Research ArticleGEOPHYSICS

The equation of motion for supershear frictional rupture fronts

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Science Advances  18 Jul 2018:
Vol. 4, no. 7, eaat5622
DOI: 10.1126/sciadv.aat5622
  • Fig. 1 Experimentally measured supershear rupture velocities.

    (A) Two PMMA blocks are used in a stick-slip friction experiment. We consider the elastic medium to be 2D with a quasi 1D frictional interface and measured the full 2D tensorial strains along and ≈3.5 mm above the frictional interface (blue squares). (B) Real contact area A(x, t) measurements along the interface, normalized before the event, show rupture nucleation at x ≈ 0, acceleration, and transition to supershear at x ≈ 50 mm. (C) Shear stress variations, τ = σxyσxyr, relative to the rupture tip arrival time, ttip, measured at x = 105 mm. Arrows denote the dynamic stress drop τ0 and peak shear strength τp. We obtain τp from the measured cohesive zone size. The gray curve schematically depicts the interfacial shear stress evolution. (D) Measured rupture velocities for two examples of rupture events in experiments with a dry interface. (B), (C), and the blue example in (D) correspond to the same rupture event.

  • Fig. 2 Comparing theoretical predictions of supershear crack velocities with numerical simulations and experimental measurements.

    (A) Top: Spatially uniform τ0 and nonuniform τp profiles are considered in simulations. The imposed τ0p profiles are shown. Bottom: Colors represent the crack velocities Cf(l) corresponding to the stress profiles in the top panel. τp is low, dashed lines, near the point of nucleation (l/l0 = 0) and increases at distances 50 for τ0p > 0.2 and at l/l0 = 130 for τ0p = 0.2 (red). Black solid lines denote predictions of Eq. 1. For sufficiently low τ0p (red example), two solution branches to Eq. 1 exist (dashed line shows the unstable solution). (B) Top: Profiles of the measured shear stress τ0 and estimated interfacial strength τp in experiments. Bottom: Measured rupture velocities for rupture events with stresses shown in the top panel. Theoretical predictions according to Eq. 1 are shown (solid lines); average τ0 values are used, l0 = μГ/τp2 = 2 mm. Dashed lines correspond to the error estimates of l0 and τp. (C) Cf, measured at l = 120 mm (averaged over ±10 mm), for multiple experiments is plotted with respect to the measured τ0 profiles averaged over the same interval. Solid dots and open circles indicate experiments performed with dry and boundary-lubricated interfaces, respectively. Black lines are the theoretical predictions with the estimated errors.

  • Fig. 3 Minimal lengths exist for supershear crack propagation under uniform loading.

    (A) Crack simulations for nonuniform τ0p profiles (see Fig. 2A, top), where the size of the weak patch is varied. Dashed and solid curves indicate propagation within the weak and strong regions, respectively. τ0p values in the figure correspond to the strong region (in the weak region, τ0p = 0.8). Supershear propagation in the strong region cannot be sustained for l < lm (lm is indicated by stars), where the supershear–to–sub-Rayleigh transition occurs. (B) lm obtained numerically (green line) from Eq. 1 shows that no minimal length exists for τ0p ≳ 0.45 (for fixed CS/CL). Normalization by l0 and τp is applied for convenient comparison to (A). Stars correspond to the values denoted in (A). The dashed line is the analytic approximation based on Eq. 2.

  • Fig. 4 Comparing theoretical predictions of supershear rupture velocities to numerical simulations for nonuniform prestress.

    (A) Top: Spatially nonuniform τ0 profiles with uniform τp profiles are considered. The imposed τ0p profiles are shown. Bottom: Rupture velocities Cf(l) corresponding to the stress profiles in the top panel. Top and bottom: A high-stress nucleation patch (τ0p = 0.8 region) facilitates a direct supershear transition. Black solid lines show theoretical predictions for the nonuniform loading described by Eq. 3. Note the difference of Cf(l) profiles with those in Fig. 2A for similar τ0p profiles. Predicted locations for supershear–to–sub-Rayleigh transition are marked by gray triangles. (B) Comparison of supershear–to–sub-Rayleigh transition positions measured in 50 simulations, with the predicted transition position based on Eq. 3. In the considered τ0(x) profiles, the size of the high-stress nucleation patch is modified [see yellow curve in (A)]. Dashed line with slope 1 and is given for reference. Colors indicate the minimal needed increase in τ0 within the low-stress region that would negate the predicted supershear–to–sub-Rayleigh transitions. Gray triangles denote the simulations shown in (A).

  • Fig. 5 The effect of sub-Rayleigh–to–supershear transition on the equation of motion of supershear cracks.

    Three different transition mechanisms are considered. Top: The first setup (brown curve) has a spatially nonuniform τ0p profile with reduced local τp for l/l0 < 50 (see main text for details). Two additional examples have spatially uniform τ0p profiles (orange and red curves). Bottom: Colors represent the crack velocities Cf(l) corresponding to the stress profiles in the top panel. Brown curve indicates continuous crack acceleration to supershear speeds (direct transition) within a weakened nucleation (nucl.) patch (high τ0p level). Orange curve indicates sub-Rayleigh rupture transitions at l/l0 ≈ 65 to supershear speed through the Burridge-Andrews (BA) mechanism (4, 5). Red curve indicates an imposed supershear seed crack leads to a self-sustained supershear crack propagation. The black dashed line denotes theoretical prediction for a spatially uniform prestress level (τ0p = 0.45).

  • Fig. 6 Calculated g(β) and B(Cf/CL)ГD(g).

    g(β) and the product of B(Cf/CL) and ГD(g), as appearing in Eq. 1, are provided for CS/CL = 0.48 (top) and CS/CL = 0.577 (bottom). ГD(g) was calculated assuming a linear cohesive zone model.

Supplementary Materials

  • Supplementary Materials

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    • Fig. S1. Comparison of theoretical predictions of supershear crack velocities with numerical simulations for various shear strength levels for v = 0.25 (plain strain).

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