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[SCG50-18] Reconsideration of the energy balance in earthquake faulting - I: The origin of the background stress field
Keywords:earthquake faulting, elastic potential energy, gravitational potential energy, background stress field, inelastic strain
The occurrence of earthquakes can be regarded as brittle shear fracture at a fault, which releases a part of the total potential energy of the earth. Since the 1950s, many researchers have studied the energy balance in earthquake faulting, but there seems to be some incoherence among these studies. The essential reason is probably in various scientific developments during the last half-century; the introduction of the new paradigm of plate tectonics, the concept of moment tensor as source representation, and the fault constitutive law governing shear rupture. So, it will be worthwhile to reconsider the energy balance in earthquake faulting from the current perspective. In the first presentation of this study, we focus on the static energy balance and discuss the origin of the background stress field in the earth.
At the end of the 1950s, Steketee (1958) introduced the elasticity theory of dislocation into geophysics and considered the static energy balance for the formation of dislocation (tangential displacement discontinuity) in a prestressed elastic body bounded by a traction-free surface. He concluded that the strain energy is always increased, independently of the initial stress state, when a dislocation is formed (Steketee's paradox). An essential question is how the prestressed state can be made in the isolated elastic body. At the end of the 1960s, Savage (1969) gave a geophysical interpretation of the interaction energy between two stress systems (Eshelby, 1956) and shown that the formation of dislocation can decrease the elastic energy of the total system including a loading machine to produce the prestressed state. To sum up, in the case of the non-gravitating earth model, the release of elastic potential energy balances with the work done for shear faulting. I will give a straightforward derivation of the energy balance equation mentioned above. The point is that the loading process must be inelastic as well as the formation of dislocation, because the earth is an isolated body.
In reality, the earth is a self-gravitating body, and so we should consider the change in gravitational potential energy as well as the change in elastic potential energy. In the case of the self-gravitating earth model, as shown by Kostrov (1974) and Dahlen (1977), the release of the total (elastic and gravitational) potential energy balances with the work done for shear faulting. In their formulations, the total release of the gravitational potential energy is represented by the volume integral of the scalar product of gravity force and coseismic displacement over the entire earth. Savage & Walsh (1978) found that this volume integral could be transformed into the surface integral of the scalar product of gravity-origin deviatoric stress vector and tangential displacement discontinuity vector over the fault. The point is that the isotropic part of gravity-origin stress is independent of the work done for shear faulting. Anyway, they evaluated the change in gravitational potential energy due to earthquake faulting in a homogeneous, isotropic, elastic earth model, and concluded as follows: For dip-slip faulting in the earth, the gravitational energy change may be several orders of magnitude greater than the energy released; that large change in gravitational energy is compensated for by a similar change in stored elastic initial energy. If it is true, to evaluate the coseismic change in elastic potential energy alone will be meaningless.
In actuality, considering the rheological property of the earth's mantle and the steady seafloor spreading and oceanic plate subduction in long time scale, the stress field caused by self-gravitation must be nearly isotropic, and so it affects the strength of faults but not the energetics of earthquake faulting substantially. In other words, we may use the energy balance equation for the non-gravitating earth model in a good approximation, where the background deviatoric stress field is tectonic origin.
At the end of the 1950s, Steketee (1958) introduced the elasticity theory of dislocation into geophysics and considered the static energy balance for the formation of dislocation (tangential displacement discontinuity) in a prestressed elastic body bounded by a traction-free surface. He concluded that the strain energy is always increased, independently of the initial stress state, when a dislocation is formed (Steketee's paradox). An essential question is how the prestressed state can be made in the isolated elastic body. At the end of the 1960s, Savage (1969) gave a geophysical interpretation of the interaction energy between two stress systems (Eshelby, 1956) and shown that the formation of dislocation can decrease the elastic energy of the total system including a loading machine to produce the prestressed state. To sum up, in the case of the non-gravitating earth model, the release of elastic potential energy balances with the work done for shear faulting. I will give a straightforward derivation of the energy balance equation mentioned above. The point is that the loading process must be inelastic as well as the formation of dislocation, because the earth is an isolated body.
In reality, the earth is a self-gravitating body, and so we should consider the change in gravitational potential energy as well as the change in elastic potential energy. In the case of the self-gravitating earth model, as shown by Kostrov (1974) and Dahlen (1977), the release of the total (elastic and gravitational) potential energy balances with the work done for shear faulting. In their formulations, the total release of the gravitational potential energy is represented by the volume integral of the scalar product of gravity force and coseismic displacement over the entire earth. Savage & Walsh (1978) found that this volume integral could be transformed into the surface integral of the scalar product of gravity-origin deviatoric stress vector and tangential displacement discontinuity vector over the fault. The point is that the isotropic part of gravity-origin stress is independent of the work done for shear faulting. Anyway, they evaluated the change in gravitational potential energy due to earthquake faulting in a homogeneous, isotropic, elastic earth model, and concluded as follows: For dip-slip faulting in the earth, the gravitational energy change may be several orders of magnitude greater than the energy released; that large change in gravitational energy is compensated for by a similar change in stored elastic initial energy. If it is true, to evaluate the coseismic change in elastic potential energy alone will be meaningless.
In actuality, considering the rheological property of the earth's mantle and the steady seafloor spreading and oceanic plate subduction in long time scale, the stress field caused by self-gravitation must be nearly isotropic, and so it affects the strength of faults but not the energetics of earthquake faulting substantially. In other words, we may use the energy balance equation for the non-gravitating earth model in a good approximation, where the background deviatoric stress field is tectonic origin.