tangential model adhesive_elasto_plastic

Purpose

The tangential model to go with the adhesive_elasto_plastic normal model (Edinburgh model).

Syntax

tangential adhesive_elasto_plastic [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)

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 [–]

Material interaction properties

coefficientRestitution ( $e$ ): The coefficient of restitution bewteen two materials [–]
coefficientFriction ( $\mu$ ): The coefficient of friction acting between two materials [–]
coefficientSlidingFriction ( $\mu_s$ ): The coefficient of sliding friction acting between two materials [–]
springPowerValue ( $n$ ): power value for loading and unloading springs [–]
pullOffForce ( $f_0$ ): constant pull of force (usually negative) [force]
tangentialStiffnessMultiplier ( $\kappa_{tm}$ ): factor relating the normal and tangential stiffnesses [–]

Description

This tangential model can only be used with the adhesive_elasto_plastic normal model.

The tangential force $F_t$ [force] is given by

$F_t = -k_t \delta_t + F_t^d &\quad \mbox{ icf } k_t \delta_t \le \|\mu_s F_{hys}\|, \\ F_t = -F_{ct} &\quad \mbox { else},$

where $k_t$ [force/length] is the tangential stiffness, $\delta_t$ [length] is the tangential overlap (tangential velocity integrated over time), $F_t^d$ [force] is the damping component of the tangential force, $\mu_s$ is the non-dimensional coefficient of sliding friction, $F_{hys}$ [force] is the hysteresis force obtained from the normal model and $F_{ct}$ [force] is the Coulomb limit force.

The tangential stiffness is given by

$k_t = \kappa_{tm} 8 G^* \sqrt{r^* \delta}$

where $G^*$ [pressure] and $r^*$ [length] are the effective shear modulus and particle radius, respectively, $\delta$ [length] is the normal overlap and $\kappa_{tm}$ is a non-dimensional scaling factor.

The Coulomb limit force is given by

$F_{ct} = \mu \|f_{hys} + k_{adh}\delta^x - f_0\|,$

where all parameters on the r.h.s. of the equation are obtained from the normal model formulation. The damping component of the tangential force is calculated by the following equation

$F_t^d = -2 \sqrt{\frac{5}{6}} \beta \sqrt{k_t m^*} v_t$

where the value of the $\beta$ parameter is also obtained from the normal model formulation, $m^*$ [mass] is the effective particle mass and $v_t$ [length/time] is the relative tangential velocity of the particle pair.

Default

disableTangentialWhenBonded = ‘off’

(Morrissey) John P. Morrissey, Ph.D. Thesis, University of Edinburgh (2013)