Fig. 3: The breakdown energy of frictional rupture.
From: Unconventional singularities and energy balance in frictional rupture
a The breakdown energy EBD(X(t); xi) as a function of X(t)/W for \(t\ > \ {t}_{{x}_{i}}\) (see text for definition), obtained in numerical simulations of rate-and-state frictional interfaces23, for four different observation points x1−4 (see legend). W is the fault/interface half-length and we set \({t}_{{x}_{i}}\ =\ 0\) for i = 1 − 4 (for presentational convenience). All curves perfectly overlap over a short lengthscale, which identifies with ℓ (see Fig. 2), defining the effective fracture energy Gc (dashed-dotted horizontal line), but branch out on larger scales. See text for additional discussion. b \({{\Delta }}{{\mathcal{E}}}_{{\rm{BD}}}(X;{x}_{i})\), corresponding to the data presented in panel a, vs. \({\mathrm{ln}}\,[X/\ell ({x}_{i})]\) for x1/W = 0.3 and x4/W = 0.6 (see legend). \({{\Delta }}{{\mathcal{E}}}_{{\rm{BD}}}(X;{x}_{i})\) follows a logarithmic behavior at an intermediate range, as highlighted by the titled dashed lines (the slope of the lower line is 0.094 and that of the upper one is 0.165, their ratio is 1.76). \({{\Delta }}{{\mathcal{E}}}_{{\rm{BD}}}(X;{x}_{i})\) presumably crosses over, roughly at \({\mathrm{ln}}\,[{x}_{i}/\ell ({x}_{i})]\) (light-gray vertical lines), to a plateau (illustrated by the horizontal dashed lines).
