Theory Manual Version 3.6
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Subsubsection 7.2.3.3: Velocities Up Section 7.2: Biphasic Contact Subsection 7.2.5: Augmented Lagrangian Scheme

### 7.2.4 Penalty Scheme

The defining characteristic of frictional stick is a lack of relative motion between points which were previously in contact. Consequently, the stick traction is obtained by penalizing relative motion between such points, where is the contact penalty parameter and we have utilized eq.(7.2.3.2-2) to define the gap vector during stick.
During slip, the normal component of the contact traction is first calculated by penalizing the normal component of the gap, given by eq.(7.2.3.1-3), The total traction vector in slip is then directly prescribed as where is the unit vector in the slip direction, given by eq.(7.2.3.3-3). We note that eq.(7.2.4-3) has the same form as the classical Coulomb friction considered in Section 7.1.9↑ and [114], with the standard Coulomb friction coefficient replaced by , as evaluated from (7.2.2-1). A trial state and return map, adapted from Section 7.1.9.7↑ [114] and presented in Section 7.2.6↓, is employed to differentiate between stick and slip.
The jump conditions for a biphasic mixture require continuity of the fluid pressure across an interface and therefore an expression for the normal fluid flux can be obtained by penalizing the fluid pressure gap between contacting points [8], where prescribes zero fluid pressure (free-draining conditions) on the portions of the boundary where no contact takes place. We define the fluid pressure gap as and is the pressure penalty parameter which has units of hydraulic permeability per unit length (e.g. m/Ns, similar to the hydraulic permeability of a membrane). Equation (7.2.4-4) is valid for both stick and slip. Note that is evaluated at a point with parametric coordinates on the primary surface, whereas the pressure on the secondary surface, , is evaluated at the parametric coordinates of the intersection of a ray issued from that primary surface point and normal to . As detailed previously [114] and summarized in Section7.2.3↑, the parametric coordinates of intersection are dependent on the stick-slip status, so care must be taken to evaluate from eq.(7.2.4-4) using the contact kinematics determined by eq.(7.2.2-2). In practice this is accomplished by calculating once the stick-slip status has been resolved for each iteration.
Subsubsection 7.2.3.3: Velocities Up Section 7.2: Biphasic Contact Subsection 7.2.5: Augmented Lagrangian Scheme