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An elliptic approximation for phase separation in a fractured material

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Published/Copyright: April 29, 2025

Abstract

We consider a free-boundary and free-discontinuity energy connecting phase separation and fracture in an elastic material. The energy excludes the contribution of phase boundaries in the cracked region, providing a heuristic approximation of the interfacial energy in the current material configuration. Our primary result shows that the sharp energy may be recovered via Γ-convergence from a modified Cahn–Hilliard energy coupled with an Ambrosio–Tortorelli-type approximation of the (linear) elastic and fracture energy.


Communicated by Ulisse Stefanelli


Award Identifier / Grant number: EXC-2047/1 – 390685813

Award Identifier / Grant number: 211504053 – SFB 1060

Award Identifier / Grant number: DMS-2108784

Award Identifier / Grant number: DMS-2136198

Funding statement: This work grew out of Solveig Wittig’s Master’s thesis at the University of Bonn. During much of the writing of the paper Kerrek Stinson was at the Institute for Applied Mathematics at the University of Bonn. Kerrek Stinson was supported by funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC-2047/1 – 390685813, the DFG project 211504053 – SFB 1060; also by the NSF (USA) under awards DMS-2108784 and DMS-2136198.

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Received: 2024-08-30
Accepted: 2025-03-28
Published Online: 2025-04-29
Published in Print: 2025-10-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

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