Mathematical Analysis of Continuum Mechanics and Industrial Applications II by Patrick van Meurs Masato Kimura & Hirofumi Notsu

Author:Patrick van Meurs, Masato Kimura & Hirofumi Notsu
Language: eng
Format: epub
Publisher: Springer Singapore, Singapore

4 Numerical Implementation for Propagating Ruptures Obeying Frictions

In this section, I will provide a slightly technical description on numerical implementation. The governing equation is given by simultaneous equations constituted by BIEs

(13)

and the boundary conditions as

(14)

where (, and denote, respectively, the pth component of the vectors of the traction change, and the initial and the residual tractions acting in the direction ( or 2) on the ith element having the normal vector of . Our problem is to evaluate the shear traction around the crack tip and to obtain the slip rate at each time step under the mixed boundary condition given in Eq. (13). In seismological applications, faults are subjected to a compressional stress regime and the residual traction is described by friction. Note that shear dislocation in one direction on a planar element does not instantaneously change the normal traction, as well as the shear traction acting in the perpendicular direction. Therefore, the current piecewise planar discretization allows us to avoid the instantaneous effect of the normal traction on the friction, meaning , where and denote, respectively, the frictional coefficient and a constant cohesive strength.

The widely used friction law for the dynamic earthquake rupture simulation is called the linear slip-weakening law [7, 15], which can reproduce seismological observations (e.g., [16]) and the first-order characteristics of rock experiments referred as the rate- and state-dependent friction law (e.g., [27]). The frictional strength is decreased as the function of the amount of the slip with the characteristic slip distance as , if or , if ; by considering this equation, Eqs. (13) and (14) become a closed simultaneous equation with the two unknowns and , which can be solved explicitly. Additionally, the fracture criterion is given by the condition outside of the area be ruptured at the next time step.

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