Non-peer reviewed preprint submitted for consideration

# Non-linear stability analysis of slip in a single-degree-of-freedom elastic
system with frictional evolution laws spanning aging to slip

Federico Ciardo  Department of Civil and Environmental Engineering, Tufts
University, Medford, MA, USA.  Robert C. Viesca  Department of Civil and
Environmental Engineering, Tufts University, Medford, MA, USA.

###### Abstract

We present a non-linear stability analysis of quasi-static slip in a spring-
block model. The sliding interface is governed by rate- and state-dependent
friction, with an intermediate state evolution law that spans between aging
and slip laws using a dimensionless parameter $\epsilon$. Our results extend
and generalize previous findings of Gu et al.,  (1984) and Ranjith and Rice,
(1999) that considered slip and aging laws, respectively. We examine the
robustness of these prior results to changes in the evolution law, including
the finding of unconditional stability of the aging law for spring stiffnesses
above a critical value. Our analysis provides analytical trajectories of slip
motion in a phase plane as function of dimensionless governing parameters. We
investigate two scenarios: a spring-block model with stationary and non-
stationary point loading rate. When the loading point is stationary, we find
that deviations from the aging law lead to only conditional stability of the
slider for spring stiffnesses above a critical value: finite perturbations can
trigger instability, consistent with prior results for the slip law. We
quantify these critical perturbations as a function of the governing
parameters. We find that, for a given supercritical stiffness, the size of the
perturbation required to induce instability grows as the state evolution law
approaches the aging law. In contrast, when the point loading rate is
stationary, our results suggest that there exists a maximum critical stiffness
above which an instability can never develop, for any perturbation size. This
critical stiffness is $\epsilon$-dependent and vanishes as the slip law is
approached $\left(\epsilon\to 0\right)$: conditional stability is then
expected in the slip law limit. For a limited extension of the results found
for a spring-slider to continuum faults, we derive relations for an effective
spring stiffness as a function of the elastic moduli and a characteristic
fault dimension or a characteristic perturbation wavelength.

Keywords: Rate- and state-dependent friction, stability and bifurcation,
spring-slider model

## Keypoints

  * •

We examine the motion and stability of a single-degree-of-freedom elastic
system with a rate- and state-dependent friction.

  * •

We allow for the state evolution law to differ from two previously considered
laws, allowing to test the robustness of prior results.

  * •

We find that unconditional stability of the aging law for large stiffnesses is
specific only to that law.

  * •

Extending our results to a continuous fault, our results suggest that
nucleation size for a dynamic instability can generally be expected to be
dependent on the strength of the perturbation.

##  1 Introduction

###  1.1 Motivation

Stability of slip on frictional interfaces in response to external
perturbations continues to pose a significant research challenge. The
conditions dictating the transition between stable and unstable frictional
sliding remain elusive for a range of realistic descriptions of frictional
strength.

Given the laboratory evidence that shows the dependence of friction on both
sliding velocity and time of nominally stationary contact (e.g., Dieterich, ,
1979; Ruina, , 1983), rate- and state-dependent frictional models have emerged
as a preferred framework for theoretical investigations of slip stability.
However, these investigations have been mostly limited to specific state
evolutions laws. Here, we perform a non-linear stability analysis of quasi-
static slip using a simplified spring-block elastic model where frictional
strength is parametrized to encompass a range of possible frictional evolution
laws, and quantify the conditions that trigger a slip instability, addressing
questions regarding the generality of prior results.

The non-linear coupling between friction and elasticity along a slipping
interface gives rise to elasto-frictional length-scales that scale critical
dimensions, determining slip dynamics. Theoretical models of a fault in an
elastic continuum with rate-and-state friction and the so-called aging-law
state evolution demonstrated the emergence of a quasi-static nucleation phase
in which slip unstably accelerates within a compact nucleation distance along
the fault, after which inertia inhibits further acceleration in this region,
leading to outward dynamic rupture (Dieterich, , 1992; Rubin and Ampuero, ,
2005; Viesca, 2016b, ; Viesca, 2016a, ). Several laboratory experiments of
frictional instabilities tend to support such a nucleation process e.g. (Okubo
and Dieterich, , 1984; Ohnaka and Shen, , 1999; Nielsen et al., , 2010; Latour
et al., , 2013; Passelègue et al., , 2017; McLaskey, , 2019; Gvirtzman and
Fineberg, , 2021; Marty et al., , 2023). However, other theoretical models
based on a different state evolution law, the slip law (which is known to give
a different frictional behavior in response to changes in slip rates that
seems to best reproduce laboratory observations (Bhattacharya et al., , 2022))
revealed that this critical dimension for dynamic slip nucleation is not
uniquely defined. Also, using a fault in an elastic continuum model, Ampuero
and Rubin,  (2008) demonstrated that under the slip law, the nucleation zone
is comparatively smaller, continually shrinking as slip unstably accelerates.
More recently, Viesca,  (2023) further showed that this nucleation zone can
vanish for slip-law frictional response, implying that a point-wise divergence
of slip rate can theoretically take place. These results are consistent with
previous findings obtained by Gu et al.,  (1984) using a spring-block model
with rate-and-state friction and the same state evolution law. Their research
highlighted that a slip instability can be triggered at any value of spring
stiffness (which inversely correlates with the dimension of a finite slipping
fault, to be demonstrated in sub-section 1.4), provided that a sufficiently
large perturbation is applied to the elastic system.

Perturbations in loading rate can be localized in space, hence in the form of
rapid fluid injection in a fault or in the form of creep converging on a stuck
brittle asperity, or can be more distributed but instantaneous in time, for
instance as a result of elastic stress transfer onto one fault due to the
sudden slip of another fault nearby. The question whether the nucleation
process of a dynamic instability is sensitive to the applied loading rate has
been the subject of many laboratory studies. Injection-induced rupture
initiation and propagation on laboratory faults have consistently revealed
that fast injection rates lead to dynamic rupture nucleation over spatial
scales much smaller than the quasi-static theoretical estimates of nucleation
sizes (Gori et al., , 2021; Ji et al., , 2022). Similar results have also been
obtained on large-scale laboratory faults when subjected to sufficiently fast
background loading rates (Xu et al., , 2018; Guérin-Marthe et al., , 2019), or
when “kicked” by a rapid increase of loading rate after a hold period (as
opposed to the spontaneous nucleation upon an increasing of loading rate with
no healing period) (McLaskey and Yamashita, , 2017; McLaskey, , 2019). This
sensitivity of nucleation size on loading rate is consistent with theoretical
and numerical models of continuum faults obeying the slip law (e.g, see
(Kaneko et al., , 2016; Garagash, , 2021; Cattania, , 2023)).

Among the questions we look to address is how general is such loading-rate
sensitivity? Do spring-block models indicate that a critical nucleation length
is uniquely defined when frictional state deviates, even slightly, from the
aging law? Can we quantify the critical perturbation in loading that can
trigger a dynamic instability for different frictional strength evolution
laws? We use the simplest model to study slip motion on frictional interfaces
of compliant bodies, i.e. the single-degree-of-freedom spring-slider elastic
system depicted in Figure 1. We investigate slip stability in response to
sudden, finite-strength perturbations in either the slider’s displacement or
point loading rate. A family of state evolution laws is considered using a so-
called intermediate law that ranges between the slip law to the aging law. We
draw analogies between the spring-block model and faults in continua and
discuss the extent to which results may be extrapolated from the former to the
latter. Given these analogies of a discrete elastic model and a continuum
fault model, here presented in details for some particular configurations, the
findings of this study have application to continuous frictional interfaces.

###  1.2 Rate-and-state friction

Rate-and-state friction, originally proposed by Dieterich,  (1978, 1979);
Ruina,  (1983), has been widely used to study stick-slip instabilities
resulting from elastic bodies in frictional contact. Driven by laboratory
friction experiments of stick-slip motion that were thought to be analogous to
earthquake ruptures in geological faults, rate-and-state friction law has been
developed based on some experimental observations: i) the evolution of the
friction coefficient from a “static” value when no slip occurs, down to a
“dynamic” value when slip (or slip rate) is non-zero (Byerlee, , 1970), ii)
the dependence of “static” friction stress on time of nominally stationary
contact, iii) the dependence of friction resistance on instantaneous slip
rate, iv) the dependence of friction on recent slip and finally v) the steady-
state feature of friction stress associated with any uniform slip (Ruina, ,
1983). The phenomenological description of all these phenomena has led to the
so-called rate-and-state friction law, in which the friction coefficient $f$
depends on both sliding velocity $V$ and state variable $\theta$.
Mathematically, this law reads

$f\left(V,\theta\right)=f_{*}+a\ln\frac{V}{V_{*}}+b\ln\frac{V_{*}\theta}{D_{c}},$
(1)

where $a$ and $b$ are two phenomenological constants of often inferred to be
order 0.01, $D_{c}$ is a characteristic slip distance, and finally $f_{*}$ and
$V_{*}$ are the value of friction coefficient and slip rate at reference
steady-state for which $\theta=D_{c}/V_{*}$. Here $f$ is a Mohr-Coulomb
friction coefficient relating the interfacial shear stress $\tau$ to the
interfacial normal stress $\sigma$ as $\tau=f\sigma$. The phenomenological
evolution equation for $\theta$ can be generally casted in the following form
(Ruina, , 1983)

$\frac{\partial\theta}{\partial t}=G\left(V,\theta\right),$ (2)

where the function $G$ is assumed to be such that the state variable evolves
towards a steady-state value $\theta_{ss}\left(V\right)$ after sufficient time
or displacement, satisfying $G=0$ at constant slip rate $V$. By considering
the state variable $\theta$ as the weighted average of some function of the
recent history of slip rate $g\left(V/V_{c}\right)$ (with $V_{c}$ being a
given dimensionless constant), Ruina,  (1983) proposed different state
evolution laws in his seminal work. If the function being averaged is the
slowness $g\left(V/V_{c}\right)=V_{c}/V$, the state variable varies in time
even at null slip rates and its corresponding time evolution follows the so-
called “aging” law (Dieterich, , 1978, 1979; Ruina, , 1983):

$\frac{\partial\theta}{\partial
t}=1-\frac{V\theta}{D_{c}}\qquad\text{(Dieterich-Ruina aging law)}$ (3)

Alternatively, if $g\left(V/V_{c}\right)=\ln(V/V_{c})$, the state variable
evolves solely with slip and its time evolution follows the so-called “slip”
law:

$\frac{\partial\theta}{\partial
t}=-\frac{V\theta}{D_{c}}\ln\frac{V\theta}{D_{c}}\qquad\text{(Ruina-Dieterich
slip law)}$ (4)

Over the past few decades, the aging law (3) has been the law of choice in
many frictional-related theoretical and numerical studies, partly because of
the inability of slip law (4) to account for the time-dependent healing of
frictional surface during stationary $(V=0)$ contact. More recent analysis of
laboratory data, however, suggests the friction coefficient is more sensitive
to slip than elapsed time, even at nearly null slip velocities, implying that
the description that best reproduces observations is one in which slip is
required for frictional strength to evolve (Bhattacharya et al., , 2017,
2022).

Both of these laws are asymptotically identical (to leading order) in the
vicinity of steady-state conditions, during which $\partial\theta/\partial
t=0$ and hence $\dfrac{V\theta}{D_{c}}=1$. Away from steady-state, these two
laws will differ, and, for example, are known to lead to different frictional
behavior in response to sudden increases or decreases of slip velocity. Under
the slip law a symmetric response to velocity changes occurs, while the
corresponding response under the aging law is asymmetric (see for instance
Figure 3). The transition between these two different responses can be
captured using an intermediate state evolution law that bridges the slip and
aging laws. For example, if $\theta$ is considered as an average of a power of
$V$, i.e. $g\left(V/V_{c}\right)=\left(V/V_{c}\right)^{-\epsilon}$ with
$\epsilon$ being a dimensionless parameter, the time evolution of the state
variable follows

$\frac{\partial\theta}{\partial
t}=\frac{1}{\epsilon}\left[\left(\frac{V\theta}{D_{c}}\right)^{-\epsilon}-1\right]\frac{V\theta}{D_{c}},\qquad\text{(Intermediate
state evolution law)}$ (5)

with the end-member aging (3) and slip (4) laws retrieved respectively with
$\epsilon=1$ and $\epsilon\to 0$. This intermediate state evolution law was
also proposed by Ruina,  (1983), but has received little attention until
recently (Viesca, , 2023; Noda and Chang, , 2023).

###  1.3 Prior linear and non-linear stability analyses

Using a single-degree-of-freedom spring-slider elastic system, such as the one
depicted in Figure 1-left, and the previously described rate-and-state
description of frictional sliding, many efforts have been devoted to study
frictional slip in response to external perturbations and the conditions that
govern stable and unstable frictional sliding (e.g Rice and Tse, , 1986; Gu
and Wong, , 1991; Helmstetter and Shaw, , 2009).

Figure 1: Left: Single-degree-of-freedom spring-slider elastic system. The
slider of mass $m$ moves along the frictional interface governed by rate- and
state-dependent friction. It slides a distance $\delta$, pulled at one
extremity of the spring with characteristic stiffness $k$ at a displacement
rate $V_{o}=d\delta_{o}/dt$. Inertia is neglected, i.e. the product
$m\cdot\dot{V}$ is negligible. Right: Frictional response to step change in
sliding velocity. The friction coefficient experiences an instantaneous
increase from reference value $f_{*}$ upon velocity jump (direct effect),
before evolving to a new steady-state value over a characteristic slip
distance $D_{c}$ (evolution effect). In this contribution, the new value of
friction coefficient is always lower than the reference value, i.e. a steady-
state rate-weakening condition is considered.

Ruina,  (1983) used linear stability analysis to show that the response to
small perturbations about steady-state sliding is unstable provided that the
spring stiffness $k$ is lower than a critical value defined as

$k_{crit}=\frac{\sigma\left(b-a\right)}{D_{c}}$ (6)

This condition holds with both slip and aging law (as well as with the
intermediate law), owing to their common linearization about steady state.
When the frictional non-linearity becomes important following the growth of
small perturbations, the form of the evolution law impacts the subsequent
development of the instability (Gu et al., , 1984; Ranjith and Rice, , 1999;
Ampuero and Rubin, , 2008; Viesca, , 2023). Using a spring-slider model with
the slip law (4), Gu et al.,  (1984) showed that an no inertia instability can
nucleate with any value of spring stiffness $k$, provided that sufficiently
large perturbations are applied to the system (with larger finite-strength
perturbations needed for values of $k$ increasing above $k_{crit}$). This lies
in contrast to the results of a subsequent non-linear stability analysis using
the aging law (3), in which unconditional stability was found for $k>k_{crit}$
(Ranjith and Rice, , 1999).

Given the simplifications inherent in assuming a single state variable
description, in addition to the assumption that a particular form of that
description is correct, one is left to wonder whether either of these results
(unconditional or conditional stability) are robust to perturbations of the
friction law itself. To test this, we extend the non-linear stability analysis
done for the aging (3) and slip (4) laws to the intervening family of laws
captured by the intermediate law (5). Although here we use a simple single-
degree-of-freedom frictional model, our results have extensions to a
frictional interface of continuous, deformable bodies, provided that mappings
from a discrete spring-slider to a continuum interface model are known. In the
following sub-section we provide a concise description of these mappings.

###  1.4 Mapping a spring-block model to a continuous frictional interface

Figure 2: Different configurations and geometries for a deformable frictional
interface (i.e. frictional fault). Cases (A) and (C) represent respectively
the scenario of an infinite planar fault embedded within an unbounded linear-
elastic medium and within a half-space, with its distance to the free-surface
denoted by $h$. Cases (B) and (D) are respectively equivalent to cases (A) and
(C), with the only difference that the frictional fault has a finite length
$2L$. Finally, case (E) denotes a circular fault embedded within a three-
dimensional unbounded medium.

The spring-block model can be considered as an idealization of a continuous
interface model, approximating the latter model with its infinite degrees of
freedom to one with a single degree of freedom.

On a continuous fault, we may decompose the shear stress as

$\tau\left(x,t\right)=\tau_{el}\left(x,t\right)+\tau_{ext}\left(x,t\right),$
(7)

where, for quasi-static motion, $\tau_{el}\left(x,t\right)$ is the
instantaneous change in shear stress along the fault owed to a continuous
distribution of slip $\delta_{c}\left(x,t\right)$ and is determined by a
linear operator acting on the spatial component of $\delta$

$\tau_{el}=\mathcal{L}\left[\delta\left(x,t\right)\right],$ (8)

and $\tau_{ext}\left(x,t\right)$ is the shear stress on the surface in the
absence of slip that may be considered as an external forcing term. To find an
equivalent stiffness for the continuum fault, we consider an idealized
scenario where slip evolves as

$\delta_{c}\left(x,t\right)=D\left(t\right)\cdot\delta\left(x\right)$ (9)

and subsequently look to find a stiffness $k$ satisfying
$\tau_{el}=k\cdot\delta$, in analogy to a spring force $F_{el}=k\cdot\delta$.
Given that the operator $\mathcal{L}$ has the property that
$\mathcal{L}\left[g(t)\delta(x)\right]=g(t)\mathcal{L}\left[\delta(x)\right]$,
finding this stiffness $k$ reduces to the solution of the eigenvalue problem

$k\cdot\delta\left(x\right)=\mathcal{L}\left[\delta(x)\right]$ (10)

with values of $k$ representing eigenstiffnesses with corresponding mode
shapes $\delta(x)$. The operator $\mathcal{L}$ depends of the mode of rupture.
For some particular configurations the eigenvalue problem (10) can be solved
analytically, but for others it can only be solved numerically or by
asymptotic approximation. In Table 1 we include a few cases, for each of which
the corresponding expressions of the elastic operator $\mathcal{L}(\delta)$
and the normalized stiffness $k/k_{*}$ are reported. Figure 2 shows the
corresponding schematics of the continuum configurations, including the
problem domain (bounded or unbounded), the mode of slip (in-plane, anti-plane,
or mixed-mode), and presence or not of a free surface. For details on how the
values of $k/k_{*}$ are obtained, we refer the reader to Appendix B.

Continuous configuration |  Elastic operator $\mathcal{L}(\delta)$ |  Characteristic stiffness $k_{*}$ |  $k/k_{*}$
---|---|---|---
(A) Infinite interface between two unbounded elastic half-spaces |  $\mathcal{L}\left(\delta\right)=\dfrac{\mu^{\prime}}{\pi}\int_{-\infty}^{\infty}\dfrac{\partial\delta/\partial s}{x-s}\,\text{ds}$ |  $\dfrac{\mu^{\prime}}{\lambda}$ |  $2\pi$
(B) Finite-length interface in an unbounded elastic medium |  $\mathcal{L}\left(\delta\right)=\dfrac{\mu^{\prime}}{\pi}\int_{-L}^{L}\dfrac{\partial\delta/\partial s}{x-s}\,\text{ds}$ |  $\dfrac{\mu^{\prime}}{L}$ |  $1.157773882\dots$ (Smallest)
(C) Infinite interface near and parallel to a free-surface of an unbounded half-space |  $\mathcal{L}\left(\delta\right)=-E^{\prime}h\dfrac{\partial^{2}\delta}{\partial x^{2}},\quad-\infty<x<\infty,\quad h\ll\lambda$ |  $\dfrac{E^{\prime}h}{\lambda^{2}}$ |  $4\pi^{2}$
(D) Finite-length interface near and parallel to a free-surface of an unbounded half-space |  $\mathcal{L}\left(\delta\right)=-E^{\prime}h\dfrac{\partial^{2}\delta}{\partial x^{2}},\quad-L\leq x\leq L,\quad h\ll L$ |  $\dfrac{E^{\prime}h}{L^{2}}$ |  $\pi^{2}/4$ (Smallest)
(E) Circular interface in an unbounded elastic medium ($\nu=0$) |  $\mathcal{L}\left(\delta\right)=\dfrac{\mu}{2\pi}\int_{0}^{R}\dfrac{\partial\delta}{\partial s}\left(\dfrac{E\left[f(r/s)\right]}{r-s}-\dfrac{F\left[f(r/s)\right]}{s+r}\right)\text{ds}$ |  $\dfrac{\mu}{R}$ |  $1.003059516\dots$ (Smallest)
Table 1: Normalized stiffness $k/k_{*}$ for different continuum frictional
interface models (see Appendix B for more details). For cases (B), (D) and
(E), only the smallest eigenstiffness is reported. For each problem
configuration and geometry, the corresponding elastic operator
$\mathcal{L}(\delta)$ is reported. $\mu^{\prime}$ is an effective shear
modulus that depends on slip-mode: $\mu^{\prime}=\mu/2(1-\nu)$ in Mode II and
$\mu^{\prime}=\mu/2$ in mode III, where $\mu$ is the shear modulus and $\nu$
is the Poisson’s ratio. Similarly, $E^{\prime}$ is an effective slip-mode-
dependent modulus that is given by $E^{\prime}=2\mu/(1-\nu)$ in mode II and
$E^{\prime}=\mu$ in mode III. Finally, in cases (C) and (D), $h$ represents
the depth of the interface from free-surface (see also Figure 2). The
condition “near” the free-surface is guaranteed if $h$ is much smaller than
the applied perturbation wavelength $\lambda$ (for case (C)) or the fault
half-length $L$ (for case (D)).

In Table 1, we note that on unbounded frictional interfaces the stiffness is
inversely related to the applied perturbation wavelength $\lambda$. In this
case, the eigenvalues are continuous and thus the stiffness varies
continuously with $\lambda$. For example, for in-plane or anti-plane slip at
interface of two unbounded half-spaces, the stiffness reads

$k=2\pi\cdot\dfrac{\mu^{\prime}}{\lambda},$ (11)

where $\mu^{\prime}$ is an effective slip-mode-dependent shear modulus:
$\mu^{\prime}=\mu/2(1-\nu)$ in mode II and $\mu^{\prime}/2$ in mode III, with
$\nu$ being the Poisson’s ratio. In contrast, for finite-sized interfaces, the
stiffness varies inversely with the surface length (here half-length $L$ or
radius $R$). Moreover, the finite problem domain leads to discrete, but
countably infinite eigenvalues. For example, for in-plane or anti-plane slip
along an interface of length $2L$ embedded within a continuum, we may write
the discrete set of eigenstiffness as

$k_{i}=\eta_{i}\cdot\frac{\mu^{\prime}}{L}\qquad i=0,1,2,\dots$ (12)

where $\eta_{0}=1.157773882\dots$ is the leading eigenvalue. As we are
concerned with the lowest possible stiffness of a system with given
$\mu^{\prime}$ and $L$ (or $R$), to compare against the critical stiffness
$k_{crit}$, the leading eigenvalue is the salient one. The presence of a
critical stiffness in a spring-block elastic system, for which slip is always
unstable when $k<k_{crit}$ regardless of the strength of the perturbation
applied (Ruina, , 1983; Gu et al., , 1984; Ranjith and Rice, , 1999),
therefore suggests that in a continuum model there exists a critical
wavelength, or a critical fault dimension, above which small perturbations to
uniform steady-state sliding grow in time. These latter can be determined by
equating the expression for $k_{crit}$(6) with the values of $k$ reported in
Table 1, obtaining for each scenario a critical dimension
$\left(\lambda_{c},L_{c},\,\text{or}\,R_{c}\right)$

$\displaystyle\begin{split}\lambda_{c}&=2\pi\cdot\frac{L_{b}}{1-a/b}\quad\text{Scenario
(A)}\\\
L_{c}&\simeq\left(1.157773882\dots\right)\cdot\frac{L_{b}}{1-a/b}\quad\text{Scenario
(B)}\\\ \lambda_{c}&=2\pi\cdot
L_{bh}\sqrt{\dfrac{1}{1-a/b}}\quad\text{Scenario (C)}\\\
L_{c}&=\dfrac{\pi}{2}\cdot L_{bh}\sqrt{\frac{1}{1-a/b}}\quad\text{Scenario
(D)}\\\
R_{c}&\simeq\left(1.003059516\dots\right)\cdot\frac{L_{b}}{1-a/b}\quad\text{Scenario
(E)}\end{split}$ (13)

where $L_{b}=\dfrac{\mu^{\prime}D_{c}}{\sigma b}$ and
$L_{bh}=\sqrt{\dfrac{E^{\prime}hD_{c}}{\sigma b}}$ are two elasto-frictional
length scales (with $\mu^{\prime}=\mu$ in scenario (E)).

As a final note, we point out the caveat that decomposing the response of a
continuum model into independent eigenmodes is only asymptotically correct in
the linear regime. Once frictional non-linearities are no longer negligible,
coupling of modes occurs, and a distinct analysis of instability in continuous
faults is required (e.g Rubin and Ampuero, , 2005; Viesca, 2016b, ; Viesca,
2016a, ; Viesca, , 2023).

##  2 Equations of motion for spring-slider elastic system

We consider the single-degree-of-freedom spring-slider elastic system depicted
in Figure 1 that undergoes frictional slip. The slider is in contact with an
elastic body and moves at a velocity $V=\dot{\delta}=d\delta/dt$, pulled by a
point loading rate $V_{o}=d\delta_{o}/dt$ imposed at one extremity of the
spring. The spring is characterized by a stiffness $k$ (per unit area of
sliding contact) whose magnitude modulates the elastic force transmitted to
the slider during slip motion. Normal stress $\sigma$ and frictional law
constitutive parameters $a$ and $b$ are assumed to remain constant on the
contact interface. Motion is assumed to be quasi-static and the block’s
inertia is neglected: i.e. the product $m\cdot\dot{V}$ is neglected, with $m$
being the block mass.

The equation of motion of the single-degree-of-freedom elastic system is given
by the following equilibrium equation

$F_{el}+F_{f}-F_{ext}=0,$ (14)

where $F_{el}=k\cdot\delta$ is taken to be the elastic force due to slip
defined as such the single-degree of freedom counterpart to $\tau_{el}$ in the
continuum model, $F_{ext}=k\cdot\delta_{o}$ is the external force proportional
to the imposed displacement $\delta_{o}$ and likewise the single-degree of
freedom counterpart to $\tau_{ext}$, and $F_{f}=\tau\cdot A$ is the frictional
force along the interface, which can be expressed, assuming a unit base area,
as the product of rate- and state-dependent friction coefficient (1) and
applied normal stress $\sigma$:

$F_{f}=\tau=f\left(V,\theta\right)\times\sigma$ (15)

In rate form, Equation (14) reads

$\frac{d\tau}{dt}=k\left(V_{o}-V\right)$ (16)

Figure 3: Frictional response of spring-slider elastic system to sudden slip
velocity increases (dark red) and decreases (light red) for a wide range of
$V_{2}/V_{1}=10^{\pm(1-4)}$, where $V_{1}$ and $V_{2}$ are respectively the
sliding velocity before and after the jump. The different panels display the
analytical change in shear stress $\Delta\tau/b\sigma$ in function of shear
displacement $\delta/D_{c}$ since the velocity jump (see Eq. 17), for
different values of frictional state parameter $\epsilon$, ranging from
$\epsilon\to 0$ (slip law, top-left) to $\epsilon=1$ (aging law, bottom-
right).

As stated beforehand, the frictional state variable $\theta$ evolves following
the evolution law (5). The dimensionless parameter $\epsilon$ ranges between
$\left[0,1\right]$: i.e. between the slip and aging-law end-members. Figure 3
shows the analytical prediction of change of frictional stress in response to
sudden increases or decreases of the sliding rate, for different values of
$\epsilon$. The changes are calculated for a block sliding at velocity $V_{1}$
at steady-state $\theta_{1}=D_{c}/V_{1}$, until a sudden jump in velocity
$V_{2}$ is imposed and eventually the new steady-state value
$\theta_{2}=D_{c}/V_{2}$ is achieved. In this case, Equation (5) can be
integrated analytically and the resulting expression can be substituted into
Eq. (15) to obtain the normalised change of frictional stress
$\Delta\tau/b\sigma=\left(\tau-\tau_{2}\right)/b\sigma$ between two
consecutive steady states:

$\Delta\tau/b\sigma=\ln\left[\left(1+\left(\left(\theta_{1}/\theta_{2}\right)^{\epsilon}-1\right)e^{-\delta/D_{c}}\right)^{1/\epsilon}\right],$
(17)

where $\delta$ is the slip accumulated since the velocity jump. As displayed
in Figure 3 for a range of velocity jumps $V_{2}/V_{1}=10^{\pm(1-4)}$, a clear
asymmetric behavior with the sign of the velocity jump is obtained for large
values of $\epsilon$ parameter, with a slip length-scale over which stress
evolves to a new steady state that depends on the magnitude of the jump itself
(as predicted by aging law, yet inconsistent with laboratory friction
experiments (Tullis and Weeks, , 1986; Blanpied et al., , 1998; Marone, ,
1998)). On the contrary, low values of $\epsilon$ parameter lead to a more
symmetric frictional response and a velocity jump-independent evolution
length-scale.

At steady state, the block moves at the same speed of that of the load point,
and

$V_{ss}=V_{o},\quad\tau_{ss}=f_{ss}\sigma=\tau_{*}-\sigma\left(b-a\right)\ln\left(\frac{V_{o}}{V_{*}}\right),$
(18)

where $\tau_{*}=f_{*}\sigma$ and the subscript ss refers to steady state. If
$b>a$, the logarithmic dependence of the steady state shear stress (or
friction coefficient) on the slip rate is negative
$\left(d\tau_{ss}/dV<0\right)$. Hence, an increase in slip rate leads to a
negative change of steady state shear stress after sufficiently slip
accumulation, a condition that is commonly referred as rate-weakening
condition. Vice-versa, a rate-strengthening interface experiences a positive
change of steady-state shear stress upon an increase in slip rate. Since rate-
strengthening interfaces are unconditionally stable, in this contribution we
restrict ourselves to rate-weakening conditions, i.e. $b/a$ is always greater
than 1.

Equations (1), (5) and (15-16) comprise a non-linear system governing the
motion of the slider. Before studying the stability of such a motion in
response to perturbations, we introduce characteristic scales and non-
dimensionalize the governing equations to identify the key parameters that
govern the motion. Similar to Ranjith and Rice,  (1999), we introduce the
dimensionless quantities:

$\displaystyle\kappa=\frac{kD_{c}}{a\sigma},\quad\psi=\frac{\tau-\tau_{*}}{a\sigma},\quad\phi=\ln\left(\frac{V}{V_{*}}\right),\quad
v_{o}=\frac{V_{o}}{V_{*}},\quad T=\frac{V_{*}t}{D_{c}}$ (19)

We then combine equations (1) and (15) with (16) and (5) to eliminate the
state variable $\theta$ and using the characteristic scales in (19), we arrive
to the dimensionless autonomous system of non-linear, first-order differential
equations

$\displaystyle\frac{d\psi}{dT}=\kappa\left(v_{o}-e^{\phi}\right)$ (20)
$\displaystyle\frac{d\phi}{dT}=\kappa\left(v_{o}-e^{\phi}\right)-\frac{b/a}{\epsilon}\left[\left(e^{\phi}\cdot
e^{\left(\psi-\phi\right)/\left(b/a\right)}\right)^{-\epsilon}-1\right]e^{\phi}$
(21)

where the evolution of both normalized shear stress $\psi$ (or friction
coefficient) and logarithmic of slip velocity $\phi$ (relative to the
reference value $V_{*}$) is only function of the following dimensionless
parameters: spring stiffness $\kappa$, rate-weakening ratio $b/a$, point
loading rate $v_{o}$ and finally the frictional state evolution parameter
$\epsilon$.

At steady state, equation (18) can be rewritten in dimensionless form as

$\psi_{ss}=\phi_{ss}\left(1-b/a\right),\quad\phi_{ss}=\ln\left(v_{o}\right)$
(22)

##  3 Non-linear stability analysis with different frictional evolution laws

Following the work of Gu et al.,  (1984) and Ranjith and Rice,  (1999) for the
slip and aging laws, we commence the non-linear stability analysis in the
phase plane $\left(\phi,\psi\right)$ by eliminating $T$ from Equations (20)
and (21) and write

$P\left(\phi,\psi\right)\text{d}\psi+Q\left(\phi,\psi\right)\text{d}\phi=0,$
(23)

where

$\displaystyle
P\left(\phi,\psi\right)=\kappa\left(v_{o}-e^{\phi}\right)-\frac{b/a}{\epsilon}\left[\left(e^{\phi}\cdot
e^{\left(\psi-\phi\right)/\left(b/a\right)}\right)^{-\epsilon}-1\right]e^{\phi}$
(24) $\displaystyle
Q\left(\phi,\psi\right)=-\kappa\left(v_{o}-e^{\phi}\right)$ (25)

The differential form in (23) is imperfect, but we may seek a perfect
differential $U$ by introducing an integrating factor
$e^{q\left(\phi,\psi\right)}$, where $q\left(\phi,\psi\right)$ remains to be
determined. In this case, requiring

$dU=\left[P\left(\phi,\psi\right)\text{d}\psi+Q\left(\phi,\psi\right)\text{d}\phi\right]e^{q\left(\phi,\psi\right)}$
(26)

to be exact implies $\dfrac{\partial
U}{\partial\phi\partial\psi}=\dfrac{\partial U}{\partial\psi\partial\phi}$,
hence

$\frac{\partial\left[e^{q\left(\phi,\psi\right)}P\left(\phi,\psi\right)\right]}{\partial\phi}=\frac{\partial\left[e^{q\left(\phi,\psi\right)}Q\left(\phi,\psi\right)\right]}{\partial\psi}$
(27)

Substituting Equations (24) and (25) into (27) we obtain the equation that $q$
must satisfy

$\kappa v_{o}\left[\frac{\partial
q\left(\phi,\psi\right)}{\partial\phi}+\frac{\partial
q\left(\phi,\psi\right)}{\partial\psi}\right]+\frac{e^{\phi}}{\epsilon}e^{\epsilon\cdot\frac{(1-b/a)\phi-\psi}{b/a}}\times\\\
\left[\frac{b}{a}\left(\epsilon-1\right)-\epsilon-\frac{b}{a}\frac{\partial
q\left(\phi,\psi\right)}{\partial\phi}+e^{\epsilon\cdot\frac{(b/a-1)\phi+\psi}{b/a}}\left(b/a-\kappa\epsilon\right)\left(1+\frac{\partial
q\left(\phi,\psi\right)}{\partial\phi}-\kappa\epsilon\frac{\partial
q\left(\phi,\psi\right)}{\partial\psi}\right)\right]=0,$ (28)

which does not lend itself to a general solution. However, for some particular
cases, $q\left(\phi,\psi\right)$ can be readily obtained and hence the
function $U$. Contours of $U=\text{const.}$ provide solutions to the system
since $dU=\dfrac{\partial U}{\partial\psi}d\psi+\dfrac{\partial
U}{\partial\phi}d\phi=0$, and tangents to the contours are
$\left.\dfrac{\partial\psi}{\partial\phi}\right|_{U=\text{const.}}=\dfrac{\partial
U/\partial\phi}{\partial U/\partial\psi}=\dfrac{d\psi/dT}{d\phi/dT}$. If an
analytical solution cannot be found, the non-linear system of equations
(20-21) can be solved numerically with initial conditions.

##  4 Non-stationary load point (arbitrary $v_{o}$)

###  4.1 Case of $\kappa=\kappa_{crit}$

We start by investigating the non-linear stability of slip for the particular
case when the load point is non-stationary ($v_{o}\neq 0$) and the spring
stiffness $\kappa$ is equal to the critical value obtained from linear
stability analysis (6), which in dimensionless form reads

$\kappa_{crit}=b/a-1$ (29)

Owing to the mapping between spring stiffness and an equivalent stiffness of a
continuum fault discussed in sub-section 1.4, we are considering either that
the dimension of a finite fault is equal to its critical value predicted by
linear stability analysis, or, for an unbounded fault, that the wavelength of
the applied perturbation is equal to its critical value. Ruina,  (1983); Rice
and Ruina,  (1983) showed that when $\kappa=\kappa_{crit}$ infinitesimal
perturbations to steady-state sliding lead to a stable and periodic slip
motion with both slip and aging law: i.e., harmonic oscillations of frictional
stress and sliding velocity, as shown for example in Figure 12 of Appendix A.
In the non-linear regime, however, the elasto-frictional response is more
complex and such a well-defined critical stiffness may not exist: systems may
be linearly stable, but non-linearly unstable, and the critical stiffness may
depend upon perturbation amplitude and type. In the following, we will derive
analytical insights on stability and quantify critical finite perturbations
that initiate unstable sliding as function of the dimensionless problem
parameters.

When $\kappa=\kappa_{crit}$ and $v_{o}\neq 0$, it can be shown that the
solution to Equation (28) is

$q\left(\phi,\psi\right)=(\psi-\phi)\left(1-\epsilon\left(1-a/b\right)\right)$
(30)

which can be used to integrate (26) and obtain the trajectories in the phase
plane $\left(\phi,\psi\right)$ in the form

$U=\frac{b}{a}\cdot e^{q\left(\phi,\psi\right)}\cdot\left[\frac{\kappa
v_{o}}{(b/a)\left(1-\epsilon\right)+\epsilon}+\frac{e^{\phi}}{\epsilon}\left(\frac{e^{\epsilon\cdot\frac{(1-b/a)\phi-\psi}{b/a}}}{\epsilon-1}+\frac{b/a-\kappa\epsilon}{(b/a)\left(1-\epsilon\right)+\epsilon}\right)\right]=\text{const.}$
(31)

(a) $\epsilon\to 0$ (Slip law) (b) $\epsilon=0.3$

(c) $\epsilon=0.6$ (d) $\epsilon=1$ (Aging law)

Figure 4: Trajectories of $U$ = const. in the phase plane
$\left(\phi,\psi\right)$ (see Equations 31-32), for a non-stationary load
point $(v_{o}=1)$ and spring stiffness $\kappa$ equal to the critical value
$\kappa_{crit}$. Panels (a-d) display trajectories for different values of
state evolution parameter $\epsilon$, a specific value of rate-weakening ratio
$b/a=5/4$ and dimensionless spring stiffness $\kappa=1/4=\kappa_{crit}$. The
panel (a) displays trajectories obtained with the slip law ($\epsilon\to 0$),
while panel (d) with the aging law ($\epsilon=1$). Panels (b) and (c), display
trajectories for two intermediate state evolution laws, respectively
$\epsilon=0.3$ and $0.6$. Finally, on each panel the blue dashed line indicate
the steady-state line, the solid red line represents the critical trajectory
that separates stable (periodic) and unstable trajectories, and the black
solid lines are trajectories obtained by numerically solving the non-linear
system of first-order differential equations (20-21) with initial conditions
consistent with an initial velocity jump.

Note that the analytical trajectories (31) are valid in the slip law limit as
well as for any intermediate state evolution law, except in the aging law
limit for which a singularity appears. Nevertheless, one can integrate (26)
using the specific solution (30) with $\epsilon=1$ and retrieve the aging-law
trajectories of Ranjith and Rice,  (1999) in the form

$U=\frac{b}{a}\left[e^{\left(\psi-\phi\right)/(b/a)}\left(v_{o}+\frac{e^{\phi}}{b/a-1}\right)\kappa-\psi\right]=\text{const.},$
(32)

Figure 4 displays examples of constant-$U$ trajectories in the phase plane
$\left(\phi,\psi\right)$, for a given value of rate-weakening ratio $b/a=5/4$,
spring stiffness $\kappa=1/4=\kappa_{crit}$ and different values of
$\epsilon$.

While the trajectories provide a complete solution, we may go on to focus on
particular scenarios. For example, consider a block initially sliding at
steady-state with the same velocity as the load point, i.e. $V(0)=V_{o}$ and
hence $\phi(0)=\ln\left(V_{o}/V_{*}\right)$ and
$\psi(0)=\phi\left(1-b/a\right)$. The initial state of the system in the phase
plane reduces to a point on the steady-state line defined by Equation (22) and
depicted by blue dashed lines in Figure 4. Since the reference constant
velocity $V_{*}$ may be chosen arbitrary, without loss of generality here we
assume that $V_{*}$ is equal to the initial loading point velocity $V_{o}$
(i.e. $v_{o}=1$), implying that the initial steady-state in the phase plane
finally reduces to the origin point $(\phi(0)=0,\psi(0)=0)$. When a
perturbation in the form of an increase of slip rate that occurs too quickly
for appreciable state evolution is introduced, the corresponding perturbation
in the phase plane has changes of $\psi$ and $\phi$ of the same amount
($\Delta\phi=\Delta\psi$), owing to the constant frictional state during the
velocity jump Such a jump follows a direction in the phase plane oriented at
$45^{\circ}$ from the horizontal.

From panels (a-d) in Fig. 4 we can observe that when sufficiently small
perturbations to steady state are applied to the system, the state in the
phase plane always jumps to a closed orbit, regardless of the value of the
state evolution parameter $\epsilon$. Closed orbits in the phase plane are
stable and characterized by $U<0$. To prove this, let us consider for instance
orbits in the slip law limit. For a given value of $U<0$ and $\psi$ (positive
or negative), Equation (31) in the limit of $\epsilon\to 0$ can be rewritten
as

$\left(b/a-1\right)e^{-\phi}=\text{const.}+b/a-(b/a-1)\phi,$ (33)

where the constant in right-hand side is always negative. From a graphical
construction, it can be easily seen that there are either two or no values of
$\phi$ that satisfy the above equation. Similarly, for specified values of
$\phi$, $b/a$ and $U<0$, we can rewrite (31) in the slip law limit as

$Ue^{-\psi}=\text{const.}+\psi,$ (34)

which shows again that it can be satisfied by either two or no values of
$\psi$. Since for a given value of a phase variable there exist two solutions
that satisfy (31), the corresponding orbit in the phase plane must be
necessarily closed. Using a similar approach, one can further show that this
is valid for any values of $\epsilon$, including the aging law limit
$\epsilon=1$. In this limit, however, equation (32) must be used (see also
Ranjith and Rice,  (1999) for more details in this case). Since closed orbits
are periodic (as further highlighted by the gray arrow in Fig. 4 indicating
the direction of periodic motion on closed orbits), a stable (bounded) and
periodic slip motion is thus expected.

If the perturbation from an initial steady-state is sufficiently large,
however, the state in the phase plane may jump from the origin point to an
unstable trajectory for $0\leq\epsilon<1$, i.e. an open trajectory
characterized by $U>0$, for which slip velocity becomes unbounded in finite
time: when $U>0$, it follows from (31) that

$e^{\epsilon\frac{\left(1-b/a\right)\phi-\psi}{b/a}}<\frac{-\kappa\epsilon\left(1-e^{-\phi}\right)+b/a}{b/a+\epsilon/\left(1-\epsilon\right)},$
(35)

and, using this inequality in (21), it can be shown that

$\frac{d\phi}{dT}>e^{\phi}+\frac{\left(b/a-1\right)\epsilon}{b/a-\left(b/a-1\right)\epsilon}$
(36)

(a)  (b)

Figure 5: Panel (a): Critical size of a sudden velocity jump $\Delta V_{crit}$
from steady-state precipitating instability, as function of $\epsilon$ and
expressed as $\Delta\phi_{crit}=\ln\left(1+\Delta V_{crit}/V_{*}\right)$. The
spring stiffness $\kappa=\kappa_{crit}=1/4$ and the rate weakening ratio is
$b/a=5/4$. The black solid line corresponds to the numerical solution to the
system of Eq. (39) and $\psi-\phi=0$. The two red dashed lines are the
asymptotic solutions (42) and (43), for the aging-law and slip-law limits.
Panel (b): Same critical jump in sliding velocity from steady state, but here
as function of rate-weakening ratio $a/b$ in the slip law limit. The dashed
red line defines the small $a/b$ scaling. The open circle indicates that the
critical velocity jump does not exist for rate neutral frictional conditions,
i.e. when $a/b=1$.

implies a finite-time instability, since the right-hand side of (36) is
strictly positive and finite for $b/a>1$ and $\epsilon\in\left[0,1\right]$,
and since $\int_{\phi}^{\infty}e^{-\phi^{\prime}}\text{d}\phi^{\prime}$ is
always bounded. Although this behavior was known to exist in the slip law
limit (Gu et al., , 1984), here we find it also occurs for any
$\epsilon\in\left[0,1\right)$, excluding the aging law limit from this
interval of $\epsilon$.

When $\epsilon=1$, indeed, the trajectories form only closed orbits in the
phase plane (Ranjith and Rice, , 1999), as shown in Fig. 4-d. This can be
easily proved by rewriting Equation (32) as (Ranjith and Rice, , 1999)

$\text{const.}=e^{-\phi/(b/a)}+\frac{e^{\phi(b/a-1)/(b/a)}}{\left(b/a-1\right)}$
(37)

where the left-hand side is constant for given values of $U<0$, $\psi$, $b/a$,
$v_{o}$ and $\kappa$. Therefore, like before, graphical construction of the
left- and right-hand side reveals that there are either two or no values of
$\phi$ that satisfy (37). A graphical construction of the function on the
right-hand side of Equation (37) indeed resembles a parabola. When
$\phi\to\pm\infty$, the value of the function is $+\infty$. The minimum of the
function instead occurs for $\phi=0$, for which the value of the function is
$1+1/(b/a-1)$. Similarly, for given values of $U<0$, $\phi$ and $b/a$, Eq.
(32) can be re-expressed as

$U+\psi\cdot\left(b/a\right)=\text{const.}\cdot e^{\psi/(b/a)},$ (38)

which is satisfied by either two or no values of $\psi$. A dynamic
instability, therefore, can never be triggered if the frictional state
evolution law follows strictly the aging law: perturbations of any size always
lead to periodic oscillations, albeit with amplitudes that increase with those
of the perturbations. On the contrary, when the state evolution deviates, even
slightly, from the aging law, there exists a critical trajectory that
separates stable (periodic) and unstable orbits, with its distance to steady
state decreasing with the parameter $\epsilon$ (reaching a minimum value in
the slip law limit $\epsilon\to 0$). This critical trajectory, represented in
panels (a-c) of Figure (4) by red solid lines, corresponds to $U=0$ and is the
curve given by, for $0<\epsilon<1$,

$\psi_{crit}\left(\phi\right)=\frac{b/a}{\epsilon}\cdot\ln\left[\frac{e^{\phi+\left(a/b-1\right)\epsilon\phi}\left(b/a-\epsilon(b/a-1)\right)}{(\epsilon-1)\left(e^{\phi}(\kappa\epsilon-b/a)-\kappa\epsilon\right)}\right],$
(39)

with its expression that further simplifies in the slip law limit
($\epsilon\to 0$) as

$\psi_{crit}\left(\phi\right)=1-\kappa\left(e^{\phi}-1\right)-\phi(b/a-1)$
(40)

The critical trajectory is still open: given a value of $\phi$, $b/a$ and
$\epsilon$ in (39), one would get only a single value of $\psi$.

We may derive the rate of divergence, since when $U=0$, the inequalities (35)
and (36) turn into equalities. Along the critical orbit, the variation of
frictional shear stress with respect to a change in slip velocity is given by

$\frac{d\psi_{crit}}{d\phi}=1+\frac{b}{a}\left(-1+\frac{\kappa}{\kappa\epsilon+e^{\phi}\left(b/a-\kappa\epsilon\right)}\right),$
(41)

whose right-hand side is always negative for any value $\phi>0$, $\epsilon<1$,
$b/a>1$ and $\kappa=\kappa_{crit}=b/a-1$. Therefore, the above equation
indicates that a continuous loss of frictional stress is expected as the slip
rate diverges.

Now, a question that arises is the following: can we determine the size of the
perturbation necessary to bring a system initially, say, at steady-state, to
instability? Let us take, for example, a perturbation in the form of a sudden,
step change in slip velocity from an initial steady state ($V=V_{o}=V_{*}$) to
a new value $V=V_{o}+\Delta V$ (where $\Delta V$ may be positive or negative).
The change is assumed to be too rapid for any appreciable evolution of state
at that instant. Can we quantify the critical jump in slip velocity $\Delta
V_{crit}$ (or equivalently $\Delta\phi_{crit}=\ln(1+\Delta V_{crit}/V_{o})$)
that would precipitate unstable slip? Since the jump in the phase plane occurs
at constant state $\theta$, $\Delta\phi=\Delta\psi$ at the instant of
perturbation, meaning the trajectory in the phase space lies along a line
$\psi=\phi+c$, where $c=0$ if initially at steady-state. Thus $\Delta
V_{crit}$ is determined by the intersection of this line with the curve
$\psi_{crit}\left(\phi\right)$, (39): i.e., replacing the left-hand side of
(39) by $\phi+c$ and solving the implicit equation for $\phi$, the solution of
which we dub $\Delta\phi_{crit}=\ln\left(1+\Delta V_{crit}/V_{*}\right)$. This
solution depends on the evolution-law parameter $\epsilon$ and rate-weakening
ratio $b/a$. We solved numerically for $\Delta\phi_{crit}$ using different
values of $\epsilon$ in the interval $\left[0,1\right)$, keeping the same
input parameters previously reported (i.e. $v_{o}=1$, $b/a=5/4$ and
$\kappa=1/4=\kappa_{crit}$), and limiting the solutions to those for which
$\Delta V>0$ (i.e. neglecting the other branch of solutions for which velocity
decreases lead to instability). The numerical results are displayed in the
log-log plot of Figure 5-a (black solid line).

Figure 6: Example of trajectory in the phase plane when rate-neutral
frictional condition is considered (i.e. $b/a=1$). The spring-slider system,
initially at steady state $\left(\phi=0,\psi=0\right)$ with a spring stiffness
$\kappa=1$, is subjected to a sudden perturbation in slip velocity, such to
increase its initial value to $\Delta\phi=\ln(1+\Delta V/V_{*})=1$ at time
$T=0^{+}$). The gray trajectory define the system transient response. The blue
dashed line represents the steady-state line.

As expected, a singularity appears when $\epsilon\to 1$: i.e. when the state
tends to follow the aging (slowness) law, unconditional stability is
retrieved. An asymptotic expansion of Equation (31) allows to obtain
analytical solutions in terms of critical velocity jump in the limits of
$\epsilon\to 1$ (aging) and $\epsilon\to 0$ (slip). With $v_{o}=1$, we obtain
respectively

$\Delta\phi_{crit}=\ln\left(2-\frac{b}{a}+\frac{1}{1-\epsilon}\right)$ (42)

in the aging law limit and

$\Delta\phi_{crit}=1+W\left(\frac{a/b-1}{e}\right)$ (43)

in the slip law limit, where $W$ is the Lambert $W$-function. As shown in
Figure 5-a, these analytical expressions (red dashed lines) agree with the
numerical solution in the two limits: a mild singularity exists as
$\epsilon\to 1$ and an $\epsilon$-independent value of critical velocity jump
occurs in the slip law limit to leading order. These results imply that a
critical perturbation always exists when the frictional state evolution
deviates from the aging law and $\kappa=\kappa_{crit}$. Furthermore, as
$\epsilon\to 1$, $\Delta V_{crit}\sim\dfrac{1}{1-\epsilon}$ and an instability
is less likely to be induced and the response to insufficient perturbation is
periodic. For decreasing values of $\epsilon$, a smaller jump in velocity is
needed to trigger unstable slip, with the lowest value achieved in the slip
law limit and defined in (43). Additionally, in the slip law limit, the lower
is the ratio $a/b$, i.e. the more pronounced is the rate-weakening condition,
the lower is $\Delta\phi_{crit}$ or $\Delta V_{crit}$: the critical
perturbation that triggers an instability becomes vanishingly small (see the
log-log plot in Figure 5-b), with an asymptotic scaling that follows
$\sim\sqrt{2\left(a/b\right)}$ in that limit (see red dashed line). Note also
that this does not apply with rate-neutral frictional condition (see exclusion
circle in Fig. 5-b). Indeed, when $a/b=1$, the critical spring stiffness (6)
is null, and is the spring stiffness $\kappa$ due to the assumption that
$\kappa=\kappa_{crit}$. In this specific case, therefore, slip stability
analysis would become irrelevant. It is worth mentioning, however, that when
the spring stiffness is not equal to $\kappa_{crit}$ and $a/b=1$, the elasto-
frictional coupling emerging from spring reaction and friction at the sliding
interface remains. In this case, if a perturbation in the form of an
instantaneous step change in slip velocity $\Delta\phi$ is applied to the
system, the frictional stress as well as the sliding velocity experience a
transient evolution from their initial (and equal) values $\Delta\psi$ and
$\Delta\phi$ respectively, to the values assumed before the velocity jump,
i.e. the reference steady-state values $(\phi=0,\psi=0)$. A typical response
is shown in Figure 6 which displays the trajectory in the phase plane obtained
with $b/a=1$, $\kappa=1>\kappa_{crit}$, $\epsilon=0.01$ and imposed initial
velocity jump $\Delta\phi=\ln(1+\Delta V/V_{*})=1$. The shear stress and
sliding velocity spiral towards their reference steady-state values
$(\phi=0,\psi=0)$, and this is valid regardless of the perturbation strength
as well as of the frictional state parameter $\epsilon$. A dynamic instability
thus can never develop with rate-neutral frictional condition and
$\kappa>\kappa_{crit}$ (unconditional stability).

###  4.2 Case of $\kappa\neq\kappa_{crit}$

Now, we move on to investigate the general elastic system response when the
spring stiffness $\kappa$ is not necessarily equal to the critical value
obtained from linear stability analysis (29), still keeping the point loading
rate non-stationary ($v_{o}\neq 0$) as well as the rate weakening ratio
$b/a>1$. Under these specific conditions, we do not find the integrating
factor that satisfies Equation (28) analytically. Instead, we solve the non-
linear system of first-order ordinary differential equations (20-21)
numerically and investigate the stability of slip subjected to sudden
perturbations in slip rate and in point loading rate.

(a)  (b)

Figure 7: Panel (a): Stability diagram in terms of log of velocity jump as a
function of spring stiffness (relative to the critical value)
$\kappa/\kappa_{crit}$ and frictional state parameter $\epsilon$. Each solid
line represents the critical slip rate jump associated with a specific
frictional state evolution law: perturbations above this line trigger a
dynamic instability, whereas stable slip occurs for perturbations below this
line. Light blue curves refer to aging-law-like evolution, while dark blue
curves to slip-law-like evolution. Panel (b): solid lines are the trajectories
in the phase plane $\left(\phi,\psi\right)$ associated with strongly
subcritical ($\Delta\phi=1$), subcritical ($\Delta\phi=1.5$), near-critical
($\Delta\phi=2.5$) and supercritical ($\Delta\phi=3$) perturbations in slip
rate. For a given value of rate weakening ratio $b/a=2$, spring stiffness
$\kappa=2\kappa_{crit}$ and frictional state parameter $\epsilon=0.25$,
velocity jumps approaching the critical value
$\Delta\phi_{crit}=\ln\left(1+\Delta V_{crit}/V_{*}\right)\approx 2.721$ from
below represent subcritical or near-critical perturbations, while a
supercritical perturbation is a velocity jump larger than the critical value
needed to initiate a dynamic instability (see panel (a)). Dashed red lines are
the orbits (31) obtained assuming $\kappa=1=\kappa_{crit}$ and provide level
lines of a Liapunov function for this particular case.

#### Perturbation in slip rate at fixed state

We first present the results of slip motion stability when the applied
perturbation is in the form of an instantaneous jump in slip rate by an amount
$\Delta V$. Similar to what discussed in section 4.1, the normalized slip rate
$\phi$ experiences an instantaneous increase from an initial steady-state
value $\phi=0$ to a new larger value $\phi=\ln\left(1+\Delta V/V_{*}\right)$.
Since the slip rate jump occurs at fixed state, the scaled shear stress $\psi$
increases by the same amount from its initial value $\psi=0$. We examine the
response assuming a rate-weakening ratio $b/a=2$ and a point loading rate
equal to $v_{o}=1$. The condition $\psi-\phi=\text{const.}$ provides the
constraint for the initial dimensionless slip rate $\phi$ and shear stress
$\psi$ at time $t=0^{+}$. We integrate numerically the governing equations
(20-21) with different values of dimensionless spring stiffness $\kappa$ and
frictional state evolution parameter $\epsilon$, and solve for the evolution
of $\phi$ and $\psi$ with dimensionless time $T=\dfrac{V_{*}t}{D_{c}}$ since
the velocity jump. A critical jump in normalized slip rate
$\Delta\phi_{crit}=\ln\left(1+\Delta V_{crit}/V_{*}\right)$ (or in shear
stress) is then identified by the jump magnitude necessary for slip motion to
become unstable.

The numerical results are summarized in the stability diagram reported in
Figure 7-a, which shows whether slip motion is stable or unstable upon a given
slip rate jump, depending on the spring stiffness ratio $\kappa/\kappa_{crit}$
and the frictional state evolution law. The blue shaded lines identify the
critical jumps in slip rate for different values of $\epsilon$ parameter, with
shades darkening towards the slip law limit. For a given value of
$\kappa/\kappa_{crit}>1$ and $\epsilon\in[0,1)$, perturbations to slip rate
below the critical value $\Delta\phi_{crit}=\ln\left(1+\Delta
V_{crit}/V_{*}\right)$ do not trigger a dynamic instability, and slip rate
jumps above the critical value lead to a dynamic slip motion. In Fig. 7-a, we
observe that the lower is the value of $\epsilon$, the wider is the interval
of $\kappa/\kappa_{crit}$ in which a dynamic instability may be triggered (for
a given slip rate perturbation). The opposite occurs when the frictional state
evolution tends to the aging law. In this limit, the boundary line becomes
vertical and centered at $\kappa/\kappa_{crit}=1$ (Ranjith and Rice, , 1999).
When $\kappa<\kappa_{crit}$, perturbations to steady-state sliding always lead
to a dynamic instability, regardless of their strength. Conditional stability,
instead, occurs when $\kappa\geq\kappa_{crit}$.

To support these results, we showcase examples of slider response in the phase
plane when sub-critical, near-critical and supercritical perturbations are
applied to the system (corresponding respectively to slip rate jumps below,
near and above the critical value $\Delta\phi_{crit}$). We fix the ratio
$\kappa/\kappa_{crit}=2$, with $\kappa_{crit}=b/a-1=1$, and assume the
frictional state evolves following an intermediate state evolution law with
$\epsilon=0.25$ (see Figure 7-a). Under these conditions, the critical jump in
slip rate that initiates a dynamic instability is $\Delta\phi_{crit}\approx
2.721$. Figure 7-b displays the system response in the phase plane
$\left(\phi,\psi\right)$. All trajectories (solid lines) start from
$45^{\circ}$ direction originated from steady-state condition (see black
dashed line) and, if the perturbation applied is sufficiently weak, they
return to the initial steady-state condition ($\phi=0$, $\psi=0$) (see
strongly sub-critical, sub-critical or near-critical trajectories) or may
diverge to infinity (see supercritical trajectory). Steady-state creep is thus
expected following a few decaying oscillations of slip rate and shear stress
on sub-critical/near-critical trajectories, as opposed to supercritical
trajectories where unstable slip motion is expected. Finally, we note the
function $U$ obtained for $\kappa=\kappa_{crit}$ and mathematically given by
(31) can be regarded as a Liapunov function for this particular case,
similarly to what Gu et al.,  (1984) obtained for the slip law. A Liapunov
function in the phase plane can be used to verify the basin of an attraction
of an equilibrium state of a non-linear dynamical system: it indicates how
close solutions need to be to equilibrium to be drawn to it. As shown in Fig.
7-b, the solid trajectories crossing the red dashed orbit $U=0$ (i.e. the
strongly sub-critical, sub-critical and near-critical trajectories) and hence
entering in the region of $U<0$ where closed dashed orbits lie, are drawn
towards the steady-state point $\left(\phi=0,\psi=0\right)$ at which $U$
reaches its minimum value. The supercritical trajectory, instead, diverges
from the $U=0$ orbit and never crosses the closed curves: unstable behavior is
then expected.

#### Perturbation in point loading rate

We also investigate slip motion stability when the applied perturbation is not
in the form of a slip rate jump, but is in the form of a jump in the point
loading rate. Let us assume that initially the block slides steadily at
velocity $V=V_{o}=V_{*}$, leading to the following initial conditions for the
variables $\phi$ and $\psi$: $\phi=0$ and $\psi=0$ at time $t=0^{+}$. Let us
impose then a sudden perturbation to the rate of the load point displacement,
increasing instantaneously its initial value from $v_{o}=1$ to $v_{o}+\Delta
v_{o}$ at time $t=0^{+}$. We fix the rate-weakening ratio to $b/a=2$ and, for
a given arbitrary value of $\kappa$ and $\epsilon\in\left(0,1\right]$, we
integrate numerically the governing equations (20-21) and solve for the
evolution of $\phi$ and $\psi$ with dimensionless time
$T=\dfrac{V_{*}t}{D_{c}}$ since the jump in loading rate (that is equivalent
to solve for slip accumulated $\delta/D_{c}$ since $t=0^{+}$). A critical
loading rate jump $\left(\Delta v_{o}/v_{o}\right)_{crit}$ from steady-state
condition is then identified, in correspondence of which slip motion becomes
unstable and frictional strength rapidly decrease.

(a)  (b)

Figure 8: Panel (a): Stability diagram in terms of loading rate jump as a
function of scaled spring stiffness $\kappa/\kappa_{crit}$ and frictional
state parameter $\epsilon$. Each solid line represents the critical point
loading rate jump associated with a specific frictional state evolution. Light
red curves refer to aging-law-like evolution, while dark red curves to slip-
law-like evolution. Unstable motion occurs for perturbations to the left of a
boundary, and stable to the right. Panel (b): trajectories in the phase plane
$\left(\phi,\psi\right)$ associated with strongly subcritical ($\Delta
v_{o}/v_{o}=4$), subcritical ($\Delta v_{o}/v_{o}=9$), near-critical ($\Delta
v_{o}/v_{o}=12$) and supercritical ($\Delta v_{o}/v_{o}=14$) perturbations in
point loading rate. For a given value of rate weakening ratio $b/a=2$, spring
stiffness $\kappa=2\kappa_{crit}$ and frictional state parameter
$\epsilon=0.25$, loading rate jumps approaching the critical value
$\left(\Delta v_{o}/v_{o}\right)_{crit}\approx 12.66$ from below represent
subcritical or near-critical perturbations, while a supercritical perturbation
is a loading rate jump larger than the critical value needed to initiate a
dynamic instability (see panel (a)).

The numerical results are summarized in the plots reported in Figure 8.
Although the stability diagram on the left-hand side looks qualitatively
similar to the one obtained for perturbations in slip rate, the response of
spring-slider in the phase plane differs. Plot (b) of Fig. 8 shows the
trajectories in the phase plane obtained with the same input parameters of the
previous case, i.e. $b/a=2$, $\epsilon=0.25$ and $\kappa/\kappa_{crit}=2$. We
observe that all the trajectories start from the initial conditions of steady-
state sliding $\left(\phi=0,\psi=0\right)$ and, depending on the strength of
the perturbation applied, the response evolves following different
trajectories. Given that the change in load point velocity occurs at fixed
state, all the trajectories share a common initial tangent. For loading rate
jumps below or near the critical value, the trajectories spiral towards a new
steady-state at a sliding rate of $v_{o}+\Delta v_{o}$ (see strongly
subcritical, subcritical and near-critical trajectories in Fig. 8-b).
Conversely, when the loading rate jump is above the critical value
(supercritical condition), the response is characterized by a rapid and
continuous loss in frictional strength and, consequently, a diverging slip
rate. Slip histories of $\phi$, $\psi$ and $\theta$ for those cases are shown
in Figure 13 of Appendix A.

##  5 Stationary load point ($v_{o}=0$)

Here, we investigate the stability of the slider under no active loading, i.e.
a stationary load point. We set $v_{o}=0$ in Eq. (28) and look for the
integrating factor $e^{q\left(\phi,\psi\right)}$ that satisfies Equation (27)
for any $\epsilon\in\left[0,1\right]$. It can be shown that the factor
$q\left(\phi,\psi\right)$ that satisfies (27) is

$q\left(\phi,\psi\right)=\frac{\phi\left(b/a\left(\epsilon-1\right)-\epsilon\right)}{b/a}+\frac{\psi\left(b/a-1\right)\left(b/a-\kappa\epsilon\right)}{\kappa\cdot(b/a)}$
(44)

which can be used to integrate (26) and define the corresponding trajectories
in the phase plane $\left(\phi,\psi\right)$ in the form

$U=\frac{e^{\psi\left(b/a-\kappa\epsilon-1\right)/\kappa}\cdot\dfrac{\kappa
b}{a}\left(1-\dfrac{b}{a}+e^{\epsilon\cdot\frac{(b/a-1)\phi+\psi}{b/a}}\left(\dfrac{b}{a}-1-\kappa\epsilon\right)\right)}{\epsilon(b/a-1)(b/a-1-\kappa\epsilon)}=\text{const.}$
(45)

Note that, unlike the non-stationary case presented in section 4, here the
analytical trajectories represented by (45) are valid for any intermediate
state evolution law, including the end-member slip and aging laws.

Similarly to the results of the previous section 4, unstable orbits are those
for which $U>0$. From (45) it follows that $U>0$ when

$e^{-\epsilon\cdot\left[\left(\psi-\phi\left(1-b/a\right)\right)/(b/a)\right]}<\frac{b/a-1-\kappa\epsilon}{b/a-1},$
(46)

which can be replaced into (21) along with the condition $v_{o}=0$ to obtain

$\frac{\partial\phi}{\partial T}>\frac{\kappa e^{\phi}}{b/a-1}$ (47)

Interestingly, the inequality (47) is the same found by Ranjith and Rice,
(1999) for the non-linear stability analysis of the spring-slider elastic
system with the aging law (3) (see their Eq. (35)). Since the right-hand side
of (47) is strictly positive and finite for $b/a>1$ (with
$\int_{\phi}^{\infty}e^{-\phi^{\prime}}\text{d}\phi^{\prime}$ bounded), then
it is clear that slip velocity becomes unbounded in finite time on
positive-$U$ trajectories.

Conversely, slip motion on trajectories characterized by $U<0$ is always
stable. To demonstrate this expectation, let us rewrite Equation (45) as

$e^{\epsilon(b/a-1)\phi/(b/a)}=\left(\dfrac{b}{a}-1\right)e^{-\epsilon\psi/(b/a)}\left[\frac{\epsilon\cdot
U}{\kappa\cdot b/a}\cdot
e^{(1+\kappa\epsilon-b/a)\psi/\kappa}-\frac{1}{1+\kappa\epsilon-b/a}\right]$
(48)

Since the right-hand side of the above equation tends to $-\infty$ when
$\psi\to-\infty$, for any $\kappa$, $\epsilon\in\left(0,1\right]$ (slip law
limit excluded), $b/a>1$ and $U<0$, there exists no value of $\phi$ that
satisfies equation (48). Therefore, a complete loss of frictional strength
never occurs on negative-$U$ trajectories when frictional state evolution
deviates from the slip law (and hence instability can never take place).
Rather, $\psi$ approaches a constant value only when $\phi\to-\infty$ and
$\epsilon$ does not vanish. Indeed, the left-hand side of (48) vanishes when
$\phi\to-\infty$ and $\epsilon>0$, and thus the limiting shear stress value
achieved at very small slip velocities (for which $V\ll V_{*}$ or $e^{\phi}$
is much smaller than 1) is given by

$\lim_{\phi\to-\infty}\psi=\left[\frac{\kappa}{(1+\kappa\epsilon-b/a)}\cdot\ln\left(\frac{b/a}{\epsilon\cdot
U}\frac{k}{(1+\kappa\epsilon-b/a)}\right)\right],$ (49)

where in the aging law limit $(\epsilon=1)$ we retrieve the previous finding
of Bhattacharya et al.,  (2017). However, unlimited weakening does occur in
the limit of small slip velocities for the slip law ($\epsilon\to 0$). To show
this, Equation (45) can be rewritten as

$\left(\frac{b}{a}-1\right)\phi=\frac{U\left(b/a-1\right)}{\kappa}\cdot
e^{-(b/a-1)\psi/\kappa}+\frac{\kappa\cdot(b/a)}{(b/a-1)}-\psi,$ (50)

whose right-hand side also tends to $-\infty$ when $\psi\to-\infty$ (for
$U<0$, $b/a>1$ and any $\kappa$). The only possible behavior that satisfies
(50) is that for which $\phi\to-\infty$, which means that also when
$\epsilon\to 0$ slip is stable, and a constant minimum value of $\psi$ is
never achieved at very small velocities.

The stability boundary that divides stable and unstable motion is the
trajectory for which $U=0$,

$\psi_{crit}(\phi)=\left(1-\frac{b}{a}\right)\phi+\frac{b}{a\epsilon}\cdot\ln\left(\frac{b/a-1}{b/a-1-\kappa\epsilon}\right)$
(51)

The critical trajectories in this case are straight lines in the phase plane,
with the ratio $b/a$ governing their slopes and the frictional state evolution
parameter $\epsilon$ affecting solely their shift from the steady state line
(22). $\psi_{crit}(\phi)$ exists if and only if

$\frac{\kappa\cdot\epsilon}{\kappa_{crit}}<1.$ (52)

When this condition is violated, all the trajectories tend towards
$\phi\to-\infty$ and are stable. This suggests there exists a stiffness
$\kappa$ above which an instability can never develop. This stiffness,
however, increases with decreasing values of $\epsilon$, and diverges as
$\epsilon\to 0$. In the slip law limit, therefore, perturbations to the
elasto-frictional system may potentially initiate a dynamic instability,
provided that their magnitudes drive the system to cross the threshold defined
in (51) for the limit $\epsilon\to 0$:
$\psi_{crit}=\left(1-\dfrac{b}{a}\right)\phi+\dfrac{\kappa(b/a)}{b/a-1}$. For
a given value of $b/a$ and spring stiffness $\kappa$, the condition (52) also
defines a critical value of $\epsilon$ above which slip motion is
unconditionally stable

$\epsilon_{crit}=\frac{\kappa_{crit}}{\kappa},$ (53)

or, similarly, a critical value of $a/b$ above which $\psi_{crit}(\phi)$ does
not exist (for a given $\epsilon$ and stiffness $\kappa$), i.e.

$(a/b)_{crit}=\left(\kappa\epsilon+1\right)^{-1}$ (54)

(a) $\epsilon\to 0$ (Slip law) (b) $\epsilon=0.3$

(c) $\epsilon=0.6$ (d) $\epsilon=1$ (Aging law)

Figure 9: Trajectories in the phase plane $\left(\phi,\psi\right)$ given by
the level-set curves of Eq. (45), for stationary load point $(v_{o}=0)$ and
spring stiffness $\kappa$ below the critical value $\kappa_{crit}$. Panels
(a-d) display trajectories for different values of frictional state evolution
parameter $\epsilon$, a specific value of rate-weakening ratio $b/a=2$ and
dimensionless spring stiffness $\kappa=0.5<\kappa_{crit}$. The panel (a)
displays trajectories obtained with the slip law ($\epsilon\to 0$), while
panel (d) with the aging law ($\epsilon=1$). Panels (b) and (c), instead,
display the elastic system response for two intermediate state evolution laws,
respectively $\epsilon=0.3$ and $0.6$. On each panel, the blue dashed line
refers to steady state line, whereas the orange dashed lines refers to
acceleration lines, which denote up to which point motion on an intersecting
trajectory accelerates. Finally, the solid red line represents the critical
trajectory that separates stable and unstable slip (see Equation 51), whereas
the black solid lines are trajectories obtained numerically by solving the
non-linear system of first-order differential equations (20-21) with initial
conditions consistent with an initial velocity jump.

(a) $\epsilon\to 0$ (Slip law) (b) $\epsilon=0.6$

(c) $\epsilon=0.7$ (d) $\epsilon=1$ (Aging law)

Figure 10: Trajectories in the phase plane $\left(\phi,\psi\right)$ (given by
Equation 45), for stationary load point $(v_{o}=0)$ and spring stiffness
$\kappa$ larger than the critical value $\kappa_{crit}$, demonstrating the
vanishing of unstable motion with $\epsilon$ increasing past a critical value
$\epsilon_{crit}=2/3$ given by (53). Panels (a-d) display trajectories for
different values of frictional state evolution parameter $\epsilon$, a value
of rate-weakening ratio $b/a=2$ and dimensionless spring stiffness
$\kappa=2>\kappa_{crit}$. The panel (a) displays trajectories obtained with
the slip law ($\epsilon\to 0$), while panel (d) with the aging law
($\epsilon=1$). Panels (b) and (c), instead, display the elastic system
response for two intermediate state evolution laws, respectively
$\epsilon=0.6$ and $0.7$. On each panel, the blue dashed line refers to steady
state line, whereas the orange dashed lines refers to acceleration lines.
Finally, the solid red line represents the critical trajectory that separates
stable and unstable slip motion (see Equation 51), whereas the black solid
lines are trajectories obtained numerically by solving the non-linear system
of first-order differential equations (20-21) with initial conditions
consistent with an initial velocity jump.

When the condition (52) is violated, slip motion is always stable. However,
the spring-slider elastic system may undergo a transient of self-driven slip
acceleration in response to perturbations. Indeed, following from (21) and the
definition of normalized state variable $\Theta$

$\Theta=\frac{b}{a}\ln\left(\frac{V_{*}\theta}{D_{c}}\right)$ (55)

for which the phase variables are related to it via $\Theta=\psi-\phi$, slip
acceleration $\left(d\phi/dT>0\right)$ occurs when the state evolution
$\dot{\Theta}$

$\dot{\Theta}<\kappa\left(v_{o}-e^{\phi}\right)$ (56)

Since $v_{o}=0$ in this case, the limit condition for slip acceleration
reduces to

$\dot{\Theta}_{a}=-\kappa e^{\phi}$ (57)

which, with the help of (5), leads to the following condition for the
dimensionless state variable $\Theta$

$\Theta_{a}=-\frac{b}{a}\phi+\frac{b}{a}\ln\left[\left(1-\frac{\epsilon\kappa}{b/a}\right)^{-1/\epsilon}\right]$
(58)

Now, since the dimensionless frictional stress $\psi$ and slip velocity jump
$\phi$ are related to each others via $\psi=\phi+\Theta$, we can obtain a
critical acceleration line in the phase plane $\left(\phi,\psi\right)$ that
reads

$\psi_{a}=\left(1-\frac{b}{a}\right)\phi+\frac{b}{a}\ln\left[\left(1-\frac{\epsilon\kappa}{b/a}\right)^{-1/\epsilon}\right]$
(59)

The acceleration line in the phase plane - when it exists - is parallel to
both the steady state line (22) and the critical trajectory (51). Moreover,
its intersection with trajectories in the phase plane occurs at points on
trajectories whose local tangent line is vertical (see for instance Figures 9
and 10).
The necessary condition for transient acceleration is

$\frac{\epsilon\kappa}{b/a}<1$ (60)

or, written dimensionally, when $k<b\sigma/(D_{c}\epsilon)$. When
$\epsilon=1$, we retrieve the results of Helmstetter and Shaw,  (2009) for the
aging law. Equation (60) implies the existence of a minimum fault dimension
(or perturbation wavelength) below which a transient of self-driven slip
acceleration can never develop. Again, such a minimum dimension (or
wavelength) decreases with decreasing values of $\epsilon$ parameter,
vanishing in the limit of $\epsilon\to 0$. In the slip law limit, therefore,
any perturbations that make the frictional stress exceed the value defined in
(59), simply given by $\psi_{a}=\left(1-b/a\right)\phi+\kappa$ when
$\epsilon\to 0$, induce a transient slip acceleration.

Now, similarly to section 4, imagine that an external perturbation alters
suddenly the motion of the slider, bringing instantaneously its slip velocity
from the steady velocity $V_{*}$ to a new, larger value $V+\Delta V$. If the
change in slip rate occurs faster than any appreciable state evolution can
occur, then the dimensionless frictional stress $\psi$ increases by the same
amount as that of the dimensionless slip velocity $\phi$. We can identify the
critical velocity perturbation as $\Delta\phi_{crit}=\ln\left(1+\Delta
V_{crit}/V_{*}\right)$ and determine its value as the distance between the
origin (if we presume initial steady-state conditions) to the critical line
$\psi_{crit}(\phi)$, given by (51), along a $45^{\circ}$ incline. Unlike the
case of a non-stationary load point, here we are able to get an analytical
expression for $\Delta\phi_{crit}$ that is valid for any
$\epsilon\in\left[0,1\right]$ and reads

$\Delta\phi_{crit}=\frac{1}{\epsilon}\cdot\ln\left(\frac{1}{1-\epsilon\cdot(\kappa/\kappa_{crit})}\right),$
(61)

with its existence that still depends on the condition (52) determining
whether $\psi_{crit}(\phi)$ exists at all.

(a)  (b)

Figure 11: For stationary loading ($v_{o}=0$), Panel (a): Critical jump in
sliding velocity leading to unstable sliding from steady-state initial
conditions as a function of frictional state parameter $\epsilon$, for a given
value of rate weakening ratio $b/a=2$. The black solid lines correspond to the
analytical solution (61) for three values of spring stiffness $\kappa$: below,
equal and above the critical spring stiffness $\kappa_{crit}$. The vertical
red dashed line refers to the critical value of $\epsilon$ parameter defined
in Eq. (53), and valid only when $\kappa>\kappa_{crit}$, i.e. when condition
(52) is satisfied. Panel (b): Same critical jump in sliding velocity from
steady-state due to external perturbation but here as a function of the rate
weakening ratio $a/b$, for a given value of stiffness $\kappa=1$ and different
fixed values of state evolution parameter $\epsilon$, ranging from aging
($\epsilon=1$) to slip law ($\epsilon\to 0$) limits. For increasing values of
$a/b$, the analytical curves asymptotically approach the critical value
defined in (54).

When $\kappa<\kappa_{crit}$, a dynamic instability can be triggered for any
value of frictional state evolution parameter $\epsilon$, with a critical jump
in velocity that is always lower compared to any case when
$\kappa\geq\kappa_{crit}$ (for the same frictional conditions). Furthermore,
the minimum value of $\Delta\phi_{crit}$ is achieved in the slip law limit and
is given by $\Delta\phi_{crit}=\kappa/\kappa_{crit}$, while the maximum value
of $\Delta\phi_{crit}$ is reached in the aging law limit, for which
$\Delta\phi_{crit}=\ln\left(\kappa_{crit}/(\kappa_{crit}-\kappa)\right)$.
Figure 11-a displays how $\Delta\phi_{crit}$ evolves as a function of
$\epsilon$, for a given rate-weakening ratio $b/a=2$ and three values of
spring stiffness $\kappa$: lower, larger, and equal to the critical value
$\kappa_{crit}=1$. Examples of constant-$U$ trajectories in Figure 9 obtained
with $\kappa<\kappa_{crit}$ confirm the above considerations. A critical
trajectory that divides stable and unstable slip motion can be clearly
distinguished for any $\epsilon\in\left[0,1\right]$ (see red solid lines in
Fig. 9 or in Fig. 2 of Suppl. Material where more results with intermediate
state evolution laws are included). Moreover, the critical jump from the
steady-state origin (denoted by the black point along the dashed blue line)
slightly increases for increasing values of $\epsilon$, as highlighted by the
length of the red arrows in panels (a-d). Finally, we can also observe that
the minimum “kick” needed to induce slip acceleration is very similar for all
the state evolution laws, although slightly larger finite-strength
perturbations are needed in the aging law compared to the slip law. Both the
critical trajectory $(U=0)$ and the acceleration line (orange dashed line)
shift away from the steady state line as $\epsilon\to 1$; the region in the
phase plane in which slip accelerates upon perturbation is wider for the aging
law.

When the spring stiffness $\kappa=\kappa_{crit}$, the variation of the
critical velocity jump $\Delta\phi_{crit}$ with $\epsilon$ parameter is very
similar to the non-stationary load point scenario (see Fig. 5): a singularity
appears in the aging law limit. When $\epsilon=1$, therefore, a dynamic
instability can never be triggered regardless of the strength of the applied
perturbation. Or, in other words, for slipping patches whose strength deviates
from the aging law, there always exists a critical perturbation that can
initiate a dynamic rupture regardless of whether the load point is stationary.
The strength of such a perturbation decreases with decreasing values of
$\epsilon$, reaching its minimum value in the slip law limit. The trajectories
in the phase plane depicted in Figure 3 of Supplementary Material again
illustrate these considerations.

Finally, when $\kappa>\kappa_{crit}$, Figure 11-a shows that the singularity
in the critical velocity jump $\Delta\phi_{crit}$ deviates from the aging law
limit and is now centered at $\epsilon=\epsilon_{crit}$ defined by Eq. (53).
This implies that there exists a range of frictional state evolution laws
$\left(\epsilon_{crit}<\epsilon\leq 1\right)$ for which slip motion is
unconditionally stable. The larger the spring stiffness is relative to the
critical value $\kappa_{crit}$, the wider is the stable interval. To
illustrate this, we show in Figure 10 how the trajectories in the phase plane
change with different values of $\epsilon$, chosen to be below and above
$\epsilon_{crit}$. We set $b/a=2$ and $\kappa=3/2>\kappa_{crit}$, such that
$\epsilon_{crit}=2/3$. As an aside note also that with these input parameters,
the necessary condition for transient slip acceleration along stable
trajectories (60) is satisfied for any $\epsilon\in\left[0,1\right]$. From
Figure 10 we can readily observe a change in elasto-frictional response when
the $\epsilon$ parameter is below or above $\epsilon_{crit}$. In panels (a)
and (b), which show trajectories associated with $\epsilon\to 0$ and
$\epsilon=0.6$ respectively, there exists a critical line that divides stable
and unstable slip motion (see the red solid line), with a jump from steady
state that considerably increases as $\epsilon$ tends to $\epsilon_{crit}$
(see also Fig. 4 in Suppl. Material). The region in the phase plane in which
stable slip accelerates transiently is larger when
$\epsilon\to\epsilon_{crit}$ (from below), and the shift of the acceleration
line from steady state increases with increasing value of $\epsilon$ (see Eq.
(59)). On the other hand, when $\epsilon>\epsilon_{crit}$ (see panels (c) and
(d)), there are no unstable trajectories. All the orbits in the phase plane
are characterized by $U<0$ and hence slip motion is always stable regardless
of the magnitude of the jump from steady state. Large enough perturbations,
therefore, can only induce transient slip acceleration in this case. Note,
however, that if a stiffer spring is considered, the acceleration condition
(60) may not be satisfied for all $\epsilon>\epsilon_{crit}$. In Figure 5 of
Supplementary Material, we present trajectories obtained with the same input
parameters as above, but now the spring stiffness is larger and equal to
$\kappa=4>\kappa_{crit}=1$. We can observe that when
$\epsilon>\epsilon_{crit}=0.25$, the spring-slider motion is always stable and
transient slip acceleration can only be induced when
$\epsilon_{crit}<\epsilon<(b/a)/\kappa=0.5$ (see panels (c-d)).

So far, we have presented how the critical jump in slip velocity in response
to perturbation varies with different state evolution laws and spring
stiffnesses, while keeping constant the rate-weakening ratio $b/a$ (or its
inverse). Frictional interfaces, however, exhibit different values of the
parameter $a$ and $b$ depending on various factors, such as type of material
of the two bodies in contact, temperature or values of effective normal stress
on the interface (e.g., Blanpied et al.,  (1998)). It is interesting to know,
therefore, how $\Delta\phi_{crit}$ changes with the ratio $a/b$ for a
particular state evolution law and spring stiffness. From Figure 11-b, we
observe that $\Delta\phi_{crit}$ decreases for more pronounced rate-weakening
interfaces and that the state evolution law does not play a role when $a/b\to
0$. All the curves, indeed, collapsed into one curve that asymptotically goes
to 0 when $a/b$ approaches to 0. An instability can thus be triggered with
small perturbations in slip motion in that limit. On the contrary, for
increasing values of $a/b$, $\Delta\phi_{crit}$ increases for any
$\epsilon\in\left[0,1\right]$. However, a particular state evolution law
defines the range of $a/b$ in which such evolution occurs, with the maximum
asymptotic value given by (54) (for a given $\epsilon$ and spring stiffness
$\kappa$). With the aging law, therefore, there exists an interval of $a/b$
for which slip motion is unconditionally stable. This interval, however,
vanishes in the slip law limit: a sufficiently large perturbation can trigger
an instability for any value of $a/b\in\left[0,1\right)$. Note that
$\Delta\phi_{crit}\to\infty$ when $a/b\to 1$, i.e. on rate-neutral interfaces
an instability can never develop regardless of the strength of the
perturbation applied.

##  6 Equivalence between the intermediate and Nagata state evolution laws

Following a series of “stress-step” experiments using a double direct shear
apparatus with pre-cut granite samples, Nagata et al.,  (2012) proposed a
revised version of rate-and-state friction law that includes a dependence on
stressing rate in the form

$\displaystyle\begin{split}&f=f_{*}+a_{N}\ln\left(\frac{V}{V_{*}}\right)+b_{N}\ln\left(\frac{V_{*}\theta}{D_{c_{N}}}\right)\\\
&\frac{\partial\theta}{\partial
t}=1-\frac{V\theta}{D_{c_{N}}}-\frac{c_{N}\theta}{\sigma
b_{N}}\frac{\partial\tau}{\partial t}\end{split}$ (62)

where the subscript N distinguishes the Nagata parameters from those of
classical rate-and-state friction law presented in sub-section 1.2. If $c=0$,
the Nagata law reduces to the aging law (3).

The Nagata evolution law is equivalent to the intermediate law (5).
Specifically, the parameters of the intermediate law
$\left(D_{c},a,b,\epsilon\right)$ are uniquely determined by the choice of the
Nagata law parameters $\left(D_{c_{N}},a_{N},b_{N},c_{N}\right)$:

$\displaystyle\begin{split}&D_{c}=\frac{D_{c_{N}}}{1+c_{N}}\\\
&a=\frac{a_{N}}{1+c_{N}}\\\ &\epsilon=1-\frac{a_{N}c_{N}}{(1+c_{N})b_{N}}\\\
&b=\left(1-\frac{a_{N}c_{N}}{(1+c_{N})b_{N}}\right)b_{N}\end{split}$ (63)

This equivalency was independently recognized by Hiroyuki Noda (personal
communication, 2018) and further discussed by Noda and Chang,  (2023).

##  7 Conclusion

We have studied non-linear slip motion stability analysis of a single-degree-
of-freedom elastic system with different frictional state evolution laws.
Using the evolution law (5) and its parametrization via a dimensionless
parameter $\epsilon$, originally proposed by Ruina,  (1983), we have
investigated analytically and numerically the elastic response of the spring-
slider system in the parameter space identified by scaling analysis. Our
results extend and generalize those of Gu et al.,  (1984) and Ranjith and
Rice,  (1999), valid respectively only for the slip and aging laws, and
provide a complete picture of slip motion stability in the non-linear regime
for a range of realistic frictional state evolutions.

We have demonstrated that the critical stiffness (and by analogy to continuum
faults, the nucleation length) for slip instability is not well-defined for a
wide range of friction descriptions and quantified the conditions that
initiate an instability as function of the governing dimensionless parameters.
Remarkably, our results show that slipping patches whose state evolve
following the slip law are more susceptible to transient states of self-driven
slip accelerations or dynamic instabilities than those whose state evolves in
an aging-law-like fashion. Moreover, a dynamic instability can nucleate on
smaller faults, provided that sufficiently large perturbations are applied to
the system and frictional state deviates, even slightly, from the aging law.
This loading rate dependency on the nucleation process agrees with
experimental observations on spatially extended frictional interfaces in which
reduced dynamic-rupture nucleation lengths have been attributed to sudden,
localized increases in loading rate associated with local strength
heterogeneities along the frictional interface or with anthropogenic fluid
injection (McLaskey and Yamashita, , 2017; Xu et al., , 2018; McLaskey, ,
2019; Guérin-Marthe et al., , 2019; Gori et al., , 2021; Ji et al., , 2022).
Despite the model simplicity and the simplifying assumptions, the analytical
results presented here may provide support to observations of laboratory
earthquake experiments and numerical findings obtained with more complex
continuum models.

## CrediT authorship contribution statement

Federico Ciardo: Formal analysis, Investigation, Software, Methodology,
Visualization, Writing - original draft, Writing - review & editing.
Robert C. Viesca: Formal analysis, Conceptualization, Funding acquisition,
Writing - review & editing, Supervision.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Data availability

No data were used for the research described in this article.

## Acknowledgments

This work was supported by the National Science Foundation (grant
EAR-1834696).

##  Appendix A Appendix: Elasto-frictional response of spring-slider system
subjected to finite perturbations

###  $\kappa=\kappa_{crit}$: periodic response

Results from linear stability analysis, valid for both slip and aging law,
showed that at neutral stability the elastic response remains quasi-static and
periodic upon small perturbations in slip motion (Ruina, , 1983; Rice and
Ruina, , 1983). Here, we show that this is also valid in the non-linear regime
with an intermediate state evolution law, provided that the applied
perturbation is not large enough to trigger a dynamic instability (as
discussed in sub-section 4.1).

We consider the single-degree-of-freedom spring-slider elastic system depicted
in Figure (1) that slides on a rate weakening interface characterized by
$b/a=2$, with frictional state evolution that follows an intermediate law
($\epsilon=0.25$). We assume that the spring stiffness $\kappa$ is equal to
the critical value $\kappa_{crit}=b/a-1=1$ and consider small perturbation in
the form of a sudden increase of load point rate from steady-state value.
Specifically, we assume a velocity jump perturbation of $\Delta
v_{o}/v_{o}=1.33$ that is slightly below the critical value $\left(\Delta
v_{o}/v_{o}\right)_{crit}=1.34$ needed to initiate a dynamic instability (see
Figure 8-a). Using as initial conditions the steady state conditions prior the
velocity jump, i.e. $\psi=0$ and $\phi=0$, we integrate numerically Equations
(20-21) for the evolution of normalized shear stress $\psi$ and log of slip
velocity $\phi$ (relative to steady state), in function of slip displacement
$\delta/D_{c}$ since velocity jump.

Numerical results are displayed in Figure 12. Panels (a), (b) and (c) display
clear periodic oscillations of shear stress $\psi$, slip velocity $\phi$ and
normalized frictional state $\Theta=(b/a)\ln\left(\theta V_{*}/D_{c}\right)$
with slip accumulation respectively. The corresponding trajectory in the phase
plane $\left(\phi,\psi\right)$ forms a closed contour (see panel (d)).

(a)  (b)

(c)  (d)

Figure 12: Example of stable and periodic elastic response of the spring-
slider when the spring stiffness $\kappa$ is equal to the critical value
$\kappa_{crit}$ and a small load point velocity perturbation is applied to the
system. In this particular example, the frictional state follows an
intermediate evolution law characterized by $\epsilon=0.25$, the rate
weakening ratio is $b/a=2$ and hence $\kappa=\kappa_{crit}=1$, and the load
point velocity perturbation is $\Delta v_{o}/v_{o}=1.33$ that is slightly
below the critical value $\left(\Delta v_{o}/v_{o}\right)_{crit}=1.34$. Panels
(a-c) display the evolution of normalized shear stress $\psi$, slip velocity
$\phi$ and frictional state $\Theta$ as function of normalized slip
accumulation $\delta/D_{c}$ since velocity jump. Panel (d), instead, displays
the corresponding elastic response in the phase plane. Figure 13: From left to
right: evolution of normalized shear stress $\psi$, slip velocity $\phi$ and
frictional state $e^{\Theta/(b/a)}=\theta V_{*}/D_{c}$ as function of slip
accumulation $\delta/D_{c}$ when subcritical, near-critical and supercritical
load point velocity perturbations are applied to the spring-slider elastic
system with stiffness $\kappa$ larger than $\kappa_{crit}$. Each row
corresponds to the specific scenario reported in Figure 8-b - note also that
the same color has been used.

###  $\kappa\neq\kappa_{crit}$: non-periodic response

Finally, we present similar responses of spring-slider system subjected to
finite perturbations in point loading rate, but now the spring stiffness
$\kappa$ is not equal to the critical value $\kappa_{crit}$. As discussed in
subsection 4.2, in this condition the elastic response is never periodic, but
can be stable (with decaying oscillations amplitude) or unstable (with
diverging slip rate) depending on the strength of the applied perturbation
(see stability diagram in Figure 8-a). Referring to the specific scenario
reported in Figure 8-b, which shows the elastic response in the phase plane
when subcritical, near-critical and supercritical perturbations are applied to
the system, we present in Figure 13 the corresponding evolution of $\psi$,
$\phi$ and $e^{\Theta/(b/a)}=\theta V_{*}/D_{c}$ as function of slip
displacement $\delta/D_{c}$. Consistently with the trajectories of Fig. 8-b,
we can observe that shear stress, the slip velocity as well as the frictional
state evolve toward new steady state values after sufficiently slip
displacement, when subcritical or near-critical velocity jumps are applied to
the elastic system. Conversely, a supercritical perturbation leads to a rapid
and continuous loss of frictional strength and - as a result - a diverging
slip rate. A new steady state condition is thus never achieved.

##  Appendix B Appendix: Equivalent stiffness in a continuum deformable
interface

Here we provide details on how an effective stiffness $k/k_{*}$ can be
determined for a continuum frictional interface via the solution of an
associated eigenvalue problem. The eigenvalues and corresponding eigenmodes
are determined by the geometry of the interface and the manner of loading. The
eigenvalues correspond to the effective stiffness of each eigenmode, and may
take on a discrete or continuous range of values, according to the geometry
considered. Generally, when the interface is unbounded, these stiffnesses will
be continuous; when the interface is finite, these will be a discrete set.

#### Case (A): Infinite interface between two unbounded elastic half-spaces

Let us consider an infinitely-long planar interface that divides two linear-
elastic half-spaces characterized by shear modulus $\mu$ and Poisson’s ratio
$\nu$ (see sketch in Figure (2)). The linear operator that determines the
elastic change in shear stress in response to a given slip distribution
$\delta(x)$ in this case is

$\mathcal{L}\left(\delta\right)=\dfrac{\mu^{\prime}}{\pi}\int_{-\infty}^{\infty}\dfrac{d\delta/ds}{x-s}\,\text{ds}$
(64)

where $\mu^{\prime}$ is an equivalent shear modulus that depends on the mode
of deformation: $\mu^{\prime}=\mu/2(1-\nu)$ in mode II (in-plane) and
$\mu^{\prime}=\mu/2$ in mode III (anti-plane). Using shorthand notation, we
can write this as

$\mathcal{L}(\delta)=\mu^{\prime}\mathcal{H}(d\delta/ds)$ (65)

where $\mathcal{H}(\cdot)$ is the Hilbert transform convolution for the
operator (64). We may write an eigenequation in the form of (10) as

$\mu^{\prime}\mathcal{H}(d\delta/ds)=k\delta,$ (66)

where $k$ is an eigenvalue (eigenstiffness) that has units of stress per unit
distance.
Given that the Hilbert transform has the property that
$\mathcal{H}\left(\mathcal{H}\left(f\right)\right)=-f$, we may rewrite (66) as

$-\mu^{\prime}\frac{d\delta}{dx}=k\mathcal{H}\left(\delta\right)$ (67)

In taking the derivative of (67) and using (66), we can re-express the
eigenvalue problem as

$\frac{d^{2}\delta}{dx^{2}}+\left(\frac{k}{\mu^{\prime}}\right)^{2}\delta=0,$
(68)

which has solutions $\delta(x)=e^{i(k/\mu^{\prime})x}$. Relating the
wavenumber $k/\mu^{\prime}$ to a wavelength $\lambda$ as
$\dfrac{k}{\mu^{\prime}}=\dfrac{2\pi}{\lambda}$, we may alternatively express
the eigenstiffness as

$k=2\pi\cdot\frac{\mu^{\prime}}{\lambda},$ (69)

which continuously varies with the choice of $\lambda$.

#### Case (B): Finite-length interface in an unbounded elastic medium

Let us consider now a similar scenario to the previous one, with the only
difference being that the sliding interface is now finite in one direction and
characterized by a half-length $L$ (see Figure 2, case (B)). The corresponding
elastic operator now is not defined over the infinite line, but it operates
over the finite interface via the following boundary integral

$\mathcal{L}\left(\delta\right)=\dfrac{\mu^{\prime}}{\pi}\int_{-L}^{L}\dfrac{d\delta/ds}{x-s}\,\text{ds}$
(70)

Upon scaling the spatial coordinates by $L$ and rearranging a few terms, the
eigenequation (10) reads

$\frac{k}{(\mu^{\prime}/L)}\delta=\frac{1}{\pi}\int_{-1}^{1}\dfrac{d\delta/ds}{x-s}\,\text{d}\text{s}$
(71)

where spatial coordinates $x,s$ are implicitly scaled by $L$. The eigenvalue
problem (71) is exactly that resulting from an instability analysis on a slip-
weakening fault under plane-strain condition, and whose solution has already
been obtained numerically by Uenishi and Rice,  (2003). Here, we solve this
eigenproblem semi-analytically using a Gauss-Chebishev quadrature with 500
integration points (see e.g. (Viesca and Garagash, , 2018)). The calculated
smallest eigenvalue is

$k=\left(1.157773882\dots\right)\cdot\frac{\mu^{\prime}}{L},$ (72)

with the associated eigenmode that is plotted in Figure 2, case (B) (see red
curve).

#### Case (C): Infinite interface near and parallel to a free-surface of an
unbounded elastic half-space

Unlike the last two scenarios, here we consider an infinitely-long planar
interface that is parallel to a free-surface of an unbounded linear-elastic
half-space (see Figure 2, case (C)). We further assume that its offset from
the free-surface $h$ is much smaller than the lengthscales associated with
variations in distance, such that this model reduces to that of a one-
dimensional compressible slab sliding on a rigid base. The elastic operator in
this case reads (see supplementary materials of Viesca, 2016b )

$\mathcal{L}(\delta)=-E^{\prime}h\frac{d^{2}\delta}{dx^{2}},$ (73)

where $E^{\prime}$ is an equivalent modulus that depends on deformation mode
($E^{\prime}=2\mu/(1-\nu)$ in mode II and $E^{\prime}=\mu$ in mode III) and
$x$ spans the interval $\left(-\infty;+\infty\right)$. In this particular
case, the eigenequation (10) reduces to

$\frac{d^{2}\delta}{dx^{2}}+\frac{k}{E^{\prime}h}\delta=0,$ (74)

which has solutions $\delta(x)=e^{i\sqrt{\frac{k}{E^{\prime}h}}x}$, and again,
relating the wavenumber of this solution to the wavelength $\lambda$ via
$\sqrt{\dfrac{k}{E^{\prime}h}}=\dfrac{2\pi}{\lambda}$, we may rewrite the
eigenstiffness as

$k=4\pi^{2}\cdot\frac{E^{\prime}h}{\lambda^{2}}$ (75)

#### Case (D): Finite-length interface near and parallel to a free-surface of
an unbounded elastic half-space

Perturbing the previous scenario, let the frictional interface be finite and
characterized by a half-length $L$. The elastic operator $\mathcal{L}(\delta)$
remains (73), as well as the eigenequation (74), except that now the slipping
coordinate spans a finite interval $\left[-L;+L\right]$. Here, however, unlike
to the infinite interface case, the finiteness of the slipping region requires
boundary conditions for $\delta$. An appropriate choice of boundary conditions
is that eigenmodes vanish at the endpoints: $\delta\left(\pm L\right)=0$.
Considering that solutions to (74) $e^{i\sqrt{\frac{k}{E^{\prime}h}}x}$ can be
decomposed into eigenmodes that are symmetric or antisymmetric about the
origin, $\cos\left(\sqrt{\dfrac{k}{E^{\prime}h}}x\right)$ or
$\sin\left(\sqrt{\dfrac{k}{E^{\prime}h}}x\right)$, imposing the boundary
conditions amounts to requiring that
$\sqrt{\dfrac{k}{E^{\prime}h}}L=n\dfrac{\pi}{2}$, with $n$ being an integer.
The smallest eigenstiffness is hence

$k=\frac{\pi^{2}}{4}\cdot\frac{E^{\prime}h}{L^{2}}$ (76)

#### Case (E): Circular interface in an unbounded elastic medium

Finally, we present the case of a circular interface with radius $R$ embedded
in a three-dimensional linear-elastic unbounded domain. The operator in this
particular scenario reads

$\mathcal{L}\left(\delta\right)=\dfrac{\mu}{2\pi}\int_{0}^{R}\dfrac{\partial\delta}{\partial
s}\left(\dfrac{E\left[f(r/s)\right]}{r-s}-\dfrac{F\left[f(r/s)\right]}{s+r}\right)\text{ds},$
(77)

where $E$ and $F$ are the complete elliptic integrals of second and first kind
respectively, and $f(U)=2\sqrt{U}/(1+U)$. This representation is valid for a
Poisson ratio $\nu=0$, but can be considered as a leading-order approximation
for a case where $\nu\neq 0$. Upon mapping the above integral from
$r,s\in\left[0;R\right]$ to $\bar{r},\bar{s}\in\left[-1;1\right]$ using the
following coordinate transformations

$s=\frac{R}{2}\left(\bar{s}+1\right)\quad
r=\frac{R}{2}\left(\bar{r}+1\right),$ (78)

and scaling slip with a characteristic scale $\delta_{*}$, the eigenequation
(10) reduces to

$\frac{k}{\mu/R}\cdot\bar{\delta}=\frac{1}{\pi}\int_{-1}^{1}\dfrac{\partial\bar{\delta}}{\partial\bar{s}}\left(\dfrac{1}{\bar{r}-\bar{s}}E\left[f\left(\frac{\bar{r}+1}{\bar{s}+1}\right)\right]-\dfrac{1}{\bar{r}+\bar{s}+2}F\left[f\left(\frac{\bar{r}+1}{\bar{s+1}}\right)\right]\right)\text{d}\bar{\text{s}},$
(79)

which can be readily solved for the eigenvalues $\dfrac{k}{\mu/R}$ and the
associated eigenmodes $\bar{\delta}=\delta/\delta_{*}$. We use Gauss-Chebyshev
quadrature for the above integral representation, considering a discretization
on a primary and a complimentary sets of nodes that correspond to the roots of
the Chebyshev polynomials of third and fourth kind respectively, and solve the
eigenvalue problem semi-analytically (Viesca and Garagash, , 2018). It is
worth noting that the elastic kernel in (79) or (77) is to the leading order
of Cauchy type. However, we acknowledge the presence of a weaker log
singularity that compromise the accuracy of the numerical solution if not
appropriately taken care of. To improve the accuracy of the quadrature, we
follow a similar approach proposed by Liu et al.,  (2019) to regularize the
logarithmic singularity. Using 500 Gauss-Chebyshev points, the smallest
eigenvalue in this case is

$k=\left(1.003059516\dots\right)\cdot\frac{\mu}{R}$ (80)

## Bibliography

  * Ampuero and Rubin, (2008) Ampuero, J.-P. and Rubin, A. M. (2008).  Earthquake nucleation on rate and state faults – aging and slip laws.  Journal of Geophysical Research: Solid Earth, 113(B1).  https://doi.org/10.1029/2007JB005082.
  * Bhattacharya et al., (2017) Bhattacharya, P., Rubin, A. M., and Beeler, N. M. (2017).  Does fault strengthening in laboratory friction experiments really depend primarily upon time and not slip?  Journal of Geophysical Research, 122:6389–6430.  https://doi.org/10.1002/2017JB013936.
  * Bhattacharya et al., (2022) Bhattacharya, P., Rubin, A. M., Tullis, T. E., Beeler, N. M., and Okazaki, K. (2022).  The evolution of rock friction is more sensitive to slip that elasped time, even at near-zero slip rates.  Earth, Atmospheric, and Planetary Science, 119(30).  https://doi.org/10.1073/pnas.2119462119.
  * Blanpied et al., (1998) Blanpied, M. L., Marone, C. J., Lockner, D. A., Byerlee, J. D., and King, D. P. (1998).  Quantitative measure of the variation in fault rheology due to fluid-rock interactions.  Journal of Geophysical Research, 103(B5):9691–9712.  https://doi.org/10.1029/98JB00162.
  * Byerlee, (1970) Byerlee, J. D. (1970).  The mechanics of stick-slip.  Tectonophysics, 9:475–486.  https://doi.org/10.1016/0040-1951(70)90059-4.
  * Cattania, (2023) Cattania, C. (2023).  A source model for earthquakes near nucleation dimension.  Bull. Seism. Soc. Am., 113(3).  https://doi.org/10.1785/0120220045.
  * Dieterich, (1978) Dieterich, J. H. (1978).  Time-dependent friction and the mechanics of stick-slip.  Pure and Applied Geophysics, 116(4-5):790–806.  https://doi.org/10.1007/BF00876539.
  * Dieterich, (1979) Dieterich, J. H. (1979).  Modeling of rock friction: 1. experimental results and constitutive equations.  Journal of Geophysical Research: Solid Earth, 84(B5):2161–2168.  https://doi.org/10.1029/JB084iB05p02161.
  * Dieterich, (1992) Dieterich, J. H. (1992).  Earthquake nucleation on faults with rate-and-state-dependent strength.  Tectonophysics, 211:115–134.  https://doi.org/10.1016/0040-1951(92)90055-B.
  * Garagash, (2021) Garagash, D. I. (2021).  Fracture mechanics of rate-and-state faults and fluid injection induced slip.  Phil. Trans. R. Soc. A, 379.  https://doi.org/10.1098/rsta.2020.0129.
  * Gori et al., (2021) Gori, M., Rubino, V., Rosakis, A. J., and Lapusta, N. (2021).  Dynamic rupture initiation and propagation in a fluid-injection laboratory setup with diagnostics across multiple temporal scales.  Earth, Atmospheric, and Planetary Science, 118(51).  https://doi.org/10.1073/pnas.2023433118.
  * Gu et al., (1984) Gu, J.-C., Rice, J. R., Ruina, A. L., and Tse, S. T. (1984).  Slip motion and stability of a single degree of freedom elastic system with rate and state dependent friction.  Journal of the Mechanics and Physics of Solids, 32(3):167–196.  https://doi.org/10.1016/0022-5096(84)90007-3.
  * Gu and Wong, (1991) Gu, Y. and Wong, T.-F. (1991).  Effects of loading velocity, stiffness, and inertia on the dynamics of a single degree of freedom spring-slider system.  Journal of Geophysical Research: Solid Earth, 96(B13):21677–21691.  https://doi.org/10.1029/91JB02271.
  * Guérin-Marthe et al., (2019) Guérin-Marthe, S., Nielsen, S., Bird, R., Giani, S., and Di Toro, G. (2019).  Earthquake nucleation size: evidence of loading rate dependence in laboratory faults.  Journal of Geophysical Research, 124(689-708).  https://doi.org/10.1029/2018JB016803.
  * Gvirtzman and Fineberg, (2021) Gvirtzman, S. and Fineberg, J. (2021).  Nucleation fronts ignite the interface rupture that initiates frictional motion.  Nature Physics, 17:1037–1042.  https://doi.org/10.1038/s41567-021-01299-9.
  * Helmstetter and Shaw, (2009) Helmstetter, A. and Shaw, B. E. (2009).  Afterslip and aftershocks in the rate-and-state friction law.  Journal of Geophysical Research, 114(B01308).  https://doi.org/10.1029/2007JB005077.
  * Ji et al., (2022) Ji, Y., Wang, L., Hofmann, H., Kwiatek, G., and Dresen, G. (2022).  High-rate fluid injection reduces the nucleation length of laboratory earthquakes on critically stressed faults in granite.  Geophys. Res. Letters, 49.  https://doi.org/10.1029/2022GL100418.
  * Kaneko et al., (2016) Kaneko, Y., Nielsen, S., and Carpenter, B. M. (2016).  The onset of laboratory earthquakes explained by nucleating rupture on a rate-and-state fault.  Journal of Geophysical Research, 121(8):6071–6091.  https://doi.org/10.1002/2016JB013143.
  * Latour et al., (2013) Latour, S., Schubnel, A., Nielsen, S., Madariaga, R., and Vinciguerra, S. (2013).  Characterization of nucleation during laboratory earthquakes.  Geophys. Res. Letters, 40:5064–5069.  https://doi.org/10.1002/grl.50974.
  * Liu et al., (2019) Liu, D., Lecampion, B., and Garagash, D. I. (2019).  Propagation of a fluid-driven fracture with fracture-length dependent apparent toughness.  Engineering Fracture Mechanics, 220:106616.  https://doi.org/10.1016/j.engfracmech.2019.106616.
  * Marone, (1998) Marone, C. (1998).  Laboratory-derived friction laws and their application to seismic faulting.  Annu. Rev. Earth Planet. Sci., 26:643–696.  https://doi.org/10.1146/annurev.earth.26.1.643.
  * Marty et al., (2023) Marty, S., Schubnel, A., Bhat, H. S., Fukuyama, E., Latour, S., Nielsen, S., and Madariaga, R. (2023).  Nucleation of laboratory earthquakes: quantitative analysis and scalings.  Journal of Geophysical Research, 128(3).  https://doi.org/10.1029/2022JB026294.
  * McLaskey, (2019) McLaskey, G. C. (2019).  Earthquake initiation from laboratory observations and implicaitons for foreshocks.  Journal of Geophysical Research: Solid Earth, 124(12):12882–12904.  https://doi.org/10.1029/2019JB018363.
  * McLaskey and Yamashita, (2017) McLaskey, G. C. and Yamashita, F. (2017).  Slow and fast ruptures on a laboratory fault controlled by loading characteristics.  Journal of Geophysical Research: Solid Earth, 122:3719–3738.  https://doi.org/10.1002/2016JB013681.
  * Nagata et al., (2012) Nagata, K., Nakatani, M., and Yoshida, S. (2012).  A revised rate- and state-dependent friction law obtained by constraining constitutive and evolution laws separately with laboratory data.  Journal of Geophysical Research, 117(B02314).  https://doi.org/10.1029/2011JB008818.
  * Nielsen et al., (2010) Nielsen, S., Taddeucci, J., and Vinciguerra, S. (2010).  Experimental observations of stick-slip instability fronts.  Geophys. J. Int., 180:697–702.  https://doi.org/10.1111/j.1365-246X.2009.04444.x.
  * Noda and Chang, (2023) Noda, H. and Chang, C. (2023).  Tertiary creep behvior for various rate- and state-dependent friction laws.  Earth and Planetary Science Letters, 619(118314).  https://doi.org/10.1016/j.epsl.2023.118314.
  * Ohnaka and Shen, (1999) Ohnaka, M. and Shen, L.-F. (1999).  Scaling of the shear rupture process from nucleation to dynamic propagation: Implications of geometric irregularity of the rupturing surfaces.  Journal of Geophysical Research, 104(B1):817–844.  https://doi.org/10.1029/1998JB900007.
  * Okubo and Dieterich, (1984) Okubo, P. G. and Dieterich, J. H. (1984).  Effects of physical fault properties on frictional instabilities produced on simulated faults.  Journal of Geophysical Research, 89(B7):5817–5827.  https://doi.org/10.1029/JB089iB07p05817.
  * Passelègue et al., (2017) Passelègue, F. X., Latour, S., Schubnel, A., Nielsen, S., Bhat, H. S., and Madariaga, R. (2017).  Influence of fault strength on precursory processes during laboratory earthquakes.  In Thomas, M. Y., Mitchell, T. M., and Bhat, H. S., editors, Fault Zone Dynamic Processes: Evolution of Fault Properties During Seismic Rupture.  https://doi.org/10.1002/9781119156895.ch12.
  * Ranjith and Rice, (1999) Ranjith, K. and Rice, J. R. (1999).  Stability of quasi-static slip in a single degree of freedom elastic system with rate and state dependent friction.  Journal of the Mechanics and Physics of Solids, 47:1207–1218.  https://doi.org/10.1016/S0022-5096(98)00113-6.
  * Rice and Ruina, (1983) Rice, J. and Ruina, A. L. (1983).  Stability of steady frictional slipping.  Journal of applied mechanics, 50(2):343–349.  https://doi.org/10.1115/1.3167042.
  * Rice and Tse, (1986) Rice, J. R. and Tse, S. T. (1986).  Dynamic motion of a single degree of freedom system following a rate and state dependent friction law.  Journal of Geophysical Research, 91(B1):521–530.  https://doi.org/10.1029/JB091iB01p00521.
  * Rubin and Ampuero, (2005) Rubin, A. M. and Ampuero, J.-P. (2005).  Earthquake nucleation on (aging) rate and state faults.  Journal of Geophysical Research: Solid Earth, 110(B11).  https://doi.org/10.1029/2005JB003686.
  * Ruina, (1983) Ruina, A. (1983).  Slip instability and state variable friction laws.  Journal of Geophysical Research: Solid Earth, 88(B12):10359–10370.  https://doi.org/10.1029/JB088iB12p10359.
  * Tullis and Weeks, (1986) Tullis, T. E. and Weeks, J. D. (1986).  Constitutive behavior and stabilty of frictional sliding of granite.  Pure Appl. Geophys., 124(3):383–414.  https://doi.org/10.1007/BF00877209.
  * Uenishi and Rice, (2003) Uenishi, K. and Rice, J. R. (2003).  Universal nucleation length for slip-weakening rupture instability under nonuniform fault loading.  Journal of Geophysical Research: Solid Earth, 108(B1).  https://doi.org/10.1029/2001JB001681.
  * (38) Viesca, R. C. (2016a).  Self-similar slip instability on interfaces with rate-and state-dependent friction.  Proc. Roy. Soc. Lond. A, 472(2192):20160254.  https://doi.org/10.1098/rspa.2016.0254.
  * (39) Viesca, R. C. (2016b).  Stable and unstable development of an interfacial sliding instability.  Physical Review E, 93(6):060202.  https://link.aps.org/doi/10.1103/PhysRevE.93.060202.
  * Viesca, (2023) Viesca, R. C. (2023).  Frictional state evolution laws and the non-linear nucleation of dynamic shear rupture.  Journal of the Mechanics and Physics of Solids, 173(105221).  https://doi.org/10.1016/j.jmps.2023.105221.
  * Viesca and Garagash, (2018) Viesca, R. C. and Garagash, D. I. (2018).  Numerical methods for coupled fracture problems.  Journal of the Mechanics and Physics of Solids, 113:13–34.  https://doi.org/10.1016/j.jmps.2018.01.008.
  * Xu et al., (2018) Xu, S., Fukuyama, E., Yamashita, F., Mizoguchi, K., Takizawa, S., and Kawakata, H. (2018).  Strain rate effect on fault slip and rupture evolution: Insigth from meter-scale rock friction experiments.  Tectonophysics, 733:209–231.  https://doi.org/10.1016/j.tecto.2017.11.039.