Keywords

• Dynamic fracture;
• Friction;
• Heterogeneous interface;
• Fracture mechanisms;
• Finite elements

1. Introduction

The propagation of slip fronts at frictional interfaces as well as classic shear cracks at weak interfaces has, in the past, often been modeled in two-dimensional systems. Neglecting the third dimension enables the use of plane-strain or plane-stress approximations and reduces the interface to a one-dimensional line. This simplifies the theoretical description of interface ruptures and decreases significantly computational cost of numerical simulations. Nevertheless, the propagation of ruptures along two-dimensional interfaces presents interesting additional aspects. The effect of non-homogeneous interfaces, for instance, is of great importance for earthquake science (because faults are not homogeneous) and engineering (with the use of “designed” materials such as composites). Most ruptures propagate along interfaces that present different types of heterogeneities at various scales. Examples are non-uniform loading, such as areas of higher or lower pre-stress, and heterogeneous interface properties, such as zones of stronger/weaker strength or rate-strengthening/weakening friction. All these types of heterogeneities affect the rupture propagation, but become particularly interesting for fronts of at least two dimensions because areas that are less easily broken can simply be circumvented (or partially circumvented and broken with delay). Even though early theoretical work (Gao and Rice, 1989) showed that a first-order perturbation analysis describes well the curved shape of a static crack around slightly tougher interface obstacles, the dynamic aspects of rupture propagation on interfaces with significant heterogeneities remain elusive.

The experimental observation of slip events at frictional interfaces is challenging and only few results of two-dimensional fronts have been reported so far. Brörmann et al. (2013) and Romero et al. (2014) showed on interfaces with discrete contacts (pillars and spherical caps, respectively) that the transition from sticking to sliding is characterized by slip fronts propagating along the interface, similar to the observations of Rubinstein et al. (2004) but at a two-dimensional interface. Latour et al. (2013) studied experimentally the effect of interface barriers on frictional slip. They showed, using different configurations of barriers, that heterogeneities can cause rupture arrest/delay as well as increase/decrease of the rupture speed. However, no systematic prediction can be provided because the behavior of a sequence of ruptures is complex and memory effects potentially cause inter-dependence of subsequent ruptures.

More experimental observations showing the effects of heterogeneities have been reported for Mode I (decohesion) ruptures. Natural heterogeneities of the interface (Måløy et al., 2006) and the bulk material (Ponson, 2009) were observed to affect the propagation speed. Måløy et al. (2006) analyzed local velocity fluctuations of an interface crack along a heterogeneous weak plane and showed that local velocities larger than the average front speed present a power law distribution. Ponson (2009) observed a depinning transition of a crack in a heterogeneous sandstone. Theoretical studies of planar (tensile) cracks in heterogeneous media suggested that elastic wave propagation is a key component to explain the observed roughness of the fracture (Ramanathan et al., 1997, Ramanathan and Fisher, 1997, Ramanathan and Fisher, 1998 and Bouchaud et al., 2002). Other experiments applied artificial heterogeneities in order to study their influence on the rupture fronts. Mower and Argon (1995) performed crack-trapping experiments showing a locally bowed configuration of a quasi-static crack between two obstacles with higher interface strength. Dalmas et al. (2009) and Chopin et al. (2011) described the shape of quasi-static and dynamic decohesion fronts, respectively, observed at interfaces with stripes of different fracture energy. Xia et al. (2012) showed in thin-film experiments that the macroscopic peel force depends on the shape of the interface heterogeneities and can vary despite having the same cumulative area.

In addition to experimental observations, some numerical studies of the propagation of frictional shear ruptures at weak interfaces have been conducted. This includes one-dimensional as well as two-dimensional interfaces with heterogeneities due to changed pre-stresses or modified frictional properties of the interface. The number of simulations with two-dimensional interfaces is limited because a set-up with a rupture far from the edges (to avoid the influence of wave reflection), with a small process zone compared to the rupture length, and with a sufficient fine discretization is computationally challenging. Early work by Day (1982) showed that the rupture speed presents large changes during the propagation along an interface with zones of higher and lower pre-stresses. Fukuyama and Olsen (2002) and Dunham et al. (2003) demonstrated with numerical simulations of two-dimensional interfaces that a circular heterogeneity of higher pre-stress or higher fracture energy can provoke a transition from sub-Rayleigh to (temporary) super-shear propagation. Three-dimensional simulations provided also the opportunity to study differences in the near-source ground motion of earthquakes when an interface rupture propagates through an area of higher strength or higher pre-stress (Page et al., 2005). Other simulations were used to study earthquake mechanisms at faults with velocity-weakening patches surrounded by velocity-strengthening areas (Rice, 1993, Ben-Zion and Rice, 1997, Madariaga and Olsen, 2000, Kaneko et al., 2008, Ariyoshi et al., 2009, Ariyoshi et al., 2012 and Kaneko and Ampuero, 2011). In these set-ups, the velocity-strengthening zone is continuously sliding and causes an energy accumulation in the system, which eventually leads to dynamic ruptures (earthquakes) propagating mostly within the velocity-weakening zones. The dynamic propagation of ruptures at two-dimensional interfaces was also numerically modeled with heterogeneous initial stresses mimicking a realistic state of a fault (Andrews, 2005 and Brietzke et al., 2009).

Even though the effect of heterogeneities on the propagation of slip fronts is different at one-dimensional interfaces, such simulations are still important tools to help understand the underlying mechanisms. Das and Aki (1977) showed that ruptures at one-dimensional interfaces can propagate through areas of higher strength with and without breaking the heterogeneity. In a different set-up with a weakening and a strengthening zone, Voisin et al. (2002) studied the arrest of frictional slip at the border of the areas of different properties and showed the presence of a self-healing slip pulse that penetrates the strong area. In a detailed study of the transition from sub-Rayleigh to inter-sonic propagation at a one-dimensional interface, Liu and Lapusta (2008) demonstrated that a single favorable heterogeneity leads to a secondary crack which accelerates to inter-sonic speeds with an abrupt jump from the Rayleigh wave speed to an inter-sonic speed. The emergence and evolution of increased pre-stress heterogeneities due to the arrest of preceding slip fronts was studied by Radiguet et al., 2013 and Radiguet et al., 2015.

In this work, we focus on the dynamic aspects of frictional in-plane shear ruptures within a heterogeneous zone of a two-dimensional interface (in contrast to studies of ruptures beyond a single interface heterogeneity (Dunham et al., 2003 and Liu and Lapusta, 2008)). We present three-dimensional finite-element simulations of a slip-front propagation along an interface with heterogeneous friction properties. We will show that two different propagation mechanisms (sub-Rayleigh and inter-sonic speed) exist in this configuration and that the propagation regime is selected by the interplay of two length-scales: size of heterogeneity vs. process-zone size. The studied set-up consists of a semi-infinite interface with a homogeneous area used as an establishing zone and an area with stripes of different slip-weakening properties (which reduces essentially to a difference in fracture energy).

2. Simulation set-up

The propagation of a slip front at a two-dimensional heterogeneous frictional interface is studied in a simple set-up as shown in Fig. 1. The two solids in contact are each of length $L=200\mathrm{mm}$ and height $H=100\mathrm{mm}$, and are chosen to be infinite in the z-direction in order to avoid edge effects. Infinity in the z-direction is modeled by periodic boundary conditions. At a perfectly homogeneous interface, this set-up results in a straight slip front. The rather large geometry of the solids ensures that reflected waves do not affect the propagation of the front over a large part of the interface. Perturbations due to reflected waves cannot be avoided close to the end of the interface because the waves ahead of the rupture are always reflected by the leading edge.

The system is loaded statically by a normal pressure $\overline{p}=5\mathrm{MPa}$ applied in the y  -direction, and an in-plane shear load $\overline{t}=2\mathrm{MPa}$ applied on all boundaries in the x–y$x\mathrm{–}y$ plane. This load configuration results in uniform contact pressure and friction tractions at the interface and separates the effect of varying interface tractions from the effects of the heterogeneity. The influence of non-uniform stress states was studied in Kammer et al., 2012 and Kammer et al., 2015.

If not indicated differently, the material and interface parameters are as follows. The bulk material is linear elastic with Young's modulus $E=2.6\mathrm{GPa}$, Poisson's ratio ν=0.37$\nu =0.37$, and density $\rho =1130\mathrm{kg}/{\mathrm{m}}^3$. The resulting shear modulus is $G=949\mathrm{MPa}$ and the wave speeds are $c_{\mathrm{d}}=2017\mathrm{m}/\mathrm{s}$, $c_{\mathrm{s}}=916\mathrm{m}/\mathrm{s}$, and $c_{\mathrm{R}}=859\mathrm{m}/\mathrm{s}$. Friction at the interface is governed by a linear slip-weakening law, as illustrated in Fig. 2(b). The frictional strength is therefore given by

equation(1)

${\tau }^{\mathrm{s}}\left(\delta \right)=\{\begin{array}{cc}{\mu }_{\mathrm{s}}\overline{p}-\frac{\delta }{d_{\mathrm{c}}}\left({\mu }_{\mathrm{s}}-{\mu }_{\mathrm{k}}\right)\overline{p}\hfill & \forall \delta <d_{\mathrm{c}}\hfill \\ {\mu }_{\mathrm{k}}\overline{p}\hfill & \forall \delta \ge d_{\mathrm{c}},\hfill \end{array}$

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$d_{\mathrm{c}}=18\mathrm{\mu }\mathrm{m}$

$\overline{p}$

${\Gamma }_{\mathrm{c}}=\overline{p}({\mu }_{\mathrm{s}}-{\mu }_{\mathrm{k}})d_{\mathrm{c}}/2=31.5\mathrm{J}/{\mathrm{m}}^2$

${\Gamma }_{\mathrm{c}}=10\mathrm{J}/{\mathrm{m}}^2$

$d_{\mathrm{c}}=10\mathrm{\mu }\mathrm{m}$

The configuration of the heterogeneous interface is shown in Fig. 2(a). The slip front is nucleated by introducing at t=0$t=0$ a notch (shown in red) which acts as a seed crack and in which the fracture energy is equal to zero by setting μ_s=μ_k=0.2${\mu }_{\mathrm{s}}={\mu }_{\mathrm{k}}=0.2$. The rupture starts at the tip of the notch and propagates in the positive x-direction. The slip front is originally straight because of the homogeneity (in the establishing zone) and the infinite width of the interface. During propagation through the establishing zone (shown in green), the rupture accelerates and constitutes an interface crack with corresponding near-tip strain fields. At the end of the establishing zone, the slip front enters the heterogeneous area in which the interface is organized in stripes of alternating frictional properties. Half of the stripes present the same properties than the establishing zone with a high fracture energy (shown also in green in Fig. 2(a)). The second half of the stripes is characterized by a lower fracture energy (shown in purple in Fig. 2(a)). The weaker and stronger stripes are always of equal width in the simulations presented here. Hence, the replication width of the simulated interface changes if the stripes width is modified.

The dynamic simulations of the slip fronts presented here are based on the finite-element method (Belytschko et al., 2000). An explicit Newmark β-method is used for time integration. Interface reactions including contact pressure and friction tractions are applied through a traction-at-split-node technique ( Andrews, 1973 and Andrews, 1999), which is particularly suitable for simulations of small slip at nominally flat interfaces. Regular meshes with up to approximately 20 million linear hexahedral elements are used to discretize the solids in contact. These meshes are sufficiently fine in order to discretize well the process zones of the modeled slip fronts and to achieve mesh convergence without any numerical damping.

3. Homogeneous reference cases

Before studying the effect of the heterogeneous interface on the propagation of a slip front, we present reference cases with homogeneous interfaces of the same frictional properties as used in the heterogeneous set-up. The aim of these homogeneous reference cases is to provide a basic understanding of the slip front propagation caused by the different frictional properties in this particular system with the given bulk material.

The propagation of a slip front at a homogeneous interface with the frictional properties corresponding to the high fracture energy is shown in Fig. 3. The configuration used for this simulation corresponds to the interface shown in Fig. 2(a) with a notch and a homogeneous interface everywhere else. The slip front nucleates at the tip of the notch and propagates in the positive x-direction. It accelerates fast over the first 20 mm and, beyond this point, the rupture speed continues increasing slowly. Over the length of the interface, the rupture approaches the Rayleigh wave speed but stays sub-Rayleigh at all times. The process zone of the slip front, the yellow-red-yellow area in Fig. 3(a), shrinks with increasing rupture speed as expected by linear elastic fracture mechanics theory ( Rice et al., 1980 and Freund, 1990).

In a second reference case, shown in Fig. 4, a slip front is nucleated by the same notch and propagates first along a homogeneous interface of the same frictional properties as for the first reference case. During this phase, the observed behavior corresponds to the rupture propagation shown in Fig. 3. However, after the establishing zone, the friction properties are modified and correspond to an interface of lower fracture energy. This set-up is similar to the configuration presented in Fig. 2(a), except that the interface is homogeneous and of low fracture energy over its entire width (only purple) in the so-called heterogeneous area. The border between the establishing zone and the area of low fracture energy is at $x=75\mathrm{mm}$ and is marked by vertical lines in Fig. 4. When the sub-Rayleigh slip front approaches the border, a secondary rupture nucleates in the area of lower fracture energy due to a shear stress peak that propagates ahead of the slip front. The secondary rupture, which merges shortly after with the main slip front, initiates directly at an inter-sonic speed just above the shear wave speed. The mechanism of a secondary crack at inter-sonic speed created within a favorable area was also observed by Liu and Lapusta (2008). After the transition, the slip front accelerates up to 0.83c_d$0.83c_{\mathrm{d}}$ at which speed it reaches a slowly evolving state. Similarly, in a set-up with homogeneous frictional properties of lower fracture energy over the entire length of the interface, the rupture accelerates directly to inter-sonic speed right after nucleation and reaches the same slowly evolving state. The peak value of the friction traction is smaller in the area of the weaker interface where ${\Gamma }_{\mathrm{c}}=10.0\mathrm{J}/{\mathrm{m}}^2$ (there is no red area in Fig. 4(a) for $x>75\mathrm{mm}$) because the static friction coefficient is smaller with μ_s=0.6${\mu }_{\mathrm{s}}=0.6$ (compared to 0.9).

In summary, the reference cases showed that the proposed interface parameters result, in the given system, in two different propagation regimes. The set-up with the higher fracture energy at the interface can be characterized, following Andrews (1976), by the seismic ratio $S=({\mu }_{\mathrm{s}}\overline{p}-\overline{t})/(\overline{t}-{\mu }_{\mathrm{k}}\overline{p})=2.5$, which causes a sub-Rayleigh propagation over the entire length of the system. The set-up with the lower fracture energy presents, however, a seismic ratio of S=1.0$S=1.0$, which results, for the applied nucleation procedure, directly in a rupture with inter-sonic speed. The effect on the propagation of a slip front by combining both friction properties in a heterogeneous configuration is analyzed in the following section.

4. Heterogeneous interface

4.1. Two distinct propagation mechanisms

The propagation of slip fronts at heterogeneous interfaces presents different behaviors depending on the width of the stripes. Various simulations were conducted with stripe widths between $W=2\mathrm{mm}$ and $W=4\mathrm{mm}(W=2,2.5,3,3.5,4\mathrm{mm})$. In all cases, the total area fraction with low fracture energy is 50% of the heterogeneous area given that the weak and strong stripes are always of the same width. In all these simulations, two distinct propagation mechanisms are observed. Snapshots of the expansion of the slip area are shown in Fig. 5 for two representative interface ruptures, which are stripe configurations with $W=2.5\mathrm{mm}$ and $W=4\mathrm{mm}$. The two distinct mechanisms are strictly separated by the width of the stripes. Slip fronts propagating at interfaces with $W\le 3\mathrm{mm}$ are equivalent to the slip front presented in the system with $W=2.5\mathrm{mm}$, whereas ruptures at interfaces with $W\ge 3.5\mathrm{mm}$ correspond to the behavior of the slip front in a set-up with $W=4\mathrm{mm}$.

As shown by the snapshots in Fig. 5, the front is flat and propagates at the same speed at both interfaces before it reaches the heterogeneous area (see at $t=60\mathrm{\mu }\mathrm{s}$). This is the consequence of the infinite width, modeled by periodic boundary conditions in the z-direction, and the homogeneous establishing zone, as presented in Fig. 2(a). As the slip front approaches the heterogeneous area starting at $x=75\mathrm{mm}(t=70\mathrm{\mu }\mathrm{s})$, a secondary rupture initiates in both systems in the stripes¹ of lower fracture energy, which corresponds to the behavior observed for the second reference case presented in Fig. 4. While the front of the secondary rupture starts propagating instantaneously at inter-sonic speed, the main rupture continues advancing and eventually merges $(t\approx 80-90\mathrm{\mu }\mathrm{s})$ with the trailing end of the secondary rupture forming one single slip event.

Up to this point in time, the slip fronts in the two interface configurations resemble each other. However, beyond this moment, the effect of the stripe size becomes noticeable. In the system with thinner stripes, the secondary rupture, which just became part of the main rupture, detaches again and distances itself progressively from the main slip front while propagating only in the stripes of lower fracture energy. In the set-up with wider stripes, the secondary rupture does not directly detach from the main rupture but pulls the front in the stronger stripes in order to catch up with the front in the weaker stripes. Nevertheless, at a later point in time, i.e., $t=130\mathrm{\mu }\mathrm{s}$, a stick front initiates at $x\approx 120\mathrm{mm}$ which creates again a main and secondary rupture. However, the secondary slip front propagates over the entire width of the interface and not only in the weaker stripes as observed for the configuration with $W=2.5\mathrm{mm}$. Another striking difference is that there is much more slip occurring in the front rupture of the wider system ($\delta \approx 60\mathrm{\mu }\mathrm{m}$ at $t=150\mathrm{\mu }\mathrm{s}$) compared to the slip pulse of the thinner set-up ($\delta \approx 1.3\mathrm{\mu }\mathrm{m}$ at $t=150\mathrm{\mu }\mathrm{s}$).

The different behaviors of the slip fronts in these two heterogeneous configurations are also recognizable by the propagation speeds shown in Fig. 6. Differences in rupture speeds can also be observed in Fig. 5 by looking at the position of the fronts at various moments. Beyond the starting point of the heterogeneous area, the main slip front in the system with $W=2.5\mathrm{mm}$ is first pulled by the secondary rupture and accelerates temporarily above the shear-wave speed. It decelerates, however, shortly after and continues propagation at sub-Rayleigh speeds (see dark green curve). The secondary rupture, which propagates only in the stripes of lower fracture energy, initiates directly at inter-sonic speeds and propagates without ever entering steady state at various speeds around the Rayleigh- and shear-wave speeds (see dark purple curve). At approximately $x=155\mathrm{mm}$, the slip pulse propagates over 10 mm continuously at the so-called forbidden speed (Freund, 1979), which is between the Rayleigh- and the shear-wave speed.

In the set-up with wider stripes, the slip front propagates in the heterogeneous area over the entire length and width of the interface at inter-sonic speeds (see bright green and purple curves). Along $x=85\mathrm{–}155\mathrm{mm}$, the slip front propagates faster in the stripes of higher fracture energy than in the weaker stripes because the (originally) main front is catching up with the front that nucleated as secondary rupture ahead of it. However, beyond $x=160\mathrm{mm}$, the front presents the same constant propagation speed over the entire interface width. The rear slip front initiating at $x=120\mathrm{mm}$ at $t=130\mathrm{\mu }\mathrm{s}$ due to the stick area appearing ahead of it, see Fig. 5(b), propagates just below the Rayleigh speed (curve not shown in Fig. 6).

It is interesting to note that these two distinct propagation mechanisms occur in systems with the same residual friction tractions and, more importantly, with the same average fracture energy. Macroscopically, these systems seem to be the same but result in different slip fronts. Clearly ruptures occur faster over wider stripes, which appear weaker.

4.2. Length scale interactions

The presence of two distinct mechanisms at interfaces with the same heterogeneous pattern and friction properties but with different stripe width suggests that an interaction between two length scales is at play. This is particularly important during the transition from the homogeneous to the heterogeneous area of the interface. Zoomed snapshots on the relevant area during this transition, shown in Fig. 7(a) and Video 1 of the supplementary material (online), illustrate that the size of the process zone, shown in brighter blue, is of the same order than the width of the stripes. The interaction between the two length scales can be expressed as

equation(2)

$\zeta =\frac{W}{w_{\mathrm{pz}}},$

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The determination of w_pz$w_{\mathrm{pz}}$ in the weaker stripes during the transition of the slip front from the homogeneous to the heterogeneous area is challenging because the rupture is inter-sonic and transient. At inter-sonic steady state, the process zone length can be computed, following Broberg (1989), by

equation(3)

$w_{\mathrm{pz}}=\frac{{\Gamma }_{\mathrm{c}}G}{{\overline{p}}^2{({\mu }_{\mathrm{s}}-{\mu }_{\mathrm{k}})}^2}T(v_{\mathrm{r}},c_{\mathrm{s}},c_{\mathrm{d}}),$

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$\overline{p}$

Considering that the secondary slip front in the weaker stripes has, in all configurations of different width, approximately the same speed during the transition, we can neglect the variation of the T function from case to case and write

equation(4)

$w_{\mathrm{pz}}\propto \frac{{\Gamma }_{\mathrm{c}}G}{{\overline{p}}^2{({\mu }_{\mathrm{s}}-{\mu }_{\mathrm{k}})}^2}.$

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${\Gamma }_{\mathrm{c}}=\overline{p}({\mu }_{\mathrm{s}}-{\mu }_{\mathrm{k}})d_{\mathrm{c}}/2$

$\varphi =\overline{p}({\mu }_{\mathrm{s}}-{\mu }_{\mathrm{k}})/d_{\mathrm{c}}$

equation(5)

$w_{\mathrm{pz}}\propto \frac{G}{\varphi }.$

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Reconsidering Eq. (2), we can now write

equation(6)

$\zeta \propto \frac{W\varphi }{G}\equiv {\zeta }_{\mathrm{a}},$

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Before considering the values of ζ_a${\zeta }_{\mathrm{a}}$ of the various systems studied so far, let us first understand how the friction weakening rate, the shear modulus and the stripe width influence the process of nucleating either of the two propagation mechanisms. When the slip front approaches the beginning of the heterogeneous area $(x=75\mathrm{mm})$, a secondary rupture initiates ahead in the weaker stripes (see snapshots at $t=70\mathrm{\mu }\mathrm{s}$). Once the main front merged with the secondary rupture, the front continues propagation over the entire width of the interface (see snapshots at $t=80\mathrm{–}90\mathrm{\mu }\mathrm{s}$). The front is considerably ahead in the weaker stripes than in the stronger ones due to the earlier initiation through the secondary rupture. During this period, there is a zone in which, at a given position x  , the interface slips in the weaker stripes but sticks in the stronger stripes. As a consequence, the maximal slip occurs on the center line of the weaker stripes and depends on its width. A wider stripe presents more slip at the center because of the larger distance from the no-slip condition at the border. This can be observed by comparing slip in the systems with $W=2.5\mathrm{mm}$ and $W=4\mathrm{mm}$ at $t=90\mathrm{\mu }\mathrm{s}$ at the center line at $x\approx 90\mathrm{mm}$ in Fig. 7(b) and in Video 2 of the supplementary material (online). Therefore, the maximal slip occurring at the center of weaker stripes surrounded by sticking stronger stripes scales as

equation(7)

$\max \delta \propto W.$

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equation(8)

$\max \mathrm{\Delta }{\tau }^{\mathrm{s}}\propto \varphi \max \delta \propto \varphi W.$

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$t=90\mathrm{\mu }\mathrm{s}$

Lower frictional strength and hence a lower friction traction at the interface has two consequences on the transition from the homogeneous to the heterogeneous area. First, a lower friction traction decelerates less the slip rate which increases the duration of slip and prevents the detachment of the slip pulse in the stripe configuration. Second, a lower friction traction enables more slip over the same period which results in a higher σ_xz shear stress in the bulk material and thus in the stronger stripes. The later are responsible for pulling the slip front in the stronger stripes into an inter-sonic propagation regime. The result of these pulling tractions is observable as red areas in Fig. 7(d). The shear modulus of the bulk material has the opposite effect. A stiffer material results in more stress transfer through σ_xz from the sticking stronger stripes to the slipping weaker stripes. Therefore, less slip occurs and the frictional strength is less reduced.

All in all, lower frictional strength at the center of the weaker stripes favors the propagation mechanism of an inter-sonic front over the entire interface width, whereas systems with a higher strength tend to cause a sub-Rayleigh rupture with a slip pulse propagating ahead in the weaker stripes. This is consistent with Eq. (6) as can be seen by writing

equation(9)

$\zeta \propto \frac{\max \mathrm{\Delta }{\tau }^{\mathrm{s}}}{G}\propto \frac{W\varphi }{G}.$

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An alternative way of explaining the non-dimensional parameter consists of idealizing slip within the weaker stripes as a two-dimensional Mode III crack of length W   that seeks to propagate into the stronger stripes. If this crack is longer than a critical length, i.e. W>W_cr$W>W_{\mathrm{cr}}$, it will slip unstably and pull the main front in the stronger stripes into the inter-sonic propagation regime. Generally, the critical length W_cr$W_{\mathrm{cr}}$ depends on the friction law and the stress conditions. However, considering fracture mechanics arguments for a crack with a cohesive zone over its entire length (another limiting case being a singular crack), it was shown that W_cr∝G/ϕ$W_{\mathrm{cr}}\propto G/\varphi$ ( Uenishi and Rice, 2003 and Chen and Knopoff, 1986), which results here in the same non-dimensional parameter as given by Eq. (6), which is

equation(10)

$\zeta \propto \frac{W}{W_{\mathrm{cr}}}\propto \frac{W\varphi }{G}.$

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The specific values of ζ_a${\zeta }_{\mathrm{a}}$ limiting the two distinct propagation behavior in the present system are deduced from the simulations presented in Section 4.1. The simulation with the largest stripe width leading to the sub-Rayleigh with slip pulse mechanism has $W=3\mathrm{mm}$ and ζ_a=0.63${\zeta }_{\mathrm{a}}=0.63$. The next larger stripe width simulated $(W=3.5\mathrm{mm})$ presented the inter-sonic slip front and is characterized by ζ_a=0.74${\zeta }_{\mathrm{a}}=0.74$.

Up to this point, the two distinct propagation mechanisms have been shown to exist in systems which differ solely by modified width of the stripes. Eq. (6), however, indicates that two additional parameters, the shear modulus of the bulk material and the friction weakening rate, affect as well the propagation mechanism occurring in a given system. Two additional simulations are performed in order to confirm the validity of Eq. (6). The set-up with $W=4\mathrm{mm}$ and ζ_a=0.84${\zeta }_{\mathrm{a}}=0.84$ resulting in an inter-sonic slip front, as presented in Section 4.1, is used as a reference. The shear modulus of the bulk and the friction weakening rate in the weaker stripes are then independently modified such that ζ_a=0.525${\zeta }_{\mathrm{a}}=0.525$. All other parameters are kept constant. This leads, as shown in Fig. 8, to the propagation mechanism of a sub-Rayleigh rupture with a slip pulse.

The observation of a sub-Rayleigh rupture in a system with $W=4\mathrm{mm}$ but with ζ_a=0.525${\zeta }_{\mathrm{a}}=0.525$ confirms that the non-dimensional parameter provides an indication of the propagation mechanism occurring in a given set-up. It is also important to note that ζ_a${\zeta }_{\mathrm{a}}$ is only an approximation to ζ because the T term is neglected in Eq. (4). For instance, if the frictional properties are modified such that the propagation speed of the secondary rupture is changed considerably during the transition into the heterogeneous area, the estimation of the process zone size could be different and the value of ζ_a${\zeta }_{\mathrm{a}}$ could indicate the wrong propagation mechanism. Nevertheless, ζ_a${\zeta }_{\mathrm{a}}$ provides fundamental insights by indicating the material and interface properties that affect the type of propagation mechanism occurring.

The interaction between the length scale of the slip front and the interface heterogeneity, observed here in a stripe configuration, is generally valid and is expected to appear also in different systems with heterogeneous areas of the order of the process zone size of the slip front. At interfaces organized with different shapes of heterogeneities, the value of ζ_a${\zeta }_{\mathrm{a}}$ limiting different propagation behaviors is potentially different from values reported here for the stripes configuration. Nevertheless, Eq. (6) is applicable and provides information on what determines the mechanism of slip front propagation.

5. Conclusion

We studied the dynamic propagation of slip fronts through a heterogeneous interface of a three-dimensional system. The heterogeneous pattern consisted of stripes with weaker and stronger frictional properties. By varying the width of the stripes from one configuration to another but keeping identical area fractions, we showed that thinner stripes result in the propagation of a sub-Rayleigh slip front preceded by a slip pulse in the stripes of lower fracture energy. This slip pulse propagated in an “unstable” manner at speeds varying around the Rayleigh- and shear-wave speed including periods of propagation at the so-called forbidden speed. At interfaces with wider stripes, the slip front transitioned to inter-sonic speeds over the entire width of the interface.

The occurrence of two distinct propagation mechanisms at the same interface with the only difference being the heterogeneous stripe width was shown to be caused by the interaction of two length scales: the process zone size of the slip front and the width of the stripes. The ratio between these two lengths was proposed as a non-dimensional parameter capable of indicating the type of slip front propagation occurring in a given system. Reducing the characteristic dimension of heterogeneities below this critical length scale results in a stronger interface with slower rupture speeds.

References

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