## 3 Modeling
### VPSC Formalism  
The VPSC crystal plasticity model, developed by [PERSON] and [PERSON] [1], is one of the most widely used models for the simulation of polycrystalline materials. The model is based on the assumption that, when a polycrystalline material is deformed, every grain behaves as a viscoplastic inclusion in a homogeneous matrix, also viscoplastic, with the properties of the polycrystal. Polycrystal and grain properties are correlated according to the interaction equation, defined for each grain:  
\[\dot{\epsilon}-\dot{\epsilon}_{g}=-\bar{M}_{g}(\sigma-\sigma_{g}) \tag{5}\]  
where \(\bar{M}_{g}\) is the interaction tensor (tensors are represented with a bold typeface), which depends on the grain shape and is found solving the corresponding Eshelby problem (the method is presented in [1] and explained in detail in [49]); \(\sigma_{g}\) and \(\dot{\epsilon}_{g}\) are the stress and strain rate of the grain, respectively; and \(\sigma\) and \(\dot{\epsilon}\) the corresponding polycrystal magnitudes, which are calculated as weighted averages  
\[\dot{\epsilon}=\sum_{g}w_{g}\dot{\epsilon}_{g} \tag{6}\] \[\sigma=\sum_{g}w_{g}\sigma_{g}\]  
with \(w_{g}\) the volume fraction of each grain.  
At the grain level, the model assumes that shear in each slip system follows the strain rate sensitivity approach of [PERSON] and [PERSON] [22]:  
\[\dot{\epsilon}_{g}=\sum_{s}\mathbf{m}_{s}\dot{\gamma}_{s}=\dot{\gamma}_{0}\sum_{ s}\mathbf{m}_{s}\bigg{(}\frac{|\mathbf{m}_{s}\sigma_{g}|}{\tau_{s}^{c}}\bigg{)}^{n_{s}} \text{sign}(\mathbf{m}_{s}\sigma_{g}) \tag{7}\]  
In this equation, \(\dot{\gamma}_{s}\) is the shear rate in the slip system \(s\), \(\dot{\gamma}_{0}\) is a normalization factor, \(\mathbf{m}_{s}\) is the symmetric Schmid tensor of each slip system (dependent on grain orientation) and \(\tau_{s}^{c}\) is the critically resolved shear stress (CRSS) associated to the corresponding deformation mode. The exponent \(n_{s}\) is equivalent to \(1/m\), where \(m\) is the exponent in Equation (1).  
After a solution is found for stress and strain rate in every grain, slip hardening, grain shape and orientation are updated. Hardening evolution is the subject of Section 3.4. The orientations of the grains are updated using the decomposition of the velocity gradient tensor \(\mathbf{L}_{g}\) into stretch and rotation rate (or spin) components with:  
\[\mathbf{L}_{g}=\dot{\epsilon}_{g}+\mathbf{\omega}_{g}=\sum_{s}\mathbf{m}_{s}\dot{\gamma}_ {s}+\sum_{s}\mathbf{q}_{s}\dot{\gamma} \tag{8}\]  
where \(\mathbf{m}_{s}\) and \(\mathbf{q}_{s}\) are the symmetric and skew-symmetric components of the Schmid tensor. Applying also the polar decomposition of the deformation gradient tensor, a relationship is found for the calculation of the derivative of the grain rotation matrix \(R_{g}\):  
\[\dot{R}_{g}=\Big{(}\mathbf{\omega}_{g}-R_{g}^{T}\mathbf{\omega}_{g}^{0}\mathbf{R}_{g}\Big{)} \mathbf{R}_{g} \tag{9}\]in which \(\omega_{g}^{\theta}\) is the spin calculated from the deformation gradient tensor in the initial crystal axis. A new orientation is calculated applying the Rodrigues rotation formula. Finally, the shape evolution is calculated from the deformation gradient tensor \(\mathbf{F}_{g}\). The value of this tensor is updated taking into account that:  
\[\dot{\mathbf{F}}_{g}=\mathbf{\textit{L}}_{g}\mathbf{\textit{F}}_{g} \tag{10}\]  
where \(\mathbf{\textit{L}}_{g}\) is given by (8). The axes lengths of the equivalent ellipsoid are given by the eigenvalues of the tensor \(\mathbf{\textit{F}}_{g}\mathbf{\textit{F}}_{g}^{T}\), while the eigenvectors indicate the orientation of these axes.