Thermoelastic instability on a frictional surface and its implication for size effect in friction experiments

<jats:title>Abstract</jats:title><jats:p>Recent laboratory friction experiments on large rock samples revealed that dynamic weakening, a remarkable reduction in the friction coefficient at elevated slip rates, occurs at lower slip rates in larger samples. There is a large difference between the sizes of natural faults and those in laboratory experiments. Therefore, it is crucial to understand the effect of size on rock friction. In the field of tribology, the interaction between frictional heating and thermoelastic effect has long been investigated. It was shown that higher slip rates than the critical value <jats:inline-formula><jats:alternatives><jats:tex-math>$${V}_{\mathrm{cr}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>V</mml:mi> <mml:mi>cr</mml:mi> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> causes growth of temperature and normal stress heterogeneity (thermoelastic instability), and <jats:inline-formula><jats:alternatives><jats:tex-math>$${V}_{\mathrm{cr}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>V</mml:mi> <mml:mi>cr</mml:mi> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> is proportional to the wavenumber of the heterogeneity. Severely heterogeneous normal stress may cause concentration of frictional power, thus locally activating dynamic weakening and leading to macroscopic weakening. Because a larger sample hosts a perturbation of a smaller wavenumber, it is expected to weaken at a lower slip rate than a smaller sample. In this study, a new numerical method was developed for analysis of thermoelastic instability based on the definition of memory variables and numerical approximation to the integration kernel, for the 2-dimensional problem of a planar fault embedded in an infinite medium. This method was advantageous over the standard integral equation method in terms of numerical costs. Numerical solutions with the new method on sinusoidal perturbations in the normal stress were compared with previously derived steady-state solution and its stability for validation. The typical thermoelastic properties of gabbro yield <jats:inline-formula><jats:alternatives><jats:tex-math>$${V}_{\mathrm{cr}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>V</mml:mi> <mml:mi>cr</mml:mi> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> in a range of experimentally adopted slip rates, indicating that the thermoelastic effect may play an important role in high-velocity friction experiments. Because the temperature rise and the resulting normal stress change smear out after the friction experiments, measurement of the temperature distribution in a sample during a friction experiment is important for further understanding the dynamic weakening and scale effect of rock friction.</jats:p> <jats:p><jats:bold>Graphical Abstract</jats:bold></jats:p>