Time-dependent strain localization in viscous media with state-dependent viscosity

The temporal evolution of viscous rheologies with dependence on an additional state variable is examined. A localization measure is introduced that quantifies the change in the degree of localization in time. Three sample rheologies are analyzed in detail, each representative of a larger class of rheologies: (1) a grain-size dependent viscosity with grain growth and diminution, (2) shear heating with temperature-dependence according to the Arrhenius law, and (3) shear heating with temperature dependence in the Frank-Kamenetzky approximation. All three rheologies display stages of temporal increase and decrease of localization, depending on the initial conditions. This localization behavior is not discernible in plots of strain rate versus strain at constant driving stress which are the typical output of creep experiments. The grain-size dependence of olivine leads to effectively non-Newtonian behavior with a stress exponent of about 5. If the laws describing the grain-size evolution are applicable to the mantle, then the lower mantle and those parts of the upper mantle that are dominated by diffusion creep thus have a stronger stress-dependence than the depth range governed by dislocation creep, provided that the timescale of deformation is greater than 105105 –106106 years.