# Vegetal-FRCM Failure under Partial Interaction Mechanism
## 5 Analytical Model  
The main failure observed was the slippage of the mesh, which led to two types of sliding. In the case of the synthetic resin, the mesh (as a unique element) slipped, cutting the mortar in two parts, while the natural coatings, with a lower interphase bond, slipped inside the mortar matrix, as shown in Figure 8.  
One of the ACK model hypotheses consists of the assumption that the matrix-fiber bond is weak [4]. However, the failure mode presented by the model is the rupture of the mesh. In this research, where the main mode consisted of the shear failure of the contact plane or the slippage of the meshes inside the mortar matrix, it indicates that the mesh is not fully in traction.  
In order to model this failure, a two-stage model approach was decided. The first stage consisted of the law of mixtures, commonly used in analytical composite models [19]. It combines [PERSON]'s modulus of each material and its volumetric fraction.  
\[\mathrm{E_{I,model}}=\mathrm{E_{m}\cdot V_{m}}+\mathrm{E_{mesh}\cdot V_{mesh}} \tag{1}\]  
where E stands for Young's modulus of the mortar and the mesh, and V stands for the volumetric fraction.  
The law of mixtures is followed until the mortar's first crack, which can be calculated as follows:  
\[\sigma_{\mathrm{mc,model}}=(\mathrm{E_{I,model}}\cdots\sigma_{\mathrm{mu}})/ \mathrm{E_{m}} \tag{2}\]  
where \(\sigma_{\mathrm{mu}}\) is the tensile strength of the mortar. The final deformation of the first stage is calculated as follows:  
\[\epsilon_{\mathrm{I,model}}=\sigma_{\mathrm{mc,model}}/\mathrm{E_{I,model}} \tag{3}\]  
The slope of the second stage consists of Hooke's law. It is combined with an effective coefficient (\(\alpha\)), defined by the type of fiber used on each specimen, that was tuned to fit the experimental curve.  
\[\mathrm{E_{II,model}}=\mathrm{E_{mesh}\cdot V_{mesh}\cdot\alpha} \tag{4}\]  
In order to calculate the model, \(\mathrm{E_{m}}=8.92\) GPa [2] and \(\sigma_{\mathrm{mu}}=2.26\) MPa were considered.  
The results of the proposed model can be found in Table 11. The experimental and modeled slopes and the efficient coefficient were used. PESHE specimens were not modeled as their failure mode was not the one proposed by the model.  
The curves comparing the FRCM experimental behaviour and the modelled behaviour can be found in Figures 11 and 12.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & & \multicolumn{3}{c}{**Two-Stage Model**} & \multicolumn{3}{c}{**Experimental**} \\ \hline
**Fiber** & **Coating** & \begin{tabular}{c} \(\sigma_{\mathrm{mc,model}}\) \\ (MPa) \\ \end{tabular} & \begin{tabular}{c} \(\mathrm{E_{I,model}}\) \\ (MPa) \\ \end{tabular} & \begin{tabular}{c} \(\mathrm{E_{II,model}}\) \\ (MPa) \\ \end{tabular} & \begin{tabular}{c} \(\alpha\) \\ (-) \\ \end{tabular} & \
\begin{tabular}{c} \(\mathrm{E_{I}}\) (MPa) \\ \end{tabular} &
\begin{tabular}{c} \(\mathrm{E_{II}\) (MPa) \\ \end{tabular} \\ \hline \multirow{4}{*}{CO} & EP & 2.18 & 8 613 & 14.29 & 0.20 & 111 & 9.93 \\  & PES & 2.18 & 8 605 & 12.62 & 0.20 & 76 & 9.80 \\  & GA & 2.18 & 8 617 & 15.10 & 0.20 & 84 & 7.23 \\  & COLTUR & 2.18 & 8 606 & 16.00 & 0.20 & 85 & 9.54 \\ \hline \multirow{4}{*}{HE} & EP & 2.36 & 9 321 & 15.39 & 0.03 & 173 & 71.93 \\  & PES & - & - & - & - & - & - \\ \cline{1-1}  & GA & 2.33 & 9 190 & 11.47 & 0.03 & 72 & 3.28 \\ \cline{1-1}  & COLAC & 2.36 & 9 328 & 15.61 & 0.03 & 112 & 0.43 \\ \hline \hline \end{tabular}
\end{table}
Table 11: Experimental and two-stage model comparison.  
As in other studies, the law of mixtures on composite materials increased the stiffness of the first stage [4]. For the second stage, the cotton specimens modeled corresponded with the experimental curves. In the case of hemp, the models do not fit the experimental curves accurately; it can be due to the low perimeter of the hemp mesh with the matrix, which does not maintain the friction between the mesh and the matrix.