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arXiv · 2507.12169

Phase-field modelling of cohesive fracture. Part I: $Γ$-convergence results

Roberto Alessi·Francesco Colasanto·Matteo Focardi
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Abstract

The main aim of this three-part work is to provide a unified consistent framework for the phase-field modeling of cohesive fracture. In this first paper we establish the mathematical foundation of a cohesive phase-field model by proving a $Γ$-convergence result in a one-dimensional setting. Specifically, we consider a broad class of phase-field energies, encompassing different models present in the literature, thereby both extending the results in \cite{ContiFocardiIurlano2016} and providing an analytical validation of all the other approaches. Additionally, by modifying the functional scaling, we demonstrate that our formulation also generalizes the Ambrosio-Tortorelli approximation for brittle fracture, therefore laying the groundwork for a unified framework for variational fracture problems. The Part~II paper presents a systematic procedure for constructing phase-field models that reproduce prescribed cohesive laws, whereas the Part~III paper validates the theoretical results with applied examples.

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Roberto Alessi, Francesco Colasanto, Matteo Focardi. 2025-10-10. Phase-field modelling of cohesive fracture. Part I: $Γ$-convergence results. https://arxiv.org/abs/2507.12169

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