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

Chemical reactions may be incorportated into a multiphasic mixture by adding a mass supply term to the equation of mass balance, $$\begin{equation} \frac{\partial\rho^{\alpha}}{\partial t}+\divg\left(\rho^{\alpha}\mathbf{v}^{\alpha}\right)=\hat{\rho}^{\alpha}\,,\label{eq150} \end{equation}$$ Where $\hat{\rho}^{\alpha}$ is the volume density of mass supply to $\alpha$ resulting from chemical reactions with all other mixture constitutents. Since mass must be conserved over all constituents, mass supply terms are constrained by $$\begin{equation} \sum\limits _{\alpha}\hat{\rho}^{\alpha}=0\,.\label{eq151} \end{equation}$$ In a mixture containing a solid constituent (denoted by $\alpha=s$ ), it is conveniemt to define the mixture domain (and thus the finite element mesh) on the solid and evaluate mass fluxes of constituents relative to the solid, $$\begin{equation} \mathbf{m}^{\alpha}=\rho^{\alpha}\left(\mathbf{v}^{\alpha}-\mathbf{v}^{s}\right)\,.\label{eq152} \end{equation}$$ Substituting (2.10-3) into (2.10-1), the differential form of the mass balance may be rewritten as $$\begin{equation} \frac{D^{s}\rho_{r}^{\alpha}}{Dt}+J\divg\mathbf{m}^{\alpha}=\hat{\rho}_{r}^{\alpha}\,,\label{eq153} \end{equation}$$ Where $D^{s}\left(\cdot\right)/Dt$ represents the material time derivative in the spatial frame, following the solid, $J=\det\mathbf{F}$ , where $\mathbf{F}$ is the deformation gradient of the solid matrix; $\rho_{r}^{\alpha}$ is the apparent density and $\hat{\rho}_{r}^{\alpha}$ is the volume density of mass supply to $\alpha$ normalized to the mixture volume in the reference configuration, $$\begin{equation} \rho_{r}^{\alpha}=J\rho^{\alpha},\quad\hat{\rho}_{r}^{\alpha}=J\hat{\rho}^{\alpha}.\label{eq154} \end{equation}$$ Since $\rho_{r}^{\alpha}$ is the mass of $\alpha$ in the current configuration per volume of the mixture in the reference configuration (an invariant quantity), this parameter represents a direct measure of the mass content of $\alpha$ in the mixture, which may thus be used as a state variable in a framework that accounts for chemical reactions. A distinction is now made between solid and solute species in the mixture, since they are often treated differential in an analysis.