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Let the normal component of the contact traction be described by the penalty function, $$\begin{equation} t_{n}=\begin{cases} \varepsilon_{n}g & g<0\\ 0 & g\geqslant0 \end{cases}\,,\label{eq633} \end{equation}$$ where $\varepsilon_{n}$ is a penalty factor associated with $t_{n}$ . Similarly, let $$\begin{equation} \begin{cases} w_{n}=\varepsilon_{p}\pi=\varepsilon_{p}\left(\tilde{p}^{\left(1\right)}-\tilde{p}^{\left(2\right)}\right) & t_{n}<0\\ \tilde{p}^{\left(i\right)}=\tilde{p}^{\ast} & t_{n}=0 \end{cases}\,,\label{eq634} \end{equation}$$ and $$\begin{equation} \begin{cases} j_{n}=\varepsilon_{c}\chi=\varepsilon_{c}\left(\tilde{c}^{\left(1\right)}-\tilde{c}^{\left(2\right)}\right) & t_{n}<0\\ \tilde{c}^{\left(i\right)}=\tilde{c}^{\ast} & t_{n}=0 \end{cases}\,,\label{eq635} \end{equation}$$ where $\varepsilon_{p}$ and $\varepsilon_{c}$ are penalty factors associated with $w_{n}$ and $j_{n}$ , respectively. It follows that $$\begin{equation} \begin{aligned}Dt_{n} & =\varepsilon_{n}\left(\Delta\mathbf{u}^{\left(2\right)}-\Delta\mathbf{u}^{\left(1\right)}+\mathbf{g}_{\alpha}^{\left(2\right)}\Delta\eta_{\left(2\right)}^{\alpha}\right)\cdot\mathbf{n}^{\left(1\right)}\\ Dw_{n} & =\varepsilon_{p}\left(\Delta\tilde{p}^{\left(1\right)}-\Delta\tilde{p}^{\left(2\right)}-\frac{\partial\tilde{p}^{\left(2\right)}}{\partial\eta_{\left(2\right)}^{\alpha}}\Delta\eta_{\left(2\right)}^{\alpha}\right)\\ Dj_{n} & =\varepsilon_{c}\left(\Delta\tilde{c}^{\left(1\right)}-\Delta\tilde{c}^{\left(2\right)}-\frac{\partial\tilde{c}^{\left(2\right)}}{\partial\eta_{\left(2\right)}^{\alpha}}\Delta\eta_{\left(2\right)}^{\alpha}\right) \end{aligned} \,.\label{eq636} \end{equation}$$ Given these relations, it can be shown that the directional derivatives of the various terms appearing in the integrand of $\delta G_{c}$ are $$\begin{equation} \begin{aligned} & D\left(t_{n}\left(\delta\mathbf{v}^{\left(1\right)}-\delta\mathbf{v}^{\left(2\right)}\right)\cdot\mathbf{g}_{1}^{\left(1\right)}\times\mathbf{g}_{2}^{\left(1\right)}\right)=\\ & -J_{\eta}^{\left(1\right)}\varepsilon_{n}\left(\delta\mathbf{v}^{\left(1\right)}-\delta\mathbf{v}^{\left(2\right)}\right)\cdot\left(\mathbf{n}^{\left(1\right)}\otimes\mathbf{n}^{\left(1\right)}\right)\cdot\left(\Delta\mathbf{u}^{\left(1\right)}-\Delta\mathbf{u}^{\left(2\right)}\right)\\ & +J_{\eta}^{\left(1\right)}t_{n}\frac{\partial\delta\mathbf{v}^{\left(2\right)}}{\partial\eta_{\left(2\right)}^{\alpha}}\cdot\left(\mathbf{n}^{\left(2\right)}\otimes\mathbf{g}_{\left(2\right)}^{\alpha}\right)\cdot\left(\Delta\mathbf{u}^{\left(1\right)}-\Delta\mathbf{u}^{\left(2\right)}\right)\\ & +t_{n}\left(\delta\mathbf{v}^{\left(1\right)}-\delta\mathbf{v}^{\left(2\right)}\right)\cdot\left(\frac{\partial\Delta\mathbf{u}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{1}}\times\mathbf{g}_{2}^{\left(1\right)}+\mathbf{g}_{1}^{\left(1\right)}\times\frac{\partial\Delta\mathbf{u}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{2}}\right) \end{aligned} \,,\label{eq637} \end{equation}$$ $$\begin{equation} \begin{aligned} & D\left(w_{n}\left(\delta\tilde{p}^{\left(1\right)}-\delta\tilde{p}^{\left(2\right)}\right)\left|\mathbf{g}_{1}^{\left(1\right)}\times\mathbf{g}_{2}^{\left(1\right)}\right|\right)=\\ & J_{\eta}^{\left(1\right)}\varepsilon_{p}\left(\delta\tilde{p}^{\left(1\right)}-\delta\tilde{p}^{\left(2\right)}\right)\left(\Delta\tilde{p}^{\left(1\right)}-\Delta\tilde{p}^{\left(2\right)}\right)\\ & -J_{\eta}^{\left(1\right)}\left[\varepsilon_{p}\left(\delta\tilde{p}^{\left(1\right)}-\delta\tilde{p}^{\left(2\right)}\right)\frac{\partial\tilde{p}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{\alpha}}\mathbf{g}_{\left(1\right)}^{\alpha}+w_{n}\frac{\partial\delta\tilde{p}^{\left(2\right)}}{\partial\eta_{\left(2\right)}^{\alpha}}\mathbf{g}_{\left(2\right)}^{\alpha}\right]\cdot\left(\Delta\mathbf{u}^{\left(1\right)}-\Delta\mathbf{u}^{\left(2\right)}\right)\\ & +w_{n}\left(\delta\tilde{p}^{\left(1\right)}-\delta\tilde{p}^{\left(2\right)}\right)\mathbf{n}^{\left(1\right)}\cdot\left(\frac{\partial\Delta\mathbf{u}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{1}}\times\mathbf{g}_{2}^{\left(1\right)}+\mathbf{g}_{1}^{\left(1\right)}\times\frac{\partial\Delta\mathbf{u}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{2}}\right) \end{aligned} \,,\label{eq638} \end{equation}$$ $$\begin{equation} \begin{aligned} & D\left(j_{n}\left(\delta\tilde{c}^{\left(1\right)}-\delta\tilde{c}^{\left(2\right)}\right)\left|\mathbf{g}_{1}^{\left(1\right)}\times\mathbf{g}_{2}^{\left(1\right)}\right|\right)=\\ & J_{\eta}^{\left(1\right)}\varepsilon_{c}\left(\delta\tilde{c}^{\left(1\right)}-\delta\tilde{c}^{\left(2\right)}\right)\left(\Delta\tilde{c}^{\left(1\right)}-\Delta\tilde{c}^{\left(2\right)}\right)\\ & -J_{\eta}^{\left(1\right)}\left[\varepsilon_{c}\left(\delta\tilde{c}^{\left(1\right)}-\delta\tilde{c}^{\left(2\right)}\right)\frac{\partial\tilde{c}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{\alpha}}\mathbf{g}_{\left(1\right)}^{\alpha}+j_{n}\frac{\partial\delta\tilde{c}^{\left(2\right)}}{\partial\eta_{\left(2\right)}^{\alpha}}\mathbf{g}_{\left(2\right)}^{\alpha}\right]\cdot\left(\Delta\mathbf{u}^{\left(1\right)}-\Delta\mathbf{u}^{\left(2\right)}\right)\\ & +j_{n}\left(\delta\tilde{c}^{\left(1\right)}-\delta\tilde{c}^{\left(2\right)}\right)\mathbf{n}^{\left(1\right)}\cdot\left(\frac{\partial\Delta\mathbf{u}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{1}}\times\mathbf{g}_{2}^{\left(1\right)}+\mathbf{g}_{1}^{\left(1\right)}\times\frac{\partial\Delta\mathbf{u}^{\left(1\right)}}{\partial\eta_{\left(1\right)}^{2}}\right) \end{aligned} \,,\label{eq639} \end{equation}$$ where $J_{\eta}^{\left(1\right)}=\left|\mathbf{g}_{1}^{\left(1\right)}\times\mathbf{g}_{2}^{\left(1\right)}\right|$ .