Γ-convergence Approximation of Fracture and Cavitation in Nonlinear Elasticity
Entity
UAM. Departamento de MatemáticasPublisher
Springer VerlagDate
2014-12-05Citation
10.1007/s00205-014-0820-3
Archive for Rational Mechanics and Analysis 216.15 (2015): 813-879
ISSN
0003-9527 (print); 1432-0673 (online)DOI
10.1007/s00205-014-0820-3Funded by
D. Henao gratefully acknowledges the Chilean Ministry of Education’s support through the FONDE-CYT Iniciación project no. 11110011. C. Mora-Corral has been supported by Project MTM2011-28198 of the Spanish Ministry of Economy and Competitivity, the ERC Starting grant no. 307179, the “Ramón y Cajal” programme and the European Social Fund. X. Xu acknowledges the funding by NSFC 11001260Editor's Version
http://dx.doi.org/10.1007/s00205-014-0820-3Subjects
MatemáticasNote
The final publication is available at Springer via http://dx.doi.org/10.1007/s00205-014-0820-3Rights
© 2014 Springer-Verlag Berlin HeidelbergAbstract
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
Files in this item
Google Scholar:Henao, Duvan A.
-
Mora Corral, Carlos
-
Xu, Xianmin
This item appears in the following Collection(s)
Related items
Showing items related by title, author, creator and subject.
-
Invertible Sobolev functions: counterexamples and applications to nonlinear elasticity
Oliva Wilkinson, Marcos de la
2017-06-23