caa a22 4500 605515069 CHVBK 20210128100707.0 cr unu---uuuuu 210128e20150601xx s 000 0 eng 10.1007/s00205-014-0820-3 doi (NATIONALLICENCE)springer-10.1007/s00205-014-0820-3 Γ -convergence Approximation of Fracture and Cavitation in Nonlinear Elasticity [Elektronische Daten] [Duvan Henao, Carlos Mora-Corral, Xianmin Xu] 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. Springer-Verlag Berlin Heidelberg, 2014 Henao Duvan Faculty of Mathematics, Pontificia Universidad Católica de Chile, Vicuña Mackenna, 4860, Santiago, Chile aut Mora-Corral Carlos Department of Mathematics, Faculty of Sciences, Universidad Autónoma de Madrid, 28049, Madrid, Spain aut Xu Xianmin LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, NCMIS, Chinese Academy of Sciences, 100190, Beijing, China aut Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 216/3(2015-06-01), 813-879 0003-9527 216:3<813 2015 216 205 https://doi.org/10.1007/s00205-014-0820-3 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00205-014-0820-3 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Henao Duvan Faculty of Mathematics, Pontificia Universidad Católica de Chile, Vicuña Mackenna, 4860, Santiago, Chile aut NATIONALLICENCE 700 1- Mora-Corral Carlos Department of Mathematics, Faculty of Sciences, Universidad Autónoma de Madrid, 28049, Madrid, Spain aut NATIONALLICENCE 700 1- Xu Xianmin LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, NCMIS, Chinese Academy of Sciences, 100190, Beijing, China aut NATIONALLICENCE 773 0- Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 216/3(2015-06-01), 813-879 0003-9527 216:3<813 2015 216 205