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The vector gap function $\mathbf{g}$ , representing the distance between the contact surfaces, is defined by $$\begin{equation} \mathbf{g}=\mathbf{x}^{\left(2\right)}-\mathbf{x}^{\left(1\right)}\,.\label{eq701-1} \end{equation}$$ The premise of a tied interface is that the parametric coordinates of $\mathbf{x}^{\left(1\right)}$ and $\mathbf{x}^{\left(2\right)}$ are both invariants (i.e., they are determined in the reference configuration and remain unchanged over time). The parametric coordinates of $\mathbf{x}^{\left(1\right)}$ correspond to the integration points on $\gamma^{\left(1\right)}$ , and those of $\mathbf{x}^{\left(2\right)}$ are evaluated once, in the reference configuration, by shooting a ray from the integration point on $\gamma^{\left(1\right)}$ to intersect $\gamma^{\left(2\right)}$ . It follows from this premise that $$\begin{equation} \begin{aligned}D\mathbf{x}^{\left(i\right)} & =\Delta\mathbf{u}^{\left(i\right)} & D\delta\mathbf{v}^{\left(i\right)} & =\mathbf{0}\\ D\tilde{p}^{\left(i\right)} & =\Delta\tilde{p}^{\left(i\right)} & D\delta\tilde{p}^{\left(i\right)} & =0\\ D\tilde{c}^{\alpha\left(i\right)} & =\Delta\tilde{c}^{\alpha\left(i\right)} & D\delta\tilde{c}^{\alpha\left(i\right)} & =0 \end{aligned} \quad i=1,2\,.\label{eq702-1} \end{equation}$$ If $\gamma^{\left(1\right)}$ and $\gamma^{\left(2\right)}$ are not initially conforming, continuity of effective fluid pressure and solute concentration, as well as normal fluid and solute fluxes, will only be enforced within the contact interface; unlike the sliding multiphasic contact interface (Section 7.4↑), ambient conditions are not set automatically on regions of $\gamma^{\left(1\right)}$ and $\gamma^{\left(2\right)}$ where a solution for $\mathbf{g}$ was not found. Therefore, these regions naturally enforce zero fluid and solute flux (impermeable boundary), unless an explicit boundary condition on the effective pressure $\tilde{p}$ or solute concentrations $\tilde{c}^{\alpha}$ are prescribed over those regions.