3.1 Flow stressstrain curves
Figure 2 and Fig. 3 are the flow stressstrain curves obtained from hot compression testing for all conditions of deformation for the VG and CE steels. As seen, both steels exhibit similar flow stress behaviour and are sensitive to the deformation temperature and strain rate. At a constant strain rate, the flow stresses decrease with an increase in the temperature of deformation. This is attributed to the increase in deformation temperature which causes an increase in the rate of vacancy diffusion, climb of edge dislocations, and crossslip of screw dislocations [26–28]. In the early stages (at strains < 0.2), work hardening prevails the deformation mechanisms causing a rapid increase in the dislocation density. The high dislocation density resists the deformation process, causing an increase in the flow stress. As the deformation progresses (to strains > 0.2), the flow stress curves display limited work hardening due to the increased softening effect. The decrease in flow stress progresses until the effect of work hardening and dynamic softening reaches an equilibrium. At this stage, the flow stress curves show steadystate flow stress where there is a relatively constant dislocation density. This characteristic flow behaviour suggests that DRV is the main softening mechanism [28].
Figure 2 Flow stressstrain curves for VG steel at different strain rates and temperatures
At constant temperature, the flow stress increases as the strain rate increases. Deformation at a higher strain rate causes a higher rate of dislocation multiplication, resulting in the dislocationstrengthening effect. This phenomenon causes higher resistance to deformation due to work hardening, which controls the deformation process at a high strain rate. This, therefore, causes deformation difficulties resulting in higher flow stresses [29]. During deformation at low strain rates, there is more time for dynamic recovery to occur, hence weakening the effect of work hardening. This relationship is observed in both steels at all temperatures and illustrated by Fig. 2d) for VG and Fig. 3d) at 1050°C.
Flow stressstrain curves reflect the effect of deformation conditions on the microstructure. Microstructural changes are related closely to the deformation mechanisms controlling the metal flow behaviour during hot forming [30]. Work hardening (WH) and flow softening mechanisms, such as dynamic recrystallization (DRX) and DRV, are reliant on the parameters during forming. The main flow softening mechanism of creepresistant steels such as X20 with high stacking fault energy is DRV. During deformation, these steels easily rearrange into polygonal subgrain structures by dislocation climb and crossslip, resulting in DRV[31]. The flow stress curves in Fig. 2 for VG and Fig. 3 for CE exhibited similar flow behaviour. It was observed that at strains below 0.2, WH dominated the deformation, and above 0.2, the curves exhibited DRV softening mechanisms until the end of the deformation process. This result implies that the hot working of the rejuvenated heat treatment CE steels within the temperature range of 950–1100°C, would allow the recycling of the steel by hot deformation into new components with comparable flow characteristics of the virgin steel.
3.2 Flow stress curve modelling
3.2.1. Constitutive modelling
Constitutive equations are applied widely in the analysis of metal flow behaviour for various metals and alloys during deformation [33–36]. These equations form a basis for developing models that are used to predict flow stress behaviour at all deformation conditions [36]. For example, the Arrhenius equation is widely applied to describe the relationship between flow stress (σ), strain rate (έ), and temperature (T) [38–41].
$$\dot{{\epsilon }}= \text{A}\text{f}\left({\sigma }\right)\text{exp}\left(\frac{\text{Q}}{\text{R}\text{T}}\right)$$ (1)
$$\text{f}\left({\sigma }\right)=\left\{\begin{array}{c}A{{\sigma }}^{{\text{n}}^{{\prime }}} for low flow stress \alpha \sigma < 0.8\\ \text{A exp}\left({{\beta }}^{{\prime }}{\sigma }\right) for high flow stress \alpha \sigma > 1.2\\ A{\left[\text{s}\text{i}\text{n}\text{h}\left({\alpha }{\sigma }\right)\right]}^{\text{n}} for all stress \sigma \end{array}\right\}$$ (2)
From Equations \(\left(1\right)\) and \(\left(2\right)\), Q is the activation energy for deformation in kJ/mol, and R is the universal gas constant [8.3145 J/(mol. K)]. A, β, n’, σ, and n are material constants. α is the stress multiplier. The constitutive constants and activation energy are derived using the general case of the Arrhenius equation [40]. Substituting f (σ) values into Equations \(\left(1\right)\) and taking natural logarithms on either side gives:
$$\text{ln}\dot{{\epsilon }}=\text{ln}\text{A}\frac{\text{Q}}{\text{R}\text{T}}+{\text{n}}^{{\prime }}\text{ln}{\sigma }$$ (3)
$$\text{ln}\dot{{\epsilon }}=\text{ln}\text{A}\frac{\text{Q}}{\text{R}\text{T}}+{\beta }{\sigma }$$ (4)
$$\text{ln}\dot{{\epsilon }}=\text{ln}\text{A}\frac{\text{Q}}{\text{R}\text{T}}+\text{nln}\text{S}\text{i}\text{n}\text{h} \left({\alpha }{\sigma }\right)$$ (5)
Material constants were obtained using the saturation stress (Qsat) in the Arrhenius type equation [41]. The saturation flow stresses were obtained directly from the stress vs strain curves at the steadystate flow stress value [34, 44]. The values of n’ and β were obtained from slopes of plots of (ln (έ) against ln σ), and (ln (έ) against σ) as presented in Fig. 4a) – b) (VG) and Fig. 5a) – b) (CE). The stress multiplier is obtained from α = β / n [43]. n (∂ ln έ against ∂ ln (sinh (ασ)) and S (∂ ln (sinh (ασ) against ∂ (1/T)) are obtained from average slope values of graphical plots of ln έ against ln (sinh (ασ)) and 1000/T against ln (sinh (ασ)), as shown in Fig. 4 (c  d) for VG and Fig. 5 (c  d) for CE. By differentiating Equation \(\left(5\right)\), the activation energy is obtained as follows [44]:
\(\text{Q}=\text{R}\text{n}\text{S}=\text{R}{{\left[\frac{\partial \text{ln}\dot{{\epsilon }}}{\partial \text{ln}\left[\text{s}\text{i}\text{n}\text{h}\left({\alpha }{\sigma }\right)\right]}\right]}_{\text{T}}\left[\frac{\partial \text{ln}\left[\text{s}\text{i}\text{n}\text{h}\left({\sigma }{\sigma }\right)\right]}{\partial \left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$\text{T}$}\right.\right)}\right]}_{{\epsilon }}\) 6
3.2.2. ZenerHollomon parameter analysis
The effects of strain rate and temperature on hot deformation behaviour relate well with Zener Hollomon parameter Z [41, 45]. This parameter characterises the influence of deformation conditions (strain rate and temperature) on the workability of the metals and alloys. The Zparameter is as Equation (\(7\)) [46].
\(\text{Z}= \dot{{\epsilon }}\text{exp}\left(\frac{\text{Q}}{\text{R}\text{T}}\right)=\text{A} {\left[\text{S}\text{i}\text{n}\text{h}\left({\alpha }{\sigma }\right)\right]}^{\text{n}}\) (7)
Taking logarithms on either side of Equation (\(7\)), the equation may be written as follows.
\(\text{ln}\text{Z}=\text{ln}\text{A}+\text{n} \text{l}\text{n}[\text{s}\text{i}\text{n}\text{h}({\alpha }{\sigma }\left)\right]\) (8)
By substituting values of Q into Equation (\(7\)), the value of Z can be obtained. The linear relationship between ln Z against ln [sinh (αα)] is shown in Fig. 6a) for VG and b) for CE. The correlation index, R2 for VG (0.98) and CE (0.99) shows a highly linear relationship between ln Z and ln [sinh (ασ)] in both steels. The high R2 values for the test conditions suggested that a single deformation mechanism controlled the process. ln A was obtained as the intercept of the plot of ln Z vs ln [sinh (αα)]. Table 2 is a summary of constitutive constants and activation energies of the VG and CE steels.
Table 2
Constitutive constants and activation energy for VG and CE steels.
X20 Steel

Ή

β

η

a (MPa− 1)

S

Q (kJmol− 1)

A

VG

8.843

0.045

6.62

0.005

6.910

380.36

5.099 x 1015

CE

7.787

0.046

5.79

0.006

9.047

435.70

2.977 x 1017

3.2.3. Material constants for VG and CE steels
Table 2. shows the constitutive constants derived from modelled equations for the VG and CE steels. The stress exponent, n, and activation energy for hot working, Q are important parameters as they provide information on plastic deformation and underlying mechanisms controlling the process. The stress exponents of the two steels were analysed and the stress exponents obtained were VG (6.6), and CE (5.8). Studies by various authors [46, 50] have shown that a stress exponent value below 5 indicates the occurrence of DRX during hot deformation. Other reports [49–51] have established that when the stress exponent is 5, the deformation process is controlled by the climb and glide dislocation mechanisms. Studies by Mirzadeh et al. [40] have suggested that in flow stress curves exhibiting DRV softening, dislocation climb and glide are the main controlling mechanisms. It is observed in Fig. 2 for VG and Fig. 3 for CE that the flow stress curves exhibit typical DRV flow softening behaviour. There are no peaks displayed followed by steadystate flow stress that is associated with DRX. It was therefore determined that the main deformation mechanism in the two steels was dislocation glide controlled by dislocation climb. A comparison of the stress exponents for the two steels showed a slight variation between the values for the VG and CE steels. The variation in the stress exponent values can be attributed to the differences in precipitation evolution during deformation for the two steels. Generally, the presence of carbides impedes motion of dislocations causing an increase in the stress exponent value [30]. Deformation at higher temperatures causes increased carbide dissolution hence, deformation occurring at the austenitic field may result in lower stress exponent values [49].
Apart from the stress exponent, the apparent activation energy for hot working is an indicator of mechanisms that control the deformation process. The activation energy obtained using Arrhenius equations does not represent the microstructural changes occurring during deformation [50]. It is therefore called ‘apparent activation energy’ as the equations assume the microstructure remains constant during deformation [50, 51]. The apparent activation energies obtained in this study were VG (380.36 kJmol− 1) and CE (435.7 kJmol− 1). These values differ greatly from the selfdiffusion activation energy (QSD) of iron in austenite (270 kJmol− 1) and for αFe in ferrite (250 kJ.mol− 1) [30].
Wang et al. [53] described the deviation of activation energy from the imagined atomic mechanisms in terms of variation of temperature with Young’s modulus [51]. McQueen and Ryan [54] explained that during hot working, the initial microstructure differs to some degree for different preheats and evolves quickly when testing. Other researchers have suggested that microalloying elements such as Cr and Mo play an important role as they increase the activation energy by solute drag effect and solid solution strengthening mechanisms [30, 53]. Further, the presence of precipitates, solutes, dispersoids, reinforcements, and inclusions also increases the activation energy by 50% [54].
As stated earlier, the activation energy is an indicator of the workability properties of a material during forming. It defines the level of resistance or difficulty to deformation a material poses to a hot working process. Comparisons of the activation energies of the two steels show, that VG steel was lower than CE. From the hot workability viewpoint, it suggests that VG steel had better workability compared to CE steel. The variation in activation energy between VG and CE may be ascribed to the variation in phase compositions during the deformation of the two steels.
The Phase compositions and precipitates influence the deformation process and therefore material constants. During hot deformation, material constants vary significantly with process parameters and the deformation temperature. The deformation temperature, in turn, influences the evolutions in microstructure and the resulting material flow stress. Flow stress directly affects the activation energy during hot deformation as flow stress increases with the activation energy for deformation [30].
The results of the apparent activation energies for the two steels indicate the energy required for initiating and maintaining the motion of dislocations within the crystals of the rejuvenation heat treatment CE steel was higher than that of the VG steel. It is concluded that the rejuvenation effect increased the materials' hot deformation resistance compared to that of the VG steel. The heat treatment CE steel would therefore have suitable application as recycled steel in an environment where resistance to plastic deformation at elevated temperatures would be desirable.
3.2.4 Constitutive models of the flow stresses
Using the saturated stresses obtained from stressstrain curves in the hyperbolic sine law; constitutive equations for the two steels are given as follows:
VG \(\dot{{\epsilon }}=5.0999\times {10}^{15}{\left[\text{s}\text{i}\text{n}\text{h}\left(0.00503{{\sigma }}_{\text{s}\text{s}}\right)\right]}^{6.62}\text{e}\text{x}\text{p}\left[\frac{380.361}{\text{R}\text{T}}\right]\) (9)
CE \(\dot{{\epsilon }}=2.9774\times {10}^{17}{\left[\text{s}\text{i}\text{n}\text{h}\left(0.00601{{\sigma }}_{\text{s}\text{s}}\right)\right]}^{5.79}\text{e}\text{x}\text{p}\left[\frac{435.699}{\text{R}\text{T}}\right]\) (10)
The material constants derived using the Arrhenius equations (Section 3.2.1) can model flow behaviour for the two steels using the Z parameter. From Equation (\(7\)), the solution of this equation gives Equation (\(11\)) [46] as:
\({\sigma }=\frac{1}{{\alpha }}\text{l}\text{n}\left\{{\left(\frac{\text{Z}}{\text{A}}\right)}^{1/\text{n}}+{\left[{\left(\frac{\text{Z}}{\text{A}}\right)}^{2/\text{n}}+1\right]}^{1/2}\right\}\) (11)
From Equation (\(11\)) the saturation flow stresses of VG and CE can be determined as follows:
VG \({\sigma }=\frac{1}{0.005}\text{l}\text{n}\left\{{\left(\frac{\text{Z}}{5.099\times {10}^{15}}\right)}^{\frac{1}{6.62}}+{\left[{\left(\frac{\text{Z}}{5.099\times {10}^{15}}\right)}^{\frac{2}{6.62}}+1\right]}^{1/2}\right\}\) (12)
CE \({\sigma }=\frac{1}{0.006}\text{l}\text{n}\left\{{\left(\frac{\text{Z}}{2.977\times {10}^{17}}\right)}^{\frac{1}{5.79}}+{\left[{\left(\frac{\text{Z}}{2.977\times {10}^{17}}\right)}^{\frac{2}{5.79}}+1\right]}^{1/2}\right\}\) (13)
3.2.5. Model validation
Verification of Equations (\(12\)) and (\(13\)) for the two steels (VG and CE) was done by comparison of flow stresses of the experimental and predicted data. The standard statistical parameters used to analyse the predictability of the equations were Pearson’s correlation coefficient (Equation\(\left(14\right)\)) and Average Absolute Relative Error (AARE) (Equation\((15\))). Figure 7 ((a) and (b)) are plots of the experimental against predicted flow stresses obtained using Equations (\(12\)) and (\(13\)). The plots had a good correlation coefficient of 0.98 for the VG steel and 0.99 for the CE steel, where a value of 1.00 is a perfect correlation. The correlation coefficient method is however liable to bias towards high or low values therefore affecting the models' accuracy. The AARE statistical analysis method complements Pearson’s correlation coefficient as it gives good accuracy and quantifies prediction deviation. In the AARE method, the calculation is through a termbyterm comparison of the relative error in the analysis of the developed equations. A lower AARE value signifies a higher accuracy of the constitutive model to predict the flow stresses and vice versa. The AARE values obtained were: VG (4.17%) and CE (9.01%). The AARE for VG indicated excellent predictability of the constitutive equation than CE. From the calculated parameters, the Arrhenius constitutive equation developed for the two steel has a high prediction accuracy of flow stress for all test conditions.
$$\text{R}=\frac{{\sum }_{\text{i}=1}^{\text{N}}\left({\text{P}}_{\text{i}}\text{P}\right)\left({\text{E}}_{\text{i}}\text{E}\right)}{\sqrt{{\sum }_{\text{i}=1}^{\text{N}}{\left({\text{P}}_{\text{i}}\text{P}\right)}^{2}}\sqrt{{\sum }_{\text{i}=1}^{\text{N}}{\left({\text{E}}_{\text{i}}\text{E}\right)}^{2}}}$$ (14)
Where Ei is the experimental flow stress, Pi is the predicted flow stress obtained from constitutive equations, E and P, are mean flow stress values of experimental data E, P is the predicted data, and N is sum of the data points.