Fig. 10 Evolutions of the recovery strain \(\ \varepsilon^{e}\) during
the entire life for different unloading rates: (a) elastic recovery
strain (b) anelastic recovery strain of unloading stage and (c)
anelastic recovery strain near valley stress.
4.3.2 Anelastic recovery strain of unloading stage\(\varepsilon_{\text{un}}^{\text{an}}\)
It should be noted that the anelasticity of Cr-Mo steels is
complex.18 To discuss the performance of anelastic
recovery strain, it is generally agreed that the anelastic displacements
are mainly related to the bowing of pinned dislocations, and the
recoverability mechanism is due to the line tension. The line tension
results in back stress that can bring dislocation lines back to their
equilibrium
position.
Thus, based on the force equilibrium condition, the anelastic slip of a
population of pinned dislocation lines during the recovery period can be
described by simplified micromechanical modelling48:
\(\dot{\varepsilon^{\text{an}}}=\frac{b}{f}(m\frac{\varepsilon_{0}^{\text{an}}-\varepsilon^{\text{an}}}{l^{2}}-\rho_{m}\tau^{\text{eff}})\)⑹
where b is the Burgers vector, m is the material factor,f is the coefficient of internal friction,\({\dot{\varepsilon}}^{\text{an}}\) is the anelastic reverse slip rate,
together with \(\varepsilon_{0}^{\text{an}}\) and\(\varepsilon^{\text{an}}\) refer to the slip displacement along applied
stress direction and anelastic reverse slip displacement of dislocation
respectively. l and \(\rho_{m}\) represent the length and the
density of the dislocation respectively. It should be noted that\(\rho_{m}\) is assumed to be constant in this law, and\(\tau^{\text{eff}}\) is the effective force to drive the deformation,
which is related to the applied stress. Moreover, the former term of
this equation is related to the back stress, which is found to be
decreasing with the increase in the anelastic reverse slip displacement.
The anelastic recovery strain includes two parts as mentioned in the
above classification. For the anelastic recovery strain of unloading
stage \(\varepsilon_{\text{un}}^{\text{an}}\), the unloading elastic
modulus \(E_{\text{un}}\) under different unloading rates are obtained
to make a comparison with the loading elastic modulus E . As shown
in Fig. 9 (b), the evolution of \(E_{\text{un}}\) displays just two
stages, which keeps stable initially with a slight decrease and finally
declines a lot before fracture. The value of unloading elastic modulus
is quite sensitive to the unloading rate, and it increases greatly with
the increase in the unloading rate. Thus, the value of\(\varepsilon_{\text{un}}^{\text{an}}\) can be calculated by Eq. (3)-(5)
and shown in Fig. 10 (b), where it decreases with the increase of
unloading rate and maintains a constant value after the first cycle. It
should be noted that when the unloading rate exceeds 20 MPa/s,\(\varepsilon_{\text{un}}^{\text{an}}\) will change to be negative. It
is due to the method of classification and calculation, rather than the
change of the dislocation slip direction. According to the Eq. (6), if
the unloading rate increases during the unloading stage, the effective
stress will fall faster correspondingly, and the anelastic reverse slip
displacement is less within a shorter time of unloading and causing more
back stress being remained. Also, according to previous
work,49 the less internal friction measured by dynamic
mechanical Analysis (DMA) was observed on 2.25Cr-1Mo steel in the case
of higher frequency, hence the friction coefficient f would
reduce with the increase of unloading rate and cause the high resulted
anelastic slip rate corresponding to the valley stress.
4.3.3
Anelastic
recovery strain near valley stress \(\varepsilon_{v}^{\text{an}}\)
The anelastic recovery strain near valley stress\(\varepsilon_{v}^{\text{an}}\) represents the further reverse slip
displacement of pinned dislocation after the unloading stage, which
corresponds to the range between the valley stress and the lower-elastic
limit. Fig. 11 shows the initial part of the reloading stress-strain
curves of the 2nd cycle under different unloading
conditions. The starting points of all curves are shifted and
superimposed at the valley stress point.
In the case with none duration of valley stress, as shown in Fig. 11
(a), the anelastic recovery strain \(\varepsilon_{v}^{\text{an}}\)increases significantly with the increase of unloading rate, which is
opposite to the trend of \(\varepsilon_{\text{un}}^{\text{an}}\). It can
also be explained by Eq. (6). According to discussions in 4.3.2, the
larger the unloading rate, the more back stress is retained after
unloading stage, when the anelastic reverse slip rate is high as well.
Therefore, the dislocation can be derived backward for a more prolonged
displacement. Moreover, the lower-elastic limit is the critical stress
of the recovery behaviour, and it is also found to grow slightly with
the increase in the unloading rate. For comparison in Fig. 11 (b), where
the unloading rates are constant, the anelastic recovery strain develops
with extended durations under valley stress. However, the increment
extent of the anelastic strain decreases when the duration of valley
stress is more than 10 min, which validates that the anelastic strain
can be saturated with time as have been reported in other
studies.18,50 Fig. 10 (c) summarizes the anelastic
recovery strain near the valley stress \(\varepsilon_{v}^{\text{an}}\)under different unloading conditions, which shows two stages, and the
values all maintain constant during most of the life. Besides, it’s
interesting to find that the \(\varepsilon_{v}^{\text{an}}\) for quite a
long duration of valley stress (\({\dot{\sigma}}^{{}^{\prime}}\)=12 MPa/s,tv =30 min) is closely equal to the instantaneous\(\varepsilon_{v}^{\text{an}}\) under large unloading rate
(\({\dot{\sigma}}^{{}^{\prime}}\)=39 MPa/s, tv =0 min). This
indicates the performance of dislocation slip displacement, which mainly
depends on the driving force and the slip time.