# Virtual Sensor for Estimating the Strain-Hardening Rate of
## 4 Discussion  
The yield strengths, \(Rp_{0.2}\) and \(Rp_{1.0}\), and also the tensile strength, \(Rm\), are mechanical parameters normally used for the comparison of the hardening level of materials in the industrial design sector for applications that request cold-forming operations. In this work, the mechanical parameter \(Rm\) is chosen as an indicator of this strain-hardening rate, and the relationship between this property and the degree of thickness reduction by cold rolling is established by LR for each ASS grade under study. The values of the coefficient \(a\) ranging between 7.2 and 17.2 with good repeatability between the different analysed samples for the same steel grade (see Figure 2a). The high values of \(R\) in Figure 2b, greater than 0.9850, confirm the good linear relationship between \(Rm\) and cold thickness reduction for the analysed austenitic grades. Therefore, it can be concluded that the coefficient \(a\) is influenced by the steel grade, i.e., the chemical composition.  
This coefficient \(a\) is used as the dependent variable in an MLR model to predict the strain-hardening rate as a function of the chemical composition. The output variable of this model is the proposed \(HRHS\) parameter which is calculated by the model as a function of the main alloying elements, rather than being experimentally determined from the cold-rolling curves. Equation (6) derived from this MLR model, revealing the effect of the 11 alloying elements (C, Si, Mn, Cr, Ni, Cu, Mo, N, Nb, Ti and S). The model's goodness fit is indicated by the \(R\) value of 0.9983 and \(MSE\) of 0.0307, both of which confirm the good accuracy of the new \(HRHS\) parameter. This is supported by Figure 3a, which shows the close alignment between the coefficients \(a\) obtained from each LR model and the \(HRHS\) value derived from the MLR model.  
The analysis of the \(p\)-values associated with each alloying element of the MLR model identifies that Cr, Mo, Si, Ni and Nb are the most significant independent variables. This means that they carry the greatest weight in the estimation of the \(HRHS\) value. A \(p\)-value lower than 0.05 for these alloying elements is reported by Figure 3b, confirming their significance. Subsequently, a new MLR model is tested with only these significant independent variables, providing a new version of the \(HRHS\) parameter, referred to as \(HRHS_{simplified}\) defined by Equation (7). This approach allows for estimating the cold hardening rate with fewer chemical elements, without the need for a more complete chemical composition  
Figure 5: Comparison of \(HRHS\) estimation from the simplified MLR (taking into account the most significant variables) and the coefficient \(a\) obtained by the LR model for all analysed ASS sheets.  
analysis of a steel grade, and yields good results in fitting the new MLR model. Figure 5 shows a good agreement between the coefficients \(a\) obtained from each LR model and the \(HRHS_{simplified}\) estimations derived from the new MLR model; both parameters are closely aligned. Additionally, the \(R\) and \(MSE\) values (0.9925 and 0.1353, respectively) are rather close to the ones obtained in the MLR model of the \(HRHS\) parameter.  
Finally, the \(HRHS\) model is validated using the LOOCV strategy, which involves splitting the original sample into training and validation sets. The results shown in Figure 4a confirm the good fit between the coefficient \(a\) obtained for each sample from the LR model and the corresponding \(HRHS\) estimated with the LOOCV model using the same sample as the validation data. The values of \(R\) and \(MSE\) for each LOOCV model shown in Figure 4b also validate the MLR model, with the \(R\) coefficient being greater than 0.9980 and the \(MSE\) values lower than 0.0318 for all LOOCV models.