# Fifty years of Schallamach waves: From rubber friction to nanoscale fracture

Koushik Viswanathan Corresponding author<EMAIL_ADDRESS>Department of
Mechanical Engineering, Indian Institute of Science, Bengaluru, India
Srinivasan Chandrasekar  Center for Materials Processing and Tribology, Purdue
University, USA

( May 30, 2022)

###### Abstract

The question of how soft polymers slide against hard surfaces is of
significant scientific interest, given its practical implications.
Specifically, such sytems commonly show interesting stick–slip dynamics,
wherein the interface moves intermittently despite uniform remote loading. The
year 2021 marked the 50th anniversary of the publication of a seminal paper by
Adolf Schallamach (_Wear_ , 1971) that first revealed an intimate link between
stick–slip and moving detachment waves, now called Schallamach waves. We place
Schallamach’s results in a broader context and review subsequent
investigations of stick–slip, before discussing recent observations of
solitary Schallamach waves. This variant is not observable in standard
contacts so that a special cylindrical contact must be used to quantify its
properties. The latter configuration also reveals the occurrence of a dual
wave—the so-called separation pulse—that propagates in a direction opposite to
Schallamach waves. We show how the dual wave and other, more general,
Schallamach-type waves can be described using continuum theory, and provide
pointers for future research. In the process, fundamental analogues of
Schallamach-type waves emerge in nanoscale mechanics and interface fracture.
The result is an on-going application of lessons learnt from Schallamach-type
waves to better understand these latter phenomena.

## 1 Introduction

The question ‘ _How does rubber slide?_ ’ marks the title of a landmark paper
by Adolf Schallamach published 50 years ago [1]. This work investigated the
basic mechanisms of sliding friction when a soft compliant object, such as
rubber, is slid against a much harder and practically rigid object, such as
glass. The question appears to be a humble one, and the 1971 paper was one of
the many pinnacles of a broad research programme on rubber friction at the
Natural Rubber Producers’ Research Association in the United Kingdom.
Testament to its success is the fact that this seemingly simple question has
thrown up a rich variety of dynamic behaviour in response, spanning length and
time scales, and occurring in completely unexpected situations [2].

In order to fully appreciate this complexity, consider a simple configuration
comprising two semi-infinte solids that are in contact along the $xy$ plane
and under a remotely applied normal force $N$ in the $z$-direction. We assume
that one solid is elastic $(z\leq 0)$ and the other rigid $(z>0)$. If, at time
$t=0$, the elastic solid is subject to a constant remote velocity $V_{0}$
parallel to its contact plane111Note that we will henceforth use the same
notation for the remote sliding velocity., how does motion at the interface
occur? Note that we have posed the problem via an applied velocity _in lieu_
of an applied remote shear force to ensure that the solids will eventually
slide against each other. Elementary mechanics tells us that if $V_{0}$ is
constant, the shear force transmitted at the interface will continuously
increase from $t=0$ until it reaches the static friction threshold $F_{s}=\mu
N$, determined by a coefficient of friction $\mu$, presumed known for this
pair of solids, and the normal force $N$. Beyond this threshold, uniform
sliding should ensue at constant interface shear force $F_{d}$, the so-called
‘dynamic’ friction force. This is the standard Coulomb model for friction [3].

Schallamach’s work dramatically exposed the shortcomings of this line of
thought. Firstly, it showed that the interface itself does not often move
uniformly but instead only via the propogation of localized wave-like
detachment zones—now aptly named Schallamach waves in his honour—that traverse
the entire length of the contact interface. Detachment waves of this nature
can best be described by analogy with the motion of a ruck in a carpet [4]: if
a carpet is to be moved by distance $\Delta x$, we could either simply
translate the entire carpet surface at once by $\Delta x$, or create a
localized slip zone by buckling—causing slip $\Delta x$—that then propagates
along the carpet, progressively causing it to slip. This buckling and
detachment type mechanism is reminiscent of that proposed by Bragg for
explaining how dislocations mediate slip in crystals [5].

The most distinguishing properties of Schallamach waves are that they cause
local interface slip, and that they propagate significantly more slowly than
any known elastic waves, e.g., Rayleigh and Stoneley waves. It is only now
being appreciated that these waves are part of a larger class of propagating
detachment fronts that arise not only in frictional interactions, but also in
nanoscale fracture [6], seismology [7, 8] and even biological locomotion [9,
10, 11].

Schallamach’s work also established the importance of elastic deformation and
adhesion in determining the frictional response of an interface. The
aforementioned Coulomb friction model for the onset of steady sliding applies
only to perfectly rigid solids. When one of the bodies is allowed to deform,
the interface dynamics can change significantly [12, 13, 14]. Likewise, when
the two bodies have significant interface adhesion, reattachment is easily
effected following local detachment, leading to localized pockets that can
then move and cause slip [15].

In this article, we revisit various facets of Schallamach waves, fifty years
after they were first reported. The first part of this manuscript presents a
topical review of Schallamach’s original contributions and their continued
impact (Sec. 2). Our experimental results are then presented in Sec. 3. The
solitary Schallamach wave is introduced, along with another detachment wave
that moves in the opposite direction—the ‘dual’ of the Schallamach wave, also
termed the separation pulse. By condensing and abstracting the basic features
of the Schallamach wave and its dual, we show how more general traveling waves
can be grouped together as ‘Schallamach-type’ waves. This potentially broadens
the applicability of lessons learned from studying waves in one situation to
an entirely different domain. Section 4 presents results from prior and on-
going theoretical investigations of Schallamach-type waves at interfaces.
Experimental observations have led to the emergence of new theoretical ideas,
even within the centuries-old framework of continuum elasticity. We conclude
our manuscript with a brief discussion and an outlook for future work in this
domain.

## 2 Schallamach’s experiments and their primary significance

Figure 1: Panels showing Schallamach waves in an adhesive contact. (Left) The
two different experimental configurations used by Schallamach comprise of a
spherical glass indenter sliding against a rubber flat (a) and a spherical
rubber indenter sliding against a glass flat (b). Note the direction of
friction force $F$ vis-á-vis wave propagation direction $w$. (Rigt) Three
frames showing the propagation of multiple Schallamach waves, sliding
direction is as shown in (a) on the left. Adapted from Ref. [1]

Schallamach’s original observation of detachment waves were made at a
rubber–glass interface, under two complementary geometries—a glass hemisphere
slid at constant speed $V_{0}$ against a rubber flat, and a rubber hemisphere
slid against a glass flat. In either case, when the sliding velocity was low
but above a critical value $V_{0}>V_{0}^{C}$, the interface did not move
uniformly at speed $V_{0}$, but instead slipped intermittently, see Fig. 1
reproduced from Schallamach’s paper [1]. The left panel in this figure shows a
schematic of the two sliding geometries. The images in the right panel
correspond to experiments with a glass indenter/rubber flat and were obtained
by directly observing the interface through the transparent glass hemisphere
and lit by a backlight source. The view here is along the normal of the plane
flat surface. The contact zone is grey and the non-contacting zones appear
darker due to the lighting.

These frames show several creases or folds traversing from right to left
within the contact zone. Each crease is a single Schallamach wave and is
comprised of a local region of detachment so that it appears dark in the
image. The frames clearly show multiple waves within the contact zone at any
time instant. Well defined individual waves, with larger detachment regions,
occur on the right side of the contact, and become increasingly
indistinguishable towards the left. The result is an intermittent motion of
the interface—each crease or wave causes a small amount of slip, and a point
on the interface remains stationary until a wave traverses it. This was
deduced by Schallamach as follows:

> Dust particles fortuitously caught in the interface were seen to be
> buffetted by successive waves, and to remain motionless in the intervals…A
> (single) wave of detachment is a fold in the rubber surface with presumably
> large strains around its three curved parts. To propagate the wave, energy
> must be supplied to cover the energy losses incurred when this strain
> configuration travels along the contact.

Strong interface adhesion between the two solids was necessary for waves to
occur. Adhesion hysteresis was identified as a source of energy loss, in
addition to that needed for changing the strain configuration in the vicinity
of the propagating wave. Schallamach also noted that the wave propagation
direction $w$ in the schematic of Fig. 1 is always against the direction of
friction force $F$ on the rubber in either contact configuration (hemisphere
or flat). This direction of propagation has important significance that will
be elaborated upon in later sections.

Finally, Schallamach also measured propagation speeds of individual waves,
along with their frequency (no. of waves per second within the contact) and
found that the latter was nearly proportional to the imposed $V_{0}$. Further,
the wave speed he measured ($\sim 0.01$ m/s) was much lesser than typical
elastic wave speeds ($\sim 100$ m/s). This is after accounting for the fact
that the wave speed was ‘more difficult to assess’. Therefore, the possibility
that these waves of detachment were associated with any known elastic surface
waves appeared remote at first sight.

## 3 Experimental observations of Schallamach waves

The most common configuration for observing Schallamach waves is the spherical
geometry described in the previous section. We first summarize experimental
observations using such a spherical indenter geometry and use them to explain
why Schallamach waves must nucleate in the first place. This is followed by a
description of a more recent configuration presented by Viswanathan _et al._
[16] that uses a cylindrical contact geometry to isolate single waves without
intervening edge effects.

### 3.1 Spherical contacts: Why do Schallamach waves nucleate?

Before we discuss the formation of Schallamach waves, it is first important to
consider the mechanics of the spherical contact in some detail. Here and
henceforth, we assume that the indenting object is rigid and the indented
object is elastically deformable. The latter is also assumed to be semi-
infinite (half-space) as an idealization. This situation is schematically
demonstrated in Fig. 2, reproduced from an early review article on Schallamach
waves [17] . When a rigid hemisphere of radius $R$ is pressed against an
elastic surface with a load $P$, the latter deforms, resulting in a contact
radius of size $a(P)$. It is well known that the depth of penetration
$\delta=a^{2}/R$ depends on the load $P$ via the Hertz contact relations [18].
Additionally, the edges of the contact zone are tangential to the spherical
surface (Fig. 2(b)).

Figure 2: Schematic comparing standard Hertzian non-adhesive contact (bottom)
and JKR-type adhesive contact (top). In the former, the rubber surface is
nearly tangential to the indenter profile. In the latter, it is normal to the
surface, due to intermolecular attraction. Adapted from Ref. [17].

In the presence of interface adhesion, the contact zone is very different
[19], see Fig. 2(a). Notably, the edge of the contact zone is now normal to
the spherical indenter and its geometry resembles that of a crack-tip [20].
This has two important consequences: firstly, the contact radius $a$ is now
larger for the same force $P$, so that $\delta$ is also larger. Secondly, and
more importantly, the surface immediately outside the contact zone is
depressed and surrounded by a region that is raised. This is because rubber is
essentially incompressible so that a compressive strain in one region must be
compensated by comparable tensile strain nearby.

Figure 3: Nucleation of a single Schallamach wave within the spherical contact
geometry. The elastic polymer is slid at constant remote velocity $V_{0}$ to
the left and initially remains stationary at the interface (frame 1). Buckling
results in a pronounced bulge on the free surface, with a corresponding
surface depression between points $A_{1}$ and $A_{2}$ (frame 2). The bulge
eventually grows enough to reattach to the indenter at point $B$ (frame 3).
Wrinkles accompanying a nucleating Schallamach wave show an initial wavelength
$\lambda_{0}$ (inset, frame 2) which doubles to $\lambda_{1}\simeq
2\lambda_{0}$ (inset, frame 3) just at the onset of wave propagation. The
inset locations are marked by white dashed lines in the corresponding frames.

With this background, the reason why a Schallamach wave must nucleate can be
clearly stated, see Fig. 3, adapted from Ref. [16]. Here, the indenter is a
glass lens and the elastic substrate is polydimethylsiloxane (PDMS). Since the
indenter and the substrate form an adhesive contact, the edge of this contact
zone forms a crack-like interface, see frame 1 and corresponding side-view
schematic below. Upon application of a remote $V_{0}$ to the elastic polymer,
the tangential force $T$ increases monotonically while adhesion causes the
interface to remain stationary. However, at a critical load, a region
immediately ahead of the contact undergoes surface buckling causing a bulge to
form on the surface and a corresponding depression between the points $A_{1}$
and $A_{2}$ (frame 2). As $T$ is increased further, the bulge grows and
eventually makes contact with the indenter at a second location $B$. This
detached region between the two contact locations $A_{1}$ and $B$ constitutes
a nucleated Schallamach wave that then begins to propagate through the
interface. Since $V_{0}$ is maintained constant, wave propagation is
accompanied by a reduction in $T$ and the entire process repeats again to
nucleate another wave. It is therefore clear from this sequence that each wave
is nucleated by a surface buckling instability. The result is intermittent
motion of the interface—commonly termed stick–slip—with some parts remaining
stationary and others (moving wave regions) slipping.

An analogous mechanism was also postulated by Schallamach:

> On applying a steadily increasing tangential stress, sliding begins when the
> compressive stress in front of the contact area has become great enough for
> buckling

It is evident from these observations that adhesion plays a key role not only
in sustaining wave motion but also in wave nucleation, both by changing the
contact geometry (making buckling more conducive) and by effecting
reattachment. The existence of the critical (buckling) load explains why
Schallamach waves are only observed after a threshold velocity $V_{0}^{C}$ is
crossed. Note that this type of constant velocity boundary loading has
historically been the most common configuration in friction experiments, as
opposed to constant shear stress or force. The reason for this is that the
transition from static to dynamic motion is best observed in this situation
[21, 22].

Compression-driven buckling has an additional side effect—the occurrence of
secondary features such as wrinkles on the detached surface [23]. A half-space
subject to compression inevitably shows wrinkles, folds and creases at the
surface [24, 25, 26]. Sample wrinkles are shown in the insets to frames 2, 3
of Fig. 3. The wrinkle pattern that forms first has an inherent wavelength
$\lambda_{0}$ that is determined by the depth of the strained surface layer
(inset, frame 2). This wavelength doubles as the tangential load $T$ is
increased so that $\lambda_{1}\sim 2\lambda_{0}$ (inset, frame 3). As the
detached zone begins to propagate, wrinkles result in imperfect readhesion in
its wake. Consequently, the size of these pockets of imperfect readhesion is
nearly equal to the final wrinkle wavelength $\lambda_{1}$.

The buckling-induced mechanism for Schallamach waves is qualitatively similar
in most situations, even, in the presence of a lubricant [27]. However, the
presence of an interfacial liquid film changes the nature of the resulting
stick–slip process quantitatively [28]. Firstly, the threshold $V_{0}^{C}$ is
lowered, perhaps because the increased contact pressure due to fluid
penetration inside the contact enhances the propensity for buckling. Note that
this possibiliy does not occur if the indenter is rigid (as in Fig. 2) since
then the contact remains dry with minimal fluid ingress. Secondly, at much
higher velocities, the picture appears to be fundamentally altered in liquids
such as water-glycerol that can wet and dewet from rubber surfaces [29]. Now
sticking and slipping occurs not due to repeated buckling instabilities but
due to a wetting-squeezing-dewetting process. These ‘dewetting waves’ are not
the same as Schallamach waves and will not be considered further in this
article. It is therefore safe to conclude that surface buckling is well
established as the underlying cause for the nucleation of Schallamach waves
[30, 31, 32].

### 3.2 Cylindrical _vs._ spherical geometry

Figure 4: Experimental schematic showing a typical cylindrical lens in
adhesive contact leading to the formation of an ‘adhesive’ channel. The
schematic on the left shows a typical configuration with a backlit source and
remote applied motion $V_{0}$. A sample image showing the contact zone (light
grey) and detached zones (dark grey) is presented in the right panel. Note
that the red, green and blue arrows represent the $x,y,z$ axes, respectively.

In contrast to the most commonly used spherical contact configuration
discussed above, a second cylindrical contact geometry is now considered, see
Fig. 4. This geometry is inspired by the need for observing Schallamach wave
propagation in a nearly planar interface without any edge effects. A fully
flat configuration has its own challenges, especially with respect to aligning
the indenter and the substrate [33]. The use of a large radius cylindrical
indenter compenstates for this since, at low load, contact is first initiated
along the cylinder axis (line contact) [34].

The contact geometry we describe here is adapted from Ref. [16]. A cylindrical
plano-convex lens (Edmund Optics, radius 16.25 mm, length 25 mm) is used as
the indenter and the substrate is again polydimethylsiloxane (as in Fig. 3).
The PDMS samples were produced mixing a base (vinyl-terminated
polydimethylsiloxane) with a curing agent (methylhydrosiloxane-
dimethylsiloxane copolymer) in the ratio 10:1 by weight. The resulting mixture
was cured for 12 hours at 60∘C in the form of slab-like specimens [16]. The
Poisson’s ratio and Young’s modulus of the final samples were 0.46 and 1 MPa,
respectively. Constant remote $V_{0}$ is effected via the same ball-screw
mechanism as before and the contact is imaged using a high-speed camera (pco
Dimax) attached to a microscope system. The right hand panel in Fig. 4
presents a typical image obtained by viewing the backlit contact through the
transparent cylindrical indenter. The lighter (darker) regions correspond to
interface contact (detachment) just as before.

In both cases, the polymer sample had dimensions along the $x,y,z$ axes (Fig.
4) of 70 mm $\times$ 22 mm $\times$ 25 mm, respectively, with the long
dimension representing the sliding direction. Prior to any experiment, the
PDMS and lens are brought into contact along the $x$-axis and maintained for a
fixed time $t=60$ s to standardize any possible contact aging effects. Contact
alignment is also checked during this period by translating the camera/light
source along the $x$-axis. The PDMS is then slid relative to the lens using
the linear stage at constant speeds $V_{0}=$10 $\mu$m/s to 20 mm/s, for a
distance of 30 mm. Concurrent with the _in situ_ imaging, sliding forces were
measured using a piezoelectric dynamometer (Kistler).

### 3.3 The solitary Schallamach wave and its opposite moving dual

One of the primary problems with the standard spherical indenter geometry used
for studying Schallamach waves is its inability to isolate single waves, as
Schallamach himself noted:

> The waves of detachment…tend to become thinner and to break up during their
> travel…various fragments sometimes join so that individual waves gradually
> lose their identity. This is one of the reasons why conditions become
> confused in the rear part of the contact

This is clearly evident in the frames reproduced in Fig. 1 (right), and is
primarily due to the nature of the spherical indenter geometry. Despite this
shortcoming, Schallamach used the spherical geometry likely because it was
(and perhaps remains!) the most commonly used contact geometry by the
tribology and contact mechanics community and thus has several analytical
solutions [18, 35, 36, 37].

Figure 5: Evolution of a single Schallamach wave under remote applied $V_{0}$
to the elastic polymer. The wave propagates from left to right, in the same
direction as $V_{0}$ and causes slip in its wake. The interface is stationary
before and after the wave has passed. This is seen by following the motion of
a dirt particle (black line) embedded on the polymer sub-surface. The scale
bar and timestamp shows that the wavespeed is much lesser than typical elastic
wave speeds.

In order to isolate single waves and study their propagation dynamics, it is
best to use the cylindrical indenter geometry described in Sec. 3.2. An
‘adhesive’ channel forms upon contact between the indenter and the substrate
and the isolated waves can be individually imaged using high-speed videography
as described earlier. Frames from a high-speed sequence showing one such wave
are reproduced in Fig. 5. Note that in this sequence, the polymer/rubber is
moved from left to right at constant $V_{0}$ far from the interface. A single
Schallamach wave traverses the interface in the same direction as $V_{0}$. It
is clear that the defining features of the Schallamach wave remain unchanged
from the spherical geometry. Firstly, the wave appears as a dark band within
the interface, due to buckling-induced interface detachment and readhesion.
Secondly, points on the interface remain stationary until the wave appears;
they then begin to move and are displaced by a fixed amount $\Delta x$.
Finally, the wave is accompanied by compression wrinkles, just as we observed
in the case of the spherical contact geometry, _cf._ Fig. 3.

In light of these single wave results, it is pertinent to revisit
Schallamach’s observations about ‘imperfections’ in the spherical contact
itself:

> …secondary effects occur at times…where a relatively large portion of the
> contact becomes detached from the main body and suddenly fuses with it
> again.

This statement refers to the rear end of the contact (taken in the sliding
direction). Just as material at the front end of the contact is subject to
compression and surface buckling, the surface just outside this rear end is
subject to tension. One can consequently imagine a local ‘tensile neck’
causing detachment from the indenter surface, followed by readhesion. These
are the ‘secondary effects’ that Schallamach refers to. They have since been
backed up by similar observations in spherical contacts of various dimensions
[17, 2].

The equivalent of this seemingly innocuous observation in our cylindrical
contact geometry of Fig. 4 is dramatic. If a tensile neck does indeed form
just outside the contact zone, resulting in a locally detached zone within the
interface, it must propagate in the opposite direction in the form of another,
very different, wave of detachment!

Figure 6: Propagation of the dual of the Schallamach wave or the separation
pulse. Remote sliding $V_{0}$ occurs in the same direction as before, but the
detachment wave moves in the opposite direction. Despite this, slip occurs in
the same direction as the applied $V_{0}$. Note that, just as with the
Schallamach wave, this separation pulse also propagates at speeds much lesser
than elastic wave speeds.

One such instance is illustrated by the set of frames taken from a high–speed
sequence shown in Fig. 6. Here again, the remote sliding direction $V_{0}$ is
from left to right, just as in Fig. 5. However, a detachment wave is now seen
to move in the opposite direction (right to left), yet causing slip in the
same direction as before. The latter fact may be ascertained by the fact that
a fortuituously trapped dirt particle within the contact is translated in the
direction of remote $V_{0}$, as is to be expected. Furthermore, this wave of
detachment is easily distinguished from the Schallamach wave by a complete
absence of compression-induced surface wrinkles—it is nucleated by a tensile
instability after all—resulting in perfect contact in its wake. This
detachment wave has been christened the separation or detachment pulse [34].
For the purposes of this article, we also refer to this as the dual of the
Schallamach wave, since it differs only in that it is tensile and propagates
in the opposite direction to the latter.

The compressive _versus_ tensile nature of the Schallamach wave and its dual,
the separation pulse, is sufficient to explain the corresponding propagation
direction observed in Figs. 5, 6, see Ref. [38]. For wave propagation at
constant speed $c_{w}$ along the $x-$axis, the tangential displacement $u_{x}$
has the form $u_{x}(x-c_{w}t)$. Hence, the $x-$derivative of the displacement
$\partial u_{x}/\partial x$ and the strain $\epsilon_{xx}$ can be written as

$\epsilon_{xx}=\frac{\partial u_{x}}{\partial x}=-c_{w}^{-1}\frac{\partial
u_{x}}{\partial t}\simeq-c_{w}^{-1}\frac{\Delta x}{T}$ (1)

where $\Delta x$ is the slip per wave and $T$ is duration of propagation.
Since the sign of $\Delta x$ is the same as $V_{0}$, the strain
$\epsilon_{xx}$ is negative (compressive) if the wave propagates in the same
direction as $V_{0}$ and is positive (tensile) if it propagates in the
opposite direction. Therefore, for the case of Schallamach waves
$\epsilon_{xx}<0$ and for separation pulses $\epsilon_{xx}>0$, as is evident
from the frames in Figs. 5 and 6.

### 3.4 Interface dynamics from space-time diagrams

Figure 7: Space–time diagrams showing propagation of a Schallamach wave (top
row) and its dual (separation pulse, bottom row). The image on the left shows
locations of the horizontal lines $P_{1},P_{2},P_{3}$, with corresponding
space–time diagrams to the right in each row. The wave appears as a dark band
in all three diagrams with constant slope (red dashed line) equal to the wave
speed. Interface slip distance, as indicated by the trajectory of the black
line marker, is as shown between parallel white dashed lines. Analogously, the
separation pulse appears as a dark band in all three diagrams (bottom row)
with constant slope (red dashed line) equal to the wave speed. Interface slip
distance, as indicated by a moving dirt particle near the interface, is marked
by parallel white dashed lines.

More quantitative information about interface dynamics accompanying wave
motion is obtained by means of space-time diagrams, that are commonly used in
wave propagation studies to obtain spatio-temporal information [39]. A space-
time diagram is constructed by stacking a fixed horizontal line in the image
sequence as a function of time. Three such fixed lines, denoted
$P_{1},P_{2},P_{3}$, and their corresponding space-time diagrams are presented
in Fig. 7 for Schallamach (top row) and separation waves (bottom row). In each
case, the wave itself appears as a dark diagonal band with a constant
inclination angle (red dashed line). Based on these diagrams, we can assert
that both the Schallamach wave and its dual propagate at constant speed
through the interface. The former is in the range $c_{w}\sim 100-300$ mm/s,
corresponding to a sliding velocity $V_{0}=50-250\,\mu$m/s and the latter is
in the range $\sim 10$ mm/s with $V_{0}=10-50\,\mu$m/s. Clearly, $c_{w}\gg
V_{0}$ in both cases, yet is much lesser than any elastic wave speed for the
elastic material ($\sim 100$ m/s).

The differences between the two waves is easily seen in the sign of the slope
in these diagrams. The negative (positive) slope is due to the corresponding
wave moving in the same (opposite) direction as applied $V_{0}$. Additionally,
the interface slip $\Delta x$ is also self-evident: for Schallamach waves,
this can be obtained by tracking the dark patch (top panel) in frame $P_{1}$
of the space-time diagram. The patch has slipped by a constant finite distance
(between white dashed lines) due to wave motion. On the other hand, a dirt
particle just below the surface may be tracked for the separation pulse (frame
$P_{3}$, bottom row) which shows the resulting slip due to wave passage (again
between white dashed lines).

### 3.5 Wave motion leading to stick–slip friction

Figure 8: Forces accompanying the propagation of a sequence of Schallamach
waves (left) and separation pulses (right). Both traces show oscillations, and
are therefore characteristic of what might be termed ‘stick–slip’ motion. Wave
nucleation occurs at the peak in each oscillation cycle, followed by a force
reduction during propagation. The magnitude of the reduction, as well as the
wave nucleation frequency differs between the Schallamach wave (left) and its
dual (right). Note that force is non-dimensionalized by the product of shear
modulus $G$ times contact area $A$ and time is non-dimensionalized by
$t_{0}=L/V_{0}$ where $L$ is the contact length and $V_{0}$ is the sliding
velocity as before.

Since the experiments described so far have been performed with a constant
remote velocity boundary condition, it is instructive to look at the
corresponding time-dependent tangential force. Forces for the Schallamach wave
and the separation pulse are shown in Fig. 8(a) and (b), respectively. Time on
the horizontal axis is non-dimensionalized by a characteristic sliding time
$t_{0}=L/V_{0}$, with $L$ being the length of the contact. Force on the
vertical axis is non-dimensionalized by the product of shear modulus $G$ and
nominal contact area $A$. It is clear at once that both force traces can be
termed as characterizing ‘stick–slip’ motion at the interface. At the start of
sliding, both traces increase from zero to a maximum value at which wave
nucleation occurs. Ensuing wave propagation triggers a sharp reduction in the
tangential force. Once a single wave has traversed the length of the contact,
the process repeats; the attendant interface motion is thus intermittent,
mediated only by the propagation of single Schallamach waves or separation
pulses. On the contrary, the two traces have very different quantitative
features, such as the frequency and reduction magnitude per cycle, see Fig. 8.

It is indeed surprising that the tensile dual of the Schallamach wave had not
been reported for a very long time, even though signs of their nucleation were
noted very early on by both Schallamach and Barquins. In fact, tensile
separation pulses have even been incorrectly reported as Schallamach waves in
the geophysics community [33]. The likely cause for this is the widespread use
of the spherical indenter geometry which obscures observations of
single/solitary wave motion.

Figure 9: A two-parameter phase diagram showing domains of occurrence of the
three Schallamach-type waves discussed in the article. Schallamach waves occur
above a critical sliding velocity, separation pulses occur a low $V_{0}$ and
normal loads while detachment-less slip pulses mediate motion at large normal
loads.

Together, these two waves are mechanically the only possibilities for moving
detachment waves at soft elastic interfaces characterized by adhesion. If the
need for physical detachment of the interface is relaxed, one other wave is
also commonly observed—the so-called slip pulse [40, 41]. This peculiar wave
is also nucleated in compression so that by Eq. 1, it travels in the same
direction as a Schallamach wave but looks nothing like it. Given the absence
of any detachment, the existence of the slip pulse can only be ascertained
using photoelastic techniques to directly measure the interface shear stress
[38, 14].

Taken together, these three waves—the Schallamach wave, its dual the
separation pulse and the slip pulse—appear to be the sole mediators of
stick–slip motion in low velocity sliding. The domains of occurrence of each
of these waves in the $V_{0}-a$ plane is shown in Fig. 9. The control
parameters in this ‘phase diagram’ are $V_{0}$ and the normal load
(equivalently contact width $a$). At low $V_{0}$ and $a$, stick–slip occurs
via the propagation of separation pulses, while beyond a certain velocity
threshold $V_{0}^{C}$ (vertical dashed line), Schallamach waves predominate.
The likely reason for this is the ease with which the corresponding nucleation
events can occur. On the contrary, with larger normal loads/contact widths
above a threshold $a^{C}$, both tensile detachment and compressive buckling
become infeasbile so that the slip pulse is the primary cause for stick–slip.
It now appears fairly certain that these three waves are the sole mediators of
low velocity stick–slip in deformable adhesive interfaces.

### 3.6 Similarities with elastic dislocations: Burgers’ vector and defect
pinning

At this junction, it is instructive to take a step back and reconsider the
basic defining feature of a Schallamach-type wave, _viz._ that it propagates
through the interface at a constant speed, and causes slip in its wake. Based
on this minimal description alone, one would be hard-pressed to exclude
crystal lattice dislocations from this set as well. Dislocations are also
similarly nucleated, propagate at finite speed, and cause (plastic) slip in
their wake.

Figure 10: Analogies between solitary Schallamach-type waves and crystal
dislocations. Time-dependent displacement curve as recorded at a single point
location on the interface. This curve shows that the point remains stationary
until a wave approaches $(\sim 13$ ms), following which slip occurs and
reaches a maximum value $\sim 250\,\mu$m. This is identical with what happens
with a gliding crystal dislocation so that the effective slip may be
identified with the magnitude of the Burgers’ vector. Inset shows the
frequency of wave generation plotted against $V_{0}$, the proportionality is
again reminiscent of the plastic strain rate relation due to dislocation
motion.

For concreteness, consider the solitary Schallamach wave introduced in Sec.
3.3. Three distinct velocity scales are evident—the imposed (remote) sliding
velocity $V_{0}$, the local material velocity $\vec{v}_{p}$ due to the wave
and the velocity $c_{w}$ of propagation of the wave itself. Directions of
$V_{0}$ and $c_{w}$ are fixed by the external loading. While $c_{w}$ is
constant as the wave propagates through the interface (see Fig. 7), the local
material velocity $\vec{v}_{p}$ varies with time from zero (as the wave
approaches), to a maximum value and finally back to zero (as the wave
departs). For example, for the data in Fig. 7, instantaneous $\vec{v}_{p}$ is
determined by the slope of the dirt particle curve at that time; $c_{w}$ is
the slope of the band (constant) and both are indeed distinct from $V_{0}$.

Based on data such as that in Fig. 7, the relative displacement of material
points on the surface can be plotted as a function of time, see Fig. 10. This
plot shows the displacement of a single point under different remote $V_{0}$
due to the propagation of a single Schallamach wave at some corresponding
speed $c_{w}$. Three features are immediately obvious from this data: firstly,
that the surface is displaced in the form of a unit step displacement
$\vec{b}$. We had earlier denoted this slip distance as $\Delta x=|\vec{b}|$
that occurs in the direction of $V_{0}$. Secondly, $|\vec{b}|$ is independent
of both $V_{0}$ and $c_{w}$ since curves for a range of values result in
similar surface slip. This is understandable within the buckling framework
discussed in Sec. 3—the buckling threshold is independent of $V_{0}$. Finally,
since the interface moves only via such slip events, the remote $V_{0}$ has to
be accommodated by waves with a frequency $n$ such that:

$V_{0}=n|\vec{b}|$ (2)

The constancy of $|\vec{b}|$ in this equation implies that $n\propto V_{0}$.
This is indeed the case, as supported by the data presented in the inset graph
to Fig. 10. With $V_{0}$ ranging from $10^{-1}-20$ mm/s, a linear dependence
between the wave generation frequency $n$ and $V_{0}$ is clearly established.

The situation summarized in Fig. 10 also applies equally well to the motion of
dislocations (via slip) on a crystal plane [42]. In this case, the
displacement magnitude is given by the dislocation Burgers vector (hence the
notation $\vec{b}$ above) and corresponds to the jump in Fig. 10. One can
hence define this as the ‘Burgers’ vector’ of a Schallamach wave. It is
determined, for a given contact geometry, by the substrate material
properties—for the experimental configuration in Sec. 3, the mean
$|\vec{b}|=255$ $\mu$m. Additionally, Eq. 2 appears analogous to the Taylor-
Orowan equation for plastic strain rate due to dislocation motion.

Figure 11: Interaction between a solitary Schallamach wave and a large dirt
particle $A$ on the interface. The wave approaches the particle and is bowed
by it (at $B$), following which a residual detachment region is left behind
$(C)$ around the dirt particle. This outcome is also analogous to that
observed during dislocation-point defect interaction.

The analogy with dislocations is further strenghtened by an interesting
phenomenon observed during propagation of Schallamach waves past stationary
dirt particles attached to the indenter, see Fig. 11. Here, the wavefront
approaches a static dirt particle $A$ in the contact region and gets ‘pinned’,
causing a bend in the wave profile (point $B$). As the front moves away from
the particle, the wave regains its original profile, leaving a residual air
pocket $C$ around the particle. This ‘bowing’ effect also occurs when
dislocations move past static obstacles (e.g., solute particles), leaving
behind Orowan loops [42]. Under dilute solute particle concentration, the bent
dislocation line eventually relaxes and regains its shape.

## 4 How can we describe Schallamach-type waves?

We now turn our attention to continuum descriptions of Schallamach-type waves
that occur at the interface between two contacting solids. Unlike in fluids,
waves at solid surfaces/interfaces can occur in various forms/guises depending
on the nature of the boundary conditions [43]. A primer for the propagation of
surface waves in elastic media is presented in the Appendix. In this section,
we explore the predictions of an elastodynamic formulation for Schallamach-
type wave propagation.

### 4.1 Mixed-boundary value problem for Schallamach-type waves

In addition to the simple boundary conditions discussed in the appendix,
interface waves often propagate under more complex, often mixed, boundary
conditions. The occurrence of net interface slip due to a wave further
complicates this situation. Slip-less detachment waves were analyzed by
Comninou and Dundurs, and found to propagate with a range of velocities, all
comparable to the Rayleigh wave speed [44, 45]. On the other hand, detachment-
free frictional waves have also been analyzed using Coulomb-friction boundary
conditions. However, these are marred by inconsistencies in enforcing the
static to dynamic transition, so that a physically valid solution does not
presently exist for any combination of materials [46, 47, 48]. In addition,
the biggest common shortcoming of all these solutions is that they cannot
account for interface slip in the wake of the wave.

It is thus clear that none of these solutions can describe the primary
features of Schallamach-type waves—slow propagation velocity and net interface
slip—with or without interface detachment. The first feature, that the wave
velocity $c_{w}\ll c_{T},c_{L}$, suggests the existence of another velocity-
scale, likely due to a velocity-dependent friction law [21, 49, 22, 50] and/or
some type of regularization [51, 52]. Yet, introducing a velocity scale via a
suitable interface friction law is non-trivial. While this is possible with
rather simplistic 1D models [53], a general description for moving detachment
waves with adhesion has hitherto remained elusive.

A recently established elastodynamic framework for describing Schallamach-type
detachment waves has, however, shown significant promise [38]. We present a 2D
version of the quasi-3D framework presented in this work, the sequence of
algebraic steps described below may be reconstructed by taking recourse to
Ref. [38]. Consider the elastic material occupying the half-space $z\leq 0$.
In order to mimic the physical conditions accompanying wave propagation, this
framework uses a mixed boundary condition formulation involving stationary and
detached zones on the contact interface $z=0$. For a wave moving along the
positive $x$-axis with velocity $c_{w}$, one can use co-moving coordinates
$\eta=k(x-c_{w}t)$ to make the wave appear stationary. The wave number $k$ is
related to both the wave width 2$\alpha$ and the parameters $V_{0}$ (sliding
speed), $c_{w}$ [38] Assuming deformation to be restricted to the $xz$-plane,
the relevant mixed boundary conditions are:

$\displaystyle\text{Detachment zone }|\eta|$
$\displaystyle<\alpha\begin{cases}&\sigma_{zz}=\sigma_{zx}=0\\\
&\dot{u}_{x}(\eta),\dot{u}_{z}(\eta)\neq 0\end{cases}$ (3)
$\displaystyle\text{Stationary/stick zone }\alpha<|\eta|$
$\displaystyle<\pi\begin{cases}&\sigma_{zz}(\eta),\sigma_{zx}(\eta)\neq 0\\\
&\dot{u}_{x}=\dot{u}_{z}=0\end{cases}$ (4)

where the $\sigma_{xz},\sigma_{zz}$ and $\dot{u}_{x},\dot{u}_{z}$ are the
interface stresses and velocities, respectively. The unknowns here are the
stresses $\sigma_{xx},\sigma_{xz}$ in the sticking zone and the velocities
$\dot{u}_{x},\dot{u}_{z}$ in the detached zone. In order to obtain expressions
for these unknowns, we begin with a commonly used _ansatz_ for surface wave
solutions in the form of a dual series representation, see for e.g., Ref.
[44]. The boundary conditions in Eq. 3 are then used to obtain coupled
singular integral equations for the interface velocities
$\dot{u}_{x},\dot{u}_{z}$ using the procedure discussed in Ref. [38]. In order
to address the slow velocity scale issue, one exploits the fact that $c_{w}\ll
c_{T},c_{L}$ (elastic wave speeds) to perform a perturbation expansion in
powers of $c_{w}/c_{L}$ and $c_{w}/c_{T}$. The result is that both Schallamach
waves and the oppositely moving separation pulse can be described within the
same elastic framework. In the limit $\nu\to 0.5$ (as with incompressible
rubbers), $\dot{u}_{x}(\eta),\dot{u}_{z}(\eta)$ can be shown to obey the
following singular integral equations:

$\displaystyle\frac{\tau_{r}}{G}+\frac{2}{\pi}\bigg{[}\text{sec}^{2}(\eta/2)\int_{-\alpha}^{\alpha}\dot{u}_{x}(\eta^{\prime})\cot\left(\frac{\eta-\eta^{\prime}}{2}\right)d\eta^{\prime}+\frac{\pi\tan(\eta/2)V_{0}}{c_{w}}\bigg{]}$
$\displaystyle=0$ (5)
$\displaystyle\frac{\sigma_{r}}{G}+\frac{2}{\pi}\bigg{[}\text{sec}^{2}(\eta/2)\int_{-\alpha}^{\alpha}\dot{u}_{z}(\eta^{\prime})\cot\left(\frac{\eta-\eta^{\prime}}{2}\right)d\eta^{\prime}\bigg{]}$
$\displaystyle=0$ (6)

where $\sigma_{r},\tau_{r}$ are the remote normal and shear stresses, $G$ is
the material’s shear modulus and $\eta=k(x-c_{w}t)$ as described earlier.

Unlike most other analytical treatments of stick–slip waves, the boundary
conditions Eq. 3 and final integral equations (Eq. 5) imply that a description
of Schallamach-type waves does not need either an explicit friction law (e.g.,
Coulomb or rate/state) or a velocity scale. Further, since the boundary
conditions in Eq. 3 are written in terms of velocities, interface slip can be
enforced by suitably setting the integration constant in the expressions for
the final displacements.

The unknowns in Eq. 5 can be obtained by inverting the singular integral
equations, resulting in expressions for the interface velocities and, upon
integration, the displacements [38]. Of particular interest are the interface
stresses, which can also be expressed in closed form using expressions for the
displacements and Hooke’s law

$\displaystyle\frac{\sigma}{\sigma_{r}}$
$\displaystyle=\frac{|\tan(\eta/2)|}{\sin(\alpha/2)}\frac{1}{\sqrt{\tan^{2}(\eta/2)-1}}$
(7)
$\displaystyle\frac{\tau}{\tau_{r}}=\frac{|\tan(\eta/2)|}{\sin(\alpha/2)}\frac{1}{\sqrt{\tan^{2}(\eta/2)-1}}$
$\displaystyle-\left(\frac{V_{0}}{c_{w}\tan(\alpha/2)}\right)\left(\frac{2G}{\tau_{r}}\right)\frac{|\tan(\eta/2)|}{\tan(\eta/2)}\frac{1}{\sqrt{\tan^{2}(\eta/2)-1}}$
(8)

and apply to both the Schallamach wave and its dual, by suitable change in the
sign of the ratio $c_{w}/V_{0}$.

The three main predictions of this theoretical treatment are as follows:
Firstly, Eqs. 7 and 8 determine interface stresses accompanying both the
Schallamach wave and its dual. In fact, Eq. 7 shows a singularity near the
detachment zone edge $\eta\to\pm\alpha$, implying that some adhesion mechanism
is operative here [19]. This is noteworthy since the entire problem, from the
boundary conditions to the final integral equations did not specity an
adhesion law _a priori_. Adhesion is hence a necessary condition for such wave
propagation. Secondly, wave motion can occur over a range of speeds
$c_{w}/V_{0}$ as opposed to a single, characteristic speed. This shows that
slow-moving Schallamach-type waves can be described without having to
introduce an explicit velocity scale.

Finally, both stress components in Eq. 7, 8 show square-root singular
behaviour near the detachment zone edges $\eta\to\pm\alpha$. The normal stress
component yields a stress intensity factor that may be readily related to the
work of adhesion. On the other hand, the tangential stress results in a (mode
II-type) stress intensity factor for wave propagation:

$K_{w}\sim(\tau_{r}+4G/\pi)\sqrt{x}$ (9)

in an appropriate singular limit. Here $x$ is the distance from the tip of the
detachment zone as measured within the stationary/stick zone. A more general
method for computing stress intensity factors such as Eq. 9 for the case
$\nu\neq 0.5$, where closed form solutions are generally not obtainable, is
presented in Ref. [54]. The consequences of Eq. 9 are discussed further in the
next section.

### 4.2 Predicting the onset of stick–slip

The occurrence of a stress singularity ahead of the detachment zone, Eq. 9,
represents a necessary condition for effecting wave motion. For a sufficient
condition, reattachment of the detached interface must be considered in more
detail. However, for the present, we contrast Eq. 9 with the analogous case of
transition from static-to-dynamic friction outlined recently [55]. According
to this work, dynamic sliding occurs when

$K_{w}\sim\tau_{r}\sqrt{x}$ (10)

reaches a critical value depending on the nature of the contact. The notation
here is the same as that in Eq. 9. While there has been much debate about the
nature of shear cracks in general adhesive contacts (see for instance, Ref.
[56] and references therein), we nonetheless explore the consequences of Eq. 9
by comparing it with Eq. 10, since the former naturally emerges as the
solution of a rigorous boundary value problem.

Based on these two equations, a phase diagram may be developed showing when
stick–slip motion via detachment waves (Eq. 9, orange curve) should occur at
the expense of uniform sliding (Eq. 10, blue curve), see Fig. 12. The dashed
curve denotes a limiting remote stress $\tau_{r}^{*}$ beyond which wave
propagation does not occur. Likewise, $\Delta x_{C}$ is the critical
detachment zone size below which wave propagation is not feasble.

Figure 12: Qualitative phase diagram comparing Schallamach-type detachment
waves with shear cracks to predict when stick–slip should occur at the expense
of steady sliding. The orange curve (d-wave) represents the equivalent stress
intensity factor for a moving detachment wave ($\nu=0.5$) while the blue curve
is for an equivalent shear crack of the same initial extent. The shaded
regions are labeled based on which phenomenon, stick–slip or steady
sliding/crack growth, is expected to occur.

The diagram is applicable to any material pair capable of forming adhesive
contact (with one material significantly stiffer than the other) and must be
interpreted as follows. Consider an adhesive interface that is loaded at
constant remote $V_{0}$ as described in Sec. 3. If a detachment zone of size
$\Delta x>\Delta x_{c}$ is nucleated, the total shear stress rises
continuously since the interface remains stationary. When it crosses the
orange curve, the necessary condition for wave motion is satisfied, however,
the interface still remains stationary so that sliding does not yet start.
When the remote $\tau_{r}$ then crosses the blue curve at the corresponding
$\Delta x$ value, a crack begins to propagate at which point the entire
interface can slip, or, if readhesion occurs, a single detachment wave
propagates through the interface. If the latter possibility, the shear stress
can reduce significantly to any point within the orange dashed region for
continued propagation. The fact that wave motion occurs at constant speed
$c_{w}$ implies that the shear stress settles at a fixed value within this
orange zone very soon after nucleation.

On the other hand, if interface adhesion is not strong enough to effect
reattachment, a single rupture will likely propagate through the interface,
and cause in the manner discussed in Ref. [55]. Hence depending on the
strength of adhesive interaction at the interface, either
possibility—stick–slip or steady sliding—can occur within the blue (cross-
hatched) zone. Once we cross the wave propagation threshold $\tau_{r}^{*}$,
detachment waves are no longer possible so that only steady sliding results.
The critical value $\Delta x_{c}$ is likewise determined by the intersection
between the threshold line and the blue curve—below this nucleation width,
wave propagation is no longer feasible.

### 4.3 Exploiting the dislocation analogy

An alternative theoretical development to the one discussed so far in this
section is to exploit the dislocation analogy presented in Sec. 3.6. This can
be used, for instance, to compute the shear force necessary for Schallamach
wave propagation to occur using force expressions for elastic dislocations
[42]. The shear force $F_{s}$ required to move the wave must be balanced by
the adhesion hysteresis in detachment and reattachment at the interface. Using
this balance principle, we obtain the relevant critical shear force as

$F_{s}^{c}=\frac{a\,L_{c}\,\Delta W}{|\vec{b}|}$ (11)

where $\Delta W$ is the adhesion hysteresis, $L_{c},a$ are the contact length
and width, respectively. Using $|\vec{b}|=255$ $\mu$m, $2a=1$ mm, $L_{c}=2.5$
cm for the data presented in Sec. 3, and $\Delta W\simeq 10$ mJ/m2 for PDMS
[57], $F_{s}^{c}$ is estimated to be $0.5$ mN. This is the minimum force
needed to propagate a single wave through the contact region. The wave pulse
can thus travel through the interface at a much lower stress than that needed
for nucleation, $F_{c}\simeq 1.6$ N for the data in Fig. 8. This thus explains
the steep tangential force reduction after wave propagation begins—as the
interface relaxes, the tangential force continues to decrease until it either
equals $F_{s}^{c}$ or the wave exits the contact region.

Note that in the above, we’ve explicitly discussed a solitary Schallamach
wave, but these considerations nonetheless apply to other Schallamach-type
waves as well.

## 5 Discussion

Having presented both experimental and theoretical results on Schallamach
waves and their properties, we now discuss several potential extensions and
some unexpected implications of our work.

### 5.1 Noteworthy siblings of the Schallamach wave

We have so far studied the underlying causes for nucleation—tensile necking,
compressive buckling and compression-induced slip—of Schallamach-type waves.
Given the generality of these mechanisms, one would not be hard pressed to
expect similar waves at any deformable adhesive interface. One such example
that has received some attention in the past is composite interfaces [15, 58].
Here adhesion is externally induced (via a thin binder layer) and relative
deformation between adjacent layers of the composite can nucleate Schallamach-
type waves at the interface. Consequently, the intriguing possibility of these
interfaces being toughened by such wave propagation events has been discussed
extensively [59, 60].

In this context, we draw attention to three analogous waves in the world of
soft-bodied locomotion [9, 10, 61, 11]. These organisms do not possess
specialized limbs for effecting motion, so must rely on elastic deformations
of their entire body structures to locomote [62]. They do so by propagating
Schallamach-type waves along the length of the zone of contact between their
body and the ground.

Direct waves—analogous to slip pulses—are compressive and used by some
polychaete worms (e.g., _Arenicola_) to effect motion within distinct ‘slip’
segments propagating from the tail to the head [63]. The equivalent of the
separation or detachment pulse is the retrograde wave, which occurs in some
Nemertines such as _Rhynchodemus_. Here, a local tensile detachment zone is
formed and propagates from the organism’s head to its posterior end. Finally,
the equivalent of the Schallamach wave is looping locomotion, seen in some
_Planaria_. Here, a detachment loop is formed at the rear end due to elastic
buckling of body segments; the loop is then propagated towards the head,
effecting slip as it moves [61].

The fact that these mechanical equivalents in biology are the only three waves
that have hitherto been recorded in the context of locomotion, begs the
question of whether the three Schallamach–type waves are the only ones capable
of mediating stick–slip motion at interfaces. The answer, as discussed in Fig.
9 earlier, appears to be in the affirmative, atleast for low $V_{0}$ and
normal loads. It is noteworthy that these biological analogues have also been
considered as conceptual models, by one of the originators of the dislocation
concept, Egon Orowan, in order to explain glide motion in metallic crystals
[64].

### 5.2 Viscoelastic and nonlinear effects on wave propagation

Viscoelastic effects can have a significant influence on surface wave motion,
yet incorporating them within the elastic framework of Sec. 4.1 remains quite
challenging. In fact, even the elementary case of Rayleigh waves becomes
remarkably complicated by the introduction of the simplest of linear
viscoelastic constitutive relations [65]. Waves corresponding to other more
complicated boundary conditions appear to be hopelessly intractable at
present, barring a couple of grossly simplified examples [53, 46, 41]. This
problem does warrant attention, given its importance from an experimental
point of view—Schallamach wave properties are influenced by the materials’
loss and storage moduli [66, 31] as well as other rate dependent properties
[67, 68]. Consequently, rubber friction does obey the Williams-Landel-Ferry
relation, typically associated with viscoelastic materials [69].

### 5.3 Other nanoscale models for Schallamach-type waves

While interface dynamics accompanying Schallamach waves is certainly
describable at the continuum scale as discussed in Sec. 4, a number of
interesting phenomena arise when a general microscopic picture of interface
friction is considered. Such a description for polymers was, unsurprisingly,
put forth by Schallamach himself in a lesser known paper from 1963 [70]. This
paper was focused on the dynamics of individual bonds at the interface, and
did not consider collective effects between several bonds at once. The latter
is the subject of a well-known model propounded by Prandtl and Tomlinson in
the late 1920s [71, 72].

Subsequently, several microscopic models have emerged, each treating bonds
between the sliding bodies at the interface in a statistical manner. For
instance, the simplest model consists of a network of horizontal and vertical
springs that interconnect a set of rigid sliders driven by an external force.
Models such as these show traveling solitary wave solutions that, at first
glance, are remarkably reminiscent of Schallamach-type waves [73]. However, a
closer look reveals that they do not explicitly account for large interface
detachment nor do they result in interface slip. Yet, these models have found
much favour within the community, likely because they are more easily amenable
to incorporating viscous damping/dissipation effects and that their reduced
dimensionality (most are quasi 1D) allows easier multi-scale analyses and
numerical solution [74]. Complexity in this approach may be introduced by
altering the nature of the springs (thereby the interaction) between the rigid
sliders to more closely represent molecular interactions [75, 76].
Unfortunately, just as with the continuum models of Sec. 4, this often comes
at the expense of tractability. Evaluating the consequences of these models
constitutes an active area of research.

The generality of this microscopic approach reveals a fundamental kinship
between the Prandtl-Tomlinson approach and two other famous discrete
models—the Frenkel-Kontorova (FK) and Burridge-Knopoff (BK) models—that have
their origins in crystal physics and tectonic fault dynamics [77, 12]. Unlike
the continuum elastic theories described in Sec. 4, statistical models of the
FK/BK type are inherently nonlinear. They consequently show very rich
dynamics, such as traveling topological defects (kinks), localized breathers
and even discrete effects [78, 13, 79]. Whether or not these solutions share
similarities with Schallamach-type waves will only be revealed by additional
experiments. We believe that the solitary Schallamach wave will play a key
role in this regard.

The microscopic approach described in this section certainly has its merits,
in that it can be applied to systems as diverse as tectonic faults and crystal
lattices. Coupled with the experimental observations and the continuum–scale
descriptions discussed in the previous section, interesting analogies between
Schallamach-type waves and other traveling wave phenomena in these diverse
systems begin to emerge.

Likewise, the addition of non-linearity—both material and geometric—to the
governing equations results in the emergence of novel wave properties [39].
When the governing wave equations are dispersive and non-linear, the
possibility of the two effects exactly balancing results in the propagation of
special particle–like waves called solitons [80]. In addition to the linear
elastic waves discussed earlier, solitons can also retain their shape while
propagating over long distances. However, the importance of nonlinear effects
has not yet been firmly established and warrants additional detailed
experiments.

### 5.4 Exploiting the fracture and dislocation analogies for Schallamach
waves

The wave-dislocation analogy raises the intriguing possibility of using
Schallamach-type waves as a model to study dynamic dislocation behavior. For
instance, one can contrive the use of interacting Schallamach waves to study
time-dependent nanoscale phenomena such as dislocation coalescence. The
existence of opposite moving detachment waves can then be used to ‘simulate’
pairs of like and unlike dislocations. At present, questions such as these
have only been treated using energetics [81] or large scale molecular dynamics
simulations with significant experimental uncertainty.

Just as with the dislocation picture, the close fracture correspondence
presented in Sec. 4.2 affords the possibility of using Schallamach-type waves
to study fracture problems and the transition from stick–slip to steady
sliding. For instance, how do moving rupture fronts interact with crack pairs?
One can also envisage the mechanical/frictional equivalent of the well-known
Rice-Thomson model of dislocation emission from crack tips [82]. If a moving
rupture front delineating stationary and slipped zones propagates through the
interface, can it become diffuse by interacting with a Schallamach wave? How
does this behaviour change between the Schallamach wave and its dual? Finally,
the length scale $\Delta x_{c}$ introduced in Fig. 12 places bounds on the
smallest Schallamach-type waves that may be observed in any system. For
elastomers, we find that $\Delta x_{c}\sim 100$ nm which raises the question
of whether nanoscale Schallamach waves can exist in any system. If so,
detailed observations of their propagation properties could help explain
stick–slip phenomena in a much wider variety of systems.

We believe that these fascinating questions can significantly enhance our
understanding of nanoscale phenomena using insights from seemingly unrelated
Schallamach-type waves. We end by quoting Schallach once more [1]

> The discussion of our experimental observations has been mostly conjecture,
> but may nevertheless suggest new approaches to the study of rubber friction.

Little did Schallamach expect that his insights have helped open doors to
linking several disparate fields together. We have every reason to believe
that the legacy of Schallamach’s discovery will endure for another fifty
years.

## 6 Summary and future outlook

As we said at the beginning of this article, Schallamach’s humble question has
opened far reaching links between rubber friction and phenomena in domains as
diverse as biological locomotion, nanoscale mechanics and interface fracture.
Let us first attempt a summary of the salient features. We’ve seen how
Schallamach waves nucleate due to a surface buckling instability, and how they
subsequently propagate through the interface at speeds much lower than any
elastic wave speed. Schallamach’s observations, followed by several others,
were all made using a spherical indenter geometry. Changing from this geometry
to a cylindrical one has allowed the isolation of single or solitary
Schallamach waves. Not only do these waves offer a clear quantitative picture
of propagation dynamics, they also reveal remarkable similarities with
propagating waves in other systems, e.g., biological locomotion. A second
major advantage of the cylindrical geometry is that it enables observation of
another slow moving detachment wave—the separation pulse or the dual of the
Schallamach wave. Together, the Schallamach wave and its dual represent the
only two possibile mechanisms for waves of detachment at soft adhesive
interfaces. A final detachment-less wave is also observed—the slip pulse—which
completed our trio of Schallamach-type waves. We noted that there are exact
analogues of these Schallamach–type waves in biological locomotion. Based on
the two key features of Schallamach–type waves, _viz._ that they move at slow
speed and result in constant slip in thier wake, we explored possible
continuum theories for explaining their propagation. The failure of
conventional elastodynamic models to explain both key features was
remedied—for the Schallamach wave and its dual atleast—by the use of a model-
free mixed boundary value problem framework. This approach has helped bridge
the gap between theory and experiment, but left several questions unanswered.
Primarily among these is the role of viscoelasticity—that appears to
completely obliterate any signs of a closed-form solution. There remains much
to be done here to explain the role of key phenomena such as the observed
Williams-Landel-Ferry relationship in rubber friction as well as the origin of
contact relaxation effects.

Moving onto smaller scales, we deliberated on the rather intimate link between
friction phenomena in general and nanoscale mechanics. Two noteworthy examples
are the correspondence between Schallamach-type waves and interface crack
pairs, and the dislocation analogy. This then raises the natural question of
whether Schallamach-type waves can be used to study the dynamics of nanoscale
mechanics and fracture phenomena. It is presently standard practice to treat
these problems using energetics, so that even rudimentary understanding of
their time-dependence has remained elusive. Little did Schallamach know that
the waves that are now named after him, could be the bridge that connects such
diverse areas. There is much yet to be unearthed in this matter and we hope
that a reader of this article will fire the first shots.

## Acknowledgements

We would like to thank Professor Kevin Kendall, FRS, for his careful reading
of a draft version of this paper, and for his constructive comments and
criticisms. This work was supported in part by a grant from the Indian
Institute of Science, Bangalore (KV) and NSF awards DMR 2104745 and CMMI
2100568 (SC)

## Appendix: A primer on elastic surface waves

The dynamic behaviour of isotropic linear elastic materials is determined by
two characteristic material velocities—the shear $c_{T}$ and longitudinal
$c_{L}$ wave speeds. The displacement field $\vec{u}$ in an elastic body is
determined as a suitable linear combination of an irrotational component
$\vec{u}_{L}$ and an incompressible component $\vec{u}_{T}$, each obeying a
linear wave equation [43]

$\displaystyle\frac{\partial\vec{u}_{L}}{\partial t^{2}}$
$\displaystyle=c_{L}^{2}\nabla^{2}\vec{u}_{L}\quad\quad\quad\nabla\times\vec{u}_{L}=0$
(12) $\displaystyle\frac{\partial\vec{u}_{T}}{\partial t^{2}}$
$\displaystyle=c_{T}^{2}\nabla^{2}\vec{u}_{T}\quad\quad\quad\nabla\cdot\vec{u}_{T}=0$

These equations must be supplemented by boundary conditions (BCs) and initial
conditions (ICs) pertinent to the particular physical problem. The BCs could
include specification of either the stresses/displacements or both on part of
or the entire surface of the elastic body. Unlike surface waves, longitudinal
and transverse waves desribed by $\vec{u}_{L},\vec{u}_{T}$ travel in the bulk
of the material (infinite space). Notably, $\vec{u}_{L},\vec{u}_{T}$ waves are
non-dispersive—their propagation speed does not depend on the wavelength.
Specifically, linear, non-dispersive waves can propagate without changing
their shape whatsoever [39]. PDEs such as Eq. 12 govern only wave propagation
and say nothing about how or why they must nucleate in the first place. For
the elastic polymer in the experiments of Sec. 3, the wave speeds
$c_{L},c_{T}\sim 100$ m/s.

The waves that can then propagate at or near an interface or free surface are
determined by specific boundary conditions. The simplest surface wave is the
Rayleigh wave and it occurs on a stress free surface of an elastic solid [83].
Consider a semi-infinite elastic solid $z\leq 0$ with the $z=0$ plane as its
bounding surface. These waves are exponentially damped into the solid $z<0$
and propagate on the surface $z=0$ at a fixed velocity $c_{R}<c_{T}<c_{L}$,
that is determined by the material’s Poisson’s ratio. Rayleigh waves are well
known free surface waves that usually accompany earthquakes; they arrive after
the first acoustic signals (since $c_{R}<c_{T},c_{L}$) and are non–dispersive.

In contrast, when two elastic materials are brought into contact, a Rayleigh
wave cannot propagate at the interface, because the surface is no longer
stress–free. On the contrary, if the two solids have continuous displacements
and stresses at the interface (so-called ‘welded contact’), a Stoneley wave
can propagate instead [84]. The existence of Stoneley waves is tied to the
elastic properties of the two contacting materials. However, when they do
exist, these waves propagate at a fixed velocity, which is a function of the
two sets of elastic properties. Just like Rayleigh waves, Stoneley waves are
also non–dispersive.

If the ‘welded’ contact described above is replaced by a frictionless contact
$(\sigma_{xz}=\sigma_{yz}=0$), then an Achenbach–Epstein (AE) wave or
generalized Rayleigh wave can propagate instead of Stoneley waves [85]. The
requisite boundary conditions now involve continuous displacements across the
interface, equal normal forces and zero tangential forces on both bodies. The
existence of AE waves at an interface between two solids depends on the
elastic properties of the two materials. Like both Rayleigh and Stoneley
waves, AE waves also propagate at a fixed velocity, dependent on the two sets
of material elastic properties, and are non–dispersive.

## References

  * [1] Schallamach A.  How does rubber slide?  Wear. 1971;17(4):301–312.
  * [2] Barquins M.  Friction and wear of rubber-like materials.  Wear. 1993;160(1):1–11.
  * [3] Bowden FP, Tabor D.  Friction: An Introduction to Tribology.  RE Krieger Publishing Company; 1973.
  * [4] Vella D, Boudaoud A, Adda-Bedia M.  Statics and inertial dynamics of a ruck in a rug.  Physical Review Letters. 2009;103(17):174301.
  * [5] Lomer WM.  Blowing bubbles with Bragg.  In: Selections and Reflections: The Legacy of Sir Lawrence Bragg. Science Reviews Limited; 1990. p. 115–118.
  * [6] Gerde E, Marder M.  Friction and fracture.  Nature. 2001;413(6853):285–288.
  * [7] Scholz CH.  Earthquakes and friction laws.  Nature. 1998;391(6662):37–42.
  * [8] Heaton TH.  Evidence for and implications of self-healing pulses of slip in earthquake rupture.  Physics of the Earth and Planetary Interiors. 1990;64(1):1–20.
  * [9] Gray J, Lissmann H.  Studies In animal locomotion: VII. Locomotory reflexes in the earthworm.  Journal of Experimental Biology. 1938;15(4):506–517.
  * [10] Trueman ER.  Locomotion of Soft-Bodied Animals.  Elsevier; 1975.
  * [11] Lai JH, del Alamo JC, Rodríguez-Rodríguez J, Lasheras JC.  The mechanics of the adhesive locomotion of terrestrial gastropods.  Journal of Experimental Biology. 2010;213(22):3920–3933.
  * [12] Burridge R, Knopoff L.  Model and theoretical seismicity.  Bulletin of the Seismological Society of America. 1967;57(3):341–371.
  * [13] Cartwright JH, Hernández-García E, Piro O.  Burridge-Knopoff models as elastic excitable media.  Physical Review Letters. 1997;79(3):527.
  * [14] Viswanathan K, Sundaram NK.  Distinct stick-slip modes in adhesive polymer interfaces.  Wear. 2017;376:1271–1278.
  * [15] Kendall K.  Preparation and properties of rubber dislocations.  Nature. 1976;261(5555):35–36.
  * [16] Viswanathan K, Mahato A, Chandrasekar S.  Nucleation and propagation of solitary Schallamach waves.  Physical Review E. 2015;91(1):012408.
  * [17] Barquins M.  Sliding friction of rubber and Schallamach waves — a review.  Materials Science and Engineering. 1985;73:45–63.
  * [18] Johnson KL.  Contact Mechanics.  Cambridge University Press; 1987.
  * [19] Johnson KL, Kendall K, Roberts AD.  Surface energy and the contact of elastic solids.  Proceedings of the Royal Society of London A Mathematical and Physical Sciences. 1971;324(1558):301–313.
  * [20] Maugis D, Barquins M.  Fracture mechanics and adherence of viscoelastic solids.  In: Adhesion and Adsorption of Polymers. Springer; 1980. p. 203–277.
  * [21] Rabinowicz E.  Stick and slip.  Scientific American. 1956;194:109–119.
  * [22] Ruina A.  Slip instability and state variable friction laws.  Journal of Geophysical Research: Solid Earth (1978–2012). 1983;88(B12):10359–10370.
  * [23] Koudine AA, Barquins M.  Formation of micro-ridges on the surface of Schallamach waves propagating in the contact area between a moving rubber sample and a glass lens.  Journal of Adhesion Science and Technology. 1996;10(10):951–961.
  * [24] Biot MA.  Surface instability of rubber in compression.  Applied Scientific Research, Section A. 1963;12(2):168–182.
  * [25] Groenewold J.  Wrinkling of plates coupled with soft elastic media.  Physica A: Statistical Mechanics and its Applications. 2001;298(1):32–45.
  * [26] Brau F, Damman P, Diamant H, Witten TA.  Wrinkle to fold transition: influence of the substrate response.  Soft Matter. 2013;9(34):8177–8186.
  * [27] Cushman JH.  Molecular-scale lubrication.  Nature. 1990;347(6290):227–228.
  * [28] Wu-Bavouzet F, Clain-Burckbuchler J, Buguin A, De Gennes PG, Brochard-Wyart F.  Stick-slip: Wet versus dry.  The Journal of Adhesion. 2007;83(8):761–784.
  * [29] Martin A, Clain J, Buguin A, Brochard-Wyart F.  Wetting transitions at soft, sliding interfaces.  Physical Review E. 2002;65(3):031605.
  * [30] Best B, Meijers P, Savkoor AR.  The formation of Schallamach waves.  Wear. 1981;65(3):385–396.
  * [31] Rand CJ, Crosby AJ.  Insight into the periodicity of Schallamach waves in soft material friction.  Applied Physics Letters. 2006;89(26):261907.
  * [32] Fukahori Y, Gabriel P, Busfield JJC.  How does rubber truly slide between Schallamach waves and stick–slip motion?  Wear. 2010;269(11):854–866.
  * [33] Yamaguchi T, Morishita M, Doi M, Hori T, Sakaguchi H, Ampuero JP.  Gutenberg-Richter’s law in sliding friction of gels.  Journal of Geophysical Research: Solid Earth. 2011;116(B12).
  * [34] Viswanathan K, Sundaram NK, Chandrasekar S.  Stick-slip at soft adhesive interfaces mediated by slow frictional waves.  Soft Matter. 2016;12(24):5265–5275.
  * [35] Mindlin RD.  Compliance of elastic bodies in contact.  Journal of Applied Mechanics. 1949;16.
  * [36] Cattaneo C.  Sul contatto di due corpi elastici: distribuzione locale degli sforzi.  Rend Accad Naz Lincei. 1938;27(6):342–348.
  * [37] Savkoor A, Briggs G.  The effect of tangential force on the contact of elastic solids in adhesion.  Proceedings of the Royal Society of London: A Mathematical and Physical Sciences. 1977;356(1684):103–114.
  * [38] Viswanathan K, Sundaram NK, Chandrasekar S.  Slow wave propagation in soft adhesive interfaces.  Soft Matter. 2016;12(45):9185–9201.
  * [39] Whitham G.  Linear and Nonlinear Waves.  North Holland; 1975.
  * [40] Baumberger T, Caroli C, Ronsin O.  Self-healing slip pulses along a gel/glass interface.  Physical Review Letters. 2002;88(7):075509.
  * [41] Caroli C.  Slip pulses at a sheared frictional viscoelastic/nondeformable interface.  Physical Review E. 2000;62(2):1729.
  * [42] Nabarro FRN.  Theory of Crystal Dislocations.  Oxford, Clarendon Press; 1967.
  * [43] Achenbach JD.  Wave propagation in elastic solids.  Elsevier; 2012.
  * [44] Comninou M, Dundurs J.  Elastic interface waves involving separation.  Journal of Applied Mechanics. 1977;44(2):222–226.
  * [45] Comninou M, Dundurs J.  Elastic interface waves and sliding between two solids.  Journal of Applied Mechanics. 1978;45(2):325–330.
  * [46] Martins JAC, Guimarães J, Faria LO.  Dynamic surface solutions in linear elasticity and viscoelasticity with frictional boundary conditions.  Journal of Vibration and Acoustics. 1995;117(4):445–451.
  * [47] Adams GG.  Steady sliding of two elastic half-spaces with friction reduction due to interface stick-slip.  Journal of Applied Mechanics. 1998;65(2):470–475.
  * [48] Ranjith K, Rice JR.  Slip dynamics at an interface between dissimilar materials.  Journal of the Mechanics and Physics of Solids. 2001;49(2):341–361.
  * [49] Dieterich JH.  Modeling of rock friction: 1. Experimental results and constitutive equations.  Journal of Geophysical Research: Solid Earth (1978–2012). 1979;84(B5):2161–2168.
  * [50] Li Q, Tullis TE, Goldsby D, Carpick RW.  Frictional ageing from interfacial bonding and the origins of rate and state friction.  Nature. 2011;480(7376):233–236.
  * [51] Cochard A, Rice JR.  Fault rupture between dissimilar materials- Ill-posedness, regularization, and slip-pulse response.  Journal of Geophysical Research. 2000;105:25.
  * [52] Baumberger T, Caroli C.  Solid friction from stick–slip down to pinning and aging.  Advances in Physics. 2006;55(3-4):279–348.
  * [53] Brener EA, Malinin S, Marchenko V.  Fracture and friction: Stick-slip motion.  The European Physical Journal E. 2005;17(1):101–113.
  * [54] Ansari MA, Viswanathan K.  Propagating Schallamach-type waves resemble interface cracks.  Physical Review E. 2022;105:045002.
  * [55] Svetlizky I, Kammer DS, Bayart E, Cohen G, Fineberg J.  Brittle fracture theory predicts the equation of motion of frictional rupture fronts.  Physical Review Letters. 2017;118(12):125501.
  * [56] Kendall K.  Energizing ASTM lap joint fracture standards.  Philosophical Transactions of the Royal Society A. 2021;379(2203):20200287.
  * [57] Chaudhury MK, Weaver T, Hui CY, Kramer EJ.  Adhesive contact of cylindrical lens and a flat sheet.  Journal of Applied Physics. 1996;80(1):30–37.
  * [58] Kendall K.  Adhesion and composites.  Composite Interfaces. 1996;4(5):299–311.
  * [59] Kendall K.  Interfacial dislocations spontaneously created by peeling.(Adhesive joint strength).  Journal of Physics D: Applied Physics. 1978;11(11):1519.
  * [60] Gittus JH.  Interfacial dislocations in frictional sliding and interfacial creep: the theory of interfaceons.  Philosophical Magazine. 1975;31(2):317–329.
  * [61] Gray J.  Animal Locomotion.  Weidenfeld & Nicolson, London; 1968.
  * [62] Chapman G.  The hydrostatic skeleton in the invertebrates.  Biological reviews. 1958;33(3):338–371.
  * [63] Wells GP.  The mechanism of proboscis movement in _Arenicola_.  Quarterly Journal of Microscopical Science. 1954;3(30):251–270.
  * [64] Orowan E.  Dislocations and mechanical properties.  In: Cohen M, editor. Dislocations in Metals. The American Institute of Mining and Metallurgical Engineers, Inc., NY; 1954. p. 69–195.
  * [65] Currie P, Hayes M, O’Leary P.  Viscoelastic rayleigh waves.  Quarterly of Applied Mathematics. 1977;35(1):35–53.
  * [66] Barquins M, Courtel R.  Rubber friction and the rheology of viscoelastic contact.  Wear. 1975;32(2):133–150.
  * [67] Roberts A, Thomas A.  The adhesion and friction of smooth rubber surfaces.  Wear. 1975;33(1):45–64.
  * [68] Roberts A, Jackson S.  Sliding friction of rubber.  Nature. 1975;257(5522):118–120.
  * [69] Grosch K.  The relation between the friction and visco-elastic properties of rubber.  Proceedings of the Royal Society of London Series A Mathematical and Physical Sciences. 1963;274(1356):21–39.
  * [70] Schallamach A.  A theory of dynamic rubber friction.  Wear. 1963;6(5):375–382.
  * [71] Tomlinson G.  CVI. A molecular theory of friction.  The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 1929;7(46):905–939.
  * [72] Prandtl L.  Ein Gedankenmodell zur kinetischen Theorie der festen Körper.  ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik. 1928;8(2):85–106.
  * [73] Braun OM, Peyrard M.  Crack in the frictional interface as a solitary wave.  Physical Review E. 2012;85(2):026111.
  * [74] Vanossi A, Manini N, Urbakh M, Zapperi S, Tosatti E.  Colloquium: Modeling friction: From nanoscale to mesoscale.  Reviews of Modern Physics. 2013;85(2):529.
  * [75] Filippov A, Klafter J, Urbakh M.  Friction through dynamical formation and rupture of molecular bonds.  Physical Review Letters. 2004;92(13):135503.
  * [76] Chernyak YB, Leonov A.  On the theory of the adhesive friction of elastomers.  Wear. 1986;108(2):105–138.
  * [77] Frenkel YI, Kontorova T.  The model of dislocation in solid body.  Zh Eksp Teor Fiz. 1938;8(1340).
  * [78] Braun OM, Kivshar YS.  Nonlinear dynamics of the Frenkel–Kontorova model.  Physics Reports. 1998;306(1-2):1–108.
  * [79] Muratov C.  Traveling wave solutions in the Burridge-Knopoff model.  Physical Review E. 1999;59(4):3847.
  * [80] Newell AC.  Solitons in Mathematics and Physics.  SIAM; 1985.
  * [81] Weertman J.  Coalescence energy of two moving dislocations of like sign.  Journal of Applied Physics. 1966;37(13):4925–4927.
  * [82] Rice JR, Thomson R.  Ductile versus brittle behaviour of crystals.  The Philosophical Magazine: A Journal of Theoretical Experimental and Applied Physics. 1974;29(1):73–97.
  * [83] Strutt JW.  On waves propagated along the plane surface of an elastic solid.  Proceedings of the London Mathematical Society. 1885;17(1):4–11.
  * [84] Stoneley R.  Elastic waves at the surface of separation of two solids.  Proceedings of the Royal Society of London Series A, Containing Papers of a Mathematical and Physical Character. 1924;p. 416–428.
  * [85] Achenbach JD, Epstein HI.  Dynamic interaction of a layer and a half-space.  Journal of the Engineering Mechanics Division. 1967;93(5):27–42.