Serge Preston

MATERIAL ELEMENT MODEL AND THE GEOMETRY OF THE ENTROPY FORM

In this work we analyze and compare the model of the material (elastic) element and the entropy form developed by Coleman and Owen with that one obtained by localizing the balance equations of the continuum thermodynamics. This comparison allows one to determine the relation between the entropy function S of Coleman–Owen and that one imported from the continuum thermodynamics. We introduce the Extended Thermodynamical Phase Space (ETPS) and realize the energy and entropy balance expressions as 1-forms in this space. This allows us to realizes I and II laws of thermodynamics as conditions on these forms. We study the integrability (closure) conditions of the entropy form for the model of thermoelastic element and for the deformable ferroelectric crystal element. In both cases closure conditions are used to rewrite the dynamical system of the model in term of the entropy form potential and to determine the constitutive relations among the dynamical variables of the model. In a related study (to be published) these results will be used for the formulation of the dynamical model of a material element in the contact thermodynamical phase space of Caratheodory and Hermann similar to that of homogeneous thermodynamics.

Uniform Materials and the Multiplicative Decomposition of the Deformation Gradient in Finite Elasto-Plasticity

Abstract In this work we analyze the relation between the multiplicative decomposition F = F e F p of the deformation gradient as a product of the elastic and plastic factors and the theory of uniform materials. We prove that postulating such a decomposition is equivalent to having a uniform material model with two configurations – total φ and the inelastic φ1. We introduce strain tensors characterizing different types of evolutions of the material and discuss the form of the internal energy and that of the dissipative potential. The evolution equations are obtained for the configurations (φ, φ1) and the material metric g. Finally, the dissipative inequality for the materials of this type is presented. It is shown that the conditions of positivity of the internal dissipation terms related to the processes of plastic and metric evolution provide the anisotropic yield criteria.

VARIATIONAL THEORY OF BALANCE SYSTEMS

In this work, we apply the Poincare–Cartan formalism of Classical Field Theory to study the systems of balance equations (balance systems). We introduce the partial k-jet bundles of the configurational bundle π : Y → X and study their basic properties: partial Cartan structure, prolongation of vector fields, etc. A constitutive relation C of a balance system is realized as the mapping between a (partial) k-jet bundle and the extended dual bundle similar to the Legendre mapping of the Lagrangian Field Theory. The invariant (variational) form of the balance system corresponding to a constitutive relation is studied. Special cases of balance systems — Lagrangian systems of order 1 with arbitrary sources and RET (Rational Extended Thermodynamics) systems are characterized in geometrical terms. The action of automorphisms of the bundle π on the constitutive mappings is studied and it is shown that the symmetry group of acts on the sheaf of solutions of balance system . A suitable version of Noether theorem for an action of a symmetry group is presented together with the special forms for semi-Lagrangian and RET balance systems. Examples of energy momentum and gauge symmetries balance laws are provided. At the end, we introduce the secondary balance laws for a balance system and classify these laws for the Cattaneo heat propagation system.

Material Uniformity and the Concept of the Stress Space

The notion of the stress space, introduced by Schaefer [14], and further developed by Kroner [7] in the context of materials free of defects, is revisited. The comparison between the Geometric Theory of Material Inhomogeneities and the Stress Space approach is discussed. It is shown how to extend Kroner’s approach to the case of the material body with inhomogeneities (defects).

On the Geometrical Characteristics of Chaotic Dynamics, I

We introduce some geometrical structures in the phase space of a dynamical system, naturally defined by the system. We study some properties of these structures and discuss their use for the constructing new characteristics of chaotic dynamical behaviour.

Continuously defective crystals with a prescribed dislocation density

We analyze some aspects of the kinematic theory of non-uniformly defective elastic crystals. Concentrating on the problem of identifying continuous defective lattices possessing the given defectiveness, as defined by the dislocation density tensor, we investigate the relation between the dislocation density tensor and the Lie algebra of vector fields associated with a defective lattice.

Conformal Laplacian and Conical Singularities

We study a behavior of the conformal Laplacian operatorLg on a manifold with tame conical singularities: when each singularity is given as a cone over a product of the standard spheres. We study the spectral properties of the operator Lg on such manifolds. We describe the asymptotic of a general solution of the equation Lgu = Qu α with 1 ≤ α ≤ n+2 n−2 near each singular point. In particular, we derive the asymptotic of a Yamabe metric near such singularity.

A Model of the Self-Driven Evolution of a Defective Continuum:

A model of the self-driven evolution law of a defective anelastic continuum is presented. Two-dimensional examples are discussed and the role of the Clausius—Duhem inequality in imposing constitutive restrictions is investigated.