*3.1. Crystal Plasticity Theory*

Crystal plasticity theory is based on the work from Taylor and his coworkers [54]. Their work suggested that plastic deformation of crystalline metals is correlated to the change of crystallographic structure. There are two major aspects contributing to the change of crystallographic structure: one is the lattice distortion, and the other is the plastic deformation from the gliding of dislocations in slip planes. There is a significant amount of dislocations in crystalline metals, which cannot be easily described by traditional continuum mechanics [54]. The large amount of dislocations in crystalline metals (≈10<sup>7</sup> per cm2 at annealed state) makes it reasonable to use the concept of continuum mechanics in the analysis of plastic deformation controlled by dislocation motion.

Assuming that dislocation gliding occurs uniformly in a grain/single crystal, Rice and Hill [55,56] proposed a kinematic formalism for the plastic deformation of crystals. They divided the deformation gradient tensor *F* for a crystal into *Fe*, representing the lattice distortion and rigid rotation, and *Fp*, representing dislocation gliding (Figure 7) as

$$F = F^{\mathfrak{e}^\*} F^{\mathfrak{p}^\*} \tag{4}$$

**Figure 7.** Deformation of a single crystal: (**a**) example of pure dislocation deformation *F* = *Fp*, and (**b**) example of pure lattice deformation *F* = *Fe*.

The plastic deformation rate tensor due to the dislocation gliding is calculated as

$$
\dot{F}^p = L^p F^p \tag{5}
$$

Here, *Lp* is the velocity gradient for plastic deformation, consisting of the contribution of the shear strain rate . <sup>γ</sup>(α) on all active slip systems as

$$L^p = \sum\_{a=1}^N \dot{\gamma}^{(a)} m^{(a)} \otimes n^{(a)} \tag{6}$$

where *m*(α) and *n*(α) are the unit slip direction and normal vectors of the slip plane for the α-th slip system, respectively, and *N* is the number of slip systems. *Lp* can be further divided into a symmetric part, i.e., plastic deformation rate tensor, *Dp*, and an antisymmetric part, i.e., spin tensor, *Wp*, as

$$D\_{ij}^P = \frac{1}{2} \sum\_{a=1}^N \dot{\gamma}^{(a)} (m\_i^{(a)} n\_j^{(a)} + m\_j^{(a)} n\_i^{(a)}) \tag{7}$$

$$\mathcal{W}\_{ij}^P = \frac{1}{2} \sum\_{a=1}^N \dot{\nu}^{(a)} (m\_i^{(a)} n\_j^{(a)} - m\_j^{(a)} n\_i^{(a)}) \tag{8}$$

The plastic deformation rate tensor, *Dp*, represents the incremental change in the deformation behavior of the material, and the spin tensor, *Wp*, represents the gliding-induced change in the crystal orientation. Equations (5)–(8) lay the foundation to establish the relationship between the deformation at macro-scale and the shear strains of individual slip systems in a grain/crystal during plastic deformation.

In addition to the kinematic relations, constitutive equations, which capture the microstructure evolution, such as the rafting and dislocation activities, and correlate stresses to strains, need to be developed in order to completely describe the deformation behavior of a grain/crystal. In the following, we present two classes of constitutive models: one is referred to as phenomenological constitutive models, and the other is referred to as physics-based constitutive models.
