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

Let the normal component of the contact traction be described by the penalty function, $$\begin{equation} t_{n}=\begin{cases} \varepsilon_{n}g & g<0\\ 0 & g\ge0 \end{cases}\,,\label{eq611} \end{equation}$$ where $\varepsilon_{n}$ is a penalty factor associated with $t_{n}$ . Similarly, let $$\begin{equation} \begin{cases} w_{n}=\varepsilon_{p}\pi=\varepsilon_{p}\left(p^{\left(1\right)}-p^{\left(2\right)}\right) & t_{n}<0\\ p^{\left(i\right)}=0 & t_{n}=0 \end{cases}\,,\label{eq612} \end{equation}$$ where $\varepsilon_{p}$ is a penalty factor associated with $w_{n}$ . It follows that $$\begin{equation} \begin{aligned}Dt_{n} & =\varepsilon_{n}\left(\Delta\mathbf{u}^{\left(2\right)}-\Delta\mathbf{u}^{\left(1\right)}+\mathbf{g}_{\alpha}^{\left(2\right)}\Delta\eta_{\left(2\right)}^{\alpha}\right)\cdot\mathbf{n}^{\left(1\right)}\,,\\ Dw_{n} & =\varepsilon_{p}\left(\Delta p^{\left(1\right)}-\Delta p^{\left(2\right)}-\frac{\partial p^{\left(2\right)}}{\partial\eta_{\left(2\right)}^{\alpha}}\Delta\eta_{\left(2\right)}^{\alpha}\right)\,. \end{aligned} \label{eq613} \end{equation}$$ Given these relations, it can be shown that the directional derivatives of the various terms appearing in the integrand of $\delta G_{c}$ are $$\begin{equation} \begin{aligned} & D\left(t_{n}\left(\delta\mathbf{v}^{\left(1\right)}-\delta\mathbf{v}^{\left(2\right)}\right)\cdot\mathbf{g}_{1}^{\left(1\right)}\times\mathbf{g}_{2}^{\left(1\right)}\right)=\\ & -J_{\eta}^{\left(1\right)}\varepsilon_{n}\left(\delta\mathbf{v}^{\left(1\right)}-\delta\mathbf{v}^{\left(2\right)}\right)\cdot\left(\mathbf{n}^{\left(1\right)}\otimes\mathbf{n}^{\left(1\right)}\right)\cdot\left(\Delta\mathbf{u}^{\left(1\right)}-\Delta\mathbf{u}^{\left(2\right)}\right)\\ & +J_{\eta}^{\left(1\right)}t_{n}\frac{\partial\delta\mathbf{v}^{\left(2\right)}}{\partial\eta_{\left(2\right)}^{\alpha}}\cdot\left(\mathbf{n}^{\left(2\right)}\otimes\mathbf{g}_{\left(2\right)}^{\alpha}\right)\cdot\left(\Delta\mathbf{u}^{\left(1\right)}-\Delta\mathbf{u}^{\left(2\right)}\right)\\ & +t_{n}\left(\delta\mathbf{v}^{\left(1\right)}-\delta\mathbf{v}^{\left(2\right)}\right)\cdot\left(\frac{\partial\Delta\mathbf{u}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{1}}\times\mathbf{g}_{2}^{\left(1\right)}+\mathbf{g}_{1}^{\left(1\right)}\times\frac{\partial\Delta\mathbf{u}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{2}}\right) \end{aligned} \,,\label{eq614} \end{equation}$$ $$\begin{equation} \begin{aligned} & D\left(w_{n}\left(\delta p^{\left(1\right)}-\delta p^{\left(2\right)}\right)\left|\mathbf{g}_{1}^{\left(1\right)}\times\mathbf{g}_{2}^{\left(1\right)}\right|\right)=\\ & J_{\eta}^{\left(1\right)}\varepsilon_{p}\left(\delta p^{\left(1\right)}-\delta p^{\left(2\right)}\right)\left(\Delta p^{\left(1\right)}-\Delta p^{\left(2\right)}\right)\\ & -J_{\eta}^{\left(1\right)}\left[\varepsilon_{p}\left(\delta p^{\left(1\right)}-\delta p^{\left(2\right)}\right)\frac{\partial p^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{\alpha}}\mathbf{g}_{\left(1\right)}^{\alpha}+w_{n}\frac{\partial\delta p^{\left(2\right)}}{\partial\eta_{\left(2\right)}^{\alpha}}\mathbf{g}_{\left(2\right)}^{\alpha}\right]\cdot\left(\Delta\mathbf{u}^{\left(1\right)}-\Delta\mathbf{u}^{\left(2\right)}\right)\\ & +w_{n}\left(\delta p^{\left(1\right)}-\delta p^{\left(2\right)}\right)\mathbf{n}^{\left(1\right)}\cdot\left(\frac{\partial\Delta\mathbf{u}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{1}}\times\mathbf{g}_{2}^{\left(1\right)}+\mathbf{g}_{1}^{\left(1\right)}\times\frac{\partial\Delta\mathbf{u}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{2}}\right) \end{aligned} \,,\label{eq615} \end{equation}$$ where $J_{\eta}^{\left(1\right)}=\left|\mathbf{g}_{1}^{\left(1\right)}\times\mathbf{g}_{2}^{\left(1\right)}\right|$ .