Dynamic Ruptures with Off-fault Plastic Yielding

This work is primarily done with my PhD advisor Yehuda Ben-Zion and Jean-Paul Ampuero from Caltech. We implemented the Spectral Element code (SEM2DPACK, developed by Ampuero) with a Mohr-Coulomb type plasticity to numerically study off-fault plastic yielding during 2-D in-plane dynamic ruptures. In the framework of commonly-used elasto-plasticity, the elastic moduli do not change with the accumulation of plastic strain while the plastic hardening modulus or other rock properties (e.g. internal friction angle, rock cohesion) can evolve with the on-going plastic deformation. For simplicity, a rate-dependent visco-plasticity with constant plastic properties is employed to produce smoothly distributed plastic strain. Detailed results on plastic yielding zone properties during dynamic ruptures and the comparison to other studies focusing on quasi-static fault propagation or fault roughness can be found in Xu et al. (2012a) and Xu et al. (2012b).

Figure 1. Plastic strain distribution for (a) a crack-like rupture and (b) a pulse-like rupture (after initial nucleation phase) under similar loading stress conditions (with the background maximum compressive stress inclined at 45o to the fault plane) but with different frictional laws. The corresponding slip and slip rate profiles plotted at different times are shown in (c) for the crack case and in (d) for the pulse case.

Dynamic Ruptures with Off-fault Brittle Damage

Unlike plasticity, the brittle damage model can account for dynamic changes of elastic moduli inside the off-fault yielding zone. The basic theory was formulated by Vladimir Lyakhovsky, Yehuda Ben-Zion and their collaborators on the basis of thermodynamics accounting for internal irreversible process (e.g. permanent deformation in rocks), and based on laboratory results that the relevant elastic modulus of rocks under compression is larger than that under tension (i.e. the effective elastic modulus depends on the density of internal distributed microcracks). The model was first used to study evolution of fault zone layers under quasi-static loadings and then was numerically implemented by Jean-Paul Ampuero and myself to focus on dynamic rupture problems. To simplify, the model assumes that the elastic moduli have a dependence on a scalar variable \alpha (the damage variable) that macroscopically describes the density of distributed microcracks inside rocks. It further assumes how \alpha can evolve under the current deformation state (described by a ratio \xi of two elastic strain invariants) by obeying the principle of non-negative entropy production. More details about this model and its applications can be found here and in our recently presented poster.

Figure 2. Schematic diagram showing the path of brittle damage evolution in (a) \xi-\alpha phase space and (b) stress-strain space. The damage growth is damped by a buffer zone near the critical state (see the grey belt regions in (a)), after which a macroscopic failure will occur associated with a considerable amount of stress (and strain) drop. The dashed arrows in (b) indicate the possible unloading path after the onset of damage accumulation, with or without reaching the ultimate stress level.

Figure 3. (a) Damage distribution for a rupture case with slip-weakening friction (SWF) over a long propagation distance. (b)-(d) Slip rate, shear and normal stress change across the fault for the case shown in (a). With the increase of rupture propagation distance, the rupture can evolve from classic crack-like to a rupture mode with a detached pulse front from the remaining freely-slipping zone, due to the progressive increase of dynamic normal stress change related with the asymmetric distribution of off-fault brittle damage near the rupture front and across the fault. It is interesting to compare this case with off-fault brittle damage with the shown cases in Fig. 1 with off-fault plasticity.

Other Problems

In addition to the above mentioned problems, I am also interested in many others. They include trapped wave imaging on fault zone structure, theoretical study on wave propagation through faults with vertical layers by using the Cagniard-de Hoop method, and on dynamic rupture propagation with mixed boundary conditions by using the Wiener-Hopf method. Thanks to the Southern California Earthquake Center (SCEC) through which I have learned a lot from others on observational seismology and geodesy, and I wish in the future I can also work on these topics.