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Regularity results for equilibria in a variational model for fracture

Published online by Cambridge University Press:  14 November 2011

Emilio Acerbi
Affiliation:
Dipartimento di Matematica, Via Massimo D'Azeglio 85/A, 43100 Parma, Italy
Irene Fonseca
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A.
Nicola Fusco
Affiliation:
Dipartimento di Matematica ‘U. Dini’, Università di Firenze, Viale Morgagni 67∕a, 50131 Firenze, Italy

Synopsis

In recent years models describing interactions between fracture and damage have been proposed in which the relaxed energy of the material is given by a functional involving bulk and interfacial terms, of the form

where Ω is an open, bounded subset of ℝN, q ≧1, gL∞ (Ω ℝN), λ, β > 0, the bulk energy density F is quasiconvex, K⊂ℝN is closed, and the admissible deformation u:Ω→ ℝN is C1 in Ω\K One of the main issues has to do with regularity properties of the ‘crack site’ K for a minimising pair (K, u). In the scalar case, i.e. when uΩ→ ℝ, similar models were adopted to image segmentation problems, and the regularity of the ‘edge’ set K has been successfully resolved for a quite broad class of convex functions F with growth p > 1 at infinity. In turn, this regularity entails the existence of classical solutions. The methods thus used cannot be carried out to the vectorial case, except for a very restrictive class of integrands. In this paper we deal with a vector-valued case on the plane, obtaining regularity for minimisers of corresponding to polyconvex bulk energy densities of the form

where the convex function h grows linearly at infinity.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1997

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