normal model adhesive_elasto_plastic

Purpose

This is a normal model with adhesion and elasto-plastic behaviour (often called Edinburgh model).

Syntax

model 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)

disableNormalWhenBonded values = 'on' or 'off'
  on = if the cohesion bond model is used, then the normal force is only added if the two particles are not bonded
  of = the normal force is always added if two particles overlap

Associated material properties

Material properties

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

Material interaction properties

coefficientRestitution ( $e$ ): coefficient of restitution of the two materials [–]
pullOffForce ( $f_0$ ): constant pull of force (usually negative) [force]
adhesionEnergy ( $\Delta\gamma$ ): contact adhesion energy [energy/length^2]
stiffnessRatio ( $\alpha$ ): ratio of $k_2 : k_1$ [–]
springPowerValue ( $n$ ): power value for loading and unloading springs [–]
adhesionPowerValue ( $x$ ): power value for adhesion branch [–]

Description

The adhesive elasto-plastic model consists of a hysteresis and a dissipative force, given by $F_{hys}$ and $F^{d}$ , respectively. The total normal force $F_{n}$ is obtained by the sum of the two components, i.e.

$F_{n} = F_{hys} + F^{d}.$

_images/normal_adhesive_elasto_plastic.png

As ilustrated in the figure above, the hysteresis force $F_{hys}$ involves three different branches, $F_{load}$ , $F_{unload}$ and $F_{adh}$ , and it is obtained from the following conditional expressions:

$F_{hys} = F_{load} \quad &\mbox{ if } F_{unload} \ge F_{load},\\ F_{hys} = F_{unload} \quad &\mbox { if } F_{load} > F_{unload} > F_{adh},\\ F_{hys} = F_{adh} \quad &\mbox { if } F_{adh} \ge F_{unload}.$

Upon first contact a spring is loaded the $F_{load}$ component is

$F_{load} = f_0 + k_1 \delta,$

where $f_0$ is the constant pull of force defined by the user, acting as an ever present force similarly to a van der Walls or electrostatic adhesion force, $k_1$ (with units force/length) is the spring stiffness for the loading branch, which is a function of $n$ , the user defined non-dimensional spring power value (see below) and $\delta$ is the normal overlap. The exponent $n$ was introduced to switch between a linear and non-linear force-overlap relationship. This branch is indicated in red in the graph above and it can only be traversed in the direction of increasing overlap. The stiffness is given by

$k_1 = \frac{4}{3} {r^*}^{2-n} {\delta}^{n-1} Y^*$

The definitions of the effective Young’s modulus $Y^*$ and effective particle radius $r^*$ are identical to the ones from the Hertz model. Note that the $F_{load}$ branch scales with $\delta^n$ and therefore the (linear) Hooke hysteresis (Luding) model and the (non-linear) Hertz model can be reproduced by using $n=1$ and $n=3/2$ , respectively.

If the overlap decreases the model switches to the unloading branch (blue) with the unloading force which is given by

$F_{unload} = f_0 + k_2(\delta^n - \delta_p^n),$

where $k_2 = \alpha k_1$ (with units force/length), $\alpha$ is a non-dimensional scaling factor defined by the user and $\delta_p$ is the plastic overlap, whose definition involves the maximum overlap $\delta_{max}$ (see the figure above) as given by

$\delta_p = \left(1 - \frac{1}{\alpha}\right)^{\frac{1}{n}} \delta_{max}.$

This branch can be traversed in both directions, i.e. if loading occurs after some unloading the branch is climbed back until the previous maximum force is reached and then the loading continues on the $k_1$ branch.

Finally, if the unloading continues and the hysteresis force would sink below the minimum force $F_{min}$ the adhesive unloading branch is followed. The minimum force is given as

$F_{min} = \frac{3}{2}\pi\, \Delta\gamma\, a,$

where $\Delta\gamma$ is the adhesion energy parameter (set by the user) and $a$ is the radius of the contact circle. The latter is defined as

$a = {r^*}^{1-x} {\delta}^{x},$

where $x$ is the non-dimensional adhesion power value parameter. The adhesive unloading force, which scales with $\delta^x$ , is defined as

$F_{adh} &= f_0 - k_{adh}\delta^x, \mbox{ with }\\ k_{adh} &= \frac{f_0 + F_{min}}{\delta_{min}^x} \mbox { and }\\ \delta_{min}^n &= \left(\delta_p^n - \frac{f_0 + F_{min}}{k_2}\right).$

This branch is only traversed in unloading direction. In case loading occurs while on this branch the force will be switched to the $k_2$ branch (as indicated by the blue dotted line in the graph above). This requires resetting $\delta_p$ to an appropriate value for continuity.

The dissipative damping force $F^d$ depends on whether the linear ( $n=1$ ) or non-linear ( $n \ne 1$ ) model is chosen. In the linear case $F^d$ is given by

$F^d = -\beta v_n,$

where $v_n$ is the relative normal velocity and

$\beta = \sqrt{\frac{4 m^* k_1}{1 + \left(\frac{\pi}{\ln(e)}\right)^2}},$

where $e$ is the coefficient of restitution.

The non-linear case defines the damping force as

$F^d &= -2\sqrt{\frac{5}{6}} \beta \sqrt{K_n\,m^*}v_n \mbox{ with,}\\ K_n &= 2 Y^* \sqrt{r^* \delta},$

and

$\beta = \frac{\ln(e)}{\sqrt{\ln^2(e) + \pi^2}}.$

The total normal force is then the sum of the hysteresis and dissipative force.

When any bond model, such as e.g. the cohesion model bond is used the disableNormalWhenBonded keyword can be used. If this parameter is set to ‘on’ then the normal model will only compute its contribution if the two neighboring particles do not have an active bond.

There is a corresponding tangential model available which can be found here.

Restrictions

If using SI units, youngsModulus must be > 5e6 If using CGS units, youngsModulus must be > 5e5

Default

disableNormalWhenBonded = ‘off’

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