Marek Elżanowski

G-Structures and Material Homogeneity

The mathematical theory of inhomogeneities in simple elastic materials is reformulated in terms of G-structures, thereby allowing for the explicit solution of the homogeneity condition for a number of special material classes.

Material Symmetry in a Theory of Continuously Defective Crystals

We propose that a particular group, which depends on the dislocation density tensor, be adopted as material symmetry group for some classes of defective crystals, and give motivation for this proposal.

Locally homogeneous configurations of uniform elastic bodies

Abstract This paper deals with one of the fundamental problems of the mathematical theory of inhemogeneities in simple elastic material bodies, namely the availability of local homogeneous configurations. Utilizing the original differential geometric approach of W. Noll and C.-C. Wang, adopted for our particular needs and restricted to the hyperelastic case, we develop a system of partial differential equations for the locally homogeneous material configurations. The existence and the form of solutions of this system are discussed by means of examples.

Connections on higher order frame bundles

In the paper we present the analysis of connections on frame bundles of higher order contact, with special emphasis on the question of local flatness.

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).

Higher Grade Material Structures

It has been often pointed out, by critics and supporters alike, that the theory of inhomogeneities of Noll [1] and Wang [2] does not enjoy the generality often demanded by those propagating the so-called lattice model. This is because in the structural approach to the theory of continuous distribution of defects it has been suggested that, although the presence of dislocations shows through the non-vanishing torsion of the material connection, disclinations are measured by the curvature of such a connection; see e.g. Anthony [3]. Since any constitutive functional associated with a simple elastic material body induces, by definition, a locally integrable parallelism it appears that the disclinations, and possibly other defects, are ruled out. A structural approach suggests also that bodies with defects, disclinations in particular, are subject to multipolar stresses. Thus, it seems natural to investigate the possibility of describing disclinations in the realm of the higher-grade materials as originally suggested by Elianowski and Epstein [4].

Another Look at the Evolution of Material Structures

The evolution of a distribution of material inhomogeneities is investigated by analyzing the evolution of the corresponding material connection. Some general relations describing how the deformation of a material G-structure modifies the material connection associated with it are derived. These relations are then analyzed for different material isotropy groups.

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.

Connection and curvature in crystals with non-constant dislocation density

Given a smooth defective solid crystalline structure defined by linearly independent ‘lattice’ vector fields, the Burgers vector construction characterizes some aspect of the ‘defectiveness’ of the crystal by virtue of its interpretation in terms of the closure failure of appropriately defined paths in the material and this construction partly determines the distribution of dislocations in the crystal. In the case that the topology of the body manifold M is trivial (e.g., a smooth crystal defined on an open set in ℝ 2 ), it would seem at first glance that there is no corresponding construction that leads to the notion of a distribution of disclinations, that is, defects with some kind of ‘rotational’ closure failure, even though the existence of such discrete defects seems to be accepted in the physical literature. For if one chooses to parallel transport a vector, given at some point P in the crystal, by requiring that the components of the transported vector on the lattice vector fields are constant, there is no change in the vector after parallel transport along any circuit based at P. So the corresponding curvature is zero. However, we show that one can define a certain (generally non-zero) curvature in this context, in a natural way. In fact, we show (subject to some technical assumptions) that given a smooth solid crystalline structure, there is a Lie group acting on the body manifold M that has dimension greater or equal to that of M. When the dislocation density is non-constant in M the group generally has a non-trivial topology, and so there may be an associated curvature. Using standard geometric methods in this context, we show that there is a linear connection invariant with respect to the said Lie group, and give examples of structures where the corresponding torsion and curvature may be non-zero even when the topology of M is trivial. So we show that there is a ‘rotational’ closure failure associated with the group structure – however, we do not claim, as yet, that this leads to the notion of a distribution of disclinations in the material, since we do not provide a physical interpretation of these ideas. We hope to provide a convincing interpretation in future work. The theory of fibre bundles, in particular the theory of homogeneous spaces, is central to the discussion.

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.