%BND_FRICTION%

Example: Coloumbs Law of Friction in cutting processes:

FrictionForce

To define the correct friction force, the relative velocity

\begin{align} {\bf v_r} = {\bf v}^{n+1} - {\bf v_p}\end{align}

at the boundary is required, where \( {\bf v_p}\) is the movement speed of the geometry. Then the normalized tangential parts

\begin{align} {\bf v_t} &= {\bf v_r} - ({\bf v_r} \cdot {\bf n}){\bf n} \\[10pt] {\bf t_1} &= \frac{\bf v_t}{\left\Vert{\bf v_t}\right\Vert}\end{align}

are used as direction. With \( {\bf t_2} = {\bf n} \times {\bf t_1}\) and the user-given friction force \( {\bf F_F}\), the boundary condition is

\begin{align} {\bf t_1} \cdot {\bf S_{s}} \cdot {\bf n} &= {\bf F_F} - {\bf t_1} \cdot {\bf S_{visc}} \cdot {\bf n} \\ {\bf t_2} \cdot {\bf S_{s}} \cdot {\bf n} &= - {\bf t_2} \cdot {\bf S_{visc}} \cdot {\bf n}\end{align} \(\bf {\bf S_{s}}\) is the deviatoric solid stress tensor and \(\bf n\) is the boundary normal.

distFactor

In cases with high friction forces, the relative velocity \( {\bf v_r}\) can be very small and noisy around zero, which might lead to incorrect forces. Therefore, a distance factor \( d\) can be set in terms of

\begin{align} {\bf x_{int}} &= {\bf x_{bnd}} + d \, h \, {\bf n}\end{align}

with smoothing length \( h\) in order to compute the relative velocity in the interior. The computation is based on Shepard interpolation and Taylor series. The algorithm itself decides, which approximation is used. The default value for the distance factor is 0.

kernel_Shepard

For the above mentioned Shepard interpolation, the kernel can be controlled with this parameter. The default value is 2.