The volume occupied by body $i$ in the reference configuration is denoted by $\Omega^{\left(i\right)}\subset\mathbb{R}^{3}$ where $i=1,2$ . The boundary of body $i$ is denoted by $\Gamma^{\left(i\right)}$ and is divided into three regions $\Gamma^{\left(i\right)}=\Gamma_{\sigma}^{\left(i\right)}\cup\Gamma_{u}^{\left(i\right)}\cup\Gamma_{c}^{\left(i\right)}$ , where $\Gamma_{\sigma}^{\left(i\right)}$ is the boundary where tractions are applied, $\Gamma_{u}^{\left(i\right)}$ the boundary where the solution is prescribed and $\Gamma_{c}^{\left(i\right)}$ the part of the boundary that will be in contact with the other body. It is assumed that $\Gamma_{\sigma}^{\left(i\right)}\cap\Gamma_{u}^{\left(i\right)}\cap\Gamma_{c}^{\left(i\right)}=\emptyset$ .

The deformation of body $i$ is defined by $\boldsymbol{\varphi}^{\left(i\right)}$ . The boundary of the deformed body $i$ , that is the boundary of $\boldsymbol{\varphi}^{\left(i\right)}\left(\Omega^{\left(i\right)}\right)$ is denoted by $\gamma^{\left(i\right)}=\gamma_{\sigma}^{\left(i\right)}\cup\gamma_{u}^{\left(i\right)}\cup\gamma_{c}^{\left(i\right)}$ where $\gamma_{\sigma}^{\left(i\right)}=\boldsymbol{\varphi}^{\left(i\right)}\left(\Gamma_{\sigma}^{\left(i\right)}\right)$ is the boundary in the current configuration where the tractions are applied and similar definitions for $\gamma_{u}^{\left(i\right)}$ and $\gamma_{c}^{\left(i\right)}$ . See the figure below for a graphical illustration of the defined regions.

Points in body 1 are denoted by $\mathbf{X}$ in the reference configuration and $\mathbf{x}$ in the current configuration. For body 2 these points are denoted by $\mathbf{Y}$ and $\mathbf{y}$ . To define contact, the location where the two bodies are in contact with each other must be established. If body 1 is the slave body and body 2 is the master body, then for a given point $\mathbf{X}$ on the slave reference contact surface there is a point $\mathbf{\bar{Y}}\left(\mathbf{X}\right)$ on the master contact surface that is in some sense closest to point $\mathbf{X}$ . This closest point is defined in a closest point projection sense: $$\begin{equation} \mathbf{\bar{Y}}\left(\mathbf{X}\right)=\arg\min\limits _{\mathbf{Y}\in\Gamma_{c}^{\left(2\right)}}\left\Vert \boldsymbol{\varphi}^{\left(1\right)}\left(\mathbf{X}\right)-\boldsymbol{\varphi}^{\left(2\right)}\left(\mathbf{Y}\right)\right\Vert \,.\label{eq564} \end{equation}$$ With the definition of $\mathbf{\bar{Y}}\left(\mathbf{X}\right)$ established the gap function can be defined, which is a measure for the distance between $\mathbf{X}$ and $\mathbf{\bar{Y}}\left(\mathbf{X}\right)$ , $$\begin{equation} g\left(\mathbf{X}\right)=-\boldsymbol{\nu}\cdot\left(\boldsymbol{\varphi}^{\left(1\right)}\left(\mathbf{X}\right)-\boldsymbol{\varphi}^{\left(2\right)}\left(\mathbf{\bar{Y}}\left(\mathbf{X}\right)\right)\right)\,,\label{eq565} \end{equation}$$ where $\boldsymbol{\nu}$ is the local surface normal of surface $\gamma_{c}^{\left(2\right)}$ evaluated at $\mathbf{\bar{y}}=\boldsymbol{\varphi}^{\left(2\right)}\left(\mathbf{\bar{Y}}\left(\mathbf{X}\right)\right)$ . Note that $g>0$ when $\mathbf{X}$ has penetrated body 2, so that the constraint condition to be satisfied at all time is $g\leqslant0$ .