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

The octahedral Lagrange strain is given by $$\begin{equation} e\left(\mathbf{U}\right)=\sqrt{\frac{2}{3}\dev\mathbf{E}:\dev\mathbf{E}}\,,\label{eq:octahedral-Lagrange-strain} \end{equation}$$ where $\dev\mathbf{E}$ is the deviatoric part of the Lagrange strain tensor, $$\begin{equation} \dev\mathbf{E}=\mathbf{E}-\frac{1}{3}\tr\left(\mathbf{E}\right)\mathbf{I}\,,\label{eq:deviatoric-Lagrange-strain} \end{equation}$$ and $\mathbf{E}$ is given in Eq.(5.6.3-2). For this damage measure we let $$\begin{equation} \Xi\left(\mathbf{U}\right)=e\left(\mathbf{U}\right)\,.\label{eq:dmg-model-OLS} \end{equation}$$ The damage surface normal can be evaluated from $$\begin{equation} \mathbf{N}=\frac{\partial e}{\partial\mathbf{E}}:\frac{\partial\mathbf{E}}{\partial\mathbf{U}}=\frac{1}{3e}\left(\dev\mathbf{E}\cdot\mathbf{U}+\mathbf{U}\cdot\dev\mathbf{E}\right)\,,\label{eq:dmg-OLS-N} \end{equation}$$ where we used $$\begin{equation} \frac{\partial e}{\partial\mathbf{E}}=\frac{2}{3e}\dev\mathbf{E}\label{eq:dmg-e-tangent} \end{equation}$$ and $$\begin{equation} \frac{\partial\mathbf{E}}{\partial\mathbf{U}}=\frac{1}{2}\frac{\partial\mathbf{U}^{2}}{\partial\mathbf{U}}=\frac{1}{2}\left(\mathbf{I}\,\underline{\overline{\otimes}}\,\mathbf{U}+\mathbf{U}\,\underline{\overline{\otimes}}\,\mathbf{I}\right)\,.\label{eq:dmg-E-U-tangent} \end{equation}$$