The onset of the frictional motion of dissimilar materials

Significance

We consider the most general type of frictional motion: frictional sliding of nonidentical bodies (“bimaterials”) with either different elastic properties or geometrical shapes. By coupling experiments, theory, and numerics, we show that, upon nucleation, rupture fronts akin to shear cracks initiate the rupture of the contacts forming a frictional interface. These ruptures, however, rapidly accelerate and transition to highly localized “slip pulses”: singular fronts unique to bimaterial interfaces, in which frictional slip is spatially confined to a small region near their leading edge. These results provide important fundamental understanding to all communities interested in frictional processes. In particular, they relate to questions of directed damage and near-field radiation patterns generated by earthquakes within natural faults bordered by different 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.

Frictional interfaces between two bodies connect surfaces that are generally rough. They are composed of discrete points of contact, whose total area depends on the normal load (1, 2). These interlocked points resist any relative motion and are the origin of frictional strength. For frictional motion to occur, contacts must first be fractured (3, 4). The fracturing process is performed by rapid rupture fronts that are essentially earthquakes (5, 6). These fronts propagate across the interface (7–10), detaching the contacts to enable their relative motion (3, 4, 8, 9, 11–13). Rupture fronts within homogenous interfaces, interfaces separating identical materials, are quantitatively described as shear cracks (12, 14, 15). Along interfaces bordered by elastically different (6, 16–18) or geometrically dissimilar (19) materials, local interface slip is coupled to normal stress variations. This purely elastic effect is called “bimaterial coupling.” Bimaterial coupling gives rise to entirely new modes of rupture: slip pulses (20, 21), pulse-like ruptures where slip is localized at their leading edge. Slip pulses contrast sharply with shear cracks; in a crack, in contrast to the localization that characterizes of slip pulses, interface slip extends over considerable distances away from the rupture tip (22).

Within homogenous interfaces, such localized ruptures can only be formed when the friction law allows strong velocity weakening of the interface (18, 23, 24). Bimaterial slip pulses, however, can result solely from bimaterial coupling; the form of the friction law is unimportant (6, 16–18). The strength of bimaterial coupling is often characterized by the material contrast, $\gamma =C_s^{stiff}/C_s^{soft}$, although a full characterization (18, 19) requires a description that involves the bimaterial mismatches between all elastic parameters. $C_S^{soft}$ and $C_S^{stiff}$ are the shear wave velocities of the elastically softer and stiffer materials, respectively. Any $\gamma \ne 1$ gives rise to bimaterial coupling. Important natural examples include strike-slip faults or subduction zones (25) that are bounded by different rock types.

Previous theoretical analysis attempted to predict the selection of slip-pulse propagation directions and limitations on their propagation velocities, $C_f$. Slip pulses only exist in the positive direction, when slip of the softer material (16, 17) is parallel to the rupture propagation direction. Values of limiting propagation velocities, $C^{lim}$, have been widely debated (17, 24, 26, 27). For $1.3>\gamma \ge 1$, $C^{lim}$ corresponds to a “generalized Rayleigh” surface wave speed, $C_{GR}$ (16, 20). At higher $\gamma$, $C_{GR}$ does not exist. Some simulations suggest that $C^{lim}$ is limited (17) by $\sim C_S^{soft}$, while others (27, 28) suggest that $C_f$ may reach transonic velocities, defined by $C_S^{soft}<C_f<C_S^{stiff}$. Note that transonic velocities are less than supershear ruptures, for which $C_f>C_S^{stiff}$. Experiments (21) have verified both the existence of bimaterial coupling and its importance for slip-pulse formation in the positive direction. Experimentally, slip pulses propagate at $C_f\sim C_{GR}$ for low (20) $\gamma$ and $C_f\sim C_S^{soft}$ for higher (21) $\gamma$. In the negative direction, when slip of the softer material opposes the rupture direction, slip pulses are neither expected (20, 21, 28) nor measured (21).

Below, we show that bimaterial slip pulses initiate as true shear cracks: singular cracks with an explicit form that depends on the bimaterial nature of the interface. We reveal that in the positive direction, bimaterial cracks lose stability to slip pulses. For $\gamma >1.3$, slip pulses propagate at transonic velocities. Our analysis correctly predicts both their unique singular form and their dynamically limited velocity. Moreover, we show that strong spatial localization of slip pulses is actually caused by frictional coupling to normal stresses.

Results

Our experiments were performed on interfaces bounded by blocks of polycarbonate (PC) and polymethylmethacrylate (PMMA) (Fig. 1A). Details are in Methods.

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.

Within bimaterial interfaces, normal stress variations can and will occur as a result of the rupture's passage. Our analytical solutions, however, assume a constant (potentially nonzero) residual shear stress behind the crack tip and therefore, will not couple interface friction to variations in ${\sigma }_{yy}$. For simplicity, we call this contact-pressure–independent strength approach a “frictionless” solution. While the analytical solutions consider only the leading singular component and a constant $C_f$, our simulations are not limited by these assumptions and solve the entire elastodynamic problem (SI Appendix) in both the frictionless (i.e., contact-pressure–independent) case and when friction is locally coupled to ${\sigma }_{yy}$ (Coulomb friction). In both the frictionless and Coulomb friction cases, the simulations introduce a cohesive zone at the rupture tip, in which the local shear stress decreases from its peak to residual value. The size of this cohesive zone is ∼1 mm and thus, of comparable order to variations in other stress fields (i.e., normal stress) (see Fig. 3D), but much smaller than the scale of the rupture.

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. 1C presents a typical rupture propagating in the positive $x$ direction at the velocity of $C_f\sim 0.6C_s$. In such crack-like ruptures, the contact area $A\left(x,t\right)$ drops like a step function to 0.8 to 0.9$A\left(x,t=0\right)$, as do homogenous shear cracks (12, 21). Dynamic strain measurements are compared with both frictionless analytic and numerical solutions in Fig. 1B for ruptures spanning the “slow” velocity range of 0.04 to 0.6$C_S^{soft}$. Including Coulomb friction in the simulations has virtually no effect on the strain signals, as shown in SI Appendix, Fig. S1. After scaled to spatial coordinates, $x-x_{tip}$, and normalized by $\mathrm{\Delta }{\stackrel{\sim }{\mathrm{\epsilon }}}_{\mathrm{xx}}^0$ (Fig. 1B), all six components of ${\epsilon }_{ij}$ (from both the stiff and soft sides of the interface), throughout this range, collapse to well-defined structures. Our singular crack solutions (SI Appendix) for each $C_f$ both agree well with the measurements and simulations and scale like the LEFM (linear elastic fracture mechanics) solutions for homogeneous interfaces (12, 22), $\mathrm{\Delta }{\epsilon }_{ij}\left(r,\theta \right)\text{∝}{\mathrm{{\rm E}}}_{\mathit{\text{ij}}}\left(C_f,\theta \right)/\sqrt{r}$ for all $C_f<C_S^{soft}$. This excellent correspondence with measured strains might be surprising in light of our neglect of the coupling of frictional strength to ${\sigma }_{yy}$. The analytic and numerical solutions at the interface, however, reveal that magnitudes of $\mathrm{\Delta }{\sigma }_{yy}$ for slow velocities are relatively small (16, 29). $A\left(x,t\right)$ measurements, reflecting $\mathrm{\Delta }{\sigma }_{yy}\left(x,y=0,t\right)$, are consistent with these predictions, having no significant variation behind the rupture tip. Further confirmation that the $1/\sqrt{r}$ singular solution dominates this velocity range is provided by measured $\mathrm{\Delta }{\epsilon }_{ij}$ at y = 3.5 and 7 mm. When normalized by $\sqrt{r}$, all $\mathrm{\Delta }{\epsilon }_{ij}$ components perfectly collapse to the predicted angular functions, ${\mathrm{{\rm E}}}_{\mathit{\text{ij}}}\left(C_f,\theta \right)$, as shown explicitly for $\mathrm{\Delta }{\epsilon }_{ij}^{soft}$ in Fig. 1B, Insets. Thus, slow bimaterial rupture fronts, up to ∼0.6$C_S^{soft}$, are singular bimaterial cracks that are well described by fracture mechanics.

Such slow ruptures are, however, rarely observed (<10%) and are generally transient. In the positive direction, bimaterial ruptures generally accelerate almost immediately (Fig. 2A) to velocities near $C_S^{soft}$ and evolve into slip pulses. Slip pulses have a unique form; $A\left(x,t\right)\to 0$ at the rupture tip and within a few millimeters, dynamically return to 0.5–0.8 $A\left(x,t=0\right)$. With propagation distance, slip-pulse amplitudes continuously increase, while pulse widths remain constant (Fig. 2B). When scaled by their amplitudes, such evolving slip pulses collapse to a well-defined shape (SI Appendix, Fig. S3B). Slip-pulse velocities were previously reported (21) as $C_f\sim C_S^{soft}$. Improved measurement accuracy (SI Appendix) now conclusively reveals that slip-pulse velocities surpass $C_S^{soft}$, attaining transonic velocities of $1.03\hspace{0.17em}\text{to}\hspace{0.17em}1.04C_S^{soft}$. As Fig. 2C demonstrates, in the narrow transonic range (purple area in Fig. 2C) of $C_S^{soft}<C_f<1.03C_s^{soft}$, strain amplitudes for slip pulses are extreme, over an order of magnitude larger than all cracks at $C_f<C_S^{soft}$.

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.

The realization that slip pulses are transonic provides crucial insight to their structure. Beyond $C_S^{soft}$, we find that singular solutions with a leading order singularity of $r^{-1/2}$ do not exist. Instead, the only existent singular solutions (Eq. 1) for transonic frictionless cracks possess a $r^{-q}$ singularity, with $q\left(C_f\right)<1/2$. As for subshear cracks, solutions for frictionless transonic cracks assume a constant residual shear stress behind the crack tip. This condition (SI Appendix) also forces $q\left(C_f\right)$ not to have a complex part. For transonic cracks, the normal stress at the interface has the form

$${\sigma }_{yy}\left(x,0\right)\equiv \left[A^{-}\left(C_f\right)H\left(-\left(x-x_{tip}\right)\right)+A^{+}\left(C_f\right)H\left(x-x_{tip}\right)\right]\cdot \frac{K}{{\left|x-x_{tip}\right|}^q},$$ [1]

where $H(.)$ is the Heaviside function, $K$ is a stress intensity factor-like coefficient, and $A^{+,-}$ are $C_f$-dependent constants (SI Appendix, Fig. S4).

The exponent $q\left(C_f\right)$ (Fig. 3B) drops precipitously from 1/2 at $C_f=C_S^{soft}$ to $\sim 0.2$ within a few percent beyond $C_S^{soft}$. Fig. 3A demonstrates that this constant $C_f$ solution for transonic cracks provides an excellent quantitative description for the leading edges of the slowly evolving transonic slip pulses described by our measurements. Ahead of x_tip (in both the stiff and soft materials), this crack-like transonic solution captures all of the measured stress components. Behind x_tip, this agreement breaks down; Eq. 1 cannot capture the slip-pulse localization. The localization is particularly apparent in the stress measurements within the soft material where $\mathrm{\Delta }{\sigma }_{ij}$ measured at y = 3.5 and 7 mm have the same width (SI Appendix, Fig. S3A). Interestingly, ${\sigma }_{ij}$ within the stiff material all collapse to well-defined angular functions, when normalized by $r^q$ (SI Appendix, Fig. S3A). What is the origin of the slip localization that defines slip pulses? Without friction, local stresses in simulations decay slowly (Fig. 3 C and D), like the transonic solutions of Eq. 1. Fig. 3 C and D demonstrates that incorporation of friction in our simulations causes localization that is similar to experimental measurements. The numerical slip pulses are qualitatively and quantitatively different from numerical cracks. Adding Coulomb friction produces interface separation behind the crack tip. This region of separation apparently corresponds to the nearly total reduction in $A\left(x,t\right)$ at slip-pulse tips observed in the experiments (21). Interface separation is coincident with 1) the rapid decay of local stresses at the tail of transonic cracks that clearly resembles the localization of experimental slip pulses and 2) significant magnification of slip-pulse amplitudes.

We believe that the extreme amplitudes of the experimental slip pulses presented in Fig. 2 result from the interface separation. While we cannot experimentally confirm complete interface separation, the complete separation observed in the numerics is certainly consistent with the extreme reduction of $A\left(x,t\right)$ observed in experiments.

The velocity dependence provided by transonic solutions of ${\sigma }_{yy}$ magnitudes behind slip-pulse tips (Eq. 1) is the key for understanding the origin of the limiting velocity, $C^{lim}$, of slip pulses. Although the transonic solution exists for all $C_S^{soft}<C_f<C_S^{stiff}$, ${\sigma }_{yy}$ behind $x_{tip}$ changes sign at a critical value, $C^{lim}$, becoming compressive all along the interface (SI Appendix, Fig. S4). This is presented in Fig. 4A, showing the transition from reduced $\left({\sigma }_{yy}>0\right)$ to increased $\left({\sigma }_{yy}<0\right)$ compression for $x<x_{tip}$ as $C_f$ crosses $C^{lim}$. Any friction, therefore, will inhibit slip-pulse propagation beyond $C^{lim}$ due to the coupling of the normal stress to the frictional resistance. The exact value of $C^{lim}$ is found in Fig. 4B by plotting the velocity dependence of the coefficient $A^{-}\left(C_f\right)$ in Eq. 1, which describes the amplitude of ${\sigma }_{yy}$ behind $x_{tip}$. For our material parameters, $A^{-}\left(C_f\right)$ changes sign precisely at the velocity $C^{lim}\approx 1.04C_S^{soft}$ where slip pulses experimentally disappear (Fig. 4C).

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.

Discussion

We have shown that explicit analytical solutions for both bimaterial cracks and bimaterial transonic ruptures correspond well with measurements. Moreover, we have empirically demonstrated that, while all bimaterial ruptures start as cracks, the vast majority of these ruptures rapidly transition to transonic slip pulses that exist in an extremely narrow range of propagation velocities. While the analytic solution provides a quantitative and detailed mechanical explanation for a dynamic selection of the limiting velocity of the slip pulses, one might ask why bimaterial ruptures are pushed to this region in the first place. The key to addressing this question is provided by comparison of the numerical solution in which we added Coulomb friction to the interface of the frictionless solution. We find that not only does the incorporation of frictional coupling to ${\sigma }_{yy}$ give rise to the spatial localization of slip near the rupture tip but that this coupling drives interface separation at transonic velocities. Paradoxically, the effect of interface separation is only felt when frictional coupling to ${\sigma }_{yy}$ exists along the interface; separated interfaces have zero frictional dissipation, by definition. We surmise that it is the near-interface separation that leads to increased slip velocity and strain amplitudes at the rupture tip (Fig. 3) relative to those realized from frictionless models. This leads to the, perhaps, counterintuitive result that incorporating friction actually seems to reduce dissipation. We anticipate that this result is general and is independent of the explicit friction law along the interface; Amontons–Coulomb friction is just the simplest friction law to implement.

Does interface separation really take place, and if so, does the LEFM solution, that assumes interface contact, remain relevant? Empirically, our experiments show that the majority of slip pulses do have finite contact (e.g., the example provided in Fig. 3 actually reaches only a 30% contact). Moreover, the highest amplitude slip pulses, for which full separation may occur, are similar to slip pulses for which contact occurs (Fig. 2B and SI Appendix, Fig. S3B, where $\mathrm{\Delta }{\epsilon }_{xx}$, which vary by a factor of two, all collapse to the same form). The leading edges of all of these are described by the analytic theory. The question of near-tip separation also raises an interesting theoretical point. LEFM is relevant only up to scales that are limited by either nonlinear elasticity (30) or the cohesive zone at the crack’s tip. For homogeneous frictional ruptures, Svetlizky and Fineberg (12) measured the cohesive zone size to be at the millimeter scale, for approximately the same experimental conditions. This is the (3- to 4-mm) scale of the possible separation region in the bimaterial case. As such, even if separation does occur within this region, the LEFM assumption of contact still remains valid.

Why are the initially crack-like ruptures empirically drawn to the transonic rupture state? We understand this as a type of positive feedback mechanism. As an initial crack-like rupture increases its propagation velocity and therefore, its slip velocity, the bimaterial coupling at its tip further reduces the normal stress. Hence, increased $C_f$ decreases the frictional resistance to motion in the tip region. This drives the crack to still higher velocities. Within homogeneous interfaces, there is a natural limiting velocity, $C_R$, which is a true asymptotic velocity that is limited by the efficiency of energy transport to the crack tip (22). This limitation may still exist when a generalized Rayleigh wave speed, $C_{GR}$, exists within bimaterial interfaces (18). When $C_{GR}$ does not exist, however, energy flow to the rupture tip is not kinematically limited; it can still flow to the rupture tip via the stiff side of the interface. This enables the rupture velocity to indeed become transonic. The transonic rupture velocity cannot increase indefinitely; the analytic solution provides us with both the mechanism and an accurate prediction for the limiting velocity (Fig. 4).

In conclusion, by combining experiments with analytical solutions and simulations, we have obtained a nearly complete understanding of the structure and development of slip pulses. Since bimaterial interfaces are the most general frictional configuration, this work provides fundamental understanding of the main vehicle that drives frictional motion in both the laboratory (20, 21) and conceivably, motion and damage (31) along numerous natural faults.

Methods

Experimental System.

For the bimaterial interface in our experiments, we used a PC block of dimensions 197 × 100 × 5.8 mm sliding on a 220 × 100 × 5.5-mm PMMA block in the x, y, and z directions, respectively (Fig. 1A). The contact faces of both blocks were diamond machined to optical flatness.

Material shear, C_S, and longitudinal, C_P, wave speeds were obtained by measuring the time of flight of 5-MHz ultrasonic pulses. Due to the small wavelength (∼0.5 mm) of the ultrasonic pulses used, compared with the dimensions of the measurement setup of the sound wave velocities, the measured C_P correspond to plane strain conditions $\left({\epsilon }_{zz}=0\right)$, $C_{L,strain}$, yielding $C_S^{PMMA}=\mathrm{1,361}\pm 13\hspace{0.17em}\mathrm{m}\hspace{0.17em}{\mathrm{s}}^{-1}$, $C_{L,strain}^{PMMA}=\mathrm{2,680}\pm 10\hspace{0.17em}\mathrm{m}\hspace{0.17em}{\mathrm{s}}^{-1}$, $C_S^{PC}=908\pm 2\hspace{0.17em}\mathrm{m}\hspace{0.17em}{\mathrm{s}}^{-1}$, and $C_{L,strain}^{PC}=\mathrm{2,191}\pm 2\hspace{0.17em}\mathrm{m}\hspace{0.17em}{\mathrm{s}}^{-1}$. These wave speed measurements were an order of magnitude more precise than the measurements used in previous work (21). This is due to improved temporal resolution (4 ns) of our time of flight ultrasonic measurements. The material sound waves corresponding to plane stress $\left({\sigma }_{zz}=0\right)$ conditions experiment are, therefore, $C_{L,stress}^{PMMA}=\mathrm{2,345}\pm 13\hspace{0.17em}\mathrm{m}\hspace{0.17em}{\mathrm{s}}^{-1}$ and $C_{L,stress}^{PC}=\mathrm{1,653}\pm 4\hspace{0.17em}\mathrm{m}\hspace{0.17em}{\mathrm{s}}^{-1}$.

The mass densities, ${\rho }^{PMMA}=\mathrm{1,170}\hspace{0.17em}\mathrm{kg}\hspace{0.17em}{\mathrm{m}}^{-3}$ and ${\rho }^{PC}=\mathrm{1,200}\hspace{0.17em}\mathrm{kg}\hspace{0.17em}{\mathrm{m}}^{-3}$, coupled to the wave speed measurements yield dynamic values for the Poisson ratio of ${\nu }^{PMMA}=0.33$ and ${\nu }^{PC}=0.39$ and dynamic Young’s moduli of $E_{dynamic}^{PMMA}=5.75\hspace{0.17em}\mathrm{GPa}$ and $E_{dynamic}^{PC}=2.76\hspace{0.17em}\mathrm{GPa}$. Note that the dynamic values of $E$ are significantly different from the static values $E_{static}^{PMMA}=3\hspace{0.17em}\mathrm{GPa}$ and $E_{static}^{PC}=2.4\hspace{0.17em}\mathrm{GPa}$. This difference is due to viscoelastic behavior of both PMMA (32) and PC (33, 34).

Loading Application.

During each experiment, the top block was rigidly clamped at its upper edge. Its lower edge was placed on the upper edge of the bottom block, whose lower edge was free to move, as it was rigidly mounted to a stiff low-friction linear translational stage. A normal force within the range $\mathrm{2,000}<F_N<\mathrm{6,000}\mathrm{N}$ was first applied to the top block, thereby pressing the two blocks together. The respective nominal (i.e., assuming a continuous contact plane) normal stress range was $2<{\sigma }_{yy}<5\hspace{0.17em}\mathrm{MPa}$.

An external shear load, $F_S$, was quasistatically applied on the translational stage after application of $F_N$, as described in ref. 21. Note that we often used an optional rigid stopper that was applied to the top block at x = 0 mm, at a controllable height 1 cm < h < 2 cm, to constrain motion of this edge in the x direction and mediate any torque applied to the system. To explore a range of external loading conditions, the experiments were conducted using two distinct ways of triggering rupture nucleation.

• 1)$F_S$ were applied to the system quasistatically, at fixed $F_N$, at loading rates between 4 and 15 N s⁻¹ until slip initiated. Using this triggering method, the ruptures usually nucleated along the quarter of the interface closest to x ∼ 0 mm, either as a result of the edge loading by the stopper or reduced local normal force in this region resulting from induced torques.
• 2)At the completion of a sequence of slip events, the applied $F_S$ was kept fixed, and $F_N$ was reduced at loading rates between 40 and 60 N s⁻¹, resulting in spontaneous rupture nucleation. This triggering method, via “unloading,” yielded a wider distribution of nucleation locations along the interface than for static application of $F_N$. In addition, these ruptures were generally more energetic than those that nucleated during the application of shear. This is due to the high shear strains that were built up within the system and clamped in place by the large applied compressive load. When these strains were released, upon unloading, normal loads were much lower with correspondingly lower values of fracture energy. As a result, ruptures nucleating via a reduction of normal load had much larger ratios of strain energy to fracture energy.

For both nucleation (triggering) methods, ruptures could simultaneously nucleate in both directions. Yet, the direction of longer propagation distance could be roughly determined by vertically inverting the compliant and stiff blocks when enabling nucleation to occur via increased shear stress application. This is apparently the result of some degree of asymmetry within our loading frame. Beyond this preference for rupture direction, our choice of whether the compliant (stiff) block was mounted on the top or bottom produced, as expected, no overall difference in our results. We note that while rupture events occurred following modification of either $F_S$ or $F_N$, the changes in $F_S$ or $F_N$ were sufficiently slow so that their values were essentially constant during the 0.1- to 0.2-ms rupture propagation period.

Real Contact Area Measurements.

The amount of the real contact area along the entire interface throughout the experiment was measured by an optical method based on total internal reflection. Basic principles were presented in detail elsewhere (12, 35, 36). An incident sheet of light illuminated the frictional interface at an angle well beyond the critical angle for total internal reflection. The light was transmitted through the interface only at the contact points and was reflected everywhere else, yielding an instantaneous light intensity that was roughly proportional to A(x,z,t) over the entire (x × z) 200 × 5.5-mm interface. We continuously imaged the trasmitted light at 580,000 frames per second and spatial resolution of 1,280 × 8 pixels using a Vision Research Phantom v711. While slip pulses were very nearly straight and oriented normal to the propagation direction, often the slow rupture fronts $\left(C_f<0.6C_S^{soft}\right)$ were tilted relative to the x axis at angles up to ∼45°, yielding an effective maximal “one-dimensional” (1D) front width of the order of the block’s width. Despite this, the block’s width (z direction) was so much smaller than the other dimensions of the block that the frictional interface could still be considered as quasi-1D. These measurements therefore provided instantaneous values of A(x,t) = <A(x,z,t)>_z along the entire 1D interface.

Rupture Front Velocity ${\mathit{C}}_{\mathit{f}}$ Calculation.

The rupture front location, x_tip, was defined for every t from A(x,t) as the point where A(x_tip,t) = 0.95A₀(x,t). $C_f\left(t\right)$ was obtained by differentiating x_tip(t). Our precision in determining $C_f\left(t\right)$ depended on our 200-μm uncertainty in x.

In the work described here, we consider “instantaneous” values of $C_f\left(t\right)$ as velocity values averaged over 15-mm intervals. This provided resolution in $C_f\left(t\right)$ of better than 5 to 10 ms⁻¹. We define “steadily propagating” ruptures with velocity $C_f$ as ruptures having no clear tendency to accelerate or decelerate. “Steady-state” values of $C_f$ used in the text were all calculated by linear fits of $C_f\left(t\right)$ values over distances of at least 50 mm.

The $C_f$ values of steadily propagating ruptures are used for two purposes. The first is to characterize the stable front propagation velocities themselves to high precision. For this analysis, we demand that instantaneous measurements of analyzed $C_f\left(t\right)$ vary by less than ±15 m s⁻¹ while traversing this (>50-mm) interval. This requirement yields precision of less than 0.5% in $C_f$ value for $C_f\sim C_S^{soft}$, which is critical for unambiguously distinguishing between $C_f>C_S^{soft}$ and $C_f<C_S^{soft}$ for the slip-pulse velocities. Only rapid ruptures, transonic slip pulses, and supershear ruptures satisfy this condition.

Our second goal is to compare the strain measurements to known analytical solutions or numeric calculations at a given propagation velocity. For these comparisons, the propagation distance considered was limited to a scale corresponding to the near-tip (putatively singular) region, of about 30 to 50 mm. For slow velocities, the strain signals are nearly independent of $C_f$, as long as the analysis is in terms of $x-x_{tip}$. As a result, the mean front propagation velocity could vary as much as ±80 m s^–1 with a negligible effect on the measured strain values. For high velocities $C_f\mathrm{\gtrsim }{C}_S^{soft}$, the strain signals strongly depend on $C_f$. For this reason, we used only ruptures with the above high-precision requirement.

Strain Measurements.

We used miniature Kulite B/UGP-1000-060-R3 rosette strain gauges for local strain measurements. Thirty such gauges were mounted at two heights, ∼3.5 and ∼7 mm, both above and beneath the interface (Fig. 1A). Three strain gauges were placed at height ∼7 mm on opposing faces at the same x locations (SI Appendix, Fig. S2) to verify that the strain measurements reflected the entire thickness dimension, z.

Each rosette strain gauge is composed of three independent active regions (each 0.4 × 0.9 mm in size)—the two side components are oriented at ±45° relative to the middle one. The centers of the individual gauges are separated by ∼0.6 ± 0.1 mm in the direction of the center component and by 0.85 ± 0.2 mm in the normal directions. During rapid rupture propagation, this distance in the direction parallel to the interface induced a small time delay that was a function of the orientation of the rosette on the block. This was taken into account for the proper calculation of ${\epsilon }_{ij}\left(t\right)$.

The rosettes were oriented so that one of the components was parallel to the interface, one was normal to the interface, and the third was oriented at 45° from the interface. The gauge factor response of these strain gauges was approximately linear $\mathrm{\Delta }R/R=80\hspace{0.17em}\mathit{ϵ}$. The effective shear sensitivity of the rosette due to the encapsulation of the strain gauges on a thin, rigid substrate was calibrated and corrected for (37). Each strain signal (60 channels) was individually amplified (gain = 11, 1-MHz bandwidth), and all strain signals were simultaneously acquired, in parallel, to 14-bit accuracy by an ACQ132 digitizer (D-tAcq Solutions Ltd) at a 1-MHz rate. This led to a sensitivity of ∼3 μStrain in ${\epsilon }_{ij}\left(t\right)$ measurements. As the measured signals are of order ∼1 mStrain, this provides a 0.3% uncertainty in ${\epsilon }_{ij}$.

Despite this relatively high precision, the overall accuracy of our strain measurements is only up to 10 to 20% because of calibration variations between different strain gauge rosettes. This lack of absolute accuracy did not affect the majority of our measurements since we are often interested in relative strain variations that were acquired at given strain gauges. When different strain gauges are compared, variations in calibration were neutralized by normalizing the strain gauge outputs relative to, for example, their initial values.

Converting Temporal to Spatial Measurements.

For steadily moving rupture fronts, ${\epsilon }_{ij}\left(x,t\right)={\epsilon }_{ij}\left(x-C_{f}t\right)$. Actually, $C_f$ generally changes slowly, and we compensate for these changes by using ${\epsilon }_{ij}\left(x,t\right)={\epsilon }_{ij}\left(x-{\displaystyle \int C_{f}dt}\right)$. By this means, we converted ${\epsilon }_{ij}\left(t\right)$ to spatial measurements ${\epsilon }_{ij}\left(x-x_{tip}\right)$ (methods in ref. 12), stresses ${\sigma }_{ij}\left(x-x_{tip}\right)$ (assuming plane stress conditions), and particle velocities ${\dot{u}}_x\left(t\right)=-{\epsilon }_{xx}\left(t\right)\cdot C_f$.

In SI Appendix, we provide details of the analytical and numerical calculations described in the text. In addition, in SI Appendix, Fig. S1, we present a numerical comparison of the near-tip strain fields for frictionless subshear fracture and simulations incorporating Coulomb friction.

We also provide additional experimental data on the scaling of the transonic ruptures in SI Appendix, Fig. S3. The concluding section explains why the exponent $q\left(C_f\right)$ must be a real function for the boundary conditions that are compatible with our experiments.

Data Availability.

All data used in this manuscript are available and have been deposited at https://data.4tu.nl/repository/uuid:aa1da044-51d4-42ae-bf7a-ea51c04ff833.