A. Flow behavior

The typical true stress–true strain curves of the alloy obtained at 1060 and 1160 °C are presented in Fig. 2. Typically, the flow stress curves exhibit similar features, i.e., an obvious work-hardening stage at the strain less than 0.2, followed by a flow softening stage and then sometimes a steady stage at high strain zone. Such flow behaviors are consistent with the previous results for nickel-based superalloys with low stacking fault energy, indicating the occurrence of DRX during hot deformation.^{Reference Guo, Li and Guo18}

FIG. 2.

True stress–true strain curves of the alloy obtained at different deformation temperatures: (a) 1060 °C and (b) 1160 °C.

In the work-hardening stage, the dislocation density continually increases and dislocations rapidly accumulate, resulting in the rapid increase of flow stress.^{Reference Cai, Xiong, Liu, Sun and Yao19} In the softening stage, the effect of work hardening can be partially neutralized by the occurrence of dynamic softening mechanisms, such as DRV and DRX,^{Reference Li and Wang20} which can lead to the decrease of flow stress. In the steady stage, the flow stress is constant due to the dynamic balance between the work hardening and dynamic softening.

B. Effect of strain on dynamic recrystallization

Figure 3 shows the microstructure evolution of the alloy deformed to various true strains ranging from 0.1 to 0.9 at 1160 °C with a strain rate of 0.1 s⁻¹. It can be found that at a strain of 0.1, the grain boundaries become serrated and bulged, indicating the initiation of DRX owing to the strain-induced boundary migration,^{Reference Rollett, Humphreys, Rohrer and Hatherly21} as shown in Fig. 3(a). Moreover, almost no newly formed DRX grains can be observed at the primary grain boundaries and the triple junctions, indicating that the thermal energy and strain at this stage are not sufficient to promote DRX. However, once the deformation degree is larger than the critical strain, the nucleation and growth of DRX grains will take place near the grain boundaries, twin boundaries, as well as the deformation bands.^{Reference Chen, Lin, Wen, Zhang and He22} At a strain of 0.3, a lot of small DRX grains are formed along the deformed grain boundaries and form the necklace structures, as shown in Fig. 3(b). With increasing strain, the initial large grains gradually disappear and more DRX grains are formed, as shown in Figs. 3(c) and 3(d). As the strain increases to 0.7, almost all the original grains have been replaced by the DRX grains. At a strain of 0.9, the specimen has been fully recrystallized, as shown in Fig. 3(e). It is seen that the microstructure is composed of uniform and fine DRX grains, which is propitious to the properties of the products. Additionally, the DRX grain size remains almost constant with increasing strain, which is in good agreement with the previous reports based on the necklace mechanism.^{Reference Rollett, Humphreys, Rohrer and Hatherly21}

FIG. 3.

The DRX evolution of the alloy at 1160 °C and 0.1 s⁻¹ with various true strains: (a) 0.1, (b) 0.3, (c) 0.5, (d) 0.7, and (e) 0.9.

More detailed information about the nucleation mechanisms of DRX for the alloy during hot deformation can be obtained from the orientation imaging microscopy (OIM) maps, which utilizes rapid, direct measurements of local lattice orientation on a grid of points fixed upon flat surfaces sectioned from polycrystals.^{Reference Adams23}Figure 4 shows the OIM maps of the alloy deformed to true strains of 0.3 and 0.7 at 1160 °C with a strain rate of 0.1 s⁻¹, in which HAGBs and low angle boundaries (LAGBs) with misorientation angles lower than 10° are represented by black lines and gray lines, respectively. Furthermore, the grain boundaries with misorientation angles between 10° and 15° are represented by white lines, which are defined as medium angle boundaries (MAGBs) here. At a true strain of 0.3, the typical characteristics of CDRX can be observed, such as a large number of subgrains with LAGBs and some individual HAGBs in the interiors of grains. Meanwhile, the serrated and bulging grain boundaries can also be observed, along which lots of DRX grains with HAGBs are formed, suggesting the occurrence of DDRX. Jia et al.^{Reference Jia, Zhang and Lu24} have reported that when the dislocation density beyond the absorption capacity of grain boundaries, dislocations will pile up and a local stress concentration form, which results in the serrated grain boundaries. When the strain increases to 0.7, the amount of subgrains with LAGBs seems to be smaller, as shown in Fig. 4(b). Moreover, it is found that twins are formed in the deformed microstructure, as shown in Figs. 4(a) and 4(b) marked by arrows. As we know, twinning can play an important role in improving the process of DRX.^{Reference Gottstein25} The nucleation of twins is mainly caused by the higher mobility of grain boundary, which occurred through the dissociation of an initial migrating grain boundary into an immobile coherent twin part, mobile incoherent twin segment, and a low angle boundary.^{Reference Mahajan, Pande, Imam and Rath26} Furthermore, the twins can also accelerate the bulging and separation of bulged parts from original grains.^{Reference Wang, Shao, Zhen, Yang and Zhang27}

FIG. 4.

OIM maps of the alloy deformed to true strains of 0.3 and 0.7 at 1160 °C with a strain rate of 0.1 s⁻¹.

Figure 5 shows the relative frequency of different misorientation angles for the alloy deformed to different strains at 1160 °C with a strain rate of 0.1 s⁻¹. It can be found that the fraction of LAGBs is relatively low in the annealed specimen. However, at the early stage of deformation, the fraction of LAGBs increases sharply due to dislocation generation and the dissociation of the original HAGBs, most of which occur in the vicinity of original grain boundaries and twin boundaries.^{Reference Li, Guo, Guo, Peng and Wu28} With further deformation, the fraction of LAGBs decreases gradually by transforming into HAGBs to eliminate the stored energy. Moreover, the fraction of MAGBs increases from 0.9% to 1.4% with the strain increasing to 0.1, but does not change significantly at larger strain. As we know, the increase of MAGBs is believed to be necessary for the nucleation of CDRX.^{Reference Ponge and Gottstein29} Therefore, the effect of CDRX strengthens rapidly at the early stage of hot deformation, which can lead to the formation of nuclei in the subgrain structures.

FIG. 5.

Relative frequency of different misorientation angles for the alloy deformed to different strains at 1160 °C with a strain rate of 0.1 s⁻¹.

TEM observation was carried out to have a clearer insight into the mechanism of CDRX at the early stage of hot deformation. TEM micrographs of the alloy deformed to a true strain of 0.3 at 1160 °C and 0.1 s⁻¹ are shown in Fig. 6. A lot of dislocations near the grain boundary can be clearly seen in Fig. 6(a). Meanwhile, the piled up dislocations rearranged in the vicinity of grain boundaries and formed low angle subgrain boundaries, which can reduce the stress concentration. Subsequently, subgrain boundaries constantly absorb dislocations in the interior of grains, which can transform into HAGBs of new grains, as shown in Fig. 6(b). Therefore, the process of CDRX includes three stages: the accumulation and rearrangement of dislocations, the formation of subgrain boundary, and the conversion to HAGBs from subgrain boundary, which is consistent with the results of previous study.^{Reference Jia, Zhang and Lu24}

FIG. 6.

TEM micrographs of the alloy deformed to a true strain of 0.3 at 1160 °C and 0.1 s⁻¹.

C. Effect of deformation temperature on dynamic recrystallization

Figure 7 shows the microstructure evolution of the alloy deformed at different deformation temperatures with a strain rate of 0.001 s⁻¹ and a true strain of 0.7. Obviously, the microstructure evolution is closely related to deformation temperature. It can be found that the volume fraction of DRX grains increased with the increase of deformation temperature, indicating that the DRX behavior occurred easily at the higher deformation temperature. Such phenomenon is related to the lower driving force and higher mobility of grain boundaries at the higher deformation temperature.^{Reference Imbert and McQueen30} On the other hand, the condition for the occurrence of DRX is thought to be dependent on the distribution and density of dislocations. The increase of deformation temperature would lead to the decrease of critical dislocation density and thus lower the critical strain for the occurrence of DRX. Moreover, it can be found that the DRX grain size increases with increasing deformation temperature, which is mainly related to the higher mobility of DRX grain boundaries at the higher deformation temperature.^{Reference Li, Guo, Guo, Peng and Wu28} Meanwhile, the dissolution of γ′ phase can also lead to the grain growth at the higher deformation temperature. It should be mentioned that a high amount of γ′ phase begins to be dissolved at 1080 °C in the alloy.^{Reference Zhang, Zhang, Lu, Zhao and Yang2} As shown in Figs. 7(c) and 7(d), the grain growth is more obvious when the deformation temperature is above 1080 °C.

FIG. 7.

The DRX evolution of the alloy at a true strain of 0.7 and 0.001 s⁻¹ with various deformation temperatures: (a) 1010 °C, (b) 1060 °C, (c) 1110 °C, and (d) 1160 °C.

To investigate the effect of deformation temperature on the orientations of grains and subgrains, the specimens deformed at different deformation temperatures were analyzed by an EBSD technique. Figure 8 shows the relative frequency of different misorientation angles for the alloy deformed to a true strain of 0.7 at different deformation temperatures with a strain rate of 0.001 s⁻¹. It can be found that the fraction of HAGBs increases with increasing deformation temperature, and the fraction of LAGBs decreases with increasing deformation temperature. It is well known that MAGBs are considered as the transition zone from low angle to high angle boundaries.^{Reference Rollett, Humphreys, Rohrer and Hatherly21} As shown in Fig. 8, the fraction of MAGBs decreases gradually with increasing deformation temperature, indicating the rapider migration rate from low angle to high angle boundaries at the higher deformation temperature. Hence, the variations of HAGBs and LAGBs with deformation temperature are mainly caused by the rapider transformation from LAGBs to HAGBs at the higher deformation temperature. As mentioned above, the large fraction of LAGBs is necessary for the occurrence of CDRX. Hence, the lower fraction of LAGBs at the higher deformation temperature will weaken the effect of CDRX.

FIG. 8.

Relative frequency of different misorientation angles for the alloy deformed to a true strain of 0.7 at different deformation temperatures with a strain rate of 0.001 s⁻¹.

D. Effect of strain rate on dynamic recrystallization

Figure 9 shows the microstructure evolution of the alloy deformed at 1160 °C/1 s⁻¹ and 1160 °C/0.01 s⁻¹ with a true strain of 0.7. Comparing Figs. 3(d) and 7(d), it can be found that the DRX process is controlled by the strain rate. In other words, the volume fraction and size of DRX grains are strongly influenced by the strain rate. The volume fraction and size of DRX grains increase with decreasing strain rate. Such phenomenon is mainly because that there is enough time for energy accumulation to reduce the driving force under the lower strain rate, which can promote the DRX process. On the other hand, DRX is considered to be driven by the potential of distortion stored inside the alloys during deformation.^{Reference Guo, Li, Peng, Guo, Hu and Du31} Hence, there is enough time for the full diffusion of the alloy atoms under the lower strain rate, resulting in the release of sufficient potentials of distortion to stimulate the nucleation of DRX process.

FIG. 9.

The DRX evolution of the alloy at a true strain of 0.7 and 1160 °C with various strain rates: (a) 1 s⁻¹ and (b) 0.01 s⁻¹.

Figure 10 shows the relative frequency of different misorientation angles for the alloy deformed to a true strain of 0.7 at 1160 °C with different strain rates. It can be found that the fraction of LAGBs increases with the increasing strain rate, indicating that the dissociations of the original HAGBs and the generations of dislocations are more effective at the higher strain rates. Meanwhile, such a trend implies that the effect of CDRX is gradually strengthened with increasing strain rate. The fraction of MAGBs increases gradually with increasing strain rate, indicating that the lower strain rate is in favor of the transition from low angle to high angle boundaries.

FIG. 10.

Relative frequency of different misorientation angles for the alloy deformed to a true strain of 0.7 at 1160 °C with different strain rates.

In addition, the combined effects of deformation temperature and strain rate on the DRX grain size can be characterized by Zener–Hollomon parameter (Z), which can be expressed as,^{Reference Balasubrahmanyam and Prasad32}

(1)$$Z = \dot \varepsilon \exp ({Q \mathord{\left/ {\vphantom {Q {RT}}} \right. \kern-\nulldelimiterspace} {RT}})\quad {\rm{,}}$$

where $\dot \varepsilon$ is the strain rate (s⁻¹), R is the gas constant, T is the absolute temperature (K), and Q is the activation energy of the alloy during hot deformation (J/mol), which has been estimated as 427,626 J/mol in authors' previous publication.^{Reference Zhang, Zhang, Lu, Zhao and Yang2} The plot of the DRX grain size versus Z parameter is shown in Fig. 11, and their relationship can be represented as:

(2)$${d_{{\rm{DRX}}}} = 5.5 \times {10^7}{Z^{ - 0.46}}\quad {\rm{.}}$$

FIG. 11.

Relationship between lnd _DRX and lnZ.

In Eq. (2), d _DRX is the DRX grain size, which can be determined accurately by using the software of TSL OIM analysis.

E. Conventional DRX kinetics model

Usually, the DRX kinetics can be described by the following exponent-type equation,^{Reference Lin, Chen, Wen and Chen33}

(3)$${X_{{\rm{DRX}}}} = 1 - \exp \left[ { - {K_{\rm{d}}}{{\left( {{{\varepsilon - {\varepsilon _{\rm{c}}}} \over {{\varepsilon _{\rm{p}}}}}} \right)}^{{n_{\rm{d}}}}}} \right]{\rm{\ }}\left( {\varepsilon \ge {\varepsilon _{\rm{c}}}} \right)\quad {\rm{,}}$$

where X _DRX is the volume fraction of DRX grains. ε_c is the critical strain for the initiation of DRX, and ε_p is the peak strain. K _d and n _d are the material constants. There are two characteristic strains (ε_c and ε_p) and two material constants (K _d and n _d) to be determined in Eq. (3), which will be calculated based on the experimental stress–strain data in the following sections.

In the authors' previous publication,^{Reference Zhang, Zhang, Lu, Zhao and Yang2} DRV and DRX are the main dynamic softening mechanisms during hot deformation for the alloy, which can occur simultaneously. As shown in Fig. 12, the curve marked with ‘DRV’ assumes that DRV is the main softening mechanism and DRX is unavailable to take place. It can be found that the flow stress increases with increasing strain and finally reaches a saturation stress (σ_sat) due to the dynamic balance between the DRV and work hardening. Meanwhile, the curve marked with ‘DRX’ assumes that DRX is the dominant softening mechanism and DRV is weak. It can be seen that the flow stress drops gradually once the deformation degree exceeds the peak strain (ε_p) and finally reaches a steady stress (σ_ss) due to the dynamic balance between the working hardening and dynamic softening (including DRV and DRX). In addition, it should be noted that DRX will take place as the deformation degree exceeds the critical strain (ε_c).

FIG. 12.

Sketch map of flow stress–strain curves.

Laasraoui and Jonas^{Reference Laasraoui and Jonas34} pointed out that when DRX is the main softening mechanism during hot deformation, the flow stress can be expressed as:

(4)$$\sigma = {\sigma _{{\rm{rec}}}} - \left( {{\sigma _{{\rm{sat}}}} - {\sigma _{{\rm{ss}}}}} \right){X_{{\rm{DRX}}}}\left( {\varepsilon \ge {\varepsilon _{\rm{c}}}} \right)\quad {\rm{,}}$$

where σ_rec is the flow stress when DRV is the main softening mechanism during hot deformation. σ_sat and σ_ss are the saturation stress and steady stress, respectively. Hence, the expression of X _DRX can also be represented as:

(5)$${X_{{\rm{DRX}}}} = {{{\sigma _{{\rm{rec}}}} - \sigma } \over {{\sigma _{{\rm{sat}}}} - {\sigma _{{\rm{ss}}}}}}\left( {\varepsilon \ge {\varepsilon _{\rm{c}}}} \right)\quad {\rm{.}}$$

In Eq. (5), there are three characteristic stresses (σ_rec, σ_sat and σ_ss) to be determined. It is well known that the critical stress (σ_c), peak stress (σ_p), saturation stress (σ_sat), and steady stress (σ_ss) can be obtained from the work hardening rate (dθ = dσ/dε) versus the flow stress (σ) curves, which can also reflect the microstructure evolution during hot deformation.^{Reference Lin, Chen, Wen and Chen33} The θ–σ curve of the alloy at 1160 °C and 0.1 s⁻¹ is plotted based on the flow stress–strain data, as shown in Fig. 13. Obviously, there are three stages in the θ–σ curve, i.e., two approximately linear stages (I and II) and one curvilinear stage (III). The value of θ decreases linearly at a high speed in stage I, which corresponds to the deformation stage from the beginning of hot deformation to the onset of subgrain formation. The subgrain formation can lower the increment speed of softening rate induced by DRV,^{Reference Chen, Lin and Ma35} which lead to the decreasing slope of the θ–σ curve. Hence, the θ–σ curve enters the second approximately linear stage. In stage II, the value of θ linearly decreases at a lower speed toward an inflection point of the curve, which is known as the critical stress (σ_c) for the onset of DRX. In other words, stage II corresponds to the deformation stage from the onset of subgrain formation to the onset of DRX. In addition, stage III corresponds to the deformation stage from the onset of DRX to the end of deformation. In this stage, the value of θ drops rapidly to zero when the flow stress increases to the peak stress (σ_p), and then the value of θ decreases to negative owing to the softening effect of DRX. As a new dynamic balance between the work hardening and dynamic softening induced by DRX and DRV is reached, the value of θ will increase to zero again and the flow stress reaches to the steady stress (σ_ss). Moreover, supposing DRX does not occur in the alloy, the curve will go along the red dotted line. The value of θ would reduce linearly to zero and the flow stress gradually increases to the saturation stress (σ_sat).

FIG. 13.

Relationship between hardening rate and flow stress under the deformation temperature of 1160 °C and strain rate of 0.1 s⁻¹.

In addition, σ_rec can be calculated by the following equation^{Reference Liu, Li and Luo36}:

(6)$${\sigma _{{\rm{rec}}}} = {\left[ {\mathop {{\sigma _{{\rm{sat}}}}}\nolimits^2 + \left( {\mathop {{\sigma _0}}\nolimits^2 - \mathop {{\sigma _{{\rm{sat}}}}}\nolimits^2 } \right){{\rm{e}}^{ - \Omega \varepsilon }}} \right]^{0.5}}\quad {\rm{,}}$$

where σ₀ is the yield stress, and Ω is the coefficient of DRV. It should be mentioned that DRV is the main softening mechanism when the true strain is smaller than the critical strain.^{Reference Chen, Lin, Wen, Zhang and He22} Therefore, the value of Ω can be calculated on the basis of the flow stress–strain curve before critical strain. Taking the logarithm on both sides of Eq. (6), and it can be written as:

(7)$$\Omega \varepsilon = \ln \left( {{{\mathop {{\sigma _{{\rm{sat}}}}}\nolimits^2 - \mathop {{\sigma _0}}\nolimits^2 } \over {\mathop {{\sigma _{{\rm{sat}}}}}\nolimits^2 - \mathop {{\sigma _{{\rm{rec}}}}}\nolimits^2 }}} \right)\quad {\rm{.}}$$

Figure 14 shows the relationship between $\ln \left[ {{{\left( {\mathop {{\sigma _{{\rm{sat}}}}}\nolimits^2 - \mathop {{\sigma _0}}\nolimits^2 } \right)} \mathord{\left/ {\vphantom {{\left( {\mathop {{\sigma _{{\rm{sat}}}}}\nolimits^2 - \mathop {{\sigma _0}}\nolimits^2 } \right)} {\left( {\mathop {{\sigma _{{\rm{sat}}}}}\nolimits^2 - \mathop {{\sigma _{{\rm{rec}}}}}\nolimits^2 } \right)}}} \right {\left( {\mathop {{\sigma _{{\rm{sat}}}}}\nolimits^2 - \mathop {{\sigma _{{\rm{rec}}}}}\nolimits^2 } \right)}\right\big]$ and ε under a deformation temperature of 1160 °C and strain rate of 0.1 s⁻¹. The value of Ω can be determined by the linear regression method. Therefore, the value of σ_rec can be obtained by taking the value of Ω into Eq. (6). Finally, X _DRX can be calculated by taking the values of σ_rec, σ_sat, and σ_ss into Eq. (5).

FIG. 14.

Relationship between $\ln \left[ {{{\left( {\mathop {{\sigma _{{\rm{sat}}}}}\nolimits^2 - \mathop {{\sigma _0}}\nolimits^2 } \right)} \mathord{\left/ \kern-\nulldelimiterspace} {\left( {\mathop {{\sigma _{{\rm{sat}}}}}\nolimits^2 - \mathop {{\sigma _{{\rm{rec}}}}}\nolimits^2 } \right)}}} \right{\big]}$ and ε under the deformation temperature of 1160 °C and strain rate of 0.1 s⁻¹.

Based on the critical stress, peak stress, and the flow strain–stress curves, the values of ε_p and ε_c can also be obtained, which are listed in Table II. It can be found that the values of ε_p and ε_c are dependent on the deformation temperature and strain rate. In other words, the values of ε_p and ε_c decrease with the increase of deformation temperature or the decrease of strain rate, indicating that the DRX behavior occurs easier at the higher deformation temperature or the lower strain rate. Moreover, the combined effects of deformation temperature and strain rate on the critical strain and peak strain of the alloy can be characterized by the parameter Z. Figure 15 shows the relationship between lnε_p and lnZ for the alloy. Therefore, ε_p can be represented as a power function of the parameter Z,

(8)$${\varepsilon _{\rm{p}}} = 0.00453{Z^{0.09518}}\quad {\rm{.}}$$

TABLE II.

Values of peak strain and critical strain.

FIG. 15.

Relationship between lnε_p and lnZ.

In addition, the relationship between ε_c and ε_p can be usually represented as^{Reference Chen, Lin and Ma35}:

(9)$${\varepsilon _{\rm{c}}} = k{\varepsilon _{\rm{p}}}\quad {\rm{,}}$$

where k is the constant between 0.6 and 0.85. Figure 16 shows the relationship between ε_c and ε_p for the alloy. The value of k can be determined as 0.6 by the linear regression method. Hence, the relationship between ε_c and ε_p of the alloy can be expressed as:

(10)$${\varepsilon _{\rm{c}}} = 0.6{\varepsilon _{\rm{p}}}\quad {\rm{.}}$$

FIG. 16.

Relationship between ε_c and ε_p.

Taking Eq. (10) into Eq. (8), and then Eq. (10) can be written as:

(11)$${\varepsilon _{\rm{c}}} = 0.00272{Z^{0.09518}}\quad {\rm{.}}$$

Taking logarithm on both sides of Eq. (3), which can be written as:

(12)$$\ln \left[ { - \ln \left( {1 - {X_{{\rm{DRX}}}}} \right)} \right] = \ln {K_{\rm{d}}} + {n_{\rm{d}}}\ln \left( {{{\varepsilon - {\varepsilon _{\rm{c}}}} \over {{\varepsilon _{\rm{p}}}}}} \right)\quad {\rm{.}}$$

Figure 17 shows the relationship between $\ln \left[ { - \ln \left( {1 - {X_{{\rm{DRX}}}}} \right)} \right]$ and $\ln \left( {{{\varepsilon - {\varepsilon _{\rm{c}}}} \over {{\varepsilon _{\rm{p}}}}}} \right)$ for the alloy under the deformation temperature of 1160 °C and strain rate of 0.1 s⁻¹. The values of n _d and K _d can be determined by the linear regression method. By applying the above method, the values of n _d and K _d under different conditions are determined and listed in Table III.

FIG. 17.

Relationship between $\ln \left[ { - \ln \left( {1 - {X_{{\rm{DRX}}}}} \right)} \right]$ and $\ln \left( {{{\varepsilon - {\varepsilon _{\rm{c}}}} \over {{\varepsilon _{\rm{p}}}}}} \right)$ under the deformation temperature of 1160 °C and strain rate of 0.1 s⁻¹.

TABLE III.

Values of K _d and n _d.

It is notable that the values of K _d are dependent on the deformation temperature and strain rate. The relationship between K _d and the parameter Z is illustrated in Fig. 18. K _d can be represented as a power function of the parameter Z,

(13)$${K_{\rm{d}}} = 0.00931{Z^{0.09784}}\quad {\rm{.}}$$

FIG. 18.

Relationship between lnK _d and lnZ.

On the other hand, from Table III it can be found that the values of n _d are mainly dependent on the deformation temperature. At 1010 °C, the values of n _d are calculated in the range of 1.077–1.135, and the average value is 1.105. Similarly, the average values of n _d at other deformation temperatures can be calculated. Figure 19 shows the relationship between the average values of n _d and the deformation temperature. Hence, n _d can be represented as a power function of deformation temperature,

(14)$${n_{\rm{d}}} = 1.76663 \times {10^{ - 14}}{T^{4.43774}}\quad {\rm{.}}$$

FIG. 19.

Relationship between lnn _d and lnT.

Summarily, the DRX kinetics equations (ε > ε_c) of the studied alloy can be expressed as,

(15)$$\eqalignb{{X_{{\rm{DRX}}}} = 1 - \exp \left[ { - {K_{\rm{d}}}{{\left( {{{\varepsilon - {\varepsilon _{\rm{c}}}} \over {{\varepsilon _{\rm{p}}}}}} \right)}^{{n_{\rm{d}}}}}} \right]\left( {\varepsilon \ge {\varepsilon _{\rm{c}}}} \right) \cr{\varepsilon _{\rm{p}}} = 0.00453{Z^{0.09518}},\;{\varepsilon _{\rm{c}}} = 0.00272{Z^{0.09518}} \cr{K_{\rm{d}}} = 0.00931{Z^{0.09784}} \cr{n_{\rm{d}}} = 1.76663 \times {10^{ - 14}}{T^{4.43774}} \crZ = \dot \varepsilon \exp ({{427,626} \mathord{\left/ {\vphantom {{427,626} {RT}}} \right. \kern-\nulldelimiterspace} {RT}})\quad {\rm{.}} \cr}$$

By applying the above DRX kinetics equations, the kinetics models of DRX at different deformation conditions can be obtained. Figure 20 shows the curves of DRX kinetics under different deformation temperatures and strain rates. It can be found that DRX did not occur when the strain was less than the critical strain. Moreover, the volume fraction of DRX grains increases with increasing strain and the DRX evolution exhibits “slow–rapid–slow” growth trend in terms of S-curves. At a given strain, the volume fraction of DRX grains is larger at the higher deformation temperature or lower strain rate. Meanwhile, Chen et al.^{Reference Chen, Lin, Wen, Zhang and He22} reported a similar DRX evolution for a typical nickel-based superalloy, which was also strengthened by the γ′ phase in the face centered cubic matrix (γ phase). However, the strengthening mechanism of the alloy is also ascribed to the γ″ phase, which is usually unstable and it can be transformed into the δ phase.

FIG. 20.

The curves of DRX kinetics under different conditions, (a) 1160 °C with different strain rates and (b) 0.001 s⁻¹ with different deformation temperatures.

The comparison between the proposed DRX kinetics model and the statistical data from the microstructure examination at 1160 °C and 0.1 s⁻¹ is shown in Fig. 21. It can be found that the theoretical results of DRX volume fraction correlate well with experimental results, which indicates that the proposed kinetics model can give an accurate estimate of DRX behavior for Ni–18.3Cr–6.4Co–5.9W–4Mo–2.19Al–1.16Ti alloy during hot deformation.

FIG. 21.

The comparison between the predicted and experimental data.