# Lifetime Prediction of Single Crystal Nickel-Based Superalloys
## 2 Experimental Procedure and Methodology
### Crystal Plasticity  
The constitutive framework of the single-crystal plasticity model, developed from [PERSON]'s crystallographic model [20], is presented in this section. This formulation \begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Ni** & **Cr** & **Co** & **Mo** & **W** & **Al** & **Ti** & **Ta** & **Re** & **Hf** \\ \hline
61.8 & 6.5 & 9 & 0.6 & 6 & 5.6 & 1 & 6.5 & 3 & 0.1 \\ \hline \end{tabular}
\end{table} 
Table 1: Chemical Composition of CMSX-4 (wt%). Figure 1: Dimensions of the notched specimen. primarily operates at mesoscales (\(10^{-7}\,\mathrm{m}\) to \(10^{-4}\,\mathrm{m}\)) by modeling the dislocation motion and latent interaction within the slip systems. At this scale, formulation captures the localized plastic strains on active slip systems. The formulation does not directly predict the macro-strain response of the material; however, the accumulation of micro-strains on active slip systems, depending on the orientation of the material and the orientation of the loading, averages the overall response of the material and hence gives the macro-response of the stress-strain distribution [21]. This model was prepared to simulate the anisotropic rate-dependent plastic behaviour of Ni-based superalloys at small strains. The linear strain tensor can be shown as \(\varepsilon=sym[\
abla\mathbf{u}]\) from the definition of the symmetric part of the displacement gradient, and since small deformation is used in this work, an additive decomposition is applied. \[\varepsilon=\varepsilon^{e}+\varepsilon^{p} \tag{2}\]  
where \(\varepsilon^{e}\) is the elastic strain and \(\varepsilon^{p}\) is the plastic part. The relationship between Cauchy stress, \(\sigma\), and elastic strain is expressed as a state law as follows:  
\[\sigma=\mathbf{C}:\varepsilon^{e} \tag{3}\]  
where \(\mathbf{C}\) is the fourth order elasticity moduli. Material to be simulated has an FCC structure. It shows cubic elastic behavior. Hence, \(\mathbf{C}\) has 3 independent constants, \(\mathbf{C_{11}}\), \(\mathbf{C_{12}}\), and \(\mathbf{C_{44}}\), as can be seen below. \[\mathbf{C}= \begin{bmatrix}C_{11}&C_{12}&C_{12}&0&0&0\\ C_{12}&C_{11}&C_{12}&0&0&0\\ C_{12}&C_{12}&C_{11}&0&0&0\\ 0&0&0&C_{44}&0&0\\ 0&0&0&0&C_{44}&0\\ 0&0&0&0&0&C_{44}\end{bmatrix} \qquad\text{where}\qquad \begin{aligned} C_{11}& =\frac{1}{3}(3K+4\mu)\\ C_{12}&=\frac{1}{3}(3K-2\mu)\\ C_{44}&=\mu+\mu^{\prime}\end{aligned}  \tag{4}\]  
where \(\mathbf{C}\) is bulk modulus, \(\mu\) is shear modulus and \(\mu^{\prime}\) is the effective shear modulus describing shear resistance in specific crystallographic directions [22]. Another state law can shown which represents isotropic hardening of the each slip system as:  
\[r^{s}=q\sum_{s=1}^{N}h^{sr}\rho^{s} \tag{5}\]  
where q is a material constant defining isotropic hardening, s is the slip system, and N is the total number of slip systems. \(h^{sr}\) is the interaction matrix that governs the self and latent hardening of the material caused by the interactions between the N number of slip systems. Literature usually suggests taking \(h^{sr}\) as the identity for single crystal Ni-based superalloys since they do not show pro-dominant latent hardening [23]. And \(\rho^{s}\) is the dislocation density, which can be defined as a nonlinear isotropic hardening rule as  
\[\rho^{s}=1-exp(-b|\gamma^{s}|) \tag{6}\]  
where b is a model parameter; hence, \(r^{s}\) becomes  
\[r^{s}=q\sum_{s=1}^{N}(1-exp(-b|\gamma^{s}|)) \tag{7}\]  
In order to define \(\gamma^{s}\) (which is plastic slip of slip system s), Schmid law must be presented here:  
\[f^{s}=|\tau^{s}|-r^{s} \tag{8}\]where \(\tau^{s}\) is the resolved shear stress of slip system s, which is the driving force for the plastic slip of the corresponding slip system. In other words, for a slip system to deform plastically, the resolved shear stress must overcome a stress barrier defined by Schmid law. Hence, [PERSON] law here can be used as a multimechanism plastic yield criterion.