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:

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.

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 2^nd 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,t_v =30 min) is closely equal to the instantaneous\(\varepsilon_{v}^{\text{an}}\) under large unloading rate (\({\dot{\sigma}}^{{}^{\prime}}\)=39 MPa/s, t_v =0 min). This indicates the performance of dislocation slip displacement, which mainly depends on the driving force and the slip time.