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

The second alternative differs more significantly from the method described above. It also begins with the definition of a single contact integral over the slave surface. $$\begin{equation} G_{c}=-\int\limits _{\gamma^{\left(1\right)}}\mathbf{t}_{c}\cdot\left(\delta\boldsymbol{\varphi}^{\left(1\right)}-\delta\bar{\boldsymbol{\varphi}}^{\left(2\right)}\right)\,da\,.\label{eq597} \end{equation}$$ But a different derivation is followed to obtain the linearization of this contact integral. The main reason for this difference is a subtly alternative definition for the gap function. In this method, it is defined as follows. $$\begin{equation} g\left(\mathbf{X}\right)=\boldsymbol{\nu}^{\left(1\right)}\cdot\left(\boldsymbol{\varphi}^{\left(1\right)}\left(\mathbf{X}\right)-\boldsymbol{\varphi}^{\left(2\right)}\left(\mathbf{\bar{Y}}\left(\mathbf{X}\right)\right)\right)\,,\label{eq598} \end{equation}$$ where, $\boldsymbol{\nu}^{\left(1\right)}$ is the normal of the slave surface (opposed to the master normal as used in the derivation above). In this case, the point $\mathbf{\bar{Y}}\left(\mathbf{X}\right)$ is no longer the closest point projection of $\mathbf{X}$ onto the master surface, but instead is the normal projection along $\boldsymbol{\nu}^{\left(1\right)}$ . The linearization of equation (7.1.8.2-2) now becomes, $$\begin{equation} \delta g=\boldsymbol{\nu}^{\left(1\right)}\cdot\left(\delta\boldsymbol{\varphi}^{\left(1\right)}\left(\mathbf{X}\right)-\delta\bar{\boldsymbol{\varphi}}^{\left(2\right)}\left(\mathbf{X}\right)-\boldsymbol{\tau}_{i}^{\left(2\right)}\delta\xi_{i}\right)\,,\label{eq599} \end{equation}$$ where, $\boldsymbol{\tau}_{i}^{\left(2\right)}=\frac{\partial\bar{\boldsymbol{\varphi}}^{\left(2\right)}}{\partial\eta_{i}}$ are the tangent vectors to the master surface at $\mathbf{\bar{Y}}\left(\mathbf{X}\right)$ . Note that since $\boldsymbol{\nu}^{\left(1\right)}$ is normal to the slave surface, equation (7.1.8.2-3) does not reduce to equation (7.1.2-7).