Critical Strain for Dynamic Recrystallisation. The Particular Case of Steels

The knowledge of the flow behavior of metallic alloys subjected to hot forming operations has particular interest for metallurgists in the practice of industrial forming processes involving high temperatures (e.g., rolling, forging, and/or extrusion operations). Dynamic recrystallisation (DRX) occurs during high temperature forming over a wide range of metals and alloys, and it is known to be a powerful tool that can be used to control the microstructure and mechanical properties. Therefore, it is important to know, particularly in low stacking fault energy materials, the precise time at which DRX is available to act. Under a constant strain rate condition, and for a given temperature, such a time is defined as a critical strain (εc). Unfortunately, this critical value is not always directly measurable on the flow curve; as a result, different methods have been developed to derive it. Focused on carbon and microalloyed steels subjected to laboratory-scale testing, in the present work, the state of art on the critical strain for the initiation of DRX is reviewed and summarized. A review of the different methods and expressions for assessing the critical strain is also included. The collected data are well suited to feeding constitutive models and computational codes.


Introduction
Recrystallisation during hot working operations of metals and metallic alloys (temperature in the range of 0.5-0.9T m , where T m is the absolute melting temperature [1][2][3]) is commonly called dynamic recrystallisation (DRX) or discontinuous dynamic recrystallisation (dDRX) and has been broadly investigated in the past decades. Jonas [4] pointed out that DRX is accepted as one of the most relevant and meaningful mechanisms available for the control of microstructure under industrial operations. DRX is able to cause a great impact on the hot flow behavior, affecting the microstructure and properties of the material after processing. The desired mechanical properties can be achieved by acting on the DRX kinetics [4]. Frequently, the presence of DRX is indicated by a well-defined peak stress value (σ p ) on the experimental true stress-true strain (σ-ε) curves. However, DRX may be initiated at strain values lower than those corresponding to the peak stress [5][6][7]. DRX can also take place in many metallic alloys, even though no clear peak stress is observed. Ignoring cases of recrystallisation of the multiple peaks type, because of the small interest from an industrial point of view, and under specific deformation conditions, the behavior of low stacking fault energy (SFE) materials differs from that of high SFE materials, and can engender two well-differentiated behaviors and DRX flow curves. In both cases, at low strains, strain hardening and softening by dynamic recovery (DRV) are the deformation On the other hand, in low SFE materials such as gamma-iron austenite (γ-Fe), the DRV kinetics is slow, allowing DRX to take place because the large amount of dislocations generated during work hardening is not annihilated. Therefore, it can be assumed that, once a certain critical dislocation density value ρ c (associated to a critical strain) is reached, DRX is activated as an additional softening mechanism. For a given material, the critical dislocation density depends on the strain rate, temperature, chemical composition, and grain size. Under low strain rate conditions, and when the critical density value is attained, DRX is initiated mainly by the bulging of pre-existing grain boundaries. In the case of high strain rates, DRX is initiated by the growth of high angle boundaries (HAB) formed by dislocation accumulation. As a consequence of this, the local differences in the dislocation density gradient over the grain boundary surrounding area act as the driving force for the nucleation of the new nuclei [9]. Then, this is followed by the long-range migration of HAB. Conventional DRX is considered to be a two-step process governed by the following: (i) nucleation and (ii) grain growth [8]. The flow curve displays a peak in stress, after which stress values gradually decrease until a steady state (σ ssDRX ) is attained. The latter state reflects the dynamic equilibrium between strain hardening and strain softening due to the formation of new grains and the associated grain boundary migration [10]. Consequently, the steady state stress can be achieved both via DRV (high SFE materials) as well as via DRX (moderate to low SFE materials). Gottstein et al. [11] point that the strain is not a state variable of crystal plasticity and therefore, in order to be precise, is more appropriate to talk about critical conditions for microstructural instability rather than to critical strain. As strain itself does not describe the current state of the material but the history of the deformation, often some models formulate the critical strain for DRX in terms of a critical dislocation density or deformation resistance for a given grain size [12].
As mentioned above, DRX is in fact initiated before the strain corresponding to the peak stress. The scientific literature reports that this particular value of strain is linked with the minimum amount of stored energy induced by deformation needed to start DRX [13,14] and is defined as the critical strain for the onset of dynamic recrystallization ε c , associated with the critical stress σ c . Consequently, as already stated, a critical dislocation density value is necessary in order to initiate DRX. Generally, this critical value is related to the nucleation by the formation of mobile high angle boundaries On the other hand, in low SFE materials such as gamma-iron austenite (γ-Fe), the DRV kinetics is slow, allowing DRX to take place because the large amount of dislocations generated during work hardening is not annihilated. Therefore, it can be assumed that, once a certain critical dislocation density value ρ c (associated to a critical strain) is reached, DRX is activated as an additional softening mechanism. For a given material, the critical dislocation density depends on the strain rate, temperature, chemical composition, and grain size. Under low strain rate conditions, and when the critical density value is attained, DRX is initiated mainly by the bulging of pre-existing grain boundaries. In the case of high strain rates, DRX is initiated by the growth of high angle boundaries (HAB) formed by dislocation accumulation. As a consequence of this, the local differences in the dislocation density gradient over the grain boundary surrounding area act as the driving force for the nucleation of the new nuclei [9]. Then, this is followed by the long-range migration of HAB. Conventional DRX is considered to be a two-step process governed by the following: (i) nucleation and (ii) grain growth [8]. The flow curve displays a peak in stress, after which stress values gradually decrease until a steady state (σ ssDRX ) is attained. The latter state reflects the dynamic equilibrium between strain hardening and strain softening due to the formation of new grains and the associated grain boundary migration [10]. Consequently, the steady state stress can be achieved both via DRV (high SFE materials) as well as via DRX (moderate to low SFE materials). Gottstein et al. [11] point that the strain is not a state variable of crystal plasticity and therefore, in order to be precise, is more appropriate to talk about critical conditions for microstructural instability rather than to critical strain. As strain itself does not describe the current state of the material but the history of the deformation, often some models formulate the critical strain for DRX in terms of a critical dislocation density or deformation resistance for a given grain size [12].
As mentioned above, DRX is in fact initiated before the strain corresponding to the peak stress. The scientific literature reports that this particular value of strain is linked with the minimum amount of stored energy induced by deformation needed to start DRX [13,14] and is defined as the critical strain for the onset of dynamic recrystallization ε c , associated with the critical stress σ c . Consequently, as already stated, a critical dislocation density value is necessary in order to initiate DRX. Generally, this critical value is related to the nucleation by the formation of mobile high angle boundaries

Literature Review
The onset of DRX is truly dynamic and cannot be easily extracted from the stress-strain curves. Then, and for simplicity, some authors state that the value of deformation needed to start DRX is proportional to the peak strain ε p [32]; ε c = aε p , where a is a coefficient of proportionality. In this regard, it should be mentioned that, usually, the peak strain and the critical strain for dynamic recrystallisation are related by a factor ranging between 0.5 and 0.90, with 0.8 being the most commonly reported value for C-Mn steels [33]. Here, this factor is defined as the critical ratio of strains R ε = ε c ε p . There are several facts that can explain the relative scatter observed in the literature for R ε . Apart from a possible dependence of this relationship on a given material, the test method employed can significantly affect the measurement of ε c [27].
As already stated, the obtaining of the strain associated to the DRX initiation may be difficult, and several evaluation methods have been proposed. Generally, it is possible to differentiate in the literature four different ways to determine the critical strain for the onset of DRX, namely, (i) through metallographic determination, (ii) by using analytical expressions for ε c or ε p (assuming R ε ), (iii) by applying methods to calculate the critical strain ε c for DRX, and (iv) by direct calculation of the critical strain ratio applying analysis of constitutive models as well as through application of constitutive physically-based models to derive the DRX nucleation. In the forthcoming paragraphs, details of each category are provided. The collected data (see Table A1 in Appendix A) comprise valuable data obtained from plain carbon steels, microalloyed steels, stainless steels, and high-alloy steels (e.g., transformation induced plasticity (TRIP) and twinning-induced plasticity steels (TWIP) steels) reported in the scientific literature. The consulted bibliography includes compression, torsion, tensile, and rolling testing data.

Metallographic Determination
The determination of the critical strain for initiation of DRX through metallography and metallographic observation techniques involves a pronounced effort on the implementation of an extensive number of tests, sample preparation, and examination. This may be a very time-consuming task. Several authors have shown the microstructure evolution during hot deformation from critical to steady state conditions. Moreover, depending on the facilities, post-dynamic phenomena can occasionally occur and, in addition to the instability of austenite at room temperature, the detection of the initiation of DRX can be not accurate enough by employing classical metallography. Additionally, cooling from hot working temperature during quenching can promote phase changes that can alter the deformed microstructure, which in turn adds difficulty to the metallographic analysis. In the following, some examples are cited.
Luton and Sellars [5], working with pure Ni and Ni-Fe alloys tested in torsion at approximately 1033 K (760 • C) to 1553 K (1280 • C) and 2.0 × 10 −3 to 4.0 s −1 , have evidenced through metallographic observation that new recrystallised grains appear when the peak stress had been attained. It is reasonable to assume that the strain value associated to the peak stress was closely related to the critical strain for dynamic recrystallisation [5,34]. Glover and Sellars [35], in torsion of ferrite tested at 773 K (500 • C) to 1073 K (800 • C) and 3.8 × 10 −4 to 6.4 s −1 , have argued that, when DRX takes place, the critical strain for the onset of dynamic recrystallisation is the most important parameter in determining the flow stress. Also, Glover and Sellars [35] have remarked that the values of critical strain for DRX lie systematically below the measured values of the strain associated to the peak strain. Sah and colleagues [20] studied the effect of the initial grain size of pure nickel under hot torsion and revealed via metallography that, before the peak torque is reached, a well-developed dynamically recrystallised microstructure was formed. Also, they showed that, when grain size decreases, the rate of recrystallisation increases markedly and is accompanied by a decrease in the critical strain for DRX. Meanwhile, Ouchi and Okita [22], working on high strength low alloy and stainless steels under hot compression testing (5.0 × 10 −4 -10.0 s −1 and 1173 K (900 • C)-1473 K (1200 • C)), developed a rapid cooling system to investigate the microstructural changes associated with DRX in order to avoid static recovery and recrystallisation. Metallographic examination revealed that dynamic recrystallisation started at a fraction of approximately 0.8 of the peak strain, originated by local bulging of grain boundaries. A similar study was performed by Maki and colleagues [36,37]. These authors analyzed the DRX behavior of austenite in an 18-8 stainless steel and an 18 Ni maraging steel by microstructural observations of the water-quenched specimens deformed under tensile deformation, applying different strain rates and temperatures, ranging from 1073 K (800 • C) to 1473 K (1200 • C) and from 1.0 × 10 −3 to 1.0 × 10 −1 s −1 , respectively. They identified the critical strain value for the occurrence of dynamic recrystallisation from the tensile hot flow curve using optical micrographs showing the progress of DRX of austenite with strain.
Additionally, the work published by Ueki et al. [38] was conducted in order to clarify the behavior of DRX in non-ferrous metals and alloys on the basis of the results obtained from mechanical tests and microstructural observations. The investigators critically discussed the factors that influence the occurrence of DRX and proposed a new way of classifying the operative dynamic restoration processes. They examined the high-temperature mechanical behavior of Cu, Cu-Al alloys, and Ni using torsional testing with hollow samples in conjunction with microstructural observations on deformed and quenched specimens using both optical and transmission electron microscopy (TEM).
More recently, Dehghan-Manshadi et al. [39,40], under torsion experiments, investigated the hot deformation behavior of an austenitic stainless steel (AISI 304) in order to characterize the evolution of the dynamically recrystallised structure at temperatures and strain rates in the domain of 1123 K (850 • C) to 1373 K (1100 • C) and 1.0 × 10 −3 to 1.0 s −1 , respectively [39]. The authors used common sample metallographic preparation and electron backscattered diffraction (EBSD) techniques to analyze the evolution of the microstructures during hot deformation from the critical to steady state. Furthermore, the rate (slope) of the work hardening rate curves was used to identify the critical strain for the initiation Metals 2020, 10, 135 7 of 58 of DRX by applying the mathematical approach proposed by Poliak and Jonas [14,29], as explained later in Section 2.4. For example, at 1173 K (900 • C) and 1.0 × 10 −2 s −1 and at strain of 0.4, Dehghan-Manshadi et al. [39] showed that formation of new grains can occur, even at lower strains than the critical strain predicted (~0.45). Also, microstructural analysis indicated that DRX was operative over an extensive domain of deformation conditions even when the curves showed continuous work hardening behavior. A similar analysis was previously reported by Stewart et al. [41], where a kinetic model of DRX for an austenitic stainless steel and a hypereutectoid plain carbon steel was derived using compression testing over the range of 1173 K (900 • C) to 1373 K (1100 • C) (at a constant strain rate of 1.0 × 10 −1 s −1 ) and electron backscatter diffraction. This technique allowed faster and more accurate measurement of the grain size and recrystallised fraction, as well as the corroboration of the critical strain for initiation of DRX and the strain to achieve maximum softening. Beladi and co-workers [42], using hot torsion tests, also reported a similar approach for an Ni-30% Fe alloy.

Analytical Expressions for ε c or ε p
Perhaps the easiest way to determine the critical strain ε c for a given material is through the assumption of a critical strain ratio and to derive the critical strain through an analytical expression for the peak stress as a function of testing conditions and microstructure. To organize and facilitate the understanding of the present paper, all the following expressions are cited in a general way, so in some cases, the name of the parameters are not exactly the same as those reported by the authors. Equation (1a), suggested by Sellars [33], represents one of the most commonly and widely reported empirical relationships for the determination of the peak strain as a function of the thermomechanical variables. In addition, in the following paragraphs, a certain number of relationships suggested in the scientific literature will be described with respect to their ability to approximate experimental data. Almost 90% of the published data in Table 1 correspond to the following power law (see Equations (1a) and (1b)): where k, K, n (usually equal to 0.5 [33]), and m are material dependent constants; d 0 is the initial grain size; and Z is the Zener-Hollomon parameter (s −1 ) [30], defined as follows: where . ε is the strain rate (s −1 ), Q hw is the activation energy for hot deformation (J·mol −1 ), R is the universal gas constant (J·mol −1 ·K −1 ), and T is the absolute temperature (K). All cases where kd n 0 = K will be referenced using the equation number without distinction between the types (a) or (b). In those cases where the initial grain size is kept constant, Equation (1a) can be simplified into Equation (1b) [43]. These expressions have been developed empirically and take into account the influence of the initial microstructure, through the grain size prior to deformation d 0 , and the deformation conditions through the Zener-Hollomon parameter Z.
Mintz et al. [44,45], working with C-Mn and C-Mn-Al steels and studying the influence of DRX on the tensile ductility at high temperature in the range of 973 K (700 • C) to 1423 K (1200 • C) and strain rates ranging between 3.0 × 10 −4 and 3.0 × 10 −2 s −1 , proposed the following empirical relation in order to take into account the effect of the initial grain size on the constant factor K from Equation (1b): where k 1 and k 2 are numerical constants. Grain size is an important factor controlling the peak strain, because it defines the number of nucleation sites for DRX. Equation (3) indicates that, for finer grain sizes, DRX is more readily attained than would be predicted by a simple d 1/2 0 relationship [44,45].  Medina et al. [46][47][48][49], working with microalloyed steels, proposed a modified relationship for the peak strain, as shown in Equation (4a). They discussed that, under hot working conditions, the Z parameter appears to be insufficient to describe with accuracy the dependence of peak strain on the strain rate and temperature. Therefore, Medina and Hernandez [47] included in their model a parameter named A (s −1 ), which depends on both the chemical composition and the activation energy.
Studying the dynamic recrystallisation of an AISI 304 stainless steel through torsion test in the temperature range of 1173 K (900 • C)-1373 K (1100 • C) and the strain rate range of 5.0 × 10 −2 -5.0 s −1 , Kim and Yoo [50] reported similar relationships for ε p and ε c . See Equations (4a) and (4b).
Kowalski et al. [51] reported, for plane strain compression testing of ultra-low C steels, that the critical strain for the onset of DRX ε c is a function of the Zener-Hollomon parameter normalized by the square of the steady state flow stress arising from the balance between work hardening and DRV (σ sDRV ), as shown in Equation (5): Several publications [28,[52][53][54][55][56], mainly dealing with microalloyed steels, reported some empirical formulations that include a corrective factor (F ), which depends on the chemical composition, as shown by Equation (6): Minami et al. [52] proposed a value of the parameter F equal to 1+20 %Nb 1.78 , for Nb steels (0.020-0.080%Nb) tested under torsion. Fernández et al. [53] investigated the dynamic recrystallisation behavior of Nb and Nb-Ti microalloyed steels using torsion tests in the temperature range between 1273 K (1000 • C) and 1473 K (1200 • C) and the strain rate range between 2.0 × 10 −2 and 5.0 s −1 . They proposed a relationship to predict the peak strain for both steels by quantifying the retardation of the DRX produced by the increase of the microalloying elements in solid solution (Nb and Ti) as the reheating temperature increases, applying the following corrective factor: 1+20(%Nb+0.02 %Ti) 1.78 . In an analogous way, Elwazri and colleagues [54], studying under torsion hypereutectoid steels (1.0%C) alloyed with high silicon and microalloying levels of vanadium, found that the constant k showed a relationship with the vanadium content that can be expressed as follows: 1.479 ln[V] + 11.782. As an extension of the early work of Fernández et al. [53], Pereda et al. [55] analysed the influence of the Mo addition on the dynamic recrystallisation behavior of Nb microalloyed steels. These latter authors claimed that Mo has a strong solute drag effect, both on the austenite grain size coarsening behavior and on the DRX kinetics. However, this effect seems to be independent of the Mo content, at least in the range analyzed. A constant corrective factor to quantify the retardation produced by Mo in solid solution, independent of Mo content, was applied to extend the validity of the pre-existing equation [53]: 1+20(%Nb+0.02 %Ti+∆0.035) 1.78 ; where ∆ is equal to 0 for Nb and Nb-Ti steels, and ∆ is equal to 1 for Nb-Mo steels.
On the basis of previous work and dealing with high temperature performance steels, Zhu and Subramanian [57], in order to obtain a homogeneous distribution of ultrafine grain size in strip finish rolling processes, have stated that the critical strain for initiation of the DRX is given by Equations (7a) and (7b): where the constants m and a took the values of 0.0299 and −0.598, respectively. San Martín et al. [58,59] on C-steels modified the classical equation for the peak strain, taking into account the influence of the activation energy Q hw for hot working, as shown in Equation (8). In this case, the authors reported an average value of −2.63 for the parameter a, for carbon contents from 0.16% to 1.06%.
Kuc et al. [60], applying extensive stereological analysis, reported an expression that included a quantitative description of the shape and size of the grains. A strong influence of the average grain size on the peak stress was observed and led to Equations (9a) and (9b): where A 0 , V 0 , and v(V 0 ) are the mean area of the initial grain plane section expressed in µm 2 , the mean grain volume expressed in µm 3 , and the variability factor of the grain volume expressed in percentage, respectively. Kuc et al. [60] also described the dependencies between peak strain ε p and the initial grain size and Zener-Hollomon parameter for Cr-Mn and Cr-Ni austenitic steels tested using a torsion plastometer. In this occasion, the authors reported values for the constants a, b, and c of 0.14, 0.08, and 0.31, respectively. Some authors report that the critical strain for dynamic recrystallisation can be expressed also as a double function of the strain rate and strain itself, as shown in Equation (10). In this sense, Panigrahi [61], reviewing the thermomechanical processing of low carbon steel plate and hot strip, and reported a value of −0.071 for the exponent m .
In order to describe the relationship between the peak or critical strain with the hot working conditions, some authors have published several mathematically equivalent expressions to Equations (1a) and (1b) [62][63][64][65][66][67]. However, in certain cases, the values of the activation energy used for the calculation of the Zener-Hollomon parameter Z (Equation (2)) do not match with the value used in the equation that represents ε p or ε c , such as Equation (11b). Equations (11a) and (11b) show some examples of these expressions, where the use or not of the R constant and/or the use of a constant prior austenite grain size can be noted. Solhjoo [68], on the basis of the model proposed by Lin et al. [69] and Mandal et al. [70], modified the Z parameter (Z m = . ε m Z, where m is a constant equal to 4 3 ) in order to improve the accuracy of the estimations of ε p , as shown in Equation (12), where a is a constant and is found to be 0.26 for AISI 304 stainless steels. In other cases, in which the prior austenite grain size is kept constant, and derivation of its effect or influence cannot be obtained or expected, another type of equation has been proposed. While several authors claim that the critical strain for the onset of DRX is also dependent on strain rate and increases according to the increase in strain rate, Senuma et al. [71], studying the recrystallisation behavior of carbon steels (0.08 wt. %C-0.81 wt. %C), reported that ε c is a solely function of the temperature, as shown in Equation (13b), where the value of the constants K and b are equal to 4.8 × 10 −4 and 8000 K (7727 • C), respectively. Similar results are reported by Anan et al. [72]. Namba et al. [73], for carbon steels (0.11%C-0.17%C), reported K and b equal 1.3 × 10 −5 and 11500 K (11227 • C), respectively, in order to predict the austenite microstructure during hot rolling.
where a and b are constants. Henceforth, in cases where kd n 0 = K and/or a RT = b T , with b = a R , will be referenced using the equation number without distinction between the types (a) or (b).
Studying the flow behavior under compression of a Nb-V high-strength low-alloy (HSLA) steel (0.13%C, 1.55%Mn, 0.028%Nb, 0.059%V) in the range temperature of 1173 K (900 • C) to 1248 K (975 • C) and strain rates ranging between 5.0 × 10 −3 to 5.0 s −1 , Wang and Lenard [74] reported the following expression for the peak strain: where K, a, and b are numerical constants. Multi-variable non-linear regression analysis of the data permits the derivation of the constitutive relations with values of K, a, b, and Q hw of 4.6 × 10 −1 , 0.542, −0.0087, and 464 J·mol −1 , respectively, obtained with a correlation coefficient of 0.99. Yang et al. [75] working with a modified CNS-II F/M steel, designed for in-core components of supercritical water cooled reactor, report another different expression for ε p : The hot flow behavior of the CNS-II F/M steel was investigated through isothermal hot compression tests at a temperature ranging from 1223 K (950 • C) to 1373 K (1100 • C) and strain rate ranging between 1.0 × 10 −3 and 10.0 s −1 . Through regression analysis, Yang et al. [75] show that a is a numerical constant, equal to 0.238.
On the other hand, Cingara and McQueen [76], McQueen et al. [77,78], and Evangelista et al. [79], among others, published a linear relationship between the peak strain and peak stress according to Equation (16): In this regard, the formers authors, working with 300 austenitic stainless type steels (torsion tests at 1173 K (900 • C)-1473 K (1200 • C) and 0.1-5.0 s −1 ), have reported values of a ranging between 0.23 and 0.39 and values of b in the range of 0.0022-0.0033 MPa −1 , respectively. Evangelista et al. [79], simulating the rolling schedule of a 41Cr4 steel by means of torsion tests at temperatures in the interval of 1123 K (850 • C) and 1323 K (1050 • C) and equivalent strain rates ranging between 0.05 and 5.0 s −1 , reported similar values of a and b (~0.11 and 0.0025 MPa −1 , respectively). McQueen and Ryan [78] also reported some values of a and b for HSLA steels analyzed in austenitic state.
Furthermore, a number of authors have developed mathematical equations for the beginning of DRX using a formulation based on the nucleation mechanisms of static recrystallization (SRX) [80][81][82]. Barnett et al. [83,84] affirm that these models fit the observed trends in DRX quite well, however, these models lack firm conclusions to be drawn in order to describe the relative kinetics of the two processes. Barnett et al. [83,84] use conventional equations for the kinetics of SRX and modified them to allow SRX to begin prior to the end of deformation. A general relationship between the kinetics of dynamic and static recrystallisation was developed as follows: where a, b (<0, related to the time taken after deformation for 50.0% recrystallisation t 0.5 ), and c (equal to 0.0145) are constants; and n is the Avrami exponent [85-88].

Methods to Calculate ε c
As already mentioned, several researchers have been studying the modeling of dynamic recrystallisation kinetics in order to predict the flow stress behavior and the microstructural changes associated. Nonetheless, owing to the complexity of the challenge (e.g., dislocation and microstructure evolution, nucleation, growth and impingement of new grains), few approaches successfully correlate the metallurgical principles of dynamic recrystallisation and behaviors.
Luton and Sellars [5] have suggested a critical strain approach to predict the flow stress behavior in terms of the incubation strain ε c for the onset of DRX and the strain required for a complete DRX cycle, ε x . This model, principally concerned with the mechanical aspects of dynamic recrystallisation, provided a significant advance in understanding the DRX phenomena. However, it does not involve the metallurgical principles of DRX and some limitations are presents. The most important one is the indefinite alternation of recrystallisation and hardening cycles [7]. In order to overcome some of the aforementioned limitations, Sah and co-workers [9,20] refined the original model to take into account the influence of grain size changes on the recrystallisation kinetics and explained the damping of the flow stress oscillations at high strains.
One of the most recognized and realistic models that address the nucleation for the onset of DRX was proposed in 1978 by Roberts and Ahlblom [81] on the basis of the model previously reported by Sandström and Lagneborg [80]. The substance of this treatment is that the reduced driving force (i.e., the stored energy difference) modifies the normal energy balance, defining the conditions for nucleation of new grains and the kinetics of nucleation. These depend on the energy of the grain boundaries and on the dislocation density difference between the new recrystallised grains and the matrix [81]. Nucleation is usually promoted by localized strain-induced grain boundary migration, and once the critical dislocation density (associated to the critical strain) is attained, the balance between driving force and surface energy is such that the largest of the bulges can grow with a continuous loss of free energy, and the recrystallisation reaction commences. The recrystallisation reaction continues until the sites at the initial grain boundaries are exhausted. Subsequently, the reaction proceeds via nucleation at the interface between unrecrystallised and recrystallised material until the regions of the latter, emanating from opposing grain boundaries, impinging at the centers of the pre-existing grains. This condition corresponds to the attainment of the steady state on the true stress-strain curve [21]. The model considers the effect of grain size changes, but is limited to the case of grain refinement and single peak flow behavior. This approach, applied to conditions of current deformation, is based upon the classical nucleation theory [89] and describes the strain hardening and dynamic recovery phenomena considering the evolution of the global dislocation density ρ, that is, neglecting the sub-grain wall density, as follows: where x represents the direction of mobility of the migration front; b is the Burgers vector; l is the mean free path of the dislocations, for example, l ∼ ρ −1/2 ; m HAB is the mobility of a high angle grain boundary; M is the mobility of dislocations; and τ is the dislocation line energy. Similar approaches to the previous equation have been presented by Bergström [90], Kocks and Mecking [91,92], Estrin and Mecking [93], and Laasraoui and Jonas [43], among a few other authors. The solution of Equation (18) gives x , where ρ s is the steady state dislocation density representing a balance between work hardening and dynamic recovery, and can be calculated as follows: Roberts and Ahlblom [81] have proposed that the nucleation of new grains is based on a critical dislocation density criterion. The new nuclei appears if the cell is located at the grain boundaries and the dislocation density within the cell reaches a critical value ρ c , depending on the deformation conditions in the following form: where S is the grain boundary energy per unit of area (J·m −2 ). Sommitsch and Mitter [94] reported a theoretical treatment for nucleation, grain growth, and dislocation evolution to model the dynamic recrystallisation of face-centered cubic materials with low stacking fault energy. The model uses the total dislocation density as a state variable and predicts the onset of DRX; the reaching critical conditions for DRX are deduced from maximizing the net free energy based on the nucleation theory of spherical grains.
Some numerical techniques of analysis have been also used to simulate the evolution of the microstructure during DRX, such as the following: Monte Carlo (MC) [95][96][97][98], cellular automata (CA) [99][100][101][102], phase field (PF) [103], vertex (front tracking) [104,105], and level set [106], among others. Nevertheless, despite all the effort put in the direction of getting a more accurate knowledge of the mechanisms of DRX for application in modelling codes, often, the implementation of these models is a complex and time-consuming operation and requires relatively more simple models, whose application on the experimental data is more practical.
Perdrix [107] studied the flow stress characteristics in hot strip mill conditions through the hot torsion technique on C-Mn-Al steels and Nb microalloyed steels. The author described in a quantitative manner the effect of strain, strain rate, and temperature on the flow stress behavior of steels. He took into account the influence of deformation conditions on structural changes: recrystallisation kinetics, recrystallised grain size, and grain growth, and used this knowledge to describe the influence of processing variables and steel parameters on the structural behavior of steels after hot deformation. The effect of temperature change between passes and the interaction between successive passes owing to partial recrystallisation was also analyzed. In his work, Perdrix [107] stated that the work hardening rate-stress (θ-σ) curve may be divided into successive domains (I, II, and III), each of these with a negative linear relationship or slope between θ and σ (see Figure 3), where θ = (∂σ/∂ε) T, . ε . He also analyzed and derived numerical expressions that reflected the influence of the chemical composition of the analyzed steels on the previous values, mainly the effect of C, Nb, and Al.
Kirihata and co-workers [108] studied the kinetics of static, dynamic, and metadynamic recrystallisation of a Cr-Mo-V-Ni-Nb (0.28%C-0.47%C) steel by means of torsion testing. In this case, the critical strain for the onset of dynamic recrystallisation was determined from the strain at which the 50.0% post-dynamic softening time (t 0.5 ) becomes independent of strain in the t 0.5 versus ε plot and from the fitting of mean flow stress data measured in hot strip mills. Reprinted from [92] with permission from Elsevier Kirihata and co-workers [109] studied the kinetics of static, dynamic, and metadynamic recrystallisation of a Cr-Mo-V-Ni-Nb (0.28%C-0.47%C) steel by means of torsion testing. In this case, the critical strain for the onset of dynamic recrystallisation was determined from the strain at which the 50.0% post-dynamic softening time (t 0.5 ) becomes independent of strain in the t 0.5 versus ε plot and from the fitting of mean flow stress data measured in hot strip mills.
While empirical relationships are recognized in order to determine the critical conditions for microstructural instability for the initiation of DRX, some authors have put effort into establishing more rigorous procedures to provide the critical strain for the onset of DRX. A mathematical approach, initially proposed by Mecking and Kocks [91], and then continued by Ryan and McQueen [78,108], was developed to derive the critical strain. As illustrated in Figure 4, the onset of DRX can be determined from slope changes in the θ-σ curves, which correspond to inflection point of these curves. Ryan and McQueen [108] defined the critical strain as the strain value at which the experimental flow stress curves deviate from the theoretical or "idealized" σ-ε curves when DRV is the only active softening mechanism [26]. Figure 5b shows that extrapolation of the second linear segment of the θ-σ curve until that θ is equal to zero determines the theoretical σ-ε curve, corresponding to DRV acting as the main restoration mechanism operating (σ sDRV ) [8]. Nevertheless, in this case, the determination of the inflection points and the followed extrapolation procedure was not well defined. While empirical relationships are recognized in order to determine the critical conditions for microstructural instability for the initiation of DRX, some authors have put effort into establishing more rigorous procedures to provide the critical strain for the onset of DRX. A mathematical approach, initially proposed by Mecking and Kocks [91], and then continued by Ryan and McQueen [78,109], was developed to derive the critical strain. As illustrated in Figure 4, the onset of DRX can be determined from slope changes in the θ-σ curves, which correspond to inflection point of these curves. Ryan and McQueen [109] defined the critical strain as the strain value at which the experimental flow stress curves deviate from the theoretical or "idealized" σ-ε curves when DRV is the only active softening mechanism [26]. Figure 5b shows that extrapolation of the second linear segment of the θ-σ curve until that θ is equal to zero determines the theoretical σ-ε curve, corresponding to DRV acting as the main restoration mechanism operating (σ sDRV ) [8]. Nevertheless, in this case, the determination of the inflection points and the followed extrapolation procedure was not well defined.
As already stated, Wray [13] highlighted that the critical strain corresponds to the application of a minimum amount of energy required to start DRX. Poliak and Jonas [14,29] have reported a theory about the initiation of DRX mainly based on previous works and energetic considerations [13,110]. This alternative approach, based on the thermodynamic laws and principles governing irreversible processes, identifies the onset of DRX by an inflection point in the θ-σ curve, and it is defined as the strain hardening rate corresponding to the appearance of an additional thermodynamic degree of freedom in the system. The authors suggest that further consideration of the stored energy threshold into the material proposed by Wray [13] is essential, but may not be sufficient [14,29]. Further, the kinetics of the process must be considered. The stored energy needs to attain a maximum value, while the latter condition demands that the energy dissipation rate reaches a minimum, and it can be quantified in terms of the appearance of a minimum in the variation of the hardening rate ( −∂θ/∂σ| T, . ε ) with the stress, as displayed in Figure 5. The approach involves multiple numerical differentiations and sometimes promotes a substantial experimental noise during the data treatment. In some cases, the quantity −∂ ln θ/∂ε| T, . ε may be used and the determination of the critical strain becomes easier. Also, it should be noted that, to precisely define the critical strain for dynamic recrystallisation, it is necessary to determine the minimum in ( −∂θ/∂σ| T, . ε ) or the minima in (−∂ ln θ/∂ ln σ) and (−∂ ln θ/∂ε) [14,29]. Moreover, this method is applicable to variable strain rate conditions and to any testing technique [111].    [14,29].
As already stated, Wray [13] highlighted that the critical strain corresponds to the application of a minimum amount of energy required to start DRX. Poliak and Jonas [14,29] have reported a theory about the initiation of DRX mainly based on previous works and energetic considerations [13,110]. This alternative approach, based on the thermodynamic laws and principles governing irreversible processes, identifies the onset of DRX by an inflection point in the θ-σ curve, and it is defined as the strain hardening rate corresponding to the appearance of an additional thermodynamic degree of   [14,29].
As already stated, Wray [13] highlighted that the critical strain corresponds to the application of a minimum amount of energy required to start DRX. Poliak and Jonas [14,29] have reported a theory about the initiation of DRX mainly based on previous works and energetic considerations [13,110]. This alternative approach, based on the thermodynamic laws and principles governing irreversible processes, identifies the onset of DRX by an inflection point in the θ-σ curve, and it is defined as the strain hardening rate corresponding to the appearance of an additional thermodynamic degree of freedom in the system. The authors suggest that further consideration of the stored energy threshold and Jonas [14,29] are identical. Thus, it is broadly recognized that the rate of variation in the slope of the σ-ε curve is a strong indicator of the microstructural changes occurring into the material during hot working [77,78,108,112]. Najafizadeh and Jonas [113] proposed that the θ-σ curve can be fitted using a third order polynomial equation applied to the normalized σ-ε curve.
Gottstein and co-workers [11,114] have emphasized that strain is not a state variable of crystal plasticity, therefore, there is no sense of a critical strain, rather there are critical conditions for microstructural instability. According to Gottstein et al. [11,114], the approach reported by Poliak and Jonas [14,29] does not offer enough details and lacks of any information about the mechanisms that lead to an instability of the microstructure, and claim that this information can be obtained from a work hardening model that defines substructure evolution and can associate the critical conditions with specific microstructural mechanisms. Accordingly, the deformed state is precisely defined using a three internal variables model (3IVM), namely, (i) the mobile dislocation density, (ii) the dislocation density stored inside the cells, and (iii) the dislocation density stored on the cell walls. The latter authors conceive the strain rate hardening behavior as a combination of different stages (I to V), and the proposed model is essentially a stage-III hardening model under the hypothesis that the deformed state comprises a cellular structure with a high dislocation density in the cell walls, which enclose cell interiors of considerably lower dislocation density [11,114]. The work hardening stage III is controlled by the cross-slip of screw dislocations and can be modeled by computing the evolution of the global dislocation density as the sole internal variable; for example, through the dislocation based strain hardening model reported by Kocks and Mecking [91,92]. Estrin and colleagues [115] have incorporated previous results [116][117][118] to reproduce and model the work hardening stages III, IV, and V. In order to taking into account the critical conditions for the appearance of the dynamic recrystallisation, Gottstein et al. [11,114] have expressed the volume fraction of dislocations in the cell walls as a function of the strain hardening rate, in agreement with Estrin and co-workers [115]. Moreover, they used different formulations for stage IV and stage V. For stage IV, they assumed that the mobility of sub-boundaries promoted a change in microstructural mechanisms that reveals the inflection point in the θ-σ curve [11,114].

Calculation of the Critical Strain Ratio by Applying Constitutive Models
Typically, several flow stress models, taking into account DRX, comprise the following: (i) a model for strain/work hardening and dynamic recovery, (ii) a nucleation criterion for dynamic recrystallisation, (iii) a function describing the dynamically recrystallised volume fraction (e.g., the well-known Kolmogorov-Johnson-Mehl-Avrami (KJMA) approach [85][86][87][88]), and finally (iv) a rule of mixture to determine the macroscopic resulting flow stress. As mentioned above, the Poliak and Jonas method [14,29] comprises multiple differentiation. The direct application of the Poliak and Jonas criterion [14,29], in order to find a relation that describes the onset of recrystallisation (e.g., the critical strain ratio, R ε = ε c ε p ) in a given constitutive model describing the strain hardening and dynamic recovery stages, is a topic that has not received much attention, and often this task does not always gives satisfactory and precise results. As can be easily seen, such an approach is not applicable in many constitutive models and equations because the second derivative criterion (SDC) [14,29], to determine the critical strain for the initiation of DRX ( ∂ − ∂θ ∂σ /∂σ T, . ε = 0), can give trivial, inaccurate, and/or incongruous solutions. An obvious violation of the SDC occurs when the model for strain hardening and dynamic recovery is incapable of showing an inflection point in the strain hardening rate as a function of the flow stress. Regarding this fact, some authors have developed mathematical approaches to determine the necessary conditions (consistency check) that the constitutive law, describing the strain hardening and the dynamic recovery, must meet to allow the applicability of the Poliak and Jonas criterion [14,29,119]. Bambach [119] shows three types of inconsistencies with the second derivative criterion [14,29] when the effect of dynamic recrystallisation is included in the flow stress model, namely, (i) the model for strain hardening and dynamic recovery do not exhibit an inflection point, (ii) the nucleation criterion for DRX is inconsistent with the inflection point in the strain hardening model, and (iii) insufficient differentiability of the function that represents the flow stress. A number of authors have made numerous attempts to predict both the critical strain for the onset of the dynamic recrystallisation and the critical strain ratio R ε . In this regard, Ebrahimi and Solhjoo [120], based on a previously published work [121], have determined the critical strain ratio of Nb steels using a constitutive equation. According to this procedure, the true stress-strain curve was modeled until the maximum stress using the model proposed and reported by Cingara and McQueen [76,122], and described in Equation (21): where C is a constant value of the material, which is weakly dependent on deformation conditions, and temperature and strain rate and must be determined for each condition [76]. Taking the logarithm of the above equation, the value of the constant C can be obtained as the slope in the linear plot representation of ln σ σ p versus 1 − ε ε p + ln ε ε p [121]. Ebrahimi and Solhjoo [120] used the Poliak and Jonas criterion [14,29] to develop an approach for determining the critical strain ratio, as follows: Although this method provides a useful mathematical expression to determine the critical strain ratio, some limitations of the approach are observed. The maximum R ε that can be obtained takes place when the value of C tends to zero, then the maximum value of R ε tends to 0.50. Despite the fact that the value of the constant C will be derived for each alloy, it was shown that a unique value could be used for a type or family of alloys (e.g., steels) with a small error [78]. For example, McQueen and Ryan [78] reported values of C ranging between approximately 0.19 and 0.22 for AISI 301, 304, 316, and 317 type steels and values of C between 0.39 and 0.73 for C steels, V steels, V-Mo steels, and Ti-Nb steels. Ebrahimi et al. [121] reported a value of C equal to 0.49 for Ti-IF steels. More recently Mirzadeh and Najafizadeh [123], on a precipitation hardening stainless steel (17-4 PH), determined the critical ratio R ε using the aforementioned approach and obtained a mean value of 0.47, with C equal to 0.24.
On the similar line of research, Solhjoo [68] assumed a linear relationship between the strain hardening rate and the strain as follows θ = ∂σ ∂ε = aε + b. The solution of the former differential equation with the following boundary conditions, σ = σ p at ε = ε p is as follows: where S is an additional parameter that contributes to improving the accuracy of the model and, as in the case of Equation (21), must be determined for each material. Taking the logarithm of Equation (23), the linear regression of the ln σ σ p versus ln ε ε p 2 − ε ε p plot gives the value of S constant for each material for each set of temperatures. Solhjoo [68] also found a negative linear relationship between S and temperature. Applying the method proposed by Poliak and Jonas [14,29], to solve ∂ ∂σ − ∂θ ∂σ = 0 with θ = Sσ 1 ε + 1 ε−2ε p , the critical strain ratio can be expressed as follows: As can be verified, when S tends to 0.5, the value of R ε tends to 1 − 1 √ 3 , that is, R ε~0 .42. In any case, and for Equations (21) and (24), the calculation of the critical ratio also exhibits some limitations.
The maximum value obtained by applying Equation (21) is less than 0.5 and, for Equation (24), the maximum value of R ε remains below 1 − √ 5 − 2; (R ε~0 .51). According to the experimental results, for an AISI 304 stainless steel tested at temperatures in the interval of 1173 K (900 • C) and 1473 K (1200 • C), it was found that the value of the parameter S varies between 0.20 and 0.41, giving values of R ε between 0.44 and 0.48 [68].
In order to contribute to the simplification of the problem, the authors of the present review state that Equation (24) can be replaced by the following expression: Lately, Solhjoo [124] proposed a new constitutive equation to predict the flow stress at high temperature up to the peak of stress-strain curve, constructed on the basis of the general form of hyperbolic tangent function, as follows: where σ 0 , ς, and K 2 are the initial stress and material constants, respectively. Solhjoo [124] claim that the use of this formulation includes an additional exponent to highly increase the accuracy of the model. Through the mathematical manipulation of the above equation, it is possible to develop a method to determine the value of the mentioned material constants. The values of ς and K 2 can be obtained as the slope in the linear plot representations of tanh −1 σ−σ 0 σ p −σ 0 1 K 2 versus 2 ε ε p and ln σ−σ 0 σ p −σ 0 versus ln tanh 2ς ε ε p , respectively. The material constants ς and K 2 are interdependent variables and can be determined using different mathematical methods. Solhjoo [124] argued that the approach represented by Equation (26) shows a powerful capability to predict the flow stress and that it is the first constitutive equation with this kind of parameter. Furthermore, using the Poliak and Jonas criterion [14,29], the author derived an equation that can be used to predict the critical strain ratio R ε , and it is expressed as follows: Unlike the above expressions (Equations (21) and (24)), Equation (26) does not exhibit upper limitations in the calculated value of the critical strain ratio R ε . However, an inadequate value of ς and/or K 2 can promote erroneous values of R ε ; for example, R ε greater than 1. Solhjoo [124] successfully applied this approach on AISI 304 austenitic stainless steel data. In this case, and in an opposing manner to the approach of Equation (24), constants ς and K 2 were independent from the hot deformation parameters (true strain rate and temperature) and mean values of~1.28 and~0.68, respectively, were determined. Finally, the reported value for the critical strain ratio was R ε~0 .38. It is useful to mention that, in this case, the domain of the function represented by Equation (27) always requires values of K 2 greater than zero and less or equal than 1.
Chen et al. [125] have presented a new model to describe the flow stress up to the peak of the σ-ε curves and, applying the SDC [14,29], the authors derive an expression to calculate the critical strain ratio R ε . In this case, the flow curve up to the peak stress was modeled using the following equation: where ψ and ζ are interdependent numerical material constants. The solution of the second derivative criterion ∂ 2 θ ∂σ 2 T, . ε = 0 [14,29] promotes the following expression for the critical strain ratio: Chen et al. [125] have developed a model used to predict the stress-strain curves of X12 an ultra-super-critical rotor steel. By applying Equation (29), R ε is determined to be equal to 0.43, which is in agreement with the experimental value obtained using the approach reported by Najafizadeh and Poliak [113].
In a relatively recent paper, Jonas et al. [126] reanalyzed previously published hot flow data [127] and contemplated the dynamic transformation (DT) as an additional softening mechanism (in addition to DRV and DRX) that contributes to the microstructural and mechanical softening responsible for the classical single-peak-shaped curve. Jonas et al. [126] applied the double differentiation method, reported previously by Poliak and Jonas [14,29], to three steels (low C steel, Nb-modified steel, and Nb-modified TRIP steel), tested under uniaxial isothermal compression at constant true strain rates ranging from 0.05 to 1.0 s −1 and temperature ranging between 1223 K ((950 • C) and 1423 K (1150 • C). Accordingly, the authors also considered and analyzed the effect of the polynomial order (2 to 15) on the sensitivity of the double differentiation method as well as on the actual values of the critical strains determined. The double differentiation method normally requires the use of polynomial fitting functions of at least the eighth order and the use polynomials of lower degrees can cause a poor accuracy in the fitting process. Using the first part of the flow curve, comprising values of strain below the peak strain of the TRIP steel curves (1423 K (1150 • C) and 0.25 s −1 ), Jonas et al. [126] stated that polynomial orders below 3 are unable to detect any minima at all, whereas polynomial order between 4 and 7 can detect the DRX minima, but are unable to identify the DT minima. On the other hand, for polynomials order above 8 and/or even higher, both minima are distinguished. Although the order has a detectable effect on the flow stress associated with the minimum, it leads to small differences in the derived critical strains. The critical strain values for DT show a small dependence on polynomial order. However, the critical strain values for DRX show some scatter and these differences can be associated to the actual value of true strain and to the slope of the flow curve after yielding. Finally, Jonas et al. [126] concluded that all the DT critical strains were within about ±0.5% of the average value and the DRX critical strains were within about ±2.0%.
The traditional way to determine the initiation of dynamic recrystallisation reported by Poliak and Jonas [14,29] comprises a relatively complex and time-consuming procedure of manipulation and processing of data, including several steps, such as smooth and filter the raw data, conversion to stress-strain data, compensate the stress drop owing to adiabatic heating or fiction (dissipation heating), second smoothing steps, determination of θ, and so on. Lohmar and Bambach [128] have developed a new concept and an alternative criterion to the Poliak-Jonas method [14,29] in the determination of the onset of DRX. They have proposed the use of a special surface interpolation method based on radial basis functions (RBFs) using a thin plate spline (TPS) kernel to approximate several flow curves at once and to smooth the course of σ over both ε and T via regularization methods [129]. This technique combines surface interpolation of various hot flow curves and smoothing in a single step. As mentioned above, and in some cases, the Poliak and Jonas method [14,29] involves the use of a multiple smoothing steps (e.g., using fast Fourier transform-based procedures, FFT). Accordingly, Jonas et al. [130] applied a seventh-order polynomial, and in some cases, a higher-order polynomial, to fit and smooth each σ-ε curve and a third-order polynomial in the description of the strain hardening rate versus flow stress data, θ-σ [113]. Lohmar and Bambach [128] stated that this type of approach, the use of an i-th order polynomial in the smoothing steps, promotes distinct roots (solution candidates) in the interval of true strains below to the peak strain, ε p . Consequently, in some cases, no unique critical point ε c and several ambiguous candidate values are obtained. According to Lohmar and Bambach [128], the explicit representation of the flow stress, as a function of strain and temperature σ(ε, T), involves the solution of the following minimization problem: where λ is a regularization parameter that describes the trade-off between interpolation and approximation. For constant strain rate and a continuous solution, σ(ε, T) is of the following form: where the thin plate spline kernel is defined as follows: By solving a system of n + 3 linear equations, the unknown coefficients a i are calculated. After solving the coefficients a i and partially deriving the TPS kernel (Equation (32) with respect to strain, the work hardening rate and the representation of the Poliak-Jonas criterion [14,29] are shown, respectively, as follows: where κ(ε) = 0 reflects the roots of the ∂ ∂σ − ∂θ ∂σ versus ε curve; for example, the critical strain candidates.
Lohmar and Bambach [128] have claimed that the developed TPS method requires a minima additional preparation of the flow strain-stress data. The improvement of this new approach lies in the convenient way to interpolate multiple strain-stress flow curves with a differentiable surface interpolant. Finally, Lohmar and Bambach [128] remarked that the TPS approach seems to be a more robust determination of the critical strain for the onset of DRX in comparison with the polynomial interpolation proposed by Jonas et al. [130], which occasionally yields multiple candidates for critical points. However, some limitations are cited, namely, at lower temperatures, the compensation of dissipation heating might affect the determination of the critical conditions for the onset of DRX and cause the detection of rather low critical strain ratios.

Influence of the Chemical Composition on the m Exponent
One of the main objectives of this review work is to reveal or expose the effect of the chemical composition of the alloy on the value of the m exponent, fundamentally for Equations (1a) and (1b). This exponent somehow reflects and controls the kinetics of the process as it is closely related to the true strain rate, for example, through the Zener-Hollomon parameter Z [30]. Regarding this, it was considered that the chemical content of the alloys is an index to evaluate the effect of the chemical composition on the DRX kinetics, and here, the equivalent carbon content (C eq ) was used as an indicator to evaluate the effect of compositions. For example, in welding, the hardenability of the steel is often expressed using a carbon equivalent content. The concentration of each solute is scaled by a coefficient, which expresses its ability, relative to C content, to retard the austenite/ferrite transformation. Bhadeshia and Honeycombe [131] have stated that the increased sensitivity of the austenite to ferrite transformation to carbon at lower concentrations leads to a decreased sensitivity to substitutional alloying elements. It is interesting that the sensitivity of transformation kinetics to carbon at low concentrations explains the need recognized widely in industry, for two carbon equivalent formulae to cover the low and high (>0.18 wt. %C) carbon steels (see Equations (35) and (36), respectively). These are in fact the two most popular expressions for C eq . The IIW formula (International Institute of Welding, Equation (36)) shows much smaller tolerance to substitutional alloying elements than Equation (35), also known as the Ito-Besseyo formula. Other similar expression can be found in the literature [43,132].
for C < 0.18% and C eq = %C + %Mn + %Si 6 + %Ni + %Cu 15 for C ≥ 0.18%. The Ito-Besseyo approach, Equation (35), has smaller coefficients for the substitutional solutes when compared with the IIW equation. It is believed to be more reliable for low-carbon steels. On the other hand, the IIW formula, Equation (36), shows much smaller tolerance to substitutional alloying elements [131].
In the previous equations, all the percentages are expressed in weight percent (wt. %). The use of the alloy content instead of the activation energy, in order to reveal the effect of the chemical composition, was preferred. Numerous authors use calculated apparent activation energy values, determined from experimental data, as an alternative to employ the self-diffusion activation energy. However, the activation energy also will be taken into account in the discussion of results as an additional criterion of analysis. It usually takes values similar or somewhat above those for self-diffusion, indicating that deformation under the discontinuous dynamic recrystallisation phenomena is thermally activated and involves self-diffusion mechanisms [1]. Table A1 in Appendix A collects the most relevant and valuable data obtained from the scientific literature from several families of steels, namely, low alloy steels (e.g., plain carbon steels), carbon steels, microalloyed steels, stainless steels, and high alloy steels (e.g., TRIP and TWIP steels). The fields of Table A1 comprise the following: the main chemical composition of the alloy expressed in weight percentage (wt. %); a brief indicator of the type of alloy (steel); the testing methodology (C: compression, T: torsion, R: rolling, TE: tension, M: multiple techniques); the mean diameter of the initial austenitic grain size d 0 expressed in microns (µm); the activation energy for hot working Q hw used to calculate Z (expressed in kJ·mol −1 ); the parameters k, K, n, and m of the aforementioned equations (Section 2.2); and the critical strain ratio R ε and the type of equation considered. The chemical composition of the alloys is represented by the C, Mn, Si, Al, Cr, Ni, Mo, Ti, Nb, V, and B contents, whereas the values of P, S, N, and O, among many others, have been omitted. When a large number of alloys are reviewed, the chemical composition is noticed using only the range of variation of the chemical elements as an interval of values (e.g., 0.12-0.25 C). When a specific chemical element does not belong to the all analyzed alloys, it is placed between parentheses (e.g., (0.005 Mo)). In some cases (especially for torsion tests), the initial grain size corresponds to the grain size in a given entry passes. The data corresponding to the critical strain equations for the onset of the DRX are shaded in grey. For simplicity, in many cases, some values in Table A1 and text were rounded and, in some specific cases, the chemical composition of steels was referenced as an average chemical composition. Figure 6 shows the relation between the m exponent and the alloy content in terms of equivalent carbon amount (C eq ). Further, Figure 7 displays the dependence of the activation energy Q hw on the equivalent carbon. The test methodology is indicated in both figures. Here, it is worth mentioning that the value of the exponent m is the exponent accompanying the Zener-Hollomon parameter Z and/or the exponent accompanying the true strain rate in the different reported equations either for peak strain and critical strain equations. See Section 2.2.    Nearly 320 experimental data were analyzed, where approximately 65.0% of them correspond to compression data, 30.0% to torsion data, and the rest to multiple techniques and tension data. See Figure 8 and Table 1. The statistical analysis of data reveals a mean value of m approximately close to 0.152 with a standard deviation of 0.07 and minimum and maximum values of 0.08 and 0.68, respectively. Moreover, the value at which the discrete probability distribution takes its maximum value (mode) has been 0.150. Regarding the activation energy Q hw , the statistical analysis of data reveals a mean value of 347 kJ·mol −1 , with a standard deviation of 81 kJ·mol −1 . The minimum and the maximum values are 129 kJ·mol −1 and 747 kJ·mol −1 , respectively. As a general trend it can be seen that the strain rate exponent m decreases slightly with the increasing carbon equivalent of the alloy, and the opposite seems to happen with the activation energy (see Figure 7). Accordingly, an increase of the carbon equivalent content would cause, though slight, slowing of the kinetics of DRX accompanied by a greater insensitivity to the deformation conditions (Z). Solute addition generally restricts the dislocation mobility and reduces the ability of the alloy to dynamically recover acting on the migration of grain boundaries, promoting a major tendency for DRX. However, as DRX also depends on the ease of the migration of grain boundaries, the rate of DRX may decrease Accordingly, an increase of the carbon equivalent content would cause, though slight, slowing of the kinetics of DRX accompanied by a greater insensitivity to the deformation conditions (Z). Solute addition generally restricts the dislocation mobility and reduces the ability of the alloy to dynamically recover acting on the migration of grain boundaries, promoting a major tendency for DRX. However, as DRX also depends on the ease of the migration of grain boundaries, the rate of DRX may decrease for higher solute values. Flow stress is increased by solute and/or fine particles, which diminish DRV, raising the dislocation density through pinning by atmospheres or through reduction in SFE. Fine particles reduce grain boundary mobility and also pin dislocations and stabilize a denser substructure, thus also retarding DRX nucleation. As the peak strain is raised by the presence of solutes and fine particles, the stress is raised more than by simple strain hardening increase, that is, just by the dislocation-dislocation interaction, thus causing a marked rise in activation energy in alloy steels [77,78]. There is scarce literature information regarding the effect of the chemical composition or alloy content on the value of m, though few authors observed that the m exponent decreases with increasing alloy content. Escobar et al. [133], for carbon steels tested at 1173 K (900 • C)-1373 K (1100 • C) and 1.0 × 10 −4 to 1.0 × 10 −1 s −1 , have argued that the onset of DRX becomes insensitive to the forming conditions for relatively high carbon (or relatively high alloy) steels and pointed to a slight decrease of m as the alloying content rises. Dealing with austenitic binary Fe-Mn alloys (1%Mn-20%Mn), Cabañas and co-workers [134], under constant initial grain size, have studied the influence of the Mn content on the constitutive equations in the temperature range of 1223 K (950 • C) to 1523 K (1250 • C) and the strain rate range of 1.0 × 10 −1 to 2.0 s −1 . The authors pointed out that the critical strain can be expressed as a function of Mn content in a similar manner to Equation (1b); however, in this case, the value of the exponent m was a linear function of the Mn content of the alloy as follows: Cabañas et al. [134] have also reported a decrease of the m exponent from 0.125 to 0.085, for increasing contents of Mn from 1.0% to 20.0% respectively. Varela-Castro [135] have analysed the hot working behavior of structural steels and the dependence with the C, Si, and Mn content. The hot flow behavior was studied by isothermal uniaxial compression in the range of 1173 K (900 • C) to 1373 K (1000 • C), applying true strain rates ranging between 5.0 × 10 −4 and 1.0 × 10 −1 s −1 . Regarding this, eight different steels were refined via the electro-slag remelting technique (ESR) with the following chemical composition range (wt. %):~0.15%C-0.45%C,~0.20%Si-0.40%Si, and~0.70%Mn-1.60%Mn, %Fe (bal.). The author proposed a complete constitutive model that describes the hot working behavior of Fe-C-Si-Mn alloys, where all the characteristic parameters are a function of the chemical composition of the alloy and where the parameters to determine the peak strain are defined as follows (see Equation (1b)) [135]: where a, b, c, and d are numerical constants, both obtained with a Pearson's coefficient of R 2~0 .97. All the percentages are expressed in wt. %. See Table 2. It can be seen from Table 2 that, for the pre-exponential parameter K, the effects caused by C and Si contents are of the equal order of magnitude and direction (equal sign) and with the influence of Mn one order of magnitude smaller and of the opposite direction. Regarding the parameter m, it can be noticed that the greater absolute effect is exerted by C, whereas Si and Mn exhibit behaviors of an order of magnitude smaller and of opposite signs. Both C and Si help to reduce the m parameter; that is, increase the insensitivity of m with Z. In this case, the Z parameter was calculated using a commonly reported value of the self-diffusion activation energy for pure iron (270 kJ·mol −1 ) [136][137][138].
Very recently, Siyasiya and Stumpf [139] have studied the relationship between the chemical composition and the Zener-Hollomon exponent in the peak strain equation for hot working of C-Mn steels. The hot deformation behavior of twelve C-Mn steels was examined applying compression tests in which the carbon and manganese contents were increased systematically between 0.035%C up to 0.52%C and 0.22%Mn to 1.58%Mn, respectively. In addition, data from other authors were analyzed and used to derive the main results of this investigation. The authors suggested that the apparent activation energy for hot working increases with an increase in C content in plain C-Mn steels, as was also found by other researchers. Also, the authors argued that this is possibly because of the retarding effect that the Mn-C complexes have on the movement of dislocations. The exponent m in the peak strain equation for hot working was found to decrease with an increase in k and Q hw (both of which increase with an increase in the C content), and also decrease with an increase in the C content according to m = 0.21 − 14 [%C], for content of carbon less than 0.8%C [139].
On the other hand, Figure 7 shows that the activation energy remains approximately constant (330 kJ·mol −1 ) for carbon equivalent values below to 1.0% and, as the alloy content increases, the reported values of hot working activation energy show an increase. Likewise, it can be noted that collected data are grouped in different areas: (i) high alloyed steels, such as stainless steels at the highest values of C eq (about 2.0 or more); (ii) medium alloyed steels, such as medium carbon and microalloyed steels at C eq between 0.2% and 1.0%; and (iii) low alloy steels at carbon equivalent values lower than 0.2%,.
The plot presented in Figure 6 also shows that there is not a clear dependence of the test type, whether compression, torsion, or tensile test, on the m exponent. Several authors have already suggested that higher values of ε p are expected in torsion (associated to k and/or K, approximately by a factor of~1.3 to 2.6), in relation to those obtained through compression testing techniques [27,108]. However, there have been no reports about the influence of the testing methodology on the DRX kinetics using expressions, as shown in Section 2.2. Figure 9 represents more than 210 experimental data, mainly for compression and torsion, where the critical strain ratio R ε is plotted against the carbon equivalent C eq . In this case, the statistical analysis reveals a mean value for R ε of approximately 0.71, with a standard deviation of 0.16 and with minimum and maximum values of 0.12 and 1.0, respectively. Furthermore, there is a slight decrease of the critical strain ratio R ε for increasing values of the alloy content and a mean value of R ε of approximately 0.62, for C eq ranging between 4.0% and 6.0%, is shown. Additionally, the plotted data do not show a clear relationship between the critical strain ratio R ε values and the testing technique. However, it appears that the data reported for torsion tests have a lower standard deviation (approximately 0.13). It can be also observed that one of the most repeated values for the critical strain ratio is around 0.80, as has been reported earlier by Sellars [33] for C-Mn steels (R ε~0 .67-0.86). Here, it is useful to clarify that many of the reported values are determined or calculated by the Poliak and Jonas method [14,29], although, for many other cases, the authors collect data directly from the reported literature.
testing technique. However, it appears that the data reported for torsion tests have a lower standard deviation (approximately 0.13). It can be also observed that one of the most repeated values for the critical strain ratio is around 0.80, as has been reported earlier by Sellars [33] for C-Mn steels (R ε 0.67-0.86). Here, it is useful to clarify that many of the reported values are determined or calculated by the Poliak and Jonas method [14,29], although, for many other cases, the authors collect data directly from the reported literature. Figure 9. Critical strain ratio R ε vs. carbon equivalent of the alloys (C eq ) using Equation (36). Figure 9. Critical strain ratio R ε vs. carbon equivalent of the alloys (C eq ) using Equation (36).
A few results were reported regarding the influence of chemical composition and alloy content on the decrease of the critical strain ratio. Siciliano and Jonas [28], for Nb containing steel, have reported a clear progressive decreasing dependence of the critical strain ratio R ε solely on the effective niobium concentration (Nb eff ), as shown in Equation (39). More recently, Xu et al. [140], for high-Nb HSLA steels, have shown a similar relation updated for higher effective Nb contents. The authors claims that the model proposed by Siciliano and Jonas [28] is only available to the relatively low effective Nb contents (i.e., smaller than 0.06%) and that there is a large deviation between predictions and measurements for the higher effective Nb content (up to 0.1%).
where a and b are constants (10.8 to 13.0 and 64.4 to 112.0, respectively [28,140]) and where Nb eff is specified by the following [28]: As a complement of the Poliak and Jonas approach [14,29], Najafizadeh and Jonas [113] have proposed that the normalized true stress-strain curve (σ/σ p vs. ε/ε p ) is suitable to apply the second derivative criterion [14,29] and the normalized strain hardening rate versus the normalized stress curve, ∂ σ/σ p /∂ ε/ε p versus σ/σ p , can be fitted using a third-order polynomial equation in order to determine the inflection points that identify the point of initiation of DRX in the range of temperature and strain rate of interest. Furthermore Najafizadeh and Jonas [113] showed that the critical stress and the critical strain ratio R σ = σ c σ p and R ε are approximately constant and independent of the Z parameter (see Figure 10). Metals 2020, 10, x FOR PEER REVIEW 26 of 53

Conclusions
After a justificative introduction about the practical interest in deriving the onset for dynamic recrystallisation, a critical review is done on the different methods reported in the literature to derive the critical strain to initiate dynamic recrystallisation. These methods cover from metallographic procedures to several empirical and theoretical ways to obtain the associated strain to the onset of DRX. All methods are introduced, and main advantages and disadvantages aspects are outlined. Particular interest is paid to get the latter strain from constitutive equations.
Finally, a large review is done on all reported values for the critical strain for DRX in steels, particularly plain carbon or microalloyed steels. An attempt is also done to derive the effect of the chemical composition on the exponent of the classical relationship between the peak strain ε p and the Zener-Hollomon parameter [30], by using a carbon equivalent parameter, considering also the type of testing conditions, whether compression, torsion, or tension.  Abbreviations ε, ε p , ε c , and ε x True strain, peak strain, critical strain and recrystallisation strain. R ε and R σ Critical ratio of strains and critical ratio of stresses [dimensionless]. Σ, σ 0 , σ p , σ c , σ sDRV , and σ ssDRX True stress, initial stress, peak stress, critical stress, saturation stress (DRV) and steady state stress (DRX) [