# Variational methods for problems of rate- and state-dependent friction

Dr. Elias Pipping

GeoSim

The model of rate- and state-dependent friction is ubiquitous in geophysics. Even though it is not derived from first principles but phenomenological in nature, it has found applications to a great many materials. Mathematically, it is poorly understood, however. To be precise, there appears to be a drastic gap:

Very simple problems like the so-called spring-block slider, where a spring is attached to a rigid block and then pulled, causing the block to slide over a rigid surface, have been thoroughly investigated and are well understood.

Then there are geometrically very complex problems, for which complex models are used, that are then solved numerically using complex algorithms (by geophysicists). A theory of existence of solutions and convergence of the corresponding algorithms does, to our knowledge, not exist.

As a part of this project, problems are studied that fall in the gap between those two areas. We consider bodies that are themselves elastic (i.e., no spring is necessary to artifically introduce elasticity and thus waves into the system). This setup gives rise to coupled variational equations (VEs), and thus objects that are, from an analyst's point of view, quite pleasant. Consequently, some results on the existence and uniqueness of solutions have been obtained. The VE-formulation furthermore leads to convex minimisation problems, for which fast and robust numerical solvers (namely Truncated Nonsmooth Newton Multigrid (TNNMG)) have been developed by the working group.

In summary, we seek to bridge the gap between theory and application through formulations that enable us to make statements about solutions to our problems, as well as to devise and implement algorithms, that allow us to solve said problems in a fast and robust manner.