Cauchy Fluxes Research Articles

Constitutive relations for polar continua based on statistical mechanics and spatial averaging

The purpose of the current work is the formulation of macroscopic constitutive relations, and in particular continuum flux densities, for polar continua from the underlying mass point dynamics. To this end, generic microscopic continuum field and balance relations are derived from phase space transport relations for expectation values of point fields related to additive mass point quantities. Given these, microscopic energy, linear momentum and angular momentum, balance relations are obtained in the context of the split of system forces into non-conservative and conservative parts. In addition, divergence–flux relations are formulated for the conservative part of microscopic supply-rate densities. For the case of angular momentum, two such relations are obtained. One of these is force-based, and the other is torque-based. With the help of physical and material theoretic restrictions (e.g. material frame-indifference), reduced forms of the conservative flux densities are obtained. In the last part of the work, formulation of macroscopic constitutive relations from their microscopic counterparts is investigated in the context of different spatial averaging approaches. In particular, these include (weighted) volume-averaging based on a localization function, surface averaging of normal flux densities based on Cauchy flux theory and volume averaging with respect to centre of mass.

Cauchy Fluxes and Gauss–Green Formulas for Divergence-Measure Fields Over General Open Sets

We establish the interior and exterior Gauss-Green formulas for divergence-measure fields in $L^p$ over general open sets, motivated by the rigorous mathematical formulation of the physical principle of balance law via the Cauchy flux in the axiomatic foundation, for continuum mechanics allowing discontinuities and singularities. The method, based on a distance function, allows to give a representation of the interior (resp. exterior) normal trace of the field on the boundary of any given open set as the limit of classical normal traces over the boundaries of interior (resp. exterior) smooth approximations of the open set. In the particular case of open sets with continuous boundary, the approximating smooth sets can explicitly be characterized by using a regularized distance. We also show that any open set with Lipschitz boundary has a regular Lipschitz deformable boundary from the interior. In addition, some new product rules for divergence-measure fields and suitable scalar functions are presented, and the connection between these product rules and the representation of the normal trace of the field as a Radon measure is explored. With these formulas at hand, we introduce the notion of Cauchy fluxes as functionals defined on the boundaries of general bounded open sets for the rigorous mathematical formulation of the physical principle of balance law, and show that the Cauchy fluxes can be represented by corresponding divergence-measure fields.

Radiative transfer and flux theory

The fundamental phenomenological equations of radiative transfer, e.g., Lambert’s cosine rule and the radiant transport equation, are derived from an analysis based on the Cauchy flux theory of continuum mechanics. For the classical case, where the radiance is distributed regularly over the unit sphere, it is shown that Lambert’s rule follows from a balance law for the transfer of radiative power in each direction u on the sphere, together with the appropriate Cauchy postulates and the additional assumption that the corresponding flux vector field ju be parallel to u. The standard radiant transport equation follows from the additional assumption that radiant flux is given as the advection of radiant energy density. A theory is also presented for the singular limit of isolated rays, where the distribution of radiance on the sphere reduces to a Borel measure.

The configuration space and principle of virtual power for rough bodies

In the setting of an n-dimensional Euclidean space, the duality between velocity fields on the class of admissible bodies and Cauchy fluxes is studied using tools from geometric measure theory. A generalized Cauchy flux theory is obtained for sets whose measure theoretic boundaries may be as irregular as flat (n − 1)-chains. Initially, bodies are modeled as normal n -currents induced by sets of finite perimeter. A configuration space comprising Lipschitz embeddings induces virtual velocities given by locally Lipschitz mappings. A Cauchy flux is defined as a real valued function on the Cartesian product of (n − 1)-currents and locally Lipschitz mappings. A version of Cauchy’s postulates implies that a Cauchy flux may be uniquely extended to an n-tuple of flat (n − 1)-cochains. Thus, the class of admissible bodies is extended to include flat n-chains and a generalized form of the principle of virtual power is presented. Wolfe’s representation theorem for flat cochains enables the identification of stress as an n-tuple of flat (n − 1)-forms representing the flat (n − 1)-cochains associated with the Cauchy flux.

Gravity Induced Diffusion: Sedimentation in Condensed Matter

In the paper the mathematical model describing the gravity induced sedimentation process in multicomponent system is presented. The model is based on the generalized interdiffusion method (bi-velocity method): (1) the volume continuity equation, (2) equation of motion, (3) Cauchy stress tensor and (4) Nernst Planck flux formulae. The method is applied to simulate the sedimentation processes of the selenium isotopes and the ternary InSbGe system.

Gauss‐Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws

AbstractWe analyze a class of weakly differentiable vector fieldsF: ℝn→ ℝnwith the property thatF∈L∞and divFis a (signed) Radon measure. These fields are calledbounded divergence‐measure fields. The primary focus of our investigation is to introduce a suitable notion of the normal trace of any divergence‐measure fieldFover the boundary of an arbitrary set of finite perimeter that ensures the validity of the Gauss‐Green theorem. To achieve this, we first establish a fundamental approximation theorem which states that, given a (signed) Radon measure μ that is absolutely continuous with respect to ℋ︁N− 1on ℝN, any set of finite perimeter can be approximated by a family of sets with smooth boundary essentially from the measure‐theoretic interior of the set with respect to the measure ||μ||, the total variation measure. We employ this approximation theorem to derive the normal trace ofFon the boundary of any set of finite perimeterEas the limit of the normal traces ofFon the boundaries of the approximate sets with smooth boundary so that the Gauss‐Green theorem forFholds onE. With these results, we analyze the Cauchy flux that is bounded by a nonnegative Radon measure over any oriented surface (i.e., an (N− 1)‐dimensional surface that is a part of the boundary of a set of finite perimeter) and thereby develop a general mathematical formulation of the physical principle of the balance law through the Cauchy flux. Finally, we apply this framework to the derivation of systems of balance laws with measure‐valued source terms from the formulation of the balance law. This framework also allows the recovery of Cauchy entropy flux through the Lax entropy inequality for entropy solutions of hyperbolic conservation laws. © 2008 Wiley Periodicals, Inc.

Cauchy’s stress theorem for stresses represented by measures

A version of Cauchy’s stress theorem is given in which the stress describing the system of forces in a continuous body is represented by a tensor valued measure with weak divergence a vector valued measure. The system of forces is formalized in the notion of an unbounded Cauchy flux generalizing the bounded Cauchy flux by Gurtin and Martins (Arch Ration Mech Anal 60:305–324, 1976). The main result of the paper says that unbounded Cauchy fluxes are in one-to-one correspondence with tensor valued measures with weak divergence a vector valued measure. Unavoidably, the force transmitted by a surface generally cannot be defined for all surfaces but only for almost every translation of the surface. Also conditions are given guaranteeing that the transmitted force is represented by a measure. These results are proved by using a new homotopy formula for tensor valued measure with weak divergence a vector valued measure.

Measure-Theoretic Analysis and Nonlinear Conservation Laws

We discuss some recent developments and trends of applying measure-theoretic analysis to the study of nonlinear conservation laws. We focus particularly on entropy solutions without bounded variation and Cauchy fluxes on oriented surfaces which are used to formulate the balance law. Our analysis employs the Gauss-Green formula and normal traces for divergence-measure fields, Young measures and compensated compactness, blow-up and scaling techniques, entropy methods, and related measuretheoretic techniques. Nonlinear conservation laws include multidimensional scalar conservation laws, strictly hyperbolic systems of conservation laws, isentropic Euler equations, two-dimensional sonic-subsonic flows, and degenerate parabolic-hyperbolic equations. Some open problems and trends on the topics are addressed and an extensive list of references is also provided.

Balanced Powers in Continuum Mechanics

An approach to weak balance laws in Continuum Mechanics is presented, involving densities with only divergence measure, which relies on the balance of power. An equivalence theorem between Cauchy powers and Cauchy fluxes is proved. As an application of this method, the construction of the stress tensor when the body is an orientable differential manifold is achieved under very general assumptions.

Cauchy's Flux Theorem in Light of Geometric Integration Theory

Cauchy's Flux Theorem in Light of Geometric Integration Theory

The Geometry of Cauchy's Fluxes

A formulation of the Cauchy theory for balance laws of scalar valued quantities is considered from a general geometric point of view. It is assumed only that the ambient space is an orientable m-dimensional manifold. The analog of the usual flux vector field is an (m−1)-differential form. Both the Cauchy theorem and differential version of the balance laws are formulated in this context.

Cauchy Fluxes Associated with Tensor Fields Having Divergence Measure

. Cauchy fluxes induced by locally summable tensor fields with divergence measure are characterized. The equivalence between integral formulations involving subsets of finite perimeter and much more restricted classes of subsets is proved.

Shadowing and the diffusionless limit in fast dynamo theory

This paper addresses the question of how the zero and small diffusivity solutions to the kinematic magnetic induction equation are related. It is shown that, in the case of perturbed linear toral automorphisms, hyperbolicity properties allow a connection between the zero diffusivity Cauchy solution and the non-zero diffusivity Wiener ensemble solution using shadowing theory. A formula is derived that calculates over finite times the small diffusivity magnetic field in terms of the local zero diffusivity magnetic field by averaging against a Gaussian density with variance proportional to diffusivity. For linear toral automorphisms, it is proven that the infinite time fast dynamo growth rate can be calculated using a local Cauchy flux average in agreement with a conjecture by Finn and Ott (1988).

The existence of the flux vector and the divergence theorem for general Cauchy fluxes

The existence of the flux vector and the divergence theorem for general Cauchy fluxes

Cauchy flux and sets of finite perimeter

Abstract : A Cauchy flux Q is a real-valued, additive, area-bounded function whose domain is the class of all Borel subsets of the reduced boundary of sets of finite perimeter. If the flux Q is also volume bounded, it is shown that Q can be represented as the integral of the normal component of some vector field. (Author)