Hysteretic Model
Introduction
The hysteretic contact model in PFC consists of a combination of the elastic portion of the Hertz model as described in the “Hertz Model” section, combined with an alternate dashpot group consisting of a nonlinear viscoelastic element in the normal direction. This model is referred to in commands and FISH as hysteretic.
Note
Revision 28 included the following additional modification:
The property inheritance mechanism that derives the contact model elastic parameters from the parameters of the contacting pieces has been changed to conform to [Mindlin1953], following the equations below . Before this modifications, the contact effective shear modulus and Poisson ratio were taken as the average of the contacting pieces shear modulus and Poisson ratio, respectively. Note that this change may change the output for models that used the property inheritance mechanism with the hysteretic model. Please refer to the documentation of the “Hertz Model” for further information.
Behavior Summary
The formulation is based on the literature review by [Machado2012]. Three different dashpot modes, which correspond to three different expressions for the dashpot force calculation, are provided.
ActivityDeletion Criteria
The hysteretic model defines an activity distance of 0.0. Contact activity is updated as:
As discussed in the “Contact Resolution” section, forcedisplacement calculations are skipped for inactive contacts.
ForceDisplacement Law
Similar to the Hertz model, the hysteretic model transmits a force and no moment. The force embeds a nonlinear elastic component \(\mathbf{F^{h}}\) and a viscous component \(\mathbf{F^{d}}\)
The nonlinear Hertz force can be decomposed into a normal and a shear contribution:
Its formulation is equivalent to that of the nonlinear Hertz force described in the Hertz contact model.
The dashpot force has a normal component only:
where the formulation of \(F_{n}^{d}\) can be expressed in terms of a hysteresis damping factor \(\chi\) as:
where \(\delta_{n} = g_c\) represents physical overlap, \(\dot{\delta}_n = \mathbf{\dot{\delta}} {\kern 1pt} \hat{\mathbf{n}}_\mathbf{c}\) is the normal component of the relative translational velocity, and the hysteresis damping factor \(\chi\) depends on the dashpot mode \(M_d\), the target restitution coefficient \(e_n\), the Hertz normal force coefficient \(h_n\) (see here), and the normal impact velocity \(\dot{\delta}^{()}_n\) with:
Note that for the case \(M_d=1\), the target restitution coefficient used in the equation above is the maximal value between \(e_n\) and a minimal value of \(e^*_n\).
Energy Partitions
The hysteretic model defines three energy partitions:
strain energy \(E_{k}\), stored in the springs;
slip work \(E_{\mu}\), defined as total energy dissipated by frictional sliding; and
damp work \(E_{\beta}\), defined as the total energy dissipated by the dashpots.
If energy tracking is activated, these energy partitions are updated as follows.
Update the strain energy:
(7)\[E_{k} = \frac{\alpha}{(\alpha+1)} \left(\frac{\left(F^{h}_n\right)^2}{k_n}\right) + \frac{1}{2} \frac{\\mathbf{F^{h}_s)}\^2}{k_s}\]Update the slip energy:
(8)\[\begin{split}\begin{array}{l} {E_{\mu } :=E_{\mu }  {\tfrac{1}{2}} \left(\left(\mathbf{F_{s}^{h}} \right)_{o} +\mathbf{F_{s}^{h}} \right)\cdot \Delta \pmb{δ}_\mathbf{s}^{\mu } } \\ {{\rm with}\qquad \Delta \pmb{δ}_\mathbf{s}^{\mu } =\Delta \pmb{δ}_\mathbf{s} \Delta \pmb{δ}_\mathbf{s}^{k} =\Delta \pmb{δ}_\mathbf{s} \left(\frac{\mathbf{F_{s}^{h}} \left(\mathbf{F_{s}^{h}} \right)_{o} }{k_{s} } \right)} \end{array}\end{split}\]where \(\left(\mathbf{F_{s}^{h}} \right)_{o}\) is the shear force at the beginning of the timestep, and the relative sheardisplacement increment of this equation of the “Contact Resolution” section has been decomposed into an elastic \(\left(\Delta \pmb{δ}_\mathbf{s}^{k} \right)\) and a slip \(\left(\Delta \pmb{δ}_\mathbf{s}^{\mu } \right)\) component. The dot product of the slip component and the average shear force occurring during the timestep gives the increment of slip energy.
Update the dashpot energy:
(9)\[E_{\beta } :=E_{\beta }  \mathbf{F^{d}} \cdot \left(\dot{\pmb{δ} }{\kern 1pt} {\kern 1pt} \Delta t\right)\]where \(\dot{\pmb{δ} }\) is the relative translational velocity of this equation of the “Contact Resolution” section.
Additional information, including the keywords by which these partitions are referred to in commands and FISH, is provided in the table below.
Keyword 
Symbol 
Description 
Range 
Accumulated 

Hertz Group: 


\(E_{k}\) 
strain energy 
\([0.0,+\infty)\) 
NO 

\(E_{\mu}\) 
total energy dissipated by slip 
\([0.0,+\infty)\) 
YES 
Dashpot Group: 


\(E_{\beta}\) 
total energy dissipated by dashpots 
\([0.0,+\infty)\) 
YES 
Properties
The properties defined by the hysteretic contact model are listed in the table below for a concise reference; see the “Contact Properties” section for a description of the information in the table columns. The mapping from the surface inheritable properties to the contact model properties is also discussed below.
Keyword 
Symbol 
Description 
Type 
Range 
Default 
Modifiable 
Inheritable 

hysteretic 
Model name 

Hertz Group: 

hz_shear 
\(G\) 
Shear modulus [stress] 
FLT 
\([0.0,+\infty)\) 
0.0 
YES 
YES 
hz_poiss 
\(\nu\) 
Poisson’s ratio [] 
FLT 
\((1.0,0.5]\) 
0.0 
YES 
YES 
fric 
\(\mu\) 
Friction coefficient [] 
FLT 
\([0.0,+\infty)\) 
0.0 
YES 
YES 
hz_mode 
\(M_s\) 
Shearforce scaling mode [] 
INT 
\(\{0;1\}\) 
0 
YES 
NO 
\(\;\;\;\;\;\;\begin{cases} \mbox{0: no scaling} \\ \mbox{1: scaling is active} \end{cases}\) 

hz_alpha 
\(\alpha\) 
Exponent [_] 
FLT 
\([0.0,+\infty)\) 
1.5 
YES 
NO 
hz_slip 
\(s\) 
Slip state [] 
BOOL 
{false,true} 
false 
NO 
N/A 
hz_force 
\(\mathbf{F^{h}}\) 
Hertz force (contact plane coord. system) 
VEC 
\(\mathbb{R}^3\) 
\(\mathbf{0}\) 
YES 
NO 
\(\left( F_n^h,F_{ss}^h,F_{st}^h \right) \quad \left(\mbox{2D model: } F_{ss}^h \equiv 0 \right)\) 

Dashpot Group: 

dp_mode 
\(M_d\) 
Dashpot mode [] 
INT 
{0;1;2} 
0 
YES 
NO 
\(\;\;\;\;\;\;\begin{cases} \mbox{0: default mode} \\ \mbox{1: mode 1} \\ \mbox{2: mode 2} \end{cases}\) 

dp_en 
\(e_n\) 
Target restitution coef. [] 
FLT 
\([0.0,1.0]\) 
1.0 
YES 
NO 
dp_enmin 
\(e^*_n\) 
Minimal restitution coef. [] 
FLT 
\([0.0,1.0]\) 
0.0 
YES 
NO 
dp_force 
\(\mathbf{F^{d}}\) 
Dashpot force (contact plane coord. system) 
VEC 
\(\mathbb{R}^3\) 
\(\mathbf{0}\) 
NO 
NO 
\(\left( F_n^d,0,0 \right)\) 
Property Inheritance
The shear modulus \(G\), Poisson’s ratio \(\nu\), and friction coefficient \(\mu\) may be inherited from the contacting pieces. The inheritance logic is similar to that of the hertz contact model described here.
Methods
No methods are defined by the hysteretic model.
Callback events
Event 
Array slot 
Value Type 
Range 
Description 

contact_activated 
Contact becomes active 

1 
C_PNT 
N/A 
contact pointer 

slip_change 
Slip state has changed 

1 
C_PNT 
N/A 
contact pointer 

2 
INT 
{0;1} 
slip change mode 

\(\;\;\;\;\;\;\begin{cases} \mbox{0: slip has initiated} \\ \mbox{1: slip has ended} \end{cases}\) 
Model Summary
An alphabetical list of the hysteretic model properties is given here.
This is a hidden link to: [Gonthier2004]
And this is a hidden link to: [Lankarani1990]
References
Y. Gonthier, J. McPhee, C. Lange, JC. Piedboeuf, “A regularized contact model with asymmetric damping and dwelltime dependent friction”, Multibody System Dynamics, vol. 11, pp. 209233 (2004).
H.M. Lankarani, P.E. Nikravesh, “A contact force model with hysteresis damping for impact analysis of multibody systems”, Journal of Mechanical Design, vol. 112, pp. 369376 (1990).
M. Machado, P. Moreira, P. Flores, H.M. Lankarani, “Compliant contact force models in multibody dynamics: Evolution of the Hertz contact theory”, Mechanism and Machine Theory, vol. 53, pp. 99121 (July 2012).
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