normal model hooke/scale_invariant

Purpose

Implement a scale-independent version of the Hooke model as proposed by Obermayr et al.

Syntax

model hooke/scale_invariant [other model_type/model_name pairs as described here ]

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

tangentialStiffness ( $k_t$ ): tangential stiffness of the two materials in contact [force/length]
normalDampingCoefficient ( $\gamma_n$ ): normal damping coefficient of the two materials in contact [force*time/length]
tangentialDampingCoefficient ( $\gamma_t$ ): tangential damping coefficient of the two materials in contact [force*time/length]

Description

The model calculates the normal force between two particles in contact as follows:

$F_n = k_n \delta n_{ij} - \gamma_n v_{n,ij}$

where $k_n$ is the normal stiffness, $\delta$ is the overlap of the two particles, $n_{ij}$ is the contact normal, $\gamma_n$ is the damping coefficient, $v_{n,ij}$ is the relative normal velocity of the two particles. The overlap of spherical particles is defined as $\delta = r_i + r_j - r$ , where $r_i$ and $r_j$ are the particles’ radii and $r$ is the centroid distance.

The normal stiffness $k_n$ is calculated from the deformation of an elastic rod with cross-section $A$ , Young’s modulus $Y$ and length $l$ connecting the centers of the particles

$k_n = \frac{Y A}{l} = \frac{Y r_{ij}^2 \pi}{2 r_{ij}} = \frac{\pi}{2} Y r_{ij}$

where $r_{ij} = (r_i + r_j)/2$ is the average radius.

This definition of the spring constant ensures that a scaling of the particles will retain the correct contact behavior, which considerably facilitates the practical usage of the model.

This model defines also the stiffness and damping coefficient of the tangential force defined as:

$F_t = k_t \delta t_{ij} - \gamma_t v_{t,ij}$

where $\delta t_{ij}$ is the tangential overlap and $v_{t,ij}$ is the relative tangential velocity. Please, refer to the documentation of the tangential models for more information.

Restrictions

If using SI units, $Y$ must be bigger than 5e6. If using CGS units, $Y$ must be > 5e5.

Coarse-graining information:

Using coarsegraining in combination with this command might lead to different dynamics or system state and thus to inconsistencies.

Literature

[1] Obermayr et al. Journal of Terramechanics 53, 93-104 (2014).