# pair_style granular command¶

## Examples¶

```
pair_style granular
pair_coeff * * hooke 1000.0 50.0 tangential linear_nohistory 1.0 0.4 damping mass_velocity
pair_style granular
pair_coeff * * hooke 1000.0 50.0 tangential linear_history 500.0 1.0 0.4 damping mass_velocity
pair_style granular
pair_coeff * * hertz 1000.0 50.0 tangential mindlin 1000.0 1.0 0.4
pair_style granular
pair_coeff * * hertz/material 1e8 0.3 0.3 tangential mindlin_rescale NULL 1.0 0.4 damping tsuji
pair_style granular
pair_coeff 1 * jkr 1000.0 500.0 0.3 10 tangential mindlin 800.0 1.0 0.5 rolling sds 500.0 200.0 0.5 twisting marshall
pair_coeff 2 2 hertz 200.0 100.0 tangential linear_history 300.0 1.0 0.1 rolling sds 200.0 100.0 0.1 twisting marshall
pair_style granular
pair_coeff 1 1 dmt 1000.0 50.0 0.3 0.0 tangential mindlin NULL 0.5 0.5 rolling sds 500.0 200.0 0.5 twisting marshall
pair_coeff 2 2 dmt 1000.0 50.0 0.3 10.0 tangential mindlin NULL 0.5 0.1 rolling sds 500.0 200.0 0.1 twisting marshall
```

## Description¶

The *granular* styles support a variety of options for the normal,
tangential, rolling and twisting forces resulting from contact between
two granular particles. This expands on the options offered by the
pair gran/* pair styles. The total computed forces
and torques are the sum of various models selected for the normal,
tangential, rolling and twisting modes of motion.

All model choices and parameters are entered in the
pair_coeff command, as described below. Unlike
e.g. pair gran/hooke, coefficient values are not
global, but can be set to different values for different combinations
of particle types, as determined by the pair_coeff
command. If the contact model choice is the same for two particle
types, the mixing for the cross-coefficients can be carried out
automatically. This is shown in the last example, where model
choices are the same for type 1 - type 1 as for type 2 - type2
interactions, but coefficients are different. In this case, the
mixed coefficients for type 1 - type 2 interactions can be determined from
mixing rules discussed below. For additional flexibility,
coefficients as well as model forms can vary between particle types,
as shown in the fourth example: type 1 - type 1 interactions are based
on a Johnson-Kendall-Roberts normal contact model and 2-2 interactions
are based on a DMT cohesive model (see below). In that example, 1-1
and 2-2 interactions have different model forms, in which case mixing of
coefficients cannot be determined, so 1-2 interactions must be
explicitly defined via the *pair_coeff 1 ** command, otherwise an
error would result.

The first required keyword for the *pair_coeff* command is the normal
contact model. Currently supported options for normal contact models
and their required arguments are:

*hooke*: \(k_n\), \(\eta_{n0}\) (or \(e\))*hertz*: \(k_n\), \(\eta_{n0}\) (or \(e\))*hertz/material*: E, \(\eta_{n0}\) (or \(e\)), \(\nu\)*dmt*: E, \(\eta_{n0}\) (or \(e\)), \(\nu\), \(\gamma\)*jkr*: E, \(\eta_{n0}\) (or \(e\)), \(\nu\), \(\gamma\)

Here, \(k_n\) is spring stiffness (with units that depend on model
choice, see below); \(\eta_{n0}\) is a damping prefactor (or, in its
place a coefficient of restitution \(e\), depending on the choice of
damping mode, see below); E is Young’s modulus in units of
*force*/*length*^2, i.e. *pressure*; \(\nu\) is Poisson’s ratio and
\(\gamma\) is a surface energy density, in units of
*energy*/*length*^2.

For the *hooke* model, the normal, elastic component of force acting
on particle *i* due to contact with particle *j* is given by:

Where \(\delta_{ij} = R_i + R_j - \|\mathbf{r}_{ij}\|\) is the particle
overlap, \(R_i, R_j\) are the particle radii, \(\mathbf{r}_{ij} = \mathbf{r}_i - \mathbf{r}_j\) is the vector separating the two
particle centers (note the i-j ordering so that \(F_{ne}\) is
positive for repulsion), and \(\mathbf{n} = \frac{\mathbf{r}_{ij}}{\|\mathbf{r}_{ij}\|}\). Therefore,
for *hooke*, the units of the spring constant \(k_n\) are
*force*/*distance*, or equivalently *mass*/*time^2*.

For the *hertz* model, the normal component of force is given by:

Here, \(R_{eff} = \frac{R_i R_j}{R_i + R_j}\) is the effective
radius, denoted for simplicity as *R* from here on. For *hertz*, the
units of the spring constant \(k_n\) are *force*/*length*^2, or
equivalently *pressure*.

For the *hertz/material* model, the force is given by:

Here, \(E_{eff} = E = \left(\frac{1-\nu_i^2}{E_i} + \frac{1-\nu_j^2}{E_j}\right)^{-1}\) is the effective Young’s
modulus, with \(\nu_i, \nu_j\) the Poisson ratios of the particles of
types *i* and *j*. Note that if the elastic modulus and the shear
modulus of the two particles are the same, the *hertz/material* model
is equivalent to the *hertz* model with \(k_n = 4/3 E_{eff}\)

The *dmt* model corresponds to the
(Derjaguin-Muller-Toporov) cohesive model, where the force
is simply Hertz with an additional attractive cohesion term:

The *jkr* model is the (Johnson-Kendall-Roberts) model,
where the force is computed as:

Here, \(a\) is the radius of the contact zone, related to the overlap \(\delta\) according to:

LAMMPS internally inverts the equation above to solve for *a* in terms
of \(\delta\), then solves for the force in the previous
equation. Additionally, note that the JKR model allows for a tensile
force beyond contact (i.e. for \(\delta < 0\)), up to a maximum of
\(3\pi\gamma R\) (also known as the ‘pull-off’ force). Note that this
is a hysteretic effect, where particles that are not contacting
initially will not experience force until they come into contact
\(\delta \geq 0\); as they move apart and (\(\delta < 0\)), they
experience a tensile force up to \(3\pi\gamma R\), at which point they
lose contact.

In addition, the normal force is augmented by a damping term of the following general form:

Here, \(\mathbf{v}_{n,rel} = (\mathbf{v}_j - \mathbf{v}_i) \cdot \mathbf{n}\ \mathbf{n}\) is the component of relative velocity along \(\mathbf{n}\).

The optional *damping* keyword to the *pair_coeff* command followed by
a keyword determines the model form of the damping factor \(\eta_n\),
and the interpretation of the \(\eta_{n0}\) or \(e\) coefficients
specified as part of the normal contact model settings. The *damping*
keyword and corresponding model form selection may be appended
anywhere in the *pair coeff* command. Note that the choice of damping
model affects both the normal and tangential damping (and depending on
other settings, potentially also the twisting damping). The options
for the damping model currently supported are:

*velocity**mass_velocity**viscoelastic**tsuji*

If the *damping* keyword is not specified, the *viscoelastic* model is
used by default.

For *damping velocity*, the normal damping is simply equal to the
user-specified damping coefficient in the *normal* model:

Here, \(\eta_{n0}\) is the damping coefficient specified for the normal
contact model, in units of *mass*/*time*.

For *damping mass_velocity*, the normal damping is given by:

Here, \(\eta_{n0}\) is the damping coefficient specified for the normal
contact model, in units of *mass*/*time* and
\(m_{eff} = m_i m_j/(m_i + m_j)\) is the effective mass.
Use *damping mass_velocity* to reproduce the damping behavior of
*pair gran/hooke/**.

The *damping viscoelastic* model is based on the viscoelastic
treatment of (Brilliantov et al), where the normal
damping is given by:

Here, *a* is the contact radius, given by \(a =\sqrt{R\delta}\)
for all models except *jkr*, for which it is given implicitly according
to \(\delta = a^2/R - 2\sqrt{\pi \gamma a/E}\). For *damping viscoelastic*,
\(\eta_{n0}\) is in units of 1/(*time***distance*).

The *tsuji* model is based on the work of (Tsuji et al). Here, the damping coefficient specified as part of
the normal model is interpreted as a restitution coefficient
\(e\). The damping constant \(\eta_n\) is given by:

For normal contact models based on material parameters, \(k_n = 4/3Ea\). The parameter \(\alpha\) is related to the restitution
coefficient *e* according to:

The dimensionless coefficient of restitution \(e\) specified as part of the normal contact model parameters should be between 0 and 1, but no error check is performed on this.

The total normal force is computed as the sum of the elastic and damping components:

The *pair_coeff* command also requires specification of the tangential
contact model. The required keyword *tangential* is expected, followed
by the model choice and associated parameters. Currently supported
tangential model choices and their expected parameters are as follows:

*linear_nohistory*: \(x_{\gamma,t}\), \(\mu_s\)*linear_history*: \(k_t\), \(x_{\gamma,t}\), \(\mu_s\)*mindlin*: \(k_t\) or NULL, \(x_{\gamma,t}\), \(\mu_s\)*mindlin/force*: \(k_t\) or NULL, \(x_{\gamma,t}\), \(\mu_s\)*mindlin_rescale*: \(k_t\) or NULL, \(x_{\gamma,t}\), \(\mu_s\)*mindlin_rescale/force*: \(k_t\) or NULL, \(x_{\gamma,t}\), \(\mu_s\)

Here, \(x_{\gamma,t}\) is a dimensionless multiplier for the normal damping \(\eta_n\) that determines the magnitude of the tangential damping, \(\mu_t\) is the tangential (or sliding) friction coefficient, and \(k_t\) is the tangential stiffness coefficient.

For *tangential linear_nohistory*, a simple velocity-dependent Coulomb
friction criterion is used, which mimics the behavior of the *pair
gran/hooke* style. The tangential force \(\mathbf{F}_t\) is given by:

The tangential damping force \(\mathbf{F}_\mathrm{t,damp}\) is given by:

The tangential damping prefactor \(\eta_t\) is calculated by scaling the normal damping \(\eta_n\) (see above):

The normal damping prefactor \(\eta_n\) is determined by the choice
of the *damping* keyword, as discussed above. Thus, the *damping*
keyword also affects the tangential damping. The parameter
\(x_{\gamma,t}\) is a scaling coefficient. Several works in the
literature use \(x_{\gamma,t} = 1\) (Marshall,
Tsuji et al, Silbert et al). The relative
tangential velocity at the point of contact is given by
\(\mathbf{v}_{t, rel} = \mathbf{v}_{t} - (R_i\mathbf{\Omega}_i + R_j\mathbf{\Omega}_j) \times \mathbf{n}\), where \(\mathbf{v}_{t} = \mathbf{v}_r - \mathbf{v}_r\cdot\mathbf{n}\ \mathbf{n}\),
\(\mathbf{v}_r = \mathbf{v}_j - \mathbf{v}_i\) .
The direction of the applied force is \(\mathbf{t} = \mathbf{v_{t,rel}}/\|\mathbf{v_{t,rel}}\|\) .

The normal force value \(F_{n0}\) used to compute the critical force
depends on the form of the contact model. For non-cohesive models
(*hertz*, *hertz/material*, *hooke*), it is given by the magnitude of
the normal force:

For cohesive models such as *jkr* and *dmt*, the critical force is
adjusted so that the critical tangential force approaches \(\mu_t F_{pulloff}\), see Marshall, equation 43, and
Thornton. For both models, \(F_{n0}\) takes the
form:

Where \(F_{pulloff} = 3\pi \gamma R\) for *jkr*, and
\(F_{pulloff} = 4\pi \gamma R\) for *dmt*.

The remaining tangential options all use accumulated tangential
displacement (i.e. contact history), except for the options
*mindlin/force* and *mindlin_rescale/force*, that use accumulated
tangential force instead, and are discussed further below.
The accumulated tangential displacement is discussed in details below
in the context of the *linear_history* option. The same treatment of
the accumulated displacement applies to the other options as well.

For *tangential linear_history*, the tangential force is given by:

Here, \(\mathbf{\xi}\) is the tangential displacement accumulated during the entire duration of the contact:

This accumulated tangential displacement must be adjusted to account for changes in the frame of reference of the contacting pair of particles during contact. This occurs due to the overall motion of the contacting particles in a rigid-body-like fashion during the duration of the contact. There are two modes of motion that are relevant: the ‘tumbling’ rotation of the contacting pair, which changes the orientation of the plane in which tangential displacement occurs; and ‘spinning’ rotation of the contacting pair about the vector connecting their centers of mass (\(\mathbf{n}\)). Corrections due to the former mode of motion are made by rotating the accumulated displacement into the plane that is tangential to the contact vector at each step, or equivalently removing any component of the tangential displacement that lies along \(\mathbf{n}\), and rescaling to preserve the magnitude. This follows the discussion in Luding, see equation 17 and relevant discussion in that work:

Here, \(\mathbf{\xi'}\) is the accumulated displacement prior to the current time step and \(\mathbf{\xi}\) is the corrected displacement. Corrections to the displacement due to the second mode of motion described above (rotations about \(\mathbf{n}\)) are not currently implemented, but are expected to be minor for most simulations.

Furthermore, when the tangential force exceeds the critical force, the tangential displacement is re-scaled to match the value for the critical force (see Luding, equation 20 and related discussion):

The tangential force is added to the total normal force (elastic plus damping) to produce the total force on the particle. The tangential force also acts at the contact point (defined as the center of the overlap region) to induce a torque on each particle according to:

For *tangential mindlin*, the Mindlin no-slip solution
is used which differs from the *linear_history* option by an additional factor
of \(a\), the radius of the contact region. The tangential force is given by:

Here, \(a\) is the radius of the contact region, given by \(a =\sqrt{R\delta}\)
for all normal contact models, except for *jkr*, where it is given
implicitly by \(\delta = a^2/R - 2\sqrt{\pi \gamma a/E}\), see
discussion above. To match the Mindlin solution, one should set
\(k_t = 8G_{eff}\), where \(G_{eff}\) is the effective shear modulus given by:

where \(G\) is the shear modulus, related to Young’s modulus \(E\)
and Poisson’s ratio \(\nu\) by \(G = E/(2(1+\nu))\). This can also be
achieved by specifying *NULL* for \(k_t\), in which case a
normal contact model that specifies material parameters \(E\) and
\(\nu\) is required (e.g. *hertz/material*, *dmt* or *jkr*). In this
case, mixing of the shear modulus for different particle types *i* and
*j* is done according to the formula above.

Note

The radius of the contact region \(a\) depends on the normal overlap.
As a result, the tangential force for *mindlin* can change due to
a variation in normal overlap, even with no change in tangential displacement.

For *tangential mindlin/force*, the accumulated elastic tangential force
characterizes the contact history, instead of the accumulated tangential
displacement. This prevents the dependence of the tangential force on the
normal overlap as noted above. The tangential force is given by:

The increment of the elastic component of the tangential force \(\mathbf{F}_{te}\) is given by:

The changes in frame of reference of the contacting pair of particles during contact are accounted for by the same formula as above, replacing the accumulated tangential displacement \(\xi\), by the accumulated tangential elastic force \(F_{te}\). When the tangential force exceeds the critical force, the tangential force is directly re-scaled to match the value for the critical force:

The same rules as those described for *mindlin* apply regarding the tangential
stiffness and mixing of the shear modulus for different particle types.

The *mindlin_rescale* option uses the same form as *mindlin*, but the
magnitude of the tangential displacement is re-scaled as the contact
unloads, i.e. if \(a < a_{t_{n-1}}\):

Here, \(t_{n-1}\) indicates the value at the previous time step. This rescaling accounts for the fact that a decrease in the contact area upon unloading leads to the contact being unable to support the previous tangential loading, and spurious energy is created without the rescaling above (Walton ).

Note

For *mindlin*, a decrease in the tangential force already occurs as the
contact unloads, due to the dependence of the tangential force on the normal
force described above. By re-scaling \(\xi\), *mindlin_rescale*
effectively re-scales the tangential force twice, i.e., proportionally to
\(a^2\). This peculiar behavior results from use of the accumulated
tangential displacement to characterize the contact history. Although
*mindlin_rescale* remains available for historic reasons and backward
compatibility purposes, it should be avoided in favor of *mindlin_rescale/force*.

The *mindlin_rescale/force* option uses the same form as *mindlin/force*,
but the magnitude of the tangential elastic force is re-scaled as the contact
unloads, i.e. if \(a < a_{t_{n-1}}\):

This approach provides a better approximation of the Mindlin-Deresiewicz
laws and is more consistent than *mindlin_rescale*. See discussions in
Thornton et al, 2013, particularly equation 18(b) of that
work and associated discussion, and Agnolin and Roux, 2007,
particularly Appendix A.

The optional *rolling* keyword enables rolling friction, which resists
pure rolling motion of particles. The options currently supported are:

*none**sds*: \(k_{roll}\), \(\gamma_{roll}\), \(\mu_{roll}\)

If the *rolling* keyword is not specified, the model defaults to *none*.

For *rolling sds*, rolling friction is computed via a
spring-dashpot-slider, using a ‘pseudo-force’ formulation, as detailed
by Luding. Unlike the formulation in
Marshall, this allows for the required adjustment of
rolling displacement due to changes in the frame of reference of the
contacting pair. The rolling pseudo-force is computed analogously to
the tangential force:

Here, \(\mathbf{v}_{roll} = -R(\mathbf{\Omega}_i - \mathbf{\Omega}_j) \times \mathbf{n}\) is the relative rolling velocity, as given in Wang et al and Luding. This differs from the expressions given by Kuhn and Bagi and used in Marshall; see Wang et al for details. The rolling displacement is given by:

A Coulomb friction criterion truncates the rolling pseudo-force if it exceeds a critical value:

Here, \(\mathbf{k} = \mathbf{v}_{roll}/\|\mathbf{v}_{roll}\|\) is the direction of the pseudo-force. As with tangential displacement, the rolling displacement is rescaled when the critical force is exceeded, so that the spring length corresponds the critical force. Additionally, the displacement is adjusted to account for rotations of the frame of reference of the two contacting particles in a manner analogous to the tangential displacement.

The rolling pseudo-force does not contribute to the total force on either particle (hence ‘pseudo’), but acts only to induce an equal and opposite torque on each particle, according to:

The optional *twisting* keyword enables twisting friction, which
resists rotation of two contacting particles about the vector
\(\mathbf{n}\) that connects their centers. The options currently
supported are:

*none**sds*: \(k_{twist}\), \(\gamma_{twist}\), \(\mu_{twist}\)*marshall*

If the *twisting* keyword is not specified, the model defaults to *none*.

For both *twisting sds* and *twisting marshall*, a history-dependent
spring-dashpot-slider is used to compute the twisting torque. Because
twisting displacement is a scalar, there is no need to adjust for
changes in the frame of reference due to rotations of the particle
pair. The formulation in Marshall therefore provides
the most straightforward treatment:

Here \(\xi_{twist} = \int_{t_0}^t \Omega_{twist} (\tau) \mathrm{d}\tau\) is the twisting angular displacement, and \(\Omega_{twist} = (\mathbf{\Omega}_i - \mathbf{\Omega}_j) \cdot \mathbf{n}\) is the relative twisting angular velocity. The torque is then truncated according to:

Similar to the sliding and rolling displacement, the angular displacement is rescaled so that it corresponds to the critical value if the twisting torque exceeds this critical value:

For *twisting sds*, the coefficients \(k_{twist}, \gamma_{twist}\)
and \(\mu_{twist}\) are simply the user input parameters that follow
the *twisting sds* keywords in the *pair_coeff* command.

For *twisting_marshall*, the coefficients are expressed in terms of
sliding friction coefficients, as discussed in
Marshall (see equations 32 and 33 of that work):

Finally, the twisting torque on each particle is given by:

The *granular* pair style can reproduce the behavior of the
*pair gran/** styles with the appropriate settings (some very
minor differences can be expected due to corrections in
displacement history frame-of-reference, and the application
of the torque at the center of the contact rather than
at each particle). The first example above
is equivalent to *pair gran/hooke 1000.0 NULL 50.0 50.0 0.4 1*.
The second example is equivalent to
*pair gran/hooke/history 1000.0 500.0 50.0 50.0 0.4 1*.
The third example is equivalent to
*pair gran/hertz/history 1000.0 500.0 50.0 50.0 0.4 1*.

LAMMPS automatically sets pairwise cutoff values for *pair_style
granular* based on particle radii (and in the case of *jkr* pull-off
distances). In the vast majority of situations, this is adequate.
However, a cutoff value can optionally be appended to the *pair_style
granular* command to specify a global cutoff (i.e. a cutoff for all
atom types). Additionally, the optional *cutoff* keyword can be passed
to the *pair_coeff* command, followed by a cutoff value. This will
set a pairwise cutoff for the atom types in the *pair_coeff* command.
These options may be useful in some rare cases where the automatic
cutoff determination is not sufficient, e.g. if particle diameters
are being modified via the *fix adapt* command. In that case, the
global cutoff specified as part of the *pair_style granular* command
is applied to all atom types, unless it is overridden for a given atom
type combination by the *cutoff* value specified in the *pair coeff*
command. If *cutoff* is only specified in the *pair coeff* command
and no global cutoff is appended to the *pair_style granular* command,
then LAMMPS will use that cutoff for the specified atom type
combination, and automatically set pairwise cutoffs for the remaining
atom types.

Styles with a *gpu*, *intel*, *kk*, *omp*, or *opt* suffix are
functionally the same as the corresponding style without the suffix.
They have been optimized to run faster, depending on your available
hardware, as discussed on the Speed packages doc
page. The accelerated styles take the same arguments and should
produce the same results, except for round-off and precision issues.

These accelerated styles are part of the GPU, USER-INTEL, KOKKOS, USER-OMP and OPT packages, respectively. They are only enabled if LAMMPS was built with those packages. See the Build package doc page for more info.

You can specify the accelerated styles explicitly in your input script by including their suffix, or you can use the -suffix command-line switch when you invoke LAMMPS, or you can use the suffix command in your input script.

See the Speed packages doc page for more instructions on how to use the accelerated styles effectively.

## Mixing, shift, table, tail correction, restart, rRESPA info¶

The pair_modify mix, shift, table, and tail options are not relevant for granular pair styles.

Mixing of coefficients is carried out using geometric averaging for
most quantities, e.g. if friction coefficient for type 1-type 1
interactions is set to \(\mu_1\), and friction coefficient for type
2-type 2 interactions is set to \(\mu_2\), the friction coefficient
for type1-type2 interactions is computed as \(\sqrt{\mu_1\mu_2}\)
(unless explicitly specified to a different value by a *pair_coeff 1 2
…* command). The exception to this is elastic modulus, only
applicable to *hertz/material*, *dmt* and *jkr* normal contact
models. In that case, the effective elastic modulus is computed as:

If the *i-j* coefficients \(E_{ij}\) and \(\nu_{ij}\) are
explicitly specified, the effective modulus is computed as:

or

These pair styles write their information to binary restart files, so a pair_style command does not need to be specified in an input script that reads a restart file.

These pair styles can only be used via the *pair* keyword of the
run_style respa command. They do not support the
*inner*, *middle*, *outer* keywords.

The single() function of these pair styles returns 0.0 for the energy of a pairwise interaction, since energy is not conserved in these dissipative potentials. It also returns only the normal component of the pairwise interaction force. However, the single() function also calculates 12 extra pairwise quantities. The first 3 are the components of the tangential force between particles I and J, acting on particle I. The fourth is the magnitude of this tangential force. The next 3 (5-7) are the components of the rolling torque acting on particle I. The next entry (8) is the magnitude of the rolling torque. The next entry (9) is the magnitude of the twisting torque acting about the vector connecting the two particle centers. The last 3 (10-12) are the components of the vector connecting the centers of the two particles (x_I - x_J).

These extra quantities can be accessed by the compute pair/local command, as *p1*, *p2*, …,
*p12*.

## Restrictions¶

All the granular pair styles are part of the GRANULAR package. It is only enabled if LAMMPS was built with that package. See the Build package doc page for more info.

These pair styles require that atoms store torque and angular velocity
(omega) as defined by the atom_style. They also
require a per-particle radius is stored. The *sphere* atom style does
all of this.

This pair style requires you to use the comm_modify vel yes command so that velocities are stored by ghost atoms.

These pair styles will not restart exactly when using the read_restart command, though they should provide statistically similar results. This is because the forces they compute depend on atom velocities. See the read_restart command for more details.

## Default¶

For the *pair_coeff* settings: *damping viscoelastic*, *rolling none*,
*twisting none*.

## References¶

**(Brilliantov et al, 1996)** Brilliantov, N. V., Spahn, F., Hertzsch,
J. M., & Poschel, T. (1996). Model for collisions in granular
gases. Physical review E, 53(5), 5382.

**(Tsuji et al, 1992)** Tsuji, Y., Tanaka, T., & Ishida,
T. (1992). Lagrangian numerical simulation of plug flow of
cohesionless particles in a horizontal pipe. Powder technology, 71(3),
239-250.

**(Johnson et al, 1971)** Johnson, K. L., Kendall, K., & Roberts,
A. D. (1971). Surface energy and the contact of elastic
solids. Proc. R. Soc. Lond. A, 324(1558), 301-313.

**Derjaguin et al, 1975)** Derjaguin, B. V., Muller, V. M., & Toporov,
Y. P. (1975). Effect of contact deformations on the adhesion of
particles. Journal of Colloid and interface science, 53(2), 314-326.

**(Luding, 2008)** Luding, S. (2008). Cohesive, frictional powders:
contact models for tension. Granular matter, 10(4), 235.

**(Marshall, 2009)** Marshall, J. S. (2009). Discrete-element modeling
of particulate aerosol flows. Journal of Computational Physics,
228(5), 1541-1561.

**(Silbert, 2001)** Silbert, L. E., Ertas, D., Grest, G. S., Halsey,
T. C., Levine, D., & Plimpton, S. J. (2001). Granular flow down an
inclined plane: Bagnold scaling and rheology. Physical Review E,
64(5), 051302.

**(Kuhn and Bagi, 2005)** Kuhn, M. R., & Bagi, K. (2004). Contact
rolling and deformation in granular media. International journal of
solids and structures, 41(21), 5793-5820.

**(Wang et al, 2015)** Wang, Y., Alonso-Marroquin, F., & Guo,
W. W. (2015). Rolling and sliding in 3-D discrete element
models. Particuology, 23, 49-55.

**(Thornton, 1991)** Thornton, C. (1991). Interparticle sliding in the
presence of adhesion. J. Phys. D: Appl. Phys. 24 1942

**(Mindlin, 1949)** Mindlin, R. D. (1949). Compliance of elastic bodies
in contact. J. Appl. Mech., ASME 16, 259-268.

**(Thornton et al, 2013)** Thornton, C., Cummins, S. J., & Cleary,
P. W. (2013). An investigation of the comparative behavior of
alternative contact force models during inelastic collisions. Powder
Technology, 233, 30-46.

**(Otis R. Walton)** Walton, O.R., Personal Communication

**(Mindlin and Deresiewicz, 1953)** Mindlin, R.D., & Deresiewicz, H (1953).
Elastic Spheres in Contact under Varying Oblique Force.
J. Appl. Mech., ASME 20, 327-344.

**(Agnolin and Roux 2007)** Agnolin, I. & Roux, J-N. (2007).
Internal states of model isotropic granular packings.
I. Assembling process, geometry, and contact networks. Phys. Rev. E, 76, 061302.