On a theory of micropolar protoelastic material bodies and constitutive equations for nonlocal micropolar elastic continua

In this paper the definition of micropolar protoclastic material bodies is given and with the help of the principle of virtual power, the variational principle of those bodies is derived. In terms of that same idea and the definition of micropolar protopotential presented here, the constitutive equations for nonlocal micropolar elastic continua are naturally derived.     

RESTUDY OF COUPLED FIELD THEORIES FOR MICROPOLAR CONTINUA (Ⅱ)-THERMOPIEZOELECTRICITY AND MAGNETOTHERMOELASTICITY

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New Principles of Power and Energy Rate of Incremental Rate Type for Generalized Continuum Field Theories

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RESTUDY OF COUPLED FIELD THEORIES FOR MICROPOLAR CONTINUA (Ⅱ)-THERMOPIEZOELECTRICITY AND MAGNETOTHERMOELASTICITY

The theories of thermopiezoelectricity and magnetoelasticity for micropolar continua have been systematically developed by W. Nowacki. In this paper, the theories are restudied. The reason why they were restricted to linear cases is analyzed. The more general conservation principle of energy, energy balance equation and Hamilton principle of thermopiezoelectricity and magnetoelasticity for micropolar continua are established. The corresponding complete equations of motion and boundary conditions as well as balance equations of energy rate for local and nonlocal micropolar thermopiezoelectricity and magnetothermoelasticity are naturally derived. By means of two new functionals and total variation the boundary conditions of displacement, microrotation, electric potential and temperature are also given. Foundation item: the National Natural Science Foundation of China (10072024); the International Cooperation Project of the NSFC (10011130235) and the DFG (51520001); the Research Foundation of the Liaoning Education Committee (990111001) Biography: DAI Tian-min (1931-)     

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 .     

Reprint of: Wave momentum in models of generalized continua

The possibility of associating the notion of quasi-particles with elastic wave modes is explored for three basic models of generalized continua: strain-gradient model (weak nonlocality), elasticity with a microstructure such as in Cosserat/micropolar materials, and a true nonlocal model involving the long-range interactions in the underlying crystal lattice. In each case a simplified one-dimensional model is considered. Approximate solutions involving small parameters and exhibiting scale effects are obtained for the Newtonian-like motion of associated quasi-particles. Interpretation for alternate wave-like and quasi-particle-like behaviors is 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. Contributed by DAI Tian-min, Original Member of Editorial Committee, AMM Foundation items: the National Natural Science Foundation of China (10072024); the Research Foundation of Liaoning Education Committee (990111001) Biography: DAI Tian-min (1931 ~)     

Size effects in two-dimensional Voronoi foams: A comparison between generalized continua and discrete models

In view of size effects in cellular solids, we critically compare the analytical results of generalized continuum theories with the computational results of discrete models. Representatives are studied from two classes of generalized continuum theories: the strain divergence theory from the class of higher-grade continua and the micropolar theory from the class of higher-order continua. In the former, the divergence of strain is proposed as an additional deformation measure, while in the latter the microrotation gradients act as the source for extra internal energy. We analytically solve a range of basic boundary value problems (simple shear, pure bending and the strain concentration around a hole) and compare the results with discrete, numerical calculations that are based on a Voronoi representation of the cellular microstructure. By comparing both the local deformation fields and the overall elastic response, we critically assess the capabilities of both theories in capturing size effects in cellular solids.     

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(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.     

Effective moduli for micropolar composite with interface effect

Size-dependence is well observed for metal matrix composites, however the classical micromechanical model fails to describe this phenomenon. There are two different ways to consider this size-dependency: the first approach is to include the nonlocal effect by idealizing the matrix material as a high order continuum (e.g., micropolar or strain gradient); the second is to take into account the interface effect. In this work, we combine these two approaches together by introducing the interface effect into a micropolar micromechanical model. The interface constitutive relations and the generalized Young–Laplace equation for micropolar material model are firstly presented. Then they are incorporated into the micropolar micromechanical model to predict the effective bulk and shear moduli of a fiber-reinforced composite. Two intrinsic length scales appear: one is related to the microstructure of the matrix material, the other comes from the interface effect. The size-dependent effective moduli due to the nonlocal effect and interface effect can be synchronized or desynchronized for nanosize fibers, depending on the nature of the interface. For the relatively large fiber size, the size-dependence is dominated by the nonlocal effect. As expected, when the fiber size tends to infinity, classical result can be recovered.     

Variational methods for the problems of nonconservative force fields in the micropolar elastodynamics

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

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.     

Micromorphic continuum Part I: Strain and stress tensors and their associated rates

Micropolar and micromorphic solids are continuum mechanics models, which take into account, in some sense, the microstructure of the considered real material. The characteristic property of such continua is that the state functions depend, besides the classical deformation of the macroscopic material body, also upon the deformation of the microcontinuum modeling the microstructure, and its gradient with respect to the space occupied by the material body. While micropolar plasticity theories, including non-linear isotropic and non-linear kinematic hardening, have been formulated, even for non-linear geometry, few works are known yet about the formulation of (finite deformation) micromorphic plasticity. It is the aim of the three papers (Parts I, II, and III) to demonstrate how micromorphic plasticity theories may be formulated in a thermodynamically consistent way.In the present article we start by outlining the framework of the theory. Especially, we confine attention to the theory of Mindlin on continua with microstructure, which is formulated for small deformations. After precising some conceptual aspects concerning the notion of microcontinuum, we work out a finite deformation version of theory, suitable for our aims. It is examined that resulting basic field equations are the same as in the non-linear theory of Eringen, which deals with a different definition of the microcontinuum. Furthermore, geometrical interpretations of strain and curvature tensors are elaborated. This allows to find out associated rates in a natural manner. Dual stress and double stress tensors, as well as associated rates, are then defined on the basis of the stress powers. This way, it is possible to relate strain tensors (respectively, micromorphic curvature tensors) and stress tensors (respectively, double stress tensors), as well as associated rates, independently of the particular constitutive properties.     

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 （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.     

Non-linear micropolar and classical continua for anisotropic discontinuous materials

Non-linear Cosserat and Cauchy anisotropic continua equivalent to masonry-like materials, like brick/block masonry, jointed rocks, granular materials or matrix/particle composites, are presented.An integral procedure of equivalence in terms of mechanical power has been adopted to identify the effective elastic moduli of the two continuous models starting from a Lagrangian system of interacting rigid elements. Non-linear constitutive functions for the interactions in the Lagrangian system are defined in order to take into account both the low capability to carry tension and the friction at the interfaces between elements. The non-linear problem is solved through a finite element procedure based on the iterative adjustment of the continuum constitutive tensor due to the occurrence of some limit situation involving the contact actions of the discrete model.Differences between the classical and the micropolar model are investigated with the aid of numerical analyses carried out on masonry walls made of blocks of different size. The capability of the micropolar continuum to discern, unlike the classical continuum, the behaviour of systems made of elements of different size is pointed out. It is also shown that for anisotropic materials, even in the elastic case, the micropolar solution in general does not tend to the classical solution when the size of the elements vanishes.     

Generalized continuum modeling of 2-D periodic cellular solids

A generalized continuum representation of two-dimensional periodic cellular solids is obtained by treating these materials as micropolar continua. Linear elastic micropolar constants are obtained using an energy approach for square, equilateral triangular, mixed triangle and diamond cell topologies. The constants are obtained by equating two different continuous approximations of the strain energy function. Furthermore, the effects of shear deformation of the cell walls on the micropolar elastic constants are also discussed. A continuum micropolar finite element approach is developed for numerical simulations of the cell structures. The solutions from the continuum representation are compared with the “exact” discrete simulations of these cell structures for a model problem of elastic indentation of a rectangular domain by a point force. The utility of the micropolar continuum representation is illustrated by comparing various cell structures with respect to the stress concentration factor at the root of a circular notch.     

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 (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 .