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REVIEW 3 major objections 6 minor 48 references

Finite element modeling of dynamic frictional rupture with rate and state friction

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Pure velocity-weakening friction cannot equilibrate a far-field load, so finite-size frictional sliding needs the strengthening-branch law.

desk verdict Useful explicit-FEM study of rate-and-state friction with a real finite-size effect, but the claim that pure velocity-weakening lacks physical validity goes beyond the convergence evidence. read the letter →

arxiv 1908.07826 v1 pith:CDHQHKDJ submitted 2019-08-03 physics.geo-ph physics.comp-ph

Roozbeh Rezakhani , Fabian Barras , Michael Brun , Jean-Francois Molinari This is my paper · ORCID
classification physics.geo-phphysics.comp-ph
keywords rateandstatefrictionfiniteelementmethodexplicitdynamicsfrictionalrupturevelocityweakeningweakening-strengtheningboundaryreflectionsstressdrop
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 asks what happens to a rate-and-state frictional interface when the sliding bodies are finite, so elastic waves bounce off the domain boundaries and return to the interface. Using an explicit finite element method with node-to-node contact, it shows that before those reflected waves arrive, both a pure velocity-weakening (VW) law and a velocity-weakening-strengthening (VWS) law reach a temporary steady sliding state with a well-defined stress drop behind the rupture front, matching the fracture analogy. After repeated reflections, the VWS law converges to a global steady state in which interface friction balances the applied far-field shear stress, while the VW law makes the blocks accelerate and never reaches equilibrium. The paper concludes that pure VW friction cannot produce interface tractions that balance the far-field load, and therefore lacks physical validity for long-term frictional sliding in finite systems. This matters because most rate-and-state simulations use infinite-domain boundary integral methods, which cannot see the boundary-reflection effects that decide whether a sliding state is stable.

What carries the argument

The central object is the rate-and-state friction law, in which the friction coefficient $f$ depends on the slip velocity $v$ and a state variable $\varphi$ (the average lifetime of load-carrying contact asperities) through $f=f_0+a\ln(v/v_*)+b\ln(\varphi/\varphi_*)$ and $\dot\varphi=1-v\varphi/D$, together with a revised law that adds a velocity-strengthening branch at high slip velocity. The numerical machinery is an explicit finite element solver with central-difference time integration and node-to-node contact, which is needed to include finite domain boundaries that boundary integral methods cannot represent. The physical mechanism that carries the argument is the return of reflected elastic waves to the interface: each reflection changes the average slip velocity and shear traction, and the paper shows that these successive changes drive VWS friction to the equilibrium point on its steady-state friction curve while driving VW friction away from any equilibrium. The paper also identifies the finite element internal-node noise that requires time steps far below the CFL stability limit for accurate rate-and-state solutions.

What would settle it

Run the velocity-weakening long-time case with successively refined interface meshes and time steps (for instance, α = 0.01 and element sizes below 1.4 mm): if the runaway acceleration disappears or a steady sliding velocity emerges as resolution improves, the claim that pure VW cannot equilibrate the far-field load would be refuted; if the runaway persists at every resolution, the claim survives.

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Extended reading notes

Core claim

The central claim is that the physically relevant distinction between the two families of rate-and-state friction laws only appears once finite boundaries are included. In an infinite-domain setting, both the pure velocity-weakening law and the velocity-weakening-strengthening law generate rupture fronts with a temporary steady state and a sharp shear-stress drop, so the fracture-mechanics analogy holds. When the blocks have finite height, the waves reflected from the top and bottom boundaries come back to the interface and repeatedly change the slip velocity and shear traction. For the VWS law these repeated impacts push the interface toward the intersection of the steady-state friction curve with the far-field loading line, so the blocks slide uniformly and the stress drop decays to zero. For the pure VW law, the same reflected waves accelerate the interface: the friction coefficient drops as velocity increases, so the interface can never produce enough traction to balance the applied shear stress, and the numerical simulation becomes unstable once reflections arrive. The paper therefore asserts that pure VW friction is not physically valid for the long-term behavior of frictional interfaces in interaction with domain boundaries, whereas VWS friction is.

Load-bearing premise

The conclusion that pure velocity-weakening friction is physically invalid for long-term sliding rests on the assumption that the runaway acceleration seen in the simulations is a real physical instability and not an artifact of the numerical noise that the paper itself shows can destabilize explicit finite element solutions.

Editorial extensions

If this is right

  • In finite-size rate-and-state models, pure velocity-weakening friction is only usable for times shorter than the wave-return time; after that it predicts runaway acceleration rather than steady sliding.
  • The fracture-mechanics analogy for frictional rupture—a stress drop behind the front—is a finite-time effect that disappears once boundary reflections homogenize the interface.
  • Long-term numerical models of frictional interfaces in finite domains should adopt velocity-weakening-strengthening friction if they aim to reproduce steady sliding under constant far-field load.
  • Explicit finite element simulation of rate-and-state friction requires time steps orders of magnitude smaller than the CFL limit, and the instability seeded by internal-node noise must be controlled before long-time physics can be extracted.
  • The agreement with boundary integral results in the pre-reflection phase establishes finite element simulations as a valid tool for capturing finite-size effects that infinite-domain methods cannot address.

Reading between the lines

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

  • If the VW runaway is physical, then long-term earthquake and laboratory sliding models on finite faults that use pure VW friction may need a high-velocity strengthening cutoff or additional regularization to avoid unbounded acceleration—a modification the paper does not explore.
  • A direct numerical test of the paper's central claim would be to refine the mesh and reduce the time step for the VW long-time case; the runaway should persist at every resolution if it is physical, and disappear if it is an artifact of internal-node noise.
  • The near-linear relation the paper observes between the average stress drop and the velocity jumps at each reflection suggests a quantitative finite-size radiation-damping law could be derived, extending infinite-domain crack theory to finite blocks.
  • Laboratory interfaces that exhibit long-term stable sliding may be those whose friction has a strengthening branch, implying that the VWS form, rather than pure VW, is the safer default for interpreting finite-size experiments.
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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 / 6 minor

Summary. This paper presents an explicit-dynamics finite element formulation for rate-and-state friction between finite elastic blocks, with node-to-node contact and central-difference time integration detailed in the appendices. The authors analyze the time-step constraint for steady sliding, attribute the need for very small time steps to noise from internal discretization nodes, and benchmark the early-time rupture response against a boundary integral method. They then study the effect of boundary reflections on long-term sliding for a velocity-weakening (VW) law and a velocity-weakening-strengthening (VWS) law. The main claims are that VWS reaches a global steady-state sliding configuration after multiple reflections, while VW accelerates and becomes unstable, so that pure VW "lacks physical validity" for long-term finite-domain studies.

Significance. If the long-term dichotomy is physical, the paper is a useful contribution to finite-size rate-and-state friction modeling: unlike boundary integral formulations, the FEM captures boundary interactions, and the paper documents a nontrivial numerical-stability requirement (time steps well below the CFL limit) in the studied configuration. The early-time FEM/BIM agreement in Figure 7b is a genuine validation, and the energy accounting and interface-averaged diagnostics in Figures 10-12 are informative. The paper also makes a falsifiable prediction: pure VW friction cannot settle into a steady sliding state after boundary reflections, whereas VWS can. The main weakness is that the key VW prediction is asserted from a single run with no convergence study, in a regime where the authors themselves show that numerical noise can trigger instability.

major comments (3)
  1. [Section 5.2 (VW perturbation analysis)] The central claim that pure VW friction lacks physical validity in finite systems is supported only by the statement that "the calculations become unstable as soon as the reflected waves reach back the frictional interface." Section 4.2 demonstrates that this explicit FEM develops non-physical oscillations for time steps alpha >= 0.05 (Figure 4) and that the onset time of such oscillations scales linearly with element size (Figure 5b). The VW reflected-wave run uses alpha = 0.02 and 250 interface elements, but no alpha- or h-refinement study is reported for this case. A convergence study (for example alpha = 0.02, 0.01, 0.005 and at least two mesh refinements, reporting the instability onset time, slip-velocity growth rate, and final state) is required to separate a physical VW instability from the discretization noise characterized in Section 4.2; without it, the paper's main conclusion is not established.
  2. [Section 5.2, final paragraph] The statement that "the pure VW friction cannot generate interface tractions that equilibrate the far field load" is imprecise and, taken literally, incorrect: for the chosen parameters, Equation (4) gives fss = 0.36 at v0_ss = 2.93627e-4 m/s (the purple star in Figure 3b), so the VW law does possess an interface traction that equilibrates the applied shear. What the simulations indicate is that this equilibrium is not stable under perturbations and after boundary reflections. The wording should be changed to "cannot stably maintain" or "does not converge to", and the discussion should distinguish between existence and stability of the equilibrium.
  3. [Section 5.1.1 and Figure 7b] The FEM/BIM validation is reported only for the maximum slip velocity and only for one discretization (250 interface elements, alpha = 0.02). Since the paper's long-term conclusions are drawn from simulations with this same discretization, a mesh- and time-step-convergence test of the rupture solution, not just of steady sliding, is needed to establish that the VWS/VW differences are not resolution effects. At minimum, a second mesh refinement and a smaller alpha should be reported for the rupture simulations.
minor comments (6)
  1. [Throughout] The manuscript contains numerous typos (e.g., "purturbed", "corresonding", "seubsequent", "caluclated", "fricition") and should be carefully proofread.
  2. [Sections 3 and 4.1] There are clear remnants of an earlier draft: orphan headings such as "3.1.2. Contact algorithm", "3.1.3. Frcition algorithm", and "3.2. Spectral element approach" appear in the text, and some equations and figures are numbered inconsistently (Equations 6-10 versus 7-9; "Figure ??" in Section 4.1). These should be cleaned.
  3. [Figure 1] Figure 1 is duplicated and the captions/panels do not match the text: one copy shows stationary friction curves with a different caption from the other copy, and the panel labels are inconsistent. Please regenerate the figure and a single correct caption.
  4. [Sections 5.1 and 5.2] The interpretation of the stress drop and radiation damping draws on refs [25,26]; reference [26] is listed as "To be submitted" and reference [25] is an arXiv preprint. The relevant statements should either be demonstrated from equations in the present paper or refer to published/archived versions.
  5. [Section 4.2] The claim that the instability originates from internal discretization nodes is an inference from the linear tcrit-versus-element-size scaling and the structured/unstructured mesh comparison (Figure 5). A direct test, such as comparing with an interface-only discretization or with nodal filtering, would make the noise-source attribution more conclusive.
  6. [Figures 4c and 10c] The axis labels in Figure 4c and the legend in Figure 10c are difficult to read; please enlarge the fonts and define the symbols (for example, the "123" marking in Figure 10b).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the simulation results are self-contained, and the authors' self-citations are used only for interpretive framing, not as load-bearing derivation.

full rationale

The paper's central results—VWS friction reaching a global steady state after boundary reflections and VW friction accelerating without equilibrating the far-field load—are obtained by explicit finite-element solution of the stated rate-and-state equations (Eqs. 1, 2, and 5) with experimentally motivated PMMA parameters. The constitutive curves are inputs, but the long-term sliding behavior is a simulated outcome, not an imposed fit. No fitted parameter is renamed as a prediction, and no equation in the derivation chain reduces to its own input. The self-citations [25,26] are used to interpret stress drops and radiation damping ('In recent work, we have shown how the stress drop and slip velocity ... can be predicted theoretically from radiation damping [25]' and 'Theoretical arguments [25] can be made to relate these jumps to order of mu/2cs, and will be investigated further in the future'), but these references are not used to define or force the VWS/VW steady-state conclusions. The strongest physical claim about VW is presented as a simulation-based inference ('In any case, the pure VW friction cannot generate interface tractions that equilibrate the far field load'), and while this is debatable and possibly conflates numerical instability with physical behavior, that is a correctness and convergence concern, not circularity. The FEM-BIM comparison uses a coauthor's thesis as a benchmark, but it validates the early-time implementation and does not define the long-term predictions. Accordingly, the derivation chain is self-contained and no circular step is exhibited.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central simulation results rest on the PMMA material parameters, the assumed rate-and-state friction laws, the small-displacement node-to-node contact formulation, and the unverified assertion that the VW long-time instability is physical rather than numerical.

free parameters (4)
  • Friction parameters f0, a, b, v*, phi*, D = f0=0.285, a=0.005, b=0.0214 (VW) / b=0.075 (VWS), v*=1e-7 m/s, phi*=3.3e-4 s, D=5e-7 m
    Inputs from PMMA experiments (Barras 2018); not fitted here, but they set the VW and VWS regimes and the steady-state operating points used in the simulations.
  • Initial steady-state slip velocity v0_ss = 3.8931e-4 m/s (VWS), 2.93627e-4 m/s (VW)
    Chosen as the intersection of fss=0.36 with the steady-state friction curve; defines the initial condition, not fitted to outcomes.
  • State perturbation amplitude eps = 1e-4
    Chosen to nucleate rupture; a numerical seed that triggers the dynamics.
  • Time step ratio alpha = 0.02 for rupture simulations
    Set below the alpha=0.04 threshold identified in the steady-state test of Section 4.2; no convergence sweep is reported for the rupture cases.
assumptions (5)
  • domain assumption The two blocks are homogeneous, linear-elastic (PMMA properties) under plane stress.
    Used in Sections 3 and 4.1; the wave speeds and reflection timing depend on this material model.
  • domain assumption The interface obeys the rate-and-state friction laws of Equations 1 and 2 (VW) or 5 (VWS), with state evolution phidot=1-v*phi/D.
    Interface constitutive assumption; not derived in this paper.
  • domain assumption The node-to-node contact algorithm is valid for small displacements.
    Stated in Appendix B; the simulated slip is small.
  • domain assumption Periodic boundary conditions on the lateral sides do not change the long-term interfacial behavior.
    Used in all simulations; the paper does not test alternative lateral boundary conditions.
  • ad hoc to paper The long-time VW instability observed in Section 5.2 is physical and not a numerical artifact of the discretization noise characterized in Section 4.2.
    This is the load-bearing premise behind the conclusion that VW lacks physical validity; the paper does not demonstrate convergence or separate physical from numerical instability.

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Pith. "Pith review of Finite element modeling of dynamic frictional rupture with rate and state friction." pith.science (2026). https://pith.science/paper/CDHQHKDJ

@misc{pith2026190807826,
  author       = {Pith},
  title        = {Pith review of: Finite element modeling of dynamic frictional rupture with rate and state friction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CDHQHKDJ}},
  note         = {Machine review of arXiv:1908.07826}
}
read the original abstract

Numerous laboratory experiments have demonstrated the dependence of the friction coefficient on the interfacial slip rate and the contact history, a behavior generically called rate and state friction. Although numerical models have been widely used for analyzing rate and state friction, in general they consider infinite elastic domains surrounding the sliding interface and rely on boundary integral formulations. Much less work has been dedicated to modeling finite size systems to account for interactions with boundaries. This paper investigates rate and state frictional interfaces in the context of finite size systems with the finite element method in explicit dynamics. We investigate the long term behavior of the sliding interface for two different friction laws: a velocity weakening law, for which the friction monotonously decreases with increasing sliding velocity, and a velocity weakening-strengthening law, for which the friction coefficient first decreases but then increases above a critical velocity. We show that for both friction laws at finite times, that is before wave reflections from the boundaries come back to the sliding interface, a temporary steady state sliding is reached, with a well-defined stress drop at the interface. This stress drop gives rise to a stress concentration and leads to an analogy between friction and fracture. However, at longer times, that is after multiple wave reflections, the stress drop is essentially zero, resulting in losing the analogy with fracture mechanics. Finally, the simulations reveal that velocity weakening is unstable at long time scales, as it results in an acceleration of the sliding blocks. On the other hand, velocity weakening-strengthening reaches a steady state sliding configuration.

Figures

Figures reproduced from arXiv: 1908.07826 by the authors.

Figure 1
Figure 1. (a) Steady state friction coefficient versus steady state sliding velocity. The blue and black curves are the steady state form of VW and VWS presented in Equations 1 and 5, respectively. (b) Evolution of friction coefficient due to a sudden change in sliding velocity for VW friction. The steady state curve of this friction model is plotted in Figure 1a in black using the same parameters presented above. The frictio… view at source ↗
Figure 2
Figure 2. Finite element representation of two solid blocks that are in contact along Γc . The two blocks are discretized with conforming mesh on the interface, and each two corresponding nodes are grouped to apply contact and frictional constraints. A zoomed view of the forces acting among the pair of nodes i is illustrated on the right at time t = tn. Γu and Γt are the portions of the domain boundary on which displacement U… view at source ↗
Figure 3
Figure 3. (a) Two elastic solids in contact and under normal and tangential far field tractions. Periodic boundary conditions are applied on lateral sides, and 0.5v 0 ss is the initial velocity assigned to both solids in opposite directions. (b) Steady state friction coefficient as function of steady state sliding velocity for the velocity weakening-strengthening friction (VWS), black curve, and the pure velocity weakening (V… view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: (a) Average velocity velocity over the interface normalized by the initial steady state velocity versus time for different α values. (b) Slip velocity over the interface plotted in time for α = 0.05, note that not all time steps are shown for clarity. (c) Variation (re…
Figure 5
Figure 5. Figure 5: (a) Finite element representation of the structured and unstructured mesh of the blocks by discretizing the interface with 50 elements. (b) For structured finite element mesh with α = 0.2, normalized time to start instability is plotted versus normalized finite element…
Figure 7
Figure 7. Figure 7: (a) Friction coefficient evolution at two points on the interface located at x = 0.125 [m] and x = 0.375 [m] is plotted in red and blue, respectively. The dash-dotted line is the far field applied boundary condition. Notice that without wave reflections from outer boun…
Figure 8
Figure 8. Figure 8: Contours of horizontal component of velocity field at four time instants during the simulation. Solid blocks shown in this figure are cut at the height of 0.55 [m] for better visualization, while their height in the simulation is 1.25 [m]. the interfacial response, the…
Figure 9
Figure 9. Figure 9: (a) Friction coefficient evolution at two points on the interface at x = 0.125 [m] and x = 0.375 [m] plotted in red and blue, respectively. The dash-dotted line is the far field applied boundary condition. Notice that, at the final simulation time, after multiple wave …
Figure 10
Figure 10. Figure 10: (a) Variation of average shear traction on the interface versus time. (b) Variation of average slip velocity on the interface versus time. (c) Average stress drop versus average velocity jump, corresponding to each times boundary-reflected waves reach back the interfa…
Figure 11
Figure 11. Figure 11: (a) Slip velocity at the interface, and (b) contour of vx in the two solid blocks at the end of the simulation with the effect of boundary reflections, showing that the two blocks move uniformly in opposite directions. can be made to relate these jumps to order of µ/2…
Figure 12
Figure 12. Figure 12: Evolution of kinetic energy Ekin, elastic energy Eelas (minus initial value due to far field loading), external work Wext, and frictional work Wfric versus time for the simulation (a) with no boundary wave reflection and (b) with boundary wave reflections. (c) Zoomed …
Figure 14
Figure 14. Figure 14: (a) Friction coefficient evolution at two points on the interface located at x = 0.125 [m] and x = 0.375 [m] plotted in red and blue, respectively. (b) Variation of maximum slip velocity on the interface minus the initial steady state velocity during the simulation ti…
Figure 15
Figure 15. Figure 15: Propagation velocity of the left moving shear rupture front is recorded for both cases of velocity weakening￾strengthening and pure velocity weakening and are plotted in (a) and (b), respectively. cs and cr are the shear and Rayleigh wave speeds. (c) Evolution of diff…
Figure 16
Figure 16. Figure 16: Contour of horizontal component of velocity field at two time instants during the initial propagation phase of shear rupture fronts and after they meet and cross each other. Solid blocks shown in this figure are cut at the height of 0.55 [m] for clarity, while their h…

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.