Physics-Based Constitutive Model to Predict Dynamic Recovery Behavior of BFe10-1-2 Cupronickel Alloy during Hot Working

Stress–strain characteristics of BFe10-1-2 cupronickel alloy during hot deformation

The true stress–strain curves obtained from the hot compression tests of BFe10-1-2 cupronickel alloy are shown in Figure 2. Although some noise is shown in the curves, it can be seen from figures that the influence of deformation temperature and strain rate on flow stress is significant. From these figures, it can be seen that the flow stress increases with decrease of deformation temperature and increase of strain rate.

Figure 2:

Flow curves of BFe10-1-2 alloy at various strain rates with temperatures of: (a) 1,023 K, (b) 1,073 K, (c) 1,123 K, (d) 1,173 K, (e) 1,223 K, (f) 1,273 K.

The variation of flow stress with the deformation temperature and the strain rate can be expressed quantitatively, and the relationship between deformation temperature, strain rate and dislocation density can be represented as the following formula [1]:

where ρm=fρ is the mobile dislocation density, f is about 0.1 representing the proportion of the mobile dislocation density in the total dislocation density ρ. A is a materials constant, ΔG∗ is the heat activity energy, and k is the Boltzmann constant. Meanwhile, the relationship between the flow stress and the dislocation density can be expressed as:

where σ∗ is the intrinsic strength of materials, b is the Burgers vector, α is the materials constant, and μ is the shear modulus. It can be found from eqs (1) and (2) that the dislocation density decreases with increasing temperatures at a given strain rate, and then the flow stress will increase with the decrease of deformation temperature. Similarly, the dislocation density increases with the increase of strain rates at a given temperature, therefore flow stress will increase with the increase of strain rates.

Meanwhile, the flow stress curves exhibit the DRV characteristics of the material without obvious peak stress. A typical DRV stress–strain curve is given in Figure 3. The flow stress rapidly increases due to WH. When the WH and DRV reach a balance, a saturation flow stress will appear and stay near constant, and the grains continue to elongate with the increase of deformation but the shape and size of subgrains remain unchanged.

Figure 3:

A schematic of true stress–strain curve characteristic of DRV behavior.

Figure 4 illustrates the metallographic microstructure of BFe10-1-2 alloy under various deformation conditions (i. e. 1,023 K, 0.001 s^–1; 1,123 K, 0.01 s^–1; 1,173 K, 1 s^–1 and 1,273 K, 10 s^–1), where the grains tend to be elongated perpendicular to the compression direction. It can be seen in Figure 2 that flow stresses drop near the strain of 0.6 when strain rate is 0.001 s^–1 and 0.01 s^–1 (such as 1,023 and 1,073 K). However, no obvious DRX grains can be detected in Figure 4(a) and (b). Similar phenomenon was reported by Park [17] in Ti-6Al-4V alloy that the flow softening at slow strain rates was attributed to the occurrence of dynamic spheroidization of the α grains or DRV rather than DRX. Meanwhile, it is difficult for DRV and DRX to occur simultaneously in hot deformation processes because the distortion energy is released by DRV and is accumulated with difficulty to the level for DRX [18]. The dislocation density, distortion energy and recrystallization locations increase with the increasing of strain rate, which leads to the fact that DRX occurs easily at high strain rates [19, 20]. Therefore, the dominant dynamic softening mechanism of BFe10-1-2 alloy is DRV.

Figure 4:

Metallographic microstructure of BFe10-1-2 alloy under the deformation condition of: (a) 1,023 K, 0.001 s^–1; (b) 1,123, 0.01 s^–1; (c) 1,173, 1 s^–1; (d) 1,273, 10 s^–1.

Activation energy for hot deformation

The flow stress is mainly influenced by temperature, strain rate and strain during hot working. Zener–Holloman parameter (Z), i. e. temperature modified strain rate, is often used to describe the effects of temperature and strain rate on material deformation behavior:

where ε˙ is the strain rate (s^–1), R is the universal gas constant (8.3145 J mol^–1 K^–1), T is the absolute temperature (K), and Q is the activation energy of hot deformation (J/mol). Then a hyperbolic sine equation can be applied predict the flow stress [21]:

where A, n and α are material constants, and α can be expressed as:

where n′ and β are the materials constants and can be obtained from following equations:

where B and C are the material constants, which are independent of the deformed temperatures. The values of n′ and β can be obtained from the slope of the lines in ln(σ)–ln(ε˙) plot and σ–ln(ε˙) plot, respectively. The relationships of ln(σ)–ln(ε˙) and σ–ln(ε˙) at the strain of 0.3 are illustrated in Figure 5. The lines are almost parallel indicating that the slope of the lines varies within a very small range. The slight variation in the slope of the lines may be attributed to scattering in the hot compression experimental data. Similar scattering was observed in 42CrMo steel [6], Ti-modified austenitic stainless steel [22], 1Cr12Ni3Mo2VNbN martensitic steel [23], and so on. Possibly, experiment error and physical characteristic lead to the data scattering. For example, a cubic spline function was often used to fit ln(σ)–ln(ε˙) for the purpose of obtaining processing maps [24, 25]. In order to reduce the error, the mean values of the slopes are taken as the values of n′ and β, and then the values of α can be calculated from eq. (5).

Figure 5:

Relationship between: (a) ln(σ) and ln(ε˙) (b) σ and ln(ε˙).

For a given strain rate, activation energy Q can be expressed as:

Then the value of Q can be derived from the slopes in the plot of ln[sinh(aσ)]–1/T, as illustrated in Figure 6. The value of Q can be obtained to be 425.3 KJ/mol by averaging the Q-values in the strain range of 0.1~0.8 and the interval of 0.1.

Figure 6:

Relationship between ln[sinh(ασ)] and 1/T.

Constitutive modeling of flow stress

The evolution of dislocation density is a result of the multiplication and annihilation of dislocations due to WH and DRV respectively and can be expressed as [4]:

where dρ/dε is the increasing rate of dislocation density with strain; U represents the WH (the dislocation storage during the deformation) and can be regarded as constant with respect to strain; Ωρ is the contribution of DRV due to dislocation annihilation and rearrangement, and Ω is the coefficient of DRV [17]. When plastic strain ε=0, dislocation density ρ=ρ0, where ρ0 is the initial dislocation density, and the corresponding flow stress is the initial yield stress σ0. Then the variations in the dislocation density during the hot deformation can be obtained by integrating eq. (9) as follows:

The contribution of the dislocation density to the flow stress can be expressed as:

where γ is Taylor constant, d is the distance between atoms in the slip direction, and G is the shear modulus. Then substituting eq. (11) into eq. (10), the flow stress during the WH and DRV can be represented as:

where σsat is the saturations tress. The combined effects of temperature and strain rate on feature parameters eq. (12), i. e. σsat, σ0 and Ω, can be characterized by the Z parameter (eq. (3)). It is usually assumed that influence of strain on flow stress is insignificant and thereby would not be considered in eq. (12). However, it can be seen from Figure 2 that some stress–strain curves show obvious WH under some deformation conditions (for example, at the temperature of 1,023 K and 1,073 K, strain rate of 1 s^–1), then no obvious σsat can be obtained from the flow stress curves. Therefore, WH should be taken into account in order to derive constitutive equations to predict the flow stress more accurately. The peak stress σpeak is introduced to replace σsat in eq. (12) for the purpose of improving precision of developed constitutive equation. Meanwhile, it can be found from Figure 3 that the value of σsat can also be regarded as that of σpeak in the typical DRV curve. Then eq. (12) can be rewritten as:

The values of σpeak under various deformation conditions can be easily obtained from the true stress–strain curves. Figure 7(a) illustrates a linear relation between σpeak and lnZ, then the values of σpeak can be expressed as:

Figure 7:

Relationship between σpeak and the lnZ: (a) linear fitting and (b) polynomial fitting.

It can be seen from eq. (13) that σpeak has considerable effects on the prediction accuracy of the developed constitutive model. Therefore, the correlation coefficient (R) and the average absolute relative error (AARE) between the predicted and measured values are employed to evaluate prediction capability of σpeak. The values of R and AARE are 0.9732 and 10.765 %, respectively. In order to further improve prediction accuracy, a third-order polynomial is used to fit the measured values of σpeak, as shown in Figure 7(b) and can be expressed as:

The values of R and AARE between the third-order polynomial predicted and measured σpeak are 0.9907 and 4.525 %, respectively. Therefore, the expression for the peak stress in the form of the third order polynomial greatly improves its prediction accuracy.

It is difficult to determine the accurate initial yield stress σ0 by the inflexion points in the actual stress–strain curves. Therefore, the initial yield stress σ0 at various deformation temperatures and strain rates are taken as the stress corresponding to a strain of 0.02 [1]. Figure 8 illustrates the relationship between the initial yield stress and lnZ, and a fourth-order polynomial is employed to fit the measured values. Then, the initial yield stress σ0 can be expressed as:

According to eq. (13), Ω can be calculated by the following formula:

Figure 8:

Relationship between the yield stress and the Zener–Hollomon parameter.

Using the true stress–strain data, the values of Ω can be determined for the whole deformation conditions. Figure 9 illustrates that a linear relation exists between lnΩ and lnZ. Meanwhile, it can be found from the figure that lnΩ decreases with the increase of lnZ, and Ω is expressed as a function of Z parameter:

Figure 9:

Relationship between Ω and the Zener–Hollomon parameter.

Therefore, the physics-based constitutive model of BFe10-1-2 cupronickel alloy during hot working can be summarized as:

Verification of the developed constitutive equations of BFe10-1-2 cupronickel alloy

In order to verify the developed physics-based constitutive equation, a comparison between the experimental and predicted flow stress data at different strains within the range of 0.15~0.75 and the interval of 0.1 (i. e. the testing data that were not used to fitting curves) was carried out in Figure 10. The predicted flow stress data from the constitutive equation could track the experimental data of BFe10-1-2 cupronickel alloy, and there is a good agreement between the experimental and predicted values under most deformation conditions. Only under some processing condition (i. e. at 1,023 K, 1,173 K in 10 s^–1 and 1,123 K, 1,273 K in 1 s^–1), a remarkable variation between experimental and computed flow stress data could be observed. The main reason of the variation may contribute to the fact that the response of flow behavior of mental materials at evaluated temperatures is highly nonlinear. Most factors affecting the flow stress are nonlinear, which make the accuracy of the predicted flow stress by the constitutive equations low and the applicable range limited [9]. Meanwhile, the fitting of material constants may lead to the variation between experimental and computed flow stress data. For example, experimental data in Figure 5(b) show some deviation. Therefore, some errors may be introduced because of the determination of the values of β and finally affect the accuracy of constitutive equation.

Figure 10:

Comparison between the experimental and predicted flow stress at the temperature of: (a) 1,023 K, (b) 1,073 K, (c) 1,123 K, (d) 1,173 K, (e) 1,223 K, (f) 1,273 K.

The predictability of the developed physics-based constitutive equation could be quantified in terms of standard statistical parameters such as R and AARE. These are expressed as following equations [26]:

where E is the experimental flow stress and P is the predicted flow stress obtained from the developed constitutive equation considering strain compensation. Eˉ and Pˉ are the mean values of E and P respectively. N is the total number of data used in this study.

R is a commonly employed statistical parameter and provides information on the strength of the linear relationship between the experimental and predicted data. And the AARE is calculated through a term by term comparison of the relative error and therefore is an unbiased statistical parameter for determining the predictability of the equation [27]. As can be seen from Figure 11, although there are some variation under some deformation conditions, the values of R and AARE are found to be 0.9833 and 6.726 % respectively, which indicate that the developed physics-based constitutive model gives an accurate and precise estimate of the DRV behavior of BFe10-1-2 cupronickel alloy.

Figure 11:

Correlation between the experimental and predicted flow stress data from the developed constitutive equation.