Speed of fast and slow rupture fronts along frictional interfaces

The transition from stick to slip at a dry frictional interface occurs through the breaking of microjunctions between the two contacting surfaces. Typically, interactions between junctions through the bulk lead to rupture fronts propagating from weak and/or highly stressed regions, whose junctions break first. Experiments find rupture fronts ranging from quasistatic fronts, via fronts much slower than elastic wave speeds, to fronts faster than the shear wave speed. The mechanisms behind and selection between these fronts are still imperfectly understood. Here we perform simulations in an elastic two-dimensional spring-block model where the frictional interaction between each interfacial block and the substrate arises from a set of junctions modeled explicitly. We find that material slip speed and rupture front speed are proportional across the full range of front speeds we observe. We revisit a mechanism for slow slip in the model and demonstrate that fast slip and fast fronts have a different, inertial origin. We highlight the long transients in front speed even along homogeneous interfaces, and we study how both the local shear to normal stress ratio and the local strength are involved in the selection of front type and front speed. Last, we introduce an experimentally accessible integrated measure of block slip history, the Gini coefficient, and demonstrate that in the model it is a good predictor of the history-dependent local static friction coefficient of the interface. These results will contribute both to building a physically based classification of the various types of fronts and to identifying the important mechanisms involved in the selection of their propagation speed.

Figure 1

Sketch of the multiscale model. (a) Slider and external loading conditions. (b) Spring-block network modeling elastodynamics. (c) Surface springs modeling friction on a block.

Figure 2

A part of the macroscopic loading curve $F_T/F_N$. The drop in a full sliding event is comprised of two parts: first, the drop associated with deformation of the slider during rupture front passage (dashed arrow); second, the drop associated with motion of the entire slider (full arrow minus dashed arrow).

Figure 3

A fast-slow-fast full sliding event. (a) Spatiotemporal plot of the fraction of pinned springs (in this simulation ${\overline{t}}_R=2\mathrm{m}\mathrm{s}$). (b) Rupture front speed $v_c$ vs front location for the event in (a). Block rupture is defined to occur when 70% of interface junctions have broken (white dashed line in the color bar). Front speed is measured as the inverse slope of the rupture line [indicated by arrows in (a)] using the end points in a five-point-wide moving stencil.

Figure 4

(a) Slip profiles from a partial slip and a full sliding event. Line without markers: slip profile for block at $x=0.16L$. Line with markers: slip profile for block at $x=0.34L$. We take $t_0$ at the beginning of each event. In both cases we chose a block located near the middle of the region where the rupture front speed was fast. For blocks closer to the fast-slow transition or to the front arrest point, the amplitude of fast slip is smaller. (b) Sketch showing how the relaxation of force associated with the junctions' relaxation from the slipping (s) to the pinned state (p) can lead to a slow slip motion of the block.

Figure 5

The force drops in the macroscopic loading curve grouped according to event type. We include partial slip events and full sliding events occurring between $t=0.63\mathrm{s}$ and $t=1.50\mathrm{s}$ (the developed stick-slip regime). The drops occurring during the passage of the rupture front have comparable amplitude among the partial slip events, fast-slow-fast events, and fast-only events. Also, the net force drop in full sliding events have comparable amplitudes for fast-slow-fast and fast-only events.

Figure 6

The fast slip and the fast front speeds scale with inertia. (a) Block slip motion for four neighboring blocks within the fast front region for four simulations of the same fast-slow-fast event. In each simulation the initial state is the same, but the mass density $\rho$ differs between the simulations. The block slip is measured from the rupture time $t_{\text{rup}}$ of each block as defined in Fig. 3. Lines with the same color (gray level) and marker are from the same simulation. Lines with the same line style represent the same block in different simulations. (b) Rescaling the time of slip with ${\rho }^{-1/2}$ collapses the data in (a). (c) The rupture front speed as a function of position for the same simulations as the data in (a) (corresponding colors and markers). The change of density modified the fast front speed, while the slow front speed remained nearly unchanged. (d) Rescaling the front speed data in (c) by ${\rho }^{-1/2}$ collapses the fast front speed measurements but splits the slow front measurements.

Figure 7

A front-type map based on simulation of homogeneous prepared interfaces. Front type observed vs initial width $\sigma$ of junction force distribution and prestress ${\overline{\tau }}_0=({\tau }_0/p-f_{\text{slip}}/f_N)/(f_{\text{thres}}/f_N-f_{\text{slip}}/f_N)$. Initial states in the forbidden region would have had some junctions stretched beyond their breaking threshold and are therefore excluded. The symbols are for simulations used in the following figures: Fig. 8: Black star and arrows. Fig. 10: Open circles (vertical, only three of the points are shown here). Fig. 11: Black filled circles, magenta (gray) filled squares, and multicolored (gray) filled circles (horizontal).

Figure 8

Changes in front characteristics upon local changes to prestress and spring stretching width are consistent with the predictions of Fig. 7. (a) The original simulation with homogeneous stress and junction distribution states in the propagation region. (b) A step increase in the initial shear stress within the propagation region leads to a slow-fast transition. (c) A step increase in the width of the junction force distribution within the propagation region also leads to a slow-fast transition. Top: Events shown as in Fig. 3 (${\overline{t}}_R=2\mathrm{m}\mathrm{s}$ in these simulations as well). Middle: Spatial distributions of initial prestress ${\tau }_0/p$ (before event triggering) is shown with drawn line. The gray region is the triggering region (see Appendix pp1). Bottom: For each block along the interface, a color coded histogram of $\varphi \left(f_T\right)$. The vertical axis shows the force level in individual springs, which extends up to $f_{\mathrm{thres}}$ (upper white line). Lower white line: $f_T=0\mathrm{N}$. The color scale denotes the fraction of springs found at each value of $f_T$ using an arbitrary bin width. Offset data: Fraction of slipping springs, taken after the triggering of the event, at $t=0.0207\mathrm{s}$.

Figure 9

Results for transient behavior of fast rupture fronts. Front speed vs position (a) and inverse position (b) for simulations where the local state is homogeneous along the propagation region ($\sigma =0, {\overline{\tau }}_0=0.4$). The magenta (gray) line with dotted markers is for a system of our reference length ($N_x=57, L_x=140\mathrm{m}\mathrm{m}$); the black line with circular markers is for a system 5 times longer. [(c) and (d)] Front speed for two systems of the same length (5 times the reference length) and the same initial state. Black line with circular markers: The same data as the black lines in (a) and (b), which was triggered by simultaneously breaking all junctions for all blocks in the triggering region as explained in Appendix pp1. Green (light gray) line with dotted markers: Front triggered by driving from the trailing edge, which modifies the stress in the loading region prior to rupture. In all panels, rupture velocity is defined as in Fig. 3. In (b) and (d), the speed for the 10 blocks closest to the leading edge are excluded from the plot to avoid the region where the edge effects dominate.

Figure 10

Rupture front speed depends on local stress state. (a) Rupture front speed vs position along the interface for a series of simulations with various homogeneous initial shear stresses in the propagation region. The junction distribution width was zero, so these simulations lie on top of the vertical axis in Fig. 7. They are all of the fast-only type. (b) The front velocity data in (a) vs the prestress value for each simulation. Dots: Mean values. Bars: Min-max values. Each dot+bar corresponds to the line in (a) with the same color (gray level)+vertical extent. The prestress values ${\overline{\tau }}_0$ are from 0.22 to 0.94 in steps of 0.06. For these simulations, a time step of $\mathrm{\Delta }t=5\times {10}^{-8}\mathrm{s}$ was used to obtain smooth measurements at high front speeds.

Figure 11

Rupture front speed depends on local strength. (a) Rupture front speed vs position along the interface for a series of simulations in which the distribution of junction forces was varied systematically while remaining homogeneous along the interface and two values of shear prestress were used [see the legend of panel (b)]. All of these events are of the fast-only type. The width $\sigma$ of the junction force distribution is increasing from the bottom to top line of each color; the values can be found in (b). For each value of $\sigma$, the front propagation speed is higher when the prestress is higher, that is, each magenta (gray) line is higher than the corresponding black line. (b) Fast front speed data from (a) vs junction force distribution width. For this data set we define, for each prestress series, the front speed at $\sigma =0$ as the reference speed $v_{\text{ref}}$. Then, for each simulation we calculate, for every position, $[v_c\left(x\right)-v_{\text{ref}}\left(x\right)]/v_{\text{ref}}\left(x\right)$. The dots are the mean of these values, the bars show the min-max values. The values on the $\sigma /(f_{\text{thres}}-f_{\text{slip}})$ axis range from 0 to $0.41$ in 14 equal steps of $0.029$. (c) Slow front speed vs junction force distribution width. The region over which the slow front extends was determined from visual inspection of the underlying data, which is shown in Fig. 19. The prestress is ${\overline{\tau }}_0=0.3$ in the triggering region and ${\overline{\tau }}_0=0.05$ in the propagation region; the junction distribution widths can be found in (d). (d) The slow front speed data in (c) vs junction force distribution width. Dots: Mean values. Bars: min-max values. Each dot+bar corresponds to the line in (c) with the same color (gray level)+vertical extent. The data values on the $\sigma /(f_{\text{thres}}-f_{\text{slip}})$ axis range from $0.12$ to $0.47$ in six equal steps of $0.059$. The simulations for this figure were selected from and are marked in the front-type map in Fig. 7.

Figure 12

Front speed is proportional to slip speed. (a) Front speed vs slip speed from simulations with controlled initial states in the regime that produces fast fronts. Taking the simulation that plots at $(0.23,0.2)$ in Fig. 7 as an arbitrary reference, we have varied the junction force distribution $\varphi \left(f_T\right)$ [magenta (gray) crosses], the prestress ${\overline{\tau }}_0$ [green (gray) dots], the mass density $\rho$ [blue (dark gray) squares], the bulk and interfacial stiffnesses (keeping $k/k_i$ constant) and $\rho$ [cyan (light gray) circles], and the normal force $F_N$ and $\rho$ [red (gray) diamonds]. (b) The data in (a) with slip speed rescaled as $v_{\text{slip,}}\text{rescaled}=v_{\text{slip}}\frac{k_i{l}_0}{{\tau }_{\text{thres}}-{\tau }_0}$. The black markers near the origin show the data for slow slip and slow fronts from Ref. [36, Fig. 3D]. [(c) and (d)] The data in (a) and (b), respectively, on logarithmic axes.

Figure 13

Rupture front speed vs prestress for several events where the prestress and the junction force distribution have been varied systematically. Different colors correspond to different widths of the junction force distribution: $\sigma /(f_{\text{thres}}-f_{\text{slip}})=0.014$ [red (gray)], $0.143$ [green (light gray)], $0.286$ [blue (dark gray)]; the bell-shaped distributions defined in Fig. 18 were used. Different markers correspond to different initial levels of prestress (before additional loading was applied): $\overline{\tau }=-0.02$ ($▾$), $0.0$ ($•$), $0.06$ ($▴$), $0.13$ ($\blacksquare$), $0.2$ ($◂$), $0.3$ ($★$), $0.4$ ($▸$), $0.5$ (♦). In terms of $\tau /p=f_{\text{slip}}/f_N+\overline{\tau }(f_{\text{thres}}/f_N-f_{\text{slip}}/f_N)$, these prestresses are 0.165, 0.170, 0.184, 0.200, 0.216, 0.239, 0.262, and 0.285, respectively. Edge effects, which can dominate visually, have been excluded by showing only the region between $x=2.5\mathrm{c}\mathrm{m}$ and $x=11\mathrm{c}\mathrm{m}$. Prestress is measured for all blocks when the first block starts slipping (as defined from the number of pinned junctions). To get a sense for the range of values on the $\tau /p$ axis, which differs from the ones in Ref. [17], consider that values much below $f_{\text{slip}}/f_N=0.17$ lead to arresting fronts and that the maximum possible value of $f_{\text{thres}}/f_N=0.4$ is reached only for narrow junction force distributions and near the loading point, so it is excluded by excluding the edges of the system. Recall from Fig. 7 that combinations of high prestress and high $\sigma$ are forbidden: This is the reason why there are fewer blue (dark gray) entries. For these simulations, a time step of $\mathrm{\Delta }t=5\times {10}^{-8}\mathrm{s}$ was used to obtain smooth measurements at high front speeds.

Figure 14

The areas that define the Gini coefficient, a measure of inequality.

Figure 15

(a) Slip profiles $x\left(t\right)$ for constant deceleration slip. The markers extend over the interval $[t_0,t_1]$. (b) Rescaled effective static friction ${\overline{\mu }}_s^{\text{eff}}$ for the slip profiles in (a) and the junction parameters in Table 1. (c) Gini coefficients of the slip profiles in (a). The line extends to the origin. The markers in (b) and (c) show the data from the slip profiles in (a) of the corresponding color (gray scale); the drawn lines are based on a denser set of slip profiles that were omitted from panel (a) for clarity. Combining (b) and (c) gives the drawn blue (dark gray) line with circular markers in Fig. 17.

Figure 16

Example slip profiles from full simulations. [(a) and (b)] Slip profiles for full sliding events. Only the slip occurring between $t_0$ and $t_1$ (defined in the text) are used to calculate the Gini coefficient. Effective static friction is measured from the junction force distribution $\varphi \left(f_T\right)$ at $t^{\prime }$. The data in (a) are for the block at $x=4\mathrm{c}\mathrm{m}$ in a fast-slow-fast event similar to that of Fig. 3 and from the same simulation, while the data in (b) are from a simulation with $f_{\text{slip}}/f_N=0.05$. [(c) and (d)] Detailed views of the slip profiles in (a) and (b), respectively, for the time interval between $t_0$ and $t_1$. Black line without markers: $x\left(t\right)$. Colored line with markers: A moving average filter of width $1\mathrm{m}\mathrm{s}$ ($={\overline{t}}_R/2$) was applied to $x\left(t\right)$ and then $N_e=10$ equally sized intervals were defined (markers indicate intervals' boundaries). The smoothing helps avoid negative increments that while possible to include are not part of the basic formulation of the Gini coefficient.

Figure 17

Rescaled effective static friction after full sliding events vs the Gini coefficient, an integrated quantifier of block slip motion. Data from full simulations, with variation in junction law parameters between simulations as shown in the legend. See Fig. 15 for an explanation of the Gini coefficient and the definition of the drawn lines, and Fig. 16 for sample block slip histories. For each simulation we show the prediction (drawn line with solid symbols) and data from two arbitrary full sliding events (solid and open symbols).

Figure 18

Scaling of effective static friction vs junction force distribution width. (a) Unscaled analytical results for uniform junction distributions. Here $f_N=0.32\mathrm{N}$ and changes were made to $f_{\text{slip}}$ and $f_{\text{thres}}$ independently. Markers indicate $f_{\text{supp}}=f_{\text{thres}}-f_{\text{slip}}$. (b) Unscaled numerical results for junction distributions that are bell-shaped polynomials with roots at $\pm a$ and the functional form $\varphi \left(\xi \right)=5/\left(4a\right)(1+3|\xi /a\left|\right){(1-|\xi /a\left|\right)}^3,\xi \in [-a,a]$. Here $f_N=1\mathrm{N}$ and changes were made to $f_{\text{slip}}$ and $f_{\text{thres}}$ independently. (c) Data collapse using exact scaling.

Figure 19

Underlying data for Fig. 11: The events that lie at $\overline{\tau }=0.05$ in Fig. 7, shown as in Fig. 3. From the top left to bottom right, the width of $\varphi \left(f_T\right)$ is $\sigma /(f_{\text{thres}}-f_{\text{slip}})=0.059, 0.088$, and then $0.117$ to $0.528$ in steps of $0.059$; the event for $\sigma =0$ is not shown, but it, too, arrests early. For each panel, the extent of the slow front was determined by visual inspection. On the left, the slow front starts after the step (ca. $x=0.04\mathrm{m}$). On the right, because the transition to a fast front is more gradual, the boundary was set where the trend of near constant front velocity is broken. The exact limits chosen can be seen from the horizontal extent of the data in Fig. 11. Panels (a)–(c) show a larger time interval than the rest of the panels. No data for Fig. 11 were extracted from the bottom right panel: Even though the front has a slow part and plots in the slow part of the front type map, the slow part is too intermixed with the transition to fast rupture for the analysis to be carried out.