# Investigation of Deformation Inhomogeneity and Low-Cycle Fatigue of a Polycrystalline Material
## 3 Measurements of Deformation Inhomogeneity and Predictions of Fatigue-Life Curve
### Fatigue Indicator Parameters (FIPs), Life Prediction and Verification
#### 3.3.2 The Deformation Inhomogeneity Growing and Fatigue Failure Occurrence Predicting
Limited to the space, only the following three conditions \(\varepsilon_{33}\leq\varepsilon_{33}^{\text{crit}}\), \(\varepsilon_{23}\leq\varepsilon_{23}^{\text{crit}}\) and \(\varepsilon_{\text{eq}}\leq\varepsilon_{\text{eq}}^{\text{crit}}\) are verified for their validity as below. Firstly, determine the minimum and maximum critical values for different FIPs (\(\varepsilon_{33}\), \(\varepsilon_{23}\) and \(\varepsilon_{\text{eq}}\)) and the critical values from the curves of \(E_{a}=0.00601\). Secondly, apply them to determine the corresponding point sequences \((N_{f}^{\text{predict}},E_{a})\); then, draw out the corresponding fatigue-life curves of the material, see Figures 9-11. Checking Figures 9b, 10b and 11b, and comparing all the predicted with the measured results, one can observe that the predicted ones are reasonable. Figure 8: Verification of fatigue indicator parameter (FIP) \(\bar{\varepsilon}_{33}+3\varepsilon_{33}\): (a) determining upper and lower of critical value and determining point sequences \((N_{f}^{\text{predict}},E_{a})\); (b) predicted fatigue–life curves and their comparison with tests. Figure 7: (a) Determining the critical value \(\varepsilon_{33}^{\text{crit}}\), by the intersection between curve of \(\bar{\varepsilon}_{33}+3\varepsilon_{33}\) (\(E_{a}=0.00601\)) and dot dash line (\(N_{f}\), test fatigue–life); determining the point sequence \((N_{f}^{\text{predict}},E_{a})\), by the intersection between each curve and the horizontal line \(\bar{\varepsilon}_{33}+3\varepsilon_{33}=0.078\); (b) predicted fatigue–life curve and its comparison with tests. Figure 11: Verification of FIP \(\ell_{\text{eq}}(E_{a})\): (**a**) determining upper and lower of critical value and determining point sequences \((N_{f}^{\text{predict}},E_{a})\); (**b**) predicted fatigue–life curves and their comparison with tests. \((N_{f}^{\text{predict}},E_{a})\). Figure 10: Verification of FIP \(\ell_{23}(E_{a})\): (**a**) determining upper and lower of critical value and determining point sequences \((N_{f}^{\text{predict}},E_{a})\); (**b**) predicted fatigue–life curves and their comparison with tests. Figure 9: Verification of FIP \(\ell_{33}(E_{a})\): (**a**) determining upper and lower of critical value and determining point sequences \((N_{f}^{\text{predict}},E_{a})\); (**b**) predicted fatigue–life curves and their comparison with tests. Therefore, it can be regarded that the growth and accumulation of deformation inhomogeneity are the main mechanism leading to the low cycle fatigue failure of materials for tension compression cycles. Thus, the fatigue-life curve can be predicted using the above method and parameters by polycrystalline RVE combined with crystal plasticity calculation. Additionally, let us see if the differential entropy is used to measure the inhomogeneity of material deformation, whether it can be used to predict the fatigue-life curve. This is like the previous discussion. Because of the similarity of differential entropy curves of all strain components in Figure 6, it can be considered \(H_{\epsilon_{ij}}\) as FIPs. That is, it can be assumed that the condition of material fatigue failure is \(H_{\epsilon_{ij}}\leq H_{\epsilon_{ij}}^{\text{crit}}\). Due to limited space, the following only discusses the rationality of the condition \(H_{\epsilon_{33}}\leq H_{\epsilon_{33}}^{\text{crit}}\) for fatigue judging. In Figure 12, it can be seen that, the minimum and maximum critical values and the critical values determined by \(E_{a}=0.00601\) cycle for \(H_{\epsilon_{33}}\) are taken respectively, then the corresponding point sequence \((N_{f}^{\text{predict}},E_{a})\) is determined (see Figure 12a) and the prediction curve of material fatigue life is drawn (see Figure 12b). The above results show that the fatigue-life curve can be reasonably predicted by using differential entropy as FIP. This further proves that it is feasible to predict the low cycle fatigue failure of materials by the accumulation of inhomogeneous deformation.