# 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  
It can be seen from Section 3.1 that under macro uniform strain loading, the distribution of longitudinal strain component in the material with respect to the tensile peak is like Gaussian (cf. Figure 4). According to the theory of Gaussian distribution, for a variable \(x>\overline{x}+3\bar{x}\), the probability is less than 0.07%. Therefore, the maximum value of \(x\) can be approximately as,  
\[x^{\text{max}}\approx\overline{x}+3\bar{x} \tag{16}\]  
Taking the normal strain \(\varepsilon_{33}\) as an example (its direction is the same as that of the macro loading axis). Its standard deviation curve \(\bar{\varepsilon}_{33}(E_{a},N)\) is shown in Figure 5a. Additionally, the curves (\(\overline{\varepsilon}_{33}+3\bar{\varepsilon}_{33}\), \(N\)) corresponding to different strain amplitude cycles are drawn in Figure 7a. For any specified strain amplitude \(E_{a}\), the mean \(\overline{\varepsilon}_{33}=E_{a}\). If the fatigue failure of the material can be determined by the maximum of the strain, this condition approximately corresponds to \(\overline{\varepsilon}_{33}+3\bar{\varepsilon}_{33}\leq\varepsilon_{33}^{ \text{crit}}\). Observing a curve in Figure 7a, for example, the curve of strain amplitude \(E_{a}=0.00601\), sign the corresponding fatigue failure point by \"\(\circ\)\" (the black circle), according to the test fatigue life \(N_{f}=3\),\(436\). The parametric critical value can be obtained and \(\varepsilon_{33}^{\text{crit}}\approx 0.078\). Then, it is applied as the critical strain value to determine the fatigue lives \(N_{f}^{\text{predict}}\) of the respective fatigue cycles with various strain amplitude \(E_{a}\). That is, obtain the point sequence \((N_{f}^{\text{predict}},E_{a})\) from the intersection between each curve in Figure 7a and the horizontal line with the ordinate value of 0.078. Thus, the fatigue-life prediction curve shown in Figure 7b can be obtained. Compared with the measured fatigue failure data, the predicted results are reasonable. Selecting any curve from Figure 7a, a critical value \(\varepsilon_{33}^{\text{crit}}\) can be determined. Then, we can use different critical value to obtain different predictions of fatigue-life curve. Hence, it is necessary to check if the different predictions obtained are all reasonable. For this purpose, all the curves (\(\overline{\varepsilon}_{33}+3\bar{\varepsilon}_{33}\), \(N\)) in Figure 7a are re-drawn in Figure 8a, and the \"\(\bigstar\)\" is marked on the corresponding curves to sign the fatigue failure points, according to the measured life value of different strain amplitude cycles. From the figure, the value range of \(\varepsilon_{33}^{\text{crit}}\) can be in interval [0.071, 0.1048]. Thus, the point sequence for fatigue-life curves corresponding to the lower bound and upper bound estimation can be obtained from Figure 8a. Comparing them with the measured data in Figure 8b, one can observe that the upper and lower bounds of prediction are reasonable. The calculated strain value of RVE model is bounded. Whereas, according to ideal Gaussian distribution, the maximum value of the statistical variable tends to infinity, and the maximum deviation from the mean value is far more than three times. Therefore, the maximum value of a statistical variable in Equation (16) may mainly depend on the standard deviation. It means that the criterion of material fatigue failure measured by the maximum of \(\varepsilon_{33}\) is equivalent to \(\varepsilon_{33}\leq\varepsilon_{33}^{\text{crit}}\). Furthermore, from the similarity of curves in Figure 5, the parameters \(\varepsilon_{ij}\)\(\varepsilon_{\text{eq}}\)\(\varepsilon_{1}\) and \(\varepsilon_{13}^{\text{M}}\) obviously can be considered as the FIP used to describe the condition of material fatigue failure.