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

Since equation (7.5.2-1) is nonlinear we need to calculate the linearization. For tied contact, this is simply given by the following equation. $$\begin{equation} \Delta W_{t}=\int\limits _{\Gamma_{c}}\varepsilon\Delta\mathbf{g}\cdot\delta\mathbf{g}\,d\Gamma\,.\label{eq683} \end{equation}$$ Where $$\begin{equation} \delta\mathbf{g}=\mathbf{w}^{\left(1\right)}-\mathbf{w}^{\left(2\right)}\label{eq684} \end{equation}$$ and $$\begin{equation} \Delta\mathbf{g}=\Delta\boldsymbol{\varphi}^{\left(1\right)}\left(\mathbf{X}\right)-\Delta\boldsymbol{\varphi}^{\left(2\right)}\left(\mathbf{\bar{Y}}\left(\mathbf{X}\right)\right)\,.\label{eq685} \end{equation}$$ We also introduced the notation $\mathbf{w}^{\left(i\right)}=\delta\boldsymbol{\varphi}^{\left(i\right)}$ .