Monday, 11 October 2004 - 4:45 PM

This presentation is part of : Horgan Symposium

Cavitation as a limit of regularised minimisers in nonlinear elasticity

Jeyabal Sivaloganathan, University of Bath, Department of Mathematical Sciences, Claverton Down, Bath, BA2 7AY, United Kingdom, Scott Spector, Southern Illinois University, Department of Mathematics, Carbondale, IL 62901, and Viveka Tilakraj, Schlumberger Geoquest, Abingdon Technology Centre, Abingdon, OX14 1DZ, United Kingdom.

Consider a hyperelastic body which occupies the domain Ω in its reference configuration and which is held in a state of tension under imposed boundary displacements. Let x0 be any given point in Ω (this represents one of possibly many flaws in the material). It is shown in [1] that there exists a minimiser of the energy in a class of deformations containing maps which may be discontinuous at x0. For weak materials it is known that any such minimiser must be discontinuous if the imposed boundary displacement is sufficiently large, this is the phenomenon of cavitation. We prove that such a discontinuous minimiser is a limit (as ε → 0) of a corresponding sequence of minimisers of regularised problems in which the body contains a pre-existing hole of radius ε (centred on x0) in its reference configuration. These results make no assumption of symmetry and involve use of the Brouwer degree and an invertibility condition introduced by Müller and Spector [2]. Finally, we indicate possible applications of these results to modelling the initiation of fracture.

[1] J. Sivaloganathan and S.J. Spector, On the existence of minimisers with prescribed singular points in nonlinear elasticity, J. Elasticity 59 (2000), 83-113.

[2] S. Müller and S.J. Spector, An existence theory for nonlinear elasticity that allows for cavitation, Arch. Rational Mech. Anal. 131 (1995), 1-66.

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