Stability and crack nucleation in variational phase-field models of fracture: Effects of length-scales and stress multi-axiality

We investigate the conditions for crack nucleation in variational gradient damage models used as phase-field models of brittle and cohesive fracture. Viewing crack nucleation as a structural stability problem, we analyze how solutions with diffuse damage become unstable and bifurcate towards localized states, representing the smeared version of cracks. We consider gradient damage models with a linear softening response, incorporating distinct softening parameters for the spherical and deviatoric modes. These parameters are employed to adjust the peak pressure and shear stress, resulting in an equivalent cohesive behavior. Through analytical and numerical second-order stability and bifurcation analyses, we characterize the crack nucleation conditions in quasi-static, rate-independent evolutions governed by a local energy minimization principle. We assess the stability of crack development, determining whether it is preceded by a stable phase with diffuse damage or not. Our results quantitatively characterize the classical transition between brittle and cohesive-like behaviors. A fully analytical solution for a one-dimensional problem provides a clear illustration of the complex bifurcation and instability phenomena, underpinning their connection with classical energetic arguments. The stability analysis under multi-axial loading reveals a fundamental non-trivial influence of the loading mode on the critical load for crack nucleation. We show that volumetric-dominated deformation mode can remain stable in the softening regime, thus delaying crack nucleation after the peak stress. This feature depends only on the properties of the local response of the material and is insensitive to structural scale effects. Our findings disclose the subtle interplay among the regularization length, the material's cohesive length-scale, structural size, and the loading mode to determine the crack nucleation conditions and the effective strength of phase-field models of fracture.