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Theory Manual Version 3.6
 Subsubsection 7.2.10.1: Linearization Up Subsection 7.2.10: Biphasic Slip Section 7.3: Biphasic-Solute Contact  

7.2.10.2 Discretization

Let the continuous variables on the primary and secondary surfaces be interpolated over each element face according to In the case of slip, the contact integral of eq.(7.2.1-4) can be split into normal and tangential parts, , such that This split will be useful for the full linearization. Discretizing this expression yields where and is given by Eqs.(7.2.4-2)-(7.2.4-3) for the penalty method and Eqs.(7.2.5-2)-(7.2.5-3) for augmented Lagrangian regularization; similarly, is given by eq.(7.2.4-4) or eq.(7.2.5-4) for penalty and augmented Lagrangian schemes, respectively. For the following linearization and discretization the normal and tangential components, representing frictionless and frictional contributions to the contact integral, will be treated separately and may then be added together as in eq.(7.2.10.2-2).
Frictionless Terms
The linearization of the frictional part of eq.(7.2.10.2-2) makes use of eq.(7.2.10.1-1) to find where we note that the virtual variables on the secondary surface now enter the linearization, as parametric coordinates on the secondary surface vary during slip according to eq.(7.2.3.1-3). The only linearization which has not been previously discretized is , and it follows from placing eq.(7.2.10.2-1) into eq.(7.2.10.1-3) that where we define and for convenience we note that Directional derivatives of virtual variables on the secondary surface will lead to expressions of the form , where In eq.(7.2.10.2-7), the pressure degrees of freedom do not enter into the linearization. In an effort to keep the expression more compact, we have not included these variables. However, this expression could easily be cast into the form of e.g. eq.(7.2.10.2-6) by adding zeros where necessary. Discretizing eq.(7.2.10.2-3) and making use of linearizations found previously [114], along with Eqs.(7.2.10.2-4) and (7.2.10.2-6), allows the resulting stiffness matrix to be expressed as where and and The expressions above are very similar to those which can be found in our frictionless biphasic contact paper [8], although the present framework is more general, as that previous study evaluated expressions in the limit as (see Zimmerman and Ateshian [114] for a detailed discussion of this assumption and the benefits of relaxing it).
Frictional Terms
Linearizing the frictional contribution follows from the second term of eq.(7.2.10.2-2) as The remaining quantity in this expression to be determined is the discretization of ; from eq.(7.2.10.1-2) and eq.(7.2.10.2-1) it follows that where we made the definition just to be used in eq.(7.2.10.2-10) for space considerations. Finally, inserting Eqs.(7.2.10.2-1) and (7.2.10.2-10) into eq.(7.2.10.2-9) yields the stiffness matrix for the frictional terms where and In eq.(7.2.10.2-12), we have defined by analogy with our definitions for and in Section 7.1.9.11↑. The stiffness matrix of the frictional contribution to the virtual work, given by eq.(7.2.10.2-11), is nonsymmetric. Summing Eqs.(7.2.10.2-8) and (7.2.10.2-11) produces the total stiffness matrix for the case of frictional biphasic slip; this total stiffness matrix is also nonsymmetric. However, as noted before [8], the stiffness matrix for biphasic materials is nonsymmetric by construction, so there is no expectation that the contact stiffness matrices be symmetric.
Equation (7.2.10.2-12) contains the solid-solid stiffness terms. Comparing eq.(7.2.10.2-12) with eq.(7.1.9.11-3) shows that the same form of the equations is recovered. Interestingly, however, some of the terms in sliding-elastic (Section 7.1.9↑) which were multiplied by the friction coefficient are now multiplied by , which indicates certain terms which are important in biphasic contact. Of course, in the absence of pressurized fluid, and we recover the sliding-elastic formulation.
 Subsubsection 7.2.10.1: Linearization Up Subsection 7.2.10: Biphasic Slip Section 7.3: Biphasic-Solute Contact