  Important physical observations in rupture dynamics such as static fault
friction, short-slip, self-healing, and supershear phenomenon in cracks are
studied. A continuum model of rupture dynamics is developed using the field
dislocation mechanics (FDM) theory. The energy density function in our model
encodes accepted and simple physical facts related to rocks and granular
materials under compression. We work within a 2-dimensional ansatz of FDM where
the rupture front is allowed to move only in a horizontal fault layer
sandwiched between elastic blocks. Damage via the degradation of elastic
modulus is allowed to occur only in the fault layer, characterized by the
amount of plastic slip. The theory dictates the evolution equation of the
plastic shear strain to be a Hamilton-Jacobi (H-J) equation, resulting in the
representation of a propagating rupture front, which is fully coupled to
elastodynamics in the whole domain. Our simulations recover static friction
laws as emergent features of our model, without putting in by hand any such
discontinuous criteria. Estimates of material parameters of cohesion and
friction angle are deduced. Short-slip and slip-weakening (crack-like)
behaviors are also reproduced as a function of the degree of damage behind the
rupture front. The long-time behavior of a moving rupture front is probed, and
it is deduced that the equilibrium profiles under no shear stress are not
traveling wave profiles under non-zero shear load. However, it is shown that a
traveling wave structure is likely attained in the limit of long times.
Finally, a crack-like damage front is driven by an initial impact loading, and
it is observed in our simulations that an upper bound to the crack speed is the
dilatational wave speed of the material unless the material is put under
pre-stressed conditions, in which case supersonic motion can be obtained.
