Bilans d'energie et de quantité de mouvement pour un continu deformable,polarisable et magnétisable

In this paper, it is shown that the different electromagnetic energy-momentum tensors proposed by various authors for a continuum interacting with an electromagnetic field all lead to the same equations of balance for energy and momentum provided the definitions of stress and internal energy are suitably related. These various tensors come out from different partitions of the total energy-momentum tensor. From a particular partition, we derive an expression of the balance of energy suitable for application to continuum thermodynamics. In the classical approximation, the corresponding equation of balance of momentum gives rise to an expression for the electromagnetic force in a polarizable and magnetizable continuum.     

Stresses and momenta in electromagnetic materials

The notion of energy–momentum, or energy–stress, pertains typically to electromagnetism. Eshelby transferred such a notion into elasticity in 1951 and, afterward, into continuum mechanics, in order to account for the force acting on a material defect. Similarities and differences between the Maxwell tensor of electromagnetism and the Eshelby tensor are shown and commented hereby. Basing on a Lagrangian approach to electromagnetic materials, canonical momenta are shown to emerge in a natural way. On the basis of these canonical quantities, one can introduce the material momentum (or pseudomomentum) along with the classical momentum and the material stress (Eshelby stress) along with the Maxwell stress, which is a Cauchy-like stress.     

On Molecular Modelling and Continuum Concepts

Continuum concepts and field values are related to local (scale-dependent) spacetime atomistic averages. Spatial averaging is effected by both weighting function and cellular localisation procedures, and resulting forms of linear momentum balance are compared. The former yields a local balance directly, with several candidate interaction stress fields. The latter results in a global balance involving a traction field expressible in terms of an interaction stress tensor field. In both approaches the Cauchy stress incorporates distinct interaction and thermokinetic contributions. Inter alia are addressed physically-distinguished choices of weighting function; the scale-dependence of the boundary of a body, its motion and material points thereof; physical interpretations of various candidate interaction stress tensors; temporal averaging and material systems whose content changes with time; and the possible relevance of the latter to investigating a molecular context for configurational forces.     

A micromorphic electromagnetic theory

This work is concerned with the determination of both macroscopic and microscopic deformations, motions, stresses, as well as electromagnetic fields developed in the material body due to external loads of thermal, mechanical, and electromagnetic origins. The balance laws of mass, microinertia, linear momentum, moment of momentum, energy, and entropy for microcontinuum are integrated with the Maxwell’s equations. The constitutive theory is constructed. The finite element formulation of micromorphic electromagnetic physics is also presented. The physical meanings of various terms in the constitutive equations are discussed.     

Convergence of Peridynamics to Classical Elasticity Theory

The peridynamic model of solid mechanics is a nonlocal theory containing a length scale. It is based on direct interactions between points in a continuum separated from each other by a finite distance. The maximum interaction distance provides a length scale for the material model. This paper addresses the question of whether the peridynamic model for an elastic material reproduces the classical local model as this length scale goes to zero. We show that if the motion, constitutive model, and any nonhomogeneities are sufficiently smooth, then the peridynamic stress tensor converges in this limit to a Piola-Kirchhoff stress tensor that is a function only of the local deformation gradient tensor, as in the classical theory. This limiting Piola-Kirchhoff stress tensor field is differentiable, and its divergence represents the force density due to internal forces. The limiting, or collapsed, stress-strain model satisfies the conditions in the classical theory for angular momentum balance, isotropy, objectivity, and hyperelasticity, provided the original peridynamic constitutive model satisfies the appropriate conditions.      

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≈)     

On the Nonlinear Continuum Theory of Dislocations: A Gauge Field Theoretical Approach

A nonlinear continuum theory of material bodies with continuously distributed dislocations is presented, based on a gauge theoretical approach. Firstly, we derive the canonical conservation laws that correspond to the group of translations and rotations in the material space using Noether’s theorem. These equations give us the canonical Eshelby stress tensor as well as the total canonical angular momentum tensor. The canonical Eshelby stress tensor is neither symmetric nor gauge-invariant. Based on the Belinfante-Rosenfeld procedure, we obtain the gauge-invariant Eshelby stress tensor which can be symmetric relative to the reference configuration only for isotropic materials. The gauge-invariant angular momentum tensor is obtained as well. The decomposition of the gauge-invariant Eshelby stress tensor in an elastic and in a dislocation part gives rise to the derivation of the famous Peach-Koehler force.     

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

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The Eshelby stress tensor,angular momentum tensor and scaling flux in micropolar elasticity

The (static) energy-momentum tensor, angular momentum tensor and scaling flux vector of micropolar elasticity are derived within the framework of Noether’s theorem on variational principles. Certain balance (or broken conservation) laws of broken translational, rotational and dilatational symmetries are found including inhomogeneities, elastic anisotropy, body forces, body couples and dislocations and disclinations present. The non-conserved J-, L- and M-integrals of micropolar elasticity are derived and discussed. We gave explicit formulae for the configurational forces, moments and work terms.     

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.     

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.     

Theory of 4D-media with stationary dislocations

In earlier studies, the authors showed that an application of classical methods of mechanics of deformable media to the study of properties of 4D-space-time continuum permit stating consistent models of nonholonomic media mechanics consistent with the first and second laws of thermodynamics. In the present paper, we show that the classical methods of continuum mechanics are also promising when modeling physical processes. It is shown that, just as in the three-dimensional theory of stationary dislocations, there exist dislocations of three types for a generalized 4D-medium. They correspond to the decomposition of the free distortion tensor into a spherical tensor, a deviator tensor, and a pseudotensor of rotations. We interpret several particular models, thus showing that the proposed model describes the spectrum of known physical interactions: electromagnetic, strong, weak, and gravitational. We show that the resolving equations include the Maxwell equations of electrodynamics and the Yukawa equations for strong interactions as subsystems.     

Continuum theory for nematic liquid crystals

This paper presents a formulation of continuum theory for nematic liquid crystals based upon the balance laws for linear and angular momentum, that derives directly expressions for stress and couple stress in these transversely isotropic liquids. This approach therefore avoids the introduction of generalised forces or torques associated with the director describing the axis of transverse isotropy.     

Configurational forces and couples in fracture mechanics accounting for microstructures and dissipation

Configurational forces and couples acting on a dynamically evolving fracture process region as well as their balance are studied with special emphasis to microstructure and dissipation. To be able to investigate fracture process regions preceding cracks of mode I, II and III we choose as underlying continuum model the polar and micropolar, respectively, continuum with dislocation motion on the microlevel. As point of departure balance of macroforces, balance of couples and balance of microforces acting on dislocations are postulated. Taking into account results of the second law of thermodynamics the stress power principle for dissipative processes is derived.Applying this principle to a fracture process region evolving dynamically in the reference configuration with variable rotational and crystallographic structure, the configurational forces and couples are derived generalizing the well-known Eshelby tensor. It is shown that the balance law of configurational forces and couples reflects the structure of the postulated balance laws on macro- and microlevel: the balance law of configurational forces and configurational couples are coupled by field variable, while the balance laws of configurational macro- and microforces are coupled only by the form of the free energy. They can be decoupled by corresponding constitutive assumption.Finally, it is shown that the second law of thermodynamics leads to the result that the generalized Eshelby tensor for micropolar continua with dislocation motion consists of a non-dissipative part, derivable from free and kinetic energy, and a dissipative part, derivable from a dissipation pseudo-potential.     

Geometric derivation of the microscopic stress: A covariant central force decomposition

We revisit the derivation of the microscopic stress, linking the statistical mechanics of particle systems and continuum mechanics. The starting point in our geometric derivation is the Doyle–Ericksen formula, which states that the Cauchy stress tensor is the derivative of the free-energy with respect to the ambient metric tensor and which follows from a covariance argument. Thus, our approach to define the microscopic stress tensor does not rely on the statement of balance of linear momentum as in the classical Irving–Kirkwood–Noll approach. Nevertheless, the resulting stress tensor satisfies balance of linear and angular momentum. Furthermore, our approach removes the ambiguity in the definition of the microscopic stress in the presence of multibody interactions by naturally suggesting a canonical and physically motivated force decomposition into pairwise terms, a key ingredient in this theory. As a result, our approach provides objective expressions to compute a microscopic stress for a system in equilibrium and for force-fields expanded into multibody interactions of arbitrarily high order. We illustrate the proposed methodology with molecular dynamics simulations of a fibrous protein using a force-field involving up to 5-body interactions.     

Micromorphic modeling of granular dynamics

This paper provides micromorphic modeling of a granular material. Micromorphic modeling treats an individual particle as a microelement and the particle composition in a representative volume element as a macroelement. By specifying the volume of a macroelement, continuum volume-type quantities such as mass density, body force, body couple, kinetic energy density, internal energy density, specific heat supply, etc., are determined by taking the averages of their discrete counterparts in a macroelement. The discrete expressions for the divergence of surface-type quantities (fluxes) are obtained with the help of discrete–continuum analogy for the discrete balance equations. We demonstrate that the discrete formulation of stress tensor in the dynamic condition, which involves both contributions from body forces and relative particle accelerations in a macroelement, can be simply expressed in terms of contact forces and branch vectors. This study constructs complete discrete-type and continuum-type balance equations for a granular material in a macroelement and at a macroscopic point, using the discrete–continuum correspondence for these field quantities.     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅵ)—CONSERVATION LAWS OF MASS AND INERTIA

The purpose is to reestablish the coupled conservation laws, the local conservation equations and the jump conditions of mass and inertia for polar continuum theories. In this connection the new material derivatives of the deformation gradient, the line element, the surface element and the volume element were derived and the generalized Reynolds transport theorem was presented. Combining, these conservation laws of mass and inertia with the balance laws of momentum, angular momentum and energy derived in our previous papers of this series, a rather complete system of coupled basic laws and principles for polar continuum theories is constituted on the whole. From this system the coupled nonlocal balance equations of mass, inertia, momentum, angular momentum and energy may be obtained by the usual 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≈)