normal model luding

Syntax

model luding [other model_type/model_name pairs as described here ] settings keyword values

zero or more keyword/value pairs may be appended after the keyword settings (after all models are specified)

tangential_damping values = 'on' or 'off'
  on = activates tangential damping
  off = no tangential damping
viscous = 'on' or 'off'
  on = restitution coefficient varies with a local Stokes number of the particle. Requires additional global properties to be specified
  off = no modification to the restitution coefficient
useCharacteristicVelocity values = 'on' or 'off'
  on = if both characteristicVelocity and elasticStiffness are defined, the characteristicVelocity will be used
  off = if both characteristicVelocity and elasticStiffness are defined, the elasticStiffness will be used [default]

Associated material properties

Material properties

youngsModulus ( $Y$ ): Young’s modulus of the material [pressure]
poissonsRatio ( $\mu$ ): Poisson’s ratio of the material [ $\cdot$ ]

Material interaction properties

coefficientRestitution ( $e$ ): coefficient of restitution of the two materials [ $\cdot$ ]
elasticStiffness ( $k_1$ ): elastic (normal) stiffness of the two materials in contact [force/length] (required only if characteristicVelocity is not defined)
coefficientMaxElasticStiffness ( $\hat{k}_2/k_1$ ): ratio between the maximum elastic stiffness $\hat{k}_2$ and the normal stiffness $k_1$ [ $\cdot$ ]
coefficientAdhesionStiffness ( $k_c/k_1$ ): ratio between the adhesion stiffness $k_c$ and the normal stiffness $k_1$ [ $\cdot$ ]
coefficientPlasticityDepth ( $\phi_f$ ): coefficient of plasticity depth [ $\cdot$ ]
pullOffForce ( $f_0$ ): constant pull of force (usually negative) [force] (optional)
FluidViscosity ( $\mu_f$ ): dynamic viscosity of the fluid between two particles [pressure*time] (required only if viscous on)
CriticalStokes ( $c_S$ ): critical Stokes number [ $\cdot$ ] (required only if viscous on)
MaximumRestitution ( $e_{max}$ ): maximum coefficient of restitution [ $\cdot$ ] (required only if viscous on)

Global scalars

characteristicVelocity ( $v_0$ ): characteristic impact velocity [length/time] (required only if elasticStiffness is not defined)

Description

This granular model implements the adhesive, elasto-plastic modification of the Hooke contact model according to Luding. Similarly to the Hooke model, it calculates the force between two granular particles in contact. In particular, this model allows to specify a hysteric behaviour, meaning there is a different behaviour between loading and unloading.

The normal force is given by

$F_n = f^{hys} - \gamma_n v_{n,ij},$

where the damping term $- \gamma_n v_{n,ij}$ is made zero with the setting tangential_damping off, and the adhesive, plastic (hysteretic) force is defined as follows:

$f^{hys} = f_0 + k_1 \delta\qquad if\quad k_2 (\delta - \delta_0) \geq k_1 \delta f^{hys} = f_0 + k_2 (\delta - \delta_0)\quad if\quad k_1 \delta > k_2 (\delta - \delta_0) > -k_c \delta f^{hys} = f_0 - k_c \delta\qquad if -k_c \delta \geq k_2 (\delta - \delta_0)$

where $k_1 \leq k_2 \leq \hat{k}_2$ (see right figure below). The lines with slope $k_1$ and $-k_c$ define the range of possible force values. Between these two extremes, unloading and reloading follow a line with slope $k_2$ , which interpolates between $k_1$ and a maximum stiffness $\hat{k}_2$ .

During initial loading the force increases linearly with the overall $\delta$ , until the maximum overlap $\delta_{max}$ is reached. The line with slope $k_1$ thus defines the maximum force possible for a given $\delta$ . During unloading the force drops on a line with slope $k_2$ , which depends, in general, on $\delta_{max}$ as follows:

$k_2 = \hat{k}_2\qquad if\quad \delta_{max} \geq \delta_{max}^* k_2 = k_1 + ( \hat{k}_2 - k_1 ) \frac{\delta_{max}}{\delta_{max}^*} \qquad if\quad \delta_{max} < \delta_{max}^*$

where $\delta_{max}^*$ is the plastic flow limit overlap, defined as:

$\delta_{max}^* = \frac{\hat{k}_2}{\hat{k}_2 - k_1} \phi_f \frac{2 a_1 a_2}{a_1 + a_2}$

where $\phi_f$ is the plasticity depth, $a_1$ and $a_2$ are the particles’ radii.

Unloading below $\delta_0$ leads to attractive adhesion forces until the minimum force $-k_c \delta_{min}$ is reached at the overlap $\delta_{min} = (k_2 - k_1) \delta_{max}/(k_2 + k_c)$ , a function of the model parameters $k_1$ , $k_2$ , $k_c$ , and the history parameter $\delta_{max}$ . Further unloading leads to attractive forces $f^{hys} = f_0 - k_c \delta$ on the adhesive branch with slope $-k_c$ .

In summary, the adhesive, plastic, hysteric normal contact model contains the four parameters $k_1$ , $\hat{k}_2$ , $k_c$ and $\phi_f$ that respectively account for 1) loading and 2) reloading stiffness and plastic deformation, 3) adhesion strength and 4) plastic overlap range of the model.

The initial spring stiffness $k_1$ is equal to $k_n$ in the original Hooke model; hence, the remaining stiffnesses $\hat{k}_2$ and $k_c$ are defined relative to $k_1$ , i.e., as $\hat{k}_2 / k_1$ and $k_c / k_1$ .

The non-contact force that results when the overlap $\delta$ is negative is

$f^{hys} = \begin{cases} f_0 + \text{sign}(f_0) k_c \delta, & |k_c| > |k_{\text{min}}| \ \text{and} \ ( CDF \ d_{ij})^2 < (r_i+r_j)^2 \\ f_0 + k_{\text{min}} \delta, & |k_c| \leq |k_{\text{min}}| \ \text{and} \ (CDF \ d_{ij})^2 < (r_i+r_j)^2 \\ 0, & (CDF \ d_{ij})^2 \geq (r_i+r_j)^2 \end{cases}$

where $d_{ij}$ [m] is the distance between the particles centers and $k_{\text{min}}$ is

$k_{\text{min}} = \frac{-f_0 \ CDF}{(r_i + r_j) (1 - \ CDF)}, \ \text{if} \ CDF \neq 1.0.$

The non-contact force is zero when $CDF = 1.0$ . $r_i$ [m] and $r_j$ [m] are the bounding radii of particles i and j, respectively. The $CDF$ [-] has a default value of 1.01 and can be modified using the contact_distance_factor keyword of the neigh_modify. For example, using neigh_modify contact_distance_factor 1.0 will make the non-contact force equal zero.

This formulation guarantees a continuous decrease of the non-contact force, from $f_0$ to zero, as the particles move away from each other.

Viscous model:

Using option viscous on, the coefficient of restitution is calculated as proposed by Legendre et al., while for viscous off no modification is performed. The viscous model option requires several additional material properties as mentioned above. The resulting coefficient of restitution is calculated as follows:

$log(e) = log(e_{max}) + \frac{c_S}{S}, S = \frac{m^* v_n}{6\pi \mu_f {r^*}^2},$

where $v_n$ is the relative normal velocity.

Influence on the tangential and rolling friction contact models:

The model implementation differs from hooke/hysteresis in the handling of tangential force, where the luding model follows the approach described in Coetzee et al.. The tangential damping coefficient and normal force (Coulomb limit) that appear in the tangential models are replaced by

and the Coulomb criterion limits the total tangential force, instead of only the elastic component.

The $F_{n, modified}$ replaces $F_n$ also in the the rolling friction models.

Restrictions

If using SI units, $Y$ must be bigger than 5e6. If using CGS units, $Y$ must be bigger than 5e5. When using viscous on, $\mu_f$ must be bigger than 0.

Coarse-graining information:

Using coarsegraining in combination with this command might lead to statistically different dynamics and system state. To the best knowledge of the developers, the cross-influence between this command and coarse-graining is unknown.

Default

pullOffForce = 0, viscous = ‘off’, tangential_damping = ‘on’

Literature

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

[2] Legendre, D., Daniel, C., & Guiraud, P. (2005). Experimental study of a drop bouncing on a wall in a liquid. Physics of Fluids, 17(9), 097105.

[3] Coetzee, C.J.. (2020). Luding’s Elasto-Plastic-Adhesion Contact - Implementation on PFC. 10.13140/RG.2.2.27080.96004.