The onset of the frictional motion of dissimilar materials

Abstract

Frictional motion between contacting bodies is governed by propagating rupture fronts that are essentially earthquakes. These fronts break the contacts composing the interface separating the bodies to enable their relative motion. The most general type of frictional motion takes place when the two bodies are not identical. Within these so-called bimaterial interfaces, the onset of frictional motion is often mediated by highly localized rupture fronts, called slip pulses. Here, we show how this unique rupture mode develops, evolves, and changes the character of the interface's behavior. Bimaterial slip pulses initiate as "subshear" cracks (slower than shear waves) that transition to developed slip pulses where normal stresses almost vanish at their leading edge. The observed slip pulses propagate solely within a narrow range of "transonic" velocities, bounded between the shear wave velocity of the softer material and a limiting velocity. We derive analytic solutions for both subshear cracks and the leading edge of slip pulses. These solutions both provide an excellent description of our experimental measurements and quantitatively explain slip pulses' limiting velocities. We furthermore find that frictional coupling between local normal stress variations and frictional resistance actually promotes the interface separation that is critical for slip-pulse localization. These results provide a full picture of slip-pulse formation and structure that is important for our fundamental understanding of both earthquake motion and the most general types of frictional processes.

Keywords: earthquake dynamics; fracture; friction; rupture fronts; seismic radiation.

Fig. 1.

Slow ruptures are bimaterial cracks. (A, Upper) Experimental system with PMMA (pink) and PC (blue) blocks. Strain gauges on block faces (z = 0 and z = 6 mm) are mounted at y locations ∼3.5 and ∼7 mm above and below the interfaces (in green); on one face at 12 locations. An additional three gauges are mounted at y = ∼7-mm height on opposing faces at the same x locations in order to verify that strain measurements are $z\hspace{0.17em}\text{independent}$ (see SI Appendix). F_N and F_S are normal and shear loads applied via load cells. (A, Lower) Instantaneous real contact area, A(x,z,t), measurements are performed along the entire interface by total internal reflection of an incident sheet of light everywhere except at contacting points (3). The transmitted light is roughly proportional to A(x,z,t). (B) Measured strain field compared to theory and simulation. The spatial structure of $\mathrm{\Delta }{\stackrel{}{\epsilon }}_{ij}\left(x-x_{tip}\right)$ at height h = 7 mm inside the stiff (Left) and soft (Right) materials; x_tip is the rupture tip location. Strain variations, $\mathrm{\Delta }{\epsilon }_{ij}$, are obtained by subtracting initial/residual values. $\mathrm{\Delta }{\stackrel{}{\epsilon }}_{ij}\left(x-x_{tip}\right)$ for both materials are normalized as $\mathrm{\Delta }{\stackrel{\sim }{\epsilon }}_{ij}=\mathrm{\Delta }{\epsilon }_{ij}\left(x-x_{tip}\right)/\mathrm{\Delta }{\epsilon }_{xx}^{0,stiff}$ where $\mathrm{\Delta }{\epsilon }_{xx}^{0,stiff}$ is the peak of the stiffer material ahead of x_tip (gray arrow). Colors: measurements for ruptures in the range C_f = 70 to 600 ms⁻¹ = 0.08 to 0.6$C_S^{soft}$. Corresponding $\mathrm{\Delta }{\stackrel{\sim }{\epsilon }}_{ij}\left(x-x_{tip}\right)$ predicted by bimaterial LEFM (black) and numerical simulations (dashed orange) for frictionless crack-like ruptures under plane stress conditions at C_f = 0.38$C_S^{soft}$ compare well with measurements. (B, Right Insets) Angular function measurements $E_{ij}\left(C_f,\theta \right)=\mathrm{\Delta }{\stackrel{\sim }{\epsilon }}_{ij}\left(r,\theta \right)\cdot \sqrt{r}$, for heights h = 3.5 and h = 7 mm within the soft material at C_f = 160 ms⁻¹ = 0.18$C_S^{soft}$ collapse to the predicted $E_{ij}\left(C_f,\theta \right)$ for bimaterial LEFM cracks (black line). (C) Space–time evolution A(x,z,t) of slow ruptures propagating at $C_f=0.6\hspace{0.17em}{C}_S^{soft}$ along the positive direction. (Left) A(x,t) = < A(x,z,t)>. (Right) Corresponding A(x,z,t) at times, t, denoted by the colored bars at the left. A(x,z,t) was normalized at t = 0.

Fig. 2.

Slip-pulse evolution. (A) A(x,z, t) of a typical slip pulse propagating at $C_f=1.024\hspace{0.17em}{C}_S^{soft}$ in the positive direction along a bimaterial interface. (Left) A(x,t) = <A(x,z,t)>. (Right) Corresponding A(x,z,t) to times, $t_i$, denoted by the colored bars at left. A(x,z, t) was normalized at t = 0. The blue arrow denotes the large drop in A(x,t) at the slip-pulse tip that is characteristic of a slip pulse. (B) Slip pulses are transonic $\left(>C_S^{soft}\right)$ and typically evolve with the propagation distance. $\mathrm{\Delta }{\epsilon }_{xx}$ of the slip pulse in A at increasing spatial positions. Note that, while the slip-pulse amplitude increases, its width remains fixed to approximately the width of the A(x,t) reduction in A. Maximal values, $\mathrm{\Delta }{\epsilon }_{xx}^{max}$ and (Inset) $\mathrm{\Delta }{\epsilon }_{yy}^{max}$, are defined by red arrows. (C) Strain amplitudes for slip pulses (purple) are up to an order of magnitude larger than those of (blue) bimaterial cracks $\left(<C_S^{soft}\right)$ despite the narrow range $C_S^{soft}<C_f<1.04\hspace{0.17em}{C}_S^{soft}$ of slip-pulse velocities compared with the large range for cracks speeds $0<C_f<C_S^{soft}$. $C_S^{soft}$ is denoted by the dashed red line. (Inset) Maximal strain amplitudes $\mathrm{\Delta }{\epsilon }_{yy}^{max}$ as a function of $\mathrm{\Delta }{\epsilon }_{xx}^{max}$. All strain units are milli-strain.

Fig. 3.

Slip pulses are bimaterial transonic cracks localized by friction. (A) Measured $\mathrm{\Delta }{\sigma }_{ij}\left(x-x_{tip}\right)$ at height h = 7 mm in stiff (Left) and soft (Right) blocks for a slip pulse at $C_f=1.035C_S^{soft}$ under applied normal loading conditions of ${\sigma }_{yy}=3.1\hspace{0.17em}\text{MPa}$. Stress fields (green) beyond the leading edge of transonic slip pulses compare well with transonic crack solutions described by (black line) the analytic solution (see SI Appendix for details) with $q=0.17$. Signals are normalized as in Fig. 1C. (B) The exponent $q$ of Eq. 1 for $C_S^{soft}<C_f<1.04\hspace{0.17em}{C}_S^{soft}$. Material properties correspond to experiments. (Inset) $q$ over the range $1<C_f/C_S^{soft}<\gamma$. (C) Including frictional coupling causes slip-pulse localization. (Green) Measurements of $\mathrm{\Delta }{\sigma }_{xx}^{soft}\left(x-x_{tip}\right)$ are compared with frictionless transonic crack solutions (dotted blue line) and a numerical solution at $C_f/C_S^{soft}=1.006$ that incorporates Coulomb friction (red line) with a kinetic friction coefficient of 0.64. Signals, ${\stackrel{\sim }{\sigma }}_{xx}^{soft}$, are scaled to have the same amplitude. (D) Incorporating friction leads to interface separation at the interface. Numerical frictionless cracks at $C_f/C_S^{soft}=1.015$ (blue line) are compared with the numerical solution incorporating friction (red line). The solutions were not rescaled. (Upper) ${\sigma }_{yy}$ at the interface $\left(y=0\right)$ for both solutions. Including friction induces ${\sigma }_{yy}\left(x,0\right)=0$, indicating separation. The interface separation coincides with slip-pulse amplitudes significantly larger than those of the transonic cracks (Lower) providing an effective localization mechanism.

Fig. 4.

The limiting velocity ${\mathit{C}}^{\mathit{lim}}$ for transonic rupture is determined by the velocity where ${\mathit{\sigma }}_{\mathit{yy}}$ at the interface behind ${\mathit{x}}_{\mathit{tip}}$ becomes compressive. (A) ${\sigma }_{yy}\left(x,0\right)$ profiles of the analytical solution at different transonic velocities. For all velocities, ${\sigma }_{yy}\left(x>x_{tip},0\right)$ is compressive (<0). Behind $x_{tip}$, ${\sigma }_{yy}\left(x<x_{tip},0\right)$ changes its sign from reduction of its background value to increased compression $\left({\sigma }_{yy}<0\right)$. (B) $A^{-}\left(C_f\right)$ (Eq. 1) describes the magnitude of ${\sigma }_{yy}$ behind $x_{tip}$. $A^{-}\left(C_f\right)$ changes its sign at a velocity of $C_f=1.041C_S^{soft}$. (C) Probability distribution of all measured ruptures in the positive direction below $C_S^{stiff}$. $C^{lim}$, which is denoted by the purple arrow, corresponds to the highest measured rupture velocity in the positive direction. The red line denotes $C_S^{soft}$. (Inset) An expanded view of the measured distribution of $C_f$ in the transonic region.