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The balance of linear momentum can be written for each of the two bodies in the reference configuration, $$\begin{equation} G^{\left(i\right)}\left(\boldsymbol{\varphi}^{\left(i\right)},\mathbf{w}^{\left(i\right)}\right)=\int\limits _{\Omega^{\left(i\right)}}\Grad\mathbf{w}^{\left(i\right)}:\mathbf{P}^{\left(i\right)}\,d\Omega-\int\limits _{\Omega^{\left(i\right)}}\mathbf{w}^{\left(i\right)}\cdot\mathbf{F}^{\left(i\right)}\,d\Omega-\int\limits _{\Gamma_{s}^{\left(i\right)}}\mathbf{w}^{\left(i\right)}\cdot\mathbf{T}^{\left(i\right)}\,d\Gamma-\int\limits _{\Gamma_{c}^{\left(i\right)}}\mathbf{w}^{\left(i\right)}\cdot\mathbf{T}^{\left(i\right)}\,d\Gamma=0\,,\label{eq566} \end{equation}$$ where $\mathbf{w}^{\left(i\right)}$ is a weighting function and $\mathbf{P}$ is the $1^{\text{st}}$ Piola-Kirchhoff stress tensor. The last term corresponds to the virtual work of the contact tractions on body $i$ . For notational convenience, the notations $\varphi$ and $w$ are introduced to denote the collection of the respective mappings $\boldsymbol{\varphi}^{\left(i\right)}$ and $\mathbf{w}^{\left(i\right)}$ (for $i=$ 1,2). In other words, $$\begin{equation} \begin{aligned}\varphi & :\bar{\Omega}^{\left(1\right)}\cup\bar{\Omega}^{\left(2\right)}\to\mathbb{R}^{3}\,,\\ w & :\bar{\Omega}^{\left(1\right)}\cup\bar{\Omega}^{\left(2\right)}\to\mathbb{R}^{3}\,. \end{aligned} \label{eq567} \end{equation}$$ The variational principle for the two body system is the sum of (7.1.2-1) for body 1 and 2 and can be expressed as, $$\begin{equation} \begin{aligned}G\left(\boldsymbol{\varphi},\mathbf{w}\right) & :=\sum\limits _{i=1}^{2}G^{\left(i\right)}\left(\boldsymbol{\varphi}^{\left(i\right)},\mathbf{w}^{\left(i\right)}\right)\\ & =\underbrace{\sum\limits _{i=1}^{2}\left\{ \int\limits _{\Omega^{\left(i\right)}}\Grad\mathbf{w}^{\left(i\right)}:\mathbf{P}^{\left(i\right)}\,d\Omega-\int\limits _{\Omega^{\left(i\right)}}\mathbf{w}^{\left(i\right)}\cdot\mathbf{F}^{\left(i\right)}\,d\Omega-\int\limits _{\Gamma_{s}^{\left(i\right)}}\mathbf{w}^{\left(i\right)}\cdot\mathbf{T}^{\left(i\right)}\,d\Gamma\right\} }_{G^{\text{int},\text{ext}}\left(\boldsymbol{\varphi},\mathbf{w}\right)}\\ & \underbrace{-\sum\limits _{i=1}^{2}\int\limits _{\Gamma_{c}^{\left(i\right)}}\mathbf{w}^{\left(i\right)}\cdot\mathbf{T}^{\left(i\right)}\,d\Gamma}_{G^{c}\left(\boldsymbol{\varphi},\mathbf{w}\right)}\,. \end{aligned} \label{eq568} \end{equation}$$ Or in short, $$\begin{equation} G\left(\boldsymbol{\varphi},\mathbf{w}\right)=G^{\text{int},\text{ext}}\left(\boldsymbol{\varphi},\mathbf{w}\right)+G^{c}\left(\boldsymbol{\varphi},\mathbf{w}\right)\,.\label{eq569} \end{equation}$$ Note that the minus sign is included in the definition of the contact integral $G^{c}$ . The contact integral can be written as an integration over the contact surface of body 1 by balancing linear momentum across the contact surface: $$\begin{equation} \mathbf{t}^{\left(2\right)}\left(\mathbf{\bar{y}}\left(\mathbf{x}\right)\right)d\Gamma^{\left(2\right)}=-\mathbf{t}^{\left(1\right)}\left(\mathbf{x}\right)\,d\Gamma^{\left(1\right)}\,.\label{eq570} \end{equation}$$ The contact integral can now be rewritten over the contact surface of body 1: $$\begin{equation} G^{c}=-\int\limits _{\Gamma_{c}^{\left(1\right)}}\mathbf{t}^{\left(1\right)}\left(\mathbf{x}\right)\cdot\left[\mathbf{w}^{\left(1\right)}\left(\mathbf{x}\right)-\mathbf{w}^{\left(2\right)}\left(\mathbf{\bar{y}}\left(\mathbf{x}\right)\right)\right]\,d\Gamma\,.\label{eq571} \end{equation}$$ In the case of frictionless contact, the contact traction is taken as perpendicular to surface 2 and therefore can be written as, $\mathbf{t}^{\left(1\right)}=t_{N}\boldsymbol{\nu}$ where $\boldsymbol{\nu}$ is the (outward) surface normal and $t_{N}$ is to be determined from the solution strategy. For example in a Lagrange multiplier method the $t_{N}$ 's would be the Lagrange multipliers.