# Strengthening Mechanisms in a Medium-Carbon Steel Subjected to Thermo-Mechanical Processing
## 4 Discussion  
Different strengthening mechanisms could be operative in low-alloy steels [5; 8; 9; 16; 17]. It is well-known that all strengthening mechanisms give additive contributions, and overall YS can be expressed as  
\[\sigma_{\text{YS}}=\sigma_{0}^{a-Fe}+\sigma_{SS}+\sigma_{P}+\sigma_{HP}+\sigma _{d}, \tag{2}\]  
where \(\sigma_{0}^{a-Fe}\) is the Peierls-Nabarro or friction stress of high-purity ferrite, \(\sigma_{\text{SS}}\) is the solid solution strengthening by substitutional and interstitial elements in ferrite, \(\sigma_{P}\) is the dispersion strengthening attributed to M\({}_{23}\)C\({}_{6}\) carbides and (V,Nb)C carbides in the present work, \(\sigma_{d}\), is the dislocation strengthening, and \(\sigma_{HP}\) is the grain size strengthening. \(\sigma_{0}^{a-Fe}\) -45 MPa is usually adopted for calculation of the YS in low-alloy steels [5; 16]. [PERSON]'s empirical equation in the form of  
\[\sigma_{\text{SS}}=32\text{ Mn}-31\text{ Cr}+11\text{ Mo}+5544\text{ Css}, \tag{3}\]  
is widely used for the calculation of solid solution strengthening originating from substitutional and interstitial solutes [16; 17]. The concentrations in Equation (3) are expressed in wt.%. The ferrite matrix is significantly depleted by carbon, chromium, and molybdenum due to precipitation of numerous carbides during tempering. Data reported by [PERSON] et al. [9] showed that dissolution of carbides due to plastic deformation is insignificant and the contribution of solid solution strengthening to overall YS is not important for strains up to about 2.67. Therefore, experimental concentrations of solute elements in the ferritic matrix could be taken from the Thermo-Calc equilibrium calculation (Table 2). The concentrations of other elements are negligible and were discarded from calculation of the solid solution strengthening. The sum of the friction stress and the solid solution strengthening, \(\sigma_{0}^{a-Fe}+\sigma_{SS}\), was calculated to be 61 and 74 MPa for the present steel at 600 and 650 \({}^{\circ}\)C, respectively. These values support the low effect of solid solution strengthening on overall YS reported by [PERSON] et al. [9]. Dispersion strengthening is generally attributed to the Orowan mechanism and can be evaluated with the following equation [5; 18]:  
\[\sigma_{\text{P}}=0.2\text{ MGb}\lambda^{-1}(\text{ln}(\text{d}^{*}/\tau_{0})+0. 7), \tag{4}\]  
where \(\lambda\) is the interparticle spacing, M is the Taylor factor taken from Table 1, G = 81 GPa is the shear modulus, b = 0.248 nm is the Burgers vector, d\({}^{*}\) depends on the ratio between d\({}_{\text{P}}\) and \(\lambda\) and can be calculated as d\({}^{*}\) = (d\({}_{\text{P}}\)\({}^{-1}\) + \(\lambda^{-1}\))\({}^{-1}\), and r\({}_{0}\) is the dislocation core dimension of approximately 2b. The interaction between the (V,Nb)C carbides precipitated in the ferritic matrix and the gliding dislocations contributes to the overall dispersion \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Temperature**, \({}^{\circ}\)C} & \multicolumn{4}{c}{**Concentrations of Elements \(\times\) 10\({}^{5}\), Mass\%**} \\ \cline{2-5}  & **C** & **Mn** & **Cr** & **Mo** \\ \hline
600 & 358 & 0.9 & 13,000 & 2500 \\
650 & 620 & 2.9 & 20,000 & 5700 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Equilibrium chemical composition of ferrite matrix of medium-carbon steel at 600 and 650 \({}^{\circ}\)C calculated with Thermo-Calc software. strengthening due to Orowan bowing in accordance with Equation (4). The interparticle spacing is calculated as \(\lambda\) = 0.2d\({}_{\rm p}\) (\(\pi\)/F\({}_{\rm v}\))\({}^{0.5}\) for these dispersoids with an average size of d\({}_{\rm P}\) and the volume fraction, F\({}_{\rm V}\), taken from Table 1. The interparticle spacing for M\({}_{23}\)C\({}_{6}\) carbides is taken as the transverse dimensions of laths or lamellae.