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Theory Manual Version 3.6
 Section 7.2: Biphasic Contact Up Section 7.2: Biphasic Contact Subsection 7.2.2: Biphasic Friction Formulation 

7.2.1 Contact Integral

See Section 2.5↑ for a review of biphasic materials, reference [12] for additional details on biphasic frictionless contact, and reference [115] for biphasic frictional contact. The presentation here follows that of [115]. The contact interface is defined between surfaces and . Due to continuity requirements on the traction and fluxes, the external virtual work resulting from contact tractions and solvent fluxes ( , may be combined into the contact integral where is the contact traction on the primary surface, which is equal and opposite to that on the secondary surface, ; is the outward normal component of the fluid flux from the primary surface, are virtual velocities, are virtual fluid pressures, and is an elemental area on the primary surface . The contact integral is then written over the invariant parametric space of , which is denoted by [28], facilitating its linearization as required for use with an iterative solution method such as Newton's method. The elemental area is given by using the covariant basis vectors where is the spatial representation of surface as it deforms in time , in terms of contravariant surface parametric coordinates . Casting eq.(7.2.1-1) into convenient matrix notation and switching the domain of integration to the parametric frame yields the invariant biphasic contact integral Since represents an invariant material frame, the linearization of can be accomplished by applying the directional derivative operator directly to the integrand, where it is understood that for any function in this biphasic analysis,
 Section 7.2: Biphasic Contact Up Section 7.2: Biphasic Contact Subsection 7.2.2: Biphasic Friction Formulation