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Theory Manual Version 3.6
 Subsection 7.2.10: Biphasic Slip Up Subsection 7.2.10: Biphasic Slip Subsubsection 7.2.10.2: Discretization 

7.2.10.1 Linearization

During slip, the contact integral over is performed over integration points with prescribed parametric coordinates . However, the point on in contact with has parametric coordinates which change with variations in and , in accordance with eq.(7.2.3.1-3). Consequently, directional derivatives of , , , and are given by where is evaluated by our modification [12] of a method proposed by Laursen and Simo [65], and may be found in Section 7.1.9.10↑. The linearization of the slip traction proceeds as before [114], with the addition of a term involving the linearization of . From eq.(7.2.2-1) it follows that where according to eq.(7.2.4-2); this term has been provided previously in eq.(7.1.9.8-8) [114]. Equation (7.2.10.1-2) was derived by recalling that and are constants. We then obtain Finally, from eq.(7.2.4-4) (evaluated at determined by eq.(7.2.3.1-1)) and eq.(7.2.10.1-1) it follows that Note that the form of given in eq.(7.2.10.1-3) contains additional terms not present in eq.(7.2.9.1-3). This is because the parametric coordinates of intersection may vary in slip, but are invariant in stick; as a consequence, whether or not the linearization of depends on is determined by the stick-slip status. Per the discussion following eq.(7.2.4-5) in the main text, the fluid flux must be calculated after the stick-slip status has been resolved.
 Subsection 7.2.10: Biphasic Slip Up Subsection 7.2.10: Biphasic Slip Subsubsection 7.2.10.2: Discretization