# Virtual Sensor for Estimating the Strain-Hardening Rate of
###### Abstract  
This study introduces a Multiple Linear Regression (MLR) model that functions as a virtual sensor for estimating the strain-hardening rate of austenitic stainless steels, represented by the Hardening Rate of Hot rolled and annealed Stainless steel sheet (\(HRHS\)) parameter. The model correlates tensile strength (\(Rm\)) with cold thickness reduction and chemical composition, evidencing a robust linear relationship with an \(R\)-coefficient above 0.9800 for most samples. Key variables influencing the \(HRHS\) value include Cr, Mo, Si, Ni, and Nb, with the MLR model achieving a correlation coefficient of 0.9983. The Leave-One-Out Cross-Validation confirms the model's generalization for test examples, consistently yielding high \(R\)-values and low mean squared errors. Additionally, a simplified \(HRHS\) version is proposed for instances where complete chemical analyses are not feasible, offering a practical alternative with minimal error increase. The research demonstrates the potential of linear regression as a virtual sensor linking cold strain hardening to chemical composition, providing a cost-effective tool for assessing strain hardening behaviour across various austenitic grades. The \(HRHS\) parameter significantly aids in the understanding and optimization of steel behaviour during cold forming, offering valuable insights for the design of new steel grades and processing conditions. a 2024 14 5508 (MRC); 11379 (MRC); 1224 14 5508 (MRC); 13790 (MRC); 1405 (MRC); 1410 (MRC); 1411 (MRC); 1412 (MRC);  
tensile testing, \(M_{d30}\) is also widely applied in rolling deformation to estimate the austenite stability and, indirectly, the expected strain hardening [6]. A higher \(M_{d30}\) temperature signifies lower austenite stability, leading to an increase in cold strain hardening due to greater martensite formation by SIM. [PERSON] et al. [7] proposed one of the most widely used expressions to estimate \(M_{d30}\), shown in Equation (1):  
\[M_{d30}(^{\circ}\mathrm{C})=551-462\cdot(C+N)-9.2\cdot Si-8.1\cdot Mn-13.7\cdot Cr -29\cdot(Ni+Cu)-18.5\cdot Mo-68\cdot Nb-1.42\cdot(GS-8) \tag{1}\]  
where \(M_{d30}\) represents the temperature of the martensite formation induced by cold deformation and is expressed in \({}^{\circ}\mathrm{C}\). GS stands for grain size, measured according to the standard ASTM E91. Additionally, each alloying element is included in the equation with its chemical symbol and unit of weight percentage (%). Various alternative expressions have been proposed for estimating \(M_{d30}\), each placing different emphasis on the impact of chemical composition elements [8]. [PERSON] et al. take into account the combined effect of C and N, while [PERSON] considers the independent effects of C and N, suggesting that nitrogen is a more effective alloying element than carbon in restraining the transformation of austenite into martensite by SIM [9,10]. However, [PERSON] et al. 's recent work [11] demonstrates that the difference in the austenite-stabilizing effects of C and N grows as the temperature increases, owing to the difference in stacking fault energy between C- and N-added steels. They also found that the relation between \(M_{d30}\) and C and N concentration is not linear, as previously assumed in other studies. The various expressions proposed to determine \(M_{d30}\) aim to accurately predict this temperature, but it remains challenging to define experimentally. Despite the importance of understanding the transformation of austenite into martensite by SIM in metastable ASS for the forming industry, \(M_{d30}\) should not be considered an exact measure of austenitic stability or an indicator of the hardening behaviour, but rather an approximation. While the TRIP effect is the most relevant one in the hardening of ASS, other mechanisms also contribute to increasing their main mechanical properties [12], such as grain refinement by recrystallization [13], an increase in \(\delta\) ferrite content [14], and solid solution strengthening [13,14].