Phase-Field Modeling of Cohesive Fracture. Part I: \({\Gamma }\)-Convergence Results

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 [Formula: see text]-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 [S. Conti, M. Focardi, and F. Iurlano, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), pp. 1033–1067] 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.