|  Shear zones in granular mixtures of hard and soft particles with high and
low friction
---|---
|  Aditya Pratap Singh, Vasileios Angelidakis, Thorsten Pöschel, and Sudeshna
Roy†
|  Granular materials show inhomogeneous flows characterized by strain
localization. When strain is localized in a sheared granular material, rigid
regions of a nearly undeformed state are separated by shear bands, where the
material yields and flows. The characteristics of the shear bands are
determined by the geometry of the system, the micromechanical material
properties, and the kinematics at the particle level. For a split-bottom shear
cell, recent experimental work has shown that mixtures of hard, frictional and
soft, nearly frictionless particles exhibit wider shear zones than samples
with only one of the two components. To explain this finding, we investigate
the shear zone properties and the stress response of granular mixtures using
discrete element simulations. We show that both interparticle friction and
elastic modulus determine the shear-band properties of granular mixtures of
various mixing ratios, but their stress response depends strongly on the
interparticle friction. Our study provides a fundamental understanding of the
micromechanics of shear band formation in granular mixtures.

##

††footnotetext: Institute for Multiscale Simulation, Friedrich-Alexander-
Universität Erlangen-Nürnberg, Cauerstrasse 3, 91058 Erlangen,
Germany††footnotetext: †<EMAIL_ADDRESS>
##  1 Introduction

Strain localization is a characteristic effect of sheared granular matter,
which is not observed in the flow of other materials, such as liquids. Once
strain localizes within a narrow shear zone (also referred to as the shear
band), the stress, strain, and strain rate of the material assume high values
within that zone. On the contrary, outside of it, the material remains nearly
undeformed. The knowledge of the properties of shear zones and the factors
that influence them is crucial for understanding the intrinsic properties of
granular materials under shear.

In many systems where particles come into contact with boundaries, narrow
shear zones appear at the interface because of particle-boundary interactions.
These shear zones are triggered by boundary effects, which hinder the accurate
characterization of bulk material properties based on the shear zone
characteristics. In our work, we present a unique setup called a split-bottom
shear cell, which is a modified version of a Couette cell. This setup involves
splitting the cell’s bottom at a specific radius ($R_{s}$), resulting in two
parts consisting of the inner disk and outer cylinder components (see Figure
1). After loading the cell with grains to a certain height ($H$), the inner
disk is set to rotational motion. This design allows granular materials to be
sheared to develop wide shear zones inside the bulk of the materials, far away
from the rigid boundaries, which allows for a comprehensive characterization
of their yielding properties and flow behavior at a macromechanical level.
This is quite different from the usual narrow shear bands that we observe
close to the boundary, which are formed due to particle-boundary interactions.
This shear cell setup has been used in experiments 1 and simulations 2 to
explore the bulk properties of granular systems.

The universal geometrical properties of shear bands in the split-bottom shear
cell have been studied extensively in the literature 3, 4, 5. The absolute
value of the macroscopic normalized angular velocity of the granular material
was measured as a function of the radial position, and it has been found to
follow closely the shape of an error function, which is characterized by two
parameters: the width $W$ and the center position $R_{c}$ of the shear zone 5,
4. The width of the shear band increases with increasing height within the
bulk of the material, while the distance of its central position $R_{c}$ from
the center of rotation decreases (that is, the shear band moves inward with
increasing height). The width of the shear band is affected by the size,
shape, and interaction properties of the particles, such as friction between
the particles, cohesion, and the shear rate 6, 7, 8, 9. The position of the
shear band, $R_{c}$, at the surface of the granular material in the shear
cell, depends on two geometrical parameters, the filling height $H$ and split
radius $R_{s}$ 10. The bulk profile of the shear band as a function of height,
$R_{c}(z)$, can be linked to a variational problem. The solution of the
function $R_{c}(z)$ that minimizes the integral representing the dissipation
rate, is related to the minimal mechanical torque and thus depends on the
material properties. Therefore, the shape of the shear band within the bulk is
not only influenced by geometric factors such as the split position ($R_{s}$)
and the filling height ($H$) but is also closely connected to the material
properties, e.g., the shear stress, and the packing fraction of the material,
as highlighted in previous studies 10, 5.

Interparticle friction and particle softness play a key role in influencing
material bulk flows, which have been widely studied. Wang et al. 11
experimentally characterized the shear zones of binary mixtures made of hard,
frictional and soft, slippery particles. Their study identified distinct shear
band characteristics specific to the mixture, distinguishing them from those
in pure-species systems. Luding 8 explored the impact of interparticle
friction on shear zones in split-bottom shear cells through numerical
simulations. Increasing friction resulted in narrower shear bands that shifted
towards the center. Ashour et al. 12 found that particle softness adds
qualitatively new features to the dynamics and the packing characteristics of
silo flows. Soft, deformable particles, such as hydrogel particles, can adapt
to local stress conditions, leading to alterations in pressure profiles and
flow fields. Hong et al. 13 highlighted the importance of particle softness in
hopper flows, where gravitational forces must be less than repulsive forces
from particle stiffness to induce clogging. Harth et al. 14 observed longer
time scales for internal rearrangement in soft hydrogel particles during silo
orifice flow compared to hard grain systems, where shorter time scales have
been reported by Unac et al. 15. This was attributed to mass flow rate
differences, roundness, and increased viscous damping of hydrogel grains. The
rate-dependent frictional regimes observed in hydrogel particle suspensions
have been attributed to their high deformability and low friction 16. Wang et
al. 17 observed experimentally that, unlike granular materials with hard
particles, soft slippery particles in a silo flow do not follow Beverloo’s 18
law, according to which the flow rate is independent of the overburden
pressure. Adding just $5\%$ hard particles to soft silo flows restored this
behavior and significantly altered system flow properties. Based on the
micromechanical observations, Saitoh et al. 19 highlighted that the evolution
of the system behavior in soft particle packings extends beyond just particle
contacts. Inter-particle gaps between nearest neighbors are also crucial and
affect the stochastic evolution of the probability distribution function of
forces. Recently, Luding et al. 20 studied the contact and force networks of
frictionless soft granular materials, revealing that the evolution of the
pressure in the system was correlated with the existence of loops formed in
the force network during particle rearrangements. Götz et al. 21 observed that
soft particles with larger deformations and increased contact areas result in
larger frictional forces and improved holding capacity of a granular gripper.
In summary, both particle friction and softness are crucial factors
influencing bulk flow behavior, and the intricate interplay between these
properties and their combined effects on bulk behavior remains a subject of
ongoing investigation.

The findings of Wang et al. 11 established that mixtures of hard and soft
particles exhibit wider shear zones than assemblies made of only one of the
individual materials (only hard or only soft particles). However, the
underlying reason for this behavior remains unclear. Our objective here is to
gain insight into the combined effect of microscopic parameters, namely
particle stiffness and inter-particle friction, on the macroscopic shear zone
characteristics for ensembles of soft, slippery particles and hard, frictional
particles sheared in a split-bottom shear cell.

##  2 Numerical setup and materials

We use an open-source Discrete Element Method (DEM) code MercuryDPM 22 to
simulate sheared systems of hard and soft particles. The numerical setup used
for this study is a split-bottom shear cell, as shown in Figure 1, where the
inner bottom plate rotates around a central axis at a constant angular
velocity, $\omega_{o}=0.52$ rad/s, while the outer boundary of the cell
remains stationary. The granular material located in the outer parts remains
static, while the material near the center of the cell rotates with the bottom
disk. The shear in this system is localized in an intermediate region,
defining a wide shear band. The position and the width of the shear zone
depend on the ratio $H/R_{s}$, where $H$ is the filling height and $R_{s}$ is
the split radius of the rotating bottom plate 23, 4. For small and moderate
filling height $(H<0.6R_{s})$, the shear zone starts from the bottom split
position, evolves through the entire height of the granular layer, and ends up
at the surface 3, as shown in Figure 1, while the inner part of the granular
material co-rotates synchronously with the bottom plate. In our numerical
setup, the filling height is $H=20$ mm and $H/R_{s}=0.42$, which is considered
a moderate filling height. This numerical setup is similar to that employed in
the experimental work by Wang et al. 11, with the difference that the
experimental configuration incorporates an additional small rod in the center
of the cell, where the shear load was imposed.

Fig. 1: Schematic of a split bottom shear cell of radius $R_{o}$ with granular
material at filling height $H$, illustrating the main system components. The
bottom plate (dark-red color) rotates around the vertical axis at constant
rotation frequency $\omega_{o}$, while the outer wall (marked in grey) stays
stationary. The differential motion triggers the formation of a shear band
(marked in orange color) starting from the split radius $R_{s}$, which widens
with increasing height. In contrast, the material outside the shear band
(marked in yellow) stays undeformed.

###  2.1 Material parameters

We consider a mixture of soft low-frictional particles and hard frictional
particles. In this Section, we describe the generation of binary granular
mixtures of such particles. We take the properties of hydrogel spheres as the
soft, low-frictional material component, and mustard seeds represent hard,
frictional particles. A reference sample of pure soft hydrogel particles
consists of $N^{0}_{A}=26795$ polydisperse particles of species A with mean
diameter, $d_{A}=2.5$ mm. Starting from this baseline sample volume, we create
mixtures by substituting $N^{0}_{A}-N^{T}_{A}$ particles of species A with
particles of species B of diameter $d_{B}=3$ mm, such that the same volume of
material A is replaced by B, i.e.
$V_{B}=(N^{0}_{A}-N^{T}_{A})(\pi/6)d_{A}^{3}=N^{T}_{B}(\pi/6)d_{B}^{3}$. For
both species A and B, we use particles of homogeneous size distribution of
mean particle diameter $d_{A}$ and $d_{B}$, respectively, with polydispersity
of $\pm 8\%$. To characterize the mixtures with different ratios of species B,
we define the mixing ratio

$X=\frac{N^{T}_{B}}{N^{T}_{A}+N^{T}_{B}}$ (1)

We analyze systems with $X\in\\{0,0.05,0.25,0.50,0.75,0.85,1.0\\}$. Table 1
presents the number of particles of the two species in each sample. Figure 2
shows some examples of different mixtures.

The properties of soft particles of species A correspond to that of hydrogel
spheres with friction coefficient $\mu_{A}=0.005$ and elastic modulus
$E_{A}=50$ kPa. The frictional properties of hard particles correspond to
mustard seeds with friction coefficient $\mu_{B}=0.20$ and the elastic modulus
$E_{B}=5000$ kPa. All relevant material and interaction properties of the two
species are given in Table 2.

Table 1:  Number of particles of the species in each sample $X$ | $0$ | $0.05$ | $0.25$ | $0.50$ | $0.75$ | $0.85$ | $1$
---|---|---|---|---|---|---|---
Species A | $26975$ | $24675$ | $17104$ | $9757$ | $4244$ | $2497$ | $0$
Species B | $0$ | $1361$ | $5695$ | $9917$ | $13160$ | $14133$ | $15465$
Table 2:  Microscopic material parameters for the model Properties | Species A | Species B
---|---|---
Mean particle diameter |  $2.5$ mm |  $3$ mm
Density |  $1017$ kg/m-3 |  $1017$ kg/m-3
Elastic modulus |  $50$ kPa |  $5000$ kPa
Poisson’s ratio | $0.40$ | $0.40$
Friction coefficient | $0.005$ | $0.20$
Fig. 2: Top view of the shear cell for granular mixtures of different
percentages of hard particles $X$, where the blue and red particles represent
species A (soft) and B (hard), respectively.

###  2.2 Contact model

In the soft-particle DEM, the evolution of particle positions and velocities
is calculated by integrating Newton’s equation of motion. When two particles
$i$ and $j$ interact, they can slightly overlap. Only viscoelastic
interactions are considered, i.e., after separating the contacting particles,
they recover their initial shape. The compression of contacting particles
$\xi$ is written as:

$\xi=R_{i}+R_{j}-|\vec{r_{i}}-\vec{r_{j}}|$ (2)

where $R_{i}$, $R_{j}$ are the particle radii and $\vec{r_{i}}$, $\vec{r_{j}}$
are the particle positions. The normal force is calculated as in 24, where an
elastic Hertzian normal force is considered in conjunction with a viscous
damper of velocity-dependent coefficient of restitution.

$\vec{F_{n}}=\min\left(0,-\rho\xi^{3/2}-\frac{3}{2}A_{n}\rho\sqrt{\xi}\dot{\xi}\right)\vec{e_{n}}$
(3)

with

$\rho=\frac{4}{3}\,E^{*}\,\sqrt{R^{*}}$ (4)

where $E^{*}$ is the effective elastic modulus,
$R^{*}=R_{i}R_{j}/\left(R_{i}+R_{j}\right)$ is the effective radius,
$\vec{e}_{n}=(\vec{r}_{i}-\vec{r}_{j})/|\vec{r}_{i}-\vec{r}_{j}|$ is the
normal unit vector, and $A_{n}=4.1\times 10^{-6}$ s is the normal dissipative
parameter, calculated as in 25, considering a coefficient of restitution of
$0.70$ for the tangential translational velocity of the rotating bottom plate
at the split radius. The effective elastic modulus $E^{*}$ is given by

$E^{*}={\bigg{(}\frac{1-\nu_{i}^{2}}{E_{i}}+\frac{1-\nu_{j}^{2}}{E_{j}}\bigg{)}}^{-1}$
(5)

where $E_{i}$, $E_{j}$ and $\nu_{i}$, $\nu_{j}$ are the elastic moduli and
Poisson ratios of particles $i$, $j$, respectively.

We model the tangential viscoelastic forces considering a path-dependent
calculation, following the no-slip solution of Mindlin 26 for the elastic part
and Parteli and Pöschel 27 for the tangential dissipative parameter
$A_{t}\approx 2A_{n}E^{*}$, which are capped by the static friction force
between two particles. The tangential force is thereby given by

$\vec{F_{t}}=-\min\left[\mu|\vec{F_{n}}|,\int_{\mathrm{path}}8G^{*}\sqrt{R^{*}\xi}\,ds+A_{t}\sqrt{R^{*}\xi}v_{t}\right]\vec{e_{t}}$
(6)

where $\mu$ is the friction coefficient, $\vec{e_{t}}$ is the unit vector in
the tangential contact direction and $G^{*}$ the effective shear modulus,
given by

$G^{*}={\bigg{(}\frac{2-\nu_{i}}{G_{i}}+\frac{2-\nu_{j}}{G_{j}}\bigg{)}}^{-1}$
(7)

The shear modulus $G_{i}=\frac{E_{i}}{2(1+\nu_{i})}$ is calculated assuming
isotropic material. For the interaction between different species, the
effective friction coefficient is considered as
$\mu^{*}=2{\bigg{(}\frac{1}{\mu_{i}}+\frac{1}{\mu_{j}}\bigg{)}}^{-1}$. It is
typical for many DEM formulations to consider the minimum coefficient of
friction when two particles of different materials come into contact. Since
the harmonic average of two values favors the smaller one, considering this
calculation of the friction coefficient for contacts between high and low
friction coefficients gives values close to considering the minimum value.
This assumption is chosen as a milder approach to simply assuming the minimum
friction coefficient of the two, and more experimental evidence is required to
understand the friction between hard and soft particles. Considering the
harmonic average here, we have three scenarios of possible contacts between
frictional-frictional, slippery-slippery and frictional-slippery (with
friction very close to slippery-slippery) particles.

##  3 Segregation effects

(a)

(b)

Fig. 3: Probability of particle number as a function of (a) radial distance
and (b) distance from the bottom surface for species A and B after $200$ s of
simulation time.

When granular mixtures with particles differing in size and/or density are
subjected to external forces such as shaking, stirring, or shearing, they
often segregate to form complex patterns. These patterns of segregation can
lead to different and inhomogeneous macroscopic behavior. In many applications
where a homogeneous packing is sought, this is undesirable, and significant
effort is often invested in attempts to avoid or control segregation during
processing and handling of granular materials. Shear-induced percolation
segregation that occurs due to kinetic sieving and squeeze expulsion is the
dominant mechanism in dense granular flows. In this study, we study a bi-
disperse mixture of two species that differ slightly in terms of particle
size. Therefore, in this section, we examine the extent to which particles
segregate in this system of hard frictional and soft slippery particles,
sheared in the split-bottom shear cell.

(a)

(b)

Fig. 4: Average (a) radial distance and (b) distance from the bottom surface
for the species A and B as a function of shear displacement $\lambda=\omega
t_{\mathrm{rot}}$.

First, we analyze the probability distribution of the particles of each
species in the radial $r$ and vertical $z$ directions for a given snapshot
after $200$ s of simulation time, as shown in Figure 3(a) and (b),
respectively. The distribution increases in the radial direction for both
species due to the increased measuring volume (the probability is calculated
using co-centric cylinders of increasing radius, aligned with the center of
the cell). Note that the distribution of the two species overlaps with each
other in the radial $r$ direction, indicating that there is no segregation
effect of the two species horizontally. The distribution in the vertical $z$
direction for species B is nearly constant. However, the distribution of
species A decreases in vertical $z$ direction from $0.065$ to $0.035$,
indicating a higher probability of species A near the bottom of the shear cell
than near the free surface. This is a signature of the segregation of species
A in the vertical direction due to kinetic sieving, and is attributed to the
smaller particle size compared to species B. To further analyze this effect,
Figure 4(a) and (b) illustrate the mean radial positions $r$ and perpendicular
positions $z$ of the two species as a function of shear displacement
$\lambda$. Here, it is to be noted that the mean position of Species A is
about $0.4$ mm less in the $r$ direction and $1.2$ mm less in the $z$
direction than that of Species B. However, the differences are negligibly
small and of less than one particle diameter.

##  4 Micro-macro transition

The current work aims to derive appropriate macroscopic fields commensurate
with the shear band features and macromechanical stress analysis, e.g., strain
rate, velocity profile, and stress, based on the given micromechanical
properties. The flow that rapidly relaxed to a steady state is purely
azimuthal and is proportional to the driving rate of the shear cell,
$\omega_{o}$. The averaging is thus performed in toroidal volumes over many
snapshots of time, leading to generic continuum fields $Q(r,z)$ as a function
of the radial and vertical positions. We choose the representative elementary
volume according to the 28 as equal to the minimum particle diameter
$d_{p}=1.37$ mm.

Shearing of a granular system leads to volume dilation and build-up of shear
stress and anisotropy in fabric. Since we are interested in the steady-state
flow profile, we probe for the minimum strain required to reach the steady-
state flow regime. We define the shear displacement $\lambda=\omega
t_{\mathrm{rot}}$ through which the system rotates (where $t_{\mathrm{rot}}$
is the simulation time) as the shear control parameter. As reported by Singh
et al. 29, both the global quantities of kinetic energy and average number of
contacts reach a steady state very fast ($\lambda\sim 5$). We analyze the
relaxation of local quantities, specifically the local velocity profiles, and
conclude that $\lambda\sim 20-30$ is the amount of shear displacement required
to form a stable shear band, which is in agreement with Ries et al. 30. The
data depicting the initial, transient evolution of local shear band properties
is not presented here, as our focus is solely on the steady-state
characteristics of the shear band. Consequently, we perform the local
averaging over almost $600$ snapshots distributed over $\lambda\sim 80$ to
$100$.

The consistency of the local averaged quantities also depends on the
accumulated local shear strain during the averaging time. We concentrate our
interest in the region where the system can be considered to reach a critical
state. The critical state is a unique steady state reached after extensive
shear, where the material deforms with applied strain without any further
change in normal stress, shear stress, and volume fraction. This is a state
where the system forgets its loading, packing, or other sample-preparation
history 31. In accordance with previous experimental and numerical results in
the same setup 32, 33, the criterion of large strain rate identifies the shear
band region, e.g., higher than a critical strain rate of
$\dot{\gamma}_{c}=0.08$ s-1. Here, the shear band center region is defined by
strain rates higher than 80% of the maximum strain rate for different heights
in the shear cell. Therefore, the critical strain rate here is a function of
height in the shear cell and is denoted as
$\dot{\gamma}_{c}(z)=0.8{}\dot{\gamma}_{\mathrm{max}}(z)$.

$R_{s}$

$X=0$

$X=0.50$

$X=1$

(a)

(b)

Fig. 5: (a) Shear band profiles for mixtures of soft and hard particles in
different mixing ratios. The color scale illustrates the normalized ratio of
macroscopic angular velocity $\omega(r,z)$ scaled by the rotation velocity of
the bottom disk $\omega_{o}$. $R_{s}$ marks the split radius, which serves as
the origin of the shear band in space. (b) Graphical representation of
macroscopic angular velocity $\omega(r)$ normalized by the bottom disk angular
velocity $\omega_{o}$, as a function of radial position $r$ for a fraction of
hard particles $X=0.50$ at different heights. The inset shows the scaled
angular velocity $\omega(r)/\omega_{o}$ vs. radial position $r$ fitted by
Equation 8, also showing the features of the shear band.

##  5 Flow profile of granular mixtures

Once the flow of the mixtures relaxes to steady state shearing, the azimuthal
velocity is proportional to the driving rate $\omega_{o}$. Figure 5(a) shows
the scaled angular velocity profile $\omega(r,z)/\omega_{o}$. Within the
limits of assumptions in our simulations, we do not see a clear difference in
the flow profiles of the three cases of pure particles and mixtures for
$X=0,0.50$, and $1$. However, we observe that the shear zone of the mixture
$(X=0.5)$ is slightly more diffused compared to the pure case simulations
involving only soft $(X=0)$ or only hard particles $(X=1)$. This indicates a
wider shear band for the mixture, although it is not clearly concluded from
this figure. A closer and more precise introspection on the shear band
properties is therefore essential to conclude the shear zone properties of the
mixture. On this ground, we analyze the dimensionless ratio of the measured
angular velocity $\omega/\omega_{o}$ as a function of the radial coordinate
$r$. The dimensionless velocity profile in a split bottom shear cell is well
approximated by an error function 5, 10

$\frac{\omega(r)}{\omega_{o}}=0.5-0.5\mathrm{erf}\bigg{(}\frac{r-R_{c}}{W}\bigg{)}$
(8)

where $R_{c}$ and $W$ are the position and width of the shear band developed
above the split at $R_{s}$, respectively. Both the position and the width of
the shear band depend on the height $z$ in the system. Therefore, we extract
the position and the width of the shear band at layers of varying height along
the filling height in the shear cell by fitting the scaled $\omega(r)$ against
the radial position $r$ data. With the goal to extract quantitative data for
the shear band features, Figure 5(b) shows values of non-dimensional angular
velocity $\omega(r)/\omega_{o}$ at different heights in the bulk of granular
materials against the radial distance.

A quantitative study of the velocity gradient leads to the strain rate
$\dot{\gamma}$, which is calculated from the shear-free sheet strain tensor
components $d_{1}$ and $d_{2}$ 34. The shear intensity in the shear plane is
thus expressed in terms of velocity gradients ${\partial v_{\theta}}/{\partial
r}$ and ${\partial v_{\theta}}/{\partial z}$ as:

$\dot{\gamma}=\sqrt{d_{1}^{2}+d_{2}^{2}}=\frac{1}{2}\sqrt{{\bigg{(}\frac{\partial
v_{\theta}}{\partial
r}-\frac{v_{\theta}}{r}\bigg{)}}^{2}+{\bigg{(}\frac{\partial
v_{\theta}}{\partial z}\bigg{)}}^{2}}$ (9)

Figure 6(a) shows the strain rate $\dot{\gamma}(r,z)$ evaluated from Equation
9 and plotted as a function of radial and vertical position for the sample
with $X=0.50$. The dashed lines overlaid on top of the contour graph indicate
the center $R_{c}$ and the edges $R_{c}+W$ and $R_{c}-W$ of the shear band,
indicated in the figure, as obtained from the fit function in Equation 8.

$R_{s}$$R_{c}$$R_{c}+W$$R_{c}-W$(a)

$R_{c}$$R_{c}+W$$R_{c}-W$$z_{\mathrm{top}}$(b)

(c)

(d)

(e)

(f)

Fig. 6: (a) Contour plot of strain rate $\dot{\gamma}(r,z)$ as a function of
the radial and vertical position for the sample with $X=0.50$. The lines
overlaid on the figure indicate the center $R_{c}$ and the edges $R_{c}+W$ and
$R_{c}-W$ of the shear band as obtained from the fit function in Equation 8.
(b) Shear band center $R_{c}$ and edges $R_{c}+W$ and $R_{C}-W$ for different
fractions of hard particles $X$, (c) scaled shear band width
$W/W_{\mathrm{top}}$ as a function of scaled height $z/H$ and (d) relative
position of shear band $R_{c}/R_{s}$ as a function of scaled height $z/R_{s}$
for samples with $X=0,0.50$ and $1$. (e) Shear band width $W$ as a function of
the fraction of hard particles $X$ and (f) shear band position $R_{c}$ as a
function of the fraction of hard particles $X$ at $z=z_{\mathrm{top}}$. The
error bars in the graph illustrate the fitting error, which was determined by
fitting the numerical data of $\omega/\omega_{o}$ vs. $r$ to the equation
presented in Equation 8. The blue $\bigcirc$, black $\diamondsuit$ and red
$\triangle$ markers highlight the shear band width and position corresponding
to $X=0,0.50$ and $1$, respectively.

An important objective of this present study is to characterize the shear zone
profiles of granular materials for ensembles of soft slippery and hard
frictional materials. Figure 6(b) shows the shear band profiles for samples
with $X=0,0.50$ and $1$. As observed from the figure, the width of shear band
corresponding to a mixture sample $X=0.50$ is wider than that of the pure
mixtures, corresponding to $X=0$ and $1$. Also, the position of the shear band
corresponding to a mixture sample $X=0.50$ is shifted closer to the center
than the pure mixtures. Note that the profile for the scaled width
$W/W_{\mathrm{top}}$ as a function of $z/H$ shown in Figure 6(c) does not
follow a quadrant shape, as described by the analysis of Ries et al. 30, for
any of the samples. The reason behind this deviation needs further
exploration. However, a possible reason for this deviation can be due to the
low $W/d$ and $z/d$ ratios, where $d$ is the mean particle diameter. The
normalized position of the shear band for the different mixtures as a function
of bulk height is shown in Figure 6(d), and the trends agree with that shown
by Unger et al. 10. From both Figure 6(c) and (d), we observe a marked
deviation in the trend for $X=0.50$ compared to the pure samples with $X=0$
and $1$. Next, we analyze the width and position of the shear band for a
horizontal slice through the axis of the shear cell at $z=z_{\mathrm{top}}=16$
mm, as represented by the dashed line in Figure 6(b), for all samples of
different fraction of hard particles $X$ and show them in Figure 6(e) and (f),
respectively. Here, we find that the shear band gets wider as well as moves
inwards in a mixture of soft and hard spheres up to a fraction $X=0.50$, which
corresponds to an intermediate mixture of $50$% soft - $50$% hard particles.
With a further increase of $X$, the shear band narrows and moves outwards in
the shear cell. The non-monotonic trend of the width of the shear band with
increasing $X$ was also observed by Wang et al. 11.

From the above observations, we conclude that the ratio of hard spheres in a
mixture of hard frictional and soft slippery particles significantly affects
the flow profiles at the bulk scale. For up to intermediate ratios of hard
particles, the shear band gets wider by up to $22\%$, and also the center
shifts inwards by up to $2\%$, compared to the values for pure soft particles.
Both of these features are influenced by the effect of inter-particle friction
and stiffness, which are dominant for frictional hard particles 8.

Wang et al. 11 observed experimentally a notable difference in the shear band
properties between samples made of pure materials and mixtures. It was
observed that the shear zones of the mixtures are wider than those of the pure
materials. Unlike the experimental observations, which show a significant
widening of the shear zone for mixtures, our simulations show a rather small
effect of widening the shear zone. However, the distinct non-monotonic trend
of the shear band width as a function of the mixing ratio is the feature
observed in experiments, which is also captured in our simulations. In
addition, we observe the shift in shear band position for the mixtures, which
were not reported in the experimental work. It is important to note that the
capillary bridges resulting from small amounts of liquid between hydrogel
particles potentially have an impact on experiments, despite not being taken
into consideration in this numerical study. For larger quantities of liquid in
the mixtures, beyond the capillary regime, the material would exhibit a more
dispersed shear behavior.

##  6 Simplified cases to explore the role of inter-particle friction and
elastic modulus

Table 3:  Simplified cases to explore the effects of friction and elastic modulus on shear band characteristics separately Cases |  Elastic modulus ($E$) |  Friction coefficient ($\mu$)
---|---|---
$E_{\mathrm{soft}}$ $\mu_{\mathrm{real}}$ |  $E=50$ kPa for both species |  $\mu_{\mathrm{soft}}=0.005$ $|$ $\mu_{\mathrm{hard}}=0.20$
$E_{\mathrm{hard}}$ $\mu_{\mathrm{real}}$ |  $E=5000$ kPa for both species |  $\mu_{\mathrm{soft}}=0.005$ $|$ $\mu_{\mathrm{hard}}=0.20$
$E_{\mathrm{real}}$ $\mu_{\mathrm{soft}}$ |  $E_{\mathrm{soft}}=50$ kPa $|$ $E_{\mathrm{hard}}=5000$ kPa |  $\mu=0.005$ for both species
$E_{\mathrm{real}}$ $\mu_{\mathrm{hard}}$ |  $E_{\mathrm{soft}}=50$ kPa $|$ $E_{\mathrm{hard}}=5000$ kPa |  $\mu=0.20$ for both species

Following the findings of this study and the qualitative agreement of our
results with the observations noted by Wang et al. 11, it becomes evident that
mixtures of soft, slippery and hard frictional grains demonstrate shear bands
of variable width for increasing mixing ratio, which evolves non-monotonically
with increasing values of the ratio. Since the materials of interest have two
independent material parameters differing distinctly, namely friction and
elastic modulus, it is interesting to look into their individual effects on
the shear band characteristics. Hence, to explore the role of inter-particle
friction and elastic modulus on the flow profile of the granular mixture
systematically and independently of each other, we vary one parameter at a
time in additional discrete element simulations of a parametric nature, while
keeping the other parameter constant. Therefore, we conducted four case
studies, as outlined in Table 3. In these studies, our primary focus is on
analyzing the shear band characteristics at a specific height, denoted as
$z=z_{\mathrm{top}}$ in Figure 6(b) as a function of mixing ratio $X$. We then
proceed to compare the obtained results with those from the real simulations.

###  6.1 Shear band characteristics of granular mixtures

Luding 8 examined the effects of friction on shear bands via simulations of a
split-bottom shear cell. He showed that the presence of inter-particle
friction does not affect the qualitative behavior of the shear band. However,
the shear band moves inwards and gets narrower in the presence of friction.
This is justified by the fact that with increasing friction, there is less
relative displacement between the particles, particles tend to move more as a
rigid block, and hence, the shear band width decreases. Here, we vary the
inter-particle friction of the mixture by varying the fraction of hard
particles $X$, keeping the elastic modulus constant in cases
$E_{\mathrm{soft}}$ $\mu_{\mathrm{real}}$ and $E_{\mathrm{hard}}$
$\mu_{\mathrm{real}}$ as implied by the red and blue curves, respectively, in
Figure 7. For both the cases, we observe that the shear band width decreases
with increasing $X$ in Figure 7(a). Also, the shear band position moves
inwards with increasing $X$ as shown in Figure 7(b). Our results agree with
the findings of Luding on the effects of friction on shear band
characteristics.

On the contrary, varying the elastic modulus and considering same friction
coefficient for the two materials leads to a wider shear band formation with
increasing fraction of hard particles $X$, as demonstrated by cases
$E_{\mathrm{real}}$ $\mu_{\mathrm{soft}}$ and $E_{\mathrm{real}}$
$\mu_{\mathrm{hard}}$. This is implied by the green and orange curves in
Figure 7(c). However, for the same friction between the two species, varying
only the elastic modulus of the species has only a small effect on the
position of the shear band as a function of $X$ as shown by the green and
orange trends in Figure 7(d).

(a)

(b)

(c)

(d)

Fig. 7: Shear band (a) width $W$ and (b) position $R_{c}$ for fixed elastic
modulus and real inter-particle friction, represented by cases
$E_{\mathrm{soft}}$ $\mu_{\mathrm{real}}$ and $E_{\mathrm{hard}}$
$\mu_{\mathrm{real}}$, as compared with the real simulations, and shear band
(c) width $W$ and (d) position $R_{c}$ for real elastic modulus and fixed
inter-particle friction, represented by cases $E_{\mathrm{real}}$
$\mu_{\mathrm{soft}}$ and $E_{\mathrm{real}}$ $\mu_{\mathrm{hard}}$, as
compared with the real simulations, where all the features are plotted as a
function of fraction of hard particles $X$. The width and position of the
shear band are obtained at $z_{\mathrm{top}}=16$ mm. The error bars represent
the fitting error, determined by fitting the numerical data for angular
velocity to Equation 8.

Overall, there is a dual effect of friction and elastic modulus that
influences the shear band properties. While increasing friction tends to
reduce the width of the shear band with increasing $X$, increasing elastic
modulus plays the opposite role in increasing the width for increasing $X$.
For smaller values of $X$, the friction of soft particles is smaller and hence
is less dominant, i.e., it is easier for inter-particle contacts to slide.
With increasing $X$, there are more contacts between hard frictional
particles, and hence friction dominates over the effect of elastic modulus.

###  6.2 Macroscopic friction coefficient of granular mixtures

The local shear stress $\tau(r,z)$ plotted against the local normal stress
$P(r,z)$ is shown in Figure 8(a). Since the system is inhomogeneous in nature,
for a given pressure, we observe a wide range of local strain rate values
$\dot{\gamma}$, and we find that shear stress $\tau$ is higher for increasing
strain rate $\dot{\gamma}$. However, for values above the critical strain
rate, i.e., $\dot{\gamma}>\dot{\gamma}_{c}$, $\tau$ becomes almost independent
of the local strain. This means that $\tau/p$ is nearly constant for all data
points with strain rate $\dot{\gamma}>\dot{\gamma}_{c}$. A linear trend is
observed for the shear stress as a function of the normal stress, which is
fitted well by:

$\tau=\mu P\,,$ (10)

where $\mu$ is the macroscopic friction coefficient obtained from linear
fitting of the local shear stress and normal stress values.

We further analyze the macroscopic friction coefficient $\mu$ for different
mixtures as a function of $X$ for the real simulations and the simplified
cases and show the results in Figure 8(b). The macroscopic friction
coefficient $\mu$ increases as a quadratic function with $X$ for the
simplified cases of $E_{\mathrm{soft}}$ $\mu_{\mathrm{real}}$ and
$E_{\mathrm{hard}}\mu_{\mathrm{real}}$, i.e., same elastic modulus for both
materials and varying inter-particle friction coefficient. On the contrary,
$\mu$ increases linearly with $X$ for cases $E_{\mathrm{real}}$
$\mu_{\mathrm{soft}}$ and $E_{\mathrm{real}}$ $\mu_{\mathrm{hard}}$, i.e.
varying elastic modulus and same inter-particle friction coefficient for both
materials. Furthermore, the macroscopic friction coefficient $\mu$ for the
real simulations also increases as a quadratic function with $X$, which
suggests that $\mu$ for the real simulations is strongly dominated by the
varying inter-particle friction coefficient between the two species. The
elastic modulus does not play a significant role in influencing the $\mu$ of
the bulk behavior. Focusing on the functional form of the quadratic function
$\mu$ vs $X$ for the real simulations, the black solid line represents the
quadratic fitting equation $\mu=\mu_{\mathrm{soft}}+aX^{2}$, where
$\mu_{\mathrm{soft}}\approx 0.14$ is the macroscopic friction coefficient
corresponding to the pure soft particle mixtures and $a=0.18$ here depends on
the inter-particle friction coefficient values of pure soft and hard species.

(a)

(b)

Fig. 8: (a) Shear stress $\tau$ plotted against normal stress $P$. The lines
represent linear regression fits given by $\tau=\mu P$. (b) Macroscopic
friction coefficient $\mu$ as a function of the fraction of hard particles $X$
for the real simulations and the simplified cases. The solid lines indicate
linear and quadratic regression fits for the different cases. The error bars
in the graph illustrate the fitting error, which was determined by fitting the
numerical data of $\mu$ Vs $P$ to the equation presented in Equation 10.

##  7 Conclusions

This work investigated the shear zone characteristics in binary mixtures of
soft low frictional and hard frictional particles, via discrete element
simulations of mixtures inside a split-bottom shear cell. The ratio of the
volume of hard particles to the soft particles was varied parametrically to
observe its effects on measures of interest, including the shear band width
$W$, shear band position $R_{c}$ and the macroscopic friction coefficient
$\mu$.

By comparing the width and position of the shear zone for samples of pure soft
low frictional and for samples of pure hard frictional particles, it becomes
evident that both systems show somewhat similar shear band features. However,
mixtures of the two materials exhibit broader shear bands and a shifting of
the shear band position towards the center of the shear cell. Both these
effects demonstrate non-monotonicity for increasing fraction of hard
particles. Although the changes in shear band characteristics for different
mixtures are small, the non-monotonic trend of the shear band features as a
function of mixing ratio is an interesting effect of the shear behavior of
binary mixtures of this nature. We attribute the effects of shear band
widening in these mixtures to an interplay of the role of inter-particle
friction and particle stiffness.

Contrary to the shear band characteristics, the macroscopic friction
coefficient shows a monotonically increasing trend as a quadratic function
with an increasing fraction of hard particles in the mixtures. The macroscopic
friction coefficient is strongly influenced by the inter-particle friction
coefficient of the species present in the mixture, and the elastic modulus
seems to play a less significant role.

This work captures critical features of shear band formation in soft-hard
particle mixtures of low-high inter-particle friction, such as the non-
monotonic evolution of the shear band width and position for increasing mixing
ratios. However, the sensitivity of these features in our simulations is minor
as compared to that observed in the experiments by Wang et al. 11. Some of the
additional factors that might play a role in the experiments but are not
captured in our simulations include the following: (i) our simulations only
consider spherical particles and do not account for non-spherical shapes or
the actual particle size distribution, (ii) the real stiffness of the mustard
seeds is not taken into account; a sufficiently stiffer material is chosen for
the mustard seeds instead, and (iii) capillary bridges due to small amounts of
liquid between hydrogel particles might play a role in experiments which are
not considered here. The material would exhibit a more dispersed shear
behavior for larger quantities of water in the mixtures beyond the pendular
regime. Even with these discrepancies, our work captures the shear band
behavior observed experimentally and the evolution of its characteristics for
varying mixing ratios of soft and hard particles.

## Author Contributions

SR and VA designed the research and planned the parametric DEM simulations
which APS carried out. SR and VA wrote the first draft of the manuscript, and
TP reviewed and wrote with SR and VA the second draft. All authors reviewed
the results and approved the final version of the manuscript. SR secured the
research funding.

## Conflicts of interest

There are no conflicts to declare.

## Acknowledgements

We acknowledge Stefan Luding for providing the code for micro-macro data
transition and carefully reviewing this work. We thank Joshua Dijksman for
stimulating discussion on this work. Financial support through the Emerging
Talents Initiative (ETI Project No. ETI-Antrag_2022-1_Tech_13_Roy) and the DFG
grant for Granular Weissenberg Effect (Project No. PO472/40-1) are
acknowledged by SR and TP, respectively.

## Notes and references

  * Wortel _et al._ 2015 G. Wortel, T. Börzsönyi, E. Somfai, S. Wegner, B. Szabó, R. Stannarius and M. van Hecke, _Soft Matter_ , 2015, 11, 2570–2576.
  * Fischer _et al._ 2016 D. Fischer, T. Börzsönyi, D. S. Nasato, T. Pöschel and R. Stannarius, _New Journal of Physics_ , 2016, 18, 113006.
  * Fenistein and Van Hecke 2003 D. Fenistein and M. Van Hecke, _Nature_ , 2003, 425, 256–256.
  * Fenistein _et al._ 2004 D. Fenistein, J. W. van de Meent and M. van Hecke, _Physical Review Letters_ , 2004, 92, 094301.
  * Dijksman and Van Hecke 2010 J. A. Dijksman and M. Van Hecke, _Soft Matter_ , 2010, 6, 2901–2907.
  * Henann and Kamrin 2013 D. L. Henann and K. Kamrin, _Proceedings of the National Academy of Sciences_ , 2013, 110, 6730–6735.
  * Singh _et al._ 2014 A. Singh, V. Magnanimo, K. Saitoh and S. Luding, _Physical Review E_ , 2014, 90, 022202.
  * Luding 2007 S. Luding, _Particulate Science and Technology_ , 2007, 26, 33–42.
  * Faroux _et al._ 2022 D. Faroux, K. Washino, T. Tsuji and T. Tanaka, _Physical Review Fluids_ , 2022, 7, 084306.
  * Unger _et al._ 2004 T. Unger, J. Török, J. Kertész and D. E. Wolf, _Physical Review Letters_ , 2004, 92, 214301.
  * Wang _et al._ 2022 J. Wang, Z. Farmani, J. A. Dijksman, C. Lübeck, O. Speck and R. Stannarius, _Granular Matter_ , 2022, 24, 103.
  * Ashour _et al._ 2017 A. Ashour, T. Trittel, T. Börzsönyi and R. Stannarius, _Physical Review Fluids_ , 2017, 2, 123302.
  * Hong _et al._ 2017 X. Hong, M. Kohne, M. Morrell, H. Wang and E. R. Weeks, _Physical Review E_ , 2017, 96, 062605.
  * Harth _et al._ 2020 K. Harth, J. Wang, T. Börzsönyi and R. Stannarius, _Soft Matter_ , 2020, 16, 8013–8023.
  * Uñac _et al._ 2012 R. O. Uñac, A. M. Vidales, O. Benegas and I. Ippolito, _Powder technology_ , 2012, 225, 214–220.
  * Rudge _et al._ 2020 R. E. Rudge, J. P. Van De Sande, J. A. Dijksman and E. Scholten, _Soft Matter_ , 2020, 16, 3821–3831.
  * Wang _et al._ 2021 J. Wang, B. Fan, T. Pongó, K. Harth, T. Trittel, R. Stannarius, M. Illig, T. Börzsönyi and R. C. Hidalgo, _Soft Matter_ , 2021, 17, 4282–4295.
  * Beverloo _et al._ 1961 W. A. Beverloo, H. A. Leniger and J. Van de Velde, _Chemical Engineering Science_ , 1961, 15, 260–269.
  * Saitoh _et al._ 2015 K. Saitoh, V. Magnanimo and S. Luding, _Soft Matter_ , 2015, 11, 1253–1258.
  * Luding _et al._ 2022 S. Luding, K. Taghizadeh, C. Cheng and L. Kondic, _Soft Matter_ , 2022, 18, 1868–1884.
  * Götz _et al._ 2022 H. Götz, A. Santarossa, A. Sack, T. Pöschel and P. Müller, _Granular Matter_ , 2022, 24, 1–9.
  * Weinhart _et al._ 2020 T. Weinhart, L. Orefice, M. Post, M. P. van Schrojenstein Lantman, I. F. Denissen, D. R. Tunuguntla, J. Tsang, H. Cheng, M. Y. Shaheen, H. Shi _et al._ , _Computer Physics Communications_ , 2020, 249, 107129\.
  * Sakaie _et al._ 2008 K. Sakaie, D. Fenistein, T. J. Carroll, M. van Hecke and P. Umbanhowar, _Europhysics Letters_ , 2008, 84, 38001.
  * Brilliantov _et al._ 1996 N. V. Brilliantov, F. Spahn, J.-M. Hertzsch and T. Pöschel, _Physical Review E_ , 1996, 53, 5382.
  * Müller and Pöschel 2011 P. Müller and T. Pöschel, _Physical Review E_ , 2011, 84, 021302\.
  * Mindlin 1949 R. D. Mindlin, _Journal of Applied Mechanics_ , 1949, 16, 259–268.
  * Parteli and Pöschel 2016 E. J. Parteli and T. Pöschel, _Powder Technology_ , 2016, 288, 96–102.
  * Lätzel _et al._ 2000 M. Lätzel, S. Luding and H. J. Herrmann, _Granular Matter_ , 2000, 2, 123–135.
  * Singh _et al._ 2015 A. Singh, V. Magnanimo and S. Luding, _New Journal of Physics_ , 2015, 17, 043028.
  * Ries _et al._ 2007 A. Ries, D. E. Wolf and T. Unger, _Physical Review E_ , 2007, 76, 051301\.
  * Wood 1990 D. M. Wood, _Soil behaviour and critical state soil mechanics_ , Cambridge University Press, 1990.
  * Luding 2008 S. Luding, _Particuology_ , 2008, 6, 501–505.
  * Szabó _et al._ 2014 B. Szabó, J. Török, E. Somfai, S. Wegner, R. Stannarius, A. Böse, G. Rose, F. Angenstein and T. Börzsönyi, _Physical Review E_ , 2014, 90, 032205.
  * Depken _et al._ 2006 M. Depken, W. van Saarloos and M. van Hecke, _Physical Review E_ , 2006, 73, 031302.