Γ-convergence for a phase-field cohesive energy
Year: 2026
Type: article
Abstract: Reproducing the key features of fracture behavior under multiaxial stress states is essential for accurate modeling. Experimental evidence indicates that three intrinsic material properties govern fracture nucleation in elastic materials: elasticity, strength, and fracture toughness (or critical energy release rate). The flexibility in introducing these features in phase-field models poses significant challenges, especially under complex loading conditions. To attain this goal, recent work introduces a new energy functional within a cohesive phase-field framework. This model introduces an internal variable to describe the inelastic response. Notably, the strength is decoupled from the internal length, that is not interpreted as a material length scale, as often done in literature, but rather as a purely variational tool. The proposed functional allows for a rigorous variational framework, enabling the use of tools from the calculus of variations. We investigate the Γ -convergence of the model to a sharp cohesive fracture energy in the one dimensional setting, using a finite element discrete formulation and exploiting the strong localization of the damage variable. Notably, unlike classical models where the elastic and fracture energies converge independently, this model exhibits a coupling of all energy terms. The limiting cohesive energy arises from the combined asymptotic behavior of the elastic energy (concentrated in a single element), the fracture energy, and the potential for the internal variable, while the remaining elastic energy converges separately. We highlight that the Γ -convergence of the model can be extended to the two-dimensional (anti-plane) setting. Finally, we present numerical simulations exploring the sensitivity of the model to mesh anisotropy, offering insight into both its theoretical robustness and its practical implementation.
Source: Mechanics of Materials
Institutions: University of Pavia, ETH Zurich
Language: English
Subfield: Mechanics of Materials
Field: Engineering
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,850
OpenAlex ID: W7123785023