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Theory Manual Version 3.6
 Subsection 7.2.10: Biphasic Slip Up Subsection 7.2.10: Biphasic Slip Subsubsection Discretization 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.( 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↑. 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.( [114]. Equation ( was derived by recalling that and are constants. We then obtain Finally, from eq.(7.2.4-4) (evaluated at determined by eq.( and eq.( it follows that Note that the form of given in eq.( contains additional terms not present in eq.( 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 Discretization