Elasticity with Disarrangements

Abstract

The subject of elasticity provides, among many other things, a continuum field theory for the dynamical evolution of bodies that undergo large deformations, that respond to changes in geometry through a stored energy function, and that experience internal dissipation in isothermal situations only during non-smooth processes. The central position of this field theory within mechanics has provided a starting point for many approaches to obtaining field theories that capture the effects of submacroscopic material structure or of submacroscopic geometrical changes on the macroscopic evolution of a body. The developments described here employ structured deformations and structured motions in order to formulate field theories for the dynamical evolution of bodies undergoing smooth geometrical changes at the macrolevel, while undergoing only piecewise smooth geometrical changes at submacroscopic levels. In keeping with elasticity, the new field equations should describe bodies that undergo large deformations and that store energy, while, in transcending elasticity in its standard form, the field equations should permit the body to experience internal dissipation during smooth dynamical processes and should provide a connection between the internal dissipation and the non-smooth geometrical changes (“disarrangements”) experienced by the body.

The author gratefully acknowledges support from the National Science Foundation, Division of Mathematical Sciences, under Grant #0102477, and thanks Amit Acharya and Morton Gurtin for valuable comments on the research described here and Gianpietro Del Piero for his careful reading and thoughtful comments on earlier versions of this article.