# Friction

Several friction formulations are available within Radioss. The simplest one, which is also the most used, is the Coulomb friction law. This formulation provides accurate results in crash analysis and requires just one parameter (Coulomb friction coefficient, $\mu$ ).
The default value for $\mu$ is 0 (no friction between surfaces). To compute the friction force, the default friction penalty formulation is a viscous one, based on the tangential velocity. During sliding penetrate the node goes from position C0 (contact point at time t) to C1 (contact position at time $t+\text{Δ}t$ ). As the contact is viscous, a viscous coefficient C is introduced to compute the adhesion force:(1)
${F}_{adh}=C\cdot {V}_{t}$
Where,
$C=VI{S}_{F}\cdot \sqrt{2KM}$
K
Instantaneous interface stiffness
VISF
Critical damping coefficient on interface friction
M
Main node mass
Once the adhesion force (Fh) is computed, if it is less than $\mu {F}_{n}$ , the friction force is unchanged equaling Fh and sticking will occur. If the adhesion force is greater than $\mu {F}_{n}$ , then the friction force is reduced and equals $\mu {F}_{n}$ .(2)
${F}_{t}=\mathrm{min}\left(\mu {F}_{n},{F}_{adh}\right)$
If sliding occurs at a very low speed (for example: quasi-static simulation), the viscous formulation will not work, as the friction force is computed upon the tangential speed. To overcome this limitation, a new friction penalty formulation is available based on tangential displacement (stiffness incremental formulation). This method introduces an artificial stiffness, K to calculate the variation of the friction force:(3)
$\text{Δ}{F}_{t}=K\cdot {V}_{t}\cdot {\delta }_{t}$
Where,
${\delta }_{t}$
Tangent displacement
Therefore, contrary to the previous formulation, the stiffness formulation is able to compute the proper friction force even at a low speed. Figure 4 illustrates this point. If an imposed displacement is applied to a part (a 3D cube) at a low speed (0.01 m/s), the viscous formulation will not work; whereas the stiffness formulation based on the tangential displacement will.
Other friction formulations are available, their principle is similar to the Coulomb friction law. Radioss first computes an adhesion force, which is then compared to $\mu {F}_{n}$ . Their differences lie in the friction coefficient ( $\mu$ ) which is not constant anymore, but function on the pressure of the normal force on the main segment and on the tangential velocity of the secondary node. Depending on the flag Ifric, three new friction formulations are available:
Generalized Viscous Friction Law
(4)
$\mu =Fric+{C}_{1}.p+{C}_{2}\cdot V+{C}_{3}.p\cdot V+{C}_{4}\cdot {p}^{2}+{C}_{5}\cdot {V}^{2}$
(5)
$\mu =Fric+{C}_{1}\cdot {e}^{\left({C}_{2}V\right)}\cdot {p}^{2}+{C}_{3}\cdot {e}^{\left({C}_{4}V\right)}\cdot p+{C}_{5}\cdot {e}^{\left({C}_{6}V\right)}$
Renard Friction Law
(6)
$\mu ={C}_{1}+\left({C}_{3}-{C}_{1}\right)\cdot \frac{V}{{C}_{5}}\cdot \left(2-\frac{V}{{C}_{5}}\right)$
if $V\in \left[0,{C}_{5}\right]$
(7)
$\mu ={C}_{3}-\left(\left({C}_{3}-{C}_{4}\right)\cdot {\left(\frac{V-{C}_{5}}{{C}_{6}-{C}_{5}}\right)}^{2}\cdot \left(3-2\cdot \frac{V-{C}_{5}}{{C}_{6}-{C}_{5}}\right)\right)$
if $V\in \left[{C}_{5},{C}_{6}\right]$
(8)
$\mu ={C}_{2}-\frac{1}{\frac{1}{{C}_{2}-{C}_{4}}+{\left(V-{C}_{6}\right)}^{2}}$
if $V\ge {C}_{6}$
(9)
$P={C}_{1}\cdot \mu +{C}_{4}\cdot \rho \cdot {C}_{\upsilon }\cdot T={C}_{1}\cdot \mu +{\alpha }_{\upsilon }\cdot T$
Note: Friction filtering is available for all friction formulations and allows you to smooth the friction force. Refer to Radioss Starter Input for more details.