7 .3 Discussion
8.3 Overview of Rupture Dynamics
Equation (8.14) does not account for non-zero pore pressures. For hydrostatic pore pressures, we simply replace the mass density at each point in equation (8.14) with the difference between the mass density and the mass density of water.
positive z Zo • Po
I
Zt • Pt Z2 • P2Zj-1 • PJ-1 z
Zj • PJ
ZN-1 • PN-1 ZN • PN
Figure 8.3: Control points that define the piecewise linear variation of the mass density with depth.
We compute the normal stresses at the location denoted by the open circle with coordinate z.
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not recover as the slip rate decreases. Of course, in the earth and our finite-element models failure occurs at a finite value which prevents the formation of a true singularity.
Stress
Slipping region
Direction of propagation
Position
Stress
Slip begins /
"Distance" / from Failure /
Initial Stress
. · 1· D~n~mi~
.t ~~:;s
Stress
,_..._...;.l ___ _
Drop " - . Slip ends
Time Figure 8.4: Diagrams of the concentration of shear stress near the rupture front as a function of space (left diagram) and as a function of time (right diagram).
The friction model controls the decrease in friction stress as slip progresses, and therefore, the dynamic stress drop. The rate of the dynamic stress drop governs the slip rate with faster decreases in shear stress leading to faster increases in slip. The dynamic stress drop and the distance from failure (the difference between the failure stress and the initial shear stress) determine the nature of the concentration of stress ahead of the leading edge of the rupture. A larger distance from failure magnifies the stress concentration, and a larger dynamic stress drop increases the rate of the decay of the stress concentration. The increase in shear stress associated with the stress concentration dictates when slip occurs at each point and, as a result, the rupture speed. Thus, the slip rate and rupture speed are related through the dynamic stress drop.
8.3.2 Energy and Rupture Dynamics
We may also study the dynamics of the rupture using energy. The increase in shear stress on the fault ahead of the rupture implies storage of strain energy in the surrounding region. As the rupture propagates, the rupture front consumes energy through sliding. We associate two forms of energy with the sliding. We call the energy dissipated during the decrease in the friction during sliding the fracture energy, because it corresponds to the fracture energy in crack models. We associate the energy dissipated through sliding at a relatively constant friction stress with the change in thermal energy. The sliding also generates the energy radiated in the seismic waves. As we increase the fracture energy, the rupture consumes more energy leaving less available for sliding. In such cases the slip rates and rupture speed decrease (Fukuyama and Madariaga 1998). Likewise, when we decrease the fracture energy, more energy is available for sliding, and the slip rates and rupture
dG ()1dD, (8.15)
where () f is the friction stress. Expanding the friction stress about the failure t:iLress (Of ail) Lu firt>L
order in dD and substituting yields
(8.16) This expression shows that the failure stress does not uniquely determine the fracture energy. We may adjust the slope of the friction model to change the fracture energy. For example, if we lower the failure stress, we may maintain the same fracture energy by reducing the rate at which the friction stress decreases with slip.
This technique plays a critical role in manipulating the dynamics of the rupture in the finite- element simulations. We want the wave propagation to govern the local element sizes. However, accurately capturing the stress concentration in shear stress near the leading edge of the rupture requires much smaller elements than those necessary to model the wave propagation (Madariaga et aL 1998). With extremely high resolution meshes, the failure stress will develop only over a very localized region. We want to capture the general behavior of such failure without modeling such localized behavior. As we increase the element size, the concentration in shear stress decreases for a given dynamic stress drop. In other words, the buildup of stress becomes distributed over a longer length, which reduces the stress concentration. Consequently, we must reduce the failure stress for the rupture to propagate. Thus, the ability of a continuous medium to generate nearly singular stresses near the leading edge of the rupture front means the fracture energy, not the precise level of the stress at failure, governs the propagation of the rupture. On the other hand, in a discrete model, such as our finite-element models, the failure stress becomes a length-scale dependent parameter, but the fracture energy continues to control the behavior of the rupture. Luckily, we may manipulate the friction model as demonstrated in figure 8.5 to maintain the same fracture energy as we change the failure stress. This means that we may use larger elements than those required to accurately capture the stress concentration and may allow the wave propagation to control the discretization size without altering the behavior of the rupture.
D D~ D~
Figure 8.5: Illustration of two sets of parameters for the slip-weakening friction model (denoted by the superscripts a and b) that have the same fracture energy with different failures stresses ( O"fail
and CJ.~ail) and characteristic slip distances (D~ and D~).