Fig. 7 Schematic diagram of classification of strains within two adjacent cycles.
As shown in Fig. 7 (b), the total recovery strain can be further divided into three parts by elastic modulus, i.e., (i)\(\ \varepsilon^{e}\), (ii)\(\ \varepsilon_{\text{un}}^{\text{an}}\), and (iii)\(\ \varepsilon_{v}^{\text{an}}\). The loading elastic modulusE and the reloading elastic modulus \(E_{\text{re}}\) correspond to the slope of the stress-strain curve in the elastic range. The unloading modulus\(\ E_{\text{un}}\), considering its nonlinearity and complexity,42,43 can be adequately defined by the straight line connecting the upper-elastic limit and the valley stress of the unloading stress-strain curve, which is similar to the fitting method of chord modulus in Kim et al.’s study.44 Based on the difference between apparent modulus \(E_{\text{un}}\) and E , the component of anelastic strain \(\varepsilon_{\text{un}}^{\text{an}}\) is separated during the unloading process. Considerably, the elastic recovery strain \(\varepsilon^{e}\) and anelastic strain\(\varepsilon_{\text{un}}^{\text{an}}\) arise simultaneously rather than separately in reality.45 Moreover, when stress is unloaded to the valley stress, it is observed that the strain can still spring back until the stress is reloaded to the lower-elastic limit. This part of the micro-plastic strain \(\varepsilon_{v}^{\text{an}}\)under the low-stress region is regarded as the major anelastic recovery in case with long duration under valley stress.13 The two parts of anelastic recovery strain indicate the existence of a two-stage spring-back phenomenon when the applied stress is removed, which is closely related to the change of elastic modulus46 and the movement of dislocations.47
The above values of strain can be obtained from experimental data, and the relationships are presented as the following:
\(\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left\{\par \begin{matrix}\&\varepsilon_{\text{un}}^{\text{an}}=\varepsilon_{2}-\varepsilon_{3}-\varepsilon^{e}\\ \&\varepsilon_{v}^{\text{an}}=\varepsilon_{3}-\varepsilon_{4}\\ \&\varepsilon^{\text{re}}=\varepsilon^{e}+\varepsilon_{\text{un}}^{\text{an}}+\varepsilon_{v}^{\text{an}}\\ \&\varepsilon^{\text{act}}=\varepsilon^{r}+\varepsilon^{c}-\varepsilon^{\text{re}}=\varepsilon_{4}-\varepsilon_{0}\\ \end{matrix}\right.\ \)
where \(\varepsilon_{0}\), \(\varepsilon_{1}\),\(\varepsilon_{2}\), \(\varepsilon_{3}\) and\(\ \varepsilon_{4}\) refer to the strains at different time points, and the values of\({(\varepsilon}_{2}-\varepsilon_{3})\) and \(\varepsilon^{e}\) can be calculated by the following equations:
\(\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ε}_{2}-\varepsilon_{3}={(\sigma}_{e}^{\max}-\sigma^{\min})/E_{\text{un}}\)
\(\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\varepsilon^{e}={(\sigma}_{e}^{\max}-\sigma^{\min})/E\)
According to the above systematic classification, it can be found that the retardation effect in strain accumulation rate under cyclic creep results from the interaction between the strain (\(\varepsilon^{r}\),\(\ \varepsilon^{c}\)) and recovery strain (\(\varepsilon^{e}\),\(\ \varepsilon_{\text{un}}^{\text{an}}\), and\(\ \varepsilon_{v}^{\text{an}}\)), or rather the anelastic recovery plays a vital role in recovering the previous inelastic deformation. Thereby, it is clear that the ultimate unrecovered strain is the actual damage created within two adjacent cycles, and the difference of retardation effect under varied unloading conditions can be studied by analyzing each part of the strain quantitatively.
Maximum strain in one cycle
4.2.1 Ratcheting strain\(\varepsilon^{r}\)
The ratcheting strain \(\varepsilon^{r}\) is defined as the total elastic and plastic strain produced by loading from the lower-elastic limit to the peak stress. The evolution of ratcheting strain during the entire life (the values come from the first three cycles and 0.1Nf, 0.2 Nf, 0.3 Nf, …, 1Nf, both here and below) under different unloading rates is shown in Fig. 8 (a). The developments of ratcheting strain follow the same trend under different unloading conditions. The evolution of ratcheting strain can be divided into three stages, where it is slightly large at first cycle, then drops to a stable value, and increases again before fracture. Moreover, the elastic component of the ratcheting strain can be obtained conveniently, which is inversely proportional to the loading elastic modulus E , as shown in Fig. 9 (a). The loading elastic modulus is hardly affected by changed unloading rate and the duration under valley stress. The evolution of loading elastic modulus also displays three stages, which shows that the value of E is large at the first cycle, then rapidly drops to a stable value, and further declines before fracture. It is found that both ratcheting strain and loading elastic modulus just enter the stable stage after the first cycle, which is related to the variation of dislocation. The degeneration of the loading elastic modulus is mainly due to the increase of dislocation density,44,46 and the multiply of dislocations can cause a hardening effect for the plastic component of the ratcheting strain. It seems that the dislocation density has significantly increased within the first cycle by the initial loading and a long-term hold of peak stress, and the dislocation density may remain almost unchanged subsequently. Hence, the slightly large ratcheting strain at the first cycle can be attributed to the considerably large plastic component and small elastic component. In addition, the specimens exhibit remarkable necking near fracture, which leads to an increase of the ratcheting strain and a further decrease of the elastic modulus.