A Stress-Driven Double-Phase–Field Framework for Tensile Fracturing Processes in Transversely Isotropic Rocks

We present a double-phase–field framework for tensile fracturing processes in transversely isotropic rocks. Two distinct phase-field variables are introduced to represent smeared approximations of tensile fractures along the weak bedding planes and through the anisotropic rock matrix, respectively. Driving forces that control fracture propagation in the phase-field framework are constructed as a stress-based formula with a recently developed tensile failure criterion that distinguishes the two failure modes in transversely isotropic rocks. For numerical implementation, we adopt a staggered integration scheme and decouple the governing equations so that the displacement field and phase-field variables can be updated in sequence for a given loading step. The finite element formulation of the proposed framework is introduced in detail in this paper and is implemented in an in-house finite element code. The numerical implementation is then validated by reproducing the uniaxial tension test results of Lyons sandstone. After that, we conduct simulations on a pre-notched square plate loaded in tension to demonstrate the features of the proposed framework. Finally, we conduct simulations of three-point bending tests of Pengshui shale and show that the proposed model can reproduce the force–displacement curves and failure patterns of specimens with different bedding plane orientations observed in laboratory experiments.