Structured deformations as energy minimizers in models of fracture and hysteresis

We review recent theories of non-classical, structured deformations and integral representations for their Helmholtz free energy. Energy minimizers for a body undergoing shearing at two di¤erent length scales and for a bar experiencing both smooth extension and macroscopic fractures then 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 microlevel, and the dichotomy between brittle and ductile fracture is related precisely to a critical length of a bar. (Running head: Structured Deformations, Fracture, and Hysteresis) Structured Deformations as Energy Minimizers in Models of Fracture and Hysteresis Contents 1 Introduction 1 2 Geometry of Structured Deformations 8 2.1 Structured Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Structured Deformations in the Context of SBV . . . . . . . . . . . . . . . . 11 3 Energetics 15 4 Applications to Hysteresis 19 4.1 Equilibrium and hysteresis within the class of two-level shears . . . . . . . . 19 4.2 Yielding and dissipation associated with two-level shears . . . . . . . . . . . 24 4.3 Two level shears and slip mechanisms in single crystals . . . . . . . . . . . . 26 5 An Application to Fracture Mechanics 29 5.1 Equilibrium con…gurations of a bar . . . . . . . . . . . . . . . . . . . . . . . 29 5.2 Metastable con…gurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40