Decomposition of elastic potential energy and a rational metric for aftershock generation

The occurrence of earthquakes can be regarded as shear fracture releasing the elastic potential energy stored in the Earth. The potential energy density of linear elastic forces is generally represented in the quadratic form of strain tensor components with the fourth-order coefficient tensor of elastic stiffness. When the material is isotropic, since the stiffness tensor is expressible as a linear combination of two independent symmetric tensors, we can decompose the elastic potential energy density into two independent parts, namely the volumetric part and the shearing part. By definition, the partial derivatives of the elastic potential energy density with respect to volumetric and shearing deformations give the corresponding generalized forces in the sense of Lagrangian mechanics: specifically, one-third of the first invariant of stress tensor (equivalent to the mean stress) to volumetric deformation and the square root of the second invariant of deviatoric stress tensor (equivalent to $\sqrt {3/2} $ times the octahedral shear stress). With these generalized forces instead of the normal and tangential stresses on a specific fault plane, we correctly represented the original concept of Coulomb's failure criterion (shear failure occurs when shearing stress is equal to shearing strength) and defined energetics-based failure stress (EFS). The change in EFS associated with the occurrence of a main fracture (ΔEFS) gives a rational metric for aftershock generation, which can be reduced to previously proposed various metrics in special cases. For example, when the level of background deviatoric stress is much higher than the magnitude of coseismic stress changes, the expression of ΔEFS is reduced to a similar form to the well-known Coulomb failure stress change (ΔCFS). Even in the energetics-based metric, the effects of pore-fluid pressure changes are essential. We theoretically examined the mechanical effects of induced and enforced pore-fluid pressure changes and elucidated that the difference between them is reflected in the focal mechanisms of aftershocks.