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

Dynamic weakening of rock friction at coseismic slip rates has been extensively studied in laboratory experiments using centimeter-scale samples. However, the potential scale dependence remains poorly understood. Recent studies suggest that the sample-size effect exists and may be explained by thermoelastic instability (TEI). In thermoelastic media, concentrated frictional heating and thermal expansion enhance normal stress heterogeneity. If this positive feedback dominates over diffusive heat transfer, the heterogeneity grows spontaneously, locally triggering dynamic weakening. The critical slip rate $${V}_{\text{c}\text{r}}$$ V cr for TEI is proportional to the wavenumber and thus inversely proportional to sample size. Here, TEI coupled with temperature-weakening friction is investigated through numerical simulations employing a newly developed method for a simplified two-dimensional periodic system. The periodic length is interpreted as the sample size, and simulation conditions are chosen to be comparable to those in laboratory experiments. When the nondimensional slip rate $$V/{V}_{\text{c}\text{r}}$$ V / V cr is smaller than unity, the behavior is independent of sample size and characterized by gradual weakening due to nearly uniform temperature rise. When $$V/{V}_{\text{c}\text{r}}$$ V / V cr exceeds unity, TEI causes growing heterogeneity, but dynamic weakening does not necessarily occur by the end of the experiments. It becomes significant only when partial contact opening develops. In this regime, macroscopic frictional resistance is strongly correlated with the macroscopic contact fraction. At sufficiently large slip rates and cumulative slip, only a small fraction of the fault remains in contact, accompanied by intense localization of temperature rise and normal stress. Although $${V}_{\text{c}\text{r}}$$ V cr depends inversely on sample size, the onset of dynamic weakening exhibits much weaker size dependence for sufficiently large systems, indicating the existence of a limiting length scale for the sample-size effect. This behavior is explained by the fact that the TEI growth rate is maximized at a wavenumber that is independent of the sample size: when the system is large enough to accommodate this mode, it dominates the evolution irrespective of sample size, whereas smaller systems are controlled by size-limited modes. These results suggest the existence of a representative length scale for constitutive description, providing a possible basis for upscaling laboratory friction laws to natural faults, although experimental validation is required. Graphical Abstract