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Theory Manual Version 3.6
 Subsubsection 7.1.9.9: Discretization Up Subsection 7.1.9: Sliding-Elastic Subsubsection 7.1.9.11: Frictional Terms 

7.1.9.10 Frictionless Terms

The linearization of the frictionless contribution makes use of eq.(7.1.9.8-5) to find Evaluating eq.(7.1.9.10-1) requires , , and . Directly inserting Eqs.(7.1.9.8-5) and (7.1.9.9-1) into Eqs.(7.1.9.8-6), (7.1.9.8-8), and (7.1.9.8-12) yields with the definitions
Applying the discretization in eq.(7.1.9.9-1) to eq.(7.1.9.10-1) and employing the linearizations of Eqs.(7.1.9.9-3) and (7.1.9.10-2) to evaluate the resulting matrix products yields the directional derivative of as where and it is apparent that the resulting frictionless stiffness matrix is also nonsymmetric. eq.(7.1.9.10-3) is very similar to that which may be found by reducing our previous frictionless biphasic contact algorithm to the case of contact between two nonporous solids (eliminating all fluid degrees of freedom) [12], although the present framework is more general, since that previous study evaluated expressions in the limit as .
 Subsubsection 7.1.9.9: Discretization Up Subsection 7.1.9: Sliding-Elastic Subsubsection 7.1.9.11: Frictional Terms