tangential model history

Purpose

This is the default tangential contact model that can be used in a wide variety of contexts.

Note

This command is supported by Aspherix GPU.

Syntax

tangential history [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)

heating_tangential_history values = 'on' or 'off'
  on = model contributes to surface heating in the frame of enable_surface_heating
  off = model does not contributes to surface heating
torsion values = 'on' or 'off'
  on = model calculates friction based on torsion
  off = model ignores rotation around the contact normal
disableTangentialWhenBonded values = 'on' or 'off'
  on = no tangential forces between bonded particles
  off = tangential forces between bonded particles

Associated material properties

Material properties

youngsModulus ( $Y$ ): The Youngs Modulus of a material, i.e. its stiffness [pressure]
poissonsRatio ( $\nu$ ): The Poisson’s ratio, i.e. the ratio of transverse to axial strain [ $\cdot$ ]

Material interaction properties

coefficientRestitution ( $e$ ): The coefficient of restitution of two materials [ $\cdot$ ]
coefficientFriction ( $\mu$ ): The coefficient of friction acting between two materials [ $\cdot$ ]

Description

This model can be used in the tangential argument of both particle_contact_model and wall_contact_model.

The tangential force component is given as

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

where $k_t$ is the tangential spring stiffness, $\gamma_t$ the tangential damping coefficient, $v_{t,ij}$ the relative tangential velocity. Finally, the tangential overlap $\delta t_{ij}$ is defined as the time integral of the relative tangential velocity at the contact point. The name history stems from this time integral and the resulting influence of past time steps on the model.

The tangential spring stiffens is calculated via

$k_t = 8 G^* \sqrt{r^* \delta_n}, \frac{1}{G^*} = \frac{2(2-\nu_i)(1+\nu_i)}{Y_i} + \frac{2(2-\nu_j)(1+\nu_j)}{Y_j}, \frac{1}{r^*} = \frac{1}{r_i} + \frac{1}{r_j},$

where $Y$ and $\nu$ are the Youngs Modulus and Poisson’s ratio, respectively.

The tangential damping coefficient is defined as

$\gamma_t = -2 \sqrt{\frac{5}{6}} \beta \sqrt{S_t m^*}, \beta = \frac{\ln{e}}{\sqrt{ln{e}^2 + \pi^2}}, S_t = 8 G^* \sqrt{r^* \delta_n}, \frac{1}{m^*} = \frac{1}{m_i} + \frac{1}{m_j},$

where $e$ and $m$ are the coefficient of restitution and particle mass, respectively.

The coefficient of friction $\mu$ is the upper limit of the tangential force through the Coulomb criterion $F_{t,spring} \leq \mu F_n$ , where $F_{t,spring}$ and $F_n$ are the tangential spring and total normal force components. Thus in the Hertzian and Hookean case, the tangential force between two particles grows according to a tangential spring and dash-pot model until $F_{t,spring}/F_n = \mu$ and is then held at $F_{t,spring} = \mu F_n$ until the particles lose contact. If the normal model Luding is used the $F_n$ and $\gamma_t$ are replaced by

where $k_c$ , $k_1$ and $f_0$ are parameters of the normal model Luding, and the Coulomb criterion limits the total tangential force, instead of only the elastic components, i.e. $k_t \delta t_{ij} - \gamma_{t, modified} v_{t,ij} \leq \mu F_{n, modified}$ .

The damping contribution is only added in time-steps where there is no slip, i.e. the Coulomb criterion is not met.

This model contributes to surface heating in the frame of enable_surface_heating if the appropriate flag is activated.

If the keyword torsion is set then the torsion will be calculated using a spring and a torque is calculated based on $T_n = k_t r \omega_n$ . The radius $r$ is taken as the radius of the spherical cap that constitutes the overlap region. For non-spherical particles (superquadric, convex) the same radius is used assuming the particle to be replaced by its bounding sphere. If the surface model convexhull/manifold is used then the manifold points are used to estimate this radius. Which allows to differentiate between corner and flat face contacts. The torque $T_n$ behaves identical to the tangential force, i.e. there is an equivalent coulomb criterion and a damping contribution if it is not met.

When the cohesion model bond is used the disableTangentialWhenBonded keyword can be used. If this parameter is set to ‘on’ then the tangential model will only compute its contribution if the two neighboring particles do not have an active bond. This switch can only be used together with disableNormalWhenBonded of gran model hertz

Coarse-graining information:

Using coarsegraining in combination with this command might lead to different dynamics or system state and thus to inconsistencies. However, the influence of this model on the global dynamics or system state might be small so in some cases the results may be valid. This has to be reviewed by a specialist on a case-by-case basis.

Default

heating_tangential_history = ‘off’, torsion = ‘off’