Toward a Field Theory for Elastic Bodies Undergoing Disarrangements

Structured deformations are used to refine the basic ingredients of continuum field theories and to derive a system of field equations for elastic bodies undergoing submacroscopically smooth geometrical changes as well as submacroscopically non-smooth geometrical changes (disarrangements). The constitutive assumptions employed in this derivation permit the body to store energy as well as to dissipate energy in smooth dynamical processes. Only one non-classical field G, the deformation without disarrangements, appears in the field equations, and a consistency relation based on a decomposition of the Piola-Kirchhoff stress circumvents the use of additional balance laws or phenomenological evolution laws to restrict G. The field equations are applied to an elastic body whose free energy depends only upon the volume fraction for the structured deformation. Existence is established of two universal phases, a spherical phase and an elongated phase, whose volume fractions are (1−γ0)3 and (1−γ0) respectively, with γ0:=(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \sqrt 5 $$ \end{document}−1)/2 the “golden mean”.