cohesion model bondRelative

Purpose

Implementing the bond model from Potyondy and Cundall with adaptations for growing or shrinking particles.

Syntax

cohesion bondRelative [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)

stressBreak values = 'on' or 'off'
  on = bond will break by exceeding maximum stresses
  off = no effect due to stress
plasticDissipation values = 'on' or 'off'
  on = plastic dissipation model is enabled
  off = plastic dissipation model is disabled

Associated material properties

Material properties

bondYm ( $Y_b$ ): Young’s modulus of the bond material [pressure]
bondPoissonsRatio ( $\nu_b$ ): Poisson’s ratio of the bond material [ $\cdot$ ]
bondRelativeYieldStressPlasticDissipation ( $\sigma_{yield}$ ): plastic yield stress of the bond material [pressure] (required only if plasticDissipation on)
bondRelativeBetaPlasticDissipation ( $\beta$ ): parameter that controls the saturation behaviour of the model [ $\cdot$ ](required only if plasticDissipation on)
bondRelativeMaxEnergyDensityPlasticDissipation ( $E_p^{crit}$ ): maximum energy density dissipated by the bond [energy/length^3] (required only if plasticDissipation on)

Material interaction properties

radiusMultiplierBond ( $\alpha_b$ ): coefficient which multiplies the minimum particle radius between bonded particles to determine the radius of the cylindrical beam representing the bond [ $\cdot$ ]
bondNormalDampingRatio ( $\xi_n$ ): ratio between the applied normal damping force and the critical normal damping force [ $\cdot$ ]
bondTangentialDampingRatio ( $\xi_t$ ): ratio between the applied tangential damping force and the critical tangential damping force [ $\cdot$ ]
maxSigmaBond ( $\sigma_{max}$ ): maximum normal stress of the bond [pressure] (required only if stressBreak on)
maxTauBond ( $\tau_{max}$ ): maximum tangential stress of the bond [pressure] (required only if stressBreak on)

Global scalars 

tsCreateBond ( $t_s$ ): time-step number at which the bonds can be created [ $\cdot$ ]

Description

This model can be used as part of pair gran and fix wall/gran

This bond model implements the parallel bond model from Potyondy and Cundall , with adaptations for growing or shrinking.

The current implementation creates the bond with zero forces and torques between the bonded particles. Due to relative motion forces and torques will act on the particles, where the normal force $F_n$ is calculated explicitly, while the tangential force $F_t$ , the normal torque $T_n$ and the tangential torque $T_t$ are calculated incrementally:

$F_n = k_n A (d_0 - d) F_t += k_t A v_t dt T_n += k_t J \omega_n dt T_t += k_n I \omega_t dt$

where $k_n$ and $k_t$ are the normal and tangential stiffness per unit area of the bond, $d_0 - d$ is the particles’ overlap, $v_t$ is the relative tangential velocity at contact, $\omega_n$ and $\omega_t$ are the relative normal and tangential angular velocity at contact. The cross-sectional area $A$ , the polar moment of inertia $J$ and the moment of inertia $I$ are defined as:

$A = \pi r_b^2 J = \frac{1}{2} \pi r_b^4 I = \frac{1}{4} \pi r_b^4$

where the the beam radius is defined as $r_b = r_{min} \alpha_b$ , i.e. the product of the minimum radius of the bonded particles and the user-defined property radiusMultiplierBond. With the default radiusMultiplierBond = 1 the beam that is located between two particles is a cylinder with radius equal to the minimum radius of the two particles. The radiusMultiplierBond allows to shrink (or grow) the beam that represents the bond between these two particles. Clearly, a thicker beam will make the system more stiff, while a thinner beam will decrease the stiffness (for constant remaining properties).

If stressBreak off, the bond breaks when the particles detach from each other. If stressBreak on, the bond breaks when the maximum bond normal stress $\sigma_{max}$ or the maximum bond tangential stress $\tau_{max}$ are reached, namely

$\sigma_b = \frac{\left | F_n \right | }{A} + \left | T_t \right | \frac{r_b}{I} = \sigma_{max} \tau_b = \frac{ \left | F_t \right | }{A} + \left | T_n \right | \frac{r_b}{J} = \tau_{max}$

Note that “normal” and “tangential” always refer to the coordinate system local to the bond. Thus, these forces and torques need to be rotated along with the bonded pair of particles to compute the forces and torques in the global frame of reference.

Warning

Aspherix® will calculate a maximum bond length / contact distance from values for maxDistanceBond or the max. sigma/tau values which is used for defining the cut-off for the neighbor list build. For unrealistically high values, this might lead to a neighbor list overflow.

In order to stabilize the system a kind of energy dissipation is required. By default the model uses the approach as suggested by Potyondy and Cundall. Thus for each degree of freedom a damping force/torque is added that is defined, in case of the normal force, as

$F_d = - \xi_n \left | F_n \right | sign(v_n)$

where $sign(v)$ provides the direction of motion (normal or tangential relative (angular) velocity). The damping coefficients $\xi_n$ and $\xi_t$ can be set by the user-defined properties bondNormalDampingRatio and bondTangentialDampingRatio, and they are defined as the ratio between the applied damping force and the critical damping force. The critical normal damping force is defined as $F_{cr}= 2 \sqrt{k_n m^*}$ , where $m^*$ is the equivalent mass.

By default bonds are created if two particles touch each other at the user-defined time-step tsCreateBond.

Plastic Dissipation Model

With the plasticDissipation switch, a model for plastic dissipation in the bond can be enabled, which will be outlined below. Primed variables always denote values updated in the current timestep, while non-primed variables are the respective values from the previous timestep.

Upon displacement of the particles, a change in force of

$\Delta F = -k \cdot \Delta x$

occurs. $k$ here depends on the Young’s modulus or shear modulus of the bond, for normal and tangential dissipation respectively. Without any dissipation, the new total force exerted by the bond is

$F^\prime = F + \Delta F$

and the stress is given by

$\sigma^\prime = F'/A_{bond}$

with $A_{bond}$ being the bond area. The same holds for the tangential stress $\tau^\prime$ .

With the dissipation model enabled, a fraction

$\alpha = \left( \frac{\sqrt{\sigma^2 + 3 \cdot \tau^2}}{\sigma_{yield}} \right)^{\beta}$

is dissipated before the forces/stresses are updated. $\sigma_{yield}$ is the plastic yield stress of the material, and $\beta$ is a parameter that controls the saturation behaviour of the model.

The model tracks the amount of dissipated plastic energy, $E_p$ , as follows:

$E_p^\prime = E_p + \alpha \cdot \Delta x \cdot A_{bond} \cdot \sqrt{\sigma^2 + 3 \cdot \tau^2}$

Note that $\Delta x$ here is the magnitude of the total displacement - normal and tangential combined.

Once the dissipated plastic energy reaches a threshold level, $E_p \geq E_p^{crit}$ , the bond is broken.

Note

The bond is also broken if the distance between the particle centers exceeds $3(r_i+r_j)$

Note

stressBreak and plasticDissipation are mutually exclusive because their combination does not make sense from a physical standpoint.

Restrictions

None.

Coarse-graining information:

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

Default

stressBreak = off, plasticDissipation = off

Literature

[1} Potyondy, D. O., & Cundall, P. A. (2004). A bonded-particle model for rock. International journal of rock mechanics and mining sciences, 41(8), 1329-1364.