Differences in breakdown work and fracture energy in slip weakening constitutive laws

Earthquakes are associated with the propagation of a dynamic rupture, which radiates elastic energy through seismic waves. The generation of seismic radiation is related to dynamic weakening of shear stress and stress drop. In modeling dynamic ruptures, shear stress evolution is commonly imposed through a constitutive law, such as the widely adopted slip weakening laws. According to these constitutive laws, shear stress evolves as a function of slip in each point of the rupturing fault, prescribing strength excess, stress drop and dynamic weakening.

Here, we compare two well-known slip weakening laws: namely, the classic Ida’s (1972) and the Ohnaka’s (1996) slip weakening laws. The former prescribes that fault stress increases from the initial stress to the peak stress with zero slip and then linearly decreases from the peak value to a residual value over a slip-distance Dc (dynamic weakening). The latter assumes that the initial stress hardening phase occurs over a non-negligible slip-distance Da and that shear stress decrease from the peak value is not linear. The Ohnaka’s law was validated with numerous laboratory experiments. The evolution of shear stress with slip allows the estimate of the breakdown work Wb, i.e. the excess of work above a minimum stress level with slip from 0 to Dc.

We collected data from high-velocity friction experiments to quantify yield, peak and residual stresses, Da and Dc distances for bare-rock samples of Carrara Marble and Gabbro deformed under various experimental conditions (room humidity, vacuum, pressurized fluids) and normal stress (from 5 to 40 MPa). The ratio Da/Dc is much lower for Carrara marble (0.015) than for Gabbro (0.12). We implemented the Ohnaka’s constitutive law in a 2D finite difference code for spontaneous dynamic ruptures characterized by a fault in a homogeneous elastic material.  We perform simulations using the two different slip weakening laws. We kept constant Dc, and we compared the results of the simulations in terms of rupture style, rupture velocity, breakdown work, and cohesive zone size. As expected both laws yield crack-like ruptures. Moreover, Ohnaka’s law in comparison to the linear slip weakening law produces:

• rupture velocity ~2 % higher;
• breakdown work (Wb) up to 60 % lower. Moreover, dividing the breakdown work into the energy dissipated between the yield stress and the peak stress over the slip-distance Da (Wb_a), we notice that Wb_a can reach up to the 30% of the total Wb in case of Gabbro (Da/Dc = 0.12).
• a cohesive zone size (defined as the portion of the fault in which the slip velocity is higher than zero and the stress is higher than its residual value) up to 50% larger.

Therefore, Ohnaka’s law generates more energetic ruptures (i.e. faster rupture velocity and peak slip-rate) despite having a larger cohesive zone due to the lower breakdown energy dissipated during rupture propagation. We discuss our results in terms of the difference between breakdown highlighting the implications on dynamic rupture propagation and earthquake energy budget. We emphasize that common interpretations of energy dissipated during rupture propagation are model-dependent.