Substantial insight into earthquake source processes has resulted from considering frictional ruptures analogous to cohesive-zone shear cracks from fracture mechanics. This analogy holds for slip-weakening representations of fault friction that encapsulate the resistance to rupture propagation in the form of breakdown energy, analogous to fracture energy, prescribed in advance as if it were a material property of the fault interface. Here, we use numerical models of earthquake sequences with enhanced weakening due to thermal pressurization of pore fluids to show how accounting for thermo-hydro-mechanical processes during dynamic shear ruptures makes breakdown energy rupture-dependent. We find that local breakdown energy is neither a constant material property nor uniquely defined by the amount of slip attained during rupture, but depends on how that slip is achieved through the history of slip rate and dynamic stress changes during the rupture process. As a consequence, the frictional breakdown energy of the same location along the fault can vary significantly in different earthquake ruptures that pass through. These results suggest the need to reexamine the assumption of predetermined frictional breakdown energy common in dynamic rupture modeling and to better understand the factors that control rupture dynamics in the presence of thermo-hydro-mechanical processes.

Fault constitutive relations that describe the evolution of shear resistance with fault motion are critical ingredients of earthquake source modeling. When coupled with the elastodynamic equations of motion, these relations provide insight into the growth and ultimate arrest of ruptures. Earthquake source processes are often considered in the framework of dynamic fracture mechanics, where the earthquake rupture may be considered as a dynamically propagating shear crack or pulse

By analogy to cohesive-zone relations for mode I opening cracks, slip-weakening laws have been commonly used to describe the dynamic decrease in shear resistance during sliding

The breakdown energy

More involved fault constitutive laws are generally required to explain a number of aspects of faulting behavior, most notably the restrengthening of faults between earthquakes. Laboratory experiments have provided significant insight into the rich behavior of shear resistance, with the frictional response at slip rates between

Many studies have attempted to infer parameters of the slip-weakening shear resistance from the strong-motion data resulting from natural earthquakes

One of the most notable features of seismologically inferred breakdown energies from natural earthquakes is that the average breakdown energy from the rupture process has been inferred to increase with the earthquake size

Several theoretical and numerical studies have demonstrated that enhanced dynamic weakening, as widely observed at relatively high slip rates (

Numerical models have shown that the incorporation of thermally activated enhanced weakening mechanisms during dynamic rupture can have profound effects on the evolution of individual ruptures, as well as the long-term behavior of fault segments, with the potential to make seemingly stable creeping regions fail violently during earthquakes

At the same time, accounting for thermo-hydro-mechanical processes during dynamic rupture can clearly weaken or even remove the analogy between frictional shear ruptures and idealized shear cracks of fracture mechanics. The analogy is based on two key assumptions: (1) that the breakdown of shear resistance is concentrated in a small region near the rupture front, referred to as small-scale yielding, and (2) that a constant residual stress level

In this study, we use numerical models of earthquake sequences with enhanced weakening due to thermal pressurization to illustrate how the inclusion of thermo-hydro-mechanical processes during dynamic shear ruptures makes breakdown energy rupture-dependent, in that the values of both local and average breakdown energy vary among ruptures on the same fault, even with spatially uniform and time-independent constitutive properties. As such, the breakdown energy is not an intrinsic fault property, but develops different values at a given location, depending on the details of the rupture process, which in part depend on the prestress before the dynamic rupture achieved as a consequence of prior fault slip history. Moreover, the local breakdown energy is not uniquely defined by the amount of slip attained during rupture, but depends on how that slip was achieved through the complicated history of slip rate and dynamic stress changes throughout the rupture process. Additional fault characteristics that we do not consider here, such as heterogeneity in fault properties and dynamically induced, evolving, inelastic off-fault damage

We conduct numerical simulations of spontaneous sequences of earthquakes and aseismic slip (SEAS) utilizing the spectral boundary integral method (BIE) to solve the elastodynamic equations of motion coupled with friction boundary conditions, including the evolution of pore fluid pressure and temperature on the fault coupled with off-fault diffusion

Our fault models adopt the laboratory-derived Dieterich–Ruina rate-and-state friction law with the state evolution governed by the aging law

During conditions of steady-state sliding (

An important, yet often underappreciated, implication of the rate- and state-dependent effects observed in laboratory experiments is that notions of static and dynamic friction coefficients, as well as the slip-weakening distance, are not well-defined and fixed quantities, as would be considered by standard linear slip-weakening laws

Laboratory experiments indicate that the standard rate-and-state laws (Eqs.

The total fault domain of size

Illustration of the rate- and state-dependence of the peak and dynamic friction coefficients,

Model parameters used in simulations of earthquakes and aseismic slip.

In the earthquake energy budget, the total strain energy change per unit source area

The dissipated energy

Seismological studies have attempted to estimate the average breakdown energy for natural earthquakes based on the standard energy partitioning diagram (Fig.

Note that the energy balance shown in Eq. (

The local slip and stress evolution are determined at every point along the fault within our simulations at all times; thus, we can calculate the local dissipation and breakdown energy throughout each rupture as well as study the evolution of these quantities in different ruptures throughout the sequence. We can also compute the average energy quantities and construct the average stress vs. slip curves for the total rupture process in a manner that preserves the overall energy partitioning

The average breakdown energy

Let us examine the spatial distribution of shear stress and breakdown energy in three ruptures of varying size within the same simulated sequence of earthquakes (Fig.

Comparison of three earthquake ruptures of different sizes nucleating over the same fault area.

Despite the fault constitutive properties being uniform and constant in time, the breakdown energy varies spatially within each event and also differs at each location for different ruptures (Figs.

The dependence of shear stress on slip for the three ruptures of Fig.

Note that the breakdown energy illustrated in Fig.

Previous theoretical work has demonstrated how the incorporation of thermo-hydro-mechanical processes such as the thermal pressurization of pore fluids can explain the inferred increase in breakdown energy with increasing event size

If slip occurs within a layer of thickness

The inclusion of thermal and hydraulic diffusion introduces a diffusion timescale to the problem, which governs the efficiency of weakening over extended slip durations. If one considers slip on a mathematical plane, a characteristic weakening timescale

Both of these thermal pressurization solutions have the convenient feature of expressing the breakdown of shear resistance as a function of slip, drawing familiarity to standard slip-weakening notions of shear fracture. As pointed out by

Comparison of accumulated slip, local shear stress vs. slip, and local slip rate vs. time for ruptures with rate-and-state (RS) friction with and without enhanced weakening due to thermal pressurization (TP). The two ruptures occur with the same initial shear stress distribution (top right), which results in a relatively small rupture in the RS-only model that is localized within the relatively highly prestressed nucleation region (top left). The inclusion of TP allows the rupture to grow and propagate over lower prestress conditions (top center). For the rupture governed by only RS (left column), the breakdown of shear resistance is generally comparable at different locations with the same fault properties, despite differences in local slip rate. This is due to the relatively mild dependence of RS friction on the slip rate. The rupture governed by RS and TP (center and right columns) exhibits a more complex evolution of local shear stress and slip rate throughout the rupture, which depends not only on the local prestress but also on the prestress and weakening behavior over the entire rupture through dynamic stress interactions.

The continued weakening with slip due to thermal pressurization is an important factor that drives rupture propagation and allows ruptures to propagate under lower (and hence less favorable) prestress conditions. Let us consider two fault models with the same initial prestress and the same rate-and-state frictional parameters, but with and without enhanced weakening due to thermal pressurization (Fig.

An important consequence of continued fault weakening is that much of the additional dissipated energy, which leads to the increase in breakdown energy with continued slip, is not concentrated near the rupture front (Fig.

Comparison of local breakdown energy for three large earthquake ruptures with nearly the same average breakdown energy and comparable average slip.

Comparison of the spatial breakdown energy distribution for the three large earthquake ruptures with nearly the same average breakdown energy and comparable average slip to Fig.

While breakdown energy does not appear to be a constant material property, one may ask if the effects of local weakening due to thermal pressurization may be adequately encapsulated into a slip-weakening formulation such as Eqs.(

The general trend of increasing breakdown energy with slip qualitatively holds for most local points within our simulated ruptures; however, there is considerable variability for individual values of

The average and local breakdown energy values for the simulated ruptures show an increasing trend with average and local slip, consistent with inferences from natural earthquakes (Fig.

For frictional ruptures, substantial slip may occur in regions that experience a net increase in shear stress, particularly in the regions near the rupture arrest (Fig.

The theoretical considerations of

While the general increase in breakdown energy with slip is qualitatively consistent among the theoretical solutions and our simulated dynamic ruptures in 2-D models with 1-D faults (Fig.

The average breakdown energy for our simulated ruptures tends to increase with increasing rupture size and average slip in a manner consistent with inferences from field observations and simplified theoretical models

The analytic formulations for the evolution of shear resistance with slip for the thermal pressurization presented by

Note that this variability in local

While we follow the assumption that most of the breakdown energy occurs on the shearing surface

The finding that the breakdown energy – as well as the weakening rate – can vary substantially along a given rupture and among subsequent ruptures, even for comparable values of slip, suggests that caution is needed in using the inferred breakdown energies from natural events for modeling of future earthquake scenarios. Some dynamic rupture simulations account for thermo-hydro-mechanical effects

Furthermore, several features of faulting in the presence of thermo-hydro-mechanical effects call into question the overall analogy with cohesive-zone dynamic fracture theory and, hence, the significance of the breakdown energy as the quantity that controls rupture dynamics. The analogy between breakdown and fracture energies, and more broadly frictional faulting and shear cracks of traditional fracture mechanics, requires that the breakdown process be confined close to the rupture tip (small-scale yielding) and that the dynamic resistance level be constant; under such conditions, the conclusions of dynamic fracture theory apply, including on the significance of breakdown energy

The data supporting the analysis and conclusions are accessible through the CaltechDATA repository (

VL and NL both contributed to developing the main ideas, designing the modeling, and producing the paper. VL carried out and analyzed the presented numerical experiments.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Thermo–hydro–mechanical–chemical (THMC) processes in natural and induced seismicity”. It is a result of the The 7th International Conference on Coupled THMC Processes, Utrecht, Netherlands, 3–5 July 2019.

Numerical simulations for this study were carried out on the High Performance Computing Center cluster of the California Institute of Technology. We thank Eric Dunham and Elisa Tinti for helpful comments and suggestions that improved the paper.

This research has been supported by the National Science Foundation (grant no. EAR 1724686), the United States Geological Survey (grant no. G19AP00059), and the Southern California Earthquake Center (SCEC; contribution no. 19085). SCEC is funded by NSF cooperative agreement (no. EAR-1033462) and USGS cooperative agreement (no. G12AC20038).

This paper was edited by Jianye Chen and reviewed by E. M. Dunham and Elisa Tinti.