Submacroscopically Stable Equilibria of Elastic Bodies Undergoing Disarrangements and Dissipation

Abstract

Elasticity is viewed here as a starting point in the description of inelastic behavior. Structured deformations and a field theory of elastic bodies undergoing disarrangements (non-smooth geometrical changes) and dissipation are used to formulate and illustrate a concept of “submacroscopically stable equilibrium configuration”. A body in a submacroscopically stable equilibrium configuration resists additional submacroscopic geometrical changes such as the occurrence of microslips, the formation of microvoids, and the appearance of localized distortions that, together, leave the macroscopic configuration of the body unchanged. Submacroscopically stable configurations represent energetically preferred phases for bodies in equilibrium, and a procedure is described here for determining the submacroscopically stable equilibria of a body. The procedure is carried out in detail here for two classes of bodies that may undergo disarrangements and experience internal dissipation. One class is characterized by its bi-quadratic free energy response function, and the requirement of submacroscopic stability reduces from five to one the number of phases available to a body that is in equilibrium under mixed boundary conditions. Boundary-value problems for the macroscopic deformation corresponding to a submacroscopically stable equilibrium configuration in a body of this class are formulated. A second class of bodies, the “near-sighted fluids”, has both a prolate and a spherical phase that may occur in equilibrium1 the submacroscopically stable equilibria of a near-sighted fluid must be stress-free, without regard to the particular phase that appears. In all considerations in this article, the term “equilibrium” is synonymous with satisfaction of balance of forces and moments in a given environment.