A Hybrid Phase-Field Method with Spherical-Spectral-Deviatoric Split for Asymmetry Tension–Compression Mixed-mode Brittle Fracture

Abstract This study proposes a novel hybrid phase-field method for modelling asymmetric tension–compression mixed-mode brittle fracture in rock-like materials. In this method, the isotropic elastic strain energy is decomposed into tensile, tensile-shear, and compressive-shear components through a combination of orthogonal decomposition and strain spectral splitting. The split components are combined with three fracture energies and integrated with the Mohr–Coulomb strength criterion to construct a new hybrid driving force for mixed-mode fracture. The driving force is then incorporated into the framework of the standard brittle phase-field model and the resultant governing equations are discretized within the finite element framework and solved using a staggered Newton–Raphson iterative method, with a history field of driving force and a bound-constrained solver to ensure damage irreversibility and boundness, respectively. The developed method was validated against experimental and numerical data through six benchmark examples of solid fracture under tensile, shear, and compressive loadings, along with an additional case in the appendix to quantitatively verify its prediction of compressive failure. It is found that the new method implicitly incorporates a practical and physically grounded energy–strength coupled failure criterion, enabling accurate simulation of complex 2D and 3D fractures under various loading conditions and thus holding great potential for structural stability and safety assessments in engineering applications.