A brief discussion on the relationship between apparent stress and slip-weakening law based on the energy partition criteria

According to the representation theorem of seismic energy radiation, we know that, at any point on the fault, the instantaneous seismic radiation energy, <i<E</i<<sub<S</sub<, the seismic moment <i<M</i<<sub<0</sub<, and the apparent stress <i<σ</i<<sub<a</sub<=<i<μ</i<<i<E</i<<sub<S</sub</<i<M</i<<sub<0</sub< (<i<μ</i< is the shear modulus on the fault plane) should be positive values at any time during an earthquake faulting. However, we have noticed that, in recent source parameter inversion scheme for deriving the critical slip-weakening distance, the apparent stress used as a constraint condition on the fault plane could be less than zero or negative value, and the negative part was considered as dissipation energy and incorporated to the fracture energy. Although the mathematical formula in such case has no influence to the final resolution, however, the earthquake dynamic source process violates obviously the basic physical law, which could results in the overestimating of radiated seismic energy. In this study, we have proposed an alternative way to take account of the apparent stress expression based on the earthquake energy partition principle, and at the same time, we have also suggested that an additional constraint regarding to the radiation energy conservation on the fault could be added into the source parameter inversion in order to estimate the critical slip-weakening distance <i<D</i<<sub<c</sub<.