D. R. J. Owen

Structured Deformations as Energy Minimizers in Models of Fracture and Hysteresis

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.

Multiscale modeling in continuum mechanics and structured deformations

Energy Minimization for Isotropic Nonlinear Elastic Bodies (M. Silhavy).- Variational Problems of Crack Equilibrium and Crack Propagation (K.C. Le).- Griffith Theory Revisited (J.-J. Marigo).- Foundations of the Theory of Structured Deformations (G. Del Piero).- Second-Order Structured Deformations: Approximation Theorems (R. Paroni).- Crystalline Plasticity and Structured Deformations (L. Deseri).- Elasticity with Disarrangements (D.R. Owen).

Energetics of Two-Level Shears and Hardening of Single Crystals

An energetic description of the hardening behavior of single crystals undergoing single slip is analyzed. Simultaneous macroscopic simple shear and mesoscopic slips are described by means of a class of structured deformations called “two-level shears,” along with recently proposed measures of separation of active slip-bands and the number of lattice cells traversed during slip. The energetics of two-level shears gives rise to a response consistent with the experimentally observed loading and unloading behavior of a single crystal in G. I. Taylor’s soft device, as well as with the Portevin-le Chatelier effect. The initial critical resolved shear stress, the flow stress, and the hardening response are obtained, and an application to aluminum single crystals is discussed.

Active slip-band separation and the energetics of slip in single crystals

This research supports recent eAorts to provide an energetic approach to the prediction of stress‐strain relations for single crystals undergoing single slip and to give precise formulations of experimentally observed connections between hardening of single crystals and separation of active slip-bands. Non‐classical, structured deformations in the form of twolevel shears permit the formulation of new measures of the active slip-band separation and of the number of lattice cells traversed during slip. A formula is obtained for the Helmholtz free energy per unit volume as a function of the shear without slip, the shear due to slip, and the relative separation of active slip-bands in a single crystal. This formula is the basis for a model, under preparation by the authors, of hardening of single crystals in single slip that is consistent with the Portevin-Le Chatelier eAect and the existence of a critical resolved shear stress. # 2000 Elsevier Science Ltd. All rights reserved.

Twin Balance Laws for Bodies Undergoing Structured Motions

Non-classical, structured motions provide additive decompositions of velocity into a part due to disarrangements and a part without disarrangements. An analogous decomposition of the stress in the context of structured motions leads to a decomposition of the power of the same type. In this article, a postulate of invariance of the power under superposed rigid motions, both with and without disarrangements, is used to derive two equations for balance of forces and two equations for balance of moments. These “twin” balance laws and the resulting, reduced expression for the power will provide a starting point for field theories of bodies undergoing disarrangements.

Elasticity with Disarrangements

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.