Anisotropy of the effective toughness of layered media

This continues the study of the effective toughness of layered materials started in Hossain et al. (2014) and Hsueh et al. (2018), with a focus on anisotropy. We use the phase-field model and the surfing boundary condition to propagate a crack macroscopically at various angles to the layers. We study two idealized situations, the first where the elastic modulus is uniform while the toughness alternates and a second where the toughness is uniform and the elastic modulus alternates. We find that in the first case of toughness heterogeneity the effective toughness displays ‘anomalous isotropy’ in that it is independent of the propagation direction and equal to that of the tougher material except when the crack propagation is parallel to the layers. In the second case of elastic heterogeneity, we find the behavior more anisotropic and consistent with the toughening effects of stress fluctuation and need for crack renucleation at the compliant-to-stiff interface. In both cases, the effective toughness is not convex in the sense of interfacial energy or Wulff shape reflecting the fact that crack propagation follows a critical path. Further, in both cases the crack path is not straight and consistent with a maximal dissipation principle. Finally, the effective toughness depends on the contrast and pinning, rather than on the extent of crack fluctuation.