  We introduce a phase-field method for continuous modeling of cracks with
frictional contacts. Compared with standard discrete methods for frictional
contacts, the phase-field method has two attractive features: (1) it can
represent arbitrary crack geometry without an explicit function or basis
enrichment, and (2) it does not require an algorithm for imposing contact
constraints. The first feature, which is common in phase-field models of
fracture, is attained by regularizing a sharp interface geometry using a
surface density functional. The second feature, which is a unique advantage for
contact problems, is achieved by a new approach that calculates the stress
tensor in the regularized interface region depending on the contact condition
of the interface. Particularly, under a slip condition, this approach updates
stress components in the slip direction using a standard contact constitutive
law, while making other stress components compatible with stress in the bulk
region to ensure non-penetrating deformation in other directions. We verify the
proposed phase-field method using stationary interface problems simulated by
discrete methods in the literature. Subsequently, by allowing the phase field
to evolve according to brittle fracture theory, we demonstrate the proposed
method's capability for modeling crack growth with frictional contact.
