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With the definition of the gap function at hand (equation (7.5.1-2)), we can define the contribution to the virtual work equation from the tied contact reaction forces. $$\begin{equation} W_{t}=\int\limits _{\Gamma_{c}}\mathbf{T}\cdot\delta\mathbf{g}\,d\Gamma\,.\label{eq681} \end{equation}$$ Here, $\mathbf{T}$ is the reaction force that enforces the constraint $\mathbf{g}\left(\mathbf{X}\right)=0$ . Since we anticipate the use of an augmented Lagrangian formalism, we can write this reaction force as follows, $$\begin{equation} \mathbf{T}=\boldsymbol{\lambda}+\varepsilon\mathbf{g}\,.\label{eq682} \end{equation}$$ The vector quantity $\boldsymbol{\lambda}$ is the Lagrangian multiplier and $\varepsilon$ is a penalty factor.