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220.127.116.11 Stick Kinematics
Equations (18.104.22.168-1) and (22.214.171.124-2) describe searching for evolving contact and are only valid during slip. In stick no relative motion occurs between contacting points, thus the intersection point on does not evolve during the iterative solution process, allowing the development of kinematics of sticking contact. Implicit in this condition is the assumption that the contact projection was previously resolved, and thus contact searching is not performed again; rather, the contact point on is given by the parametric coordinates of intersection found from the previous time point, which will be denoted as , with the subscripted referring to the previous time. Thus, we write the spatial position of in stick as where is a vectorial gap function, Here, is the vectorial distance, at the current time , between material points which were in contact at the previous time step; for perfect stick we must have . In a finite element implementation however, it is important to note that the intersection point is not in general the point that would be found from a ray directed from along the unit outward normal . This is because stick will not be enforced exactly, so the points and will separate slightly when using a penalty method for enforcing this constraint. How to minimize this separation and enforce perfect stick behavior is the subject of the forthcoming sections.