2021

[1]Wataru Iwashita

[1]Department of Mechanical Science and Bioengineering, Osaka University, 1-3
Machikaneyama, Toyonaka, 560-8531, Osaka, Japan

2]Department of Physical Sciences, Aoyama Gakuin University, 5-10-1 Fuchinobe,
Sagamihara, 252-5258, Kanagawa, Japan

# Control of Static Friction by Designing Grooves on Friction Surface

<EMAIL_ADDRESS>Hiroshi Matsukawa  Michio Otsuki  *  [

###### Abstract

This study numerically investigated the friction of viscoelastic objects with
grooves. A 3D viscoelastic block with grooves on a rigid substrate is slowly
pushed from the lateral side under uniform pressure on the top surface. The
local friction force at the interface between the block and the substrate
obeys Amontons’ law. Numerical results obtained using the finite element
method reveal that the static friction coefficient decreases with increasing
groove width and depth. The propagation of the precursor slip is observed
before bulk sliding. Furthermore, bulk sliding occurs when the area of slow
precursor slip reaches a critical value, which decreases with increasing
groove size. A theoretical analysis based on a simplified model reveals that
the static friction coefficient is related to the critical area of the
precursor, which is determined by the instability of the precursor. A scaling
law for the critical area is theoretically predicted, and it indicates that
the decrease in the effective viscosity due to the formation of the grooves
leads to a decrease in the static friction coefficient. The validity of the
theoretical prediction is numerically confirmed.

###### keywords:

Static friction coefficient, Groove design, Precursor slip, Amontons’ law,
Viscoelastic object

### Graphical Abstract

##  1 Introduction

Friction forces occur in different situations, such as sliding parts of
machines and contact surfaces between tires and the ground, and prevent the
relative motion between two objects in contact. Friction forces are desirable
in applications requiring low slippage, whereas they are undesirable in
sliding parts of machines due to energy loss. Therefore, the control of
friction forces is important in engineering Bowden1950 ; Persson2000 ;
Popov2017_text ; Rabinowicz1995 ; Dowson1998 ; Bhushan2013 ; Baumberger2006 .
One of the known methods to control friction forces is designing friction
surfaces by forming grooves. Generally, grooves on the surfaces of tires and
sliding parts of machines are formed to reduce undesirable lubrication in wet
conditions, which leads to a decrease in the friction coefficient and results
in accidental slippage Li2004 ; Li2005 ; Li2006 ; Yamaguchi2012 . However, the
dependence of friction force on grooves in dry conditions has not been
established clearly.

Generally, for friction between solids in dry conditions, Amontons’ law is
expected to hold Bowden1950 ; Persson2000 ; Popov2017_text ; Rabinowicz1995 ;
Dowson1998 ; Bhushan2013 ; Baumberger2006 . According to Amontons’ law, the
friction coefficient does not depend on the external pressure or size and
shape of the object. However, the phenomenological explanation of Amontons’
law is based on the adhesion of microscopic asperities at the friction
interface Bowden1950 ; Persson2000 ; Popov2017_text ; Rabinowicz1995 ;
Dowson1998 ; Bhushan2013 ; Baumberger2006 ; Archard1957 ; Dieterich1996 , and
implicitly assumes the uniformity of the stress field. For macroscopic objects
associated with the non-uniform stress field, Amontons’ law may not hold.
Therefore, the friction coefficient may depend on the shape of the macroscopic
objects in dry conditions.

In fact, recent studies on the friction of objects with flat friction surfaces
have shown that Amontons’ law is not satisfied when the non-uniformity of the
stress field is significant Bouissou1998 ; Ben-David2011 ; Otsuki2013 ;
Katano2014 ; IwashitaSciRep2023 . In Refs. 16; 18, numerical simulations and
analysis of a simplified model have revealed the mechanism of breakdown of
Amontons’ law in viscoelastic materials. The analysis clarified that the local
precursor slip before bulk sliding due to the non-uniform stress field leads
to the breakdown of Amontons’ law, and that the static friction coefficient
exhibits characteristic load dependence. The relationship between the
precursor slip and breakdown of Amontons’ law and the load dependence of the
static friction coefficient have been verified in experiments on acrylic glass
blocks Katano2014 . Precursor slip relates to earthquake Yamaguchi2011 ;
Obara2016 ; Kato2021 ; Petrillo2020 ; Xu2023  and fracture Svetlizky2014 ;
Bayart2016 ; Svetlizky2017 ; Berman2020 ; Gvirtzman2021 ; Kammer2015 ;
Castellano2023 , and has been extensively studied in experiments Katano2014 ;
Yamaguchi2011 ; Xu2023 ; Rubinstein2007 ; Ben-David2011 ; Svetlizky2014 ;
Bayart2016 ; Svetlizky2017 ; Berman2020 ; Gvirtzman2021 ; Maegawa2010  and
numerical simulations Otsuki2013 ; IwashitaSciRep2023 ; Petrillo2020 ;
Maegawa2010 ; Burridge1967 ; deSousa1992 ; Braun2009 ; Capozza2012 ; Ozaki2014
; Scheibert2010 ; Amundsen2012 ; Tromborg2014 ; Radiguet2013 ; Kammer2015 ;
Castellano2023 ; Albertini2021 ; Taloni2015 ; deGeus2019 . However, many of
these studies have considered only flat friction surfaces, and the effects of
grooves in the friction surface on frictional properties and precursor slip
are yet to be discovered.

Recently, several studies have been conducted to reveal the dependence of the
friction coefficient on the macroscopic shape of the friction surface. In
Refs. 45; 46; 47; 48; 49; 50, shapes of friction surfaces were represented by
a spatial dependence of the local friction coefficient in 1D or 2D spring-
block models. The frictional properties of the models vary with the spatial
pattern of the local friction coefficient Capozza2015 ; Costagliola2016 ;
Costagliola2017 ; Costagliola2018 ; Costagliola2022IJSS ; Maegawa2017 , and
these results have been applied to experiments of macroscopic objects
Berardo2019 ; Balestra2022 . However, it is unclear to what extent the results
of the spring-block models with a spatial pattern of local friction
coefficient reflect the effect of the actual surface shape. Experiments with
rubber and gel blocks have also revealed the dependence of the friction
coefficient on the macroscopic shape of the friction surface Maegawa2016 ;
Gao2023 . However, it is unclear whether the results for relatively soft
objects such as rubber and gel can be applied to harder materials where
Amontons’ law is locally satisfied.

In this study, using the finite element method (FEM), we numerically
investigate the friction of a 3D viscoelastic material with grooves in a dry
condition, where the friction force locally obeys Amontons’ law. The
dependence of the static friction coefficient on the groove shape is
investigated. We find that the static friction coefficient is a decreasing
function of the groove width and depth. We also observe that local precursor
slip occurs before bulk sliding of the viscoelastic material. The bulk sliding
occurs when the area of the precursor slip reaches a critical value. The
static friction coefficient is scaled by the normalized critical area of the
precursor slip. The propagation of the precursor slip is analytically studied
based on a simplified model. We derive the conditions for the onset of bulk
sliding and the dependence of the static friction coefficient on the groove
shape. The results show that the static friction coefficient decreases due to
the decrease in effective viscosity as the groove width and depth increase.
Both pillars in the friction surface and main body supporting them play an
important role. Our results aid in the improvement of sliding interface design
by making grooves for both wet and dry conditions.

##  2 Model and Methods

Figure 1:  Schematic of the system. (a) Grooved viscoelastic block moving on a
rigid substrate. (b) Cross-section perpendicular to the $y$ direction
indicated as I in (a). (c) The bottom of the block indicated as II in (a). The
blue region represents the contact area between the rigid substrate and the
block.

We consider grooved viscoelastic blocks on a rigid substrate under a uniform
external pressure $P_{\mathrm{ext}}$ with width $W$, length $L$, and height
$H$ along the $x$, $y$, and $z$ axes, respectively, as shown in Fig. 1. The
rigid substrate is at $z=0$. A rigid plate with a width of $W$ and a height of
$0.5H$ pushes the side of the block at $y=0$ and $0.5H\leq z\leq H$ with a
slow constant velocity $V$ along the $y$ direction. This study considers a
longitudinal groove parallel to the $y$ direction, as shown in Fig. 1. The
number of pillars in the friction surface of the block is denoted by $n_{x}$.
The pillars are equally spaced with width $l_{\mathrm{g}}$. The height and
width of the pillar are denoted by $d$ and $W/n_{x}-l_{\mathrm{g}}$,
respectively. The cross-section perpendicular to the $y$ direction is
symmetrical, as shown in Fig. 1b. The pillar is in contact with the rigid
substrate, as shown in Fig. 1c. The ratio $\phi$ of the area of the non-
contact surface to the area of the bottom surface is given by
$\phi=l_{\mathrm{g}}n_{x}/W$, and the contact area of the friction surface is
given by $LW(1-\phi)$. Here, $\phi=0$ corresponds to a rectangular block
without grooves.

The equation of motion for the viscoelastic object is given by

$\rho\ddot{u}_{i}=\sum_{j}\partial_{j}\sigma_{ij}$ (1)

with density $\rho$, displacement vector $\bm{u}$, and stress tensor
$\bm{\sigma}$; where $\sigma_{ij}$ is the $ij$ component of $\bm{\sigma}$,
${u}_{i}$ is the $i$ component of $\bm{u}$, and $\ddot{u}_{i}$ is its second-
order time derivative. The stress $\bm{\sigma}$ is given by
$\bm{\sigma}=\bm{\sigma}^{\mathrm{(E)}}+\bm{\sigma}^{\mathrm{(V)}}$ with the
elastic stress $\bm{\sigma}^{\mathrm{(E)}}$ obeying Hooke’s law and the
viscous stress $\bm{\sigma}^{\mathrm{(V)}}$ proportional to strain rate. We
assume that the viscoelastic material of the block is isotropic. The $ij$
component of the elastic stress tensor $\sigma^{(\mathrm{E})}_{ij}$ is given
by

$\sigma^{\mathrm{(E)}}_{ij}=\frac{E}{1+\nu}\epsilon_{ij}+\frac{\nu
E}{(1+\nu)(1-2\nu)}\sum_{k}\epsilon_{kk}\delta_{ij}$ (2)

with Young’s modulus $E$, Poisson’s ratio $\nu$, Kronecker’s delta
$\delta_{ij}$, and strain tensor $\epsilon_{ij}$. The $ij$ component of the
viscous stress tensor $\sigma^{(\mathrm{V})}_{ij}$ is given by

$\sigma^{\mathrm{(V)}}_{ij}=\eta_{1}\dot{\epsilon}_{ij}+\eta_{2}\sum_{k}\dot{\epsilon}_{kk}\delta_{ij}$
(3)

with the two viscosity coefficients $\eta_{1}$ and $\eta_{2}$ and the strain
rate tensor $\dot{\epsilon}_{ij}$ Landau1986 .

The boundary conditions on the top surface of the block at $z=H$ are given by
$\sigma_{zz}=-P_{\mathrm{ext}}$ and $\sigma_{xz}=\sigma_{yz}=0$. At surfaces
except the top and bottom of the block, free boundary conditions
($\sum_{j}\sigma_{ij}n_{j}=0$) are applied, where $n_{j}$ is the $j$ component
of the normal vector $\bm{n}$ to the surface. The boundary conditions at the
contact surface with the rigid plate at $y=0$ are given by
$\sigma_{xy}=\sigma_{zy}=0$ and $\dot{u}_{y}=V$, where $\dot{u}_{y}$ is the
velocity along the $y$ direction. We set $V$ sufficiently small to push the
block quasi-statically.

The friction between the block bottom and substrate obeys Amontons’ law
locally. Since the substrate is rigid, the $z$-direction displacement $u_{z}$
satisfies $u_{z}\geq 0$. At the bottom, the tangential stress vector
$\bm{t}(x,y)=(\sigma_{xz},\sigma_{yz})$ at the position $(x,y)$ is given by

$\bm{t}=-\frac{\bm{v}}{v}\,\sigma^{\mathrm{(fric)}}\ ,$ (4)
$\sigma^{\mathrm{(fric)}}(x,y)=\mu(v(x,y))\,p(x,y)\ ,$ (5)

where $\sigma^{\mathrm{(fric)}}$ is the frictional stress,
$\bm{v}(x,y)=(\dot{u}_{x},\dot{u}_{y})$ is the slip velocity vector with
velocities along the $i$ direction $\dot{u}_{i}$, and
$v(x,y)=\lvert\bm{v}\rvert$ is the slip velocity ccm2006 . The bottom pressure
$p(x,y)=-\sigma_{zz}(x,y,z=0)$ is set to satisfy $u_{z}\geq 0$, where $p=0$
for $u_{z}>0$. Here, $\mu(v)$ is the local friction coefficient depending on
$v$. In the static region with $v(x,y)=0$, $\mu(v)$ is lower than
$\mu_{\mathrm{S}}$, and set to balance the local internal shear stress with
the frictional stress. In the slip region with $v(x,y)>0$, $\mu(v)$ is given
by

$\mu(v)=\left\\{\begin{array}[]{ll}\mu_{\mathrm{S}}-\left(\mu_{\mathrm{S}}-\mu_{\mathrm{K}}\right)v/v_{\mathrm{c}},{}&0<v<v_{\mathrm{c}}\\\
\mu_{\mathrm{K}},{}&v\geq v_{\mathrm{c}}\end{array}\right.\ ,$ (6)

where $\mu_{\mathrm{S}}$ and $\mu_{\mathrm{K}}$ are the local static and
dynamic friction coefficients, respectively. Here, $v_{\mathrm{c}}$ is the
characteristic velocity. The local Amontons’ law is expected to hold when a
local region considered in the interface contains a sufficiently large number
of real contact points, and has a negligibly small spatial variation in
internal stress Archard1957 ; Dieterich1996 ; Dieterich1994 .

We numerically solve Eq. (1) using FEM. The viscoelastic block is divided into
cubes with length $\Delta x$, comprising six tetrahedrons. The displacements
of its nodes are evolved based on Eq. (1), and the displacement and velocity
within each element are approximated using linear interpolation. The local
friction coefficient $\mu(v)$ is approximately given as

$\mu(v)=\left\\{\begin{array}[]{lll}\mu_{\mathrm{S}}\,v/v_{\mathrm{e}},{}&0\leq
v\leq v_{\mathrm{e}}\\\
\mu_{\mathrm{S}}-\left(\mu_{\mathrm{S}}-\mu_{\mathrm{K}}\right)v/v_{\mathrm{c}},{}&v_{\mathrm{e}}<v<v_{\mathrm{c}}\\\
\mu_{\mathrm{K}},{}&v\geq v_{\mathrm{c}}\end{array}\right.$ (7)

with a sufficiently small velocity scale $v_{\mathrm{e}}$. The state with
$0\leq v\leq v_{\mathrm{e}}$ corresponds to the static region, and the state
with $v>v_{\mathrm{e}}$ corresponds to the slip region. We set
$v_{\mathrm{e}}/V=2.5\times 10^{-2}$ to satisfy $v_{\mathrm{e}}\ll
V,v_{\mathrm{c}}$, and use $\Delta x/H=1/48$, $\Delta
t/(H\sqrt{\rho/E})\thickapprox 10^{-6}$, and $V\sqrt{\rho/E}=2.83\times
10^{-5}$. In our simulation, we first apply a uniform pressure
$P_{\mathrm{ext}}$ to the top surface and relax the system to an equilibrium
state. From the time $t=0$ after the relaxation, the rigid plate pushes the
side of the block with a constant velocity $V$, and the calculation continues
until a periodic stick-slip is observed.

We set the length and width of the block to $L/H=4$ and $W/H=1$, respectively.
Qualitatively similar results are obtained for $L/H=2$, as shown in Appendix
A. We adopt $\nu=0.34$, $\eta_{1}/(H\sqrt{\rho E})=2.83$,
$\eta_{2}/\eta_{1}=1$, $\mu_{\mathrm{S}}=0.38$, $\mu_{\mathrm{K}}=0.1$, and
$v_{\mathrm{c}}\sqrt{\rho/E}=4.81\times 10^{-4}$ following previous
simulations Otsuki2013 ; IwashitaSciRep2023 . We select the number of pillars
in the friction surface as $n_{x}=3$, and confirm that the dependence of the
numerical results on $n_{x}$ is small, as shown in Appendix B. In this study,
we investigate the dependence on the external pressure $P_{\mathrm{ext}}$,
fraction of non-contact area $\phi$, and groove depth $d$.

##  3 Results

###  3.1 Numerical Simulation

Figure 2:  Ratio of friction force $F_{\mathrm{T}}$ to applied normal force
$F_{\mathrm{N}}$ against displacement of the rigid plate $U$ for
$P_{\mathrm{ext}}/E=0.003$ and $d/H=0.5$. The thin and thick solid lines
represent the results for $\phi=0$ and $\phi=0.5$, respectively. The thin and
thick horizontal solid lines represent macroscopic static friction coefficient
$\mu_{\mathrm{M}}$ for $\phi=0$ and $\phi=0.5$, respectively. The dotted and
dashed lines represent $\mu_{\mathrm{S}}$ and $\mu_{\mathrm{K}}$,
respectively.

Figure 2 shows the friction force $F_{\mathrm{T}}$ against the displacement of
the rigid plate $U=Vt$ at time $t$ for $P_{\mathrm{ext}}/E=0.003$ and
$d/H=0.5$. Here, $F_{\mathrm{T}}$ is given by the force on the rigid plate in
the $y$ direction. In Fig. 2, $F_{\mathrm{T}}$ is normalized by the normal
load $F_{\mathrm{N}}=P_{\mathrm{ext}}LW$ applied to the top surface of the
block. The thin and thick solid lines represent the results for $\phi=0$ and
$\phi=0.5$, respectively. For each $\phi$, $F_{\mathrm{T}}/F_{\mathrm{N}}$
increases approximately linearly with $U$, and rapidly decreases after
reaching a peak value. When the rapid decrease occurs, the entire system
slides, and the block returns to a static state after reaching a minimum value
close to the local dynamic friction coefficient $\mu_{\mathrm{K}}$. The
increase and decrease in $F_{\mathrm{T}}/F_{\mathrm{N}}$ repeat periodically,
which corresponds to stick-slip motion. We define the maximum value of
$F_{\mathrm{T}}/F_{\mathrm{N}}$ in the periodic stick-slip region as the
macroscopic static friction coefficient $\mu_{\mathrm{M}}$, which is lower
than the local static friction coefficient $\mu_{\mathrm{S}}$. Figure 2 shows
that $\mu_{\mathrm{M}}$ for the block with grooves is lower than that for the
flat block.

In Fig. 3, we plot the macroscopic static friction coefficient
$\mu_{\mathrm{M}}$ against the groove depth $d$ for different values of $\phi$
with $P_{\mathrm{ext}}/E=0.003$ and $0.006$. Note that the results for
$\phi=0$ are independent of $d$. For each $P_{\mathrm{ext}}$,
$\mu_{\mathrm{M}}$ is a decreasing function of $d$. As $d$ approaches 0,
$\mu_{\mathrm{M}}$ converges to that for $\phi=0$. The macroscopic static
friction coefficient $\mu_{\mathrm{M}}$ is a decreasing function of $\phi$.
These results indicate that the static friction force decreases as the size of
the groove increases. Comparing Fig. 3a and b, we find that $\mu_{\mathrm{M}}$
is a decreasing function of $P_{\mathrm{ext}}$, which is consistent with the
results of previous studies on rectangular blocks without grooves Otsuki2013 ;
Katano2014 ; IwashitaSciRep2023 .

Figure 3:  Macroscopic static friction coefficient $\mu_{\mathrm{M}}$ against
$d$ for different values of $\phi$ with (a) $P_{\mathrm{ext}}/E=0.003$ and (b)
$P_{\mathrm{ext}}/E=0.006$. The dotted and dashed lines represent
$\mu_{\mathrm{S}}$ and $\mu_{\mathrm{K}}$, respectively.  Figure 4:  Spatial
distributions of the slip velocity $v$ in the friction surface at $z=0$ for
$U=U_{1},U_{2},U_{3}$, and $U_{4}$ shown in Fig. 2 for
$P_{\mathrm{ext}}/E=0.003$, $\phi=0.5$, and $d/H=0.5$. The blue area
represents the static region. The yellow-green and yellow areas represent the
slip regions with $v\leq V$ and $v>V$, respectively. The rigid plate pushes
the block at $y=0$. The white area represents the groove region.

In Fig. 4, we present the spatial distributions of the slip velocity $v$ in
the friction surface at $z=0$ for the displacements $U=U_{1},U_{2},U_{3}$, and
$U_{4}$ shown in Fig. 2 with $P_{\mathrm{ext}}/E=0.003$, $\phi=0.5$, and
$d/H=0.5$. Here, we select $U_{1}/L=4.6\times 10^{-3}$, $U_{2}/L=5\times
10^{-3}$, $U_{3}/L=5.4\times 10^{-3}$, and $U_{4}/L=5.57\times 10^{-3}$ in the
periodic stick–slip region. In Fig. 4, the blue area represents the static
region, and the yellow-green and yellow areas represent the sliding regions
with $v\leq V$ and $v>V$, respectively. The quasi-static precursor slip with
$v\leq V$ begins to propagate from the region near the rigid plate at $y=0$
for $U=U_{1}$, and the area of precursor slip expands quasi-statically as $U$
increases to $U_{2}$ and $U_{3}$. After $U_{3}$, the area of precursor slip
develops rapidly, and the entire system begins to slide, leading to bulk
sliding at $U_{4}$ (see Supplementary Videos). During bulk sliding, the slip
velocity $v$ exceeds $V$. We confirm that the displacement due to these slips
is approximately along the $y$ direction for all parameters.

Figure 5:  Normalized precursor slip area $\tilde{S}$ against $U/L$ for
$P_{\mathrm{ext}}/E=0.003$ and $d/H=0.5$. The thin and thick lines represent
the results for $\phi=0$ and $\phi=0.5$, respectively. The thin and thick
horizontal lines represent the normalized critical area of precursor slip
$\tilde{S}_{\mathrm{c}}$ for $\phi=0$ and $\phi=0.5$, respectively.

Figure 5 shows the normalized slip area $\tilde{S}=S/[LW(1-\phi)]$ against
$U/L$ for $P_{\mathrm{ext}}/E=0.003$ and $d/H=0.5$. Here, the precursor slip
area $S$, defined by the sum of the yellow-green and yellow areas in Fig. 4,
is normalized by the contact area in friction surface $LW(1-\phi)$. When
$\tilde{S}=0$, the entire friction surface is static, while $\tilde{S}=1$
indicates bulk sliding where the entire friction surface is sliding. The thin
and thick solid lines represent the results for $\phi=0$ and $\phi=0.5$,
respectively. The normalized precursor slip area $\tilde{S}$ increases
gradually with $U$. When $\tilde{S}$ reaches a threshold value
$\tilde{S}_{\mathrm{c}}$, the propagation speed of $\tilde{S}$ suddenly
increases, and $\tilde{S}$ reaches unity, which corresponds to the bulk
sliding. The oscillation of $\tilde{S}$ in Fig. 5 and small drops of
$F_{\mathrm{T}}/F_{\mathrm{N}}$ in Fig. 2 before bulk sliding is caused by the
sequence of the bounded rapid precursors (BRPs) Otsuki2013 . Each BRP reduces
the stress and $\tilde{S}$, but they both recover quickly due to a slight
increase in the driving force. The BRP becomes significant depending on the
values of the parameters. We evaluate the critical area
$\tilde{S}_{\mathrm{c}}$ as the maximum value of $\tilde{S}$ in the sequence
of the BRP. Figure 5 shows that $\tilde{S}_{\mathrm{c}}$ decreases with
increasing $\phi$.

Figure 6:  Normalized critical area of precursor slip $\tilde{S}_{\mathrm{c}}$
against $d$ for different values of $\phi$ with (a) $P_{\mathrm{ext}}/E=0.003$
and (b) $P_{\mathrm{ext}}/E=0.006$.

In Fig. 6, we show the normalized critical area of the precursor slip
$\tilde{S}_{\mathrm{c}}$ against the groove depth $d$ for different values of
$\phi$ with $P_{\mathrm{ext}}/E=0.003$ and $0.006$. We find that
$\tilde{S}_{\mathrm{c}}$ is a decreasing function of $d$. As $d$ approaches 0,
$\tilde{S}_{\mathrm{c}}$ approaches that for $\phi=0$. We also find that
$\tilde{S}_{\mathrm{c}}$ decreases with increasing $\phi$. Comparing Fig. 6a
and b, we see that $\tilde{S}_{\mathrm{c}}$ is a decreasing function of
$P_{\mathrm{ext}}$, which is consistent with the results of previous studies
on blocks without grooves Otsuki2013 ; Katano2014 ; IwashitaSciRep2023 .

Figure 7:  Macroscopic static friction coefficient $\mu_{\mathrm{M}}$ against
$\tilde{S}_{\mathrm{c}}$ for different values of $\phi$ and $d$. The filled
and open symbols represent the results for $P_{\mathrm{ext}}/E=0.003$ and
$P_{\mathrm{ext}}/E=0.006$, respectively. The solid line represents the
analytical results given by Eq. (20). The dotted and dashed lines represent
$\mu_{\mathrm{S}}$ and $\mu_{\mathrm{K}}$, respectively.

The dependence of the macroscopic static friction coefficient
$\mu_{\mathrm{M}}$ on $\phi$ and $d$ shown in Fig. 3 is similar to that of
$\tilde{S}_{\mathrm{c}}$ shown in Fig. 6. This similarity indicates a close
relation between $\mu_{\mathrm{M}}$ and $\tilde{S}_{\mathrm{c}}$. In fact, as
shown in Fig. 7, $\mu_{\mathrm{M}}$ is an almost linear function of
$\tilde{S}_{\mathrm{c}}$ for different values of $\phi$ and $d$ with
$P_{\mathrm{ext}}/E=0.003$ and $0.006$. Figure 7 also shows that
$\mu_{\mathrm{M}}$ lies between $\mu_{\mathrm{S}}$ and $\mu_{\mathrm{K}}$.
This scaling of $\mu_{\mathrm{M}}$ using $\tilde{S}_{\mathrm{c}}$ is
consistent with the results of previous studies on blocks without grooves
Otsuki2013 ; Katano2014 ; IwashitaSciRep2023 .

Figure 8:  Spatial distribution of stress in the friction surface at $U=U_{3}$
for $P_{\mathrm{ext}}/E=0.003$, $\phi=0.5$, and $d/H=0.5$. (a) Spatial
distribution of ratio of frictional stress $\sigma^{\mathrm{(fric)}}$ to
bottom pressure $p$. (b) Spatial distribution of $p$. The rigid plate pushes
the block at $y=0$. The white area in (a) represents the region without
contact. The white area in (b) represents the groove area.

Figure 8a shows the spatial distribution of the ratio of the frictional stress
$\sigma^{\mathrm{(fric)}}$ to the bottom pressure $p$ in the friction surface
at $U=U_{3}$ for $P_{\mathrm{ext}}/E=0.003$, $\phi=0.5$ and $d/H=0.5$. Here,
$U=U_{3}$ indicates the state just before bulk sliding, as shown in Figs. 2
and 5. Note that the local static friction in the static state with $v=0$
takes any values for $0<\sigma^{\mathrm{(fric)}}/p<\mu_{\mathrm{S}}$. As shown
in Supplementary Videos and previous studies Otsuki2013 ; IwashitaSciRep2023 ,
the ratio $\sigma^{\mathrm{(fric)}}/p$ returns to the value near
$\mu_{\mathrm{K}}$ in the entire area just after the bulk sliding. As the
block is pushed, $\sigma^{\mathrm{(fric)}}/p$ reaches the local static
friction coefficient $\mu_{\mathrm{S}}$ near the region pushed by the rigid
plate. The area with $\sigma^{\mathrm{(fric)}}/p\thickapprox\mu_{\mathrm{S}}$
gradually increases as $U$ increases. The region with
$\sigma^{\mathrm{(fric)}}/p\thickapprox\mu_{\mathrm{S}}$ corresponds to the
slip region at $U=U_{3}$ in Fig. 4, while $\sigma^{\mathrm{(fric)}}/p$ remains
near $\mu_{\mathrm{K}}$ in the static region.

Figure 8b shows the spatial distribution of the bottom pressure $p$ at
$U=U_{3}$ for $P_{\mathrm{ext}}/E=0.003$, $\phi=0.5$ and $d/H=0.5$. Although a
uniform pressure $P_{\mathrm{ext}}$ is applied at the top surface, the spatial
average of $p$ becomes $P_{\mathrm{ext}}/(1-\phi)$, because the contact area
in the friction surface, $LW(1-\phi)$, is smaller than the area of the top
surface, $LW$, due to the grooves. We confirm that the bottom pressure is
$p\approx P_{\mathrm{ext}}/(1-\phi)$ in most areas except for the regions near
$y=0$ and $L$. The spatial distribution of $p$ is almost independent of the
time $t$, as shown in Supplementary Videos.

###  3.2 Theoretical Analysis

We theoretically analyze the effect of the longitudinal grooves shown in Sect.
3.1 based on a simplified model Otsuki2013 ; IwashitaSciRep2023 . The
precursor slip is approximately uniform in the $x$ direction and propagates
toward the $y$ direction, as shown in Fig. 4. Therefore, we neglect
displacements in the $z$ and $x$ directions and consider only the
$y$-dependent displacement along the $y$ direction. Since the bottom pressure
$p$ at $z=0$ is approximately uniform, as shown in Fig. 8b, we assume
$p=P_{\mathrm{ext}}/(1-\phi)$. Additionally, since the deformation is
significant in the region near the bottom before bulk sliding in our 3D
simulations, as shown in Appendix C, we focus on the slip and deformation in
the region $0\leq z/H\leq\alpha$ with a constant $\alpha$, as shown in the red
shaded area in Fig. 9.

Figure 9:  (a) Schematic of the derivation of the simplified model for grooved
viscoelastic block. (b) Cross-section perpendicular to the $y$ direction
indicated as I in (a). The red shaded areas represent the region from the
bottom to the height $z=\alpha H$. The dotted rectangle represents the element
with infinitesimal width $\mathrm{d}y$.

We consider the equation of motion for a thin element at $y$ with small width
$\mathrm{d}y$ indicated by the dotted rectangle in Fig. 9a. The mass of the
element is given by $\rho A(\phi,d)\mathrm{d}y$, where $A(\phi,d)$ is the
cross-sectional area of the red region in Fig. 9b excluding the groove. In
Fig. 9a, the forces acting on the left and right surfaces of that element are
given by $A(\phi,d)\sigma_{yy}(y,t)$ and
$A(\phi,d)\sigma_{yy}(y+\mathrm{d}y,t)$, respectively. Here, the normal stress
in the $y$ direction is denoted by $\sigma_{yy}$. The friction force acting on
the bottom is given by $\mu P_{\mathrm{ext}}W\mathrm{d}y$. The equation of
motion for the displacement $q_{y}(y,t)$ of the thin element along the $y$
direction is given by

$\rho
A(\phi,d)\mathrm{d}y\,\ddot{q}_{y}(y,t)=A(\phi,d)\left[\sigma_{yy}(y+\mathrm{d}y,t)-\sigma_{yy}(y,t)\right]-\mu(\dot{q}_{y}(y,t))P_{\mathrm{ext}}W\mathrm{d}y\
,$ (8)

where $\dot{q}_{y}(y,t)$ and $\ddot{q}_{y}(y,t)$ are the first and second-
order time derivatives of $q_{y}(y,t)$, respectively. We assume a plane stress
state, where the normal stress $\sigma_{yy}(y,t)$ is given by

$\sigma_{yy}(y,t)=E_{1}\frac{\partial q_{y}(y,t)}{\partial
y}+\eta_{\mathrm{t}}\frac{\partial\dot{q}_{y}(y,t)}{\partial y}$ (9)

with the elastic modulus $E_{1}=E/[(1+\nu)(1-\nu)]$ and viscous modulus
$\eta_{\mathrm{t}}=\eta_{1}(\eta_{1}+2\eta_{2})/(\eta_{1}+\eta_{2})$.

The cross-sectional area $A(\phi,d)$ is given by

$A(\phi,d)=A_{0}[1-\kappa(\phi,d)]$ (10)

with the cross-sectional area $A_{0}=\alpha HW$ for $\phi=0$. Here,
$\kappa(\phi,d)$ is the decreasing rate of the cross-sectional area by the
groove,

$\kappa(\phi,d)=\left\\{\begin{array}[]{lll}\phi\,d/(\alpha H),{}&0\leq
d\leq\alpha H\\\ \phi,{}&\alpha H<d\leq H\end{array}\right.\ .$ (11)

Substituting Eqs. (9) and (10) into Eq. (8) and taking the limit of
$\mathrm{d}y\rightarrow 0$, we obtain

$\rho(1-\kappa)\ddot{q}_{y}(y,t)=(1-\kappa)\left[E_{1}\frac{\partial^{2}q_{y}(y,t)}{\partial
y^{2}}+\eta_{\mathrm{t}}\frac{\partial^{2}\dot{q}_{y}(y,t)}{\partial
y^{2}}\right]-\frac{\mu(\dot{q}_{y}(y,t))P_{\mathrm{ext}}}{\alpha H}\ .$ (12)

The boundary conditions are given by $\partial q_{y}(L,t)/\partial y=0$ for
the free boundary at $y=L$ and $q_{y}(0,t)=U$ for the fixed boundary at $y=0$.

We set $t=0$ just after the bulk sliding, where the friction coefficient is
given by $\mu=\mu_{\mathrm{K}}$. When a precursor slip occurs with the
normalized slip area $\tilde{S}$ for $U>0$, the friction coefficient is given
by $\mu=\mu_{\mathrm{S}}$ in the region $0\leq y/L\leq\tilde{S}$, because the
slip distances of the precursors are significantly smaller than that in bulk
sliding. In the other regions, $\mu$ remains $\mu_{\mathrm{K}}$ due to the
frictional stress drop after the bulk sliding. This is confirmed by direct
numerical calculations of Eq. (12) and qualitatively consistent with the
results in Sect. 3.1. For sufficiently slow driving with
$\ddot{q}_{y}\thickapprox 0$ and $\dot{q}_{y}\thickapprox 0$, the quasi-static
solution of $q_{y}$ in Eq. (12) is analytically derived as described in
Appendix D. In this quasi-static solution $q_{\mathrm{a}}(y)$, $\tilde{S}$ is
given as an increasing function of $U$.

We conduct a stability analysis based on Eq. (12) following the procedure in
the previous studies Otsuki2013 ; IwashitaSciRep2023 . Substituting
$q_{y}(y,t)=q_{\mathrm{a}}(y)+\delta q(y,t)$ into Eq. (12) with the
perturbation $\delta q(y,t)$, we obtain the equation for $\delta q(y,t)$ as

$\rho(1-\kappa)\delta\ddot{q}(y,t)=(1-\kappa)\left[E_{1}\frac{\partial^{2}\delta
q(y,t)}{\partial
y^{2}}+\eta_{\mathrm{t}}\frac{\partial^{2}\delta\dot{q}(y,t)}{\partial
y^{2}}\right]-\frac{(\mu_{\mathrm{S}}-\mu_{\mathrm{K}})P_{\mathrm{ext}}}{v_{\mathrm{c}}\alpha
H}\delta\dot{q}(y,t)\ .$ (13)

Note that $\delta q(y,t)$ has a non-zero value in the region
$0<y/L<\tilde{S}$, and $\delta q(y,t)$ remains zero in the other region due to
static friction. Since the perturbation $\delta q(y,t)$ is zero for $y=0$ and
$\tilde{S}<y/L<1$, $\delta q(y,t)$ is expressed as

$\delta q(y,t)=\sum_{m}q_{m}e^{\lambda_{m}t}\sin{k_{m}\xi}\ ,$ (14)

where $m$ is a positive integer, $q_{m}$ is a constant, $\lambda_{m}$ is the
eigenvalue of the time evolution operator with $k_{m}=m\pi$ and
$\xi=y/(\tilde{S}L)$. Substituting Eq. (14) into Eq. (13), multiplying by
$2\sin{k_{n}\xi}$ with positive integer $n$, and integrating in
$0<y<\tilde{S}L$, we obtain

$(1-\kappa)\rho
L^{2}\lambda_{m}^{2}+(1-\kappa)E_{1}\frac{k_{m}^{2}}{\tilde{S}^{2}}+(1-\kappa)\eta_{\mathrm{t}}\frac{k_{m}^{2}}{\tilde{S}^{2}}\lambda_{m}-\frac{(\mu_{\mathrm{S}}-\mu_{\mathrm{K}})P_{\mathrm{ext}}L^{2}}{v_{\mathrm{c}}\alpha
H}\lambda_{m}=0\ .$ (15)

The perturbation $\delta q(y,t)$ is unstable in the case of
$\real\lambda_{m}>0$ and $\imaginary\lambda_{m}\neq 0$. The latter condition,
$\imaginary\lambda_{m}\neq 0$, indicates the oscillatory motion. However, the
backward motion associated with the oscillation is inhibited by the static
friction. Then, the instability induces only forward motion, that is, the BRP,
and does not increase further. Therefore, the perturbation develops and causes
bulk sliding only in the case of $\real\lambda_{m}>0$ and
$\imaginary\lambda_{m}=0$. In Eq. (15), we find that the stability conditions
of the system are generally determined by the competition between the
viscosity represented by the third term and velocity-weakening friction
represented by the fourth term on the left-hand side. These terms are
considered as the stabilizing and destabilizing factors, respectively. The
stabilizing factor decreases due to $\tilde{S}^{-2}$ in the third term as the
precursor slip area $\tilde{S}$ increases. When $\tilde{S}$ reaches the
critical area $\tilde{S}_{\mathrm{c}}$, the destabilizing factor overwhelms
the stabilizing factor, and the perturbation $\delta q(y,t)$ becomes unstable.
Therefore, $\tilde{S}$ increases rapidly just after reaching
$\tilde{S}_{\mathrm{c}}$, as shown in Fig. 5, and bulk sliding occurs. The
viscous term is proportional to $1-\kappa$. The velocity-weakening friction
term is proportional to the load on the top of the block but independent of
$\kappa$. Thus, if $\kappa(\phi,d)$ increases by increasing $\phi$ and $d$,
the viscosity becomes effectively smaller, which leads to the decrease of
$\tilde{S}_{\mathrm{c}}$.

As $\tilde{S}$ increases, the mode with $m=1$ in Eq. (14) becomes unstable
first, which determines $\tilde{S}_{\mathrm{c}}$. By definition, the maximum
value of $\tilde{S}_{\mathrm{c}}$ does not exceed $1$, and for
$\tilde{S}_{\mathrm{c}}<1$, $\tilde{S}_{\mathrm{c}}$ satisfies

$\pi^{2}(1-\kappa)\eta_{\mathrm{t}}\tilde{S}_{\mathrm{c}}^{-2}+2\pi(1-\kappa)L\sqrt{\rho
E_{1}}\tilde{S}_{\mathrm{c}}^{-1}=\frac{\left(\mu_{\mathrm{S}}-\mu_{\mathrm{K}}\right)P_{\mathrm{ext}}L^{2}}{v_{\mathrm{c}}\alpha
H}\ ,$ (16)

which is derived from Eq. (15). Therefore, $\tilde{S}_{\mathrm{c}}$ is given
by

$\tilde{S}_{\mathrm{c}}=\min(\tilde{S}_{\mathrm{c}}^{*},1)\ ,$ (17)

where $\min(a,b)$ is a function that takes the smaller value between $a$ and
$b$, and $\tilde{S}_{\mathrm{c}}^{*}$ is the solution of Eq. (16) given by

$\tilde{S}_{\mathrm{c}}^{*}=\dfrac{\pi\eta_{\mathrm{t}}}{L\left(-\sqrt{\rho
E_{1}}+\sqrt{\rho
E_{1}+\frac{\mu_{\mathrm{S}}-\mu_{\mathrm{K}}}{1-\kappa}\,\frac{P_{\mathrm{ext}}\eta_{\mathrm{t}}}{v_{\mathrm{c}}\alpha
H}}\right)}\ .$ (18)

For $\tilde{S}_{\mathrm{c}}\ll 1$, $\tilde{S}_{\mathrm{c}}$ is approximately
given by

$\tilde{S}_{\mathrm{c}}\simeq\pi\left\\{\frac{[1-\kappa(\phi,d)]\alpha}{\mu_{\mathrm{S}}-\mu_{\mathrm{K}}}\,\frac{\eta_{\mathrm{t}}v_{\mathrm{c}}}{P_{\mathrm{ext}}H}\right\\}^{\frac{1}{2}}\frac{H}{L}\
.$ (19)

This result indicates that the normalized critical area of the precursor slip
$\tilde{S}_{\mathrm{c}}$ is a decreasing function of $P_{\mathrm{ext}}$ and
the size of grooves, because $\kappa(\phi,d)$ in Eq. (19) increases with
$\phi$ and $d$, as described in Eq. (11). These analytical results are
qualitatively consistent with those of the FEM simulations shown in Fig. 6.

The macroscopic static friction coefficient $\mu_{\mathrm{M}}$ can be
analytically derived in our simplified model Otsuki2013 ; IwashitaSciRep2023 .
Since the ratio of the local frictional stress to bottom pressure is
$\mu_{\mathrm{S}}$ in the slip region and $\mu_{\mathrm{K}}$ in the static
region, $\mu_{\mathrm{M}}$ just before the bulk sliding is given by

$\mu_{\mathrm{M}}=\mu_{\mathrm{K}}+(\mu_{\mathrm{S}}-\mu_{\mathrm{K}})\tilde{S}_{\mathrm{c}}\
.$ (20)

This result is qualitatively consistent with the FEM simulations shown in Fig.
7, where Eq. (20) is represented by the solid line.

Substituting Eq. (20) into Eq. (19), we obtain

$\mu_{\mathrm{M}}-\mu_{\mathrm{K}}\simeq\pi\left\\{(\mu_{\mathrm{S}}-\mu_{\mathrm{K}})[1-\kappa(\phi,d)]\alpha\,\frac{\eta_{\mathrm{t}}v_{\mathrm{c}}}{P_{\mathrm{ext}}H}\right\\}^{\frac{1}{2}}\frac{H}{L}\
.$ (21)

This equation, together with Eq. (11), implies that the macroscopic static
friction coefficient $\mu_{\mathrm{M}}$ is a decreasing function of
$P_{\mathrm{ext}}$, $\phi$ and $d$. The analytical results are qualitatively
consistent with the FEM simulations shown in Fig. 3.

Figure 10:  Normalized critical area of precursor slip
$\tilde{S}_{\mathrm{c}}$ against $\kappa(\phi,d)$ for different values of
$\phi$ and $d$ with (a) $P_{\mathrm{ext}}/E=0.003$ and (b)
$P_{\mathrm{ext}}/E=0.006$. The symbols represent the results of the FEM
simulations. The solid lines represent the analytical results given by Eq.
(17) with $\alpha=0.2$.  Figure 11:  Macroscopic static friction coefficient
$\mu_{\mathrm{M}}$ against $\kappa(\phi,d)$ for different values of $\phi$ and
$d$ with (a) $P_{\mathrm{ext}}/E=0.003$ and (b) $P_{\mathrm{ext}}/E=0.006$.
The symbols represent the results of the FEM simulations. The solid lines
represent the analytical results given by Eqs. (17) and (20) with
$\alpha=0.2$. The dotted and dashed lines indicate $\mu_{\mathrm{S}}$ and
$\mu_{\mathrm{K}}$, respectively.

Equations (19) and (21) indicate that $\tilde{S}_{\mathrm{c}}$ and
$\mu_{\mathrm{M}}$ for different $\phi$ and $d$ are scaled by the decreasing
rate of cross-sectional area $\kappa(\phi,d)$. Figures 10 and 11 respectively
show $\tilde{S}_{\mathrm{c}}$ and $\mu_{\mathrm{M}}$ obtained from the FEM
simulations against $\kappa(\phi,d)$. Both $\tilde{S}_{\mathrm{c}}$ and
$\mu_{\mathrm{M}}$ are scaled by $\kappa(\phi,d)$ and decrease with increasing
$\kappa(\phi,d)$. The solid line in Fig. 10 represents the analytical result
given by Eq. (17). The solid line in Fig. 11 represents the result given by
Eqs. (17) and (20). Here, we set $\alpha=0.2$ to semi-quantitatively reproduce
the results of the FEM simulations in Sect. 3.1. The deformation before bulk
sliding is significant only in the region $z/H<0.5$, as shown in Appendix C.
This result is consistent with the estimate of $\alpha=0.2$. The numerical
results of FEM are semi-quantitatively consistent with the theoretical
analysis.

These results explain the decreases of $\mu_{\mathrm{M}}$ and
$\tilde{S}_{\mathrm{c}}$ with the increases of $\phi$ and $d$ in Figs. 3 and
6. Here, $\phi$ represents the decrease in the contact area of the pillars,
and $d$ represents the decrease in the size of the main body. These values
determine the decreasing rate of the cross-sectional area $\kappa(\phi,d)$.
Thus, it can be concluded that $\mu_{\mathrm{M}}$ and $\tilde{S}_{\mathrm{c}}$
decrease with increasing $\phi$ or $d$ because of the decrease in effective
viscoelasticity due to the reduction of the cross-sectional area, leading to
the decline of the stability and bulk sliding with a smaller size of the
precursor slip.

##  4 Discussion

Generally, grooves on friction surfaces are designed to control lubrication
properties in wet conditions. It is considered that the friction coefficient
at the wet interface increases with the width and depth of the grooves,
because they can eject more lubricant from the friction interface Li2004 ;
Li2005 ; Li2006 ; Yamaguchi2012 . However, this study reveals that the groove
size also affects friction in dry conditions. The static friction coefficient
in dry conditions decreases with increases in groove width and depth. This is
opposite to the usual consideration for the friction in the wet case. Even in
wet conditions, the friction force at the solid-solid interface determines the
total friction force after the ejection of the lubricant. These results should
aid in improving the design of sliding interfaces with grooves for both wet
and dry conditions.

The influence of the groove shape on friction in dry conditions has recently
been investigated based on different models, where the effect of grooves is
represented by the spatial distribution of the local friction coefficient
Capozza2015 ; Costagliola2016 ; Costagliola2017 ; Maegawa2017 ;
Costagliola2018 ; Costagliola2022IJSS ; Berardo2019 ; Balestra2022 . These
previous works report a decrease in the static friction coefficient by forming
longitudinal grooves, which is consistent with our results. However, the
effect of the depth of the longitudinal grooves is ignored in their models,
while our study is based on a realistic 3D system and reveals its importance.
Moreover, previous studies have investigated different patterns of grooves
including transversal grooves. Their results have suggested that complex
shapes in the friction surface used in various industrial products such as
shoe soles and tires and adopted on the surface of living things such as
snakes Baum2014BJN ; Baum2014TL ; Costagliola2017  affect the frictional
properties. Therefore, the extension of our study to these complex shapes will
lead to more efficient guiding principles for groove design.

In this study, the parameter values for virtual materials are adopted to
reduce the computational load, which does not correspond to those for real
materials. These are selected to compare the results with those in previous
simulations Otsuki2013 ; IwashitaSciRep2023 . However, the mechanism of
changes in the friction coefficient revealed by our theory in Sect. 3.2 is
universal and independent of specific parameter values. The numerical results
for flat friction surfaces in Ref. 16 have been reproduced in experiments of
acrylic glass Katano2014 . Therefore, we expect our results will be
experimentally verified in future work.

##  5 Conclusion

Friction surfaces of products such as shoe soles, tires, and sliding parts of
machines have grooves. Several studies on grooves have focused on controlling
lubrication properties via their design. However, it has been empirically
known that grooves affect friction even in dry conditions, although no
theoretical explanation exists. In this study, we have performed numerical
simulations of viscoelastic objects using 3D FEM to clarify the effect of
longitudinal grooves on static friction in a dry condition. We have revealed
that the static friction coefficient is a decreasing function of the groove
size, and that precursor slip occurs before bulk sliding. The static friction
coefficient is scaled by the normalized critical area of the precursor slip.
Based on the simplified model, we have theoretically derived the equation for
the static friction coefficient depending on the groove size. The theoretical
result indicates that the static friction coefficient decreases with the
decreasing rate of the cross-sectional area in the viscoelastic object. The
decrease in the cross-sectional area reduces the effective viscosity, which
enhances the instability of the precursor slip and decreases the static
friction coefficient. Our results provide new guiding principles for groove
design for static friction control beyond the empirical laws for both wet and
dry conditions. Investigation of different types of grooves and their effect
on dynamic friction will be the subject of future studies.

## Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## References

  *   * (1) Bowden, F.P., Tabor, D.: The Friction and Lubrication of Solids. Oxford University Press, New York (1950)
  * (2) Persson, B.N.J.: Sliding Friction: Physical Principles and Applications, 2nd edn. Springer, Berlin (2000)
  * (3) Popov, V.L.: Contact Mechanics and Friction: Physical Principles and Applications, 2nd edn. Springer, Berlin (2017)
  * (4) Rabinowicz, E.: Friction and Wear of Materials, 2nd edn. John Wiley & Sons, New York (1995)
  * (5) Dowson, D.: History of Tribology, 2nd edn. John Wiley & Sons, New York (1998)
  * (6) Bhushan, B.: Principles and Applications of Tribology, 2nd edn. John Wiley & Sons, New York (2013)
  * (7) Baumberger, T., Caroli, C.: Solid friction from stick-slip down to pinning and aging. Advances in Physics 55(3-4), 279–348 (2006). https://doi.org/10.1080/00018730600732186
  * (8) Li, K.W., Chen, C.J.: The effect of shoe soling tread groove width on the coefficient of friction with different sole materials, floors, and contaminants. Applied Ergonomics 35, 499–507 (2004). https://doi.org/10.1016/j.apergo.2004.06.010
  * (9) Li, K.W., Chin, J.C.: Effects of tread groove orientation and width of the footwear pads on measured friction coefficients. Safety Science 43, 391–405 (2005). https://doi.org/10.1016/j.ssci.2005.08.006
  * (10) Li, K.W., Wu, H.H., Lin, Y.-C.: The effect of shoe sole tread groove depth on the friction coefficient with different tread groove widths, floors and contaminants. Applied Ergonomics 37, 743–748 (2006). https://doi.org/10.1016/j.apergo.2005.11.007
  * (11) Yamaguchi, T., Umetsu, T., Ishizuka, Y., Kasuga, K., Ito, T., Ishizawa, S., Hokkirigawa, K.: Development of new footwear sole surface pattern for prevention of slip-related falls. Safety Science 50, 986–994 (2012). https://doi.org/10.1016/j.ssci.2011.12.017
  * (12) Archard, J.F.: Elastic deformation and the laws of friction. Proceedings of the Royal Society of London A 243(1233), 190–205 (1957). https://doi.org/10.1098/rspa.1957.0214
  * (13) Dieterich, J.H., Kilgore, B.D.: Imaging surface contacts: power law contact distributions and contact stresses in quartz, calcite, glass and acrylic plastic. Tectonophysics 256(1-4), 219–239 (1996). https://doi.org/10.1016/0040-1951(95)00165-4
  * (14) Bouissou, S., Petit, J.P., Barquins, M.: Normal load, slip rate and roughness influence on the polymethylmethacrylate dynamics of sliding 1. Stable sliding to stick-slip transition. Wear 214(2), 156–164 (1998). https://doi.org/10.1016/S0043-1648(97)00242-1
  * (15) Ben-David, O., Fineberg, J.: Static friction coefficient is not a material constant. Physical Review Letters 106(25), 254301 (2011). https://doi.org/10.1103/PhysRevLett.106.254301
  * (16) Otsuki, M., Matsukawa, H.: Systematic breakdown of Amontons’ law of friction for an elastic object locally obeying Amontons’ law. Scientific Reports 3, 1586 (2013). https://doi.org/10.1038/srep01586
  * (17) Katano, Y., Nakano, K., Otsuki, M., Matsukawa, H.: Novel friction law for the static friction force based on local precursor slipping. Scientific Reports 4, 6324 (2014). https://doi.org/10.1038/srep06324
  * (18) Iwashita, W., Matsukawa, H., Otsuki, M.: Static friction coefficient depends on the external pressure and block shape due to precursor slip. Scientific Reports 13, 2511 (2023). https://doi.org/10.1038/s41598-023-29764-w
  * (19) Yamaguchi, T., Morishita, M., Doi, M., Hori, T., Sakaguchi, H., Ampuero, J.-P.: Gutenberg-Richter’s law in sliding friction of gels. Journal of Geophysical Research: Solid Earth 116(B12), 12306 (2011). https://doi.org/10.1029/2011JB008415
  * (20) Obara, K., Kato, A.: Connecting slow earthquakes to huge earthquakes. Science 353(6296), 253–257 (2016). https://doi.org/10.1126/science.aaf1512
  * (21) Kato, A., Ben-Zion, Y.: The generation of large earthquakes. Nature Reviews Earth and Environment 2, 26–39 (2021). https://doi.org/10.1038/s43017-020-00108-w
  * (22) Petrillo, G., Lippiello, E., Landes, F.P., Rosso, A.: The influence of the brittle-ductile transition zone on aftershock and foreshock occurrence. Nature Communications 11, 3010 (2020). https://doi.org/10.1038/s41467-020-16811-7
  * (23) Xu, S., Fukuyama, E., Yamashita, F., Kawakata, H., Mizoguchi, K., Takizawa, S.: Fault strength and rupture process controlled by fault surface topography. Nature Geoscience 16, 94–100 (2023). https://doi.org/10.1038/s41561-022-01093-z
  * (24) Svetlizky, I., Fineberg, J.: Classical shear cracks drive the onset of dry frictional motion. Nature 509, 205–208 (2014). https://doi.org/10.1038/nature13202
  * (25) Bayart, E., Svetlizky, I., Fineberg, J.: Fracture mechanics determine the lengths of interface ruptures that mediate frictional motion. Nature Physics 12, 166–170 (2016). https://doi.org/10.1038/nphys3539
  * (26) Svetlizky, I., Kammer, D.S., Bayart, E., Cohen, G., Fineberg, J.: Brittle fracture theory predicts the equation of motion of frictional rupture fronts. Physical Review Letters 118(12), 125501 (2017). https://doi.org/10.1103/PhysRevLett.118.125501
  * (27) Berman, N., Cohen, G., Fineberg, J.: Dynamics and properties of the cohesive zone in rapid fracture and friction. Physical Review Letters 125(12), 125503 (2020). https://doi.org/10.1103/PhysRevLett.125.125503
  * (28) Gvirtzman, S., Fineberg, J.: Nucleation fronts ignite the interface rupture that initiates frictional motion. Nature Physics 17, 1037–1042 (2021). https://doi.org/10.1038/s41567-021-01299-9
  * (29) Kammer, D.S., Radiguet, M., Ampuero, J.-P., Molinari, J.-F.: Linear elastic fracture mechanics predicts the propagation distance of frictional slip. Tribology Letters 57, 23 (2015). https://doi.org/10.1007/s11249-014-0451-8
  * (30) Castellano, M., Lorez, F., Kammer, D.S.: Nucleation of frictional slip: A yielding or a fracture process? Journal of the Mechanics and Physics of Solids 173, 105193 (2023). https://doi.org/10.1016/j.jmps.2022.105193
  * (31) Rubinstein, S.M., Cohen, G., Fineberg, J.: Dynamics of precursors to frictional sliding. Physical Review Letters 98(22), 226103 (2007). https://doi.org/10.1103/PhysRevLett.98.226103
  * (32) Maegawa, S., Suzuki, A., Nakano, K.: Precursors of global slip in a longitudinal line contact under non-uniform normal loading. Tribology Letters 38, 313–323 (2010). https://doi.org/%****␣sn-article.bbl␣Line␣525␣****10.1007/s11249-010-9611-7
  * (33) Burridge, R., Knopoff, L.: Model and theoretical seismicity. Bulletin of the Seismological Society of America 57(3), 341–371 (1967). https://doi.org/10.1785/BSSA0570030341
  * (34) de Sousa Vieira, M.: Self-organized criticality in a deterministic mechanical model. Physical Review A 46(10), 6288 (1992). https://doi.org/10.1103/PhysRevA.46.6288
  * (35) Braun, O.M., Barel, I., Urbakh, M.: Dynamics of transition from static to kinetic friction. Physical Review Letters 103(19), 194301 (2009). https://doi.org/10.1103/PhysRevLett.103.194301
  * (36) Capozza, R., Urbakh, M.: Static friction and the dynamics of interfacial rupture. Physical Review B 86(8), 085430 (2012). https://doi.org/10.1103/PhysRevB.86.085430
  * (37) Ozaki, S., Inanobe, C., Nakano, K.: Finite element analysis of precursors to macroscopic stick-slip motion in elastic materials: analysis of friction test as a boundary value problem. Tribology Letters 55, 151–163 (2014). https://doi.org/10.1007/s11249-014-0343-y
  * (38) Scheibert, J., Dysthe, D.K.: Role of friction-induced torque in stick-slip motion. Europhysics Letters 92(5), 54001 (2010). https://doi.org/10.1209/0295-5075/92/54001
  * (39) Amundsen, D.S., Scheibert, J., Thøgersen, K., Trømborg, J., Malthe-Sørenssen, A.: 1D model of precursors to frictional stick-slip motion allowing for robust comparison with experiments. Tribology Letters 45, 357–369 (2012). https://doi.org/10.1007/s11249-011-9894-3
  * (40) Trømborg, J.K., Sveinsson, H.A., Scheibert, J., Thøgersen, K., Amundsen, D.S., Malthe-Sørenssen, A.: Slow slip and the transition from fast to slow fronts in the rupture of frictional interfaces. Proceedings of the National Academy of Sciences of the U.S.A. 111(24), 8764–8769 (2014). https://doi.org/10.1073/pnas.1321752111
  * (41) Radiguet, M., Kammer, D.S., Gillet, P., Molinari, J.-F.: Survival of heterogeneous stress distributions created by precursory slip at frictional interfaces. Physical Review Letters 111(16), 164302 (2013). https://doi.org/10.1103/PhysRevLett.111.164302
  * (42) Albertini, G., Karrer, S., Grigoriu, M.D., Kammer, D.S.: Stochastic properties of static friction. Journal of the Mechanics and Physics of Solids 147, 104242 (2021). https://doi.org/10.1016/j.jmps.2020.104242
  * (43) Taloni, A., Benassi, A., Sandfeld, S., Zapperi, S.: Scalar model for frictional precursors dynamics. Scientific Reports 5, 8086 (2015). https://doi.org/10.1038/srep08086
  * (44) de Geus, T.W.J., Popović, M., Ji, W., Rosso, A., Wyart, M.: How collective asperity detachments nucleate slip at frictional interfaces. Proceedings of the National Academy of Sciences of the U.S.A. 116(48), 23977–23983 (2019). https://doi.org/10.1073/pnas.1906551116
  * (45) Capozza, R., Pugno, N.M.: Effect of surface grooves on the static friction of an elastic slider. Tribology Letters 58, 35 (2015). https://doi.org/10.1007/s11249-015-0510-9
  * (46) Costagliola, G., Bosia, F., Pugno, N.M.: Static and dynamic friction of hierarchical surfaces. Physical Review E 94(6), 063003 (2016). https://doi.org/10.1103/PhysRevE.94.063003
  * (47) Costagliola, G., Bosia, F., Pugno, N.M.: Hierarchical spring-block model for multiscale friction problems. ACS Biomaterials Science and Engineering 3(11), 2845–2852 (2017). https://doi.org/10.1021/acsbiomaterials.6b00709
  * (48) Costagliola, G., Bosia, F., Pugno, N.M.: A 2-D model for friction of complex anisotropic surfaces. Journal of the Mechanics and Physics of Solids 112, 50–65 (2018). https://doi.org/%****␣sn-article.bbl␣Line␣775␣****10.1016/j.jmps.2017.11.015
  * (49) Costagliola, G., Bosia, F., Pugno, N.M.: Correlation between slip precursors and topological length scales at the onset of frictional sliding. International Journal of Solids and Structures 243, 111525 (2022). https://doi.org/10.1016/j.ijsolstr.2022.111525
  * (50) Maegawa, S., Itoigawa, F., Nakamura, T., Matsuoka, H., Fukui, S.: Effect of tangential loading history on static friction force of elastic slider with split contact surface: model calculation. Tribology Letters 65, 37 (2017). https://doi.org/10.1007/s11249-017-0811-2
  * (51) Berardo, A., Costagliola, G., Ghio, S., Boscardin, M., Bosia, F., Pugno, N.M.: An experimental-numerical study of the adhesive static and dynamic friction of micro-patterned soft polymer surfaces. Materials and Design 181, 107930 (2019). https://doi.org/10.1016/j.matdes.2019.107930
  * (52) Balestra, S., Costagliola, G., Pegoraro, A., Picollo, F., Molinari, J.-F., Pugno, N.M., Vittone, E., Bosia, F., Sin, A.: Experimental and numerical study of the effect of surface patterning on the frictional properties of polymer surfaces. Journal of Tribology 144(3), 031704 (2022). https://doi.org/%****␣sn-article.bbl␣Line␣850␣****10.1115/1.4052777
  * (53) Maegawa, S., Itoigawa, F., Nakamura, T.: Effect of surface grooves on kinetic friction of a rubber slider. Tribology International 102, 326–332 (2016). https://doi.org/10.1016/j.triboint.2016.05.019
  * (54) Gao, Y., Wang, Y.: Research on the dynamic behaviors in the sliding friction of silicone. Tribology Letters 71, 15 (2023). https://doi.org/10.1007/s11249-022-01675-3
  * (55) Landau, L.D., Lifshitz, E.M., Kosevich, A.M., Pitaevskii, L.P.: Theory of Elasticity, 3rd edn. Butterworth-Heinemann, Oxford (1986)
  * (56) Wriggers, P.: Computational Contact Mechanics, 2nd edn. Springer, Berlin (2006)
  * (57) Dieterich, J.H., Kilgore, B.D.: Direct observation of frictional contacts: New insights for state-dependent properties. Pure and Applied Geophysics 143, 283–302 (1994). https://doi.org/10.1007/BF00874332
  * (58) Baum, M.J., Heepe, L., Gorb, S.N.: Friction behavior of a microstructured polymer surface inspired by snake skin. Beilstein Journal of Nanotechnology 5, 83–97 (2014). https://doi.org/10.3762/bjnano.5.8
  * (59) Baum, M.J., Kovalev, A.E., Michels, J., Gorb, S.N.: Anisotropic friction of the ventral scales in the snake Lampropeltis getula californiae. Tribology Letters 54, 139–150 (2014). https://doi.org/10.1007/s11249-014-0319-y

*Acknowledgments This research used computational resources of Fugaku at the RIKEN Center for Computational Science (Project: hp220372), Yukawa-21 at YITP, Kyoto University, SQUID at the Cybermedia Center, Osaka University, ohtaka and kugui at ISSP, the University of Tokyo, the supercomputers at RCCS, Okazaki, Japan (Project: 23-IMS-C126), and JSS3 at JAXA. We would like to thank Editage (www.editage.com) for English language editing.

*Funding This work was supported by the Japan Society for the Promotion of Science KAKENHI (Grant Numbers JP21H01006, JP22KJ2190, JP23K03248, and JP23K03252).

## Author information

*Contributions All authors contributed to the study conception and design. The FEM simulation and analysis based on a simplified model are performed by WI. The first draft of the manuscript was written by WI, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

## Ethics declarations

*Competing interests The authors declare no competing interests.

## Supplementary information

The online version contains supplementary material.

##  Appendix A Results for $L/H=2$

Figure 12:  (a) Normalized critical area of precursor slip
$\tilde{S}_{\mathrm{c}}$ and (b) macroscopic static friction coefficient
$\mu_{\mathrm{M}}$ against $d$ for different values of $\phi$ with $L/H=2$ and
$P_{\mathrm{ext}}/E=0.003$. The dotted and dashed lines indicate
$\mu_{\mathrm{S}}$ and $\mu_{\mathrm{K}}$, respectively.

In this appendix, we show the results for $L/H=2$. Figure 12a and b
respectively show $\tilde{S}_{\mathrm{c}}$ and $\mu_{\mathrm{M}}$ against $d$
for different values of $\phi$ with $P_{\mathrm{ext}}/E=0.003$. We find that
$\tilde{S}_{\mathrm{c}}$ and $\mu_{\mathrm{M}}$ are decreasing functions of
$\phi$ and $d$, which is consistent with the results for $L/H=4$, as shown in
Figs. 3 and 6. Compared to Figs. 3a and 6a for the identical pressure, we find
that $\tilde{S}_{\mathrm{c}}$ and $\mu_{\mathrm{M}}$ increase with decreasing
$L$.

Figure 13a and b respectively show $\tilde{S}_{\mathrm{c}}$ and
$\mu_{\mathrm{M}}$ against $\kappa(\phi,d)$ with $P_{\mathrm{ext}}/E=0.003$.
We find that $\tilde{S}_{\mathrm{c}}$ and $\mu_{\mathrm{M}}$ are decreasing
functions of $\kappa(\phi,d)$. The solid lines show the analytical results for
$\alpha=0.2$ based on Eqs. (17) and (20). It is shown that $\kappa(\phi,d)$
for $\alpha=0.2$ can scale $\tilde{S}_{\mathrm{c}}$ and $\mu_{\mathrm{M}}$
even for the system with $L/H=2$. The decreases in $\tilde{S}_{\mathrm{c}}$
and $\mu_{\mathrm{M}}$ by increasing $L$ is consistent with Eqs. (19) and (21)
for the theoretical analysis in Sect. 3.2.

Figure 13:  (a) Normalized critical area of precursor slip
$\tilde{S}_{\mathrm{c}}$ and (b) macroscopic static friction coefficient
$\mu_{\mathrm{M}}$ against $\kappa(\phi,d)$ for different values of $\phi$ and
$d$ with $L/H=2$ and $P_{\mathrm{ext}}/E=0.003$. The solid lines represent the
analytical results of Eqs. (17) and (20) with $\alpha=0.2$. The dotted and
dashed lines indicate $\mu_{\mathrm{S}}$ and $\mu_{\mathrm{K}}$, respectively.

##  Appendix B Dependence on Number of Pillars $n_{x}$ in Friction Surface

Figure 14:  (a) Normalized critical area of precursor slip
$\tilde{S}_{\mathrm{c}}$ and (b) macroscopic static friction coefficient
$\mu_{\mathrm{M}}$ against $n_{x}$ for $P_{\mathrm{ext}}/E=0.003$ and $0.006$
with $L/H=4$, $d/H=0.5$, and $\phi=0.5$. The dotted and dashed lines represent
$\mu_{\mathrm{S}}$ and $\mu_{\mathrm{K}}$, respectively.

In this appendix, we verify the dependence of the $\tilde{S}_{\mathrm{c}}$ and
$\mu_{\mathrm{M}}$ on the number of pillars of the block $n_{x}$ for $L/H=4$,
$d/H=0.5$, and $\phi=0.5$. Figure 14a and b show $\tilde{S}_{\mathrm{c}}$ and
$\mu_{\mathrm{M}}$ against $n_{x}$ for $P_{\mathrm{ext}}/E=0.003$ and $0.006$,
respectively. In Fig. 14, we find that $\tilde{S}_{\mathrm{c}}$ and
$\mu_{\mathrm{M}}$ are almost independent of $n_{x}$. According to the
theoretical results given by Eqs. (19) and (21), $\tilde{S}_{\mathrm{c}}$ and
$\mu_{\mathrm{M}}$ depend on the decreasing rate of the cross-sectional area
$\kappa(\phi,d)$, which is independent of $n_{x}$, as described in Eq. (11).
This analytical result explains the behaviors of $\tilde{S}_{\mathrm{c}}$ and
$\mu_{\mathrm{M}}$ in Fig. 14.

##  Appendix C Acceleration Distribution

Figure 15 demonstrates the spatial distribution of the $y$-direction
acceleration $\ddot{u}_{y}$ for $P_{\mathrm{ext}}/E=0.003$, $d/H=0.5$, and
$\phi=0.5$ in the cross-section at $x=0.5W$ with $U/L=5.43\times 10^{-3}$,
just before bulk sliding. The critical length in the $y$ direction of the
precursor slip is estimated to be $\tilde{S}_{\mathrm{c}}L$, because the
precursor slip propagates uniformly in the $x$-direction, as shown in Fig. 4.
The acceleration around the region of $y/H=0$ and $0.5\leq z/H\leq 1$ is
small, because the displacement is fixed at the rigid plate position at the
region. Moreover, the acceleration is negligible in the static region at the
bottom ($z=0$) with $\sigma^{\mathrm{(fric)}}/p\thickapprox\mu_{\mathrm{K}}$
for $\tilde{S}_{\mathrm{c}}L\leq y\leq L$, as shown in Fig. 8a. Therefore, the
region with significant acceleration before bulk sliding is concentrated in
the region for $0\leq y\leq\tilde{S}_{\mathrm{c}}L$ near the bottom, as shown
in Fig. 15.

Figure 15:  Spatial distribution of acceleration along the $y$ direction,
$\ddot{u}_{y}$, on cross-section at $x=0.5W$ for $P_{\mathrm{ext}}/E=0.003$,
$d/H=0.5$, and $\phi=0.5$ at $U/L=5.43\times 10^{-3}$, just before bulk
sliding. The gray thick line for $y/H=0$ and $0.5\leq z/H\leq 1$ indicates the
position of the rigid plate.

##  Appendix D Quasi-static Solution

Using the equation of motion (12), boundary conditions, and assumption of
friction coefficient $\mu$ in Sect. 3.2, we obtain the quasi-static solution
$q_{0}(y)$ at $U=0$ as

$q_{0}(y)=\frac{\mu_{\mathrm{K}}P_{\mathrm{ext}}}{2(1-\kappa)E_{1}\alpha
H}\left(y^{2}-2Ly\right)\ .$ (22)

The quasi-static solution $q_{\mathrm{a}}(y)$ for $U>0$ is obtained as

$q_{\mathrm{a}}(y)=\left\\{\begin{array}[]{cc}q_{1}(y),&~{}~{}~{}0\leq
y\leq\tilde{S}L\\\ q_{0}(y),&~{}~{}~{}\tilde{S}L\leq y\leq
L\end{array}\right.\ .$ (23)

With the assumption of $\mu$ and connectivity condition at $y=\tilde{S}L$,
$q_{1}(\tilde{S}L)=q_{0}(\tilde{S}L)$ and
$\mathrm{d}q_{1}(\tilde{S}L)/\mathrm{d}y=\mathrm{d}q_{0}(\tilde{S}L)/\mathrm{d}y$,
we obtain $q_{1}$ as

$q_{1}(y)=q_{0}(y)+\frac{(\mu_{\mathrm{S}}-\mu_{\mathrm{K}})P_{\mathrm{ext}}}{2(1-\kappa)E_{1}\alpha
H}\left(y^{2}-2\tilde{S}Ly+\tilde{S}^{2}L^{2}\right)\ .$ (24)

To satisfy the boundary condition at $y/L=0$, $\tilde{S}$ is derived as

$\tilde{S}=\left[\frac{2(1-\kappa)E_{1}\alpha
HU}{(\mu_{\mathrm{S}}-\mu_{\mathrm{K}})P_{\mathrm{ext}}L^{2}}\right]^{\frac{1}{2}}\
.$ (25)