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Transverse isotropy can be introduced by adding a vector field representing the material preferred direction explicitly into the strain energy [81]. We require that the strain energy depends on a unit vector field $\mathbf{A}$ , which describes the local fiber direction in the undeformed configuration. When the material undergoes deformation, the vector $\mathbf{A}\left(\mathbf{X}\right)$ may be described by a unit vector field $\mathbf{a}\left(\boldsymbol{\varphi}\left(\mathbf{X}\right)\right)$ . In general, the fibers will also undergo length change. The fiber stretch, $\lambda$ , can be determined in terms of the deformation gradient and the fiber direction in the undeformed configuration, $$\begin{equation} \lambda\mathbf{a}=\mathbf{F}\cdot\mathbf{A}\,.\label{eq95} \end{equation}$$ Also, since $\mathbf{a}$ is a unit vector, $$\begin{equation} \lambda^{2}=\mathbf{A}\cdot\mathbf{C}\cdot\mathbf{A}\,.\label{eq96} \end{equation}$$ The strain energy function for a transversely isotropic material, $\Psi\left(\mathbf{C},\mathbf{A},\mathbf{X}\right)$ is an isotropic function of $\mathbf{C}$ and $\mathbf{A}\otimes\mathbf{A}$ . It can be shown [73] that the following set of invariants are sufficient to describe the material fully: $$\begin{equation} I_{1}=\tr\mathbf{C}\,,\quad I_{2}=\frac{1}{2}\left[\left(\tr\mathbf{C}\right)^{2}-\tr\mathbf{C}^{2}\right]\,,\quad I_{3}=\det C=J^{2}\,,\label{eq97} \end{equation}$$ $$\begin{equation} I_{4}=\mathbf{A}\cdot\mathbf{C}\cdot\mathbf{A}\,,\quad I_{5}=\mathbf{A}\cdot\mathbf{C}^{2}\cdot\mathbf{A}\,.\label{eq98} \end{equation}$$ The strain energy function can be written in terms of these invariants such that $$\begin{equation} \Psi\left(\mathbf{C},\mathbf{A},\mathbf{X}\right)=\Psi\left(I_{1}\left(\mathbf{C}\right),I_{2}\left(\mathbf{C}\right),I_{3}\left(\mathbf{C}\right),I_{4}\left(\mathbf{C},\mathbf{A}\right),I_{5}\left(\mathbf{C},\mathbf{A}\right)\right)\,.\label{eq99} \end{equation}$$ The second Piola-Kirchhoff can now be obtained in the standard manner: $$\begin{equation} \mathbf{S}=2\frac{\partial\Psi}{\partial\mathbf{C}}=2\sum\limits _{i=1}^{5}\frac{\partial\Psi}{\partial I_{i}}\frac{\partial I_{i}}{\partial\mathbf{C}}\,.\label{eq100} \end{equation}$$ In the transversely isotropic constitutive models described in Chapter 5↓ it is further assumed that the strain energy function can be split into the following terms: $$\begin{equation} \Psi\left(\mathbf{C},\mathbf{A}\right)=\Psi_{1}\left(I_{1},I_{2},I_{3}\right)+\Psi_{2}\left(I_{4}\right)+\Psi_{3}\left(I_{1},I_{2},I_{3}I_{4}\right)\,.\label{eq101} \end{equation}$$ The strain energy function $\Psi_{1}$ represents the material response of the isotropic ground substance matrix, $\Psi_{2}$ represents the contribution from the fiber family (e.g. collagen), and $\Psi_{3}$ is the contribution from interactions between the fibers and matrix. The form (2.4.4-7) generalizes many constitutive equations that have been successfully used in the past to describe biological soft tissues e.g. [38, 40, 41]. While this relation represents a large simplification when compared to the general case, it also embodies almost all of the material behavior that one would expect from transversely isotropic, large deformation matrix-fiber composites.