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

A solid material may consist of a heterogeneous mixture of various solid constituents that are constrained to move together. If each constituent is denoted by the superscript $\sigma$ , a constrained mixture satisfies $\mathbf{v}^{\sigma}=\mathbf{v}^{s}$ for all $\sigma$ , where $\mathbf{v}^{s}$ is the velocity of the solid mixture. For example, a fiber-reinforced material may consist of a mixture of fibers and a ground matrix. In general, the constitutive relation for such a constrained mixture of solids may be a complex function of the mass fraction of each constituent as well as the ultrastructure of the constituents and their mutual interactions. The mass fraction of each constituent may be represented by the apparent density $\rho_{r}^{\sigma}$ , which is the ratio of the mass of $\sigma$ to the volume of the mixture in the reference configuration, in an elemental region. In the framework of hyperelasticity, the general representation for the strain energy density for such a solid mixture may have the form $$\begin{equation} \Psi=\Psi\left(\mathbf{F}^{\left(1\right)},\cdots,\mathbf{F}^{\left(n\right)},\rho_{r}^{\left(1\right)},\cdots,\rho_{r}^{\left(n\right)}\right)\,,\label{eq135} \end{equation}$$ where $\mathbf{F}^{\sigma}$ is the deformation gradient of constituent $\sigma$ and $n$ is the number of solid constituents in the mixture. Though the solid constituents are constrained to move together, their deformation gradients are not necessarily the same, depending on how the various solid constituents of a constrained mixture were assembled [6].