Prev Subsubsection 184.108.40.206: Stick Kinematics Up Subsection 7.1.9: Sliding-Elastic Subsubsection 220.127.116.11: Coulomb Frictional Contact Next
The kinematics developed above can be used to formulate velocities. The formulation for stick developed in Section 18.104.22.168↓ does not rely on velocity constraints, therefore we are only concerned with velocities of opposing contact points in slip. The development of velocities presented below anticipates the need for a relative velocity as required by Coulomb's law of kinetic friction, which aligns the friction force with this slip direction.
Since the parametric coordinates of integration points represent material points on , the velocity of these points on the primary surface is evaluated from the material time derivative in the material frame, In contrast, since material may convect through the intersection point of the ray with , the total velocity at the intersection point on needs to be evaluated using the material time derivative in the spatial frame, Here, represents the velocity of the intersection point on , whereas are the contravariant components of the convective velocity of material passing through this intersection point. In effect, represents the relative (slip) velocity between the material on and that on . Importantly, as noted below when performing time discretization and linearization, by definition is evaluated while keeping constant.
We now use these relations to produce a more practical formulation of the slip velocity for our frictional contact implementation. Taking the material time derivative of eq.(22.214.171.124-1) and using the contact persistency condition  produces . Substituting Eqs.(126.96.36.199-1)-(188.8.131.52-2) into this expression yields a frame-invariant measure of relative velocity between and [81, 37], where is the material time derivative of in the parametric material frame of , evaluated from eq.(7.1.9-3) as Here, denotes the material time derivative of in the material frame, and we define the tangential plane projection tensor as
Previous authors have utilized directly instead of the right-hand-side of eq.(184.108.40.206-3); this expression requires the evaluation of time derivatives of parametric coordinates [112, 65, 89], which necessitates special integration algorithms to handle crossing element boundaries . The relative velocity measure on the right-hand-side of eq.(220.127.116.11-3) obviates the need for any such special treatment. In particular, the choice of ensures that element boundaries will never be crossed when calculating this velocity, since it is evaluated while keeping constant.
The tangential frictional traction in slip depends only on the tangential component of the relative velocity, therefore we may define the slip direction as the unit vector of the projection of onto the tangent plane of , These definitions of contact kinematics may now be used to formulate frictional contact.