A variational phase-field model for anisotropic fracture accounting for multiple cohesive lengths

We propose a novel variational phase-field model for fracture in anisotropic materials. The model is specifically designed to allow a more flexible calibration of crack nucleation than existing anisotropic fracture formulations, while avoiding the introduction of multiple damage variables. In addition to the classical components of anisotropic phase-field models based on a single damage variable -- namely, anisotropic elasticity and the extension of the fracture energy density via a second-order structural tensor -- the proposed approach introduces fracture anisotropy through a cohesive degradation function with potentially distinct cohesive lengths along the principal material directions. For this reason, we refer to it as multi-cohesive model. This feature enables independent control of the critical stresses governing crack nucleation in each material direction. We analyze the homogeneous solution and its second-order stability, and we compare the resulting strength surfaces with those of two representative anisotropic phase-field models available in the literature. Finally, numerical simulations in two and three dimensions demonstrate the capability of the proposed model to independently control crack nucleation and propagation in anisotropic fracture problems of increasing complexity.