A note about the rate-and-state-dependent friction model in a thermodynamic framework of the Biot-type equation

The conventional, phenomenological rate-and-state-dependent friction model of Dieterich-Ruina's type is discussed and slightly modified so that, after introducing an artificial internal variable (formally in a position like effective temperature) on the fault, it is driven by a free and a dissipative energies. In contrast to the original model, it thus allows for a formulation in the framework of rational thermodynamics, including the energy balance, and for rigorous numerical analysis. This also suggests an analogous rate-and-state-dependent plastic bulk model using damage/temperature as the state variable controlling the plastic yield stress.