the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 25 Nov 2024
Stress drops and earthquake nucleation in the simplest pressure-sensitive ideal elasto-plastic media
Abstract. This study explores stress drops and earthquake nucleation within the simplest elasto-plastic media using two-dimensional simulations, emphasizing the critical role of temporal and spatial resolutions in accurately capturing stress evolution and strain fields during seismic cycles. Our analysis reveals that stress drops, triggered by plastic deformation once local stresses reach the yield criteria, reflect fault rupture mechanics, where accumulated strain energy is released suddenly, simulating earthquake behavior. Finer temporal discretization leads to sharper stress drops and lower minimum stress values, while finer spatial grids provide more detailed representations of strain localization and stress redistribution. Our analysis reveals that displacement accumulates gradually during interseismic periods and intensifies during major stress drops, reflecting natural earthquake cycles. Furthermore, the initial wave field patterns during earthquake nucleation are complex, with high-amplitude shear components.
The histogram of stress drop amplitudes shows a non-Gaussian distribution, characterized by a sharp peak followed by a gradual decay, where small stress drops are more frequent, but large stress drops still occur with significant probability. This "solid turbulence" behavior suggests that stress is redistributed across scales, with implications for understanding the variability of seismic event magnitudes.
Our results demonstrate that high-resolution elasto-plastic models can reproduce key features of earthquake nucleation and stress drop behavior without relying on complex frictional laws or velocity-dependent weakening mechanisms. These findings emphasize the necessity of incorporating plasticity into models of fault slip to better understand the mechanisms governing fault weakening and rupture. Furthermore, our work suggests that extending these models to three-dimensional fault systems and accounting for material heterogeneity and fluid interactions could provide deeper insights into seismic hazard assessment and earthquake mechanics.
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RC1: 'Comment on egusphere-2024-3237', Anonymous Referee #1, 10 Jan 2025
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This paper introduces a numerical method to jointly simulate long-term and short-term evolution of faults, including dynamic rupture and fault localization and growth, in elasto-plastic media with viscous regularization. Such models have emerged in recent years to tackle important questions at the interface between earthquake research and long-term crustal deformation research. This work is an interesting contribution to those efforts. The results illustrate how models with ideal plasticity (constant friction) can generate earthquakes, despite the absence of explicit weakening of fault friction. Â
My main suggestions are
- Parts of the text and results (e.g. line 5, “Finer temporal discretization leads to sharper stress drops …”) give the impression that the simulations have not reached numerical convergence yet. If that is the case, I think you should keep refining the space and time discretization until the results converge (i.e. until there is negligible changes upon further refinement) and discuss only converged results. A focus on converged results can have a substantial impact on the statistics of stress drops and other physical quantities. If this requires new and more expensive simulations, it qualifies as major revision.
But I wonder if this lack of convergence is only apparent. With each refinement, are you also changing the value of the artificial viscosity (regularization parameter)? If that is the case, maybe you should instead keep the viscosity fixed in convergence studies. Unless there is a good reason to scale the viscosity to the mesh size, but that should be explained in the paper and it should be done in a way that guarantees convergence. - I found it very interesting that a bulk plasticity model with constant friction can generate earthquakes, because this is in contrast to fault friction models that are common in the computational earthquake dynamics community (earthquakes on pre-existing faults cannot be simulated without frictional weakening). This is not new, though, and it would be great to make more connections to existing related theoretical results. In particular, I find the work by Le Pourhiet (2013 https://doi.org/10.2113/gssgfbull.184.4-5.357) contains very insightful explanations of “structural weakening” in plastic models and plenty of useful references.
Minor comments:
- Line 34, “Recent studies ….”: You can also cite old seminal studies by Joe Andrews. In the 1976 paper (https://doi.org/10.1029/JB081i020p03575) where he basically opened the era of computational earthquake dynamics by introducing slip-weakening rupture simulations, he also realized that friction models were insufficient and introduced simulations with plasticity in the bulk. He was clearly very far ahead of his time. Renewal of this topic had to wait his 2005 paper (https://doi.org/10.1029/2004JB003191), which motivated the papers in computational earthquake dynamics that you cite. There is also important literature on plastic fracture dynamics in the fracture mechanics community; you can find many cited in Ben Freund’s book and in Gabriel et al (2013).
- Line 42: You can cite the earliest 3D studies of dynamic rupture with plasticity, e.g. Ma (2008 https://doi.org/10.1029/2008GC002231), Ma and Andrews (2010, https://doi.org/10.1029/2009JB006382)
- Line 59: It would be useful to emphasize in this sentence that the friction coefficient is assumed constant (no softening/hardening, “ideal plasticity”).
- Line 69: the word “static” can be removed (one could misinterpret the sentence as implying that there is a dynamic coefficient and it’s not constant).
- Line 114: should equation 13 involve the elastic strain instead of the total strain? Are you assuming plasticity also during dynamic stages of the simulation? If not, this assumption needs to be justified.
- Line 162, “This re-scaling process is iterated over "pseudo-time" …”: explain this in more detail. Make sure the description of the methods is complete enough to guarantee reproducibility.
- Line 165: show also the continuum equations describing the modified rheology assumed, so that readers don’t have to go look for it in previous papers.
- Line 166: explain the rationale to set the viscosity value.
- Line 196, “integrated stress”: define this quantity (integrated in space? in time? over what domain?)
- Line 237, “fault gouge … fault plane”: Which gouge? Which fault? These objects are not explicitly introduced in the model, I think you just mean “shear band” or “plastic zone” here.
- Section 4.5: Do you change the viscosity when you change N? Clarify.
- Line 254-255, “simulations with sufficient resolution produce stress drops and their amplitudes are similar”: The shapes of the curves are still different. Can you try even larger values of N to show convergence convincingly?
- Line 297, “These results highlight the sensitivity of fault behavior to the dilatation angle”: relate to published results, or instanc Templeton and Rice (2008)
- Line 308: Figure 14 seems to show lack of convergence. Clarify.
- Line 310, “simulation with fine temporal resolution and the lowest regularization”: This suggests that you are changing systematically the viscosity when you refine the simulations. Please clarify, explain that in detail.
- Line 316, “dynamic rupture events, akin to the rapid stress release observed during seismic slip”: Are these events as fast as earthquakes (slip rate of m/s, rupture speeds of few km/s)? Is the inertial term important during these events?
- Line 360, “characterized by a sharp peak followed by a gradual decay”: I see instead a broad peak and two long tails on both sides.
- Line 357, “This insight aligns with the Gutenberg-Richter law”: but here the distribution is truncated at low values too. Show a log-log plot to check if the upper tail is really a power law analogous to the G-R law.
- Section 5.1, “the nature of stress drops”: relate your results to insights from existing theory, e.g. Le Pourhiet (2013 https://doi.org/10.2113/gssgfbull.184.4-5.357)
- Lines 395+, “3D simulations … with zero regularization …. convergence tests performed in 3D”: are you suggesting that simulations converge even without regularization? If so, why is regularization needed? Clarify.
- Line 409, “closely mirrors the earthquake cycle seen in nature”: Do you find multiple stress drop happening on the same “fault” (shear band) or do they occur each time on a different segment of the fault?
- Section 5.6: there is redundancy with previous sections, which could be avoided.
- Line 469, “that plasticity should be considered alongside traditional frictional models in future earthquake simulations”: This is already the case in published work, e.g. Erickson et al (2017 https://doi.org/10.1016/j.jmps.2017.08.002), Preuss et a (2020 https://se.copernicus.org/articles/11/1333/2020/), Simpson (2023 https://doi.org/10.1016/j.tecto.2023.230089). Rephrase and add references.
Citation: https://doi.org/10.5194/egusphere-2024-3237-RC1 - Parts of the text and results (e.g. line 5, “Finer temporal discretization leads to sharper stress drops …”) give the impression that the simulations have not reached numerical convergence yet. If that is the case, I think you should keep refining the space and time discretization until the results converge (i.e. until there is negligible changes upon further refinement) and discuss only converged results. A focus on converged results can have a substantial impact on the statistics of stress drops and other physical quantities. If this requires new and more expensive simulations, it qualifies as major revision.
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