Physics of the Earth and Planetary Interiors 157 (2006) 151–163 Time-dependent strain localization in viscous media with state-dependent viscosity C.F. Hieronymus ∗ Institute of Geophysics, ETH Z ¨ urich, Switzerland Received 24 April 2005; received in revised form 15 March 2006; accepted 20 March 2006 Abstract 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 10 5 –10 6 years. ©2006 Elsevier B.V. All rights reserved. Keywords: Rheology; Shear localization; Ductile creep; Transient creep; Shear instability 1. Introduction Localization is a common geological phenomenon occurring over a broad range of scales, from the concen- tration of tectonic deformation along plate boundaries down to shear bands in hand samples of rock and even below the crystal scale. The mere fact that deformation is often concentrated into narrow zones hints that such zones are of great dynamical importance, and it is thus not surprising that shear localization has been the subject of numerous studies. ∗ Present address: Department of Earth Sciences (Geophysics) Uppsala University, Villav¨ agen 16, SE-752 36, Sweden. Fax: +46 18 501 110. E-mail address: christoph.hieronymus@geo.uu.se (C.F. Hieronymus). In recent years, there have been significant advances in geodynamical modeling of shear zones (Bercovici and Karato, 2003; Regenauer-Lieb and Yuen, 2003; Tackley, 1998; Fleitout and Froidevaux, 1980; Yuen et al., 1978). But while we are approaching our goals of dynamically and self-consistently modeling localization processes such as continental rifting and plate tectonics, there is much disagreement about the appropriate rheological laws. The most successful models in terms of matching the geological observations typically employ ad-hoc as- sumptions about material softening. These models may be seen as an inverse approach of using observations of localization behavior to determine the mathematical form of a suitable rheological law. The other approach is to use empirically determined rheological parameteriza- tions. Recent studies demonstrate that these rheologies 0031-9201/$ – see front matter ©2006 Elsevier B.V. All rights reserved. doi:10.1016/j.pepi.2006.03.020