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Let the specific free energy stored in intact bonds be represented by $\psi\left(\mathbf{F}\right)$ ; that of broken bonds is zero. Therefore, the free energy density of the mixture is $$\begin{equation} \Psi_{r}\left(\mathbf{F}\right)=\rho_{r}^{i}\psi\left(\mathbf{F}\right)\,.\label{eq:dmg-FED} \end{equation}$$ We may define the mass fraction $w^{\alpha}$ of bond species $\alpha$ as $$\begin{equation} w^{\alpha}=\frac{\rho_{r}^{\alpha}}{\rho_{r}}\,.\label{eq:dmg-mass-fraction} \end{equation}$$ Now, the mixture mass balance in Eq.(5.6.1-2) may be rewritten as $\sum_{\alpha}w^{\alpha}=1$ , or more specifically, $$\begin{equation} w^{i}+w^{b}=1\,.\label{eq:dmg-massfraction-balance} \end{equation}$$ We may also rewrite the mixture free energy density in Eq.(5.6.2-1) as $$\begin{equation} \Psi_{r}\left(\mathbf{F}\right)=w^{i}\rho_{r}\psi\left(\mathbf{F}\right)=\left(1-w^{b}\right)\rho_{r}\psi\left(\mathbf{F}\right)\,,\label{eq:dmg-FED-redux} \end{equation}$$ where we have made use of Eq.(5.6.2-3). The corresponding Cauchy stress may be evaluated using the standard hyperelasticity formula, $$\begin{equation} \boldsymbol{\sigma}=J^{-1}\frac{\partial\Psi_{r}}{\partial\mathbf{F}}\cdot\mathbf{F}^{T}=\left(1-w^{b}\right)\frac{\rho_{r}}{J}\frac{\partial\psi}{\partial\mathbf{F}}\cdot\mathbf{F}^{T}.\label{eq:dmg-stress} \end{equation}$$ These relation show that the free energy density and stress of a damaged material are scaled by the mass fraction $w^{i}=1-w^{b}$ of remaining intact bonds. Comparing these formulas to those of classical damage mechanics [43, 64, 26, 51, 50, 71], it becomes immediately apparent that the classical damage variable $D$ appearing in those theories is equivalent to the mass fraction $w^{b}$ of broken bonds, $$\begin{equation} D\equiv w^{b}.\label{eq:dmg-variable} \end{equation}$$ To further clarify this equivalence, we may let $\Psi_{0}\equiv\rho_{r}\psi$ represent the free energy density of an intact elastic solid, such that Eq.(5.6.2-4) may be rewritten as $\Psi_{r}=\left(1-D\right)\Psi_{0}$ . Similarly, Eq.(5.6.2-5) may be rewritten as $\boldsymbol{\sigma}=\left(1-D\right)\boldsymbol{\sigma}_{0}$ , where $\boldsymbol{\sigma}_{0}$ is the stress in the intact elastic solid, derived from the hyperelasticity relation $\boldsymbol{\sigma}_{0}=J^{-1}\left(\partial\Psi_{0}/\partial\mathbf{F}\right)\cdot\mathbf{F}^{T}$ .