<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg" display="inline" id="d1e573"> <mml:mi>Γ</mml:mi> </mml:math> -convergence for a phase-field cohesive energy

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. • Cohesive fracture. • Γ -convergence. • Finite element. • Mesh bias.