7.7 Tied Multiphasic Contact

See Section Triphasic and Multiphasic Materials for a review of multiphasic materials, and ¹ for additional details on contact interfaces involving solutes. 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 \(\mathbf{t}^{\left(i\right)}\), solvent fluxes \(w_{n}^{\left(i\right)}\) and solute fluxes \(j_{n}^{\alpha\left(i\right)}\) (\(i=1,2)\) may be combined into the contact integral

\[ \begin{equation} \begin{aligned}\delta G_{c} & =\int_{\gamma^{\left(1\right)}}\left(\delta\mathbf{v}^{\left(1\right)}-\delta\mathbf{v}^{\left(2\right)}\right)\cdot\mathbf{t}^{\left(1\right)}\,da^{\left(1\right)}\\ & +\int_{\gamma^{\left(1\right)}}\left(\delta\tilde{p}^{\left(1\right)}-\delta\tilde{p}^{\left(2\right)}\right)w_{n}^{\left(1\right)}\,da^{\left(1\right)}\\ & +\sum_{\alpha}\int_{\gamma^{\left(1\right)}}\left(\delta\tilde{c}^{\alpha\left(1\right)}-\delta\tilde{c}^{\alpha\left(2\right)}\right)j_{n}^{\alpha\left(1\right)}\,da^{\left(1\right)}\,. \end{aligned} \label{eq696-1} \end{equation} \]

Note that the summation in \eqref{eq696-1} is performed only over solutes that are present on both sides of the contact interface. No special treatment is needed for solutes that only belong to one side, since the natural boundary condition for these solutes enforces zero normal flux across the contact interface.

To evaluate and linearize \(\delta G_{c}\), define the covariant basis vectors on each surface as

\[ \begin{equation} \mathbf{g}_{\alpha}^{\left(i\right)}=\frac{\partial\mathbf{x}^{\left(i\right)}}{\partial\eta_{\left(i\right)}^{\alpha}},\quad\alpha=1,2,\label{eq697-1} \end{equation} \]

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

\[ \begin{equation} \mathbf{n}^{\left(i\right)}=\frac{\mathbf{g}_{1}^{\left(i\right)}\times\mathbf{g}_{2}^{\left(i\right)}}{\left|\mathbf{g}_{1}^{\left(i\right)}\times\mathbf{g}_{2}^{\left(i\right)}\right|}\,.\label{eq698-1} \end{equation} \]

Now the contact integral may be rewritten as

\[ \begin{equation} \begin{aligned}\delta G_{c} & =\int_{\gamma^{\left(1\right)}}\left(\delta\mathbf{v}^{\left(1\right)}-\delta\mathbf{v}^{\left(2\right)}\right)\cdot\mathbf{t}\,\left|\mathbf{g}_{1}^{\left(1\right)}\times\mathbf{g}_{2}^{\left(1\right)}\right|\,d\eta_{\left(1\right)}^{1}d\eta_{\left(1\right)}^{2}\\ & +\int_{\gamma^{\left(1\right)}}\left(\delta\tilde{p}^{\left(1\right)}-\delta\tilde{p}^{\left(2\right)}\right)w_{n}\left|\mathbf{g}_{1}^{\left(1\right)}\times\mathbf{g}_{2}^{\left(1\right)}\right|\,d\eta_{\left(1\right)}^{1}d\eta_{\left(1\right)}^{2}\\ & +\sum_{\alpha}\int_{\gamma^{\left(1\right)}}\left(\delta\tilde{c}^{\alpha\left(1\right)}-\delta\tilde{c}^{\alpha\left(2\right)}\right)j_{n}^{\alpha}\left|\mathbf{g}_{1}^{\left(1\right)}\times\mathbf{g}_{2}^{\left(1\right)}\right|\,d\eta_{\left(1\right)}^{1}d\eta_{\left(1\right)}^{2} \end{aligned} \,,\label{eq699-1} \end{equation} \]

where \(\mathbf{t}\equiv\mathbf{t}^{\left(1\right)}\), \(w_{n}\equiv w_{n}^{\left(1\right)}\) and \(j_{n}^{\alpha}\equiv j_{n}^{\alpha\left(1\right)}\). The linearization \(D\delta G_{c}\) of \(\delta G_{c}\) has the form

\[ \begin{equation} D\delta G_{c}=\sum\limits_{i=1}^{2}D\delta G_{c}\left[\Delta\mathbf{u}^{\left(i\right)}\right]+D\delta G_{c}\left[\Delta\tilde{p}^{\left(i\right)}\right]+\sum_{\alpha}D\delta G_{c}\left[\Delta\tilde{c}^{\alpha\left(i\right)}\right]\,.\label{eq700-1} \end{equation} \]

Gap Function

The vector gap function \(\mathbf{g}\), representing the distance between the contact surfaces, is defined by

\[ \begin{equation} \mathbf{g}=\mathbf{x}^{\left(2\right)}-\mathbf{x}^{\left(1\right)}\,.\label{eq701-1} \end{equation} \]

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 Multiphasic Contact), 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.

Penalty Method

Let the tied contact traction be described by the penalty function,

\[ \begin{equation} \mathbf{t}=\varepsilon_{n}\mathbf{g}\,,\label{eq703-1} \end{equation} \]

where \(\varepsilon_{n}\) is a penalty factor associated with \(\mathbf{t}\). Similarly, let

\[ \begin{equation} w_{n}=\varepsilon_{p}\pi=\varepsilon_{p}\left(\tilde{p}^{\left(1\right)}-\tilde{p}^{\left(2\right)}\right)\,,\label{eq704-1} \end{equation} \]

where \(\varepsilon_{p}\) is a penalty factor associated with \(w_{n}\), and

\[ \begin{equation} j_{n}^{\alpha}=\varepsilon_{c}\chi^{\alpha}=\varepsilon_{c}^{\alpha}\left(\tilde{c}^{\alpha\left(1\right)}-\tilde{c}^{\alpha\left(2\right)}\right)\,,\label{eq704-1b} \end{equation} \]

where \(\varepsilon_{c}^{\alpha}\) is a penalty factor associated with \(j_{n}^{\alpha}\). It follows that

\[ \begin{equation} \begin{aligned}D\mathbf{t} & =\varepsilon_{n}\left(\Delta\mathbf{u}^{\left(2\right)}-\Delta\mathbf{u}^{\left(1\right)}\right)\,,\\ Dw_{n} & =\varepsilon_{p}\left(\Delta\tilde{p}^{\left(1\right)}-\Delta\tilde{p}^{\left(2\right)}\right)\,,\\ Dj_{n}^{\alpha} & =\varepsilon_{c}^{\alpha}\left(\Delta\tilde{c}^{\alpha\left(1\right)}-\Delta\tilde{c}^{\alpha\left(2\right)}\right)\,. \end{aligned} \label{eq705-1} \end{equation} \]

Given these relations, it can be shown that the directional derivatives of the various terms appearing in the integrand of \(\delta G_{c}\) are

\[ \begin{equation} \begin{aligned} & -D\left(J_{\eta}^{\left(1\right)}\left(\delta\mathbf{v}^{\left(1\right)}-\delta\mathbf{v}^{\left(2\right)}\right)\cdot\mathbf{t}\right)=\\ & \left(\delta\mathbf{v}^{\left(1\right)}-\delta\mathbf{v}^{\left(2\right)}\right)\cdot\varepsilon_{n}J_{\eta}^{\left(1\right)}\mathbf{I}\cdot\left(\Delta\mathbf{u}^{\left(1\right)}-\Delta\mathbf{u}^{\left(2\right)}\right)\\ & +\left(\delta\mathbf{v}^{\left(1\right)}-\delta\mathbf{v}^{\left(2\right)}\right)\cdot\mathbf{t}\otimes\left[\left(\mathbf{g}_{1}^{\left(1\right)}\times\mathbf{n}^{\left(1\right)}\right)\cdot\frac{\partial\Delta\mathbf{u}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{2}}-\left(\mathbf{g}_{2}^{\left(1\right)}\times\mathbf{n}^{\left(1\right)}\right)\cdot\frac{\partial\Delta\mathbf{u}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{1}}\right] \end{aligned} \,,\label{eq706-1} \end{equation} \]

\[ \begin{equation} \begin{aligned} & -D\left(w_{n}\left(\delta p^{\left(1\right)}-\delta p^{\left(2\right)}\right)J_{\eta}^{\left(1\right)}\right)=\\ & -J_{\eta}^{\left(1\right)}\varepsilon_{p}\left(\delta p^{\left(1\right)}-\delta p^{\left(2\right)}\right)\left(\Delta\tilde{p}^{\left(1\right)}-\Delta\tilde{p}^{\left(2\right)}\right)\\ & +w_{n}\left(\delta p^{\left(1\right)}-\delta p^{\left(2\right)}\right)\mathbf{n}^{\left(1\right)}\cdot\left(\mathbf{g}_{2}^{\left(1\right)}\times\frac{\partial\Delta\mathbf{u}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{1}}-\mathbf{g}_{1}^{\left(1\right)}\times\frac{\partial\Delta\mathbf{u}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{2}}\right) \end{aligned} \,,\label{eq707-1} \end{equation} \]

\[ \begin{equation} \begin{aligned} & -D\left(j_{n}^{\alpha}\left(\delta\tilde{c}^{\alpha\left(1\right)}-\delta\tilde{c}^{\alpha\left(2\right)}\right)J_{\eta}^{\left(1\right)}\right)=\\ & -J_{\eta}^{\left(1\right)}\left(\delta\tilde{c}^{\alpha\left(1\right)}-\delta\tilde{c}^{\alpha\left(2\right)}\right)\varepsilon_{c}^{\alpha}\left(\Delta\tilde{c}^{\alpha\left(1\right)}-\Delta\tilde{c}^{\alpha\left(2\right)}\right)\\ & +j_{n}^{\alpha}\left(\delta\tilde{c}^{\alpha\left(1\right)}-\delta\tilde{c}^{\alpha\left(2\right)}\right)\mathbf{n}^{\left(1\right)}\cdot\left(\mathbf{g}_{2}^{\left(1\right)}\times\frac{\partial\Delta\mathbf{u}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{1}}-\mathbf{g}_{1}^{\left(1\right)}\times\frac{\partial\Delta\mathbf{u}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{2}}\right) \end{aligned} \,,\label{eq707-1b} \end{equation} \]

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

\[ \begin{equation} \delta G_{c}=\sum\limits_{e=1}^{n_{e}^{\left(1\right)}}\sum\limits_{k=1}^{n_{\mbox{int}}^{\left(e\right)}}W_{k}J_{\eta}^{\left(1\right)}\left[\left(\delta\mathbf{v}^{\left(1\right)}-\delta\mathbf{v}^{\left(2\right)}\right)\cdot\mathbf{t}+w_{n}\left(\delta p^{\left(1\right)}-\delta p^{\left(2\right)}\right)+\sum_{\alpha}j_{n}^{\alpha}\left(\delta\tilde{c}^{\alpha\left(1\right)}-\delta\tilde{c}^{\alpha\left(2\right)}\right)\right]\,.\label{eq708-1} \end{equation} \]

The variables may be interpolated over each element face according to

\[ \begin{equation} \begin{aligned}\delta\mathbf{v}^{\left(1\right)} & =\sum\limits_{a=1}^{m^{\left(1\right)}}N_{a}^{\left(1\right)}\delta\mathbf{v}_{a}^{\left(1\right)} & \delta\mathbf{v}^{\left(2\right)} & =\sum\limits_{b=1}^{m^{\left(2\right)}}N_{b}^{\left(2\right)}\delta\mathbf{v}_{b}^{\left(2\right)}\\ \Delta\mathbf{u}^{\left(1\right)} & =\sum\limits_{c=1}^{m^{\left(1\right)}}N_{c}^{\left(1\right)}\Delta\mathbf{u}_{c}^{\left(1\right)} & \Delta\mathbf{u}^{\left(2\right)} & =\sum\limits_{d=1}^{m^{\left(2\right)}}N_{d}^{\left(2\right)}\Delta\mathbf{u}_{d}^{\left(2\right)}\\ \delta\tilde{p}^{\left(1\right)} & =\sum\limits_{a=1}^{m^{\left(1\right)}}N_{a}^{\left(1\right)}\delta\tilde{p}_{a}^{\left(1\right)} & \delta p^{\left(2\right)} & =\sum\limits_{b=1}^{m^{\left(2\right)}}N_{b}^{\left(2\right)}\delta\tilde{p}_{b}^{\left(2\right)}\\ \Delta\tilde{p}^{\left(1\right)} & =\sum\limits_{c=1}^{m^{\left(1\right)}}N_{c}^{\left(1\right)}\Delta\tilde{p}_{c}^{\left(1\right)} & \Delta\tilde{p}^{\left(2\right)} & =\sum\limits_{d=1}^{m^{\left(2\right)}}N_{d}^{\left(2\right)}\Delta\tilde{p}_{d}^{\left(2\right)}\\ \delta\tilde{c}^{\alpha\left(1\right)} & =\sum\limits_{a=1}^{m^{\left(1\right)}}N_{a}^{\left(1\right)}\delta\tilde{c}_{a}^{\alpha\left(1\right)} & \delta c^{\alpha\left(2\right)} & =\sum\limits_{b=1}^{m^{\left(2\right)}}N_{b}^{\left(2\right)}\delta\tilde{c}_{b}^{\alpha\left(2\right)}\\ \Delta\tilde{c}^{\alpha\left(1\right)} & =\sum\limits_{c=1}^{m^{\left(1\right)}}N_{c}^{\left(1\right)}\Delta\tilde{c}_{c}^{\alpha\left(1\right)} & \Delta\tilde{c}^{\alpha\left(2\right)} & =\sum\limits_{d=1}^{m^{\left(2\right)}}N_{d}^{\left(2\right)}\Delta\tilde{c}_{d}^{\alpha\left(2\right)} \end{aligned} \,.\label{eq709-1} \end{equation} \]

Then,

\[ \begin{equation} \begin{aligned}\delta G_{c} & =\sum\limits_{e=1}^{n_{e}^{\left(1\right)}}\sum\limits_{k=1}^{n_{\mbox{int}}^{\left(e\right)}}W_{k}J_{\eta}^{\left(1\right)}\left(\sum\limits_{a=1}^{m^{\left(1\right)}}\left[\begin{array}{cccc} \delta\mathbf{v}_{a}^{\left(1\right)} & \delta\tilde{p}_{a}^{\left(1\right)} & \delta\tilde{c}^{\alpha\left(1\right)} & \delta\tilde{c}^{\beta\left(1\right)}\end{array}\right]\cdot\left[\begin{array}{c} \mathbf{f}_{a}^{\left(1\right)}\\ w_{a}^{\left(1\right)}\\ j_{n}^{\alpha\left(1\right)}\\ j_{n}^{\beta\left(1\right)} \end{array}\right]\right.\\ & \left.+\sum\limits_{b=1}^{m_{k}^{\left(2\right)}}\left[\begin{array}{cccc} \delta\mathbf{v}_{b,k}^{\left(1\right)} & \delta\tilde{p}_{b,k}^{\left(1\right)} & \delta\tilde{c}_{b,k}^{\alpha\left(1\right)} & \delta\tilde{c}_{b,k}^{\beta\left(1\right)}\end{array}\right]\cdot\left[\begin{array}{c} \mathbf{f}_{b,k}^{\left(1\right)}\\ w_{b,k}^{\left(1\right)}\\ j_{b,k}^{\alpha\left(1\right)}\\ j_{b,k}^{\beta\left(1\right)} \end{array}\right]\right) \end{aligned} \,,\label{eq710-1} \end{equation} \]

where

\[ \begin{equation} \begin{aligned}\mathbf{f}_{a}^{\left(1\right)} & =N_{a}^{\left(1\right)}\mathbf{t} & \mathbf{f}_{b,k}^{\left(2\right)} & =-N_{b}^{\left(2\right)}\mathbf{t}\\ w_{a}^{\left(1\right)} & =N_{a}^{\left(1\right)}w_{n} & w_{b,k}^{\left(2\right)} & =-N_{b}^{\left(2\right)}w_{n}\\ j_{a}^{\gamma\left(1\right)} & =N_{a}^{\left(1\right)}j_{n}^{\gamma} & j_{b,k}^{\gamma\left(2\right)} & =-N_{b}^{\left(2\right)}j_{n}^{\gamma} \end{aligned} \,.\label{eq711-1} \end{equation} \]

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}{cccc} \delta\mathbf{v}_{a}^{\left(1\right)} & \delta\tilde{p}_{a}^{\left(1\right)} & \delta\tilde{c}_{a}^{\alpha\left(1\right)} & \delta\tilde{c}_{a}^{\beta\left(1\right)}\end{array}\right]\cdot\left(\sum\limits_{c=1}^{m^{\left(1\right)}}\left[\begin{array}{cccc} \mathbf{K}_{ac}^{\left(1,1\right)} & \mathbf{0} & \mathbf{0} & \mathbf{0}\\ \mathbf{k}_{ac}^{\left(1,1\right)} & k_{ac}^{\left(1,1\right)} & 0 & 0\\ \mathbf{k}_{ac}^{\alpha\left(1,1\right)} & 0 & k_{ac}^{\alpha\alpha\left(1,1\right)} & 0\\ \mathbf{k}_{ac}^{\beta\left(1,1\right)} & 0 & 0 & k_{ac}^{\beta\beta\left(1,1\right)} \end{array}\right]\cdot\left[\begin{array}{c} \Delta\mathbf{u}_{c}^{\left(1\right)}\\ \Delta\tilde{p}_{c}^{\left(1\right)}\\ \Delta\tilde{c}_{c}^{\alpha\left(1\right)}\\ \Delta\tilde{c}_{c}^{\beta\left(1\right)} \end{array}\right]\right.\right.\\ & +\left.\sum\limits_{d=1}^{m_{k}^{\left(2\right)}}\left[\begin{array}{cccc} \mathbf{K}_{ad,k}^{\left(1,2\right)} & \mathbf{0} & \mathbf{0} & \mathbf{0}\\ \mathbf{0} & k_{ad,k}^{\left(1,2\right)} & 0 & 0\\ \mathbf{0} & 0 & k_{ad,k}^{\alpha\alpha\left(1,2\right)} & 0\\ \mathbf{0} & 0 & 0 & k_{ad,k}^{\beta\beta\left(1,2\right)} \end{array}\right]\cdot\left[\begin{array}{c} \Delta\mathbf{u}_{d}^{\left(2\right)}\\ \Delta\tilde{p}_{d}^{\left(2\right)}\\ \Delta\tilde{c}_{d}^{\alpha\left(2\right)}\\ \Delta\tilde{c}_{d}^{\beta\left(2\right)} \end{array}\right]\right)\\ & +\sum\limits_{b=1}^{m_{k}^{\left(2\right)}}\left[\begin{array}{cccc} \delta\mathbf{v}_{b,k}^{\left(2\right)} & \delta\tilde{p}_{b,k}^{\left(2\right)} & \delta\tilde{c}_{b,k}^{\alpha\left(2\right)} & \delta\tilde{c}_{b,k}^{\beta\left(2\right)}\end{array}\right]\cdot\left(\sum\limits_{c=1}^{m^{\left(1\right)}}\left[\begin{array}{cccc} \mathbf{K}_{bc,k}^{\left(2,1\right)} & \mathbf{0} & \mathbf{0} & \mathbf{0}\\ \mathbf{k}_{bc,k}^{\left(2,1\right)} & k_{bc,k}^{\left(2,1\right)} & 0 & 0\\ \mathbf{k}_{bc,k}^{\alpha\left(2,1\right)} & 0 & k_{bc,k}^{\alpha\alpha\left(2,1\right)} & 0\\ \mathbf{k}_{bc,k}^{\beta\left(2,1\right)} & 0 & 0 & k_{bc,k}^{\beta\beta\left(2,1\right)} \end{array}\right]\cdot\left[\begin{array}{c} \Delta\mathbf{u}_{c}^{\left(1\right)}\\ \Delta\tilde{p}_{c}^{\left(1\right)}\\ \Delta\tilde{c}_{c}^{\alpha\left(1\right)} \end{array}\right]\right.\\ & +\left.\left.\sum\limits_{d=1}^{m_{k}^{\left(2\right)}}\left[\begin{array}{cccc} \mathbf{K}_{bd,k}^{\left(2,2\right)} & \mathbf{0} & \mathbf{0} & \mathbf{0}\\ \mathbf{0} & k_{bd,k}^{\left(2,2\right)} & 0 & 0\\ \mathbf{0} & 0 & k_{bd,k}^{\alpha\alpha\left(2,2\right)} & 0\\ \mathbf{0} & 0 & 0 & k_{bd,k}^{\beta\beta\left(2,2\right)} \end{array}\right]\cdot\left[\begin{array}{c} \Delta\mathbf{u}_{d}^{\left(2\right)}\\ \Delta\tilde{p}_{d}^{\left(2\right)}\\ \Delta\tilde{c}_{d}^{\alpha\left(2\right)}\\ \Delta\tilde{c}_{d}^{\beta\left(2\right)} \end{array}\right]\right)\right)\,, \end{aligned} \label{eq712-1} \end{equation} \]

where

\[ \begin{equation} \begin{aligned}\mathbf{K}_{ac}^{\left(1,1\right)} & =N_{a}^{\left(1\right)}\left(\varepsilon_{n}N_{c}^{\left(1\right)}\mathbf{I}+\mathbf{t}\otimes\mathbf{a}_{c}^{\left(1\right)}\right)\\ \mathbf{K}_{ad,k}^{\left(1,2\right)} & =-\varepsilon_{n}N_{a}^{\left(1\right)}N_{d}^{\left(2\right)}\mathbf{I}\\ \mathbf{K}_{bc,k}^{\left(2,1\right)} & =-N_{b}^{\left(2\right)}\left(\varepsilon_{n}N_{c}^{\left(1\right)}\mathbf{I}+\mathbf{t}\otimes\mathbf{a}_{c}^{\left(1\right)}\right)\\ \mathbf{K}_{bd,k}^{\left(2,2\right)} & =\varepsilon_{n}N_{b}^{\left(2\right)}N_{d}^{\left(2\right)}\mathbf{I} \end{aligned} \,,\label{eq713-1} \end{equation} \]

\[ \begin{equation} \begin{aligned}\mathbf{k}_{ac}^{\left(1,1\right)} & =N_{a}^{\left(1\right)}w_{n}\mathbf{a}_{c}^{\left(1\right)}\\ \mathbf{k}_{bc,k}^{\left(2,1\right)} & =-N_{b}^{\left(2\right)}w_{n}\mathbf{a}_{c}^{\left(1\right)}\\ \mathbf{k}_{ac}^{\alpha\left(1,1\right)} & =N_{a}^{\left(1\right)}j_{n}^{\alpha}\mathbf{a}_{c}^{\left(1\right)}\\ \mathbf{k}_{bc,k}^{\alpha\left(2,1\right)} & =-N_{b}^{\left(2\right)}j_{n}^{\alpha}\mathbf{a}_{c}^{\left(1\right)} \end{aligned} \,,\label{eq714-1} \end{equation} \]

\[ \begin{equation} \begin{aligned}k_{ac}^{\left(1,1\right)} & =-\varepsilon_{p}N_{a}^{\left(1\right)}N_{c}^{\left(1\right)}\\ k_{ad,k}^{\left(1,2\right)} & =\varepsilon_{p}N_{a}^{\left(1\right)}N_{d}^{\left(2\right)}\\ k_{bc,k}^{\left(2,1\right)} & =\varepsilon_{p}N_{b}^{\left(2\right)}N_{c}^{\left(1\right)}\\ k_{bd,k}^{\left(2,2\right)} & =-\varepsilon_{p}N_{b}^{\left(2\right)}N_{d}^{\left(2\right)} \end{aligned} \,,\label{eq715-1} \end{equation} \]

\[ \begin{equation} \begin{aligned}k_{ac}^{\gamma\gamma\left(1,1\right)} & =-\varepsilon_{c}^{\gamma}N_{a}^{\left(1\right)}N_{c}^{\left(1\right)}\\ k_{ad,k}^{\gamma\gamma\left(1,2\right)} & =\varepsilon_{c}^{\gamma}N_{a}^{\left(1\right)}N_{d}^{\left(2\right)}\\ k_{bc,k}^{\gamma\gamma\left(2,1\right)} & =\varepsilon_{c}^{\gamma}N_{b}^{\left(2\right)}N_{c}^{\left(1\right)}\\ k_{bd,k}^{\gamma\gamma\left(2,2\right)} & =-\varepsilon_{c}^{\gamma}N_{b}^{\left(2\right)}N_{d}^{\left(2\right)} \end{aligned} \label{eq715-1b} \end{equation} \]

and

\[ \begin{equation} \mathbf{a}_{c}^{\left(1\right)}=\frac{1}{J_{\eta}^{\left(1\right)}}\mathbf{n}^{\left(1\right)}\times\left(\mathbf{g}_{2}^{\left(1\right)}\frac{\partial N_{c}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{1}}-\mathbf{g}_{1}^{\left(1\right)}\frac{\partial N_{c}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{2}}\right)\,.\label{eq716-1} \end{equation} \]

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1. Ateshian, Gerard A; Maas, Steve; Weiss, Jeffrey A. "Solute transport across a contact interface in deformable porous media." J Biomech, vol. 45, pp. 1023-7 (2012). ↩