Theory Manual Version 3.6
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Subsubsection 7.1.9.6: Augmented Lagrangian Scheme Up Subsection 7.1.9: Sliding-Elastic Subsubsection 7.1.9.8: Linearization

#### 7.1.9.7 Stick-Slip Algorithm

Determination of whether stick or slip is active is accomplished by a trial state and return map, and follows the same procedure for both penalty and augmented Lagrangian regularizations. We begin by calculating a trial traction assuming stick, utilizing either eq.(7.1.9.5-1) or eq.(7.1.9.6-1). The trial normal and tangential components and are evaluated from and inserted into the slip criterion , Based on the slip criterion and trial traction vector, we perform a return mapping and obtain the traction vector as where is given by either eq.(7.1.9.5-2) or eq.(7.1.9.6-2). In the case of first contact, a trial stick traction cannot be calculated, as stick requires a previous intersection point. Consequently, first contact is treated as a case of slip in this framework. The alternative of treating first contact as frictionless is unsatisfying, as the lack of friction at the first iteration can lead to premature failure and thus precludes the modeling of certain problems that rely on frictional forces for stability (such as load-control analyses). After one iteration, the traction can be evaluated via the return map described above.
Computationally, care must be taken to ensure that augmentation does not unnecessarily change the stick-slip status. For normal contact, the update will augment the normal traction until the non-penetration contact constraint is adequately satisfied, with no adverse consequence if slightly overshoots the final target value during an intermediate augmentation. For tangential contact however, augmentation of the tangential traction that overshoots the final target value may cross the boundary between stick and slip, thus changing the nature of the solution. For example, at the first augmentation step, the stick traction is and the multiplier is augmented from its initial zero value to according to eq.(7.1.9.6-3). Thus, at the start of the next iteration, when has not yet changed, the traction is calculated as according to eq.(7.1.9.6-1), which essentially counts the gap function twice and can in some cases shift the contact from stick to slip. To circumvent potential error introduced by this step, our implementation freezes the stick-slip status until the completion of the first iteration following an augmentation step. After the first iteration, the gap function has been reduced by the augmentation and the traction is split appropriately between the remaining gap and the multiplier. In this way, the double-counting of the gap function does not unnecessarily shift the contact status, but augmentation is able to modify the stick-slip status if accurate enforcement of the contact constraints requires it.
Subsubsection 7.1.9.6: Augmented Lagrangian Scheme Up Subsection 7.1.9: Sliding-Elastic Subsubsection 7.1.9.8: Linearization