RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CON-TINUUM THEORIES (Ⅱ)—MICROMORPHIC CONTINUUM THEORY AND COUPLE STRESS THEORY

The purpose is to reestablish the balance laws of momentum, angular momentum and energy and to derive the corresponding local and nonlocal balance equations for micromorphic continuum mechanics and couple stress theory. The desired results for micromorphic continuum mechanics and couple stress theory are naturally obtained via direct transitions and reductions from the coupled conservation law of energy for micropolar continuum theory, respectively. The basic balance laws and equations for micromorphic continuum mechanics and couple stress theory are constituted by combining these results derived here and the traditional conservation laws and equations of mass and microinertia and the entropy inequality. The incomplete degrees of the former related continuum theories are clarified. Finally, some special cases are conveniently derived. Foundation items: the National Natural Science Foundation of China (10072024); the Research Foundation of Liaoning Education Committee (990111001) Biography: DAI Tian-min (1931≈)     

Analysis of micropolar elastic beams

In this paper, a linear theory for the analysis of beams based on the micropolar continuum mechanics is developed. Power series expansions for the axial displacement and micro-rotation fields are assumed. The governing equations are derived by integrating the momentum and moment of momentum equations in the micropolar continuum theory. Body couples and couple stresses can be supported in this theory. After some simplifications, this theory can be reduced to the well-known Timoshenko and Euler–Bernoulli beam theories. The nature of flexural and longitudinal waves in the infinite length micropolar beam has been investigated. This theory predicts the existence of micro-rotational waves which are not present in any of the known beam theories based on the classical continuum mechanics. Also, the deformation of a cantilever beam with transverse concentrated tip loading has been studied. The pattern of deflection of the beam is similar to the classical beam theories, but couple stress and micro-rotation show an oscillatory behavior along the beam for various loadings.     

EXTENSION OF POINCARE’S NONLINEAR OSCILLATION THEORY TO CONTINUUM MECHANICS (I)-BASIC THEORY AND METHOD

In this paper we extend Poincare’s nonlinear oscillation theory of discrete system to continuum mechanics. First we investigate the existence conditions of periodic solution for linear continuum system in the states of resonance and non-resonance. By applying the results of linear theory, we prove that the main conclusion of Poincare’s nonlinear oscillation theory can be extended to continuum mechanics. Besides, in this paper a new method is suggested to calculate the periodic solution in the states of both resonance and nonresonance by means of the direct perturbation of partial differential equation and weighted integration.     

Second law violations,continuum mechanics,and permeability

The violations of the second law are relevant as the length and/or time scales become very small. The second law then needs to be replaced by the fluctuation theorem and mathematically, the irreversible entropy is a submartingale. First, we discuss the consequences of these results for the axioms of continuum mechanics, arguing in favor of a framework relying on stochastic functionals of energy and entropy. We next determine a Lyapunov function for diffusion-type problems governed by stochastic rather than deterministic functionals of internal energy and entropy, where the random field coefficients of diffusion are not required to satisfy the positive definiteness everywhere. Next, a formulation of micropolar fluid mechanics is developed, accounting for the lack of symmetry of stress tensor on molecular scales. This framework is then applied to employed to show that spontaneous random fluctuations of the microrotation field will arise in Couette—and Poiseuille-type flows in the absence of random (turbulence-like) fluctuations of the classical velocity field. Finally, while the permeability is classically modeled by the Darcy law or its modifications, besides considering the violations of the second law, one also needs to account for the spatial randomness of the channel network, implying a modification of the hierarchy of scale-dependent bounds on the macroscopic property of the network.     

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.     

Extensions of nondissipative continuum mechanics toward complex fluids and complex solids

Nondissipative classical continuum mechanics, in both Eulerian and Lagrangian formulations, is extended to complex fluids and complex solids. The extensions involve extra fields characterizing microstructure, preserve the Hamiltonian structure of the classical theory, and are rigorously derived from Hamiltonian formulations of the particle mechanics and statistical mechanics.     

A screw dislocation by nonlinear continuum mechanics

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On universal relations in continuum mechanics

This paper is devoted to a systematic study of local universal relations in continuum mechanics. We show that it is possible to determine the complete set of independent universal relations whose characterization is obtained by linear universal rules. A historical review of the literature on the topic and various significant examples are given. Received: August 16, 1996     

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 continuum micromechanical theory of overall plasticity for particulate composites including particle size effect

Classical continuum micromechanics cannot predict the particle size dependence of the overall plasticity for composite materials, a simple analytical micromechanical method is proposed in this paper to investigate this size dependence. The matrix material is idealized as a micropolar continuum, an average equivalent inclusion method is advanced and the Mori–Tanaka's method is extended to a micropolar medium to evaluate the effective elastic modulus tensor. The overall plasticity of composites is predicted by a new secant moduli method based on the second order moment of strain and torsion of the matrix in a framework of micropolar theory. The computed results show that the size dependence is more pronounced when the particle's size approaches to the matrix characteristic length, and for large particle sizes, the prediction coincides with that predicted by classical micromechanical models. The method is analytical in nature, and it can capture the particle size dependence on the overall plastic behavior for particulate composites, and the prediction agrees well with the experimental results presented in literature. The proposed model can be considered as a natural extension of the widely used secant moduli method from a heterogeneous Cauchy medium to a micropolar composite.     

Kinematic variables bridging discrete and continuum granular mechanics

It is known that there is wide, and at present, unbridgeable, gap between discrete and continuum granular mechanics. In this contribution, first, microscopic kinematic variables neglected in classical continuum granular mechanics are investigated based on the kinematics of discs in contact. Then, a kinematic variable called the averaged pure rotation rate (APR) is proposed for an assembly of circular discs of different sizes, which is then used to produce another two kinematic tensors with one equal to the deformation rate tensor and the other unifying the spin tensor and the APR. As an example, the kinematic variables are incorporated into the unified double-slip plasticity model. Finally, these theoretical analyses are verified using a two-dimensional discrete element method. The study shows that these kinematic variables can be used to bridge discrete and continuum granular mechanics.     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CON-TINUUM THEORIES (Ⅱ)—MICROMORPHIC CONTINUUM THEORY AND COUPLE STRESS THEORY

IntroductionThispaperisadirectcontinuationofRef.[1 ] .InitthecoupledconservationlawofenergypresentedinRef.[2 ]wasextendedandtherathercompletesystemsofbasicbalancelawsandequationsformicropolarcontinuumtheoryhavebeenconstitutedbycombiningtherenewedresultsandthetraditionalconservationlawsofmassandmicroinertiaandtheentropyinequality .Thepurposeofthispaperistorestablishthesystemsofbasicbalancelawsandequationsformicromorphiccontinuumtheoryandcouplestresstheoryviadirecttransitionsandreductionsfromth…     

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

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Non-local continuum mechanics and fractional calculus

The present work introduces fractional calculus into the continuum mechanics area describing non-local constitutive relations. Considering a one-dimensional body and assuming total stored energy depending not only upon the local strain but also upon a fractional derivative of the stain, an elastic model with non-local stress–strain behavior is introduced. Fractional calculus provides a natural framework for describing non-local constitutive relations and requires no assumptions for the interval of non-local influence. Furthermore, the proposed method works in finite intervals contrary to the existing theories requiring infinite domains.     

On the thermodynamic consistency of the equivalence principle in continuum damage mechanics

We consider theories of continuum damage mechanics involving damage effect variables of different tensorial ranks. It turns out that orthotropic damage together with the use of Lemaitre's equivalence principle for the elastic part does not allow thermodynamic potentials such as the free enthalpy to exist. As the existence of these potentials is, however, a strict thermodynamic requirement, a theory employing orthotropic damage in this way is inconsistent. We show that the use of a rank-4 damage effect variable allows a consistent use of the equivalence principle.