purpose is to reestablish rather complete basic balance equations and boundary conditions for polar thermomechanical continua based on the restudy of the traditional theories of micropolar thermoelasticity and thermopiezoelectricity . The equations of motion and the local balance equation of energy rate for micropolar thermoelasticity are derived from the rather complete principle of virtual power. The equations of motion, the balance equation of entropy and all boundary conditions are derived from the rather complete Hamilton principle . The new balance equations of momentum and energy rate which are essentially different from the existing results are presented. The corresponding results of micromorphic thermoelasticity and couple stress elastodynamics may be naturally obtained by the transition and the reduction from the micropolar case , respectively . Finally , the results of micropolar thermopiezoelectricity are directly given .     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅳ)--SURFACE COUSERVATION LAWS

The purpose is to reestablish rather complete surface conservation laws for micropolar thermomechanical continua from the translation and the rotation invariances of the general balance law. The generalized energy-momentum and energy-moment of momentum tensors are presented. The concrete forms of surface conservation laws for micropolar thermomechanical continua are derived . The existing related results are naturally derived as special cases from the results proposed in this paper . The incomplete degrees of the existing surface conservation laws are clearly seen from the process of the deduction. The surface conservation laws for nonlocal micropolar thermomechanical continua may be easily obtained via localization .     

Renewal of basic laws and principles for polar continuum theories (Ⅺ)—consistency problems

Some consistency problems existing in continuum field theories are briefly reviewed.Three arts of consistency problems are clarified based on the renewed basic laws for polar continua.The first art discusses the consistency problems between the basic laws for polar continua.The second art discusses the consistency problems between the basic laws for polar continua and for other nonpolar continua.The third art discusses the consistency problems between the basic laws for micropolar continuum theories and the dynamical equations for rigid body.The results presented here can help us to get a deeper understanding the structure of the basic laws for various continuum theories and the interrelations between them.In the meantime,these results obtained show clearly that the consistency problems could not be solved in the framework of traditional basic laws for continuum field theories.     

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.     

A micropolar mixture theory of multi-component porous media

A mixture theory is developed for multi-component micropolar porous media with a combination of the hybrid mixture theory and the micropolar continuum theory. The system is modeled as multi-component micropolar elastic solids saturated with multi- component micropolar viscous fluids. Balance equations are given through the mixture theory. Constitutive equations are developed based on the second law of thermodynamics and constitutive assumptions. Taking account of compressibility of solid phases, the volume fraction of fluid as an independent state variable is introduced in the free energy function, and the dynamic compatibility condition is obtained to restrict the change of pressure difference on the solid-fluid interface. The constructed constitutive equations are used to close the field equations. The linear field equations are obtained using a linearization procedure, and the micropolar thermo-hydro-mechanical component transport model is established. This model can be applied to practical problems, such as contaminant, drug, and pesticide transport. When the proposed model is supposed to be porous media, and both fluid and solid are single-component, it will almost agree with Eringen＇s model.     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅲ) —NOETHER’S THEOREM

The existing various couple stress theories have been carefully restudied.The purpose is to propose a coupled Noether’s theorem and to reestablish rather complete conservation laws and balance equations for couple stress elastodynamics. The new concrete forms of various conservation laws of couple stress elasticity are derived. The precise nature of these conservation laws which result from the given invariance requirements are established. Various special cases are reduced and the results of micropolar continua may be naturally transited from the results presented in this paper.     

Axisymmetric free vibrations of infinite micropolar thermoelastic plate

The propagation of axisymmetric free vibrations in an infinite homogeneous isotropic micropolar thermoelastic plate without energy dissipation subjected to stress free and rigidly fixed boundary conditions is investigated. The secular equations for homogeneous isotropic micropolar thermoelastic plate without energy dissipation in closed form for symmetric and skew symmetric wave modes of propagation are derived. The different regions of secular equations are obtained. At short wavelength limits, the secular equations for symmetric and skew symmetric modes of wave propagation in a stress free insulated and isothermal plate reduce to Rayleigh surface wave frequency equation. The results for thermoelastic, micropolar elastic and elastic materials are obtained as particular cases from the derived secular equations. The amplitudes of displacement components, microrotation and temperature distribution are also computed during the symmetric and skew symmetric motion of the plate. The dispersion curves for symmetric and skew symmetric modes and amplitudes of displacement components, microrotation and temperature distribution in case of fundamental symmetric and skew symmetric modes are presented graphically. The analytical and numerical results are found to be in close agreement.     

Asymptotic behavior of the Taylor-expansion method of moments for solving a coagulation equation for Brownian particles

The evolution equations of moments for the Brownian coagulation of nanoparticles in both continuum and free molecule regimes are analytically studied. These equations are derived using a Taylor-expansion technique. The self-preserving size distribution is investigated using a newly defined dimensionless parameter, and the asymptotic values for this parameter are theoretically determined. The dimensionless time required for an initial size distribution to achieve self-preservation is also derived in both regimes. Once the size distribution becomes self-preserving, the time evolution of the zeroth and second moments can be theoretically obtained, and it is found that the second moment varies linearly with time in the continuum regime. Equivalent equations, rather than the original ones from which they are derived, can be employed to improve the accuracy of the results and reduce the computational cost for Brownian coagulation in the continuum regime as well as the free molecule regime.     

The traction problem in elastodynamics

In continuum mechanics, Cauchy's six equations are incomplete, and the famous Cauchy's laws of motion where ρ(?), ρb, T and divT are continuous are also incomplete. The first six equations are incomplete because the geometrical representation of deformation at a given point is as yet incomplete, and the last two laws are incomplete because b,T and divT are frame-indifferent, but ρ(?) is not, and T is a symmetric, as Cauchy interpreted himself. Therefore, we say, the last two laws can't accommodate to the asymmetric tensor. The purpose of this paper is to complete Cauchy's laws of motion by postulating an asymmetric tensor for the underlying traction field of 3-dimensional space on a general framing.     

New discrete element models for elastoplastic problems

The discrete element method （DEM） has attractive features for problems with severe damages, but lack of theoretical basis for continua behavior especially for nonlinear behavior has seriously restricted its application. The present study proposes a new approach to developing the DEM as a general and robust technique for modeling the elastoplastic behavior of solid materials. New types of connective links between elements are proposed, the interelement parameters are theoretically determined based on the principle of energy equivalence and a yield criterion and a flow rule for DEM are given for describing nonlinear behavior of materials. Moreover, a numerical scheme, which can be applied to modeling the behavior of a continuum as well as the transformation from a continuum to a discontinuum, is obtained by introducing a fracture criterion and a contact model into the DEM. The elastoplastic stress wave propagations and the tensile failure process of a steel plate are simulated, and the numerical results agree well with those obtained from the finite element method （FEM） and corresponding experiment, and thus the accuracy and efficiency of the DEM scheme are demonstrated.     

Constitutive Theories for Nonlocal Micropolar Continua with Implicity and with Multiple Interactions

In this paper the constitutive theory for nonlocal micropolar continuawhich was proposed by A.C.Eringen is extended to the cases for nonlocalmicropolar continua with implicity and with multiple interactions.Herenonlocal micropolar thermoelastic solids with implicity and with multipleinteractions are cited as instances to illustrate the procedure for the estab-lishment of their constitutive theories as well as two relevant theoremsconcerning the constitutive theories for those solids are given.     

Melting heat transfer effects on stagnation point flow of micropolar fluid saturated in porous medium with internal heat generation （absorption）

The effect of melting heat transfer on the two dimensional boundary layer flow of a micropolar fluid near a stagnation point embedded in a porous medium in the presence of internal heat generation/absorption is investigated. The governing non-linear partial differential equations describing the problem are reduced to a system of non-linear ordinary differential equations using similarity transformations solved numerically using the Chebyshev spectral method. Numerical results for velocity, angular velocity and temperature profiles are shown graphically and discussed for different values of the inverse Darcy number, the heat generation/absorption parameter, and the melting parameter. The effects of the pertinent parameters on the local skin-friction coefficient, the wall couple stress, and the local Nusselt number are tabulated and discussed. The results show that the inverse Darcy number has the effect of enhancing both velocity and temperature and suppressing angular velocity. It is also found that the local skin-friction coefficient decreases, while the local Nusselt number increases as the melting parameter increases.     

Accuracy analysis of predicted velocity profiles of laminar duct flow with entropy generation method

The objective of this work is to estimate the accuracy of a predicted velocity profile which can be gained from experimental results, in comparison with the exact ones by the methodology of entropy generation. The analysis is concerned with the entropy generation rate in hydrodynamic, steady, laminar, and incompressible flow for Newtonian fluids in the insulated channels of arbitrary cross section. The entropy generation can be calculated from two local and overall techniques. Adaptation of the results of these techniques depends on the used velocity profile. Results express that in experimental works, whatever the values of local and overall entropy generation rates are close to each other, the results are more accuracy. In order to extent the subject, different geometries have been investigated. Also, the influence studied, and the distribution of volumetric geometries is drawn. of geometry on the entropy generation rate is local entropy generation rate for the selected geometries is drawn.     

THE APPLICATION OF DISCRETE ELEMENT METHOD IN SOLVING THREE-DIMENTIONAL IMPACT DYNAMICS PROBLEMS

A three-dimensional discrete element model of the connective type is presented. Moreover, a three- dimensional numerical analysis code, which can carry out the transitional process from connective model (for continuum) to contact model (for non-continuum), is developed for simulating the mechanical process from continuum to non-continuum. The wave propagation process in a concrete block (as continuum) made of cement grout under impact loading is numerically simulated with this code. By comparing its numerical results with those by LS-DYNA, the calculation accuracy of the model and algorithm is proved. Furthermore, the failure process of the concrete block under quasi-static loading is demonstrated, showing the basic dynamic transitional process from continuum to non-continuum. The results of calculation can be displayed by animation. The damage modes are similar to the experimental results. The two numerical examples above prove that our model and its code are powerful and efficient in simulating the dynamic failure problems accompanying the transition from continuum to non-continuum. It also shows that the discrete element method (DEM) will have broad prospects for development and application.     

Effect of double dispersion on non-Darcy mixed convective flow over vertical surface embedded in porous medium

A numerical study of a non-Darcy mixed convective heat and mass transfer flow over a vertical surface embedded in a dispersion, melting, and thermal radiation is porous medium under the effects of double investigated. The set of governing boundary layer equations and the boundary conditions is transformed into a set of coupled nonlinear ordinary differential equations with the relevant boundary conditions. The transformed equations are solved numerically by using the Chebyshev pseudospectral method. Comparisons of the present results with the existing results in the literature are made, and good agreement is found. Numerical results for the velocity, temperature, concentration profiles, and local Nusselt and Sherwood numbers are discussed for various values of physical parameters.     

New exact solutions for free vibrations of rectangular thin plates by symplectic dual method

The separation of variables is employed to solve Hamiltonian dual form of eigenvalue problem for transverse free vibrations of thin plates, and formulation of the natural mode in closed form is performed. The closed-form natural mode satisfies the governing equation of the eigenvalue problem of thin plate exactly and is applicable for any types of boundary conditions. With all combinations of simplysupported （S） and clamped （C） boundary conditions applied to the natural mode, the mode shapes are obtained uniquely and two eigenvalue equations are derived with respect to two spatial coordinates, with the aid of which the normal modes and frequencies are solved exactly. It was believed that the exact eigensolutions for cases SSCC, SCCC and CCCC were unable to be obtained, however, they are successfully found in this paper. Comparisons between the present results and the FEM results validate the present exact solutions, which can thus be taken as the benchmark for verifying different approximate approaches.     

Isotropicalized spline integral equation method for the analysis of anisotropic plates

In the view of Reissner’s and Kirchhoff’s theories,respectively,we formulate theisotropicalized governing equations for the anisotropic plates,and give the proof of theequivalence relation between these two plate-models for the simply-supported rectangularorthotropic plates.The well-known fundamental solutions of the isotropic plates are utlizedfor the spline integral equation analysis of anisotropic plates.Even with sparse meshes thesatisfactory results can be obtained.The analysis of plates on two-parameter elasticfoundation is so simple as the common case that only a few terms should be added to theformulas of fictitious loads.     

A new micromechanical approach of micropolar continuum modeling for 2-D periodic cellular material

In this paper, we present a new united approach to formulate the equivalent micropolar constitutive relation of two-dimensional (2-D) periodic cellular material to capture its non-local properties and to explain the size effects in its structural analysis. The new united approach takes both the displacement compatibility and the equilibrium of forces and moments into consideration, where Taylor series expansion of the displacement and rotation fields and the extended aver-aging procedure with an explicit enforcement of equilibrium are adopted in the micromechanical analysis of a unit cell. In numerical examples, the effective micropolar constants obtained in this paper and others derived in the literature are used for the equivalent micropolar continuum simulation of cellular solids. The solutions from the equivalent analysis are compared with the discrete simulation solutions of the cellu-lar solids. It is found that the micropolar constants developed in this paper give satisfying results of equivalent analysis for the periodic cellular material.     

THE EXTREMITY LAWS OF HYDRO-THERMODYNAMICS

This paper presents the law of maximum rate of energy dissipa-tion in hydrodynamics and also in general continuum dynamicsas an addition to the classical conservation laws expressed inthe equation of continuity and the equations of motion.Thecorollary of the law is B(?)langer-B(?)ss theorem of minimum reser-ved specific energy in applied hydraulics.The mechanical energy dissipated is transformed into heatreserved in the substance.The rate of energy dissipation ata time at a given temperature gives rise to the increase in en-tropy production.Hence the maximum rate of energy dissipationsuggests itself the idea of reformulation of the second law ofthermodynamics that the rate of entropy production in mech-anical motion is always the maximum possible.The proposed extremity law in continuum dynamics has beenderived from the variational principle and the reformulatedsecond law of thermodynamics analyzed microscopically in thepaper.The two laws together form the extremity laws of hydro-thermodynamics.