Flow Softening-based Formation of Widmanstätten Ferrite in a 0.06%C Steel Deformed Above the Ae₃

Abstract

Compression tests were carried out at a strain rate of 1 s^–1 on a 0.06%C-0.3%Mn-0.01%Si steel over two temperature ranges: i) 920°C to 980°C, and ii) 500 to 750°C. Optical and scanning electron microscopy images indicated that significant volume fractions of Widmanstätten ferrite were formed dynamically above the Ae₃ temperature. The ferrite plates coalesced into polygonal grains during straining. The double differentiation method was applied to the stress-strain curves, providing average values for the dynamic transformation (DT) and dynamic recrystallization (DRX) critical strains of 0.12 and 0.20, respectively. These results are interpreted in terms of the flow softening-based transformation model by calculating both the driving forces promoting the transformation as well as the energy barriers that oppose it. The model predicts the temperature range over which DT can occur as well as the observed critical strains.

1. Introduction

Yada and co-workers showed in the 1980’s that austenite can be transformed into ferrite by deformation at temperatures above the Ae₃. By means of rolling simulations and compression tests, they demonstrated that such dynamic transformation (DT) can take place at temperatures as much as 166°C above the Ae₃.^1,2) By applying accumulated strains of as much as 3 to 4, they were able to produce ferrite volume fractions of 70% to 80% with grain sizes in the range 1 μm to 2 μm. In a follow-up study, they provided real time evidence for DT by carrying out in-situ X-ray diffraction experiments in which ferrite peaks appeared during hot deformation in the austenite region.³⁾ Further experimental evidence for DT was reported during the decade that followed,^4,5,6) accompanied by Monte Carlo simulations^7,8) in which it was shown that DT is generally expected to precede dynamic recrystallization (DRX) under finish rolling conditions.

This unusual phenomenon was initially justified in terms of the stored energy of the dislocations introduced into the deformed austenite. For example, Sun et al.⁹⁾ arrived at a free energy increase of 22.3 J/mol. Hanlon et al.¹⁰⁾ converted such energies into Ae₃ temperature increases of up to 10°C, values that are too low given Yada’s observations regarding the increase in the transformation temperature.^1,2)

For this reason, Ghosh et al.¹¹⁾ extended Hanlon’s analysis by allowing for the dislocation density distribution to be inhomogeneous, increasing the driving force in this way to as much as 197 J/mol. However, this method is based on the application of very large strains, significantly greater than the critical strains of about 0.10 that have been reported to be associated with the initiation of the transformation.^12,13) This limitation was later resolved by introducing the concept of mechanical or stress activation.¹⁴⁾ According to this view, the flow stress developed at the critical strain acts as the driving force that overcomes the various obstacles to the transformation. In this way, the model was able to account for the rapid forward displacive transformation,^11,12) which stands in sharp contrast to the much slower, diffusional reverse transformation.

In the present work, the driving force is taken instead as the net softening associated with the transformation.¹⁵⁾ This is defined as the difference between the austenite flow stress at the critical strain and the yield stress of the fresh Widmänstatten ferrite that replaces it. In this way, the driving force consists only of the softening that takes place during the transformation. This force is opposed by the sum of the chemical free energy difference between the austenite and the Widmänstatten ferrite and the dilatation work and shear accommodation work that are associated with the process.

For this purpose, compression tests were carried out on a 0.06wt%C-0.30wt%Mn-0.01wt%Si steel and the driving force required to initiate DT was determined experimentally from the difference between the work hardened austenite flow stress and the yield stress of the Widmänstatten ferrite projected into the same temperature range. The latter was estimated from that of polygonal ferrite specimens deformed below the Ae₁, over the range 500 to 750°C. The polygonal ferrite values were in turn corrected for the difference in flow stress between Widmanstätten and polygonal ferrite, as determined by means of hardness tests carried out at room temperature.¹⁵⁾ In this way, the critical strains predicted by the model could be compared with the experimental values obtained by means of the double differentiation method,¹⁶⁾ described in more detail below.

2. Experimental Procedure

2.1. Compression Testing

A 0.06wt%C steel containing 0.01 wt% Si and 0.30 wt% Mn was investigated in the current study. This was received in the form of hot rolled plates. The complete chemical composition, including the corresponding paraequilibrium and orthoequilibrium Ae₃ and Ae₁ temperatures, is listed in Table 1. The transformation temperatures were calculated using the FSstel database of the FactSage thermodynamic software.¹⁷⁾ All the analyses that follow are based on paraequilibrium conditions, under which only the interstitials (in this case C) are able to partition between the product and parent phases.¹⁸⁾

Table 1. Chemical composition (mass%) and equilibrium transformation temperatures (°C).

C	Mn	Si	Ae1	Paraequilibrium Ae3	Orthoequilibrium Ae3
0.06	0.30	0.01	724°C	870°C	877°C

Cylindrical compression specimens, 6 mm in diameter and 9 mm long were machined from the hot rolled plates with their axes parallel to the rolling direction. Hot compression tests were conducted using a 100 kN MTS servo-hydraulic testing machine equipped with a vertical radiation furnace and a temperature controller. Two sets of experiments were carried out. In the first, the yield stress of the specimens was established in the ferrite region below the Ae₁ temperature (Fig. 1(a)): in the second, the flow stress of the austenite phase was measured above the Ae₃ (Fig. 1(b)). In the experiments of Fig. 1(a), the samples were heated at the rate of 1°C/s up to test temperatures in the range 500°C to 750°C; they were then held for 20 min prior to deformation.

Fig. 1.

Compression test schedules employed to determine a) the yield stress of ferrite below the Ae₁ and b) the flow stress of work hardened austenite above the Ae₃.

The same heating rate was employed in the tests of Fig. 1(b), in which case the specimens were heated up to 1000°C and then held for 20 minutes to ensure that the transformation to austenite was complete. The samples were then cooled at 1°C/s down to test temperatures in range 920°C to 980°C and held for 10 minutes before straining. In this case, much larger strains of 0.7 were applied at the same strain rate. Prior to testing, the top and bottom surfaces of the specimens were covered with Mo sheets and lubricated by applying a mixture of boron nitride and ethanol. In this way, relatively homogeneous deformation was achieved, as shown in Fig. 2. In order to minimize oxidation and decarburization during testing, an argon protective atmosphere was employed in all cases. The compression samples were rapidly water quenched (in about 1 s) after testing so as to preserve the deformation microstructure. As static (post deformation) transformations involve alloying element partitioning by substitutional diffusion, such quenches are considered sufficiently fast to prevent the formation of polygonal ferrite during cooling.¹⁸⁾

Fig. 2.

Optical micrograph showing the homogeneous nature of the deformation in a sample compressed to a strain of 0.7 at 1 s^–1 and 980°C.

2.2. Metallography

Cross-sections containing the longitudinal axis were cut from the deformed compression samples. These were then mounted using a phenolic hot mounting resin. Rough polishing was carried out up to 1200 grit SiC paper. Both 3 μm and 1 μm diamond suspensions were used for final polishing before etching with 2% nital to reveal the microstructure for optical and scanning electron microscopy. The volume fractions of dynamically formed ferrite were estimated from the optical micrographs by means of area measurement using the ImageJ software.

3. Results

3.1. Stress-Strain Curves

Some typical true stress - true strain curves determined in the austenite phase field are shown in Fig. 3. These samples were deformed as much as 110°C above the paraequilibrium Ae₃ temperature. The stress levels of the curves increase with decrease in temperature in the normal way. Here there is softening after the peak stress is attained, leading to a drop of around 10% in the stress. Such softening involves both dynamic transformation as well as dynamic recrystallization, as discussed elsewhere.¹⁹⁾ The sample temperatures measured during deformation are also shown in this diagram and can be read on the right hand axis. It can be seen that these temperatures remained substantially constant during the tests carried out over the range from 920°C to 980°C. These temperatures are 50°C to 100°C above the paraequilibrium Ae₃. Note that the work done per unit volume can be deduced from the experimental stress-strain curves. When converted into temperature increases, these lead to maximum increments of only 2–3°C at the critical strain. Thus transformation was essentially initiated at the temperatures shown.

Fig. 3.

Stress-strain curves of the 0.06wt%C-0.3wt%Mn-0.01wt%Si steel compressed to a strain of 0.7 at 1 s^–1. Stress drops of approximately 10% are observed after the stress peaks. The horizontal lines display the temperature of the sample (on the right hand axis) during deformation.

Fig. 4.

(a) Plot of strain hardening rate (θ) versus σ derived from the stress-strain curve of the compression test carried out at 920°C and 1 s^–1. Each inflection point corresponds to the initiation of a particular softening mechanism. (b) The inflection points in (a) are identified more clearly by the minima in the –(dθ/dσ) versus σ curve.

In order to identify the moments of initiation of these two softening mechanisms, the double differentiation method^16,19) was used. For this purpose, the flow curves of Fig. 3 were fitted with 8th order polynomials using the MatLab software. In order to determine the values of the critical stresses associated with these mechanisms, θ versus σ curves were prepared, as shown in Fig. 3(a), where θ is the strain hardening rate (δσ/δε) ${}_{\dot{\epsilon }}$ . The moment of initiation of each mechanism is identified by a particular inflection point in the θ-σ plot and corresponds to the condition:   

$$\frac{\delta }{\delta \sigma }\left(\frac{\delta \theta }{\delta \sigma }\right)=0$$(1)

The two minima in the –δθ/δσ versus σ curve of Fig. 3(b) identify the onsets of first DT and then DRX. As shown earlier by means of microstructural analysis,^12,13) the critical strain for DT is significantly lower than that for DRX.

The dependences on temperature of the critical strains are illustrated in Fig. 5(a). Here the average critical strain for DT is about 0.12 while that for DRX is roughly 0.20. The critical stresses that correspond to these strains are displayed in Fig. 5(b). Here it can be seen that the DRX stress depends on temperature in the manner expected for a thermally activated mechanism. By contrast, DT, being a displacive and athermal process, displays no obvious temperature dependence. The critical strains and stresses of Fig. 5(a) will be employed below in order to calculate the mechanical driving force for the initiation of DT.

Fig. 5.

(a) The critical strains for dynamic transformation (DT) and dynamic recrystallization (DRX) determined by the double differentiation method. (b) The critical stresses associated with the critical strains of Fig. 3(a). Those of DRX are temperature dependent while those of DT show no dependence.

3.2. Microstructure

It was shown earlier⁶⁾ that the volume fraction of dynamically formed ferrite increases with strain and time and decreases with the temperature interval above the Ae₃. This phenomenon involves the displacive/diffusional transformation of austenite to ferrite. The nucleation of Widmanstätten ferrite takes place by a displacive mechanism²⁰⁾ characterized by an invariant-plane strain with a large shear component.²¹⁾ However, the growth of the plates involves the diffusion of carbon,²⁰⁾ but not of the substitutional alloying elements. Because of these characteristics, the overall process can be described as a paraequilibrium displacive transformation.²¹⁾

Some typical optical micrographs illustrating this phenomenon in the current material are presented in Fig. 6. The martensite (retained austenite) phase appears dark while the ferrite is light. The microstructure of the material deformed at 980°C (110°C above the Ae₃) to a strain of 0.7 (Fig. 6(a)) contains approximately 40% ferrite. The same strain and strain rate were used to produce the microstructure of Fig. 6(b) but straining was carried out at 960°C (i.e. 90°C above the Ae₃). In this case, there is around 60% of ferrite present, illustrating the temperature dependence of ferrite formation. As will be seen in more detail below, the driving force increases with decreasing temperature, while the obstacle free energy barrier decreases as the Ae₃ temperature is approached. Thus both trends contribute to the increased formation of Widmanstätten ferrite as the temperature interval above the Ae₃ is reduced.

Fig. 6.

Optical micrographs of the 0.06wt%C-0.3wt%Mn-0.01wt%Si steel deformed to a strain of 0.7 at a strain rate of 1 s^–1 at (a) 980°C and (b) 960°C. The volume fraction of dynamically formed ferrite is higher at the lower temperature.

The appearance of the displacive ferrite plates that form during straining is illustrated in Fig. 7. These are approximately 200 nm in width²²⁾ and gradually coalescence into polygonal ferrite grains during straining.²²⁾ The Widmanstätten ferrite produced in this way differs significantly from that formed conventionally below the Ae₃. Here dynamic plate formation precedes the appearance of polygonal ferrite, whereas static Widmanstätten is normally nucleated on the allotriomorphic ferrite that appeared first. The Widmanstätten ferrite plates produced by mechanical activation form in self-accommodating pairs of near-identical orientation.²³⁾

Fig. 7.

EBSD images of 0.09%C steel strained to 0.5 at 856°C (20°C above the Ae₃). (a) and (b) secondary electron images of Widmanstätten ferrite plates, (c) orientation image map of the ferrite phase, [001] is along the longitudinal axis, and (d) orientation image map of ferrite after 90° rotation around [010].²²⁾

4. Discussion

4.1. Mechanical Driving Force

In what follows, the driving force is evaluated as the difference between the flow stress of the austenite work hardened up to the critical strain and the yield stress of the fresh Widmanstätten ferrite that takes its place. As a first step, the yield stress of polygonal ferrite (YS-α) at temperatures below the Ae₁ was taken from the experimental curves (not shown here) by applying the 0.2% offset rule. These values are plotted against inverse absolute temperature in Figs. 8(a) and 8(b). The flow stresses of the work hardened austenite at strains of 0.05, 0.10 and 0.12 are displayed in Fig. 8(a), while those for strains of 0.002, 0.05, 0.10 and 0.20 are illustrated in Fig. 8(b). The various transformation temperatures are also indicated in these diagrams.

Fig. 8.

(a) Dependence of the austenite flow stress on inverse absolute temperature. The net softening associated with DT is evaluated by subtracting the yield stress of Widmanstätten ferrite (YS-Wα) from the flow stress of the austenite work hardened to the DT critical strain of 0.12. (b) The driving force for DRX can be estimated from the difference between the flow stress of the work hardened austenite at the DRX critical strain of 0.20 and the yield stress of undeformed austenite grains (YS-γ).

The linear dependence on inverse temperature is based on the so-called exponential stress law for creep:²⁴⁾   

$$\dot{\epsilon }=Bexp\left(C\sigma \right)exp\left(\frac{-Q}{RT}\right)$$(2)

which can also be expressed as   

$$\sigma =\frac{ln\left(\dot{\epsilon }/B\right)}{C}+\frac{Q}{CR}\left(\frac{1}{T}\right)$$(3)

where B and C are material constants, R is the universal gas constant and Q is the activation energy of flow. This kind of linear relationship has been observed to apply to numerous steels²⁵⁾ and is commonly used to determine the T_nr in rolling mills.

In order to estimate the yield stress of Widmanstätten ferrite at temperatures above the Ae₃, two assumptions were made here. The first is that the flow stress of the displacive ferrite is also linearly dependent on inverse temperature. The second is that this stress is 18% higher than that of conventional ferrite, an estimate that is based on hardness measurements carried out in an earlier investigation.¹⁵⁾

Once the various flow stresses displayed in Fig. 8(a) are available, the mechanical driving force for DT can be deduced by subtracting the Widmanstätten yield stress (YS-Wα) from the austenite flow stress at a strain of 0.12 (the critical strain for DT). This difference is highest, about 480 J/mol, at temperatures close to the paraequilibrium Ae₃ and decreases linearly as the inverse temperature is decreased. The driving force per unit volume, (W/V)_{driving force}, also varies with grain orientation, as it depends on the Schmid factor (m), as given by:¹⁴⁾   

$${\left(\text{W}/\text{V}\right)}_{\text{driving force}}=\mathrm{\Delta }\text{Flow Stress X}\mathrm{\hspace{0.17em} }m$$(4)

Here m is the Schmid factor and ΔFlow Stress is the difference between the flow stress of work hardened austenite at a strain of 0.12 and the yield stress of the fresh Widmanstätten ferrite. The flow stresses are converted into J/mol (1 MPa = 7.2 J/mol) for comparison with the energy obstacles introduced in the next section.

An interesting application of the present approach is that the mechanical driving force for DRX can also be deduced in this way. Although DRX is diffusion-controlled and therefore thermally activated, it is the stored energy of the dislocations introduced by straining that provides the driving force. This can be estimated from the net softening resulting from the motion of a grain boundary while it is annihilating dislocations. This has the effect of reducing the local flow stress from its current work hardened value down to the dislocation-free yield stress. It is accordingly given by the vertical distance between the flow stress at ε = 0.20 in Fig. 8(b) (the critical strain for DRX) and the austenite yield stress. Here it falls in the range 320 to 60 J/mol, decreasing with increasing temperature. By contrast, DT is an entirely athermally-activated phenomenon; the mechanical driving force of Eq. (4) will now be compared with the energy barriers opposing the transformation.

4.2. Free Energy Obstacles to Transformation

The free energy obstacle to the formation of ferrite at temperatures above the Ae₃ (ΔG_γ-α) is illustrated in Fig. 9 (solid line). At room temperature, Widmanstätten ferrite, a non-equilibrium phase, has a stored energy of around 50 J/mol²⁶⁾ when compared with conventional ferrite. Here it is assumed that, at the high temperatures of interest in this work, namely 870–1020°C, this difference decreases to approximately 15 J/mol (at 870°C), declining further to 5 J/mol at 1020°C. This additional free energy gap is added to ΔG_γ-α in Fig. 9 to provide an estimate of the free energy barrier opposing the formation of displacive ferrite (ΔG_γ-Wα) in this temperature range, shown as a broken line.

Fig. 9.

Chemical free energies of polygonal (solid line) and Widmanstätten (broken line) ferrite as a function of the temperature interval ΔT (experimental temperature – Ae_3(p)). The free energy barrier increases with ΔT because of the increasing instability of ferrite at elevated temperatures.

The transformation also requires the expenditure of work to accommodate shearing of the plates as well as the dilatation that accompanies ferrite formation.¹⁴⁾ A shear strain of γ = 0.36 is associated with the transformation of austenite into Widmanstätten ferrite,^21,27) accompanied by a dilatation of about 3%, perpendicular to the habit plane.²⁷⁾ In order to calculate the work done, the transformation habit plane was taken to be (0.506, 0.452, 0.735) and the shear direction to be (–0.867, 0.414, 0.277).^21,27) These values were employed to evaluate the Schmid and orientation factors, m and λ, required in these calculations. The values of the shear accommodation, (W/V)_SA, and dilatation, (W/V)_D, work per unit volume were then calculated using the following relations:¹⁴⁾   

$${\left(\mathrm{W}/\text{V}\right)}_{\text{SA}}=\text{m}\times {\sigma }_{0.12}\times 0.36$$(5)

  

$${\left(\mathrm{W}/\text{V}\right)}_{\text{D}}=\lambda \times {\sigma }_{0.12}\times 0.03$$(6)

Here σ_0.12 is the flow stress at the DT critical strain of 0.12.

The Schmid factors governing the formation of Widmanstatten ferrite were calculated using the method of Ref. 28) applied to the transformation habit plane and shear direction specified above. These are displayed in inverse pole figure (IPF) form in Fig. 10. Here the maximum and minimum m-values are 0.477 and 0.203, respectively. Note that the habit plane normal and the shear vector are not orthogonal because of the rigid body rotation and dilatation associated with the present type of displacive transformation.

Fig. 10.

The Schmid factors pertaining to transformation on the (0.506 0.452 0.735) habit plane and along the [-0.867 0.414 0.277] shear direction displayed in inverse pole figure (IPF) form.

The dependence of the flow stress on temperature interval above the Ae_3(p) (required to solve Eqs. (5) and (6)) is illustrated in Fig. 11 (solid line). The values of the shear accommodation work (broken line) and dilatation work (dotted line) obtained from these equations are also displayed in Fig. 11 for m values of 0.5 and 0.2 and orientation factors, λ = √m, of 0.707 and 0.447. The former quantities represent the maximum amount of work that can be done by the applied stress in an optimally oriented grain (m = 0.5; λ = 0.707). The lowest possible Schmid factor as shown above is 0.203. Here m = 0.2 is used to calculate the work performed in the least well-oriented grain. These values are used below to estimate the extent to which DT can occur at a strain of 0.12 in grains of specific orientations.

Fig. 11.

Austenite flow stress at ε = 0.12 determined in compression displayed as a function of ΔT (right-hand axis). The shear accommodation work (SA) and dilatation work (DW) as a function of ΔT (left-hand axis). m-values of 0.5 and 0.2 are employed to illustrate the effect of grain orientation on the energy barriers.

4.3. Driving Forces Versus Free Energy Barriers

The mechanical driving forces developed at a constant strain of 0.12 in grains with m values in the range 0.2 to 0.5 are illustrated in Fig. 12(a) (dotted lines). Here they are compared with the total free energy barriers (solid lines) as given by the sum of the chemical free energy difference (ΔGγ–Wα), shear accommodation work and dilatation work:   

$$\text{Total Energy Barrier}=\mathrm{\Delta }{G}_{\gamma -\text{W}\alpha }+{\left(\mathrm{W}/\text{V}\right)}_{\text{SA}}+{\left(\mathrm{W}/\text{V}\right)}_{\text{D}}$$(7)

The intersections in the plot, labeled here with points, indicate the maximum temperature up to which DT can take place in grains of particular orientations. For example, poorly (m = 0.2) and well (m = 0.5) oriented grains can undergo transformation at temperatures up to 7 and 54°C above the Ae₃, respectively.

Fig. 12.

(a) Dependence of the driving force (DF) at ε = 0.12 on temperature interval and Schmid factor in the range 0.2 ≤ m ≤ 0.5 (dotted lines). The total barriers to transformation over the same temperature range (solid lines). The intersections (labeled with points) correspond to the maximum temperatures up to which dynamic transformation can take place in grains of particular m-values. (b) At a lower strain (i.e. ε = 0.05), the absence of intersections indicates that transformation is not possible.

The effect of loading the material to a stress level below that of the critical stress is depicted in Fig. 12(b). Taking the flow stress at a strain of ε = 0.05 as an example (Fig. 8(a)), it can be seen that the driving force is too low to overcome the barrier to the transformation even in the most ideally oriented grains (m = 0.5). This state of affairs also applies to strains of 0.06 and 0.07.

4.4. Limitations of the Present Model

According to the current simplified model, see Fig. 12, dynamic transformation can be initiated at temperatures up to 54°C above the Ae_3(p). However, these predictions conflict with experimental observations, according to which DT can be induced as much as 200°C above this temperature. As there are no adjustable parameters in this model, it seems reasonable to question the accuracy of the experimental data employed in the calculations. Although measurements of the flow stresses are subject to error, these are generally small, even with regard to the extrapolations employed for the Widmanstätten ferrite. The problem is probably associated instead with the simplifying assumptions employed in the model.

The most direct way to correct for this discrepancy is to set aside the assumption that the critical strain has an approximately constant value of 0.12 and to allow for the observation that it increases with temperature, as displayed in Fig. 5(a). This is consistent with the observations of other workers.¹⁹⁾ Concurrently, the critical stress remains approximately constant over the experimental temperature range, see Fig. 5(b), in keeping with the nature of a displacive process. If the driving forces derived using Fig. 8(a) (based on a constant critical strain) are now modified to allow for a constant critical stress instead, it can be seen that the former actually increase slightly with temperature (because the Widmanstätten ferrite yield stress decreases with increasing temperature). For the present purpose, for simplicity however, the driving force can be taken as approximately constant over the current temperature range.

The effect of driving forces that are essentially independent of temperature is depicted in Fig. 13. Here it can be seen that the driving force and total obstacle curves do not intersect at all for m = 0.5, indicating that dynamic transformation can take place at temperatures as much as 150°C and more above the Ae₃. For grains with m values of 0.2 or 0.35, the intersection temperatures have all increased when compared with those of Fig. 12(a).

Fig. 13.

Illustration of the effect of assuming that the driving force for the formation of Widmanstätten ferrite remains essentially independent of temperature. The driving force and obstacle curves for m = 0.5 no longer intersect, while the intersection temperatures for m = 0.35 and m = 0.2 have increased when compared with those of Fig. 12(a).

5. Conclusions

(1) The net softening during transformation can be considered as the driving force for phase change. This is given by the difference between the flow stress of the work hardened austenite at the critical strain and the yield stress of the fresh Widmanstätten ferrite that takes its place. The obstacle to the transformation consists of three parts: i) the free energy difference between austenite and Widmanstätten ferrite, ii) the shear accommodation work that is imposed on the neighborhood by the transformation, and iii) the dilatation work associated with making room for the lower density ferrite.

(2) The mechanical driving force for DT can be readily evaluated by plotting the flow stresses and yield stresses against inverse absolute temperature.

(3) In the present material, dynamic transformation begins at a critical strain of about 0.11 (at 920°C), increasing to about 0.14 at 980°C. At lower strains, the driving force is too low to permit the obstacles to the transformation to be overcome.

(4) The use of a constant critical strain in the simplified model presented here accounts for the occurrence of dynamic transformation at temperatures up to 54°C above the Ae₃. However, if allowance is made for the increase in the critical strain with temperature (together with an approximately constant critical stress), then the dynamic transformation observations reported in the literature at temperatures as high as 150–200°C above the Ae₃ can be readily justified.

(5) The current transformation model also leads to evaluation of the stored energy that is the driving force for dynamic recrystallization. This falls in the range 60 to 320 J/mol.

Acknowledgements

The authors thank Professor In-Ho Jung of McGill University for providing access to the FactSage software. They are grateful to Professors Carlos Sergio da Costa Viana, Leo Kestens and John Ågren as well as Doctor Dorien De Knijf for helpful comments. They also acknowledge with gratitude funding received from the McGill Engineering Doctoral Award (MEDA) program and the Natural Sciences and Engineering Research Council of Canada.

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