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

The axiom of mass balance in a reactive constrained mixture reduces to $$\begin{equation} \dot{\rho}_{r}^{\alpha}=\hat{\rho}_{r}^{\alpha}\,,\label{eq:dmg-mass-balance-alpha} \end{equation}$$ where $\dot{\rho}_{r}^{\alpha}$ is the material time derivative of $\rho_{r}^{\alpha}$ and $\hat{\rho}_{r}^{\alpha}$ is a function of state representing the referential mass supply density to constituent $\alpha$ due to reactions with all other constituents. In the damage framework, the above relations show that $$\begin{equation} \begin{aligned}\rho_{r}^{i} & =\rho_{r}\left(1-F\left(\Xi_{m}\right)\right)\\ \rho_{r}^{b} & =\rho_{r}F\left(\Xi_{m}\right) \end{aligned} \,.\label{eq:dmg-mass-densities} \end{equation}$$ Substituting these expressions into Eq.(5.6.4-1) shows that the referential mass density supplies are given by $$\begin{equation} \begin{aligned}\hat{\rho}_{r}^{i} & =-\rho_{r}\dot{F}\left(\Xi_{m}\right)\\ \hat{\rho}_{r}^{b} & =\rho_{r}\dot{F}\left(\Xi_{m}\right) \end{aligned} \,,\label{eq:dmg-mass-supplies} \end{equation}$$ where $$\begin{equation} \dot{F}\left(\Xi_{m}\right)=\begin{cases} \left.f\left(\Xi\right)\dot{\Xi}\right|_{\Xi_{m}} & \text{advancing damage}\\ 0 & \text{otherwise} \end{cases}\,.\label{eq:dmg-CDF-mtd} \end{equation}$$ In these expressions, the damage is advancing when $\Xi_{m}$ increases over two consecutive time points. In this expression for $\dot{F}$ we need to evaluate $$\begin{equation} \dot{\Xi}\left(\mathbf{U}\right)=\frac{\partial\Xi}{\partial\mathbf{U}}:\dot{\mathbf{U}}=\mathbf{N}:\dot{\mathbf{U}}\,,\label{eq:dmg-Xsi-dot} \end{equation}$$ where we defined $$\begin{equation} \mathbf{N}\equiv\frac{\partial\Xi}{\partial\mathbf{U}}\,\label{eq:dmg-surface-normal} \end{equation}$$ to represent the tensorial normal to the damage hypersurface, which needs to be evaluated at $\Xi_{m}$ .