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Phase-field approximation for a class of cohesive fracture energies with an activation threshold

Phase-field approximation for a class of cohesive fracture energies with an activation threshold AbstractWe study the Γ-limit of Ambrosio–Tortorelli-type functionals Dε⁢(u,v){D_{\varepsilon}(u,v)}, whose dependence on the symmetrised gradient e⁢(u){e(u)}is different in 𝔸⁢u{\mathbb{A}u}and in e⁢(u)-𝔸⁢u{e(u)-\mathbb{A}u}, for a ℂ{\mathbb{C}}-elliptic symmetric operator 𝔸{\mathbb{A}}, in terms of the prefactor depending on the phase-field variable v.The limit energy depends both on the opening and on the surface of the crack, and is intermediate between the Griffith brittle fracture energy and the one considered by Focardi and Iurlano[Asymptotic analysis of Ambrosio–Tortorelli energies in linearized elasticity,SIAM J. Math. Anal. 46 2014, 4, 2936–2955].In particular, we prove that G(S)BD functions with bounded 𝔸{\mathbb{A}}-variation are (S)BD. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Calculus of Variations de Gruyter

Phase-field approximation for a class of cohesive fracture energies with an activation threshold

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Publisher
de Gruyter
Copyright
© 2021 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1864-8266
eISSN
1864-8266
DOI
10.1515/acv-2019-0018
Publisher site
See Article on Publisher Site

Abstract

AbstractWe study the Γ-limit of Ambrosio–Tortorelli-type functionals Dε⁢(u,v){D_{\varepsilon}(u,v)}, whose dependence on the symmetrised gradient e⁢(u){e(u)}is different in 𝔸⁢u{\mathbb{A}u}and in e⁢(u)-𝔸⁢u{e(u)-\mathbb{A}u}, for a ℂ{\mathbb{C}}-elliptic symmetric operator 𝔸{\mathbb{A}}, in terms of the prefactor depending on the phase-field variable v.The limit energy depends both on the opening and on the surface of the crack, and is intermediate between the Griffith brittle fracture energy and the one considered by Focardi and Iurlano[Asymptotic analysis of Ambrosio–Tortorelli energies in linearized elasticity,SIAM J. Math. Anal. 46 2014, 4, 2936–2955].In particular, we prove that G(S)BD functions with bounded 𝔸{\mathbb{A}}-variation are (S)BD.

Journal

Advances in Calculus of Variationsde Gruyter

Published: Oct 1, 2021

Keywords: Free discontinuity problems; special functions of bounded deformation; cohesive fracture; 49J45; 26A45; 49Q20; 74R99; 35Q74

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