2:45 PM - 3:00 PM
[SCG56-16] Reconsideration of the energy balance in earthquake faulting – III: Rupture growth rate
Keywords:Earthquake faulting, wave energy radiation, strain energy release, slip-weakening friction law, rapture growth rate
The occurrence of earthquakes can be regarded as brittle shear fracture at a fault, which releases a part of the elastic 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 them. The purpose of this study is to reconsider the energy balance in earthquake faulting from the current perspective.
The earth is a self-gravitating body, and so we should include a change in gravitational potential energy together with a change in elastic potential energy in the calculation of energy balance in earthquake faulting (Kostrov, 1974; Dahlen, 1977). In Part I of this study (SCG50-18, JpGU2021), we considered the origin of the background stress field in the earth in the framework of plate tectonics and obtained the following conclusions. The stress field caused by self-gravitation is nearly isotropic, and so it does not affect energetics in shear faulting substantially. Then, in quasi-static problems, the decrease of elastic potential energy balances with the work done for shear faulting.
In Part II (SCG52-24, JpGU2022), we considered the mechanical energy balance of a non-gravitating earth model in dynamic shear faulting and obtained the following conclusions. The general expression of energy balance can be derived from basic equations in continuum mechanics as K (kinetic energy) = ΔE (released elastic potential energy) – ΔW (work done for shear faulting). So far, a simplified expression, K/S = ½(τi+τf)Df - ∫0Dfτ(D)dD, has been used for evaluating the radiated seismic energy. However, this expression is misleading, because the direct evaluation of K based on elastic dislocation theory shows its dependence on the rupture growth rate.
In Part III, we consider the physical process of rupture growth governed by a slip-weakening friction law (e.g., Matsu'ura et al, 1992). After the quasi-static nucleation process, fault slip with strength weakening starts and gradually accelerates. The fault slip generates dynamic disturbance, categorized into near-, intermediate-, and far-field terms, in its surrounding region. The far-field term radiates from the source as traveling waves, but the near- and intermediate-field terms remain as permanent deformation, after the disturbance died down, and cause static stress changes. In linear elasticity, the change in stress is independent of the background stress field, but the change in elastic potential energy is not. The important thing is that the growth rate of shear rupture is controlled by the inflow rate of released shear strain energy into the rupture front (e.g., Aki & Richards, 1980). When the inflow rate of shear strain energy is greater (smaller) than the fracture surface energy, the growth of rupture is accelerated (decelerated). In other words, ΔW in the general expression depends on the time history of rupture growth with fault-slip acceleration and deceleration. As a special case, we consider quasi-static rupture growth. In this case, the work done for shear faulting must always balance with the released shear strain energy because of no seismic wave radiation. This means that we can use the shear stress–fault slip curve τ=fss(D) in the quasi-static case as a reference. In the case of normal earthquakes, the dynamic rupture accelerated to a terminal velocity is forcibly arrested by the existence of strong barriers. Even after the arrest of dynamic rupture, the shear strain energy released at a distant place in the past will continue to flow into the source region for a short while. As a result, the stress level inside the ruptured area decreases gradually, and the fault slip overshoots (e.g., Madariaga, 1976). Including this adjustment process, we represent the shear stress–fault slip curve in the dynamic case by τ=fdy(D). Then, we can evaluate the radiated seismic energy as -K/S = ∫0Df[fdy(D)-fss(D)]dD, which means the wave energy outgoing from the mechanical system.
The earth is a self-gravitating body, and so we should include a change in gravitational potential energy together with a change in elastic potential energy in the calculation of energy balance in earthquake faulting (Kostrov, 1974; Dahlen, 1977). In Part I of this study (SCG50-18, JpGU2021), we considered the origin of the background stress field in the earth in the framework of plate tectonics and obtained the following conclusions. The stress field caused by self-gravitation is nearly isotropic, and so it does not affect energetics in shear faulting substantially. Then, in quasi-static problems, the decrease of elastic potential energy balances with the work done for shear faulting.
In Part II (SCG52-24, JpGU2022), we considered the mechanical energy balance of a non-gravitating earth model in dynamic shear faulting and obtained the following conclusions. The general expression of energy balance can be derived from basic equations in continuum mechanics as K (kinetic energy) = ΔE (released elastic potential energy) – ΔW (work done for shear faulting). So far, a simplified expression, K/S = ½(τi+τf)Df - ∫0Dfτ(D)dD, has been used for evaluating the radiated seismic energy. However, this expression is misleading, because the direct evaluation of K based on elastic dislocation theory shows its dependence on the rupture growth rate.
In Part III, we consider the physical process of rupture growth governed by a slip-weakening friction law (e.g., Matsu'ura et al, 1992). After the quasi-static nucleation process, fault slip with strength weakening starts and gradually accelerates. The fault slip generates dynamic disturbance, categorized into near-, intermediate-, and far-field terms, in its surrounding region. The far-field term radiates from the source as traveling waves, but the near- and intermediate-field terms remain as permanent deformation, after the disturbance died down, and cause static stress changes. In linear elasticity, the change in stress is independent of the background stress field, but the change in elastic potential energy is not. The important thing is that the growth rate of shear rupture is controlled by the inflow rate of released shear strain energy into the rupture front (e.g., Aki & Richards, 1980). When the inflow rate of shear strain energy is greater (smaller) than the fracture surface energy, the growth of rupture is accelerated (decelerated). In other words, ΔW in the general expression depends on the time history of rupture growth with fault-slip acceleration and deceleration. As a special case, we consider quasi-static rupture growth. In this case, the work done for shear faulting must always balance with the released shear strain energy because of no seismic wave radiation. This means that we can use the shear stress–fault slip curve τ=fss(D) in the quasi-static case as a reference. In the case of normal earthquakes, the dynamic rupture accelerated to a terminal velocity is forcibly arrested by the existence of strong barriers. Even after the arrest of dynamic rupture, the shear strain energy released at a distant place in the past will continue to flow into the source region for a short while. As a result, the stress level inside the ruptured area decreases gradually, and the fault slip overshoots (e.g., Madariaga, 1976). Including this adjustment process, we represent the shear stress–fault slip curve in the dynamic case by τ=fdy(D). Then, we can evaluate the radiated seismic energy as -K/S = ∫0Df[fdy(D)-fss(D)]dD, which means the wave energy outgoing from the mechanical system.