See Section Biphasic Material for a review of biphasic materials, and for additional details on biphasic contact. The contact interface is defined between surfaces \(\gamma^{\left(1\right)}\) and \(\gamma^{\left(2\right)}\). Due to continuity requirements on the traction and fluxes, the external virtual work resulting from contact tractions \(\mathbf{t}^{\left(i\right)}\) and solvent fluxes \(w_{n}^{\left(i\right)}\) (\(i=1,2)\) may be combined into the contact integral
\[ \begin{equation} \begin{aligned}\delta G_{c} & =\int_{\gamma^{\left(1\right)}}\left(\delta\mathbf{v}^{\left(1\right)}-\delta\mathbf{v}^{\left(2\right)}\right)\cdot\mathbf{t}^{\left(1\right)}da^{\left(1\right)}\\ & +\int_{\gamma^{\left(1\right)}}\left(\delta p^{\left(1\right)}-\delta p^{\left(2\right)}\right)w_{n}^{\left(1\right)}da^{\left(1\right)}\,. \end{aligned} \label{eq696} \end{equation} \]
To evaluate and linearize \(\delta G_{c}\), define the covariant basis vectors on each surface as
\[ \begin{equation} \mathbf{g}_{\alpha}^{\left(i\right)}=\frac{\partial\mathbf{x}^{\left(i\right)}}{\partial\eta_{\left(i\right)}^{\alpha}},\quad\alpha=1,2,\label{eq697} \end{equation} \]
where \(\mathbf{x}^{\left(i\right)}\) represents the spatial position of points on \(\gamma^{\left(i\right)}\), and \(\eta_{\left(i\right)}^{\alpha}\) represent the parametric coordinates of that point. The unit outward normal on each surface is then given by
\[ \begin{equation} \mathbf{n}^{\left(i\right)}=\frac{\mathbf{g}_{1}^{\left(i\right)}\times\mathbf{g}_{2}^{\left(i\right)}}{\left|\mathbf{g}_{1}^{\left(i\right)}\times\mathbf{g}_{2}^{\left(i\right)}\right|}\,.\label{eq698} \end{equation} \]
Now the contact integral may be rewritten as
\[ \begin{equation} \begin{aligned}\delta G_{c} & =\int_{\gamma^{\left(1\right)}}\left(\delta\mathbf{v}^{\left(1\right)}-\delta\mathbf{v}^{\left(2\right)}\right)\cdot\mathbf{t}\,\left|\mathbf{g}_{1}^{\left(1\right)}\times\mathbf{g}_{2}^{\left(1\right)}\right|\,d\eta_{\left(1\right)}^{1}d\eta_{\left(1\right)}^{2}\\ & +\int_{\gamma^{\left(1\right)}}\left(\delta p^{\left(1\right)}-\delta p^{\left(2\right)}\right)w_{n}\left|\mathbf{g}_{1}^{\left(1\right)}\times\mathbf{g}_{2}^{\left(1\right)}\right|\,d\eta_{\left(1\right)}^{1}d\eta_{\left(1\right)}^{2} \end{aligned} \,,\label{eq699} \end{equation} \]
where \(\mathbf{t}\equiv\mathbf{t}^{\left(1\right)}\) and \(w_{n}\equiv w_{n}^{\left(1\right)}\). The linearization \(D\delta G_{c}\) of \(\delta G_{c}\) has the form
\[ \begin{equation} D\delta G_{c}=\sum\limits_{i=1}^{2}D\delta G_{c}\left[\Delta\mathbf{u}^{\left(i\right)}\right]+D\delta G_{c}\left[\Delta p^{\left(i\right)}\right]\,.\label{eq700} \end{equation} \]
Gap Function
The vector gap function \(\mathbf{g}\), representing the distance between the contact surfaces, is defined by
\[ \begin{equation} \mathbf{g}=\mathbf{x}^{\left(2\right)}-\mathbf{x}^{\left(1\right)}\,.\label{eq701} \end{equation} \]
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
\[ \begin{equation} \begin{aligned}D\mathbf{x}^{\left(1\right)} & =\Delta\mathbf{u}^{\left(1\right)} & D\mathbf{x}^{\left(2\right)} & =\Delta\mathbf{u}^{\left(2\right)}\\ Dp^{\left(1\right)} & =\Delta p^{\left(1\right)} & Dp^{\left(2\right)} & =\Delta p^{\left(2\right)}\\ D\delta\mathbf{v}^{\left(1\right)} & =\mathbf{0} & D\delta\mathbf{v}^{\left(2\right)} & =\mathbf{0}\\ D\delta p^{\left(1\right)} & =0 & D\delta p^{\left(2\right)} & =0 \end{aligned} \,.\label{eq702} \end{equation} \]
If \(\gamma^{\left(1\right)}\) and \(\gamma^{\left(2\right)}\) are not initially conforming, continuity of fluid pressure and normal flux will only be enforced within the contact interface; unlike the sliding biphasic contact interface (Section Biphasic Contact), free-draining 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 flux (impermeable boundary), unless an explicit boundary condition on the pressure \(p\) is prescribed over those regions.
Penalty Method
Let the tied contact traction be described by the penalty function,
\[ \begin{equation} \mathbf{t}=\varepsilon_{n}\mathbf{g}\,,\label{eq703} \end{equation} \]
where \(\varepsilon_{n}\) is a penalty factor associated with \(\mathbf{t}\). Similarly, let
\[ \begin{equation} w_{n}=\varepsilon_{p}\pi=\varepsilon_{p}\left(p^{\left(1\right)}-p^{\left(2\right)}\right)\,,\label{eq704} \end{equation} \]
where \(\varepsilon_{p}\) is a penalty factor associated with \(w_{n}\). It follows that
\[ \begin{equation} \begin{aligned}D\mathbf{t} & =\varepsilon_{n}\left(\Delta\mathbf{u}^{\left(2\right)}-\Delta\mathbf{u}^{\left(1\right)}\right)\,,\\ Dw_{n} & =\varepsilon_{p}\left(\Delta p^{\left(1\right)}-\Delta p^{\left(2\right)}\right)\,. \end{aligned} \label{eq705} \end{equation} \]
Given these relations, it can be shown that the directional derivatives of the various terms appearing in the integrand of \(\delta G_{c}\) are
\[ \begin{equation} \begin{aligned} & -D\left(J_{\eta}^{\left(1\right)}\left(\delta\mathbf{v}^{\left(1\right)}-\delta\mathbf{v}^{\left(2\right)}\right)\cdot\mathbf{t}\right)=\\ & \left(\delta\mathbf{v}^{\left(1\right)}-\delta\mathbf{v}^{\left(2\right)}\right)\cdot\varepsilon_{n}J_{\eta}^{\left(1\right)}\mathbf{I}\cdot\left(\Delta\mathbf{u}^{\left(1\right)}-\Delta\mathbf{u}^{\left(2\right)}\right)\\ & +\left(\delta\mathbf{v}^{\left(1\right)}-\delta\mathbf{v}^{\left(2\right)}\right)\cdot\mathbf{t}\otimes\left[\left(\mathbf{g}_{1}^{\left(1\right)}\times\mathbf{n}^{\left(1\right)}\right)\cdot\frac{\partial\Delta\mathbf{u}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{2}}-\left(\mathbf{g}_{2}^{\left(1\right)}\times\mathbf{n}^{\left(1\right)}\right)\cdot\frac{\partial\Delta\mathbf{u}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{1}}\right] \end{aligned} \,,\label{eq706} \end{equation} \]
\[ \begin{equation} \begin{aligned} & -D\left(w_{n}\left(\delta p^{\left(1\right)}-\delta p^{\left(2\right)}\right)J_{\eta}^{\left(1\right)}\right)=\\ & -J_{\eta}^{\left(1\right)}\varepsilon_{p}\left(\delta p^{\left(1\right)}-\delta p^{\left(2\right)}\right)\left(\Delta p^{\left(1\right)}-\Delta p^{\left(2\right)}\right)\\ & +w_{n}\left(\delta p^{\left(1\right)}-\delta p^{\left(2\right)}\right)\mathbf{n}^{\left(1\right)}\cdot\left(\mathbf{g}_{2}^{\left(1\right)}\times\frac{\partial\Delta\mathbf{u}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{1}}-\mathbf{g}_{1}^{\left(1\right)}\times\frac{\partial\Delta\mathbf{u}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{2}}\right) \end{aligned} \,,\label{eq707} \end{equation} \]
where \(J_{\eta}^{\left(1\right)}=\left|\mathbf{g}_{1}^{\left(1\right)}\times\mathbf{g}_{2}^{\left(1\right)}\right|\).
Discretization
The contact integral may be discretized as
\[ \begin{equation} \delta G_{c}=\sum\limits_{e=1}^{n_{e}^{\left(1\right)}}\sum\limits_{k=1}^{n_{\mbox{int}}^{\left(e\right)}}W_{k}J_{\eta}^{\left(1\right)}\left[\left(\delta\mathbf{v}^{\left(1\right)}-\delta\mathbf{v}^{\left(2\right)}\right)\cdot\mathbf{t}+w_{n}\left(\delta p^{\left(1\right)}-\delta p^{\left(2\right)}\right)\right]\,.\label{eq708} \end{equation} \]
The variables may be interpolated over each element face according to
\[ \begin{equation} \begin{aligned}\delta\mathbf{v}^{\left(1\right)} & =\sum\limits_{a=1}^{m^{\left(1\right)}}N_{a}^{\left(1\right)}\delta\mathbf{v}_{a}^{\left(1\right)} & \delta\mathbf{v}^{\left(2\right)} & =\sum\limits_{b=1}^{m^{\left(2\right)}}N_{b}^{\left(2\right)}\delta\mathbf{v}_{b}^{\left(2\right)}\\ \Delta\mathbf{u}^{\left(1\right)} & =\sum\limits_{c=1}^{m^{\left(1\right)}}N_{c}^{\left(1\right)}\Delta\mathbf{u}_{c}^{\left(1\right)} & \Delta\mathbf{u}^{\left(2\right)} & =\sum\limits_{d=1}^{m^{\left(2\right)}}N_{d}^{\left(2\right)}\Delta\mathbf{u}_{d}^{\left(2\right)}\\ \delta p^{\left(1\right)} & =\sum\limits_{a=1}^{m^{\left(1\right)}}N_{a}^{\left(1\right)}\delta p_{a}^{\left(1\right)} & \delta p^{\left(2\right)} & =\sum\limits_{b=1}^{m^{\left(2\right)}}N_{b}^{\left(2\right)}\delta p_{b}^{\left(2\right)}\\ \Delta p^{\left(1\right)} & =\sum\limits_{c=1}^{m^{\left(1\right)}}N_{c}^{\left(1\right)}\Delta p_{c}^{\left(1\right)} & \Delta p^{\left(2\right)} & =\sum\limits_{d=1}^{m^{\left(2\right)}}N_{d}^{\left(2\right)}\Delta p_{d}^{\left(2\right)} \end{aligned} \,.\label{eq709} \end{equation} \]
Then,
\[ \begin{equation} \begin{aligned}\delta G_{c} & =\sum\limits_{e=1}^{n_{e}^{\left(1\right)}}\sum\limits_{k=1}^{n_{\mbox{int}}^{\left(e\right)}}W_{k}J_{\eta}^{\left(1\right)}\left(\sum\limits_{a=1}^{m^{\left(1\right)}}\left[\begin{array}{cc} \delta\mathbf{v}_{a}^{\left(1\right)} & \delta p_{a}^{\left(1\right)}\end{array}\right]\cdot\left[\begin{array}{c} \mathbf{f}_{a}^{\left(1\right)}\\ w_{a}^{\left(1\right)} \end{array}\right]\right.\\ & \left.+\sum\limits_{b=1}^{m_{k}^{\left(2\right)}}\left[\begin{array}{cc} \delta\mathbf{v}_{b,k}^{\left(1\right)} & \delta p_{b,k}^{\left(1\right)}\end{array}\right]\cdot\left[\begin{array}{c} \mathbf{f}_{b,k}^{\left(1\right)}\\ w_{b,k}^{\left(1\right)} \end{array}\right]\right) \end{aligned} \,,\label{eq710} \end{equation} \]
where
\[ \begin{equation} \begin{aligned}\mathbf{f}_{a}^{\left(1\right)} & =N_{a}^{\left(1\right)}\mathbf{t} & \mathbf{f}_{b,k}^{\left(2\right)} & =-N_{b}^{\left(2\right)}\mathbf{t}\\ w_{a}^{\left(1\right)} & =N_{a}^{\left(1\right)}w_{n} & w_{b,k}^{\left(2\right)} & =-N_{b}^{\left(2\right)}w_{n} \end{aligned} \,.\label{eq711} \end{equation} \]
Similarly,
\[ \begin{equation} \begin{aligned}-D\delta G_{c} & =\sum\limits_{e=1}^{n_{e}^{\left(1\right)}}\sum\limits_{k=1}^{n_{\mbox{int}}^{\left(e\right)}}W_{k}J_{\eta}^{\left(1\right)}\\ & \times\left(\sum\limits_{a=1}^{m^{\left(1\right)}}\left[\begin{array}{cc} \delta\mathbf{v}_{a}^{\left(1\right)} & \delta p_{a}^{\left(1\right)}\end{array}\right]\cdot\left(\sum\limits_{c=1}^{m^{\left(1\right)}}\left[\begin{array}{cc} \mathbf{K}_{ac}^{\left(1,1\right)} & \mathbf{0}\\ \mathbf{k}_{ac}^{\left(1,1\right)} & k_{ac}^{\left(1,1\right)} \end{array}\right]\cdot\left[\begin{array}{c} \Delta\mathbf{u}_{c}^{\left(1\right)}\\ \Delta p_{c}^{\left(1\right)} \end{array}\right]\right.\right.\\ & +\left.\sum\limits_{d=1}^{m_{k}^{\left(2\right)}}\left[\begin{array}{cc} \mathbf{K}_{ad,k}^{\left(1,2\right)} & \mathbf{0}\\ \mathbf{0} & k_{ad,k}^{\left(1,2\right)} \end{array}\right]\cdot\left[\begin{array}{c} \Delta\mathbf{u}_{d}^{\left(2\right)}\\ \Delta p_{d}^{\left(2\right)} \end{array}\right]\right)\\ & +\sum\limits_{b=1}^{m_{k}^{\left(2\right)}}\left[\begin{array}{cc} \delta\mathbf{v}_{b,k}^{\left(2\right)} & \delta p_{b,k}^{\left(2\right)}\end{array}\right]\cdot\left(\sum\limits_{c=1}^{m^{\left(1\right)}}\left[\begin{array}{cc} \mathbf{K}_{bc,k}^{\left(2,1\right)} & \mathbf{0}\\ \mathbf{k}_{bc,k}^{\left(2,1\right)} & k_{bc,k}^{\left(2,1\right)} \end{array}\right]\cdot\left[\begin{array}{c} \Delta\mathbf{u}_{c}^{\left(1\right)}\\ \Delta p_{c}^{\left(1\right)} \end{array}\right]\right.\\ & +\left.\left.\sum\limits_{d=1}^{m_{k}^{\left(2\right)}}\left[\begin{array}{cc} \mathbf{K}_{bd,k}^{\left(2,2\right)} & \mathbf{0}\\ \mathbf{0} & k_{bd,k}^{\left(2,2\right)} \end{array}\right]\cdot\left[\begin{array}{c} \Delta\mathbf{u}_{d}^{\left(2\right)}\\ \Delta p_{d}^{\left(2\right)} \end{array}\right]\right)\right)\,, \end{aligned} \label{eq712} \end{equation} \]
where
\[ \begin{equation} \begin{aligned}\mathbf{K}_{ac}^{\left(1,1\right)} & =N_{a}^{\left(1\right)}\left(\varepsilon_{n}N_{c}^{\left(1\right)}\mathbf{I}+\mathbf{t}\otimes\mathbf{a}_{c}^{\left(1\right)}\right)\\ \mathbf{K}_{ad,k}^{\left(1,2\right)} & =-\varepsilon_{n}N_{a}^{\left(1\right)}N_{d}^{\left(2\right)}\mathbf{I}\\ \mathbf{K}_{bc,k}^{\left(2,1\right)} & =-N_{b}^{\left(2\right)}\left(\varepsilon_{n}N_{c}^{\left(1\right)}\mathbf{I}+\mathbf{t}\otimes\mathbf{a}_{c}^{\left(1\right)}\right)\\ \mathbf{K}_{bd,k}^{\left(2,2\right)} & =\varepsilon_{n}N_{b}^{\left(2\right)}N_{d}^{\left(2\right)}\mathbf{I} \end{aligned} \,,\label{eq713} \end{equation} \]
\[ \begin{equation} \begin{aligned}\mathbf{k}_{ac}^{\left(1,1\right)} & =N_{a}^{\left(1\right)}w_{n}\mathbf{a}_{c}^{\left(1\right)}\\ \mathbf{k}_{bc,k}^{\left(2,1\right)} & =-N_{b}^{\left(2\right)}w_{n}\mathbf{a}_{c}^{\left(1\right)} \end{aligned} \,,\label{eq714} \end{equation} \]
\[ \begin{equation} \begin{aligned}k_{ac}^{\left(1,1\right)} & =-\varepsilon_{p}N_{a}^{\left(1\right)}N_{c}^{\left(1\right)}\\ k_{ad,k}^{\left(1,2\right)} & =\varepsilon_{p}N_{a}^{\left(1\right)}N_{d}^{\left(2\right)}\\ k_{bc,k}^{\left(2,1\right)} & =\varepsilon_{p}N_{b}^{\left(2\right)}N_{c}^{\left(1\right)}\\ k_{bd,k}^{\left(2,2\right)} & =-\varepsilon_{p}N_{b}^{\left(2\right)}N_{d}^{\left(2\right)} \end{aligned} \,,\label{eq715} \end{equation} \]
and
\[ \begin{equation} \mathbf{a}_{c}^{\left(1\right)}=\frac{1}{J_{\eta}^{\left(1\right)}}\mathbf{n}^{\left(1\right)}\times\left(\mathbf{g}_{2}^{\left(1\right)}\frac{\partial N_{c}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{1}}-\mathbf{g}_{1}^{\left(1\right)}\frac{\partial N_{c}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{2}}\right)\,.\label{eq716} \end{equation} \]