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

At each material point $\mathbf{X}$ in the continuum, damage occurs when a scalar damage (or failure) measure $\Xi\left(\mathbf{F}\right)$ achieves a critical value $\Xi_{m}$ over the loading history, $$\begin{equation} \Xi_{m}\left(\mathbf{X}\right)=\max_{-\infty<s\le t}\Xi\left(\mathbf{F}\left(\mathbf{X},s\right)\right)\,.\label{eq:dmg-critical-measure} \end{equation}$$ The scalar damage measure $\Xi\left(\mathbf{F}\right)$ must be invariant to orthogonal transformations $\mathbf{Q}$ that preserve material symmetry, or else the damage formulation would not be observer-independent. For example, for isotropic materials, $\Xi\left(\mathbf{F}\right)$ must be an isotropic function of the deformation, in which case it should be expressed as $\Xi\left(\mathbf{U}\right)$ or $\Xi\left(\mathbf{E}\right)$ , where $\mathbf{U}$ is the right stretch tensor in the polar decomposition $\mathbf{F}=\mathbf{R}\cdot\mathbf{U}$ of the deformation gradient, and $$\begin{equation} \mathbf{E}=\frac{1}{2}\left(\mathbf{F}^{T}\cdot\mathbf{F}-\mathbf{I}\right)=\frac{1}{2}\left(\mathbf{U}^{2}-\mathbf{I}\right)\label{eq:Lagrange-strain-tensor} \end{equation}$$ is the Green-Lagrange strain tensor. It follows that $$\left(\frac{\partial\mathbf{E}}{\partial\mathbf{U}}\right)_{ijmn}=\frac{1}{2}\left(\frac{1}{2}\left(\delta_{im}U_{nj}+\delta_{in}U_{mj}\right)+\frac{1}{2}\left(U_{im}\delta_{jn}+U_{in}\delta_{jm}\right)\right)\,.$$ For anisotropic materials where $\mathbf{a}$ is the unit normal to a symmetry plane and $\mathbf{A}=\mathbf{a}\otimes\mathbf{a}$ , the damage measure $\Xi$ must satisfy $$\begin{equation} \Xi\left(\mathbf{U},\mathbf{A}\right)=\Xi\left(\mathbf{Q}\cdot\mathbf{U}\cdot\mathbf{Q}^{T},\mathbf{Q}\cdot\mathbf{A}\cdot\mathbf{Q}^{T}\right)\,,\label{eq:dmg-criterion-invariance} \end{equation}$$ for transformations $\mathbf{Q}$ that satisfy $\mathbf{Q}\cdot\mathbf{A}\cdot\mathbf{Q}^{T}=\mathbf{A}$ (or $\mathbf{Q}\cdot\mathbf{a}=\mathbf{a}$ ). We may replace $\mathbf{U}$ with $\mathbf{E}$ in the above expression.