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 limit_damping
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:
If two particles are moving away from each other while in contact, there is a possibility that the particles could experience an effective attractive force due to damping. If the optional limit_damping keyword is used, this option will zero out the normal component of the force if there is an effective attractive force. This keyword cannot be used with the JKR or DMT models.
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, INTEL, KOKKOS, OPENMP and OPT packages, respectively. They are only enabled if LAMMPS was built with those packages. See the Build package 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 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 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.