Elements of Field Theory in Inhomogeneous and Defective Materials

Abstract

So-called configurational forces, also called material forces in modern continuum mechanics, and more generally energetic driving forces, are those « forces »which are associated by duality to the displacement or motion of whatever may be considered a defect in a continuum field theory. Conceptually simple examples of such « defects » are dislocations in ordered crystals, disclinations in liquid crystals, vortices in fluid mechanics, cracks and cavities in materials science, propagating fronts in phase-transition problems, shock waves in continuum mechanics, domain walls in solid-state science, and more generally all manifestations, smooth or abrupt, of changes in material properties. In such a framework, the material symmetry of the physical system is broken by the presence of a field singularity of a given dimensionality (point, line, surface, volume). Until very recently all these domains were studied separately but a general framework emerged essentially through the works of the authors and co-workers, basing initially on inclusive ideas of J.D.Eshelby (deceased 1985) — hence the coinage of Eshelbian mechanics by the authors for the mechanics of such forces. In this framework which is developed in a somewhat synthetic form, all configurational forces appear as forces of a non-Newtonian nature, acting on the material manifold (the set of points building up the material whether discrete or continuous) and not in physical space which remains the realm of Newtonian forces and their more modern realizations which usually act per quantity of matter (mass or electric charge). That is, configurational forces act on spatial gradients of properties, on field singularities, etc. They acquire a true physical meaning only in so far as the associated expanded power is none other than a dissipation; accordingly, configurational forces are essentially used to formulate criteria of progress of defects in accordance with the second law of thermodynamics. Within such a general vision, in fact, many irreversible properties of matter (e.g., damage, plasticity, magnetic hysteresis, phase transition, growth) are seen as irreversible local rearrangements of matter (material particles in an ordered crystal, spin layout in a ferromagnetic sample, director network in a liquid crystal) that are represented by pure material mappings. This is where some elements of modern differential geometry enter the picture following earlier works by Kroner, Noll, and others.