Monday, 11 October 2004 - 5:10 PM

This presentation is part of : Horgan Symposium

Cavitation and Configurational Forces in Nonlinear Elasticity

Scott Spector, Southern Illinois University, Department of Mathematics, Carbondale, IL 62901 and Jeyabal Sivaloganathan, University of Bath, Department of Mathematical Sciences, Bath, BA2 7AY, England.

Experiments on elastomers indicate that large tensile stress can induce cavitation, that is, the appearance of voids in the material that were not previously evident. This phenomenon can be viewed as either the growth of pre-existing infinitesimal holes in the material or, alternatively, as the spontaneous creation of new holes in an initially perfect body. In this lecture our approach is to adopt both views concurrently within the framework of the variational theory of nonlinear elasticity. An elastomer is modeled, on a macroscale, as a void-free material and, on a microscale, as a material containing certain defects that are the only points at which hole formation can occur. Mathematically, this is accomplished by the use of deformations whose point singularities are constrained. One consequence of this viewpoint is that cavitation may then take place at a point that is not energetically optimal. We show that this disparity will generate configurational forces, a type of force identified previously in dislocations in crystals, in phase transitions in solids, in solidification, and in fracture mechanics.

As an application of this approach we consider the energetically optimal point for a solitary hole to form in a two-dimensional homogeneous and isotropic elastic disk subject to radial boundary displacements. We show that the center of the disk is the unique optimal point. Our analysis utilizes the energy-momentum tensor, the asymptotics of an equilibrium solution with an isolated singularity, and the linear theory of elasticity at the stressed configuration that the material occupies immediately prior to cavitation.

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