\section{Introduction}
In this report, a hexagonally packed carbon fiber composite with an epoxy matrix will be studied. The effect of the volume fraction the transverse modulus of the carbon fibers on the composite will be studied and discussed. Finally, some simple micro mechanical models will be compared to the finite element analysis (FEA) and the Halpin-Tsai model will be fit to the results from the FEA.

Finite element analysis is done in Abaqus. The materials were generated using the material properties displayed in Table \ref{tab:material_props}. The carbon fibers are assumed to be transversely isotropic, while the epoxy matrix is assumed to be isotropic. These assumptions imply that the following relations are valid.

\begin{align*}
    E_{2f} &= E_{3f} & E_{ijm} &= E_m\\
    \nu_{12f} &= \nu_{13f} & \nu_{ijm} &= \nu_{m}\\
    G_{12f} &= G_{13f} & G_{ijm} =\ &G_m = \frac{E_m}{2(1 + \nu_m)} \\
    G_{23f} &= \frac{E_{2f}}{2(1 + \nu_{23f})}
\end{align*}

\begin{table}[hb!]
    \centering
    \begin{tabular}{c c c c c c c}
    \toprule
         $E_{1f}$ [MPa]& $E_{2f}$ [MPa]& $\nu_{12f}$ [-]& $\nu_{23f}$ [-]& $G_{12f}$ [MPa] & $E_m$ [MPa]& $\nu_m$ [-]\\
    \midrule
         233 000 & 15 000 & 0.2 & 0.35 & 9000 & 4100 & 0.37\\
    \bottomrule
    \end{tabular}
    \caption{Material properties}
    \label{tab:material_props}
\end{table}


