Introduction to Sabber’s JGR manuscript
The energy conservation principle in its general form can consistently describe a wide range of geological and geodynamic phenomena, because the general form accounts for the energetics of deformation as well as the usual heat advection and diffusion. For instance, Hunt et al. (1991) show that materials softening in proportion to strain rate have a non-convex Helmholtz energy with strain as an independent variable. According to their analysis, non-periodic localized folding can be viewed as a superposition of folding modes that correspond to multiple local minima of the non-convex energy function. Also concerned about folding, Hobbs et al. (2011) propose that the feedback between shear heating and temperature-dependent viscosity can explain folding occurring in layers with small viscosity contrast, where the classical Biot’s theory (19XX?) predicts that folding would not occur. Dissipative energetics has been considered in the lithospheric scale as well. Regenauer-Lieb et al. (2006) consider the feedback between the energy dissipated due to inelastic deformation and changes in viscous and plastic material properties due to temperature changes resulting from the energy dissipation. They found that the two-way feedback process can make the brittle-ductile transition (BDT) zone weaker than other parts of lithosphere although the classical strength envelopes predict the BDT zone of lithosphere to be the strongest (Ranalli et al., 1987; Goetze et al., 1979; Brace et al., 1980). Energy dissipated in the form of shear heating is shown to promote a necking in the subducting slab and ultimately lead to slab detachment (Gerya et al., 2004). In the whole-mantle scale, Ita et al. (1994) explicitly include dissipative heating in the energy balance along with variable thermodynamic properties (thermal expansion coefficient, heat capacity, and latent heat) to show that vertical flow across the 660-km phase boundary can be significantly increased. Similarly, Yuen et al. (1987) emphasize that feedback between rheology and dissipative energy in mantle convection is potentially an important mechanism that can warm up the mantle by several hundred degrees above the incompressible profile.
Several theoretical works have shown how to derive a set of governing equations for a thermo-mechanical system from the general form of the thermodynamic principles. The common procedure, to be detailed later, is to relate the rate of change of the internal energy appearing in the statement of energy balance to that of thermodynamic potentials such as the Helmholtz free energy and the Gibbs free energy. Thermodynamic potentials quantify the capacity to do mechanical work in addition to heat content. For instance, the Helmholtz free energy is “the portion of the internal energy available for doing work at constant temperature” (p.263 in Malvern, 1969). The main difference between the the Helmholtz and the Gibbs free energy is whether strain is an independent variable as in the former or stress is as in the latter. Since the definitions of these thermodynamic potentials involve the product of temperature and entropy, the energy balance principle takes an intermediate form involving the time derivative of entropy. The last step in deriving the temperature evolution equation is to express the time derivative of entropy in terms of that of temperature and other variables. Regenauer-Lieb et al. (2003)a start from the energy conservation principle stated in terms of the Helmholtz free energy to derive the partial differential equation for temperature evolution as well as other equations that are coupled with it (e.g., mass conservation equation and constitutive relations) for shear zone-developing systems in geological and planetary sciences. Their final system of equations can consistently describe the feedback processes among energy, rheology and other variables such as grain size and water content in shear zone formation. Similarly, Lyakhovsky et al. (1997) show how an evolution equation for elastic damage can be derived from the energy conservation principle. They start from expressing the energy balance equation in terms of the Helmholtz free energy that has a non-dimensional variable quantifying the amount of damage along with temperature and infinitesimal elastic deformation as independent variables. By analyzing an entropy source term due to damage process in the intermediate equation for entropy evolution, they derive a damage evolution equation that is proportional to the rate of free energy change with damage.
Computational long-term tectonic modeling (LTM), concerned about the formation and long-term evolution of geological structures of various scales, has yet to fully embrace deformation energetics. Simplifications commonly made in LTM preclude consistent thermo-mechanical coupling. For instance, energy balance is considered only in terms of heat energy and thermo-mechanical feedback is realized only through temperature-dependent viscosity. In addition, long-term tectonic models often require mechanical assumptions or rheological models that have been ignored or rejected in the existing theoretical work on thermo-mechanical coupling. Reconciling the differences between the modeling conventions in LTM and the general thermodynamic frameworks is desirable.
We identify three features that are commonly adopted in LTM but still need to be incorporated into a thermodyanmic framework.
Firstly, we note that the elastic or plastic deformations are frequently assumed to be incompressible although neither required nor well-justified by the thermodynamic principles (Regenauer-Lieb et al., 2003; Regenauer-Lieb et al., 2006; Regenauer-Lieb et al., 2008; Connolly, 2009; Hobbs et al., 2011). Volumetic strain can have a significant effect on energy budget (Hunsche, 1991; Zinoviev et al., 1994) and in general, non-negligible during brittle deformations (e.g. Brace et al., 1966; Choi et al., 2013) and phase transformations (Hyndman 2003, Hetényi 2011).
Secondly, thermal stresses are often ignored even though they turn out to be a significant source of transient stresses and associated deformation (Choi et al., 2008; Korenaga, 2007). For instance, a tem