*3.2. Phenomenological Constitutive Models*

Several internal state variables are used in the development of phenomenological constitutive models. The deformation state of a material is determined by the variations of the internal state variables with thermal and mechanical loading. The phenomenological constitutive models can be categorized into power-law type (medium stress condition), hyperbolic-sine type (medium to high stress condition) and linear type or combination of linear and power-law types (low to medium stress condition). Chowdhury et al. [57] compared the applications of these constitutive models in the study of crystal plasticity. For engineering practices, the analysis of the plastic deformation of turbine blades is usually based on the power-law constitutive models. The power-law constitutive models have been widely used in the stress analysis due to that it is cost-effective in determining material parameters in the constitutive models and they are applicable in the analysis of the steady-state creep deformation of turbine blades at elevated temperature over a wide range of stresses. However, the pre-factor of the power-law constitutive models is a function of temperature due to different rate mechanisms.

For nickel-based superalloys, dislocation motion in both γ and γ' phases are the dominant mechanism for plastic deformation at elevated temperature [58,59]. In the power-law constitutive models, the ratio or difference between the resolved shear stress, τ(α), and the slip resistance, *g*(α), determines the slipping activity of the slip systems. A general mathematic relationship between the shear strain rate, the resolved shear stress, and the slip resistance can be expressed as [60]

$$
\dot{\gamma}^a = f(\pi^{(a)}, \mathcal{g}^{(a)}) \tag{9}
$$

There are three different types of power-law constitutive models for creep deformation as [57]

$$\dot{\boldsymbol{\gamma}}^{(a)} = \dot{\boldsymbol{\gamma}}\_0 \left| \frac{\boldsymbol{\tau}^{(a)}}{\mathcal{g}^{(a)}} \right|^n \text{sgn}(\boldsymbol{\tau}^{(a)}) \tag{10}$$

$$\dot{\boldsymbol{\gamma}}^{(a)} = \dot{\boldsymbol{\gamma}}\_0 \left| \frac{\boldsymbol{\pi}^{(a)} - \boldsymbol{X}^{(a)}}{\mathcal{S}^{(a)}} \right|^n \text{sgn}(\boldsymbol{\pi}^{(a)} - \boldsymbol{X}^{(a)}) \tag{11}$$

$$\dot{\gamma}^{(a)} = \left\langle \frac{\left| \tau^{(a)} - X^{(a)} \right| - R^{(a)}}{K} \right\rangle^n \text{sgn}(\tau^{(a)} - X^{(a)}) \tag{12}$$

where . γ<sup>0</sup> and *n* are the shear rate at the reference state and the rate sensitivity parameter, respectively, *g*(α) and *R*(α) are two threshold stresses (both evolve with the activity of crystallographic slips [57]), *X*(α) is internal stress or the back stress, and *K* is temperature-dependent material parameter. The angle bracket "<*y*>" in Equation (12) represents the positive part of *y*, and the "*sgn*(•)" is the sign function.

For the power-law constitutive model of Equaton (10), the activity of crystallographic slips is determined by the resolved shear stress, τ(α). The slip resistance, *g*(α), is the only internal state variable as

$$\dot{\mathbf{g}}^{(\alpha)} = \sum\_{\beta=1}^{N} h\_{\alpha\beta} \left| \dot{\mathbf{y}}^{(\beta)} \right| \tag{13}$$

with *h*αβ as the hardening matrix, which determines the hardening effect of the α-th slip system on the β-th slip system [61]. For α = β, the hardening parameter, *h*αα, represents the self-hardening modulus as [61]

$$h\_{\alpha\alpha} = h(\mathcal{Y}) = h\_0 \text{sech}^2 \left| \frac{h\_0 \mathcal{Y}}{\tau\_s - \tau\_0} \right| \tag{14}$$

for α β, the hardening parameter, *h*αβ, represents the latent hardening modulus as

$$h\_{a\beta} = qh(\gamma), (\alpha \not\simeq \beta) \tag{15}$$

Here, *h*<sup>0</sup> is the initial hardening modulus, τ*<sup>s</sup>* and τ<sup>0</sup> are the saturated shear stress and initial yield stress, respectively, and *q* is a constant. The cumulative shear strain, γ, is calculated as

$$\gamma = \sum\_{\alpha} \int\_{0}^{t} |\gamma^{(\alpha)}| dt \tag{16}$$

Note that the hardening matrix, *h*αβ, is also reported in a power-law form as [60,62]

$$h\_{\alpha\beta} = q\_{\alpha\beta} h\_0 \Big(1 - \frac{\tau\_0}{\tau\_s}\Big)^{n\_0} \tag{17}$$

with *n*<sup>0</sup> as a material constant. For the self-hardening modulus, *q*αβ = 1, and for the latent hardening modulus, *q*αβ = 1.4.

For the constitutive models of Equations (11) and (12), the activity of crystallographic slips is determined by the effective stress <sup>τ</sup>(α) <sup>−</sup> *<sup>X</sup>*(α) and τ(α) <sup>−</sup> *<sup>X</sup>*(α) <sup>−</sup> *<sup>R</sup>*(α), respectively. The internal stress, *X*(α), can be calculated by the following equation [63]

$$X^{(a)} = \mathbb{C}^{(a)} a^{(a)} \tag{18}$$

$$\dot{a}^{(a)} = \phi(\nu^{(a)})\dot{\gamma}^{(a)} - \left| \dot{\gamma}^{(a)} \right| d^{(a)} a^{(a)} \tag{19}$$

$$
\phi(\nu^{(a)}) = \phi\_0 + (1 - \phi\_0)e^{-\delta\nu^{(a)}} \tag{20}
$$

$$\nu^{(a)} = \int\_0^t \left| \dot{\boldsymbol{\nu}}^{(a)} \right| dt \tag{21}$$

In Equation (18), *C*(α) is the determinate internal stress at a temperature, and *a*(α) is a control variable [57]. The constant *d* is a recovery parameter. The flow accumulation function of φ(ν(α)) is calculated from the accumulated shear strain of the α-th slip system of ν(α) and the constants of δ and φ<sup>0</sup> in Equation (20). It needs to be pointed out that some works [12,19] replaced the term of φ(ν(α)) . γ(α) with | . γ(α) <sup>|</sup> in Equation (19) and did not consider the flow accumulation function of <sup>φ</sup>(ν(α)).

In general, the use of the effective stress instead of the resolved shear stress for the constitutive models of Equations (11) and (12) makes it possible to analyze the plastic deformation in materials consisting of complex microstructures involving multiphases [64,65]. This is because the internal stresses from the complex microstructures do not directly contribute to the activities of crystallographic slips and needs to be deducted. Additionally, the increments of the internal state variables in Equations (11) and (12) have provided the basis to accurately describe crystallographic slips and determine the threshold stress. All of these suggest that the power-law constitutive models of Equations (11) and (12) can likely provide better correlations between stresses and strains for the analysis of the plastic deformation of the nickel-based superalloys.
