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In a Newton-Raphson implementation the contact integral must be linearized with respect to the current configuration: $$\begin{equation} \Delta G^{c}\left(\boldsymbol{\varphi},\mathbf{w}\right)=\int\limits _{\Gamma_{c}^{\left(1\right)}}\Delta\left(t_{N}\delta g\right)\,d\Gamma\,.\label{eq574} \end{equation}$$ Examining the normal contact term first, the directional derivative of $t_{N}$ is given (for the case of the penalty regularization) by: $$\begin{equation} \begin{aligned}\Delta t_{N} & =\Delta\left\{ \varepsilon_{N}\left\langle g\right\rangle \right\} \\ & =H\left(g\right)\varepsilon_{N}\Delta g \end{aligned} \,,\label{eq575} \end{equation}$$ where $\varepsilon_{N}$ is the penalty factor and $H\left(g\right)$ is the Heaviside function. The quantity $\Delta\left(\delta g\right)$ is given by $$\begin{equation} \begin{aligned}\Delta\left(\delta g\right) & =g\left[\boldsymbol{\nu}\cdot\delta\varphi_{,\gamma}^{\left(2\right)}\left(\mathbf{\bar{Y}}\left(\mathbf{X}\right)\right)+\kappa_{\alpha\gamma}\left(\mathbf{\bar{Y}}\left(\mathbf{X}\right)\right)\delta\bar{\xi}_{\alpha}\right]m^{\gamma\beta}\\ & \left[\boldsymbol{\nu}\cdot\Delta\varphi_{,\beta}^{\left(2\right)}\left(\mathbf{\bar{Y}}\left(\mathbf{X}\right)\right)+\kappa_{\alpha\beta}\left(\mathbf{\bar{Y}}\left(\mathbf{X}\right)\right)\Delta\bar{\xi}^{\alpha}\right]\\ & +\delta\bar{\xi}^{\beta}\boldsymbol{\nu}\cdot\left[\Delta\varphi_{,\beta}^{\left(2\right)}\left(\mathbf{\bar{Y}}\left(\mathbf{X}\right)\right)\right]+\Delta\bar{\xi}^{\beta}\boldsymbol{\nu}\cdot\left[\delta\boldsymbol{\varphi}_{,\beta}^{\left(2\right)}\left(\mathbf{\bar{Y}}\left(\mathbf{X}\right)\right)\right]\\ & +\kappa_{\alpha\beta}\left(\mathbf{\bar{Y}}\left(\mathbf{X}\right)\right)\delta\bar{\xi}^{\beta}\Delta\bar{\xi}^{\alpha}\,. \end{aligned} \label{eq576} \end{equation}$$