Thermal runaway and the challenges of rapid localization

Thermal runaway is a ductile weakening mechanism that describes the feedback loop of shear heating, temperature-dependent viscosity and localization. It has been linked to deep-focus earthquakes which are unlikely to be caused by brittle failure due to the large lithostatic pressure. We present one- and two-dimensional (1D and 2D) numerical, thermomechanical models that investigate the occurrence, nucleation and temporal evolution of thermal runaway in a simple shear setting. The models are characterized by a visco-elastic rheology where viscous creep is accommodated with a composite rheology of diffusion and dislocation creep as well as low-temperature plasticity. We implement the model in the Julia programming language and utilize the pseudo-transient iterative method to solve this nonlinear system of equations. Graphical processing unit (GPU) computing (by making use of the package ParallelStencil.jl) allows us to achieve high resolution models in 2D. Like brittle plasticity, thermal runaway presents the challenge of modeling a very thin shear band in a continuum mechanics approach with finite resolution. To address this issue, we tested two different regularization techniques to provide stable solutions and introduce a grid-independent shear zone width. A viscosity regularization and a second-order gradient regularization achieve similar results. Loading and heating time scales of thousands of years in combination with relaxation time scales on the order of seconds also require an adaptive time stepping scheme that can span 12 orders of magnitude. We achieve this by adjusting time steps to the maximum gradients in temperature and stress, and by dynamically rescaling variables during computation to minimize rounding errors. Combining the aforementioned techniques allows us to cover loading and heating on geological time scales as well as near-instantaneous stress drop and local temperature surge in one model framework. 2D experiments show that thermal runaway allows highly localized ductile ruptures to nucleate at small heterogeneities and propagate like brittle fractures. The ruptures accelerate during propagation and reach the highest velocities when two tips link up. Rupture trajectories are usually parallel to the direction of background deformation but bend in the vicinity of other ruptures to allow for a link up.