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Γ-convergence Approximation of Fracture and Cavitation in Nonlinear Elasticity

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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, 2010). 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.

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Authors and Affiliations

  1. Faculty of Mathematics, Pontificia Universidad Católica de Chile, Vicuña Mackenna, 4860, Santiago, Chile

    Duvan Henao

  2. Department of Mathematics, Faculty of Sciences, Universidad Autónoma de Madrid, 28049, Madrid, Spain

    Carlos Mora-Corral

  3. LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, NCMIS, Chinese Academy of Sciences, Beijing, 100190, China

    Xianmin Xu

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  1. Duvan Henao
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  2. Carlos Mora-Corral
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  3. Xianmin Xu
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Correspondence to Carlos Mora-Corral.

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Communicated by A. Braides

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Henao, D., Mora-Corral, C. & Xu, X. Γ-convergence Approximation of Fracture and Cavitation in Nonlinear Elasticity. Arch Rational Mech Anal 216, 813–879 (2015). https://doi.org/10.1007/s00205-014-0820-3

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