A physical model for earthquakes: 3. Thermodynamical approach and its relation to nonclassical theories of nucleation

If a theory of earthquakes, sliding, and frictional slip is to have general validity, it should ideally arise out of fundamental thermodynamical considerations. We construct just such a theory using the principle that physical systems, including fault systems, tend toward a state of minimal Helmholtz free energy. If the system is not in a state of minimal free energy, thermodynamic forces arise which drive the system toward a minimum. Allowed minima in the free energy may be either global, representing static (absolute) equilibrium, or local, representing states of metastable equilibrium. Within the context of this theory, fault systems are understood to be in states of metastable equilibrium, kept there by action of the far-field plate-driving stresses, which continually keep the far-field displacement ahead of the state of slip on the fault. Earthquakes therefore represent a sudden decay of the fault system from a state of metastable equilibrium to a state closer to static (absolute) equilibrium. The major difficulty is the formulation of an appropriate expression for the Helmholtz free energy of the system. The solution of this problem is facilitated by starting from classical nucleation theory, which was first formulated by Gibbs for general thermodynamical systems and later applied by Griffith for the specific case of the fracture of solids. Much more recently, significant advances have been made in understanding the physics of nucleation. In accordance with these ideas, in this paper the free energy is therefore written as a sum of two terms: one which describes the elastic potential energy contained by the solid (i.e., the internal energy) and one which describes the energy necessary to rupture and/or reconnect atomic and cohesive bonds (i.e., an entropy term). The former term describes thermodynamically reversible processes, and the latter more generally describes thermodynamically irreversible processes. In the Griffith theory of fracture the latter term is a surface energy term, since the creation of an element of surface requires the rupture of atomic or cohesive bonds. Together, these two terms most generally define a nonclassical free energy functional, which, when minimized, leads to two equations. The first equation is the classical equation obtained by Gibbs and Griffith, which relates the size of the slip region to the applied field. The second equation is a nonclassical Euler-Lagrange equation, which can be solved to obtain the distribution of slip on the slip surface. In addition to developing these ideas, it is shown that the assumption of thermodynamic irreversibility, that is, nonnegative entropy production, leads to a specific requirement for the kinetic growth of slip and the growth of the slip region. A discussion is then given of how these ideas relate to the kind of hysteresis effects commonly observed in frictional experiments and of how these effects may be interpreted in terms of the physics of states of metastable equilibrium.