The phase-field model of fracture incorporating Mohr-Coulomb, Mogi-Coulomb, and Hoek-Brown strength surfaces

Abstract

Classical phase-field theories of brittle fracture capture toughness-controlled crack growth but do not account for the material's strength surface, which governs fracture nucleation in the absence of cracks. The phase-field formulation of Kumar et al. (2020) proposed a blueprint for incorporating the strength surface while preserving toughness-controlled propagation by introducing a nucleation driving force and presented results for the Drucker-Prager surface. Following this blueprint, Chockalingam (2025) recently derived a general driving-force expression that incorporates arbitrary strength surfaces. The present work implements this driving force within a finite-element framework and incorporates representative strength surfaces that span diverse mathematical and physical characteristics-the Mohr-Coulomb, 3D Hoek-Brown, and Mogi-Coulomb surfaces. Through simulations of canonical fracture problems, the formulation is comprehensively validated across fracture regimes, capturing (i) nucleation under uniform stress, (ii) crack growth from large pre-existing flaws, and (iii) fracture governed jointly by strength and toughness. While the strength surfaces examined here already encompass a broad range of brittle materials, the results demonstrate the generality and robustness of the proposed driving-force construction for materials governed by arbitrary strength surfaces.