## 2 The Numerical Tool Development
### Mathematical Models: Development of the Numerical Model's Logic
#### 2.2.2 The Cure Shrinkage of the Resin  
The MARC FEM tool relies on two powerful mathematical models to deal with complex simulations of materials' behavior:  
1. [leftmargin=*]
2. Boghetti and Gillespie [31]. The dependence between the cure shrinkage and the degree of cure is represented as a second-order polynomial: \[V_{r}^{S}=0.0;\ \alpha<\alpha_{\text{C1}}\] (7) \[V_{r}^{S}=A\cdot\alpha_{s}+\left(V_{r}^{S_{\infty}}-A\right) \cdot\alpha_{S}^{2};\ \alpha_{\text{C1}}\leq\alpha<\alpha_{\text{C2}}\] (8) \[V_{r}^{S}=V_{r}^{S_{\infty}};\ \alpha\geq\alpha_{\text{C2}}\] (9) \[\alpha_{s}=\frac{\alpha-\alpha_{\text{C1}}}{\alpha_{\text{C2}}- \alpha_{\text{C1}}}\] (10)  
The \(V_{r}^{S_{\infty}},\alpha_{\text{C1}},\alpha_{\text{C2}}\), \(A\) parameters are explained in Table 3. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Variable** & **Description** & **Units** \\ \hline \(V_{r}^{S}\) & Resin volumetric cure shrinkage & - \\ \hline \(V_{r}^{S_{\infty}}\) & Total volumetric resin shrinkage from 0 to 1 & - \\ \hline \(\alpha_{s}\) & Degree of cure shrinkage & - \\ \hline \(\alpha_{\text{C1}}\) & Degree of cure after which the resin & - \\  & shrinkage begins & - \\ \hline \(\alpha_{\text{C2}}\) & Degree of cure after which the resin & - \\  & shrinkage stops & - \\ \hline A & \
\begin{tabular}{c} Polynomial cure shrinkage coefficient \\ ([PERSON] and [PERSON]) \\ \end{tabular} & - \\ \hline B &
\begin{tabular}{c} Exponential cure shrinkage coefficient ([PERSON] \\ and [PERSON]) \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters used in the resin cure shrinkage model. Figure 10: Graphical comparison of (**a**) equation linearization results in the case of the numerical simulation and (**b**) Johnston–Hubert results. [PERSON] and [PERSON] [32]. The curing shrinkage as a function of the degree of cure is represented as an exponential dependence:  
\[V_{r}^{S}=V_{r}^{S\infty}*10^{B(\alpha-a_{C})};\;\alpha\leq a_{C} \tag{11}\]  
\[V_{r}^{S}=V_{r}^{S\infty};\;\alpha>a_{C} \tag{12}\]  
The equations have similar parameters: \(V_{r}^{S};a_{C};B\). The cure shrinkage strain is calculated according to the volumetric shrinkage due to the curing process. The resin degree of cure shrinkage is defined as the ratio of the volumetric cure shrinkage (\(V_{r}^{s}\)) and the maximum volumetric cure shrinkage (\(V_{r}^{S\infty}\)) of the resin material as follows:  
\[\alpha_{s}=\frac{V_{r}^{S}}{V_{r}^{S\infty}} \tag{13}\]  
Considering [PERSON]'s mathematical formulation of cure shrinkage and the known values from Table 4, the resin cure behavior can be plotted over the entire cure cycle, as presented in Figure 11. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Units** & **Value** \\ \hline \(\alpha_{C1}\) & - & 0 \\ \hline \(\alpha_{C2}\) & - & 1 \\ \hline \(V_{r}^{S\infty}\) & - & 0.02 (from [PERSON]’s experimental results) \\ \hline \multirow{3}{*}{A} & \multirow{3}{*}{-} & 0.001 \\ \cline{3-3}  & & 0.02 \\ \cline{3-3}  & & 0.04 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Starting values for the Bogetti-Gillespie cure shrinkage model. Figure 11: Variations in resin volumetric shrinkage with cure based on the Bogetti–Gillespie equation. As defined by [31], the equations that describe the model are as follows:  
\[V_{r}^{S}=0.0\;when\;\alpha<\alpha_{\text{C1}}; \tag{14}\]  
and  
\[V_{r}^{S}=V_{r}^{\text{S}\infty}\;when\;\alpha<\alpha_{\text{C2}}; \tag{15}\]  
\[V_{r}^{S}=A\cdot\alpha_{S}+(V_{r}^{\text{S}\infty}-A)\;when\;\alpha_{\text{C1}} \leq\alpha\leq\alpha_{\text{C2}}; \tag{16}\]  
with  
\[\alpha_{S}=\frac{\alpha-\alpha_{\text{C1}}}{\alpha_{\text{C2}}-\alpha_{\text{ C1}}} \tag{17}\]  
In the simulation, the following values were considered: \(\alpha_{\text{C1}}=0\) and \(\alpha_{\text{C2}}=1\)--see Tables 3 and 4. In the Bogetti-Gillespie model implemented in MARC, the \"A\" parameter was defined to tune the resin shrinkage versus the degree of cure dependence over the experimental data.