\section{Load tests}
\label{sec:load_tests}

For all load tests, the unit cell described in section \ref{sec:unit_cell} was used, together with the material constants given in Table \ref{tab:material_props}. Specific parameters for each simulation are found in appendix \ref{sec:load_test_params}.

\subsection{Mesh size convergence}

\begin{figure}[hb!]
    \centering
    \includegraphics[width=.7\textwidth]{figs/convergence.png}
    \caption{Calculated properties as a function of inverse mesh size.}
    \label{fig:mesh_converge}
\end{figure}

In this section we want to see for our model how the precision of the mesh influence the precision of the results of the simulation. The mesh used for the load test simulations was gradually refined. The composite moduli computed from the load test results converged around a mesh size of 0.02 (arb. units) as shown in Figure \ref{fig:mesh_converge}.

As we can see, further that an certain point, the results doesn't change anymore. Only the simulation time increase, not the precision. For the rest of this assignment we will use the mesh size of the convergence point determine here. A relatively coarse mesh may have a minor influence on the results.


\subsection{Variation of the model depending on $ E_{2f} $}

The transverse modulus of the fiber, $E_{2f}$ is estimated and associated with significant uncertainty. The theoretical variation was be computed from equation \eqref{eq:theoretical_Et}. Additionally, $E_{2f}$ was varied in the finite element model, the results are displayed in Figure \ref{fig:pertubations}. As expected, the variation is more prominent at higher volume fractions of fiber, it is clear that the analytical model under-predicts the variation. Especially for a high volume fraction of fiber one can observe that the response of the composite transverse modulus to a change in the transverse fiber modulus is highly non-linear, and increases rapidly as the pertubation surpasses $\approx \pm 2\%$.

\begin{figure}[h]
    \centering
    \includegraphics[width=.8\textwidth]{figs/pertubations.png}
    \caption{Relative change in the transverse modulus as a function of relative change in the transverse modulus of the fiber. Solid lines show theoretical variation, dashed lines show results from FEA. $E_t^\circ$ indicates the composite transverse modulus at $E_{2f} = E_{2f}^\circ = \SI{15000}{\mega\pascal}$}
    \label{fig:pertubations}
\end{figure}

\subsection{Simulation results}
Now we can add on the graphics the results obtain with the Abaqus simulation, load testing simulated in Abaqus with a mesh size of $ \frac{1}{50}$ was compared to theoretical values computed as described in section. Our result with the FEA model are displayed in Figure \ref{fig:properties_comparison}.

\begin{figure}[h]
    \centering
    \includegraphics[width=.8\textwidth]{figs/theoretical_moduli.png}
    \caption{Comparison of composite moduli calculated by the theoretical models in section \ref{sec:theoretical_properties} and by FEA.}
    \label{fig:properties_comparison}
\end{figure}

As we can see, the simulation fit really well the longitudinal modulus, on the contrary of the transverse modulus where the error is large. Therefore, another model is required if we want to use theoretical calculus. The comparison shown in Figure \ref{fig:properties_comparison}, shows that the simple micro-mechanical model is not accurate to the real value of the transverse modulus. Another model is required.

\subsection{The Halpin-Tsai model}

Using these results, we can refine our theoretical model with the Halpin-Tsai semi-empirical method. Now we want to try to find the right $ \xi_{1} $ value with the results of our simulation. The fit resulting from a non-linear least squares regression computed using \code{scipy.optimize.curve\_fit()} is shown in Figure \ref{fig:properties_comparison}

\begin{figure}[h]
    \centering
    \includegraphics[width=.7\textwidth]{figs/halplin_tsai.png}
    \caption{Comparison of composite moduli calculated by the theoretical models in section \ref{sec:theoretical_properties} and by FEA.}
    \label{fig:properties_comparison}
\end{figure}

We have now a model that fit with a few error the simulation. This model could be use for different engineering application or for further calculus.