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REVIEW 3 major objections 5 minor 80 references

Hydrodynamics of fault gouges from constitutive modelling to the physics of friction

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper claims that a tensorially generalised hydrodynamic clay model yields a friction law whose thickness, normal-stress, and solid-fraction dependence reproduces laboratory velocity steps and stick-slip earthquake cycles.

desk verdict A genuinely new physics-based friction law with one solid holdout prediction, but the earthquake-cycle extension rests on an acknowledged and unquantified uniform-thickness assumption. read the letter →

arxiv 2411.14660 v3 pith:YZAXLW6L submitted 2024-11-22 physics.geo-ph cond-mat.mtrl-sciphysics.class-ph

Filippo Masi , Itai Einav This is my paper · ORCID
classification physics.geo-phcond-mat.mtrl-sciphysics.class-ph
keywords faultgougefrictionrate-and-statehydrodynamicconstitutivemodelmeso-temperatureclaystick-slipearthquakenucleationdilatancy
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that the frictional behaviour of clay-rich fault gouges, including its rate-and-state character, dilation, and dependence on layer thickness, normal stress, and solid fraction, can be derived rather than fitted. The route is to take the Terracotta hydrodynamic constitutive model for clay, which was built for triaxial loading, generalise it to tensorial form, and then specialise it to direct shear. The resulting Terracotta friction law has two explicit terms, one elastic and one viscous, with the meso-temperature of the clay as the state variable that controls transients. Validated against velocity-stepping experiments on illite, smectite, and kaolinite, the law retains the triaxial model's predictive power and explains dilatancy through evolution of gouge thickness. Coupled to a spring-dashpot host rock, it produces elastic stress buildup, stick-slip nucleation, and periodic seismicity whose stress drops and recurrence intervals match laboratory and field trends, so a sympathetic reader would take away that rate-and-state friction is not a standalone empirical law but a special case of a broader material physics.

What carries the argument

The carrying object is the Terracotta friction law, Eq. (24), produced by the hydrodynamic machinery of two-scale temperatures: a thermal temperature for atomic fluctuations and a meso-temperature $T_m$ for the kinetic energy of clay platelets and aggregates, with energy flowing from meso to micro scale through a sink term. Around that object, the argument uses four pieces: thermodynamically reciprocal transport coefficients that couple volumetric and deviatoric plastic strain rates; a critical-state compression line $p_c(\phi) = p_I \phi^{\lambda}$ that ties solid fraction to pressure; the tensorial generalisation of the Terracotta elastic energy with elastic instability criterion $q^e/p^e \le M^e$; and direct-shear boundary conditions, $v = \dot{\gamma} h$ and constant $\sigma_n$, that reduce the tensorial model to the two-term friction law. The evolution of $T_m$, Eq. (18), controls the transient and steady-state viscous response, while the evolution of $h$, Eq. (25d), makes dilatancy and solid-fraction changes part of the state. This machinery converts what are usually empirical rate-and-state parameters into measurable hydrodynamic, elastic, and critical-state quantities.

What would settle it

Resolve the shear strain profile inside a clay gouge layer during direct shear while measuring stress drops and recurrence intervals for initial thicknesses from about 0.01 mm to 10 mm; if a persistent shear band much thinner than the nominal layer carries the slip, or if the peak-and-decrease pattern of stress drop versus thickness is absent, the uniform-strain foundation of Eq. (24) fails.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central discovery is a closed friction law, Eq. (24): $\tau = (A/h^6)\,\varepsilon_v^e\,\gamma^e + (B/h)\,T_m\,v$, where $h$ is gouge thickness, $\varepsilon_v^e$ and $\gamma^e$ are elastic volumetric and shear strains, $T_m$ is the meso-temperature, and $A$, $B$ lump elastic and rheological constants. The first term is an elastic stress that builds before rupture and carries the model's density- and pressure-dependent elasticity; the second is a viscous stress controlled by the fluctuating kinetic energy of clay aggregates and platelets. The law is obtained by writing the Terracotta clay model in full tensorial form and imposing constant-normal-stress direct-shear kinematics, so the thickness $h$ evolves with compaction and dilation and the solid fraction follows $\phi = \phi_0 h_0 / h$. The paper shows that this single law reproduces velocity-step experiments on illite shale, smectite, and kaolinite, including transient peaks, isotach behaviour, dilation and compaction, and normal-stress dependence without recalibrating material constants. When the law is attached to a spring-dashpot host rock, the combined Terracotta-spring model predicts an elastic loading phase, a first earthquake, subsequent periodic stick-slip events, and eventual convergence to aseismic sliding, with stress drops and recurrence statistics consistent with laboratory and natural seismicity data.

Load-bearing premise

The load-bearing assumption is that the gouge shears uniformly through its full thickness, so the strain rate is exactly $v/h$; if slip localises into a thinner band whose width changes during an earthquake, the predicted stress drops, thickness scaling, and recurrence intervals would change.

Editorial extensions

If this is right

  • Rate-and-state friction parameters acquire physical meaning: they follow from elastic moduli, critical-state constants, and rheological coefficients that can be calibrated in triaxial or isotropic tests, so the same material description works under non-fault loadings.
  • Gouge thickness becomes a dynamic state variable rather than a fixed surface property, so the model predicts how stress drops and return periods vary with layer thickness, including a critical thickness that maximises stress drop.
  • The model provides an analytical account of elastic shear-stress buildup before the first rupture, a precursor phase that phenomenological rate-and-state laws cannot represent.
  • Coupled to a spring-dashpot host rock, the model generates complete seismic cycles, including an isolated first event, periodic stick-slip, and eventual stable sliding, with dilation-compaction cycles synchronised to the meso-temperature and matching observed trends.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the derivation is as general as claimed, the same hydrodynamic route could produce friction laws for saturated, chemically active, or higher-temperature gouges by adding the corresponding degrees of freedom, since the underlying balance and entropy-production structure already accommodates multiphysics couplings.
  • The steady-state law's quadratic dependence on slip velocity, Eq. (38), is a concrete, testable signature that distinguishes it from logarithmic rate-and-state laws; velocity-step experiments spanning several decades of $v$ would separate the two.
  • Because solid fraction was used as a fitting parameter in the validations, direct measurements of $\phi_0$ and of the strain profile across the layer, rather than inferred values, would provide the sharpest independent test of the model's thickness and volumetric predictions.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper derives a friction law for clay-rich fault gouges by starting from the Terracotta constitutive model for clay, generalizing it to tensorial form, and then specializing it to direct shear boundary conditions. The resulting Terracotta friction law, Eq. (24), expresses the shear stress as an elastic term proportional to ε_v^e γ^e / h^6 and a viscous term proportional to T_m v / h, with state evolution equations for elastic strains, meso-temperature, and thickness. The authors calibrate the model to published direct-shear experiments on illite, smectite, and kaolinite gouges, compare the predictions against additional normal-stress experiments, and then couple the friction law to a spring-dashpot host-rock model to simulate earthquake nucleation, stick-slip cycles, and recurrence statistics. The paper claims that the resulting framework is a physics-based alternative to empirical rate-and-state friction laws, with explicit dependence on gouge thickness, normal stress, and solid fraction.

Significance. If the central claims are accepted, the paper would offer a useful route from a thermodynamically grounded clay constitutive model to a friction law that naturally includes layer thickness, density state, and dilatancy, and it would connect laboratory friction behavior to a spring-slider seismicity model. The strengths of the manuscript are the analytic steady-state reduction in Appendix B, the availability of the numerical code, and the out-of-sample normal-stress comparison with Ashman and Faulkner (2023), which is genuinely predictive because no additional parameter adjustment is made for that dataset. However, the significance is limited by two load-bearing issues: most of the validation is in-sample calibration rather than independent prediction, and the field-scale earthquake predictions rest on the explicit assumption of uniform shear across the full gouge thickness, an assumption the authors themselves acknowledge neglects shear localization. These issues do not invalidate the derivation, but they need to be addressed or clearly bounded before the broader claims can be accepted.

major comments (3)
  1. [§3 and Table 1] The validation protocol is largely in-sample calibration, not independent prediction. The opening paragraph of Section 3 states that the parameters are 'iteratively updated to fit the experimental curves,' and Section 3.1 additionally recalibrates c, a, η, and M for the 100 MPa illite and smectite tests (c = 45·10^6 Ks, η = 53·10^3 K^-1s^-1, M = 0.86 for illite, with different values for smectite). Consequently, the agreement in Figures 4 and 5 is expected, and the claim that the law 'validates' rate-and-state behavior is overstated. The only truly out-of-sample test is the Ashman and Faulkner (2023) normal-stress comparison in Figure 7. Please report the full parameter set for each protocol, report misfit or uncertainty measures, and clearly distinguish fitted curves from predictive curves throughout Sections 3 and 6.
  2. [§2.3.2 and Eq. (26a)] The uniform-strain assumption γ̇ = v/h with h equal to the full gouge thickness is load-bearing for the earthquake predictions. The paper explicitly acknowledges in Section 2.3.2 that this 'neglects the coseismic evolution of shear localisation from a full gouge thickness to a localised band,' but the friction law Eq. (24) and the steady-state expression Eq. (38) scale with 1/h and 1/h_s^2, respectively. All thickness-dependent results in Section 5.2.1 and Figures 12-13, including the critical thickness separating stick-slip from stable sliding, the stress-drop amplitudes, and the recurrence intervals, therefore depend directly on the choice of full-layer h. In clay-rich gouges the localized band is often one to two orders of magnitude thinner than the h0 = 0.1-10 mm used in the simulations, and the model has no internal equation for the active band thickness because the second-gradient meso-temperature term is explicitly set aside in Section 2.2.1. Please quantify the sensitivity of the Section 5 predictions to h, or reformulate the model so that the active thickness is a dynamic field, before presenting the field-scale stick-slip and recurrence results as quantitative predictions.
  3. [§B, Eq. (38)] The steady-state velocity dependence is not independently predicted: the steady-state friction coefficient in Eq. (38) contains B, c, η, M, and h_s, and h_s in Eq. (35) depends on φ0, h0, ω, p_I, and λ. All of these quantities are calibrated, either directly or indirectly, from the same velocity-step experiments used for the validation in Section 3. In other words, the rate-strengthening or rate-weakening behavior reproduced in Figures 4-6 is largely a consequence of the fitted transport coefficients rather than a parameter-free consequence of the hydrodynamic framework. To make the predictive claim credible, please show which parameters can be fixed from independent triaxial, isotropic compression, or elastic-wave tests, and which ones necessarily require direct-shear velocity steps, and discuss how parameter uncertainty propagates into the steady-state friction law.
minor comments (5)
  1. [§3.1 and Figure 5] The parameter values used for the 100 MPa illite and smectite predictions are reported only in the text and are not entered in Table 1, which makes the Figure 5 results difficult to reproduce; please add them to the table or to a supplementary table.
  2. [Appendix A] The numerical implementation uses the initial thickness h0 rather than the current thickness h in the constraint (26b), with the justification that the normal strain remains below about 5%. Please state this approximation directly in the main text where Eq. (26b) is introduced, rather than only in the appendix, and justify that the 5% bound holds for all simulations including the parametric sweeps in Section 5.
  3. [§2.3, Eq. (24)] The notation in Eq. (24), with the underbrace labels 'state' and 'rate', is visually unclear; the variables v and v_n are rates, but T_m and the elastic strains are also evolving state variables. A cleaner grouping of arguments would improve readability.
  4. [§6] The conclusion states that the friction law 'explains for the first time analytically the buildups of an elastic stress towards first rupture.' The elastic buildup is already evident from the elastic component in Eq. (24) and from the spring-slider integration, but the claim of analytic explanation should be supported by a direct analytic expression for the pre-rupture stress evolution, or softened.
  5. [§5.1 and Figure 16] The energy budget quantities in Figure 16(a) are defined visually rather than by explicit formulas in the text; please provide the equations used to compute E_R, E_G, and E_F so that the reported values E_G = 0.21 J/m^2, E_F = 170 J/m^2, and E_R = 0.04 J/m^2 can be verified.

Circularity Check

2 steps flagged · score 6.0 of 10

Rate- and state-dependent 'validation' is calibrated to the same velocity-step experiments; thickness evolution is also calibrated to the thickness data used for validation.

  1. fitted input called prediction [Section 2.3.1, Section 3.1, and Appendix B, Equations (18), (24), (38)]
    "The remaining three rheological parameters, η, a, and c, require calibration thorough rate-dependent experiments such as velocity stepping or slide-hold-slide tests, as those considered below. ... an initial guess is made based on previous values obtained by Wiebicke and Einav (2024), which are then iteratively updated to fit the experimental curves ... Calibration of the material parameters for this particular protocol yields the following values: c=45·10^6 Ks, a≡c, η=53·10^3 K^-1s^-1, and M=0.86 for illite."

    The velocity dependence presented as validation is controlled by the rheological constants c, a, η, and (in the 100 MPa protocol) M, which are fitted to the same velocity-step experiments used for comparison. The meso-temperature evolution in Equation (18) and the viscous stress in Equation (21a) make the friction response depend on these fitted constants, and Appendix B's steady-state friction coefficient, µs(hs)≈M/√3[1+(B/(Mσn))√(c/η)(v/hs)^2], explicitly contains √(c/η). Agreement with the calibration data is therefore a consistency check rather than an independent prediction of rate-and-state behaviour.

  2. fitted input called prediction [Section 3.2, Bedford et al. (2022) validation, Figure 6]
    "The new friction law is calibrated based on the time evolution of the gouge thickness, h, presented in Figure 6(b), before the application of velocity steps that initiates after slip displacement of 1.5 mm."

    The thickness evolution is the quantity later shown as evidence that the friction law captures the volumetric/dilational behaviour of the gouge, but the same h(t) data are used to calibrate the model. The predicted thickness and normal-velocity curves in Figure 6(b,c) therefore follow by construction from the calibration target. The friction-coefficient comparison provides some independent information, but the volumetric validation is same-data consistency.

full rationale

The tensorial generalization of Terracotta and its specialization to direct shear are self-contained algebraic steps, and the derivation of Equation (24) from the hydrodynamic equations is not circular. The significant circularity is in the validation protocol: the parameters that govern the rate dependence, c, a, η, and in the 100 MPa velocity-step runs also M, are iteratively fitted to the very same Saffer & Marone and Bedford et al. experiments against which the law is then said to 'predict' velocity-stepping and rate-and-state friction; Appendix B shows the steady-state velocity term is √(c/η)(v/hs)^2, so that prediction is a restatement of fitted inputs. Likewise, the Bedford thickness evolution used to corroborate the volumetric nature is itself a calibration target. The thickness and normal-stress dependencies retain structural content, because they enter through γ̇=v/h and the φ0h0/h factor rather than through fitted fault-friction parameters, so the paper is not wholly circular. Self-citations to the original Terracotta model (Wiebicke and Einav, 2024) are not themselves counted as circular here, since that triaxial model was calibrated to independent triaxial and critical-state data and does not assume the fault-friction result; however, the present calibration inherits those parameter values without re-derivation. The explicitly acknowledged uniform-strain simplification γ̇=v/h with the full gouge thickness is a substantive correctness risk for the field-scale earthquake predictions, but it is a modelling approximation rather than a circular step.

Assumptions & free parameters 6 free parameters · 7 assumptions · 1 invented entities

The central claim rests on the hydrodynamic framework, the specific energy expression, the meso-temperature evolution, and a set of material parameters calibrated to triaxial and direct shear data. The form of the friction law is derived, but the rate-dependent constants are fitted to the same experiments used for validation, so the derivation is not parameter-free.

free parameters (6)
  • tilde_K, tilde_G (elastic stiffnesses) = Illite 1040/400 MPa, smectite 260/100 MPa, kaolinite 15.6/12.5 MPa
    Elastic moduli calibrated to fit stress-strain responses, inherited from Terracotta and iteratively updated for each validation dataset.
  • M, omega (critical state parameters) = M=0.96, omega=0.6 (illite); M=0.42, omega=0.5 (smectite); M=0.44, omega=0.4 (kaolinite)
    Critical state line parameters obtained from triaxial or direct shear tests; omega=0.5 is used as a default in the absence of density measurements.
  • lambda, phi_I (compressibility parameters) = lambda=10, phi_I=0.20 (illite); lambda=4.5, phi_I=0.05 (kaolinite)
    Parameters of the isotropic compression line, calibrated from compression tests or post-peak slopes, and used in p_c(phi) = p_I phi^lambda.
  • a, c, eta (rheological constants) = Table 1: illite a=7 Ks, c=7 Ks, eta=7.2e4 K^-1 s^-1; 100 MPa recalibration gives c=45e6 Ks, eta=53e3 K^-1 s^-1
    Rheological transport and energy-sink coefficients fitted to rate-dependent velocity-step and slide-hold-slide experiments; large differences between calibrations indicate path dependence.
  • phi_0 (initial solid fraction) = 0.57 and 0.71 for Saffer-Marone runs; 0.65 for Bedford and Ashman kaolinite; 0.71 for the earthquake benchmark
    Explicitly treated as a fitting parameter when the initial solid fraction was not measured in the source experiments.
  • h_0 (initial gouge thickness) = 2.25 to 2.75 mm for Saffer-Marone; 1 mm for Bedford; 0.1 mm for the earthquake benchmark
    System parameter set from experimental initial thickness or assumed shear band thickness; choices markedly affect the predicted earthquake dynamics.
assumptions (7)
  • domain assumption The internal energy of the gouge has the polynomial form u_e = phi^6 (tilde_K/6 epsilon_v^{e3} + tilde_G epsilon_v^e e_ij e_ij) plus a meso-entropy term, with no thermal temperature dependence.
    Inherited from Terracotta (Eq 8, generalised in Eq 13); this specific functional form controls the pressure- and density-dependent elasticity and is not derived in this paper.
  • domain assumption Meso-temperature T_m is a valid non-equilibrium state variable obeying T_m_dot = a eps_v^2 + (2/3)c e_ij e_ij - eta T_m^2 with quadratic relaxation.
    The rate dependence of the friction law enters exclusively through T_m; its existence and evolution are assumed from granular hydrodynamics, not independently measured here.
  • domain assumption The Onsager transport coefficients for plastic flow and viscous stress take the specific forms in Eqs 20-21, calibrated to normal consolidation and critical state asymptotes.
    These forms were calibrated in Wiebicke and Einav (2024) to recover critical state behaviour, and are transferred here to direct shear without re-derivation.
  • domain assumption The solid particles are incompressible and the gouge is dry, so rho_s is constant and phi = phi_0 h_0 / h.
    Invoked in Section 2.1.1 before Eq 3; this excludes pore fluid effects that are important in saturated fault zones.
  • domain assumption The gouge deforms homogeneously with a uniform shear strain rate gamma_dot = v/h across the full layer thickness, with no shear localization.
    Stated in Section 2.3.2; the thickness dependence and earthquake nucleation results rely on this uniform-shear assumption.
  • domain assumption Constant normal stress is maintained by adjusting the normal velocity v_n, and the numerical implementation uses h0 instead of current h in Eq 26b.
    Section 2.3 and Appendix A; justified only by the normal strain remaining below about 5 percent in the simulations.
  • domain assumption Thermal temperature is decoupled and the process is isothermal, ignoring thermal pressurization, flash heating, and frictional melting.
    The model is isothermal; the authors note these mechanisms are outside scope in Section 5.2.1.
invented entities (1)
  • Meso-temperature T_m independent evidence
    purpose: Non-equilibrium state variable representing the kinetic energy of velocity fluctuations of clay platelets and aggregates; it controls viscous stress and plastic strain rates, giving the rate and state dependence of friction.
    Borrowed from granular hydrodynamics (Jiang and Liu 2009; Alaei et al. 2021) and claimed to be inferable from particle trajectories or kinetic pressure, but not directly measured in this paper.

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Cite this review

Pith. "Pith review of Hydrodynamics of fault gouges from constitutive modelling to the physics of friction." pith.science (2026). https://pith.science/paper/YZAXLW6L

@misc{pith2026241114660,
  author       = {Pith},
  title        = {Pith review of: Hydrodynamics of fault gouges from constitutive modelling to the physics of friction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YZAXLW6L}},
  note         = {Machine review of arXiv:2411.14660}
}
read the original abstract

The development of rate- and state-dependent friction laws offered important insights into the key physical mechanisms of the frictional behaviour of fault gouges and their seismic cycle. However, past approaches were specifically tailored to address the problem of fault shearing, leaving questions about their ability to comprehensively represent the gouge material under general loading conditions. This work establishes an alternative approach for developing a physical friction law for fault gouges that is grounded on the rigour of the hydrodynamic procedure with two-scale temperatures through Terracotta, a thoroughly robust constitutive model for clay in triaxial loading conditions. By specifying the model for direct shearing, the approach yields an alternative friction law that readily captures the frictional dynamics of fault gouges, including explicit dependencies on gouge layer thickness, normal stress, and solid fraction. Validated against available laboratory experiments, the friction law retains the original predictive capabilities of Terracotta in triaxial conditions and explains the rate-and-state, dilatational behaviour of fault gouges in direct shear conditions. Finally, when the Terracotta friction law is connected to a spring-dashpot representation of the host rock, the combined model predicts an elastic buildup precursor to the onset of and subsequent seismicity, with results closely reflecting experimental evidence and field observations. While this study focuses on clay-rich gouges, the approach and findings are expected to offer much wider implications to a variety of materials.

Figures

Figures reproduced from arXiv: 2411.14660 by the authors.

Figure 1
Figure 1. Hydrodynamics of fault gouges: from constitutive to friction law. A fault gouge (a) sub [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Schematic outline of the range of behaviours captured by the Terracotta constitutive model [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Summary of the mathematical development from (a) the original Terracotta model for tri [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: Shear response of illite shale and smectite gouges under constant normal stress [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: Detailed view of velocity steps on illite shale (a-c) and smectite (d-f) under constant nor [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: Comparison between experiments (Bedford et al. 2022) and the Terracotta friction law for [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: The variations in the friction coefficient, [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: The impact of the rheological parameters on the model response under velocity steps (cf. [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: The role of the meso-temperature in establishing the short-term, peak friction response. [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]
Figure 10
Figure 10. Figure 10: The impact of the elastic parameters on the model response under velocity steps (cf. Figure [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]
Figure 11
Figure 11. Figure 11: Response of the Terracotta-spring model: (a) friction coefficient [PITH_FULL_IMAGE:figures/full_fig_p021_11.png]
Figure 12
Figure 12. Figure 12: Influence of the non-dimensional gouged shear zone thickness, [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]
Figure 13
Figure 13. Figure 13: Influence of the non-dimensional gouged shear zone thickness, [PITH_FULL_IMAGE:figures/full_fig_p024_13.png]
Figure 14
Figure 14. Figure 14: Influence of the solid fraction, ϕ 0 , on earthquake dynamics and characteristics under con￾stant normal stress σn = 40 MPa: (a) phase portrait of the friction versus the non-dimensional slip velocity and limit cycle, (b) friction drops, and (c) return period in terms…
Figure 15
Figure 15. Figure 15: Influence of the normal stress, σn, on earthquake dynamics and characteristics under a constant overconsolidation ratio p/pc ∼ 4: (a) stick-slip limit cycles, (b) friction drops, and (c) return period in terms of σn. Results refer to v∞ = 1 cm/year, and initial condit…
Figure 16
Figure 16. Figure 16: Influence of the normal stress on fault energetics and statistics: (a) schematic representation [PITH_FULL_IMAGE:figures/full_fig_p026_16.png]
Figure 17
Figure 17. Figure 17: Influence of the far-field velocity, v∞ on fault energetics and statistics: (a) stick-slip limit cycles, (b) friction drops, and (c) return period versus v∞. Results refer to initial conditions of h 0 = 0.1 mm, ϕ 0 = 0.71, and σn = 40 MPa [PITH_FULL_IMAGE:figures/ful…
Figure 18
Figure 18. Figure 18: Influence of the far-field velocity, v∞: relationships between v∞ and return period, tr. shear stress is obtained by time differentiation of Equation (24): τ˙ = ∂τ ∂εe v ε˙ e v + ∂τ ∂γe γ˙ e + ∂τ ∂Tm T˙m + ∂τ ∂h vn + ∂τ ∂v v˙, (32) where ε˙ e v , γ˙ e , T˙m, and vn (o…

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