# Investigation of Deformation Inhomogeneity and Low-Cycle Fatigue of a Polycrystalline Material
## 4 Discussion and Conclusions  
In this paper, deformation inhomogeneity evolution at the grain level of a superalloy GH4169 under macro uniform tension compression cyclic loading and temperature 500 \({}^{\circ}\)C is studied by using polycrystalline RVE associated with crystal plasticity simulation. Based on the statistical parameters of standard deviation and differential entropy \(\bar{\varepsilon}_{33}\ +\ 3\varepsilon_{33}\varepsilon_{33}\varepsilon_{23} \varepsilon_{\text{eq}}\) and \(H_{\varepsilon_{33}}\), the various FIPs are taken into account as the measurement of deformation inhomogeneity. Additionally, the relationship between the FIPs and low cycle fatigue of materials is discussed. These monotonically increased parameters reflect the irreversible inhomogeneity of the material deformation and the evolution of material microstructure with the number of cycles. Furthermore, criteria using these parameters' critical values are applied to predict the strain fatigue-life curve of the material. Based on these investigations, we conclude:  
1. At grain level, the standard deviation \(\hat{\varepsilon}_{ij}\) and differential entropy \(H_{\varepsilon_{ij}}\) of the respective strain tensor component \(\varepsilon_{ij}\) increase monotonously with the cycle. The standard deviation values of each strain component are almost the same, and the law of their growth with the increase in cycle number is similar. The values of components of differential entropy \(H_{\varepsilon_{ij}}\) are close to each other, and the law of that with the number of cycles is also similar. 2. The respective standard deviations of the effective strain \(\hat{\varepsilon}_{\text{eq}}\), first principal strain \(\hat{\varepsilon}_{1}\) and maximum principal shear strain \(\hat{\varepsilon}_{13}^{\text{M}}\) are similar in numerical and growth law with the number of cycles. 3. The parameters \(\bar{\varepsilon}_{33}+3\varepsilon_{33}\varepsilon_{33}\varepsilon_{23} \varepsilon_{\text{eq}}\) and \(H_{\varepsilon_{33}}\) can be used as fatigue index parameters, and the corresponding critical values can be determined by a single strain amplitude cycle fatigue test, based on which the fatigue-life curve can be predicted. It should be pointed out that (1) from conclusions 1 and 2, all components for \(\hat{\varepsilon}_{ij}\), and \(\hat{\varepsilon}_{1}\) and \(\hat{\varepsilon}_{13}^{\text{M}}\) can be used as FIPs and for use for fatigue-life curve prediction, but the detailed results are not shown due to the limited space. This result implies that the proposed method may be also effective under the loading condition of complicated stress state. (2) The FEM is used to analyze, and the calculation results may be affected by the size of mesh to a certain extent. The numerical results may be different with mesh sizes varying, but the regularity of the results will preserve unchanged [23]. Figure 14: The curves of SD at strain amplitude 0.00601 vs. cycle number under different conditions: loading on axis-3 and on axis-1 and loading on axis-1 adopting periodic boundary conditions. **Author Contributions:** Conceptualization, M.-H.Z. and K.-S.Z. ; methodology, K.-S.Z. ; software, L.H. ; validation, L.H. and K.-S.Z. ; formal analysis, L.H. and K.-S.Z. ; investigation, M.-H.Z. and L.H. ; resources, X.-H.S., and K.-S.Z. ; data curation, L.H., M.-H.Z. and K.-S.Z. ; writing--original draft preparation, L.H., M.-H.Z. and K.-S.Z. ; writing--review and editing, L.H., M.-H.Z., X.-H.S., and S.Z. ; visualization, L.H. and K.-S.Z. ; supervision, X.-H.S. and K.-S.Z. ; project administration, X.-H.S. and K.-S.Z. ; funding acquisition, X.-H.S. and K.-S.Z. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was funded by the National Natural Scientific Foundation of China, grant number 11632007, 62031021 and 11472085. The APC was funded by X.-H.S. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable.