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- Title
Γ-convergence Approximation of Fracture and Cavitation in Nonlinear Elasticity.
- Authors
Henao, Duvan; Mora-Corral, Carlos; Xu, Xianmin
- Abstract
Our starting point is a variational model in nonlinear elasticity that allows for cavitation and fracture that was introduced by Henao and Mora-Corral (Arch Rational Mech Anal 197:619-655, ). The total energy to minimize is the sum of the elastic energy plus the energy produced by crack and surface formation. It is a free discontinuity problem, since the crack set and the set of new surface are unknowns of the problem. The expression of the functional involves a volume integral and two surface integrals, and this fact makes the problem numerically intractable. In this paper we propose an approximation (in the sense of Γ-convergence) by functionals involving only volume integrals, which makes a numerical approximation by finite elements feasible. This approximation has some similarities to the Modica-Mortola approximation of the perimeter and the Ambrosio-Tortorelli approximation of the Mumford-Shah functional, but with the added difficulties typical of nonlinear elasticity, in which the deformation is assumed to be one-to-one and orientation-preserving.
- Subjects
STOCHASTIC convergence; APPROXIMATION theory; CAVITATION; NONLINEAR elastic fracture; FINITE element method; INTEGRALS; PERIMETERS (Geometry)
- Publication
Archive for Rational Mechanics & Analysis, 2015, Vol 216, Issue 3, p813
- ISSN
0003-9527
- Publication type
Article
- DOI
10.1007/s00205-014-0820-3