## 2 The Numerical Tool Development
### Mathematical Models: Development of the Numerical Model's Logic
#### 2.2.1 Cure Kinetics  
Initial numerical and experimental investigations conducted on small-scale coupons and specimens highlighted the importance of correlating numerical simulations with an appropriate cure kinetics model. Various methodologies exist for characterizing the cure kinetics of resin materials, including embedded cure kinetics models, tabular definitions,  
Figure 8: Epoxy resin RMT6 typical cycle monitoring results. DC disposable sensors can measure different events occurring during resin curing. and user subroutines. In the context of curing analysis, the cure rate is determined at each time increment, with the assumption that the cure rate is a function of both the degree of cure and the temperature:  
\[\frac{d\alpha}{dt}=f(\alpha,T) \tag{3}\]  
The integration of the degree of cure is performed using the backward Euler method:  
\[\alpha_{n}^{i}=\Delta t\cdot f(\alpha_{n}^{i-1},T)+\alpha_{n-1} \tag{4}\]  
where \(i\) is the iteration number for the cure, \(n\) is the current increment number, \(n-1\) is the previous increment number, \(f\) is a function defined by the cure kinetics model, and \(\Delta t\) is the increment's time step size. In MSC MARC, the cure kinetics defines the cure rate as a function of the degree of cure and the temperature of resin materials. Different approaches are used to determine the cure kinetics of resin materials: embedded cure kinetics models, table definitions, and user subroutines. The four cure kinetics models implemented in Marc are as follows:  
* Cure kinetics model 1: by [PERSON], [PERSON], and [PERSON] (1982);
* Cure kinetics model 2: by [PERSON] (1991);
* Cure kinetics model 3: by [PERSON], [PERSON], and [PERSON] (1992);
* Cure kinetics model 4: by [PERSON] and [PERSON] (1996). The Johnston-Hubert model was selected for the present modeling and simulation strategy due to its specific advantages:  
* This model represents one of the latest advancements in the field;
* It incorporates a sophisticated array of parameters in its design;
* The accompanying documentation for this mathematical model is publicly available and readily accessible. Within this documentation, the authors have included empirical data derived from experiments conducted on materials analogous to those utilized in the ELADINE project;
* According to the findings presented in the article [30], there is a strong correlation between the experimental results and the data generated by the model, as illustrated in Figure 9.  
Among the four cure kinetics models available in MARC, we selected the Johnston and Hubert model for our analysis [30]. To evaluate the preferred model for our simulation,  
Figure 9: Comparison between (**a**) the numerically simulated degree of cure and (**b**) the Johnston–Hubert results. we compared the degree of cure plots with the results published by [PERSON] and [PERSON]. The resulting graphs exhibit a striking resemblance and are presented in Figure 9.  
The mathematical formulation proposed by [PERSON] and [PERSON] is expressed through the subsequent equations:  
\[\frac{d\alpha}{dt}=\frac{Ka^{m}(1-\alpha)^{n}}{1+e^{C\{a-(a_{C0}+a_{CT}T)\}}} \tag{5}\]  
\[K_{i}=A_{i}e^{-\Delta E_{i}/(RT)} \tag{6}\]  
The \(A,\Delta E,m,n,C,\alpha_{C0},a_{CT},H_{R}\) parameters are explained in Table 1.  
In Table 2 are given the parameters used for HEXCEL 8552 8552 epoxy resin cure kinetics model [22].