*3.3. Physics-Based Constitutive Models*

In the phenomenological constitutive models, the threshold stresses, *g*(α) and/or *R*(α), whose evolution follows a hardening rule, are used to represent the contribution of dislocation motion. However, it is very difficult, if not impossible, to experimentally determine the hardening rule and to validate the hardening rule under service-like conditions. Additionally, these constitutive models fail to capture the orientation dependence of the mechanical behavior of single crystals [66] and the hardening rule of materials at the micron scale [67]. There is a great need to develop physics-based constitutive models, which use dislocation density as an important internal state variable.

There are various physics-based constitutive models, which incorporate dislocation density in the theory of plasticity for the analysis of the plastic deformation of crystalline materials [68–71]. In the heart of the physics-based constitutive models is the evolution of statistically stored dislocation (SSD) and geometrically necessary dislocation (GND) during plastic deformation. The SSD is associated with the "homogeneous" deformation, while the GND is associated with the "inhomogeneous" deformation in single crystals only at small length scale [68].

SSDs are quantified by dislocation density, ρ, (line length per unit volume), Burgers vector, *b*, and unit vector of dislocation-line segment, *t*. The magnitude of the Burgers vector is discrete and related to the lattice constant of crystal. For simplification, the unit vector of dislocation-line segment is sometimes limited to a finite set, which makes it easy to include dislocations in numerical calculation.

The Orowan relationship instead of the phenomenological constitutive relationship is used to correlate the plastic shear rate with dislocation density in a physics-based constitutive model as [60]

$$
\dot{\varphi}^{(a)} = \rho\_m^{(a)} b \nu^{(a)} \tag{22}
$$

where ρ*<sup>m</sup>* is the density of mobile dislocations and is calculated from the SSD density, ρ*SSD*, and *b* and ν are the magnitude of Burgers vector and average velocity of the mobile dislocations, respectively. One specific form of Equation (22) is [68,69]:

$$\dot{\boldsymbol{\gamma}}^{(a)} = \left( \rho^{(a)}\_{c+} \overline{\boldsymbol{\nu}}^{(a)}\_{c+} + \rho^{(a)}\_{c-} \overline{\boldsymbol{\nu}}^{(a)}\_{c-} + \rho^{(a)}\_{s+} \overline{\boldsymbol{\nu}}^{(a)}\_{s+} + \rho^{(a)}\_{s-} \overline{\boldsymbol{\nu}}^{(a)}\_{s-} \right) \mathbf{b} \mathbf{b} \tag{23}$$

where the subscripts, *e* and *s*, represent edge and screw dislocations, respectively, and the symbols, "+" and "−", represent the polarity of the SSD density. The density of mobile dislocation, ρ*m*, in Equation (22) can also be divided into two portions: ρ*<sup>P</sup>* for the dislocations parallel to slip planes, and ρ*<sup>F</sup>* for the dislocations perpendicular to slip planes as [70,71]

$$
\rho\_m^{(a)} = BT \sqrt{\rho\_P^{(a)} \rho\_F^{(a)}} \tag{24}
$$

$$B = \frac{2k\_B}{c\_1 \mu b^3} \tag{25}$$

$$\rho\_p^{(a)} = \sum\_{\beta=1}^N \chi^{(a)(\beta)} \rho\_{SSD}^{(\beta)} \left| \sin(\mathfrak{n}^{(a)}, \mathfrak{k}^{\beta}) \right| \tag{26}$$

$$\rho\_F^{(a)} = \sum\_{\beta=1}^N \chi^{(a)(\beta)} \rho\_{SSD}^{(\beta)} |\cos(\mathfrak{n}^{(a)}, \mathfrak{t}^{\beta})| \tag{27}$$

where *kB* is Boltzmann constant, *c*<sup>1</sup> is a constant for the evolution of dislocation density, and χ(α)(β) represents the interaction strength between different slip systems. The dislocation interaction plays an important role in determining the plastic deformation in single crystal superalloys. There are nucleation and annihilation of dislocations during plastic deformation, which determine the evolution of dislocation densities [68,70].

The SSD alone is not enough to describe the plastic deformation of crystalline materials. If heterogeneous two-phase microstructure exists in a material, a local strain gradient related to the activity of GNDs is always generated between the strengthening precipitates, such as γ' phases in nickel-based superalloys [72]. In crystal plasticity models, GNDs can be obtained from slip gradients, and they are further divided into the edge dislocation, ρ*GND*,*e*, along the slip direction and the screw dislocation, ρ*GND*,*s*, perpendicular to the slip direction as [67]

$$
\rho\_{\rm GND,c}^{(a)} b = -\nabla \gamma^{(a)} \cdot \mathfrak{m}^{(a)} \tag{28}
$$

$$\circlearrowleft\_{\text{GND}\lrcorner s}^{(a)} b = \nabla \circlearrowleft^{(a)} \circ \mathbf{p}^{(a)}\tag{29}$$

where *<sup>p</sup>*(α) = *<sup>m</sup>*(α) <sup>×</sup> *<sup>n</sup>*(α), *<sup>m</sup>* is a unit vector parallel to the Burgers vector, and *<sup>n</sup>* is the unit normal to the slip plane.

With the SSD density and GND density, the internal stress or back stress, *X*(α), and the slip resistance, *g*(α), in phenomenological constitutive models can be calculated as [67]

$$\log^{(a)} = \mu b \sqrt{\sum\_{\beta} H^{(a)(\beta)} \rho^{(\beta)}} \tag{30}$$

$$X^{(a)} = \frac{\mu b \mathbb{R}^2}{8} \left| \frac{1}{(1 - \nu)} \nabla \rho\_{\text{GND}, \boldsymbol{x}}^{(a)} \cdot \mathbf{m}^{(a)} - 2 \sum\_{\beta} \delta\_s^{(a)(\beta)} (\nabla \rho\_{\text{GND}, \boldsymbol{s}}^{(a)} \cdot \mathbf{p}^{(a)}) \right| \tag{31}$$

where υ is Poisson's ratio, R is a length scale in the calculation model [73], *H*(α)(β) is the interaction matrix between two slip systems of α and β with six types of interaction in the matrix [74]. δ (α)(β) *<sup>s</sup>* is the interaction coefficient as [73]

$$
\delta\_s^{(a)(\not\ge)} = \begin{cases}
1 & \text{for}(a,\not\ge) = (4,13), (6,18), (8,17), (9,15), (10,16), (11,14), \\
& (1,16), (2,17), (3,18), (5,14), (7,13), (12,15) \\
& 0 & \text{otherwise}
\end{cases}
\tag{32}
$$

Here, α (=1 to 12) represents the edge dislocation, and α (=13 to 18) represents the screw dislocation. ρ(β) in Equation (30) is the total dislocation density on the β-th slip system. Note that Tinga et al. [51] considered the contributions of the SSD density and GND density to the total dislocation density, respectively.

Some works incorporated dislocation dynamics, such as discrete dislocation dynamics (DDD) [16,17] and continuum dislocation dynamics (CDD) [18], in the analysis of plastic deformation and microstructural evolution. The DDD models are based on discrete description of dislocation motion and require sufficiently fine grid spacing and great computational cost in the simulation of dislocation activities. The CDD models are based on a continuum quantity (dislocation density) instead of individual dislocations and need much less computational cost [18]. However, the simulation with either type of dislocation dynamics models still costs more computational effort than that with the crystal plasticity models, which incorporate the dislocation activity in a phenomenological or empirical form [19,20]. Thus, crystal plasticity models are widely used to account for the microstructure evolution and plastic deformation and to provide valuable information for engineering applications [57,75]. Table 2 presents the comparisons of the plasticity models used in the rafting analysis.


**Table 2.** Comparisons of different plasticity models used in the rafting analysis.
