We present a class of models for elastoplastic material behavior for
geodynamical processes in the Earth's crust and upper mantle. The rock
material can be treated as a viscoelastoplastic fluid (with internal
variables), where velocities range in the regime of millimeters per year. For
modeling the slow evolution of crustal faults between different lithospheric
plates, these plates can be understood as prescribed boundary data.
In a 3D model (studied with Th. Eiter and K. Hopf, 2022) we obtain weak
solutions for a coupled system of Navier-Stokes equations with a convected
internal viscoelastoplastic stress tensor. For simplified 1D model (studied
with T. Roubicek, 2024) we show existence and obtain localization and
instability of steady shear flows. Numerical simulations show periodic behavior
where sticking phases alternate with slipping phases quite similar to the
so-called "episodic tremor and slip".