  Cracking of rocks and rock-like materials exhibits a rich variety of patterns
where tensile (mode I) and shear (mode II) fractures are often interwoven.
These mixed-mode fractures are usually cohesive (quasi-brittle) and frictional.
Although phase-field modeling is increasingly used for rock fracture
simulation, no phase-field formulation is available for cohesive and frictional
mixed-mode fracture. To address this shortfall, here we develop a
double-phase-field formulation that employs two different phase fields to
describe cohesive tensile fracture and frictional shear fracture individually.
The formulation rigorously combines the two phase fields through three
approaches: (i) crack-direction-based decomposition of the strain energy into
the tensile, shear, and pure compression parts, (ii) contact-dependent
calculation of the potential energy, and (iii) energy-based determination of
the dominant fracturing mode in each contact condition. We validate the
proposed model, both qualitatively and quantitatively, with experimental data
on mixed-mode fracture in rocks. The validation results demonstrate that the
double-phase-field model -- a combination of two quasi-brittle phase-field
models -- allows one to directly use material strengths measured from
experiments, unlike brittle phase-field models for mixed-mode fracture in
rocks. Another standout feature of the double-phase-field model is that it can
simulate, and naturally distinguish between, tensile and shear fractures
without complex algorithms.
