Thermoelastic Instability Sets a Limit to Sample-Size Effects in Rock Friction

Why do laboratory measurements of rock friction sometimes fail to describe how real faults behave during earthquakes? A new theoretical study published in Earth, Planets and Space argues that part of the answer lies in the size of the rock sample itself, and that this size effect does not grow without bound. Once a sample is large enough to contain a particular fastest-growing disturbance wavelength, further enlargement no longer changes when dynamic weakening begins. The finding offers a possible route for converting centimeter-scale laboratory friction laws into constitutive rules that can be used in continuum models of natural faults, although the author emphasizes that experimental validation is still required.

Dynamic weakening of rock friction at coseismic slip rates has been studied for decades, mostly with centimeter-scale specimens. Whether the results scale up has remained poorly understood. Recent work suggested that a sample-size effect exists and may be explained by thermoelastic instability, or TEI, a phenomenon long studied in engineering. In a thermoelastic medium, concentrated frictional heating causes local thermal expansion, which raises the normal stress where the surfaces press together most tightly. That higher normal stress generates still more heating, a positive feedback. If this feedback outruns the smoothing effect of diffusive heat transport, small perturbations in normal stress grow spontaneously, and dynamic weakening can be triggered locally. The critical slip rate for this instability is proportional to the wavenumber of the disturbance and therefore inversely proportional to the sample size, which is precisely why a larger sample should become unstable more easily.

The new study goes beyond the earlier linearized analysis by simulating the full nonlinear evolution of the instability. Noda developed a new spectral method for computing the on-fault temperature, complementing an existing method for the normal stress, and coupled both to a temperature-weakening friction law in which the local friction coefficient drops linearly as temperature rises. The model is a deliberately simplified two-dimensional system: a planar frictional surface embedded in an infinite thermoelastic medium, with anti-plane slip, quasistatic elasticity, and periodic boundary conditions. The periodic length of the domain, denoted W, serves as a proxy for the sample size, with values ranging from 0.1 millimeters to 100 millimeters. The thermoelastic properties were taken from gabbro, and the mean initial normal stress was set to 6.7 megapascals, matching the conditions of large-scale high-velocity friction experiments reported by Yamashita and colleagues in 2015.

The simulations were designed to mirror laboratory experiments in other respects as well. Each numerical experiment ran to a total slip of 0.4 meters, comparable to the large-specimen experiments, with slip rates between 1 millimeter per second and 0.1 meters per second. The initial normal stress distribution was generated as a spatially correlated Gaussian random field whose amplitude is independent of sample size while its wavelength scales linearly with W, reflecting the reasonable expectation that larger samples carry larger topographic fluctuations. The friction parameters, an initial coefficient of 0.75 and a temperature-weakening coefficient of minus 0.4 per 1000 kelvin, were based on high-speed friction experiments on dolerite. The numerical scheme used an adaptive exponential time-differencing integrator, and the method was validated against a benchmark problem with a constant friction coefficient, achieving errors of about 0.003 megapascals.

The results divide cleanly into two regimes separated by a nondimensional slip rate, v, defined as the actual slip rate divided by the critical slip rate for thermoelastic instability. When v is below one, the behavior is independent of sample size. Thermal diffusion homogenizes the temperature, the fault heats up nearly uniformly, the normal stress barely evolves, and the macroscopic friction coefficient declines only modestly. In one representative case with a 1-millimeter sample slipping at 0.1 meters per second, the friction coefficient fell from 0.75 to 0.676 over 0.4 meters of slip. Simulations with 0.1-millimeter and 1-millimeter samples produced nearly identical results, indicating that these cases approach a small-sample limit in which size no longer matters at all.

Above the critical slip rate, the picture changes dramatically. Thermoelastic instability drives growing heterogeneity in both temperature and normal stress, but dynamic weakening does not necessarily follow within the duration of an experiment. Significant weakening develops only when the feedback becomes strong enough to open parts of the fault, so that contact is lost over patches of the surface. In the most unstable case examined, a 100-millimeter sample slipping at 0.1 meters per second, dynamic weakening began at less than 0.1 meters of slip and the macroscopic friction coefficient dropped to 0.436. By the end of that simulation, only a small fraction of the fault remained in contact, with local temperature rises approaching 992 kelvin and local normal stresses reaching 194 megapascals, a stress concentration factor of nearly 30. Seven contact patches roughly 1 millimeter long survived, with maximum opening just under 8 micrometers.

Across the full two-dimensional parameter study, the macroscopic friction coefficient correlated strongly with the macroscopic contact fraction, and the data from all simulated cases collapsed onto a single master curve relating the two quantities. The critical slip rate line, where v equals one, marked the onset of dynamic weakening only for relatively small samples below about 10 millimeters. For larger samples, the onset of weakening showed no significant dependence on size, even though the critical slip rate itself keeps falling as samples grow. The author explains this saturation through the structure of the linearized growth rate: among all disturbance wavelengths that a sample can accommodate, one particular wavelength grows fastest, at a wavenumber equal to about 0.472 times the critical wavenumber. Once a sample is long enough to contain that mode, the fastest-growing disturbance is the same regardless of size, so the evolution of the instability becomes effectively size-independent. Smaller samples, by contrast, are controlled by the longest wavelength they can hold, which does depend on size.

This saturation is more than a numerical curiosity. If laboratory friction measurements depended on sample size without limit, they could not be used as point-wise material properties in continuum mechanics at all. The existence of a limiting length scale is a necessary condition for a representative elementary scale and therefore for a constitutive law that can be transferred from the laboratory to natural faults. Using the gabbro properties and a nondimensional onset parameter of five, the study derives an expression for the slip accumulated when TEI-driven weakening begins in the large-sample limit: at a seismic slip rate of 0.1 meters per second, shear traction averaged over a region wider than about 2.6 millimeters is predicted to begin weakening at roughly 6 centimeters of slip, in addition to any weakening from the overall temperature rise. Notably, this predicted onset slip is independent of the applied normal stress, in contrast to the slip-weakening distance associated with uniform heating, which decreases as normal stress increases.

The study also connects its findings to flash heating, the classical mechanism in which microscopic asperities heat up almost instantaneously during sliding. In the simulations, the extreme concentration of temperature and stress onto small contact patches emerges from the macroscopic thermoelastic model itself, and the peak temperatures correspond to hot-spot temperatures rather than the flash temperatures measured in metal friction experiments. The calculated contact patches, on the order of a millimeter or larger, may exceed the real contact areas typical of laboratory samples, and at still higher slip rates the continuum approximation would eventually break down as discrete asperity effects become important.

Important caveats remain. The model neglects variations in slip direction, inertial effects, geometric finiteness, and wear processes, all of which are likely to matter for quantitative comparison with experiments. The simulated onset of weakening somewhat underestimates the slip rates observed in the large-sample experiments, and the limitation of the sample-size effect has not been confirmed experimentally, since building meter-scale or larger apparatus is technically and financially demanding. The steady-state friction in the model also depends on system-level factors such as the surrounding apparatus, because heat conducted away through metal components can suppress the instability. The author suggests that spatial measurements of temperature distribution during friction experiments would help identify whether overall heating or thermoelastic instability dominates weakening in any given test, a distinction that determines how laboratory results should be scaled up to the much larger length scales of natural faults.

Subject of Research: Sample-size dependence of dynamic rock friction weakening caused by thermoelastic instability

Article Title: Sample-size effect and its limiting length scale in dynamic weakening of rock friction: insights from thermoelastic instability coupled with temperature-weakening friction

Article References: Noda, H. (2026). Sample-size effect and its limiting length scale in dynamic weakening of rock friction: insights from thermoelastic instability coupled with temperature-weakening friction. Earth, Planets and Space, 78(1), Article 234. https://doi.org/10.1186/s40623-026-02556-y

Image Credits: AI Generated

DOI: 10.1186/s40623-026-02556-y

Keywords: rock friction, thermoelastic instability, dynamic weakening, sample-size effect, earthquake fault mechanics, high-velocity friction experiments, temperature-weakening friction, fault constitutive law, frictional heating, seismic slip, numerical simulation, upscaling laboratory friction laws