# 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
#### 3.3.2.1 Validation  
It can be viewed in Section 3.3.2 that after cycle simulation, according to the life data measured by the test of one strain amplitude cycles test, one point sequence of fatigue life and strain amplitude \((N_{f}^{\text{predict}},E_{a})\), that is, one fatigue-life curve, can be predicted. Therefore, different strain amplitude tests will give different fatigue curve predictions. For example, three prediction curves for each FIP are determined in the section above. Including upper and lower bound curve prediction, and curve prediction based on the test at strain amplitude of 0.00601, see Figures 8b, 9b, 10b, 11b and 12b, respectively. The predicted values (ordinate) and measured values (abscissa) in these figures are plotted in Figure 13c, and the predictions made with different FIPs are given, respectively: (a) the prediction according to the test at strain amplitude 0.00601; (b) the upper bound prediction; (c) the lower bound prediction. They, respectively, show the deviation between the predicted and the measured fatigue life. It can be seen from the figure that the predicted results are within the interval factor of 2, except for one point predicted by the upper bound of the fatigue failure criterion, see Figure 13b. It implies that this method is able to rationally predict fatigue-life curves, according to the FIP critical value, which determined using fatigue tests of any single specified strain amplitude. Figure 12: Verification of FIP \(H_{\epsilon_{33}}\): (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. In the above study, only one RVE model was used, so it is necessary to discuss the dependency of RVE of the method. It involves the reproducibility of this method, that is, whether similar results can be reproduced by another random model generated by the same method. In reference [31], the RVEs with the same grain number but finite element number of 8000, 27,000 and 64,000 were calculated, respectively. Although the local stress and strain in RVE vary with the finite element size, the statistical mean strain value of the model remains unchanged, and the statistical standard deviation only increases slightly with the decrease in the element size. To further discuss this problem, let the loading of RVE changed from the axis-3 to axis-1 direction. This change makes the polycrystalline structure and grain orientation in the model vary greatly. Taking the fatigue cycle analysis with a strain amplitude of 0.01003 as an example, the difference in results between the two simulations is compared. As can be seen from the results in Figure 14, the statistical standard deviation only slightly changes in value, while the law of the result curve does not change. Therefore, under the condition of this paper, the change in the above model has no obvious effect on the fatigue judgment. In the calculation above, boundary conditions of RVE deformation by Equation (1) are shown. In the following, we use the periodic boundary conditions described by Equation (2) for calculation and compare the results with the above. Similarly, fatigue cycle analysis with a strain amplitude of 0.01003 is taken as an example, and it is loaded in the direction  
Figure 13: Validation of fatigue lives predicted by applying FIPs \(\bar{\epsilon}_{33}+3\bar{\epsilon}_{33}\), \(\hat{\epsilon}_{33}\), \(\hat{\epsilon}_{23}\), \(\hat{\epsilon}_{\text{eq}}\) and \(H_{\varepsilon_{20}}\): (**a**) the critical values determined according to the fatigue test at the strain amplitude cycle of 0.00601; (**b**) the upper prediction; (**c**) the lower prediction. axis 1. The calculated statistical standard deviation curve is also presented in Figure 14, from which it can be observed that the statistical standard deviation difference between the results by two boundary conditions is also very small.