\section{Theory}
\input{unit_cell}
\subsection{Simple micro-mechanical models}
\label{sec:theoretical_properties}
When a load is applied to a UDFC in the fiber direction, hereby denoted the 1 direction, the fiber strain will be equal to the matrix strain which will equal the composite strain. Additionally, stress on the fibers can be assumed to be proportional to the area fraction of the fibers, which is equal to the volume fraction of the fibers. From this, an expression relating the material moduli and volume fractions to the composite modulus can be derived.
\begin{equation}
    \begin{split}
        \sigma_1 &= V_f\sigma_{f,1} + V_m \sigma_{m,1}\\
        E_1 \epsilon_1 &= V_f E_{1,f} \epsilon_{1,f} + V_m E_{m,1} \epsilon_{m,1}\\
        E_1 &= V_f E_{1,f} + V_m E_{1,m}
    \end{split}
    \label{eq:theoretical_E1}
\end{equation}

Conversely, for a load in the transverse direction, it may be assumed that the strain is equal to the volume weighted average of the fiber strain and matrix strain, with the stress experienced by the fiber and matrix being equal. By these assumptions, an expression relating the material properties and volume fractions to the composite properties in the transverse directions can be derived. Denoting $\phi_2 = \phi_3 = \phi_t$ for properties $\phi$ in the transverse direction of the composite
\begin{equation}
    \begin{split}
        \epsilon_t = V_f \epsilon_f + V_m \epsilon_m\\
        \frac{\sigma_t}{E_t} = V_f \frac{\sigma_{f,t}}{E_{f,t}} + V_m \frac{\sigma_{m,t}}{E_{m,t}}\\
        \frac{1}{E_t} = \frac{V_f}{E_{f,t}} + \frac{V_m}{E_{m,t}}
    \end{split}
    \label{eq:theoretical_Et}
\end{equation}

From the expression in equations \eqref{eq:theoretical_E1} and \eqref{eq:theoretical_Et} and the corresponding assumptions, one can derive an expression for the Poisson ratio of the composite,
\begin{equation}
    \nu_{12} = V_{f}\nu_{12f}+V_{m}\nu_{m}.
\end{equation}

\subsection{The Halpin-Tsai model}
\label{section:Halpin-Tsai}
The Halpin-Tsai model is a semi-empirical model that is used to approximate the transverse modulus and shear modulus of a transversely orthotropic material. The model is derived by the self-consistent field approach and contains a fitting parameter $\xi$. The transverse modulus is approximated as 
\begin{equation}
    E_2 = E_m \frac{1 + \xi_1 \eta_1 V_f}{1 - \eta_1 V_f},\hspace{1cm} \eta_1 = \frac{E_{2,f} - E_m}{E_{2,f} + \xi_1 E_m}.
\end{equation}
The shear modulus $G_{12}$ is approximated as 
\begin{equation}
    G_{12} = G_m\frac{1 + \xi_2 \eta_2 V_f}{1 - \eta_2 V_f}, \hspace{1cm} \eta_2 = \frac{G_{12,f} - G_m}{G_{12,f} + \xi_2 G_m}
\end{equation}
The parameters $\xi_1$ and $\xi_2$ can be fitted to experimental data or data from a finite element analysis by non-linear regression. 

The problem with this model is that it's a semi-empirical mode. Simulation or experimentation are therefore required to determine the $\xi$-parameter. Additionally, the value of the parameter has little physical interpretation outside of parametrising the increased strength of the composite resulting from the interaction between the matrix and fibers.
