# The dynamics of unsteady frictional slip pulses

Anna Pomyalov1 Fabian Barras2 Thibault Roch3 Efim A. Brener4,5 e.brener@fz-
juelich.de Eran Bouchbinder1<EMAIL_ADDRESS>1Chemical and
Biological Physics Department, Weizmann Institute of Science, Rehovot 7610001,
Israel
2The Njord Centre, Departments of Physics and Geosciences, University of Oslo,
0316 Oslo, Norway
3Civil Engineering Institute, Materials Science and Engineering Institute,
Ecole Polytechnique Fédérale de Lausanne, Station 18, CH-1015 Lausanne,
Switzerland
4Peter Grünberg Institut, Forschungszentrum Jülich, D-52425 Jülich, Germany
5Institute for Energy and Climate Research, Forschungszentrum Jülich, D-52425
Jülich, Germany

###### Abstract

Self-healing slip pulses are major spatiotemporal failure modes of frictional
systems, featuring a characteristic size $L(t)$ and a propagation velocity
$c_{\rm p}(t)$ ($t$ is time). Here, we develop a theory of slip pulses in
realistic rate-and-state dependent frictional systems. We show that slip
pulses are intrinsically unsteady objects, yet their dynamical evolution is
closely related to their unstable steady-state counterparts. In particular, we
show that each point along the time-independent
$L^{\mbox{\tiny{(0)}}}\\!(\tau_{\rm d})\\!-\\!c^{\mbox{\tiny{(0)}}}_{\rm
p}\\!(\tau_{\rm d})$ line, obtained from a family of steady-state pulse
solutions parameterized by the driving shear stress $\tau_{\rm d}$, is
unstable. Nevertheless, and remarkably, the $c^{\mbox{\tiny{(0)}}}_{\rm
p}[L^{\mbox{\tiny{(0)}}}]$ line is a dynamic attractor such that the unsteady
dynamics of slip pulses — whether growing ($\dot{L}(t)\\!>\\!0$) or decaying
($\dot{L}(t)\\!<\\!0$) — reside on the steady-state line. The unsteady
dynamics along the line are controlled by a single slow unstable mode. The
slow dynamics of growing pulses, manifested by $\dot{L}(t)/c_{\rm
p}(t)\\!\ll\\!1$, explain the existence of sustained pulses, i.e. pulses that
propagate many times their characteristic size without appreciably changing
their properties. Our theoretical picture of unsteady frictional slip pulses
is quantitatively supported by large-scale, dynamic boundary-integral method
simulations.

Frictional systems, such as geological faults and the tectonic plates that
form them, undergo interfacial failure through various spatiotemporal rupture
modes that mediate rapid sliding [1, 2]. One major rupture mode is self-
healing slip pulses, which are solitonic structures that feature a
characteristic size (compact support) $L(t)$ and a propagation velocity
$c_{\rm p}(t)$, where $t$ is time [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. The existence of self-healing
slip pulses — having no counterparts in the tensile failure of materials —
crucially depends on the nonequilibrium nature of frictional interfaces, most
notably on the ability of contact interfaces to recover their strength after
undergoing sliding-induced weakening, i.e. to self heal [26, 27, 28, 29, 30,
31, 32].

Frictional slip pulses pose significant physical and mathematical challenges;
even obtaining steady-state pulse solutions for realistic, laboratory-derived
interfacial constitutive relations and studying their dynamic stability remain
major open challenges. Yet, there exist some indications that steady-state
slip pulses are intrinsically unstable objects [5, 21, 22]. At the same time,
experimental and numerical observations show that slip pulses can be
dynamically sustained, i.e. to propagate over distances that are much larger
than their characteristic size without appreciably changing their shape and
propagation velocity [4, 17, 21, 18]. This combined evidence may appear
intrinsically conflicting, reflecting the lack of current understanding of the
dynamics of unsteady frictional slip pulses. This problem is of prime
importance in fields such as nonequilibrium and nonlinear physics, materials
physics, tribology and geophysics.

Here, we develop a theoretical picture of unsteady frictional slip pulses, and
reveal a surprising intrinsic relation between them and their unstable steady-
state counterparts. Generic rate-and-state dependent frictional systems admit
a family of steady-state slip pulse solutions parameterized by the driving
shear stress $\tau_{\rm d}$, featuring a size-velocity
$L^{\mbox{\tiny{(0)}}}(\tau_{\rm d})\\!-\\!c^{\mbox{\tiny{(0)}}}_{\rm
p}(\tau_{\rm d})$ relation (the superscript ${}^{\mbox{\tiny{(0)}}}$ refers to
time-independent steady-state pulse solutions) [25]. We show that each point
along the $c^{\mbox{\tiny{(0)}}}_{\rm p}[L^{\mbox{\tiny{(0)}}}]$ line is
unstable. That is, slightly perturbed steady-state pulses either grow in time
($\dot{L}(t)\\!>\\!0$) or decay ($\dot{L}(t)\\!<\\!0$) such that steady-state
pulses correspond to a saddle configuration. Yet, the resulting unsteady slip
pulses do not follow arbitrary dynamics in the $L(t)\\!-\\!c_{\rm p}(t)$ plane
as a function of time, but are rather attracted toward and closely adhere to
the $c^{\mbox{\tiny{(0)}}}_{\rm p}[L^{\mbox{\tiny{(0)}}}]$ line. Consequently,
one is able to understand the dynamics of unsteady slip pulses.

Furthermore, the unsteady dynamics along the $c^{\mbox{\tiny{(0)}}}_{\rm
p}[L^{\mbox{\tiny{(0)}}}]$ line are controlled by a single slow unstable mode.
In particular, growing pulses feature $\dot{L}(t)/c_{\rm p}(t)\\!\ll\\!1$ and
are nothing but sustained pulses that propagate many times their
characteristic size without appreciably changing their properties. The
$c^{\mbox{\tiny{(0)}}}_{\rm p}[L^{\mbox{\tiny{(0)}}}]$ line is a dynamic
attractor for unsteady slip pulses independently of their initial distance
from the line, i.e. the theory applies to general initial conditions that lead
to unsteady slip pulses. Our theoretical picture of unsteady frictional slip
pulses is quantitatively supported by large-scale, dynamic boundary-integral
method simulations for various initial conditions.

The dynamic stability of steady-state pulse solutions.— The frictional
strength $\tau$ of contact interfaces depends on the time rate of change of
the tangential displacement discontinuity across the interface, i.e. on the
slip velocity $v(x,t)$ (here $x$ is the coordinate along the interface), and
on the structural state of the interface. As macroscopic contact interfaces
are never smooth on all scales [27, 29, 31, 33], a central structural state
field is the real contact area, composed of an ensemble of contact asperities,
which is typically orders of magnitude smaller than the nominal contact area
[27, 33]. Moreover, this scale separation between the real and nominal contact
area implies that contact asperities experience intense loads and typically
age with time. The aging time $\phi(x,t)$ is commonly used to quantify the
nonequilibrium structural state of the interface [29, 31].

Figure 1: (main) The steady frictional resistance $\tau[v,\phi\\!=\\!D/v]$
(solid brown line, in units of the normal stress $\sigma$) vs. slip velocity
$v$ (in units of the $v_{\rm min}$, the velocity at the local minimum). A
steady-state slip pulse corresponds to a closed (homoclinic) orbit
$(v_{{}_{\rm ss}}\\!(\xi),\tau_{{}_{\rm ss}}\\!(\xi))$ (solid green line)
parameterized by the co-moving space-time coordinate
$\xi\\!=\\!x-c^{\mbox{\tiny{(0)}}}_{\rm p}t$, where
$c^{\mbox{\tiny{(0)}}}_{\rm p}$ is the steady pulse propagation velocity. The
closed orbit starts and ends at $(v_{\rm stick},\tau_{\rm d})$, where
$\tau_{\rm d}$ is the driving shear stress and $v_{\rm stick}$ is extremely
small. The existence of a minimum $(v_{\rm min},\tau_{\rm min})$ of the
friction curve, whose values are used to normalize various quantities, plays
no important roles in this work. (upper inset) $v_{{}_{\rm ss}}\\!(\xi)$,
which features a maximal value $v_{\rm m}$ and a characteristic size
$L^{\mbox{\tiny{(0)}}}$ (not marked, see [34]), is plotted ($\xi$ is expressed
in units of $L_{\rm n}$, a natural normalization length [34]). (lower inset)
The size-velocity relation $c^{\mbox{\tiny{(0)}}}_{\rm
p}[L^{\mbox{\tiny{(0)}}}]$ (in units of the shear wave-speed $c_{\rm s}$) for
a family of steady-state pulse solutions parameterized by $\tau_{\rm d}$ is
plotted (solid line). A solution for a given $\tau_{\rm d}$ is marked by the
diamond, and its expected stability properties are represented by the arrows
(see text for discussion).

Consequently, the frictional strength is a functional
$\tau[v(x,t),\phi(x,t)]$, whose form is extracted from experimental
measurements and theoretical considerations [29, 31, 35], see Fig. 1. Finally,
contact aging in the absence of sliding ($v\\!=\\!0$) and sliding-induced
($v\\!>\\!0$) rejuvenation (anti-aging) are described as
$\partial_{t}\phi(x,t)\\!=\\!1\\!-\\!v(x,t)\,\phi(x,t)/D$, where the
transition between the two rheological behaviors is controlled by a
characteristic slip displacement $D$ [36, 37, 38, 29, 31]. The frictional
strength $\tau[v(x,t),\phi(x,t)]$ balances the shear stress emerging from the
deformation of the bodies forming the contact interface, at any point $x$ and
time $t$. The deformable bodies extend in the $y$ direction (the interface is
located at $y\\!=\\!0$), and are assumed to be translationally invariant along
the third direction $z$, making the problem considered here two-dimensional.
When the bodies forming the contact interface are linear elastic and of
infinite extent (or when the dynamics of interest are such that waves
reflected from finite boundaries do not have time to affect the interface),
the interfacial shear stress can be expressed in terms of the elastodynamic
Green’s function.

Considering then a frictional system driven anti-symmetrically by a shear
stress $\tau_{\rm d}$ applied far from the interface, interfacial stress
balance takes the form [39]

$\tau[v(x,t),\phi(x,t)]=\tau_{\rm d}-\frac{\mu}{2c_{s}}v(x,t)+s(x,t)\ .$ (1)

Here, $\mu$ and $c_{s}$ are the shear modulus and wave-speed of the bodies,
respectively, and $s(x,t)$ is a spatiotemporal convolutional integral that
accounts for the long-range interaction of different parts of the interface,
mediated by bulk deformation. In general, $s(x,t)$ does not admit an explicit
real-space representation, and its specific form is different for anti-plane
shear symmetry (i.e. slip displacement in the $z$ direction) and in-plane
shear symmetry (i.e. slip displacement in the $x$ direction) [40]. Together
with the $\partial_{t}\phi(x,t)$ equation stated above and an explicit
expression for $\tau[v(x,t),\phi(x,t)]$ [34] (see Fig. 1), the coupled bulk-
interface frictional dynamics are described by the specified set of nonlinear
integro-differential equations.

Self-healing slip pulses correspond to solutions to the set of nonlinear
integro-differential equations, which propagate at an instantaneous velocity
$c_{\rm p}(t)$ and feature a finite length $L(t)$. The compact support $L(t)$,
over which slip velocities are large, separates two spatially homogeneous,
nearly quiescent, identical interfacial states. The latter are characterized
by a vanishingly small slip velocity $v_{\rm stick}\\!\to\\!0$ (‘stick’
denotes the fact that the state is nearly quiescent). It is obtained by
solving $\tau_{\rm d}\\!=\\!\tau[v_{\rm stick},D/v_{\rm stick}]$ for a given
driving shear stress $\tau_{\rm d}$, where $\phi\\!=\\!D/v$ corresponds to
$\partial_{t}\phi\\!=\\!0$. In the language of dynamical systems, self-healing
slip pulses correspond to closed (homoclinic) orbits in the $v\\!-\\!\phi$
plane, which both starts and ends at $(v_{\rm stick},D/v_{\rm stick})$.
Alternatively, they correspond to closed (homoclinic) orbits in the $v-\tau$
plane, which both start and end at $(v_{\rm stick},\tau_{\rm d})$, see Fig. 1.

Steady-state slip pulses correspond to solutions that also satisfy $c_{\rm
p}(t)\\!=\\!c^{\mbox{\tiny{(0)}}}_{\rm p}$, i.e. which propagate steadily at a
time-independent velocity $c^{\mbox{\tiny{(0)}}}_{\rm p}$. Under such
conditions, all fields depend on the co-moving space-time coordinate
$\xi\\!\equiv\\!x-c^{\mbox{\tiny{(0)}}}_{\rm p}t$ and one has
$\partial_{t}\\!=\\!-c^{\mbox{\tiny{(0)}}}_{\rm p}\partial_{\xi}$ (applied to
the $\partial_{t}\phi(x,t)$ equation). Equation (S1) then takes the form [41]

$\tau[v_{{}_{\rm ss}}\\!(\xi),\phi_{{}_{\rm ss}}\\!(\xi)]=\tau_{\rm
d}-\frac{\mu\,{\cal F}(c^{\mbox{\tiny{(0)}}}_{\rm
p}/c_{s})}{c^{\mbox{\tiny{(0)}}}_{\rm
p}}\\!\int_{-\infty}^{\infty}\frac{v_{{}_{\rm ss}}\\!(z)}{z-\xi}dz\ ,$ (2)

where the integral (which should be understood in the Cauchy principal value
sense) is the steady-state limit of the two last terms on the right-hand-side
of Eq. (S1) and ${\cal F}(c^{\mbox{\tiny{(0)}}}_{\rm p}/c_{s})$ is a known
function (different for anti-plane and in-plane shear symmetries) [41].

A complete family of steady-state slip pulse solutions, i.e. $v_{{}_{\rm
ss}}\\!(\xi)$, $\phi_{{}_{\rm ss}}\\!(\xi)$ and $c^{\mbox{\tiny{(0)}}}_{\rm
p}$ parameterized by $\tau_{\rm d}$, has been very recently obtained [25], see
an example in the upper inset in Fig. 1. The properties of the solutions and
their theoretical understanding have been discussed in [25]. Most relevant for
our purposes here is that the family of steady-state slip pulse solutions
features a size-velocity $L^{\mbox{\tiny{(0)}}}(\tau_{\rm
d})\\!-\\!c^{\mbox{\tiny{(0)}}}_{\rm p}(\tau_{\rm d})$ relation that is
parameterized by $\tau_{\rm d}$, where $L^{\mbox{\tiny{(0)}}}$ is the
characteristic size of each pulse [34]. This relation is monotonically
increasing and is plotted in the lower inset in Fig. 1 for anti-plane shear
symmetry (and a representative set of frictional constitutive parameters
[34]).

In principle, the physical relevance of the obtained steady-state slip pulse
solutions should be judged based on their dynamic stability. The above
formulated problem, which corresponds to bodies of infinite height $H$ (i.e.
$H\\!\to\\!\infty$), has been previously solved in the limit of small $H$
(where long-range elastodynamic interactions become local) [21]. It has been
shown that steady-state slip pulses in this limit are intrinsically unstable,
i.e. that $L^{\mbox{\tiny{(0)}}}(\tau_{\rm d})+\delta{L}$ perturbations (with
$\delta{L}\\!>\\!0$) lead to growing pulses, while
$L^{\mbox{\tiny{(0)}}}(\tau_{\rm d})-\delta{L}$ perturbations lead to decaying
pulses. If the two problems are smoothly connected by continuously increasing
$H$, we expect slip pulses in the $H\\!\to\\!\infty$ limit to be intrinsically
unstable as well, i.e. to correspond to a saddle configuration for each
$\tau_{\rm d}$. This expectation is in line with additional evidence in the
literature [5, 22] and is tested next.

Figure 2: Dynamic stability analysis of a steady-state pulse with $\tau_{\rm
d}\\!=\\!1.03\tau_{\rm min}\\!\approx\\!0.351\sigma$ (featuring
$c^{\mbox{\tiny{(0)}}}_{\rm p}\\!=\\!0.82c_{\rm s}$) using the $\phi(x,t)$
field $\epsilon$-perturbation scheme described in the text, with
$\epsilon\\!=\\!-0.05$ (brown data) and $\epsilon\\!=\\!-0.15$ (blue data).
(a) $L(t)/L^{\mbox{\tiny{(0)}}}$ vs. $t/{\cal T}$. (b)
$c^{\mbox{\tiny{(0)}}}_{\rm p}/c_{\rm s}$ vs. $t/{\cal T}$. (c) $v_{\rm
m}(t)/v_{\rm min}$ vs. $t/{\cal T}$, with ${\cal
T}\\!=\\!L^{\mbox{\tiny{(0)}}}/c^{\mbox{\tiny{(0)}}}_{\rm p}$. For all 3
quantities, $\epsilon\\!=\\!-0.05$ (brown data) leads to growth and
$\epsilon\\!=\\!-0.15$ (blue data) to decay, demonstrating the unstable nature
of the steady-state pulse. The timescale of instability is similar in all
quantities and is set by ${\cal T}$. In the inset of panel (a), we test the
predicted unstable linearized dynamics in Eq. (3). Since $\epsilon$ is not
exactly centered around $\epsilon_{\rm c}\\!=\\!0$, we consider the solution
of Eq. (3) in the form
$L_{\pm}(t)\\!=\\!L^{\mbox{\tiny{(0)}}}\\!\pm\\!\Delta{L}\exp[a_{\pm}(t-t_{1})/{\cal
T}]$ for $t\\!>\\!t_{1}$, with $t_{1}\\!\simeq\\!8{\cal T}$, where $\pm$
correspond to the upper/lower branches, respectively. The results with
$\Delta{L}/L^{\mbox{\tiny{(0)}}}\\!\simeq\\!0.01$ (quantifying the deviation
from steady state at $t_{1}$) are superimposed, for the upper branch with
$a_{+}\\!=\\!0.13$ (green dashed-dotted line) and for the lower one with
$a_{-}\\!=\\!0.17$ (black dashed-dotted line), supporting the prediction in
Eq. (3).

To test this expectation, we use the steady-state pulse solutions —
$v_{{}_{\rm ss}}\\!(\xi)$, $\phi_{{}_{\rm ss}}\\!(\xi)$ and
$c^{\mbox{\tiny{(0)}}}_{\rm p}$ for a given $\tau_{\rm d}$ — as initial
conditions for the entire history $-\infty\\!<\\!t\\!<\\!0$ in Eq. (S1) and
for the $\partial_{t}\phi(x,t)$ equation, slightly perturb them in both
directions ($\pm\delta{L}$) at $t\\!=\\!0$ and follow the subsequent dynamics
using large-scale, dynamic boundary-integral method simulations [39, 42],
using the open-source library cRacklet [43], see details in [34]. The
perturbation at $t\\!=\\!0$ takes the form
$\phi(x,t\\!=\\!0)\\!=\\!(1+\epsilon)\phi(x,t\\!=\\!0^{-})$. That is, we
perturb the state field $\phi(x,t)$, where $\epsilon\\!<\\!0$ corresponds to
$+\delta{L}$ perturbations and $\epsilon\\!>\\!0$ to $-\delta{L}$ ones. The
results are presented in Fig. 2, where $L(t\\!>\\!0)$, $c_{\rm p}(t\\!>\\!0)$
and the maximal slip velocity $v_{\rm m}(t\\!>\\!0)$ (see upper inset in Fig.
1) are plotted for $\epsilon\\!=\\!-0.15$ and $\epsilon\\!=\\!-0.05$. It is
observed that for $\epsilon\\!=\\!-0.15$ the pulse grows, while for
$\epsilon\\!=\\!-0.05$ it decays, where transition from decay to growth takes
place at $\epsilon_{\rm c}$ in between. While theoretically we expect the
transition to occur exactly at $\epsilon_{\rm c}\\!=\\!0$, the numerical
transition point is shifted by a few percent due to finite size effects and
computational constraints (which limit the history duration, see [34]).

The results in Fig. 2 show that steady-state pulses are intrinsically unstable
and correspond to a saddle configuration, separating growing from decaying
pulses. The characteristic timescale of variation of all 3 quantities ($L(t)$,
$c_{\rm p}(t)$ and $v_{\rm m}(t)$), to be denoted by ${\cal T}$, appears to be
similar. Focusing, for example, on the time evolution of $L(t)$ near
$L^{\mbox{\tiny{(0)}}}(\tau_{\rm d})$, we expect it to follow the linearized
dynamics

$\dot{L}(t)\\!=\\!\frac{L(t)-L^{\mbox{\tiny{(0)}}}(\tau_{\rm d})}{{\cal
T}(\tau_{\rm
d})}\,\,\quad\hbox{for}\quad\,\,\frac{L(t)\\!-\\!L^{\mbox{\tiny{(0)}}}(\tau_{\rm
d})}{L^{\mbox{\tiny{(0)}}}(\tau_{\rm d})}\\!\ll\\!1\ .$ (3)

Moreover, a dimensional estimate suggests that ${\cal T}(\tau_{\rm
d})\\!\sim\\!L^{\mbox{\tiny{(0)}}}(\tau_{\rm d})/c^{\mbox{\tiny{(0)}}}_{\rm
p}(\tau_{\rm d})$. These predictions are verified in the inset of Fig. 2a.
Taken together, our results indicate that slip pulses are unsteady objects and
that their dynamics near steady state is characterized by a single unstable
mode, featuring an instability rate $\sim\\!1/{\cal T}(\tau_{\rm d})$.

Theory of unsteady frictional slip pulses.— The unstable nature of steady-
state slip pulses in large systems, established above, may suggest that
steady-state slip pulse solutions are largely irrelevant for understanding the
unsteady dynamics of slip pulses in realistic frictional systems. That is, the
results show that slip pulses (under a fixed $\tau_{\rm d}$) that are
initially close to $(L^{\mbox{\tiny{(0)}}}(\tau_{\rm
d}),c^{\mbox{\tiny{(0)}}}_{\rm p}(\tau_{\rm d}))$ are dynamically driven away
from this point. Moreover, the obtained results appear to indicate that pulses
that are unrelated to $(L^{\mbox{\tiny{(0)}}}(\tau_{\rm
d}),c^{\mbox{\tiny{(0)}}}_{\rm p}(\tau_{\rm d}))$ to begin with, will reveal
no special relation to the $c^{\mbox{\tiny{(0)}}}_{\rm
p}[L^{\mbox{\tiny{(0)}}}]$ line, corresponding to unstable steady-state slip
pulse solutions. If true, then a theory of unsteady frictional slip pulses
should be developed independently of steady-state slip pulse solutions (which
is a formidable task). We show that this is in fact not the case. That is,
while all of the above statements regarding the unstable steady-state point
$(L^{\mbox{\tiny{(0)}}}(\tau_{\rm d}),c^{\mbox{\tiny{(0)}}}_{\rm p}(\tau_{\rm
d}))$ for a given $\tau_{\rm d}$ are valid, the entire
$c^{\mbox{\tiny{(0)}}}_{\rm p}[L^{\mbox{\tiny{(0)}}}]$ line — parameterized by
$\tau_{\rm d}$ — is of fundamental importance for unsteady frictional slip
pulses.

To understand this, let us consider pulse dynamics in the $L(t)\\!-\\!c_{\rm
p}(t)$ plane, where the steady-state $c^{\mbox{\tiny{(0)}}}_{\rm
p}[L^{\mbox{\tiny{(0)}}}]$ line is defined (see lower inset of Fig. 1). That
is, while slip pulses are characterized by the fields $v(x,t)$ and
$\phi(x,t)$, i.e. by infinitely many degrees of freedom, we are looking for a
reduced-dimensionality description based on a small number of coarse-grained
discrete variables, in particular two such variables. The specific choice of
$L(t)$ and $c_{\rm p}(t)$ is physically motivated, as the pulse size and
propagation velocity are of fundamental importance, yet it is by no means
unique (in fact, we show that $L(t)$ and $v_{\rm m}(t)$ could have been used
as well [34]). The two-dimensional dynamical system for $\dot{L}(t)$ and
$\dot{c}_{\rm p}(t)$ (under a given $\tau_{\rm d}$) is unknown and is in
general nonlinear arbitrarily far from the point
$(L^{\mbox{\tiny{(0)}}}(\tau_{\rm d}),c^{\mbox{\tiny{(0)}}}_{\rm p}(\tau_{\rm
d}))$.

Yet, the short-time dynamics near $(L^{\mbox{\tiny{(0)}}}(\tau_{\rm
d}),c^{\mbox{\tiny{(0)}}}_{\rm p}(\tau_{\rm d}))$ follow linearized equations
as in Eq. (3) along its unstable direction. As the dynamical system is two-
dimensional and there is one unstable mode (formally corresponding to a
positive eigenvalue), the other mode is stable (corresponding to a negative
eigenvalue). It is natural to expect the direction of the unstable mode in the
$L(t)\\!-\\!c_{\rm p}(t)$ plane to be along the local tangent to
$c^{\mbox{\tiny{(0)}}}_{\rm p}[L^{\mbox{\tiny{(0)}}}]$ line at
$(L^{\mbox{\tiny{(0)}}}(\tau_{\rm d}),c^{\mbox{\tiny{(0)}}}_{\rm p}(\tau_{\rm
d}))$ and the stable direction to be perpendicular to it, as is illustrated by
the outgoing and ingoing arrows in the lower inset in Fig. 1. Plotting the
results of Fig. 2a-b in the $L(t)\\!-\\!c_{\rm p}(t)$ plane (see [34]) indeed
show that the unstable short-time dynamics near
$(L^{\mbox{\tiny{(0)}}}(\tau_{\rm d}),c^{\mbox{\tiny{(0)}}}_{\rm p}(\tau_{\rm
d}))$ evolve along the local tangent to $c^{\mbox{\tiny{(0)}}}_{\rm
p}[L^{\mbox{\tiny{(0)}}}]$ line at this point.

Since the above results apply to any point along the
$c^{\mbox{\tiny{(0)}}}_{\rm p}[L^{\mbox{\tiny{(0)}}}]$ line, a rather
remarkable picture emerges. Invoking the continuity of dynamical trajectories
in $L(t)\\!-\\!c_{\rm p}(t)$ plane and assuming $c^{\mbox{\tiny{(0)}}}_{\rm
p}[L^{\mbox{\tiny{(0)}}}]$ to be the only steady-state line, we expect all
dynamical trajectories that are away from the $c^{\mbox{\tiny{(0)}}}_{\rm
p}[L^{\mbox{\tiny{(0)}}}]$ line to be attracted to it and once they hit it,
say at time $t_{0}$, to evolve along the $c^{\mbox{\tiny{(0)}}}_{\rm
p}[L^{\mbox{\tiny{(0)}}}]$ line, whether corresponding to decaying or growing
pulses. The latter implies that if one treats $L(t)$ as the “independent
variable”, we expect $c_{\rm p}(t)$ to be determined according to

$c_{\rm p}(t)\simeq c^{\mbox{\tiny{(0)}}}_{\rm p}[L(t)]\qquad\hbox{for}\qquad
t>t_{0}\ ,$ (4)

where the right-hand-side of Eq. (4) is the steady-state relation
$L^{\mbox{\tiny{(0)}}}[c^{\mbox{\tiny{(0)}}}_{\rm p}]$ and its argument is the
dynamic $L(t)$. Moreover, if $L(t_{0})\\!>\\!L^{\mbox{\tiny{(0)}}}(\tau_{\rm
d})$, then the pulse will grow ($\dot{L}(t\\!>\\!t_{0})\\!>\\!0$), and if
$L(t_{0})\\!<\\!L^{\mbox{\tiny{(0)}}}(\tau_{\rm d})$, it will decay
($\dot{L}(t\\!>\\!t_{0})\\!<\\!0$).

The emerging physical picture of unsteady slip pulses, to be quantitatively
tested below, reveals that while each point
$(L^{\mbox{\tiny{(0)}}}\\!(\tau_{\rm d}),c^{\mbox{\tiny{(0)}}}_{\rm
p}\\!(\tau_{\rm d}))$ along the steady-state $c^{\mbox{\tiny{(0)}}}_{\rm
p}[L^{\mbox{\tiny{(0)}}}]$ line is unstable in itself, i.e. the dynamics are
repelled from it (as indicated by the arrows in the lower inset in Fig. 1),
the entire steady-state $c^{\mbox{\tiny{(0)}}}_{\rm p}[L^{\mbox{\tiny{(0)}}}]$
line is a dynamic attractor for unsteady slip pulses, whether decaying or
growing.

Sustained slip pulses.— Growing pulses are of particular importance as they
are accompanied by large slip and radiated energy, and hence would play major
roles in the failure of frictional interfaces. Indeed, many experimental and
numerical works demonstrate the existence of such unsteady slip pulses [3, 4,
5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. It
is observed that while these pulses are indeed out of steady-state (as our
theory predicts), they are nevertheless “sustained” in the sense that they
propagate many times their characteristic size without appreciably changing
their properties. Our theory of unsteady pulses may offer a natural
explanation for sustained slip pulses. Clearly, we focus on pulses that
correspond to $L(t_{0})\\!>\\!L^{\mbox{\tiny{(0)}}}(\tau_{\rm d})$, leading to
growth, i.e. $\dot{L}(t)\\!>\\!0$ for $t\\!>\\!t_{0}$. If these are indeed
sustained pulses, then the unstable mode (discussed above) that controls their
growth should be a slow mode.

To quantify the possible slowness of the growing unstable mode, note that the
time it takes a growing pulse to propagate a distance comparable to its own
size is $\Delta{t}\\!\simeq\\!L(t)/c_{\rm p}(t)$. The latter is valid if the
change in size over this time $\Delta{L}(t)\\!\simeq\\!\Delta{t}\,\dot{L}(t)$
is much smaller than $L(t)$, which is a dimensionless measure of the slowness
of the unstable mode. The expectation that $\Delta{L}(t)\\!\ll\\!L(t)$ amounts
to

$\dot{L}(t)/c_{\rm
p}(t)\\!\ll\\!1\,\,\qquad\hbox{for}\qquad\,\,\dot{L}(t)\\!>\\!0$ (5)

for $t\\!>\\!t_{0}$, which will be also quantitatively tested below.

Boundary-integral method simulational support.— Our next goal is to
quantitatively test the theoretical predictions discussed above in large-
scale, dynamic boundary-integral method simulations [43, 34]. This is achieved
in two steps. First, we aim at testing the predictions that: (i) pulses that
are initially close to the steady-state $c^{\mbox{\tiny{(0)}}}_{\rm
p}[L^{\mbox{\tiny{(0)}}}]$ line remain close to it at later times and (ii)
that the position of the initial point relative to
$(L^{\mbox{\tiny{(0)}}}\\!(\tau_{\rm d}),c^{\mbox{\tiny{(0)}}}_{\rm
p}\\!(\tau_{\rm d}))$ determines whether a pulse grows or decays along the
line.

Figure 3: (main) $L(t)\\!-\\!c_{\rm p}(t)$ trajectories corresponding to
unsteady slip pulses generated from perturbed steady-state pulse solutions
(various $\tau_{\rm d}$ and $\epsilon$ perturbation values). It is observed
that under all conditions, unsteady pulses (whether growing, marked in hot
colors, or decaying, marked in cold colors) closely follow the steady-state
$c^{\mbox{\tiny{(0)}}}_{\rm p}[L^{\mbox{\tiny{(0)}}}]$ line (solid black
line), if they are initially near it, as predicted in Eq. (4). Shown are 9
datasets, with $\tau_{\rm d}/\tau_{\rm min}$ in the range $1.025\\!-\\!1.07$
and $\epsilon$ in the range $-0.41\\!-\\!0$ [34]. The results for $\tau_{\rm
d}\\!=\\!1.05\tau_{\rm min}$ with $\epsilon\\!=\\!0$ (decaying pulse, blue
squares) and $\epsilon\\!=\\!-0.3$ (growing pulse, brown squares) are shown in
the upper inset. The steady-state $(L^{\mbox{\tiny{(0)}}}\\!(\tau_{\rm
d}),c^{\mbox{\tiny{(0)}}}_{\rm p}\\!(\tau_{\rm d}))$ point (black square)
indeed separates growing from decaying pulses, as predicted. (lower inset)
$\dot{L}(t)/c_{\rm p}(t)$ vs. $t\\!-\\!t_{0}$, verifying the prediction in Eq.
(5) for growing pulses (see the scale of the y-axis).

Since steady-state pulse solutions reside on the $c^{\mbox{\tiny{(0)}}}_{\rm
p}[L^{\mbox{\tiny{(0)}}}]$ line (by definition), using perturbed steady-state
pulse solutions as initial conditions is most suitable in the present context.
We employ the same $\phi(x,t)$ field $\epsilon$-perturbation scheme used above
for the dynamic stability analysis (cf. Fig. 2 and the discussion related to
it). In the upper inset of Fig. 3, we present the results for two initial
conditions for $\tau_{\rm d}\\!=\\!1.05\tau_{\rm min}$ (a different value
compared to Fig. 2), one (brown points) that features an up perturbation
relative to the point $(L^{\mbox{\tiny{(0)}}}\\!(\tau_{\rm
d}),c^{\mbox{\tiny{(0)}}}_{\rm p}\\!(\tau_{\rm d}))$ (large black square) and
another (blue points) that features a down perturbation relative to it. It is
observed that the former becomes a growing pulse (increasing $L(t)$ and
$c_{\rm p}(t)$) and the latter becomes a decaying pulse (decreasing $L(t)$ and
$c_{\rm p}(t)$), as predicted. Moreover, and crucially, in both cases the
trajectory $L(t)\\!-\\!c_{\rm p}(t)$ closely follows the steady-state
$c^{\mbox{\tiny{(0)}}}_{\rm p}[L^{\mbox{\tiny{(0)}}}]$ line (solid black line)
at any time, as predicted theoretically in Eq. (4).

Next, in the main panel of Fig. 3, we superimpose $L(t)\\!-\\!c_{\rm p}(t)$
trajectories for pulses obtained from steady-state solutions at various
$\tau_{\rm d}$ values and various $\epsilon$ perturbations (see figure
caption). Growing pulses appear in hot colors and decaying ones in cold
colors. It is observed that independently of the value of $\tau_{\rm d}$ and
independently of whether the resulting pulse is growing or decaying, all
unsteady pulse trajectories reside on the steady-state
$c^{\mbox{\tiny{(0)}}}_{\rm p}[L^{\mbox{\tiny{(0)}}}]$ line (solid black line)
at any time, as predicted theoretically in Eq. (4). Finally, in the lower
inset we test the slow unstable growth prediction for growing pulses
($\dot{L}(t)\\!>\\!0$, hot colors). It is observed that $\dot{L}(t)/c_{\rm
p}(t)\\!\ll\\!1$ during the entire pulse propagation, as expected according to
Eq. (5). These results demonstrate that growing pulses along the
$c^{\mbox{\tiny{(0)}}}_{\rm p}[L^{\mbox{\tiny{(0)}}}]$ line are indeed
sustained pulses.

In the second test of the theory, an even more stringent test, we aim at
generating slip pulses that are initially arbitrarily far away from the
steady-state $c^{\mbox{\tiny{(0)}}}_{\rm p}[L^{\mbox{\tiny{(0)}}}]$ line. This
will allow us to test the prediction that pulse trajectories in the
$L(t)\\!-\\!c_{\rm p}(t)$ plane are attracted to the
$c^{\mbox{\tiny{(0)}}}_{\rm p}[L^{\mbox{\tiny{(0)}}}]$ line, and once they hit
it (say at time $t_{0}$) they remain close to the line at later times.
Moreover, the resulting pulses are predicted to be growing or decaying
depending on the position of the hitting point $(L(t_{0}),c_{\rm p}(t_{0}))$
relative to $(L^{\mbox{\tiny{(0)}}}(\tau_{\rm d}),c^{\mbox{\tiny{(0)}}}_{\rm
p}\\!(\tau_{\rm d}))$. The latter two predictions have already been verified
in Fig. 3 for slip pulses that are initially close to the
$c^{\mbox{\tiny{(0)}}}_{\rm p}[L^{\mbox{\tiny{(0)}}}]$ line.

Slip pulses that are initially arbitrarily far from the
$c^{\mbox{\tiny{(0)}}}_{\rm p}[L^{\mbox{\tiny{(0)}}}]$ line are generated by
introducing perturbations to a nearly quiescent frictional state (a state that
slides at $v_{\rm stick}$, i.e. at an extremely slow rate). The perturbations
are initially stationary (i.e. non-propagating), and may correspond to remote
triggering processes and/or some interfacial heterogeneity. By varying the
size of the perturbation, as explained in detail in [34], slip pulses of
various properties can be generated.

Figure 4: The same as Fig. 3, but for 6 slip pulses that are initially far
from the $c^{\mbox{\tiny{(0)}}}_{\rm p}[L^{\mbox{\tiny{(0)}}}]$ line (solid
black line), under a fixed $\tau_{\rm d}\\!=\\!1.04\tau_{\rm min}$. Slip
pulses are excited as detailed in the text and in [34], and their first time
point is marked by a larger symbol in each case. It is observed that all pulse
trajectories are attracted to the $c^{\mbox{\tiny{(0)}}}_{\rm
p}[L^{\mbox{\tiny{(0)}}}]$ line, and once hitting the line (at a time denoted
as $t_{0}$), they remain close to it at any $t\\!>\\!t_{0}$ time. Moreover,
the steady-state $(L^{\mbox{\tiny{(0)}}}\\!(\tau_{\rm
d}),c^{\mbox{\tiny{(0)}}}_{\rm p}\\!(\tau_{\rm d}))$ point (black square)
separates growing (hot colors) from decaying (cold colors) pulses, as
predicted. The prediction in Eq. (5) for growing pulses is verified in the
inset (see the scale of the y-axis).

In Fig. 4, we present results for 6 different pulse trajectories in the
$L(t)\\!-\\!c_{\rm p}(t)$ plane for $\tau_{\rm d}\\!=\\!1.04\tau_{\rm min}$.
The first time at which well-defined slip pulse size $L$ and propagation
velocity $c_{\rm p}$ exist in each case is marked by a larger symbol. It is
observed that while all pulse trajectories are initially far from the
$c^{\mbox{\tiny{(0)}}}_{\rm p}[L^{\mbox{\tiny{(0)}}}]$ line, they are
dynamically attracted to it. Interestingly, the time it takes each pulse from
formation to hitting the line is of order ${\cal T}(\tau_{d})$ [34],
indicating that the latter is a basic timescale in the pulse problem in a way
that goes beyond the linearized dynamics of Eq. (3). Once each trajectory hits
the line (at time $t_{0}$), it remains close to it at any time. Moreover, the
position of the hitting point relative to $(L^{\mbox{\tiny{(0)}}}\\!(\tau_{\rm
d}),c^{\mbox{\tiny{(0)}}}_{\rm p}\\!(\tau_{\rm d}))$ (large black square)
approximately determines whether the dynamics along the line correspond to
decaying pulses (cold colors) or to growing pulses (hot colors). For the
latter, we find that $\dot{L}(t)/c_{\rm p}(t)\\!\ll\\!1$ for $t\\!>\\!t_{0}$
(see inset of Fig. 4), as expected according to Eq. (5), demonstrating again
that the growing pulses are indeed sustained pulses.

Summary and discussion.— A comprehensive theoretical picture of the dynamics
of unsteady frictional slip pulses has been developed. We showed that steady-
state slip pulses are intrinsically unstable objects, yet the steady-state
line $L(\tau_{\rm d})\\!-\\!c_{\rm p}(\tau_{\rm d})$ (parameterized by the
driving stress $\tau_{\rm d}$) is of fundamental importance for unsteady pulse
dynamics. In particular, while each point along the
$c^{\mbox{\tiny{(0)}}}_{\rm p}[L^{\mbox{\tiny{(0)}}}]$ line is unstable, the
line itself is a dynamic attractor for pulses under a given $\tau_{\rm d}$.

Once a general pulse $L(t)\\!-\\!c_{\rm p}(t)$ trajectory hits the
$c^{\mbox{\tiny{(0)}}}_{\rm p}[L^{\mbox{\tiny{(0)}}}]$ line, it remains close
to it at any time. The position of the hitting point relative to
$(L^{\mbox{\tiny{(0)}}}\\!(\tau_{\rm d}),c^{\mbox{\tiny{(0)}}}_{\rm
p}\\!(\tau_{\rm d}))$ determines whether the dynamics along the line
correspond to decaying or growing pulses. Finally, pulse dynamics along the
$c^{\mbox{\tiny{(0)}}}_{\rm p}[L^{\mbox{\tiny{(0)}}}]$ line are controlled by
a single slow unstable mode, which for growing pulses is manifested by
$\dot{L}(t)/c_{\rm p}(t)\\!\ll\\!1$. The latter corresponds to sustained
pulses, which are of prime importance for the failure dynamics of frictional
systems [21]. The comprehensive theoretical picture has been quantitatively
supported by large-scale, dynamic boundary-integral method simulations.

The emerging theory suggests that the knowledge of the family of steady-state
pulse solutions, in particular the steady-state line $L(\tau_{\rm
d})\\!-\\!c_{\rm p}(\tau_{\rm d})$, allows to understand and predict many of
the properties of unsteady slip pulses. The deep relation between unsteady
slip pulses and their steady-state counterparts is valid for other discrete,
coarse-grained variables (such as $L(t)$ and $v_{\rm m}(t)$) and in fact also
for the entire $v(x,t)$ and $\phi(x,t)$ fields themselves, as shown in [34].
The theory also predicts the linearized dynamics near each unstable point, cf.
Eq. (3), with a characteristic timescale ${\cal T}(\tau_{\rm
d})\\!\sim\\!L^{\mbox{\tiny{(0)}}}(\tau_{\rm d})/c^{\mbox{\tiny{(0)}}}_{\rm
p}(\tau_{\rm d})$. In fact, the timescale ${\cal T}(\tau_{\rm d})$ also
characterizes the approach to the $c^{\mbox{\tiny{(0)}}}_{\rm
p}[L^{\mbox{\tiny{(0)}}}]$ line. Hence, ${\cal T}(\tau_{\rm d})$ — which is
also obtained from steady-state pulse solutions — appears to play the role of
a basic timescale in the problem.

In addition, while the theory does not separately predict the general
nonlinear dynamics of $\dot{L}(t)$ and $\dot{c}_{\rm p}(t)$ along the line,
their ratio is given as $\dot{L}(t)/\dot{c}_{\rm
p}(t)\\!=\\!(dc^{\mbox{\tiny{(0)}}}_{\rm p}[L(t)]/dL(t))^{-1}$. Finally, Eq.
(4) may be viewed as an equation of motion for unsteady pulses, where their
velocity $c_{\rm p}(t)$ instantaneously follows $L(t)$. This interpretation
bears similarities to the equation of motion of ordinary cracks in bulk
materials, where the crack’s tip is viewed as a massless defect whose velocity
instantaneously follows the crack length [44].

The numerical support to the theory, as presented above, employed a generic
rate-and-state dependent friction law and anti-plane symmetry conditions. We
have no obvious reasons to expect any major/qualitative differences in the
theoretical picture once different rate-and-state dependent friction laws (or
frictional constitutive parameters) and in-plane symmetry conditions are
considered. Yet, future work should test the emerging picture in a wider range
of physical situations. In general, knowledge of the interfacial constitutive
relation (friction law) allows one to come up with concrete predictions for
unsteady pulse dynamics, which can be experimentally tested (e.g. in
laboratory experiments).

Finally, in a broader context, the theory of the dynamics of unsteady slip
pulses in frictional systems contributes to our general understanding of the
dynamics of nonlinear spatially-extended dissipative systems. This
particularly pertains to the reduced-dimensionality description in terms of a
two coarse-grained dynamical variables, such as $L(t)$ and $c_{\rm p}(t)$,
whose dynamics are controlled by a single slow unstable mode. Such a
theoretical picture bears some similarities to the dynamics of equilibrium
first-order phase transitions, such as solidification [45], involving the
formation of critical nuclei and front propagation in which a stable phase
invades an unstable one (e.g. crystal growth in the solidification problem).
In such cases, the dynamics are also dominated by a single slow unstable mode,
which corresponds to the size of the growing nucleus.

Author contributions E.B. and E.A.B. designed the research, conceptualized the
research, developed the theory and guided the numerical simulations, A.P.
performed all numerical simulations and their analysis, F.B. and T.R. helped
with the numerical simulations, E.B. and A.P. wrote the manuscript.

Acknowledgements This work has been supported by the Israel Science Foundation
(grant no. 1085/20). E.B. acknowledges support from the Ben May Center for
Chemical Theory and Computation and the Harold Perlman Family.

Supplemental Materials for: “The dynamics of unsteady frictional slip pulses”

The goal of this document is to provide some technical details that allow to
reproduce the results presented in the manuscript and to offer some additional
supporting data.

##  S-1 The interfacial constitutive relation (friction law)

The coupled bulk-interface frictional problem we solved is formulated using
the boundary integral approach, where the shear stress at the interface
(located at $y\\!=\\!0$) is given on the right-hand-side (RHS) of [39]

$\tau[v(x,t),\phi(x,t)]=\tau_{\rm d}-\frac{\mu}{2c_{s}}v(x,t)+s(x,t)\ ,$ (S1)

as presented in Eq. (1) in the manuscript. Here, $\tau_{\rm d}$ is the driving
stress applied far from the interface and $(\mu/2c_{\rm s})v(x,t)$ ($\mu$ and
$c_{\rm s}$ are the shear modulus and wave-speed of the bulks, respectively)
is the so-called radiation damping term, which depends on the local slip
velocity $v(x,t)$. $s(x,t)$ is a spatiotemporal convolutional integral that
accounts for the long-range interaction of different parts of the interface,
mediated by bulk deformation. In general, $s(x,t)$ does not admit an explicit
real-space representation [40], but is rather expressed in terms of the
Fourier components of $v(x,t)$ (in fact, it involves the slip displacement
$u(x,t)\\!=\\!\int\\!v(x,t)\,dt$, obtained from $v(x,t)$ through time
integration). Its specific form is different for anti-plane shear symmetry
(i.e. slip displacement in the $z$ direction), which is used in our computer
simulations, and for in-plane shear symmetry (i.e. slip displacement in the
$x$ direction).

The interfacial shear stress (RHS of Eq. (S1)) is balanced at each point in
time and interfacial position by the frictional strength
$\tau[v(x,t),\phi(x,t)]$, which depends in addition to $v(x,t)$ on the
internal state field $\phi(x,t)$ (see manuscript for discussion). The
frictional strength we employed, which is based on extensive laboratory data
[29, 31, 35], takes the form [21]

$\displaystyle\tau(v,\phi)/\sigma$ $\displaystyle=$
$\displaystyle\left[1+b\log\left(1+\phi/\phi^{\star}\right)\right]\times$
$\displaystyle\left[f_{0}/\sqrt{1+\left(v^{\star}\\!/v\right)^{2}}+\alpha\,\log\left(1+{\left|v\right|}/v^{\star}\right)\right]\
.$

The frictional parameters appearing in Eq. (S-1) are given in Table S-1, along
with the linear elastic bulk parameters of Eq. (S1) and the value of the
normal stress $\sigma$ [21]. As explained in the manuscript, the internal
state field $\phi(x,t)$ satisfies a well-established local (in space)
differential equation (known as the ‘aging-law’ [29, 31])

$\partial_{t}\phi(x,t)\\!=\\!1-\\!\frac{v(x,t)\,\phi(x,t)}{D}\,.$ (S3)

The latter include the characteristic slip distance $D$, whose value is also
given in Table S-1.

$\sigma$ | $1$ | MPa
---|---|---
$\mu$ | $9$ | GPa
$c_{\rm s}$ | $2739$ | m/s
$D$ | $5\times 10^{-7}$ | m
$v^{*}$ | $10^{-7}$ | m/s
$\phi^{*}$ | $3.3\times 10^{-4}$ | s
$\alpha$ | $0.005$ | –
$b$ | $0.075$ | –
$f_{0}$ | $0.28$ | –
Table S-1: The normal stress $\sigma$, linear elastic bulk parameters and
frictional parameters (cf. Eqs. (S1)-(S-1)) used in the computer simulations.

Under persistent sliding at a slip velocity $v$, Eq. (S3) attains a steady
state of the form $\phi\\!=\\!D/v$. When the latter is substituted in
$\tau(v,\phi)$ of Eq. (S-1), the steady-state friction curve of Fig. 1 in the
manuscript (solid brown line therein) is obtained. It is generally N-shaped,
featuring a local minimum at $(v_{\rm min},\tau_{\rm min})$ (cf. Fig. 1 in the
manuscript). While the local minimum emerges from basic physical
considerations and is supported by experimental data [31, 35], its existence
does not play a fundamental role in the slip pulses problem we solved. Yet,
the characteristic slip velocity $v_{\rm min}\\!=\\!0.005971$ m/s and
characteristic frictional strength $\tau_{\rm min}\\!=\\!0.3406679$ MPa can be
used to nondimensionalize various quantities, as done in the manuscript. In
particular, the driving stress we employed was varied in the range $\tau_{\rm
d}/\tau_{\rm min}\\!\in\\![1.015,1.07]$. Moreover, using the previously-
calculated effective fracture energy $G_{\rm c}\\!=\\!0.65$ J/m2 of slip
pulses for the frictional parameters in Table S-1 [25], one can construct a
characteristic pulse length as $L_{\rm n}\\!=\\!(c_{\rm s}/v_{\rm
min})^{2}(G_{\rm c}/\mu)\\!=\\!15.2$ m. The latter is used to normalize
quantities of length dimension in the manuscript.

##  S-2 Dynamic boundary-integral method simulations using the open-source
library cRacklet

The mathematical problem defined by Eqs. (S1)-(S3) is solved using the open-
source library cRacklet [43]. The latter is based on a spectral formulation of
linear elastodynamics for two semi-infinite solids based on the relevant
Green’s function (cf. RHS of Eq. (S1)) [40]. Its implementation is specially
tailored for modeling dynamic rupture propagation along a planar interface
bonding the two half-spaces. In particular, it allows to implement an
interfacial constitutive relation $\tau[v(x,t),\phi(x,t)]$ (cf. LHS of Eq.
(S1) and Eq. (S-1)) and accompanying differential equations such as Eq. (S3).

The spectral formulation, which involves a Fourier representation of the
relevant fields, implies that all fields are periodic in the spatial
integration domain of size $W$. That is, in our case we have
$v(x,t)\\!=\\!v(x+W,t)$ and $\phi(x,t)\\!=\\!\phi(x+W,t)$ at any time $t$.
Moreover, initial conditions for the two fields should be provided, i.e.
$v(x,t\\!=\\!0)$ and $\phi(x,t\\!=\\!0)$. Finally, past frictional dynamics
(i.e. for $t\\!<\\!0$) can be incorporated through the slip displacement
history $u(x,t\\!\leq\\!0)$. The latter implies a non-vanishing contribution
to the interfacial stress at $t\\!=\\!0$, $s(x,t\\!=\\!0)$. Next, we discuss
the two types of initial conditions employed in our numerical simulations
using cRacklet.

##  S-3 Two types of initial conditions

In the manuscript, two types of initial conditions are employed, depending on
the physical question of interest. Here, we briefly describe both.

###  S-3.1 Initial conditions based on steady-state slip pulses

A family of steady-state slip pulse solutions, parameterized by the driving
stress $\tau_{\rm d}$, has been recently obtained numerically in [25]. Such
solutions take the form $v_{{}_{\rm ss}}\\!(\xi)$ and $\phi_{{}_{\rm
ss}}\\!(\xi)$, with $\xi\\!=\\!x-c^{\mbox{\tiny{(0)}}}_{\rm p}t$ being a co-
moving space-time coordinate and $c^{\mbox{\tiny{(0)}}}_{\rm p}$ is the steady
pulse propagation velocity. For reasons that are explained in detail in the
manuscript, one is interested in using these steady-state pulse solutions (or
a slightly perturbed version of them, see below) as initial conditions to the
set of integro-differential equations defined in Eqs. (S1)-(S3).

To achieve this, we proceed in two steps. First, one needs to conform with the
periodic boundary conditions employed in solving Eqs. (S1)-(S3) using
cRacklet. In principle, steady-state slip pulse solutions correspond to a
closed (homoclinic) orbit $(v_{{}_{\rm ss}}\\!(\xi),\phi_{{}_{\rm
ss}}\\!(\xi))$ in the $v-\phi$ plane (the corresponding closed orbit in the
$v-\tau$ plane is presented in Fig. 1 in the manuscript, cf. solid green line
therein). Consequently, these solutions essentially satisfy periodic boundary
conditions. Yet, since slip pulses feature long, non-exponential tails [25],
the actual numerical solutions may not strictly correspond to closed orbits.
Whenever a difference in the $v$ and $\phi$ values at both ends of a steady-
state pulse solution exists, we imposed periodicity by exponentially
extrapolating the fields between the different values at the tail (we checked
that a linear extrapolation for the velocity field does not affect the results
for large enough $W$) and by padding the values ahead of the pulse to be
accommodated into the cRacklet numerical grid of overall size $W$.
Interpolation is then invoked to assign values of the fields at each grid
point.

Second, with $W$-periodic steady-state pulse solutions at hand, one needs to
generate slip history that corresponds to steady-state pulses that propagated
steadily at a velocity $c^{\mbox{\tiny{(0)}}}_{\rm p}$ indefinitely in the
past, $-\infty\\!<\\!t\\!<\\!0$. This condition, however, is clearly in
conflict with the spatial periodic boundary conditions used. That is,
extending the history (past propagation) beyond the domain boundary implies
that the past spuriously affects future dynamics. Therefore, we require that
the past propagation distance of the steady-state pulse, denoted by $w$, is
much smaller than the spatial periodicity domain $W$, $w\\!\ll\\!W$, see Fig.
S1. Moreover, the position of the pulse at $t\\!=\\!0$ should be also
sufficiently far from the domain boundary, as is demonstrated in Fig. S1 as
well.

On the other hand, to approximately mimic steady-state propagation, $w$ has to
be much larger than the characteristic slip pulse size
$L^{\mbox{\tiny{(0)}}}$. The latter is defined, following [25], as the width
at the half-height of a logarithmic representation of the slip velocity field,
corresponding to the solutions of $\log(1+v(\xi)/v^{*})/\log(1+v_{\rm
m}/v^{*})\\!=\\!1/2$. Here, $v_{\rm m}$ is the maximal slip velocity of the
pulse, cf. the inset of Fig. 1 in the manuscript. This operative pulse size
definition is uniformly applied in the entire manuscript, also to time-
dependent unsteady pulses, resulting in $L(t)$ (the corresponding pulse
propagation velocity $c_{\rm p}(t)$ is determined by the time rate of change
of the spatial position of $v_{\rm m}(t)$). Different operational definitions
of the pulse width did not affect any of the reported results. Consequently,
past steady-state pulse propagation can be mimicked by choosing
$w\\!\gg\\!L^{\mbox{\tiny{(0)}}}$, see Fig. S1.

Taken together, using initial conditions based on steady-state slip pulses
entails the resolution of the following condition
$L^{\mbox{\tiny{(0)}}}\\!\ll\\!w\\!\ll\\!W$, which is computationally
demanding (see additional discussion of this point below). Once a given slip
history is generated, it is used to calculate the spatiotemporal convolutional
integral and the resulting field $s(x,t\\!=\\!0)$ is used in Eq. (S1) as an
initial condition. The latter can be supplemented with a perturbation in the
initial internal state field, $\phi(x,t\\!=\\!0)$, as will be further
discussed below.

Figure S1: An illustration of the procedure by which slip history based on
steady-state slip pulses is generated. A system with spatial periodicity $W$
in the $x$ direction is considered. A steady-state pulse solution $v_{{}_{\rm
ss}}\\!(x-c^{\mbox{\tiny{(0)}}}_{\rm p}t)$ is placed at $t\\!=\\!0$ not close
to the periodic boundaries of the system (see the rightmost field in the main
panel and note that the $y$-axis is truncated for better visibility). Moving
time backwards (i.e. into the past, $t\\!<\\!0$), $v_{{}_{\rm
ss}}\\!(x-c^{\mbox{\tiny{(0)}}}_{\rm p}t)$ is “dragged” in space according to
its propagation velocity $c^{\mbox{\tiny{(0)}}}_{\rm p}$, representing steady-
state pulse propagation in the time interval $-w/c^{\mbox{\tiny{(0)}}}_{\rm
p}\\!<\\!t\\!<\\!0$. The spatial extent of steady-state propagation $w$ is
selected to be much smaller than $W$, i.e. $w\\!\ll\\!W$. The latter ensures
that the boundary on the left ($x\\!=\\!0$) is not reached. Had it reached the
left boundary, the pulse would have emerged from the right boundary (due to
the spatial periodicity), implying a spurious future effect, which must be
avoided. In addition, to mimic steady-state pulse propagation in the past, $w$
has to be much larger than $L^{\mbox{\tiny{(0)}}}$, i.e.
$w\\!\gg\\!L^{\mbox{\tiny{(0)}}}$. The latter is illustrated in the inset,
where the slip velocity snapshots in the main panel are re-plotted on a semi-
logarithmic scale, which allows to mark $L^{\mbox{\tiny{(0)}}}$ by the tiny
orange marker (corresponding to $L^{\mbox{\tiny{(0)}}}/L_{\rm n}\\!=\\!0.96$).
Overall, the figure illustrates the requirement
$L^{\mbox{\tiny{(0)}}}\\!\ll\\!w\\!\ll\\!W$, as discussed in Sect. S-3.1.

###  S-3.2 Stationary perturbations of a nearly quiescent frictional state

Another form of initial condition employed in the manuscript is obtained by
perturbing at $t\\!=\\!0$ a nearly quiescent frictional state. The latter
corresponds to a spatially homogeneous frictional state at a slip velocity
$v_{\rm stick}$, which is the smallest $v$ solution to the equation
$\tau[v,\phi\\!=\\!D/v]\\!=\\!\tau_{\rm d}$ (cf. Fig. 1 in the manuscript).
Since $v_{\rm stick}$ is an extremely small slip velocity, the corresponding
frictional state is essentially quiescent (i.e. non-sliding/sticking).

At $t\\!=\\!0$, we introduce a slip velocity perturbation such that
$v(x,t\\!=\\!0)$ takes the form

$\displaystyle v(x,t\\!=\\!0)-v_{\rm stick}=\,$ (S4)
$\displaystyle\,\,\Delta{v}\left(1+{\rm
erf}\\!\left[s\,\frac{\displaystyle(x-x_{0})}{\displaystyle\sqrt{2}\,\Delta{L}}\right]\right)\exp\\!\left[-\frac{\displaystyle(x-x_{0})^{2}}{\displaystyle
2\,(\Delta{L})^{2}}\right]\,,$

where $\Delta{v}$ is the amplitude of the perturbation, $x_{0}$ is the
position of the maximum of the perturbation, $\Delta{L}$ is the width of the
perturbation and $s$ is the dimensionless skewness/shape parameter. We used
the skewed/asymmetric Gaussian perturbations of Eq. (S4) so as to favor the
generation of a single slip pulse (while symmetric perturbations may give rise
to a pair of pulses [21]). We set $\Delta{v}\\!=\\!5\times 10^{-4}$ m/s and
$s\\!=\\!5$ in all of the calculations reported in the manuscript in relation
to Fig. 4 therein. We found that for a range of perturbation widths
$\Delta{L}$, the above-described initial condition leads to the excitation of
slip pulses (see additional information in Fig. S3 below).

##  S-4 Dynamic evolution of unstable steady-state pulses

As explained at length in the manuscript, the first application of the initial
conditions described in Sect. S-3.1 is in studying the dynamic stability of
steady-state pulses using cRacklet. To that aim, as is also explained in the
manuscript, we supplemented the slip history generated based on steady-state
pulse solutions (as detailed in Sect. S-3.1) with perturbations of the $\phi$
field at $t\\!=\\!0$. Specifically, after the slip history is generated, we
introduced $\phi(x,t\\!=\\!0^{-})$ (also corresponding to the steady-state
pulse solution) with an amplitude perturbation of the form
$\phi(x,t\\!=\\!0)\\!=\\!(1+\epsilon)\phi(x,t\\!=\\!0^{-})$, quantified by
$\epsilon$.

Performing dynamic stability analysis of steady-state pulses amounts to using
small values of $|\epsilon|$ and following the dynamics in cRacklet over
relatively short periods of time $t\\!>\\!0$. Moreover, as explained in the
manuscript, we theoretically expect steady-state slip pulses to be
categorically unstable, where $\epsilon\\!<\\!0$ leads to growing pulses and
$\epsilon\\!>\\!0$ leads to decaying pulses. That is, we theoretically expect
a critical $\epsilon_{\rm c}\\!\approx\\!0$ to separate growing and decaying
pulses.

A very accurate test of this theoretical expectation is somewhat challenging
in view of the requirement $L^{\mbox{\tiny{(0)}}}\\!\ll\\!w\\!\ll\\!W$, due to
the finite system size, and the computational limitations associated with it.
The quality of resolution of this condition determines the degree by which we
represent steady pulse propagation in the past ($t\\!<\\!0$), and hence is
expected to affect the quantitative value of $\epsilon_{\rm c}$. For any
reasonable resolution of this condition, we found that indeed steady-state
pulses are categorically unstable and that there exists a critical
$\epsilon_{\rm c}$ that separates growing and decaying pulses, as expected
theoretically.

The actual value of $\epsilon_{\rm c}$ was found to be finite and negative. To
demonstrate that it is indeed a finite size effect, we extracted
$\epsilon_{\rm c}$ for calculations in which we systematically improved the
resolution of the condition $L^{\mbox{\tiny{(0)}}}\\!\ll\\!w\\!\ll\\!W$. We
found that, as expected, $|\epsilon_{\rm c}|$ decreases when
$L^{\mbox{\tiny{(0)}}}\\!\ll\\!w\\!\ll\\!W$ is better resolved. In the
manuscript, we show (cf. Fig. 2 therein) that for
$L^{\mbox{\tiny{(0)}}}/L_{\rm n}\\!=\\!0.96$, $w/L_{\rm n}\\!\approx\\!25$ and
$W/L_{\rm n}\\!\approx\\!255$, we have $\epsilon_{\rm c}\\!\approx\\!-0.1$.
Further reducing $\epsilon_{\rm c}$ would necessitate very long calculations,
which are not needed in view of the trend of convergence of $\epsilon_{\rm c}$
toward zero.

Additional support to the expected unstable nature of slip pulses is given in
the inset of Fig. 2a in the manuscript, where the linearized dynamics of Eq.
(3) therein are verified. Yet another strong supporting evidence for the
theoretical predictions and their numerical validation is provided in Fig. S2,
where the short time dynamics of $L(t)$ and $c_{\rm p}(t)$ of Fig. 2a-b in the
manuscript, for both growing (brown data) and decaying (blue data), are
plotted in the $L\\!-\\!c_{\rm p}$ plane, superimposed on the steady-state
$c^{\mbox{\tiny{(0)}}}_{\rm p}[L^{\mbox{\tiny{(0)}}}]$ line. It is observed
that the two pulse trajectories initiate close to the corresponding steady-
state $(L^{\mbox{\tiny{(0)}}}(\tau_{\rm d}),c^{\mbox{\tiny{(0)}}}_{\rm
p}(\tau_{\rm d}))$ point (black square) and evolve along the local tangent to
the steady-state $c^{\mbox{\tiny{(0)}}}_{\rm p}[L^{\mbox{\tiny{(0)}}}]$ line
at that point. As discussed in the manuscript, this result supports the
expectation that the single unstable mode in the problem is directed along the
local tangent.

Figure S2:  The data shown in Fig. 2a-b in the manuscript for $L(t)$ and
$c_{\rm p}(t)$ are re-plotted (for a growing pulse and a decaying pulse, the
same colors and symbols) in the $L\\!-\\!c_{\rm p}$ plane, parameterized by
$t$ for short times. Superimposed are the steady-state
$c^{\mbox{\tiny{(0)}}}_{\rm p}[L^{\mbox{\tiny{(0)}}}]$ line (solid black line)
and the steady-state $(L^{\mbox{\tiny{(0)}}}(\tau_{\rm
d}),c^{\mbox{\tiny{(0)}}}_{\rm p}(\tau_{\rm d}))$ point (black square). It is
observed, as predicted, that the two pulse trajectories evolve along the local
tangent to the steady-state line at that point.

A second application of the initial conditions described in Sect. S-3.1 is in
generating slip pulses that are initially relatively close to their
corresponding steady-state $(L^{\mbox{\tiny{(0)}}}\\!(\tau_{\rm
d}),c^{\mbox{\tiny{(0)}}}_{\rm p}\\!(\tau_{\rm d}))$ point (and to the steady-
state $c^{\mbox{\tiny{(0)}}}_{\rm p}[L^{\mbox{\tiny{(0)}}}]$ line in general),
though not insisting on being very close to it (as was the goal in the dynamic
stability analysis discussed above). Therefore, we use a reasonable resolution
of $L^{\mbox{\tiny{(0)}}}\\!\ll\\!w\\!\ll\\!W$ (though not a stringent one)
and a wider range of $\epsilon$ values. In particular, in Fig. 3 in the
manuscript we present results for $\tau_{\rm d}/\tau_{\rm min}\\!=\\!1.05$
with $\epsilon\\!=\\!0$ (blue squares) and $\epsilon\\!=\\!-0.3$ (brown
squares), shown also in the inset therein, for $\tau_{\rm d}/\tau_{\rm
min}\\!=\\!1.07,\,\epsilon\\!=\\!0$ (filled blue left triangle), for
$\tau_{\rm d}/\tau_{\rm min}\\!=\\!1.04,\,\epsilon\\!=\\!0$ (large open blue
squares), for $\tau_{\rm d}/\tau_{\rm min}\\!=\\!1.04,\,\epsilon\,=-0.3$
(small blue filled squares), for $\tau_{\rm d}/\tau_{\rm
min}\\!=\\!1.03,\,\epsilon\\!=\\!0$ (light open blue down triangles), for
$\tau_{\rm d}/\tau_{\rm min}\\!=\\!1.03,\,\epsilon\\!=\\!-0.3$ (light blue
filled down triangles), for $\tau_{\rm d}/\tau_{\rm
min}\\!=\\!1.025,\,\epsilon\\!=\\!0$ (light blue open diamonds), for
$\tau_{\rm d}/\tau_{\rm min}\\!=\\!1.04,\,\epsilon\\!=\\!-0.41$ (filled yellow
squares). All these simulations were carried out using $w/L_{\rm
n}\\!\approx\\!15$.

##  S-5 Additional supporting results

Here, we present a few additional results that further support the theoretical
picture of unsteady slip pulses developed in the manuscript.

###  S-5.1 Dynamics in the $L\\!-\\!v_{\rm m}$ plane

In Fig. 4 in the manuscript, we present the dynamics of unsteady slip pulses
(generated through stationary perturbations of a nearly quiescent frictional
state as explained in Sect. S-3.2) as trajectories in the $L\\!-\\!c_{\rm p}$
plane. This reduced-dimensionality description in terms of discrete, coarse-
grained variables turned out to be most useful. In particular, the attraction
of the various pulse trajectories to the steady-state
$c^{\mbox{\tiny{(0)}}}_{\rm p}[L^{\mbox{\tiny{(0)}}}]$ line (which are
initially far from it), the adherence of the various trajectories to the
steady-state line after they hit it and the sorting of the different
trajectories to growing/decaying pulses based on the position of the hitting
point relative to the steady-state $(L^{\mbox{\tiny{(0)}}}(\tau_{\rm
d}),c^{\mbox{\tiny{(0)}}}_{\rm p}(\tau_{\rm d}))$ point all provide great
support to the developed theoretical picture of unsteady slip pulses.

Yet, the developed theoretical picture indicates that unsteady pulse dynamics
are controlled by a single slow unstable mode and hence implies that while the
choice of $L(t)$ and $c_{\rm p}(t)$, which is clearly (strongly) physically
motivated, is not unique. That is, it implies that a pair of other discrete,
coarse-grained variables could have revealed the same picture. To test this
prediction, we use the very same simulations that gave rise to the results in
Fig. 4 in the manuscript and present them in Fig. S3 in the $L\\!-\\!v_{\rm
m}$ plane, along with the steady-state $v_{\rm
m}^{\mbox{\tiny{(0)}}}[L^{\mbox{\tiny{(0)}}}]$ line (solid black line) and the
steady-state $(L^{\mbox{\tiny{(0)}}}(\tau_{\rm d}),v_{\rm
m}^{\mbox{\tiny{(0)}}}(\tau_{\rm d}))$ point. The results are essentially the
same as in Fig. 4 in the manuscript, lending additional support to the theory.
Note that the values of the perturbation widths $\Delta{L}/L_{\rm n}$ (cf. Eq.
(S4) and the discussion around it) that generated the unsteady pulses are
reported in the caption of Fig. S3. In addition, in the manuscript it is
stated that the time it takes each pulse from formation to hitting the line is
of order ${\cal T}(\tau_{d})$. The actual times are reported in the caption of
Fig. S3.

Figure S3:  The same data as Fig. 4 in the manuscript (same colors and
symbols), but for $v_{\rm m}(t)$ against $L(t)$ for each pulse. Pulses are
generated using Eq. (S4) (see discussion around it) with perturbation widths
with $\Delta{L}/L_{\rm
n}\\!=\\!36.77,\,42.62,\,52.05,\,53.69,\,63.11,\,121.72$ (from bottom to top).
The steady-state $v_{\rm m}^{\mbox{\tiny{(0)}}}[L^{\mbox{\tiny{(0)}}}]$ line
(solid black line) and the steady-state $(L^{\mbox{\tiny{(0)}}}(\tau_{\rm
d}),v_{\rm m}^{\mbox{\tiny{(0)}}}(\tau_{\rm d}))$ point (large black square)
are added. The results are essentially the same as in Fig. 4 in the
manuscript, see text for discussion. In addition, the time it takes each pulse
— from the moment it is formed to hitting the line — is
$0.62,1.5,3.64,1.57,2.32$ and $5{\cal T}$, respectively (again, from bottom to
top). Indeed, as stated in the manuscript, this time is ${\cal O}({\cal T})$.

###  S-5.2 Full fields comparison

The results presented in Figs. 3-4 in the manuscript and in Fig. S3 here
suggest that the deep relation between unsteady slip pulses and their steady-
state counterparts is valid not just at the level of discrete, coarse-grained
variables that characterize the dynamical fields in the problem, but rather to
the fields $v(x,t)$ and $\phi(x,t)$ (which of course include infinitely many
degrees of freedom) themselves. That is, we expect instantaneous snapshots of
$v(x,t)$ and $\phi(x,t)$ of unsteady pulses (whether growing or decaying) to
be very similar to their steady-state counterparts $v_{{}_{\rm ss}}\\!(\xi)$
and $\phi_{{}_{\rm ss}}\\!(\xi)$ for some $\tau_{\rm d}$, once the latter is
selected such that it is the $\tau_{\rm d}$ that corresponds to the point on
the steady-state $c^{\mbox{\tiny{(0)}}}_{\rm p}[L^{\mbox{\tiny{(0)}}}]$ line
at which the unsteady pulse trajectory resides at time $t$. This expectation
is verified in Fig. S4 for two unsteady pulses, one growing and one decaying,
which happened to reside close to the steady-state $c^{\mbox{\tiny{(0)}}}_{\rm
p}[L^{\mbox{\tiny{(0)}}}]$ line at some time during their dynamics.

Figure S4:  The steady-state slip velocity field $v_{{}_{\rm ss}}\\!(\xi)$
(with $\xi\\!\to\\!x)$ for $\tau_{\rm d}/\tau_{\rm min}\\!=\\!1.04$ (solid
green line). Superimposed are instantaneous snapshots of $v(x,t)$ generated
with $\tau_{\rm d}/\tau_{\rm min}\\!=\\!1.03,\,\epsilon\\!=\\!-0.3$ (dashed
blue line, a decaying pulse) and $\tau_{\rm d}/\tau_{\rm
min}\\!=\\!1.05,\,\epsilon\\!=\\!-0.3$ (dash-dotted brown line, a growing
pulse) at times $t$ for which each unsteady pulse trajectory coincides with
the steady-state point $(L^{\mbox{\tiny{(0)}}}(\tau_{\rm
d}\\!=\\!1.04\tau_{\rm min}),\,c^{\mbox{\tiny{(0)}}}_{\rm p}(\tau_{\rm
d}\\!=\\!1.04\tau_{\rm min}))$. (inset) The same for $\phi(x,t)$, on a semi-
logarithmic scale (the $x$-axis is truncated at large negative values). The
results reveal great quantitative agreement between the fields of the unsteady
pulses (both growing and decaying) and their steady-state counterpart.

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