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, $\mu$ ).

Figure 1. Normal and Tangential Forces Applied to a Node

Figure 2. Adhesion Force Computation

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

VIS_F

Critical damping coefficient on interface friction

M

Main node mass

Once the adhesion force (F_h) is computed, if it is less than $\mu F_n$ , the friction force is unchanged equaling F_h 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)

$$F_t=\min \left(\mu F_n,F_{adh}\right)$$

Figure 3. Friction Force Computation

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.

Figure 4. Viscous Formulation vs Stiffness Formulation

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 I_fric, three new friction formulations are available:

Generalized Viscous Friction Law

(4)

$$\mu =Fric+C_1.p+C_2\cdot V+C_3.p\cdot V+C_4\cdot p^2+C_5\cdot V^2$$

Modified Darmstad Friction Law

(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 (7)

$$V\in \left[C_5,C_6\right]$$

(8)

$$\mu =C_3-\left(\left(C_3-C_4\right)\cdot {\left(\frac{V-C_5}{C_6-C_5}\right)}^2\cdot \left(3-2\cdot \frac{V-C_5}{C_6-C_5}\right)\right)$$

if (9)

$$V\in \left[C_5,C_6\right]$$

(10)

$$\mu =C_2-\frac{1}{\frac{1}{C_2-C_4}+{\left(V-C_6\right)}^2}$$

if (11)

$$V\ge C_6$$

(12)

$$P=C_1\cdot \mu +C_4\cdot \rho \cdot C_{\upsilon }\cdot T=C_1\cdot \mu +{\alpha }_{\upsilon }\cdot T$$

Figure 5. Graphical Representation of Renard Friction Model