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:
\(\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left\{\par
\begin{matrix}\&\varepsilon_{\text{un}}^{\text{an}}=\varepsilon_{2}-\varepsilon_{3}-\varepsilon^{e}\\
\&\varepsilon_{v}^{\text{an}}=\varepsilon_{3}-\varepsilon_{4}\\
\&\varepsilon^{\text{re}}=\varepsilon^{e}+\varepsilon_{\text{un}}^{\text{an}}+\varepsilon_{v}^{\text{an}}\\
\&\varepsilon^{\text{act}}=\varepsilon^{r}+\varepsilon^{c}-\varepsilon^{\text{re}}=\varepsilon_{4}-\varepsilon_{0}\\
\end{matrix}\right.\ \) ⑶
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:
\(\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ε}_{2}-\varepsilon_{3}={(\sigma}_{e}^{\max}-\sigma^{\min})/E_{\text{un}}\)⑷
\(\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\varepsilon^{e}={(\sigma}_{e}^{\max}-\sigma^{\min})/E\)⑸
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.