Incorporating elastic and plastic work rates into energy balance for long-term tectonic modeling

Abstract:

Deformation-related energy budget is usually considered in the simplest form or even completely omitted from the energy balance equation. We derive an energy balance equation that accounts not only for heat energy but also for elastic and plastic work. Such a general description of the energy balance principle will be useful for modeling complicated interactions between geodynamic processes such as thermoelastisity, thermoplasticity and mechanical consequences of metamorphism. Following the theory of large deformation plasticity, we start from the assumption that Gibbs free energy (g) is a function of temperature (T), the second Piola-Kirchhoff stress (S), density (ρ) and internal variables (q_j, j=1…n). In this formulation, new terms are derived, which are related to the energy dissipated through plastic work and the elastically stored energy that are not seen in the usual form of the energy balance equation used in geodynamics. We then simplify the generic equation to one involving more familiar quantities such as Cauchy stress and material density assuming that the small deformation formulation holds for our applications. The simplified evolution equation for temperature is implemented in DyanEarthSol3D, an unstructured finite element solver for long-term tectonic deformation. We calculate each of the newly derived terms separately in simple settings and compare the numerical results with a corresponding analytic solution. We also present the effects of the new energy balance on the evolution of a large offset normal fault.