Structured Deformations as Energy Minimizers in Models of Fracture and Hysteresis

Abstract

This paper reviews recent theories of nonclassical, structured deformations and integral representations for their Helmholtz free energy. Energy minimizers for a body undergoing shearing at two different length scales and for a bar experiencing both smooth extension and macroscopic fractures are determined, and applications to the shearing of single crystals and to the cohesive fracture of solids are discussed. Yield, hysteresis, and the associated dissipation in two-level shears are shown to arise from instabilities at the micro level, and the dichotomy between brittle and ductile fracture is related precisely to a critical length of a bar.