1. Introduction

2. Materials and Methods

Figure 3. The optical micrographs for a) steel A b) steel B.

Figure 3. The optical micrographs for a) steel A b) steel B.

3. Results

3.1. Flow stress-strain curves

Flow stress-strain curves provide information on metal deformation behaviour during forging. Flow stress-strain curves obtained from the uniaxial compression test are affected by the interfacial friction between the tool and the workpiece. Studies have shown that the interfacial friction at the sample/die interface modifies the force required during deformation [30]. At high strain, the friction effect was more significant, causing inhomogeneous deformation. Hence, barrelling of the sample occurs. Therefore, the flow stress-strain curves should be friction corrected before further analysis. [30], [31].

After the hot compression test, the barrelling coefficient was calculated using the sample geometry size. According to Roebuck et al. [32], Equation(1) provides an efficient way of determining the barrelling coefficient, b. The allowable values for barrelling should be within the range of 1 ˂ b ≤ 1.1 [33]. These barrelling range values are negligible [34]. Figure 4 shows plots of uncorrected and friction-corrected flow stress curves for the two P91 steels at a temperature of 950°C and strain rate of 0.01 s⁻¹, 0.1 s⁻¹, 1s⁻¹, and 10s⁻¹. The results show that the measured flow stress values were the highest. This flow stress-strain curve variation can be due to the interfacial friction effect experienced at the test sample and anvil interface.

$$b=\frac{h_f{d}_f^2}{h_o{d}_o^2}$$

(1)

Where, h_o and h_f are initial and final height of the sample and d_o and d_f are the initial and final diameter of the sample.

Figure 4. Plots of Uncorrected and Corrected flow stress against True strain.

Figure 4. Plots of Uncorrected and Corrected flow stress against True strain.

Figure 5 (steel A) and Figure 6 (steel B) shows friction-corrected flow stress-strain curve at various deformation conditions. The flow stress-strain curves show that the flow stress depends on the strain rate and deformation temperature. For a given strain rate, flow stresses decrease with an increase in the deformation temperature of the two steels investigated. These metal flow patterns may be due to the increase in mobility of grain boundaries as deformation temperature increases [35]. At a given temperature, flow stress increase with an increase in the strain rate. The flow stress-strain curves (Figure 5 and Figure 6) show that at the beginning of the deformation process, the flow stress increased rapidly up to a strain of ∼0.2. This characteristic flow behaviour occurs due to the rapid generation of dislocation density resulting in a work-hardening deformation mechanism. [36]. At higher strain (>0.2), most flow stress-strain curves for the two steels reached a steady-state condition. This characteristic behaviour of flow curves indicates a balance between work hardening and softening mechanisms, especially dynamic recovery (DRV for these steels [37]. The flow stress-strain curves for the two steels studied did not exhibit a clear peak and subsequent steady-state flow stress caused by DRX [33], [38]. Therefore, for most deformation conditions, the flow curves had increasing flow stress until the start of the saturation stress (σ_sat) region. The results show that DRV was the softening mechanism. Similar flow stress-strain curves behaviour results are available in the literature for creep resistant steels [39], [40].

3.2. Constitutive model and material constants

Constitutive models accurately predict the flow stress behaviour of metals and alloys during deformation [41]. For example, the Arrhenius equation is widely used to show the relationship between stress, temperature and strain rates during forming [23]. The effect of temperature and strain rate can also be expressed using the temperature-compensated strain rate, the Zener-Hollomon parameter (Z) [42].

$$\dot{\epsilon }=\mathit{Af}\left(\sigma \right)\mathit{exp}\left(\frac{-Q}{RT}\right)$$

(2)

$$Z=\dot{\epsilon }\mathit{exp}\left(\frac{Q}{RT}\right)$$

(3)

Where, σ is flow stress, έ is strain rate, T is absolute temperature (K) and Q is activation energy for hot deformation (kJ/mol), A is a material constant and R is the universal gas constant (8.314 Jmol⁻¹ K⁻¹).

$$Z=\dot{\epsilon }\mathit{exp}\left(\frac{Q}{RT}\right)=f\left(\sigma \right)=A{\sigma }^{n^{\prime }}$$

(4)

$$Z=\dot{\epsilon }\mathit{exp}\left(\frac{Q}{RT}\right)=f\left(\sigma \right)=A\mathit{exp}\left(\beta \sigma \right)$$

(5)

$$Z=\dot{\epsilon }\mathit{exp}\left(\frac{Q}{RT}\right)=f\left(\sigma \right)=A{\left[Sinh\left(\alpha \sigma \right)\right]}^n$$

(6)

The relationship between the Zener-Hollomon parameter and flow stress, is expressed by Equations (4) to (6). The terms in these equations: n’ β, n, and α, are material constants. Equation (4) is the power law applied for low stresses (ασ ˂ 0.8) as it breaks down at higher flow stress. Equation (5) is the exponential law used for higher stress (ασ > 1.2) and breaks down at strain rate under 1s⁻¹ and high temperature. Equation (6) is a hyperbolic sine function equation developed by Sellars and Tegart [43] to be applicable to a wide range of stresses (both high and low). Equations (4)-(6) can also give Equations (7) and (9):

$$\dot{\epsilon }=A{\sigma }^{n^{\prime }}\mathit{exp}\left(\frac{-Q}{RT}\right)$$

(7)

$$\dot{\epsilon }=A\mathit{exp}\left(\beta \sigma \right)\left(\frac{-Q}{RT}\right)$$

(8)

$$\dot{\epsilon }=A{\left[\mathit{sinh}\left(\mathit{\alpha \sigma }\right)\right]}^{n}exp\left(\frac{-Q}{RT}\right)$$

(9)

The constants n’ and β are determined by taking logarithms on both sides of Equations (7) and (8). These are presented as Equations (10) and (11). The plots of ln έ versus ln σ (Equation 10) and ln έ versus σ (Equation 11) are used to calculate the material constants n’ and β respectively. Figure 7 ((a) and (b)) is a representation of the plots based on flow stress data to a true strain of 0.6.

$$\ln \dot{\epsilon }=\ln A+n^{\prime }\ln \sigma -\frac{Q}{RT}$$

(10)

$$\ln \dot{\epsilon }=\ln A+\beta \sigma -\frac{Q}{RT}$$

(11)

The stress multiplier α is an adjustable constant. This α-value is obtained from: α = β/n. The stress exponent, n, and activation energy, Q, are determined by taking logarithms on both sides of Equation (9), which gives Equations (12) and (13). These equations are plotted as ln έ versus ln (sinh (σα)) and ln (sinh (σα)) versus (1000)/T, as shown in Figure 8 (a and b). The average value of the slopes obtained in these plots gives n and S values for calculating the activation energy Q.

$$\frac{1}{n}={\left[\frac{\partial \ln \left[sinh\left(\sigma \sigma \right)\right]}{\partial \left(ln\dot{\epsilon }\right)}\right]}_T$$

(12)

$$Q=RnS=Rn{\left[\frac{\partial \ln \left[sinh\left(\sigma \sigma \right)\right]}{\partial \left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$T$}\right.\right)}\right]}_{\epsilon }$$

(13)

The relationship between the flow stress and the Zener-Holloman parameter is, as given in Equation (6). The structure factor (A) is determined by taking logarithms on both sides of Equation (6), as in Equation (14). The A value is the intercept of the best fit line obtained by plotting ln (Z) versus ln (sinh(ασ)), as shown in Figure 9.

$$\ln Z=\ln A+n\ln Sinh\left(\alpha \sigma \right)$$

(14)

Figure 7. (a and b) Plots of ln (έ) versus ln σ and ln (έ) versus σ for determination of n’ steel A and steel B and (c and d) for determine of β in steel A and steel B.

Figure 7. (a and b) Plots of ln (έ) versus ln σ and ln (έ) versus σ for determination of n’ steel A and steel B and (c and d) for determine of β in steel A and steel B.

Figure 8. ((a) and (b)) plots of ln έ versus σ and ln (sinh (σα)) for determination of n and ((c) and (d)) are plots of ln (sinh (σα)) versus (1000)/T to determine S value for steel A and steel B respectively.

Figure 8. ((a) and (b)) plots of ln έ versus σ and ln (sinh (σα)) for determination of n and ((c) and (d)) are plots of ln (sinh (σα)) versus (1000)/T to determine S value for steel A and steel B respectively.

Figure 9. Plots of ln (Z) vs. ln (sinh(ασ)) for a) Steel A and b) for steel B.

Figure 9. Plots of ln (Z) vs. ln (sinh(ασ)) for a) Steel A and b) for steel B.

The calculated material constants for steel A and steel B for the constitutive equations are as given in Table 1.

3.5. Verification of constitutive models

The accuracy of constitutive equations (18) and (19) to predict flow stress was evaluated by comparing predicted to experimental data. Figures 10 a) and b) show graphical plots for the two equations obtained at different strain rates and temperatures. The graphs indicate a close correlation between the flow stresses (predicted and experimental) of these two steels investigated.

The accuracy of the developed equations to predict flow stress accurately was analysed using statistical parameters such as Pearson’s correlation coefficient, R and Average Absolute Relative Error (AARE). Pearson’s correlation coefficient, R, ranges between 0 and 1 and is determined using Equation (20) [54]. The proximity of R to 1 signifies a higher capability for the model to predict flow stress and vice versa. The R-value obtained for steel A was 0.97, and steel B was 0.98. The R-value obtained for both materials indicated a good correlation and a higher ability of the models to predict flow stress. These values compare well with the reports of other authors, such as Mwema et al.[55] and Samantaray et al. [22], with an R-value of 0.994.

Pearson’s correlation coefficient, R, is liable to bias towards greater or lower values, hence resulting in misrepresenting the accuracy of the model to predict flow stresses[55]. Using the Average Absolute Relative Error (AARE) complements Pearson’s correlation method and is less subject to bias[56]. AARE analyses errors for each predicted flow stress against experimental value on a case-by-case basis. Therefore, AARE is applied for each deformation condition to realise the statistical efficiency of the models developed. Therefore, AARE offers better reliability and presents unbiased statistical analysis[33]. Equation (21) provides means to calculate AARE as a percentage. A lower percentage value indicates a greater ability to predict flow stresses and vice versa. AARE obtained for the steel A was 7.62%, and for heat-treated creep-exhausted steel was 6.54%. Reports by other authors indicated marginally lower AARE findings for hot deformation of P91. Mwema et al. [55] obtained an AARE of 2.999%, and Samantaray et al. [22] had 5.27%. These variations are due to the differences in hot deformation parameters in the respective studies. The study by Mwema et al. [55] was for hot deformation temperature between 900 °C and 1200 °C and strain rate of 1s⁻¹, 5s⁻¹, 10s⁻¹ and 15s⁻¹. In Samantaray et al. [22], the temperature for hot deformation was between 850 °C and 1100 °C and strain rates of 0.001s⁻¹, 0.1s⁻¹ and 100s⁻¹. Validation by both models thus signified satisfactory levels of confidence in the respective constitutive equations to predict flow stresses.

$$R=\frac{{\sum }_{i=1}^N\left(P_i-P\right)\left(E_i-E\right)}{\sqrt{{\sum }_{i=1}^N{\left(P_i-P\right)}^2}\sqrt{{\sum }_{i=1}^N{\left(E_i-E\right)}^2}}$$

(20)

$$AARE=\frac{1}{N}\sum _{i=1}^N\left|\frac{E_i-P_i}{E_i}\right|\times 100\%$$

(21)

Where E_i is the flow stress from the experiment, Pi is the flow stress predicted from constitutive equations, E and P, are average flow stress values of experimental data E, and the predicted data P. N is the sum of data points.

Figure 10. Prediction of flow stress of mathematical models for a) Steel A and b) steel B.

Figure 10. Prediction of flow stress of mathematical models for a) Steel A and b) steel B.

4. Conclusion

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