## 3 Modeling
### Work Hardening  
Several authors have attempted to describe work hardening on steels using a hardening law based on physical principles that makes use of dislocation theory [16; 17; 51; 52]. In this work, the work hardening law presented in [17], based on a model by [PERSON] [16], is extended to take into account additional microstructural factors in order to better adapt to the description of the behavior of the baintic steels under study. Grain (and lath) size is considered for the calculation of the initial CRSS, the effect of carbides in the hardening behavior is included and latent hardening phenomena are taken into account. Moreover, a (partially) implicit integration scheme is introduced in order to mitigate the effect of the time step size on the results of the model. The initial CRSS of the slip system \(s\) at reference strain rate, \(\tau_{c}^{s}(\dot{\epsilon}_{0},0)\), is calculated as the addition of two terms: a static yield stress \(\tau_{0}\), which is the same for all the slip systems of every grain of each phase, and a second term \(\tau_{d}^{s}(\rho_{s})\), which depends on dislocation density and, therefore, on the deformation history of the slip system \(s\). The \(\tau_{0}\) term is calculated on the basis of a phenomenological expression for the estimation of macroscopic yield stress from chemical composition and other physical quantities [53] (a similar method is also applied in [51]). It is assumed that yield stress, \(\sigma_{0}\), results from the sum of the hardening effect of the crystal lattice, \(\sigma_{l}\) (which for pure iron has been measured as 77 MPa [54]), and of the elements in solid solution, \(\sigma_{ss}\). A phenomenological expression for \(\sigma_{ss}\) was given in [55]. Moreover, the boundary strengthening is considered through an additional term given by the Hall-Petch relationship [56; 57] or Hall-Petch-like relationships for phases with a lath structure [18]. The final expression for \(\sigma_{0}\) is:  
\[\sigma_{0}=\sigma_{l}+\sigma_{ss}+\sigma_{d} \tag{15}\]  
with  
\[\sigma_{l} =77\,\text{MPa} \tag{16a}\] \[\sigma_{\text{ss}} =37W_{\text{Mn}}+678W_{\text{P}}+60W_{\text{Si}}+80W_{\text{Cu}} +45W_{\text{Ni}}+60W_{\text{Cr}}+11W_{\text{Mo}}+\sigma_{\text{C}}\] (16b) \[\sigma_{\text{C}} =5000(W_{\text{C}}+W_{\text{N}})\text{ for ferrite/baintic, }3065(W_{\text{C}}+W_{\text{N}})-161\text{ for martensite}\] (16c) \[\sigma_{d} = \begin{cases}K_{\text{HP}}/\sqrt{d}&\text{with }d\equiv\text{ Grain diameter}\\ K_{\text{HP}}M\mu b/d&\text{with }d\equiv\text{Lath width}\end{cases}  \tag{16d}\]  
where \(W_{\text{X}}\) is the weight percentage of the element X, \(M\) is the Taylor factor, \(\mu\) is the elastic shear modulus (calculated as \(\frac{1}{2}\sqrt{C_{44}C_{11}-C_{12}}\), with \(C_{ij}\) the element at row \(i\) and column \(j\) of \(C_{sc}\), and consistent with the value in [58]) and \(b\) is the cell size (0.287 nm for steel). These phenomenological relationships have been developed to calculate the yield stress. In order to apply them to the CRSS in slip systems, it is necessary to multiply the obtained \(\sigma_{y}\) expression by a proportionality constant, \(C_{1}\), which should be of the same order of magnitude of the Taylor factor:  
\[\tau_{0}=\frac{\sigma_{0}}{C_{1}} \tag{17}\]  
The hardening induced by dislocations is added to the static term given by Equation (15) according to [PERSON] equation [15], after modifying it to take the latent hardening effect into account:  
\[\tau_{c}^{s}=\tau_{0}+\alpha\mu b\sqrt{\sum_{s^{\prime}}a_{ss^{\prime}}\rho_{ s^{\prime}}} \tag{18}\]  
where \(\alpha\) depends on the structure of dislocations and \(\mu\) and \(b\) are the same magnitudes in (16d). The coefficients \(a_{ss^{\prime}}\) account for the hardening in system \(s\) caused by the dislocations in system \(s^{\prime}\).