## 4 Crystal Plasticity Model
### Kinematic Relations  
The mesoscale behavior of polycrystalline metallic alloys can be well-described by the CPFE method [50; 51; 52], which postulates that plastic deformation occurs through the sliding of dislocations on preferential slip systems with respect to the crystal lattice orientation in each material grain. To represent the anisotropic plastic strain and the potentially large lattice rotation, various kinematic treatments of the constitutive equations have been proposed using hyperelastic (strain energy-based) [51; 53] or hypoelastic (objective stress rate-based) [50; 54] approaches. In this case, the Green-Naghdi hypoelastic approach is adopted from [47] due to its advantage of formulating the stress update formulas in the unrotated frame [55], which separates the large strain kinematics from the shear stress to slip rate relations, allowing for the incorporation of multiple models in a unified implementation. The Green-Naghdi objective rate of the Cauchy stress tensor \(\sigma\) is expressed through the material time derivative \((\bullet)=\dfrac{\partial(\bullet)}{\partial t}\) as  
\[\check{\sigma}=\check{\sigma}+\sigma\mathbf{\Omega}-\mathbf{\Omega}\sigma. \tag{6}\]  
The right polar decomposition of the deformation gradient \(\mathbf{F}=\mathbf{R}\mathbf{U}\) into rotation tensor \(\mathbf{R}\) and stretch tensor \(\mathbf{U}\) is used to define the skew-symmetric angular velocity tensor \(\mathbf{\mathsf{\blacksquare}}=\dot{\mathbf{R}}\mathbf{R}^{T}\). For crystal plasticity of metals and minerals with stiff elastic modulus, the elastic strains \(\varepsilon\ll\mathbf{I}\) are quite small compared to the identity tensor \(\mathbf{I}\). Therefore, the traditional multiplicative split of \(\mathbf{F}\) into elastic and plastic parts is assumed to take a simplified form through polar decomposition of each component and retention of the small elastic strains, i.e.,  
\[\mathbf{F}=\mathbf{F}^{\mathsf{e}}\mathbf{F}^{\mathsf{p}}\approx(\mathbf{I}+ \varepsilon)\mathbf{R}^{\mathsf{e}}\mathbf{R}^{\mathsf{p}}\mathbf{U}^{\mathsf{ p}}. \tag{7}\]  
This treatment accommodates features of plastic anisotropy on slip systems and finite elastic rotation \(\mathbf{R}^{\epsilon}\). The derivations in [47] provide the substitution of (7) into (6) and mapping to the unrotated intermediate configuration to yield the rate equation for the unrotated Cauchy stress \(\mathbf{t}=\mathbf{R}^{T}\sigma\mathbf{R}\) provided by  
\[\dot{\mathbf{t}}=\mathbf{C}_{0}:\left(\mathbf{d}-\mathbf{d}^{(\mathsf{p})} \right)+\mathbf{R}\mathbf{w}^{\mathsf{p}}\mathbf{R}^{T}\mathbf{t}-\mathbf{t} \mathbf{R}\mathbf{w}^{\mathsf{p}}\mathbf{R}^{T}. \tag{8}\]  
Figure 20: PolyCISAMR mesh for the polycrystalline microstructure shown in Figure 12.  
The first term in this equation involves the constant anisotropic elastic fourth-order elasticity tensor \(\mathbf{C}_{0}\) in the lattice frame multiplying the elastic strain rate \(\dot{\mathbf{\epsilon}}\). This is additively decomposed into the total unrotated rate of deformation \(\mathbf{d}=\frac{1}{2}\big{(}\mathbf{U}\mathbf{U}^{-1}+\mathbf{U}^{-1}\mathbf{U} \big{)}\) and the plastic deformation rate \(\bar{\mathbf{d}}^{\text{P}}\). The other two terms involving the plastic vorticity \(\bar{\mathbf{w}}^{\text{P}}\) are correcting terms due to plastic anisotropy. The major postulate of crystal plasticity theory is that plastic straining occurs due to the accumulation of slip of dislocations on preferred crystallographic planes.