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Theory Manual Version 3.6
 Subsubsection 7.1.9.4: Coulomb Frictional Contact Up Subsection 7.1.9: Sliding-Elastic Subsubsection 7.1.9.6: Augmented Lagrangian Scheme 

7.1.9.5 Penalty Scheme

During stick, no relative motion may occur between surfaces; thus, points on and that were in contact at the previous time must remain in contact. The stick traction may be obtained by penalizing any relative motion between such points, where is the penalty parameter and we have employed eq.(7.1.9.2-2).
During slip, we first calculate the normal component of the contact traction by penalizing the normal gap of eq.(7.1.9.1-2), The total traction vector is then directly obtained as Here we have achieved an exact expression for the tangential traction in slip and remark that since is strictly negative in contact, the frictional contact traction acting on is in the direction opposing the motion of relative to . A trial state and return map, presented in Section 7.1.9.7↓, is employed to differentiate between stick and slip.
 Subsubsection 7.1.9.4: Coulomb Frictional Contact Up Subsection 7.1.9: Sliding-Elastic Subsubsection 7.1.9.6: Augmented Lagrangian Scheme