Screen reader users, click here to load entire articleThis
page uses JavaScript to progressively load the article content as a
user scrolls. Screen reader users, click the load entire article button
to bypass dynamically loaded article content.
Please note that Internet Explorer version 8.x will not be supported as of January 1, 2016. Please refer to this blog post for more information.
Close
Civil
Engineering Institute, Materials Science and Engineering Institute,
Ecole Polytechnique Fédérale de Lausanne (EPFL), Station 18, CH-1015
Lausanne, Switzerland
Received 3 September 2015, Revised 24 November 2015, Accepted 20 December 2015, Available online 24 December 2015
We
present three-dimensional finite-element simulations showing the
propagation of slip fronts at striped heterogeneous interfaces. The
heterogeneous area consists of alternating stripes of weaker and
stronger frictional properties, which is equivalent to a lower and
higher fracture energy, respectively. By comparing the slip front
propagation at interfaces that differ solely by the length scale of the
heterogeneous pattern, we illustrate that two different propagation
regimes exist. Interfaces with wide stripes present slip fronts with
propagation speeds that transition from sub-Rayleigh to inter-sonic.
Thinner stripes are, however, characterized by the propagation of
sub-Rayleigh slip fronts, which are preceded by slip pulses of
negligible slip in the weaker stripes. From a macroscopic point of view,
an interface with a smaller heterogeneous pattern appears to be
stronger than the equivalent coarser interface even though both have the
same average properties. The numerical results as well as a theoretical
approach based on fracture-mechanics considerations suggest that the
origin of these two distinct propagation mechanisms lies in the
interaction between the length scales of the cohesive zone and the
heterogeneous configuration. We further show by estimating the relevant
length scales that the occurring propagation mechanism is influenced by
the friction weakening rate of the interface as well as the shear
modulus of the bulk material.
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 and height , 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.
Fig. 1.
Three-dimensional set-up of simulations. Two solids each of length L and height H are in contact and loaded by normal pressure and shear traction . Only a finite strip is modeled but periodic boundary conditions in the z-direction ensure an infinite width (with replication width 2W ). A slip front is nucleated by a notch (red area) introduced at t=0 and propagates in the x-direction.
The slipping (broken) and sticking areas of the interface are shown in
blue and gray, respectively. (For interpretation of the references to
color in this figure caption, the reader is referred to the web version
of this paper.)
The system is loaded statically by a normal pressure applied in the y -direction, and an in-plane shear load applied on all boundaries in the x–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 , Poisson's ratio ν=0.37, and density . The resulting shear modulus is and the wave speeds are , , and . 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
with static and kinetic friction coefficients μs and μk, respectively, characteristic weakening length dc, and interface slip δ . Two different sets of parameters are applied to create a heterogeneous pattern at the interface. Strong areas have μs=0.9, μk=0.2 and , which results for the applied normal load in a fracture energy of , as shown by the gray area in Fig. 2(b). Other areas have a low fracture energy of originating from μs=0.6, μk=0.2, and .
Note that both sets of parameters have the same kinetic friction
coefficient, which results in a uniform stress state behind the slip
front. The weaker interface, however, has a lower static friction
coefficient and a smaller characteristic weakening length. The material
and interface properties do not represent a particular material but are,
nevertheless, realistic values for polymeric material. Emphasis is
given to the necessity of enabling mesh-converged simulations of an
interface rupture with a process zone of small size compared to the
rupture length. Smaller fracture energies would lead to a smaller
process zone and higher computational cost, whereas larger fracture
energies could violate the small process zone hypothesis.
Fig. 2.
(a) Configuration of heterogeneous interface (not to scale) of length L in the x-direction and infinite width in the z-direction.
Periodic Boundary Conditions (PBC) are used to model the infinite
width. Duplicates of the simulated interface are shown in lighter colors
in order to provide an overall view. This figure corresponds to a view
of the interface shown in Fig. 1.
The interface is separated into three areas: notch, establishing zone,
and heterogeneous area. In the notch, the fracture energy is set to
zero, which causes the nucleation of the slip front. In the establishing
zone, before reaching the heterogeneous area, the rupture accelerates
and approaches a steady state with a straight front. In the
heterogeneous part, the interface consists of stripes of width W
(which corresponds to half of the replication width). Purple and green
areas indicate frictional properties of high and low fracture energy,
respectively. (b) Linear slip-weakening friction law with friction
threshold and residual friction . The weakening process is characterized by the friction weakening rate ϕ and the length scale dc. The resulting fracture energy Γc
is shown as gray area. (For interpretation of the references to color
in this figure caption, the reader is referred to the web version of
this paper.)
The configuration of the heterogeneous interface is shown in Fig. 2(a). The slip front is nucleated by introducing at 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 μs=μk=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).
Fig. 3.
The propagation of a slip front at a homogeneous interface with fracture energy . Data behind the seed crack as well as close to the edge
is hidden in order to focus attention to a rupture that is not affected
by wave reflections. (a) The friction traction at the interface is
shown in color in a space-time map. The slip front starts at and propagates in the positive x -direction.
The front is located in the red area, where the friction traction is
maximal. The initiation phase of the rupture is invisible due to low
data acquisition frequency during the beginning of the simulations .
(b) The rupture speed measured at each point of the space
discretization (black points), which is also reported by an averaged
value (red line), indicates that the slip front propagates at
sub-Rayleigh speed. Values are normalized with respect to the
dilatational wave speed cd
of the bulk material. (For interpretation of the references to color in
this figure caption, the reader is referred to the web version of this
paper.)
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 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 0.83cd
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 (there is no red area in Fig. 4(a) for ) because the static friction coefficient is smaller with μs=0.6 (compared to 0.9).
Fig. 4.
The
propagation of a slip front at a homogeneous interface crossing the
boundary from an area of high to low fracture energy, with and ,
respectively. (a) The space-time map of the friction tractions presents
a sudden change at the boundary from the high to low fracture energy
area (indicated by a vertical white line). The maximal friction traction
observed in the area of lower fracture energy is smaller than in the
establishing zone because the static friction coefficient is smaller.
(b) The rupture speed follows first the same trend as in Fig. 3(b) until it reaches the area of lower fracture energy, in which it accelerates instantaneously to inter-sonic speed.
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 ,
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,
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 and .
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 and . The two distinct mechanisms are strictly separated by the width of the stripes. Slip fronts propagating at interfaces with are equivalent to the slip front presented in the system with , whereas ruptures at interfaces with correspond to the behavior of the slip front in a set-up with .
Fig. 5.
Interface snapshots of slip fronts at heterogeneous interfaces with stripe width (a+c) and (b+d) . (a–b) Slip area is shown in blue and (c–d) slip rate in red where maximal velocity is indicated by dark colors. The heterogeneous area starts in all cases at and consists of stripes, as presented in Fig. 2(a). The stripe with lower fracture energy (purple stripe in Fig. 2(a)) is located in the center of the shown interface. The same scale is applied to the x- and z -axes
in order to preserve the correct aspect ratio of the interface. The
system with thinner stripes (a+c) results in a slip pulse in the weaker
stripes propagating ahead of the main front. The configuration with
wider stripes (b+d), however, presents a slip front propagating over the
entire width. At approximately , a “stick” front initiates at and follows the slip front in the positive x-direction.
(For interpretation of the references to color in this figure caption,
the reader is referred to the web version of this paper.)
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 ). 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 , 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 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., , a stick front initiates at
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 . Another striking difference is that there is much more slip occurring in the front rupture of the wider system ( at ) compared to the slip pulse of the thinner set-up ( at ).
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
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 , 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.
Fig. 6.
Propagation speed of slip fronts shown in Fig. 5.
The rupture speed is reported for the first slip front at the center
line of each stripe. The slip pulse in the weaker stripe of the
configuration with ,
shown in dark purple, propagates with varying speed around the
Rayleigh- and shear-wave speed. The main slip front in this system,
shown in dark green, propagates at sub-Rayleigh speed with a short
inter-sonic period at the beginning of the heterogeneous area. The slip
front propagating at the interface with stripe width
is inter-sonic over the entire width. The rupture speed in the stripes
of higher and lower fracture energy is depicted by bright green and
bright purple line, respectively. The thin gray lines show the rupture
speed of the homogeneous reference cases presented in Fig. 3 and Fig. 4.
The vertical black line indicates the position at which the
heterogeneous area starts. (For interpretation of the references to
color in this figure caption, the reader is referred to the web version
of this paper.)
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 ,
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 , the front presents the same constant propagation speed over the entire interface width. The rear slip front initiating at at 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
where wpz is the characteristic size of the process zone in the weaker stripes, and W is the width of the stripes.
Fig. 7.
Interface
snapshots for various fields shown for instants in time when the slip
front enters the heterogeneous area. The configuration with stripe width
and , as presented in Fig. 5, are illustrated in the left and right column, respectively. The same scale is applied to the x- and z -axes
in order to preserve the correct aspect ratio of the interface. The
snapshots show the (a) slip area in dark blue with the process zone in
brighter color, (b) slip (distance) with scale: and , (c) frictional strength with scale: and (μs
of high fracture energy), and (d) friction traction with the same scale
as the frictional strength. (For interpretation of the references to
color in this figure caption, the reader is referred to the web version
of this paper.)
The determination of wpz
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
where Γc is the fracture energy of the interface, μs and μk are the static and kinetic friction coefficients, is the normal pressure, G is the bulk's shear modulus, and T is a function depending on the propagation speed vr and the dilatational cd and shear cs wave speeds.
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
Substituting the fracture energy with and by defining the slope of the friction weakening process as , as shown in Fig. 2(b), Eq. (4) is reduced to two basic parameters:
which
provides, in first approximation, a fundamental understanding of the
parameters affecting the preferred propagation mechanism. From the
simulations presented in Section 4.1, we know that lower values of ζ lead to sub-Rayleigh ruptures with slip pulses propagating in the weaker stripes ahead, whereas higher values of ζ result in an inter-sonic slip front propagating over the entire width of the interface.
Before considering the values of ζa
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 , a secondary rupture initiates ahead in the weaker stripes (see snapshots at ).
Once the main front merged with the secondary rupture, the front
continues propagation over the entire width of the interface (see
snapshots at ).
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 and at at the center line at 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
The
resulting reduction of the frictional strength at the center of the
weaker stripes depends on the friction law as well as on the quantity of
slip. In the case of a linear slip-weakening friction, the strength
decreases for both, increasing slip δ and increasing weakening rate ϕ.
Therefore, the maximal strength reduction at the center of the weaker
stripes is given, if the adjoining stronger stripes are sticking, by
Note
that these are not linear relations because slip and frictional
strength are inter-dependent. More slip at the center line of the weaker
stripes leads to lower strength, and lower strength resists less
against the external load, which results in more slip. The difference in
the maximal frictional-strength reduction is well observable in Fig. 7(c) at and in Video 3 of the supplementary material (online).
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 σxz 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 σxz
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
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>Wcr,
it will slip unstably and pull the main front in the stronger stripes
into the inter-sonic propagation regime. Generally, the critical length Wcr
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 Wcr∝G/ϕ ( 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
The specific values of ζa 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 and ζa=0.63. The next larger stripe width simulated presented the inter-sonic slip front and is characterized by ζa=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 and ζa=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 ζa=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.
Fig. 8.
Propagation speed for slip fronts at different interfaces with heterogeneous stripe width . The inter-sonic rupture shown in Fig. 6
is reported in gray as a reference. The propagation speed of two
additional slip fronts is shown in color. The rupture illustrated by
cyan curves has a modified friction weakening rate by setting (instead of ), which leads to a fracture energy of . The magenta curves present the propagation of a rupture in a system of modified shear modulus by setting the shear modulus (compared to ). Both slip fronts present at an interface with stripe width
the sub-Rayleigh propagation mechanism with a slip pulse ahead. The
thicker lines represent the speed of the main front, whereas the thinner
lines indicate the slip pulse speed. (For interpretation of the
references to color in this figure caption, the reader is referred to
the web version of this paper.)
The observation of a sub-Rayleigh rupture in a system with but with ζa=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 ζa 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 ζa could indicate the wrong propagation mechanism. Nevertheless, ζa
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 ζa
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.
Acknowledgments
The
authors thank the anonymous reviewer for pointing out the alternative
explanation for the non-dimensional parameter based on a Mode III crack
consideration. The research described in this article was supported by
the European Research Council (ERCstg UFO-240332). DSK acknowledges support from Cornell University.
Time evolution of slip area in dark blue with process zone in brighter color shown for stripe configuration with W= 2.5 mm (top) and W=4 mm (bottom). Axes x and bbbbz
are scaled equally in order to preserve the correct aspect ratio of the
interface. Snapshots of segments with a single heterogeneity (with
replication width 2 W) are shown in Fig. 7(a).
Time evolution of slip (distance) shown for stripe configuration with W= 2.5 mm (top) and W=4 mm (bottom). The color code is: blue = 0 mm and red = 0.03 mm. Axes x and z
are scaled equally in order to preserve the correct aspect ratio of the
interface. Snapshots of segments with a single heterogeneity (with
replication width 2 W) are shown in Fig. 7(b).
Time evolution of frictional strength shown for stripe configuration with W= 2.5 mm (top) and W=4 mm (bottom). The color code is: residual strength in blue and peak strength of high fracture energy in red. Axes x and z
are scaled equally in order to preserve the correct aspect ratio of the
interface. Snapshots of segments with a single heterogeneity (with
replication width 2 W) are shown in Fig. 7(c).
K. Ariyoshi, T. Matsuzawa, J.-P. Ampuero, R. Nakata, T. Hori, Y. Kaneda, R. Hino, A. Hasegawa
Migration
process of very low-frequency events based on a chain-reaction model
and its application to the detection of preseismic slip for megathrust
earthquakes
The
plural form of stripe is used because, even though only one weaker
stripe and two stronger half-stripes are shown in the figures, there is
an infinite number of stripes due to the periodic boundary conditions.
Qingyan Tang, Mingjie Zhang, Chusi Li, Ming Yu, Liwu Li
The chemical compositions and abundances of volatiles in the Siberian large igneous province: Constraints on magmatic CO2 and SO2 emissions into the atmosphere
Chemical Geology, Volume 339, 15 February 2013, Pages 84–91