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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 $J_{\eta}^{\left(1\right)}=\left|\mathbf{g}_{1}^{\left(1\right)}\times\mathbf{g}_{2}^{\left(1\right)}\right|$ .