$\newcommand{\Dev}{\operatorname{Dev}}$ $\newcommand{\dev}{\operatorname{dev}}$ $\newcommand{\grad}{\operatorname{grad}}$ $\newcommand{\Grad}{\operatorname{Grad}}$ $\newcommand{\divg}{\operatorname{div}}$ $\newcommand{\Ei}{\operatorname{Ei}}$ $\newcommand{\tr}{\operatorname{tr}}$ $\newcommand{\Divg}{\operatorname{Div}}$ $\newcommand{\cay}{\operatorname{cay}}$ $\newcommand{\rot}{\operatorname{rot}}$

The contact integral may be discretized as $$\begin{equation} \delta G_{c}=\sum\limits _{e=1}^{n_{e}^{\left(1\right)}}\sum\limits _{k=1}^{n_{\mbox{int}}^{\left(e\right)}}W_{k}J_{\eta}^{\left(1\right)}\left[t_{n}\left(\delta\mathbf{v}^{\left(1\right)}-\delta\mathbf{v}^{\left(2\right)}\right)\cdot\mathbf{n}^{\left(1\right)}+w_{n}\left(\delta\tilde{p}^{\left(1\right)}-\delta\tilde{p}^{\left(2\right)}\right)+\sum\limits _{\gamma}j_{n}^{\gamma}\left(\delta\tilde{c}^{\gamma\left(1\right)}-\delta\tilde{c}^{\gamma\left(2\right)}\right)\right]\,.\label{eq668} \end{equation}$$ The variables may be interpolated over each element face according to $$\begin{equation} \begin{aligned}\delta\mathbf{v}^{\left(1\right)} & =\sum\limits _{a=1}^{m^{\left(1\right)}}N_{a}^{\left(1\right)}\delta\mathbf{v}_{a}^{\left(1\right)} & \delta\mathbf{v}^{\left(2\right)} & =\sum\limits _{b=1}^{m^{\left(2\right)}}N_{b}^{\left(2\right)}\delta\mathbf{v}_{b}^{\left(2\right)}\\ \Delta\mathbf{u}^{\left(1\right)} & =\sum\limits _{c=1}^{m^{\left(1\right)}}N_{c}^{\left(1\right)}\Delta\mathbf{u}_{c}^{\left(1\right)} & \Delta\mathbf{u}^{\left(2\right)} & =\sum\limits _{d=1}^{m^{\left(2\right)}}N_{d}^{\left(2\right)}\Delta\mathbf{u}_{d}^{\left(2\right)}\\ \delta\tilde{p}^{\left(1\right)} & =\sum\limits _{a=1}^{m^{\left(1\right)}}N_{a}^{\left(1\right)}\delta\tilde{p}_{a}^{\left(1\right)} & \delta\tilde{p}^{\left(2\right)} & =\sum\limits _{b=1}^{m^{\left(2\right)}}N_{b}^{\left(2\right)}\delta\tilde{p}_{b}^{\left(2\right)}\\ \Delta\tilde{p}^{\left(1\right)} & =\sum\limits _{c=1}^{m^{\left(1\right)}}N_{c}^{\left(1\right)}\Delta\tilde{p}_{c}^{\left(1\right)} & \Delta\tilde{p}^{\left(2\right)} & =\sum\limits _{d=1}^{m^{\left(2\right)}}N_{d}^{\left(2\right)}\Delta\tilde{p}_{d}^{\left(2\right)}\\ \delta\tilde{c}^{\gamma\left(1\right)} & =\sum\limits _{a=1}^{m^{\left(1\right)}}N_{a}^{\left(1\right)}\delta\tilde{c}_{a}^{\gamma\left(1\right)} & \delta\tilde{c}^{\gamma\left(2\right)} & =\sum\limits _{b=1}^{m^{\left(2\right)}}N_{b}^{\left(2\right)}\delta\tilde{c}_{b}^{\gamma\left(2\right)}\\ \Delta\tilde{c}^{\gamma\left(1\right)} & =\sum\limits _{c=1}^{m^{\left(1\right)}}N_{c}^{\left(1\right)}\Delta\tilde{c}_{c}^{\gamma\left(1\right)} & \Delta\tilde{c}^{\gamma\left(2\right)} & =\sum\limits _{d=1}^{m^{\left(2\right)}}N_{d}^{\left(2\right)}\Delta\tilde{c}_{d}^{\gamma\left(2\right)} \end{aligned} \,.\label{eq669} \end{equation}$$ Then, $$\begin{equation} \begin{aligned}\delta G_{c} & =\sum\limits _{e=1}^{n_{e}^{\left(1\right)}}\sum\limits _{k=1}^{n_{\mbox{int}}^{\left(e\right)}}W_{k}J_{\eta}^{\left(1\right)}\left(\sum\limits _{a=1}^{m^{\left(1\right)}}\left[\begin{array}{cccc} \delta\mathbf{v}_{a}^{\left(1\right)} & \delta\tilde{p}_{a}^{\left(1\right)} & \delta\tilde{c}_{a}^{\alpha\left(1\right)} & \delta\tilde{c}_{a}^{\beta\left(1\right)}\end{array}\right]\cdot\left[\begin{array}{c} \mathbf{f}_{a}^{\left(1\right)}\\ w_{a}^{\left(1\right)}\\ j_{a}^{\alpha\left(1\right)}\\ j_{a}^{\beta\left(1\right)} \end{array}\right]\right.\\ & +\left.\sum\limits _{b=1}^{m^{\left(2\right)}}\left[\begin{array}{cccc} \delta\mathbf{v}_{b}^{\left(2\right)} & \delta\tilde{p}_{b}^{\left(2\right)} & \delta\tilde{c}_{b}^{\alpha\left(2\right)} & \delta\tilde{c}_{b}^{\beta\left(2\right)}\end{array}\right]\cdot\left[\begin{array}{c} \mathbf{f}_{b,k}^{\left(2\right)}\\ w_{b,k}^{\left(2\right)}\\ j_{b,k}^{\alpha\left(2\right)}\\ j_{b,k}^{\beta\left(2\right)} \end{array}\right]\right)\,, \end{aligned} \label{eq670} \end{equation}$$ where $$\begin{equation} \begin{aligned}\mathbf{f}_{a}^{\left(1\right)} & =t_{n}N_{a}^{\left(1\right)}\mathbf{n}^{\left(1\right)} & \mathbf{f}_{b,k}^{\left(2\right)} & =-t_{n}N_{b}^{\left(2\right)}\mathbf{n}^{\left(1\right)}\\ w_{a}^{\left(1\right)} & =w_{n}N_{a}^{\left(1\right)} & w_{b,k}^{\left(2\right)} & =-w_{n}N_{b}^{\left(2\right)}\\ j_{a}^{\gamma\left(1\right)} & =j_{n}^{\gamma}N_{a}^{\left(1\right)} & j_{b,k}^{\gamma\left(2\right)} & =-j_{n}^{\gamma}N_{b}^{\left(2\right)} \end{aligned} \,.\label{eq671} \end{equation}$$ 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\right.\\ & \left(\sum\limits _{c=1}^{m^{\left(1\right)}}\left[\begin{array}{cccc} \mathbf{K}_{ac}^{\left(1,1\right)} & 0 & 0 & 0\\ \mathbf{g}_{ac}^{\left(1,1\right)} & g_{ac}^{\left(1,1\right)} & 0 & 0\\ \mathbf{h}_{ac}^{\alpha\left(1,1\right)} & 0 & h_{ac}^{\alpha\alpha\left(1,1\right)} & 0\\ \mathbf{h}_{ac}^{\beta\left(1,1\right)} & 0 & 0 & h_{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.\\ & +\left.\sum\limits _{d=1}^{m_{k}^{\left(2\right)}}\left[\begin{array}{cccc} \mathbf{K}_{ad,k}^{\left(1,2\right)} & 0 & 0 & 0\\ \mathbf{g}_{ad,k}^{\left(1,2\right)} & g_{ad,k}^{\left(1,2\right)} & 0 & 0\\ \mathbf{h}_{ad,k}^{\alpha\left(1,2\right)} & 0 & h_{ad,k}^{\alpha\alpha\left(1,2\right)} & 0\\ \mathbf{h}_{ad,k}^{\beta\left(1,2\right)} & 0 & 0 & h_{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)} & 0 & 0 & 0\\ \mathbf{g}_{bc,k}^{\left(2,1\right)} & g_{bc,k}^{\left(2,1\right)} & 0 & 0\\ \mathbf{h}_{bc,k}^{\alpha\left(2,1\right)} & 0 & h_{bc,k}^{\alpha\alpha\left(2,1\right)} & 0\\ \mathbf{h}_{bc,k}^{\beta\left(2,1\right)} & 0 & 0 & h_{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)}\\ \Delta\tilde{c}_{c}^{\beta\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)} & 0 & 0 & 0\\ \mathbf{g}_{bd,k}^{\left(2,2\right)} & g_{bd,k}^{\left(2,2\right)} & 0 & 0\\ \mathbf{h}_{bd,k}^{\alpha\left(2,2\right)} & 0 & h_{bd,k}^{\alpha\alpha\left(2,2\right)} & 0\\ \mathbf{h}_{bd,k}^{\beta\left(2,2\right)} & 0 & 0 & h_{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{eq672} \end{equation}$$ where $$\begin{equation} \begin{aligned}\mathbf{K}_{ac}^{\left(1,1\right)} & =N_{a}^{\left(1\right)}\left(\varepsilon_{n}N_{c}^{\left(1\right)}\mathbf{N}^{\left(1\right)}+t_{n}\mathbf{A}_{c}^{\left(1\right)}\right)\\ \mathbf{K}_{ad,k}^{\left(1,2\right)} & =-\varepsilon_{n}N_{a}^{\left(1\right)}N_{d}^{\left(2\right)}\mathbf{N}^{\left(1\right)}\\ \mathbf{K}_{bc,k}^{\left(2,1\right)} & =-N_{c}^{\left(1\right)}\left(\varepsilon_{n}N_{b}^{\left(2\right)}\mathbf{N}^{\left(1\right)}+t_{n}\mathbf{M}_{b}^{\left(2\right)}\right)-t_{n}N_{b}^{\left(2\right)}\mathbf{A}_{c}^{\left(1\right)}\\ \mathbf{K}_{bd,k}^{\left(2,2\right)} & =N_{d}^{\left(2\right)}\left(\varepsilon_{n}N_{b}^{\left(2\right)}\mathbf{N}^{\left(1\right)}+t_{n}\mathbf{M}_{b}^{\left(2\right)}\right) \end{aligned} \,,\label{eq673} \end{equation}$$ $$\begin{equation} \begin{aligned}\mathbf{g}_{ac}^{\left(1,1\right)} & =N_{a}^{\left(1\right)}\left(\varepsilon_{p}N_{c}^{\left(1\right)}\mathbf{p}^{\left(1\right)}-w_{n}\mathbf{A}_{c}^{\left(1\right)}\cdot\mathbf{n}^{\left(1\right)}\right)\\ \mathbf{g}_{ad,k}^{\left(1,2\right)} & =-\varepsilon_{p}N_{a}^{\left(1\right)}N_{d}^{\left(2\right)}\mathbf{p}^{\left(1\right)}\\ \mathbf{g}_{bc,k}^{\left(2,1\right)} & =N_{c}^{\left(1\right)}\left(-\varepsilon_{p}N_{b}^{\left(2\right)}\mathbf{p}^{\left(1\right)}+w_{n}\mathbf{m}_{b}^{\left(2\right)}\right)+w_{n}N_{b}^{\left(2\right)}\mathbf{A}_{c}^{\left(1\right)}\cdot\mathbf{n}^{\left(1\right)}\\ \mathbf{g}_{bd,k}^{\left(2,2\right)} & =N_{d}^{\left(2\right)}\left(\varepsilon_{p}N_{b}^{\left(2\right)}\mathbf{p}^{\left(1\right)}-w_{n}\mathbf{m}_{b}^{\left(2\right)}\right) \end{aligned} \,,\label{eq674} \end{equation}$$ $$\begin{equation} \begin{aligned}g_{ac}^{\left(1,1\right)} & =-\varepsilon_{p}N_{a}^{\left(1\right)}N_{c}^{\left(1\right)} & g_{ad,k}^{\left(1,2\right)} & =\varepsilon_{p}N_{a}^{\left(1\right)}N_{d}^{\left(2\right)}\\ g_{bc,k}^{\left(2,1\right)} & =\varepsilon_{p}N_{b}^{\left(2\right)}N_{c}^{\left(1\right)} & g_{bd,k}^{\left(2,2\right)} & =-\varepsilon_{p}N_{b}^{\left(2\right)}N_{d}^{\left(2\right)} \end{aligned} \,,\label{eq675} \end{equation}$$ $$\begin{equation} \begin{aligned}\mathbf{h}_{ac}^{\gamma\left(1,1\right)} & =N_{a}^{\left(1\right)}\left(N_{c}^{\left(1\right)}\varepsilon_{c}\mathbf{q}^{\gamma\left(1\right)}-\mathbf{A}_{c}^{\left(1\right)}\cdot j_{n}^{\gamma}\mathbf{n}^{\left(1\right)}\right)\\ \mathbf{h}_{ad,k}^{\gamma\left(1,2\right)} & =-N_{a}^{\left(1\right)}N_{d}^{\left(2\right)}\varepsilon_{c}\mathbf{q}^{\gamma\left(1\right)}\\ \mathbf{h}_{bc,k}^{\gamma\left(2,1\right)} & =N_{b}^{\left(2\right)}\left(-N_{c}^{\left(1\right)}\varepsilon_{c}\mathbf{q}^{\gamma\left(1\right)}+\mathbf{A}_{c}^{\left(1\right)}\cdot j_{n}^{\gamma}\mathbf{n}^{\left(1\right)}\right)+N_{c}^{\left(1\right)}j_{n}^{\gamma}\mathbf{m}_{b}^{\left(2\right)}\\ \mathbf{h}_{bd,k}^{\gamma\left(2,2\right)} & =N_{b}^{\left(2\right)}N_{d}^{\left(2\right)}\varepsilon_{c}\mathbf{q}^{\gamma\left(1\right)}-N_{d}^{\left(2\right)}j_{n}^{\gamma}\mathbf{m}_{b}^{\left(2\right)} \end{aligned} \,,\label{eq676} \end{equation}$$ $$\begin{equation} \begin{aligned}h_{ac}^{\gamma\gamma\left(1,1\right)} & =-\varepsilon_{c}N_{a}^{\left(1\right)}N_{c}^{\left(1\right)}\\ h_{ad,k}^{\gamma\gamma\left(1,2\right)} & =\varepsilon_{c}N_{a}^{\left(1\right)}N_{d}^{\left(2\right)}\\ h_{bc,k}^{\gamma\gamma\left(2,1\right)} & =\varepsilon_{c}N_{b}^{\left(2\right)}N_{c}^{\left(1\right)}\\ h_{bd,k}^{\gamma\gamma\left(2,2\right)} & =-\varepsilon_{c}N_{b}^{\left(2\right)}N_{d}^{\left(2\right)} \end{aligned} \,,\label{eq677} \end{equation}$$ and $$\begin{equation} \begin{aligned}\mathbf{N}^{\left(1\right)} & =\mathbf{n}^{\left(1\right)}\otimes\mathbf{n}^{\left(1\right)} & \mathbf{A}_{c}^{\left(1\right)} & =\frac{1}{J_{\eta}^{\left(1\right)}}\boldsymbol{\mathcal{A}}\left\{ \frac{\partial N_{c}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{1}}\mathbf{g}_{2}^{\left(1\right)}-\frac{\partial N_{c}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{2}}\mathbf{g}_{1}^{\left(1\right)}\right\} \\ \mathbf{M}_{b}^{\left(2\right)} & =\mathbf{n}^{\left(2\right)}\otimes\mathbf{m}_{b}^{\left(2\right)} & \mathbf{m}_{b}^{\left(2\right)} & =\frac{\partial N_{b}^{\left(2\right)}}{\partial\eta_{\left(2\right)}^{\alpha}}\mathbf{g}_{\left(2\right)}^{\alpha}\\ \mathbf{p}^{\left(1\right)} & =\frac{\partial\tilde{p}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{\alpha}}\mathbf{g}_{\left(1\right)}^{\alpha} & \mathbf{q}^{\gamma\left(1\right)} & =\frac{\partial\tilde{c}^{\gamma\left(1\right)}}{\partial\eta_{\left(1\right)}^{\alpha}}\mathbf{g}_{\left(1\right)}^{\alpha} \end{aligned} \,.\label{eq678} \end{equation}$$