Nonlinear multivariate constitutive equations for modeling hot deformation behavior

Nonlinear constitutive equations are proposed to model variations in flow stress as a function of strain rate and temperature during hot deformation. Modified constitutive are applied to seventy data sets about hot deformation of alloys. Two modifications to conventional constitutive models are introduced, viz. (1) nonlinear and (2) multivariate models with the fitting of flow stress simultaneously with two variables. The predictive accuracy of constitutive equations was evaluated using three statistical parameters and compared with a conventional Arrhenius-like model. It is shown that nonlinear constitutive equations have improved predictive accuracy for variations in flow stress during hot deformation. The advantages of multivariate models include less computation and material parameters that are constants independent of temperature or strain rate. In another type of multivariate model, flow stress is expressed as linear and nonlinear polynomial functions of the Zener-Holloman parameter. This approach gives a single value of the activation energy of hot deformation. The results have indicated that a generalized second-order multivariate constitutive equation can be used to better predict flow stress, as a function strain rate and temperature, during hot deformation.


Introduction
Hot working is a deformation of materials carried out at a combination temperature and strain rate such that substantial recovery occurs [1].It is essential to select a suitable set of temperature and strain rate to impart considerable strain without appreciable strain hardening.Optimized sets of temperature and strain rate result in desirable mechanical properties.Owing to its importance for controlling final mechanical properties, hot deformation of alloys is an extensively studied field of materials processing [2][3][4][5][6][7][8].Appropriate knowledge of material flow and microstructural behavior during hot working is crucial for selecting suitable temperatures and strain rates.It requires accurate modeling of variations in flow stress with temperature and strain rate (at different levels of strains) using constitutive equations.Such modeling helps in determining optimized hot working conditions and understanding underlying metallurgical processes like work hardening (WH), dynamic recovery (DRV), and dynamic recrystallization (DRX) [9].
Arrhenius-type model is used extensively in studying hot deformation.For investigating the combined effect of strain rate and temperature on flow stress, the Zener-Holloman parameter Z (temperature-compensated strain rate parameter) is used [31,32] by treating hot deformation as a thermally activated process.The Zener-Holloman parameter is related to flow stress by following equations.It can be described similarly to strain rate dependence on activation energy and temperature in creep studies. is valid for all values of stresses.The activation energy, Q d , obtained from the above three expressions is expected to be comparable.Typically, material parameters in equation (1) are obtained at peak flow stress [33][34][35] which is considered as independent of strain.For better prediction, temperature-dependent self-diffusion coefficient and Young's modulus are also incorporated sometimes in the hyperbolic representation of flow stress in equation (1) [36,37].
All representations of flow stress are collectively denoted by the symbol z.In addition, strain rate is represented as x ln  e = and temperature as y = 1/T .According to equation (1), apparent activation energy Q d is given by Calculation of materials constants is based on assumptions that plots z versus x and z versus y are parallel straight lines so that the Arrhenius relationship can be used to obtain Q d .In other words, activation energy Q d and other material constants that appear in equation (1) should be independent of strain rate or temperature.This also implies that fitted straight lines of z versus x and z versus y should be parallel so that the slopes are true constants.However, this is usually not the case, and material constants and activation energy Q d are generally reported as average values.It is also argued that since the role of microstructure is not considered, the fitting parameters are apparent materials constants [36].
It is observed that the variation of z with x and y can be nonlinear, particularly at lower or higher values of Z. Intermediate values of Z give a largely linear relationship with flow stress.One way to overcome this is to fit the entire temperature range using different segments so the individual segment can still be a straight line even if the entire temperature range shows nonlinear behavior.Each segment has a separate set of material constants, including activation energy.This approach was adopted in [38].The second approach is the nonlinear fitting of flow stress with temperature and strain rate.This is done by assuming material parameters and activation energy Q d in equation (1) as functions of ln  e and 1/T, for example, see [39].In this approach, the activation energy is a function of strain rate and temperature, unlike in the case of linear fitting.The primary objective of this investigation is to apply the multivariate fitting of variations in flow stress such that the materials constants in these models are independent of T and  e over the entire experimental range.This contrasts conventional constitutive models in which flow stress is modeled separately with T and  e.Two kinds of multivariate nonlinear fitting approaches are proposed, one in which activation energy is a function of temperature and strain rate and the other in which activation energy is independent of temperature and strain rate.

Conventional and modified constitutive models
It may be noted that the primary focus of this investigation is to model variations in flow stress; equation (1) is used in the following form.
Conventionally, z is fitted with x and y separately, hence, linearly fitted curves of z versus y (for different x) and z versus x (for different y) can be expressed as Since material parameters in equation ( 6) is obtained for a given y, the values of a 0 and b 0 are typically different for different values of y.Similarly, parameters a 1 and b 1 depend on x.The above linear models are collectively denoted A. For a better fitting, nonlinear fitting can be applied instead of the linear fitting given above

( ) = + +
The above models are the quadratic models for single-variable fitting, collectively designated as B. As mentioned for equation (6), a 0 , b 0 , and c 0 depend on y and a 1 , b 1 , and c 1 depend on x.Activation energy Q d is calculated by taking the ratio of the average values of slopes in equation (2).It may be emphasized that apparent activation energy for all constitutive models in the present investigation is calculated using equation (2).Equations ( 6) and ( 8) are examples of single-variable fitting meaning that curves z versus x and z versus y are fitted separately for a given value of y and x, respectively.Since z is a function of x and y, for a given alloy condition, multivariate fitting can also be applied; that is, z can be fitted simultaneously with x and y.In such cases, all fitting parameters will be independent of x and y.
Collectively equation (1) can be written as Here, a is a set of coefficients of x, b is a set of coefficients y, and c is a set of constant terms of equation (1).For the above equation, the apparent activation energy, according to equation (2), is given by This equation represents z as a linear function of Zener-Holloman parameter, represented as . As in the case of single-variable models, this can be extended to a quadratic relationship.The linear and nonlinear dependence of flow stress (z) on the Zener-Holloman parameter can be written as, Equation (11) represents a 3D plane and equation (12) represents 3D surfaces; these models are known denoted by F. There are 3 material constants in equation (11) as against 4 in equation (6) and 4 material constants for equation (12) as against 6 in single-variable quadratic fitting given in equation (8).Separate two-dimensional plots (z versus x and z versus y) based on equation (11) will be parallel straight lines.For models given in equation (11) and equation (12), Q d and other material parameters are independent of x and y.The generalized homogeneous second-order polynomial of x and y can be written as, z a a x a y a xy a x a y G: As in the case of F models, material parameters for equation (13) are independent of x and y.However, the apparent activation energy for equation (13) is not constants but function of x and y.It can be calculated as an average value, as in the case of single-variable fitting, for all experimental data point pairs (x, y).
The following notations are used for partial derivatives of z with respect to x and y.
Accordingly, the expressions for m s and b s for single-variable and multivariate models are given by Thus, for linear models F 0 , A e , B e , A t , and B t , the slopes m s and b s are constants, independent of x and y.On the other hand, for nonlinear models, the slopes are functions of x and y.Furthermore, except for models F 0 and F 1 , Q d is also a function of x and y.Based on the above two equations, values of Q d can be calculated according to equation (2) by taking the ratio of the average values of b s and m s .Another method is calculating Q d at each pair of (x, y) and then taking the average.
Several sample data sets  were chosen from the published literature to compare the prediction accuracy of modified constitutive models mentioned above.Sample data sets were randomly chosen and consisted of only metallic materials, both ferrous and non-ferrous alloys.All fitting in the present investigation were done using Scipy [86] open-source library of Python programming language [87].
Three statistical parameters are used to determine the prediction accuracy and compare different models.The correlation coefficient R 2 is useful for estimating the predictive capability of linear models; however, a large R 2 value may not always indicate a better-fitting model.The coefficient of determination is given by [88] where N is the number of data points, z i act is an experimental data point, and z i pred is the corresponding predicted value for a given model.In above equation, z act ¯is the average value of all experimental flow stress of a given sample data and z pred ¯is the average value of the corresponding predicted values.Another error parameter, root mean square error (δ r ) is given by Another statistical parameter, average absolute relative error (δ a ), is calculated using the following formula.

Comparison of prediction accuracy
The average values of three statistical parameters are plotted in figure 1 for two forms of flow stress, viz.z ln s = and z = σ; R 2 is shown in figure 1(a), δ a is shown in figure 1(b), and δ r is shown in figure 1(c).Based on higher values of R 2 and lower values of δ a , δ r , it is clear that models B and G have the highest accuracy among all models.Model F 0 showed the lowest accuracy among all models.For z ln s = , the average values of R 2 for A, B, F 0 , F 1 , and G models are 0.9851, 0.9965, 0.9727, 0.9924, and 0.9954, respectively.For z = σ, corresponding values are 0.9845, 0.9965, 0.9785, 0.9933, and 0.9955, respectively.Thus, both forms of flow stress have almost equal prediction accuracy in terms of R 2 (figure 1(a)).For z ln s = , the average values of δ a for A, B, F 0 , F 1 , and G models are 4.76, 1.95, 8.73, 4.49, and 3.49%, respectively.The corresponding values for z = σ are 6.34, 2.23, 14.08, 5.81, and 5.13%, respectively.Hence, when δ a is taken as a measure of prediction accuracy, logarithmic form (z ln s = ) of flow stress has better prediction accuracy as compared to z = σ.This evident from figure 1(b).In terms of δ r , both forms of flow stress have nearly equal prediction accuracy except for model F 0 ; for F 0 , z ln s = has lower prediction accuracy (δ r = 21.76%)than z = σ (δ r = 14.16%).
Figure 2 shows histograms of statistical parameters for all the sample data sets for z ln s = .The distribution of values of statistical parameters shown in figure 2 also confirms that models B and G have the highest prediction accuracy while linear models A and F 0 have the lowest accuracy.
The distribution of the values of R 2 is identical for z = σ (figure 3(a)) and z ln s = (figure 2(a)).It is also obvious that model B showed the highest accuracy for among all models (figure 3), similar to z ln s = (figure 2).
In terms of parameter δ a , the number of sample data sets showing values in the range of 5%-10% is higher for z = σ (figure 3

Linear variations in flow stress
Hot deformation of two experimental Al-Mg-Si alloys was investigated using uni-axial hot compression tests in the temperature range of 683-823 K and strain rate in the range of 1-125 s −1 [40].Among these two alloys, an experimental dataset corresponding to an Al-Mg-Si alloy with minor additions of Mn and Cr was chosen (figure 5(a) in [40]) for the application of conventional and modified constitutive equations described in section 2. The flow stress was reported as the mean stress value between the strain of 0.3 and 0.7.Experimental data extracted was for z ln s = from which the corresponding data for z = σ was computed.This sample data set will be referred to as D 1 .
Three statistical parameters viz.R 2 , δ a , and δ r , for all forms of flow stress, are listed in table 1 for D 1 .It is observed that the value of R 2 is greater than 0.99 for linear and nonlinear models; the values of R 2 for all models are nearly identical.As is the case with the overall average values of δ a and δ r (described in the previous section), F 0 shows the highest values for z ln s = and z = σ (table 1).For z ln sinh( ) as = , F 1 and G exhibit marginally higher values of δ a than F 0 .Also, as can be seen from table 1, multivariate models F 0 , F 1 , and G have lower prediction accuracy than single-variable linear and quadratic models (as indicated by relatively higher values of δ a and δ r ).
As per δ a , nonlinear single-variable model B show 36, 45, and 12% improvement in the prediction accuracy over linear single-variable model A for z ln s = , z = σ, and z ln sinh( ) as = , respectively.Model F 1 has 29, 44, and −9% improvement in the prediction accuracy (in terms of δ a ) over linear single-variable model A for , respectively.Thus, the highest improvement in prediction accuracy is observed for z = σ, and, improvement in accuracy of nonlinear single-variable and multivariate models are identical for z = σ and z ln s = .Multivariate models F 1 and G show around 10% reduction in the prediction accuracy over planar multivariate model F 0 .
For all forms of flow stress, material parameters are listed in table 2 as a function of the Zener-Holloman parameter for F and G models.Unlike for A and B models, F models directly give relationships between flow stress and the Zener-Holloman parameter.
Relationships given in table 2 are plotted in figure 4 for three forms of flow stress; flow stress is plotted as functions of Zener-Holloman parameter.It is clear that for z ln sinh( ) as = , fitted curves are linear in nature which is not the case for z = σ and z ln s = .It is generally observed that the variations in flow stress for all values are linear for hyperbolic representation.For z ln s = and z = σ, the prediction of flow stress is relatively less accurate for lower and higher values of Z (figure 4).
The results of single-variable fitting of D 1 is shown in figure 5.It is obvious that the variations in flow stress with T and  e are almost linear.For z ln s = , linear and nonlinear fitted curves are nearly identical for z versus x and z versus y and showed almost linear behaviour, particularly at higher temperatures and lower strain rates.For z = σ, non-linearity is higher compared to the other two forms of flow stress.The relative values of statistical parameters (see table 1) for three forms of flow stress also confirm that z = σ has lower prediction accuracy for linear and nonlinear models.Nonlinear behaviour is typically observed for higher strain rates and lower temperatures.For z versus x for different y, fitted straight lines are nearly parallel, except for 683 K and 125 s −1 (figure 5).
Variations in flow stress, modelled using F and G are shown in figure 6 for z ln s = and z = σ.The prediction of flow stress for z = σ is less accurate for low T, high  e (that is, high Z values) as well as for high T and low  e (that is, low Z values).The accuracy of fitting for planar multivariate model F 0 is higher for z ln s = (figure 6   Table 1.Statistical parameters for all constitutive models for D 1 .

Nonlinear variations in flow stress
To illustrate fitting using proposed nonlinear constitutive equations, hot deformation data from [57] was selected.Hot deformation behaviour of alloy D9 was studied using isothermal compression in the temperature range of 1123-1523K and strain rate in the range of 10 −3 -10 2 s −1 .Flow stress at a true strain of 0.1 was considered to evaluate material parameters for constitutive models.This sample data set is designated as D 2 .Table 3 lists the values of statistical parameters R 2 , δ a , and δ r for D 2 for all forms of flow stress.It is clear from the table that, for each model, the accuracy of fitting for D 3 is considerably less than the corresponding values for D 1 (table 1).The increase in prediction accuracy for B over A are 76, 59, and 58% for z ln s = , z = σ, and z ln sinh( ) as = , respectively.Thus, as in the case of D 1 , flow stress in the logarithmic form has the greatest improvement in accuracy among all three forms The improvement in fitting accuracy for F 1 as compared F 0 are 75, 18, and 43% for z ln s = , z = σ, and z ln sinh( ) as = , respectively.Thus, the single-variable quadratic model B shows significant improvement for all three forms of flow stress.Conversely, the prediction accuracy of z = σ for multivariate models have not improved much.Further, the increase in prediction accuracy for G over F 1 is not significant in comparison to the improvement for F 1 over F 0 .As in the case of D 1 , B has the highest prediction accuracy among all models.
The fitted curves for single-variable models A and B shown in figure 7 indicate that the variations in flow stress with x and y is not as linear as that observed for D 1 (see figure 5).Non-linearity is more pronounced for z ln s = and z = σ as compared to z ln sinh( ) as = . Also, it is clear from figure 5 that fitted curves are almost parallel only for z ln sinh( ) as = , except for 1123 K and 10 and 100 s −1 strain rates (figure 7).The fitted curves for A (linear) and B are nearly identical except at 1123 K and 10, 100 s −1 strain rates (figure 7).
For D 2 , constitutive equations for multivariate models are tabulated in table 4 for all three forms of flow stress.Figure 8 shows graphical representations of constitutive equations along with experimental data (shown as solid-symbols); flow stress is plotted as a function of Z.It is clearly seen that the variation in flow stress as function of Z is not linear unlike that for D 1 (shown in figure 4).For z ln s = , it is observed that increase in flow stress with Z decreases sharply for higher values Z (figure 8(a)).For z ln sinh( ) as = , flow stress reduces with the increase in Z (figure 8(b)), however, the reduction is not as sharp as in the case of z ln s = (figure 8(c)).On the other hand, flow stress increases continuously for z = σ are more.The fitting of flow stress using the linear function of Z is not accurate for D 2 .
Poor-fitting using conventional Arrhenius model at lower (<0.1 s −1 ) and higher (>1 s −1 ) strain rate was reported in [57], as shown in figures  deformation conditions except at 1123 K with strain rates of 10 and 100 s −1 [57].Inaccurate prediction of flow stress for higher strain rates was attributed to adiabatic heating during hot deformation leading to flow softening [57].Three-dimensional fitted curves obtained using multivariate models F 0 , F 1 , and G are shown in figure 9 for z ln s = and z = σ.It is obvious that the planar multivariate model F 0 do not provide accurate fitting (figures 9(a) and (b)), as confirmed by higher values of δ a for F 0 (see table 3).Comparatively lesser predictive accuracy for z ln s = as compared to z = σ for F 0 is also evident from the fitted curves in figures 9(a) and (b).As shown in figure 9, the improvement in accuracy for multivariate model F 1 over F 0 is substantial for z ln s = (figure 9(c)) as compared to z = σ (figure 9(d)).This is also confirmed by the corresponding values of δ a (table 3).Though, there is marginal improvement in the value of δ a for G as compared to F 1 , no visible difference exists between fitted curves for G (figure 9 In contrast to modified constitutive equation proposed in [57], nonlinear single-variable (B, see figure 7) model and multivariate (F 1 , G, see figure 9) models accurately predict flow stress, particularly for z ln s = , at all temperatures and strain rates investigated in [57].The nonlinear models proposed in the present investigation successfully predict softening even at higher strain rates.
6. Material parameters for multivariate models

Gaussian distribution of parameters
To understand variations in their numerical values, the material parameters of F 0 , F 1 , and G are fitted using the following form of Gaussian distribution.
Here, μ is the expected value of the normal distribution of parameter u, and p is the corresponding probability.The probability p is calculated as a fraction of a particular range of a given material parameter among the entire distribution.The probability corresponding to the expected value of μ is p 0 + A while Full Width Half Maximum (FWHM) is given by w 2 ln4.The probability p for material parameters for models F 0 , F 1 , and G are shown in figures 10 through 14 for all three forms of flow stress.For z ln s = and F 0 , bimodal distribution of m s has two expected values of 0.13 and 0.20 with corresponding probability of 0.28 and 0.20, respectively (figure 10    For b s corresponding to z ln sinh( ) as = and F 0 , the value of R 2 for fitting using equation ( 21) is 0.91, the expected value is μ = 8774 with the corresponding probability of p = 0.27 (figure 10(b)).Of all values, 85% lie in the range of 5038-14038.
In contrast to z ln s = and z ln sinh( ) as = , the spread of m s and b s show better normal distribution for z = σ as indicated by higher values of R 2 for both m s (R 2 = 0.96) and b s (R 2 = 1.00), as shown in figure 10(b).Also, among all forms of flow stress, for model F 0 , the probability corresponding to expected values of m s and b s is the highest for z = σ.The expected value of m s for z = σ for model F 0 is μ = 13.3217 and the corresponding probability is p = 0.36.Of all values of b s , 69% lie in the range of 11-23, the corresponding range for β (=1/m s for z = σ, according to equation ( 4)) is 0.0434-0.091.The average value of R 2 for the fitting of m s and b s using equation ( 21) are 0.95, 0.94, and 0.98 for z ln s = , z ln sinh( ) as = , and z = σ, respectively.Thus, m s and b s for z = σ have the best fitting and the highest probability for the expected value among all forms of flow stress.
The distribution of parameters b¢ and c¢ for model F 1 (see equation ( 12)) for different forms of flow stress is almost identical to the parameters m s and b s for F 0 ; parameters for z = σ have the highest probability for at the expected values as compared to the other two forms for flow stress (figure 11).The average value of R 2 for the fitting of m s and b s using equation ( 21) are 0.96, 0.92, and 0.97 for z ln s = , z ln sinh( ) as = , and z = σ, respectively.Thus, as in the case of F 0 , the parameters of F 1 for z = σ show better fitting using normal distribution (equation ( 21)).The expected values and corresponding probabilities for b¢ are μ = 0.50, p = 0.36, μ = 0.21, p = 0.27, and μ = − 16.71, p = 0.41 for z ln s = , z ln sinh( ) as = , and z = σ, respectively.Whereas, the expected values and corresponding probabilities for c¢ are μ 1 = − 0.009, p 1 = 0.14, μ 2 = − 0.004, p 2 = 0. , and z = σ, respectively.These values are comparable with the corresponding values for models F 0 and F 1 .Also, the probabilities associated with the expected values of parameters are higher comparatively higher for model G in comparison to F 0 and F 1 .It can be observed (figures 12 through 14 ) that the parameters associated with temperature (a 2 and a 5 ) have higher probabilities corresponding to the expected values as compared to the parameters associated with strain rate (a 1 , a 3 , a 4 ) for all forms for flow stress.
For z ln s = , of all sample data sets, 89% values of a 1 lie in the range of 0.3-0.7,97% values of a 2 lie in −8 × 10 4 to 8 × 10 4 , 90% values of a 3 lie in the range of −800 to −200, 81% values of a 4 lie the range of −0.016 to −5 × 10 −4 , and 93% values of a 5 lie in the range of −2.5 × 10 7 to 7.5 × 10 7 .
For z ln sinh( ) as = , of all sample data sets, 80% values of a 1 lie in the range of −0.05-0.55,97% values of a 2 lie in −1 × 10 5 to 1 × 10 5 , 67% values of a 3 lie in the range of −250 to 350, 77% values of a 4 lie the range of −0.0125 to −0.0025, and 97% values of a 5 lie in the range of −4 × 10 7 to 1.2 × 10 8 .
For z = σ, of all sample data sets, 77% values of a 1 lie in the range of −75 to 25, 96% values of a 2 lie in −1 × 10 7 to 1 × 10 7 , 73% values of a 3 lie in the range of 2 4 to 8 × 10 4 , 83% values of a 4 lie the range of −0.95 to 0.85, and 89% values of a 5 lie in the range of −2.5 × 10 9 to 4.5 × 10 9 .

Parameters derived from constitutive models
Three parameters m s , b s , and Q d are derived from material parameters of constitutive models.It is observed that the average values of these parameters are comparable for different models for a given form of flow stress.In addition, the apparent activation energy Q d is comparable for different models and different forms of flow stress.The difference between the values of the three derived parameters for various models can be estimated by considering the values corresponding to Arrhenius models (A e and A t ) as baseline values and calculating statistical parameter δ a (using equation (20)), considering all sample data sets) for the remaining models.For example, for a given data set i, if the value of m s corresponding to Arrhenius model is m i and m B i for model B, the the δ a for this combination is given by Similar expressions are used for b s and Q d .
For z ln s = .The values of δ a for m s , with respect A, for B, F 1 , and G are 0.15, 0.86, and 0.16%, respectively; the smallest difference between the values of m s for different models, for a given data set, is observed for B and G. Though, models B, F 1 , and G are quadratic, the difference is between the values of m s is greater for F 1 .Similar observations are made for the values of δ a for b s and Q d ; the values of δ a for b s and Q d are in the range of 2.00%-    For z ln sinh( ) as = . The values of δ a for m s are 0.05, 0.48, and 0.06% for B, F 1 , and G respectively; these values are practically nil values.Further, as in the case if z ln s = , the values of δ a for b s and Q d are comparable and marginally lower (in the range of 1-1.40%) than the corresponding values for z ln s = .For z = σ.The values δ a for m s and b s are higher as compared to the other two forms of flow stress.For Q d , the value of δ a is higher than that for z ln sinh( ) as = and nearly equal to that for z ln s = .The maximum difference between the values of Q d , for the Arrhenius model, is 36%, observed for F 1 .When compared with z ln s = and the other forms of flow stress, the value of δ a for Q d is the highest (δ a = 8.30%) for model , the values of m s are the exactly same for all models for D 1 (table 5) whereas the values of b s are also almost identical for different models for all forms of flow stress.The values of m s marginally differs for z = σ for D 1 (table 5).Also, the values of Q d are the exactly same for z ln sinh( ) as = and nearly identical for z ln s = and z = σ for all models.The difference in the value of Q d despite nearly identical values of m s and b s is because Q d is a model parameter for multivariate models F 0 and F 1 .In contrast, for the other models, it is represented as the ratio of the average values of m s and b s .Thus, the values of Q d for all flow stress and all models are in a very close range of 178-183 kJ mol −1 .The reported value of Q d is 177 kJ mol −1 [40].The calculated value of n (in equation (1)) for D 1 in the present investigation is 5.56 whereas the reported value is 5.16 [40].
As in the case of D 1 , the values of m s are the same or almost identical for different models for a given form of flow stress (table 6).However, n contrast to D 1 (table 5), the values of b s and Q d are, though comparable, differs slightly for different models and different forms of flow stress (table 6).The values of Q d is in the range of 464-485, 504-507, and 482-493 kJ mol −1 for z ln s = , z = σ, and z ln sinh( ) as = , respectively.Thus, unlike in the case of D 1 , the values of Q d differ for different forms of flow stress.For z = σ, the values of Q d are in the closest range as compared z ln s = and z ln sinh( ) as = . For D 2 , the parameter n is common for ln s versus ln  e and ln sinh( ) as versus ln  e plots [57].The reported value of n is 8.8415 [57] and the calculated value in the current investigation is 9.09, while, the reported value of β in equation (1) is 0.07445 [57] and in the current investigation in 0.0784.The calculated values of n and β are the same for all models in the present investigation.However, the reported value of Q d is 638.571kJmol −1 [57], and in the current investigation, it is in the range of 482-507 kJmol −1 .Thus, the calculated value of Q d and the reported value differ significantly.It is worth noting that alloy D9 is austenitic Type 321 SS (Stainless Steel), and reported values of austenitic SS are typical in the range of 420-500 kJ mol −1 [35,36,62].Thus, the average value of Q d obtained in the present investigation is closer to the reported values.The deviation in values of Q d for the proposed models could be due to adiabatic heating at lower temperatures and grain boundary movements or change in dislocation mobility at higher temperature.) and temperature (y = 1/T) corresponding to multivariate models for z ln s = .For multivariate models, variations in slopes m s and b s with strain rate and temperature are shown in figure 15  It is worth noting that m s represents the rate of change in flow stress (z) with strain rate (x) at constant temperature (y).Thus, a higher value of m s indicates higher rates of increase in flow stress.It is observed that for D 1 (Figure 15(a)) and D 2 (figure 15(c)); the values of m s are the highest for higher temperatures and lower strain rates.The decrease in temperature and the increase in strain rate reduce m s for D 1 and D 2 .The nature of variations in b s is identical to that of m s (figures 15(b) and (d)).Therefore, the increase in strain rate and decrease in temperature reduces the rate of increase in flow stress.
For multivariate models, F 0 and F 1 , the value of Q d is constant, while for G, it is a function of strain rate and temperature.Variations in Q d for different forms of flow stress are shown in figure 16 for D 1 and D 2 .The nature of change in the values of Q d with strain rate and temperature is identical for all forms of flow stress for D 1 (figures 16(a), 16(c), 16(e)); higher values of Q d are observed lower temperatures and lower strain rates.For a given temperature, the increase in strain rate reduces Q d , while for a given strain rate, the decrease in temperature increases Q d .Since Q d is regarded as resistance to hot deformation, it is expected that the values of Q d will be higher for lower temperatures.The ranges of Q d for D 1 are 177-183, 175-182, and 170-183 kJ mol −1 for z ln s = , z ln sinh( ) as = , and z = σ, respectively.Thus, the value of Q d remains practically constant for the entire range of hot deformation investigated in [40].
In contrast to D 1 , the variations in Q d is different for different forms of flow stress; for z ln s = , a sharp increase in the values was observed for lower temperatures and higher strain rates (figure 16(b)).For flow stress in hyperbolic form, in contrast to D 1 , the values of Q d for D 2 are slightly higher for higher strain rates as compared to lower strain rates (figure 16(d)).On the other hand, for z = σ, the nature of change in the values of Q d is similar to that of D 1 ; the values of Q d are higher for lower temperatures (figure 16(f)).
The ranges of Q d for D 2 are −27-1217, 334-769, and 343-807 kJ mol −1 for z ln s = , z ln sinh( ) as = , and z = σ, respectively.Thus, the range of Q d for each form of flow stress is significantly wider for D 2 than for D 1 .Further, it is observed that, for D 2 , the values of Q d differ at the highest strain rate of 100 s −1 (for all temperatures) for different forms of flow stress.The wider range of D 2 for Q d is probably due to nonlinear variations of flow stress with strain rate and temperature as compared to D 1 .Larger variations in slope due to non-linearity are expected, leading to variations in Q d .
For single-variable models A and B, Q d cannot be expressed as an explicit expression.For multivariate models F 0 and F 1 , Q d remains constant for a given strain rate and temperature.The expression for Q d for model G is given by, Based on the above equation, it is difficult to predict the dependence of Q d on strain rate and temperature.Mathematical expressions for Q d at a given temperature and strain rate are reported earlier [39,89] by differentiating the Arrhenius model with sine hyperbolic representation given in equation (5).These expressions of Q d (given in [39,89]) are complex polynomials of strain rate and temperature, different than equation (22).
The main assumption of the Arrhenius-like model presented in equation ( 1) is linearity, which means that flow stress is proportional to temperature and strain rate.Q d is independent of strain rate and temperature in this case.However, because the slope of nonlinear curves is not constant, the incorporation of nonlinear models naturally introduces the dependence of Q d on strain rate and temperature.
The observation of departure from predicted linear behaviour at lower and higher temperatures implies that the mechanism has changed.It is probable that at these temperatures, dislocation interaction varies due to variations in mobility (decrease at lower temperatures, increase at higher degrees).This can cause a general change in flow behaviour.

Comparison of proposed models
According to figures 1 through 3, the model with the least number of parameters (F 0 ) has the lowest prediction accuracy.In contrast, the quadratic model B shows the highest prediction accuracy.Thus, the models analysed in the present investigation can be ranked in increasing accuracy as F 0 < A < F 1 < G < B. Though the number of material parameters for A is always higher than for F 1 , the latter shows comparatively higher accuracy.Thus, all nonlinear quadratic models have higher prediction accuracy than the linear models A and F 0 .This also confirmed by plots of flow stress versus Zener-Holloman parameters, shown in figures 4 and 8.A better fit means that a model reflects the underlying physical reality more correctly.The curvature introduced by quadratic models properly describes hot deformation, especially at low and high temperatures.Arrhenius-like linear models are only valid in a relatively small range of temperature and strain rate; the key advantage of the presented models is that they are valid throughout a greater range of temperature and strain rate.
A strain-compensated constitutive equation can be obtained by fitting parameters of equation (13) as a function of strain.Though the prediction accuracy of G is lower than that of B, the number of material parameters associated with G is less than that for B. The computational time associated with G is less than that for B because the former is a multivariate model.Further, for quadratic multivariate models, it is not required to fit flow stress in three different forms In addition, the term xy in G implies that flow behaviour is interdependent on time and temperature unlike the other models wherein flow behaviour with can be modelled separately for temperature and time.Thus, G describes physical phenomenon more accurately.Therefore, models B clearly does not reflect physical phenomena as precisely as G. Thus, for improved prediction accuracy, model G can be used to model flow stress variations during hot deformation.

Discussion
For hot deformation, activation energy Q d is referred as 'apparent activation energy', since hot deformation is a complex phenomenon involves simultaneous processes of generation of point line defects that are constantly annihilated by dynamic recovery and dynamic recrystallization, respectively.Usually, activation energy is compared with self-diffusion of a single element and it is because of this reason most of the time the reported values of Q d can be varied for a given alloy in the same hot deformation range.Hence, since many mechanisms operate simultaneously during hot deformation, Q d cannot be compared with typical activation energy associated with a single process (that is, single mechanism) which is usually done in reaction rate theory.
The apparent activation energy for hot deformation does not remain same for every strain levels, most of the hot deformation investigations fit activation energy as a function of strain.It does not remain constant over the entire hot deformation process.
In addition, it is also worth noting that it is not necessary that the Arrhenius rate equation is always followed, there are several reports in which the linearity of Arrhenius equation is violated, see for example [90].

Conclusions
A total of 70 sample data sets of hot deformation behaviour of alloys were analysed using a conventional Arrhenius-like model, single-variable quadratic model, and multivariate linear and quadratic models.The prediction accuracy of each model was evaluated in terms of coefficient of determination (R 2 ), Average Absolute Relative Error (AARE), and Root Mean Square Error (RMSE).Based on the present investigation, the following conclusions are drawn.
d is the activation energy of deformation and R = 8.314 J/mol • K is the universal gas constant.In equation (1), A 1 , A 2 , and A 3 are scaling factors, exponents n¢, n, and β, α are material parameters with n  a b ¢.It is generally observed that the expression j (b)) as compared to z ln s = (figure 2(b)).On the other hand, the distribution of δ r for all samples data sets are nearly identical for z = σ (figure 3(c)) and z ln s = (figure 2(c)).

Figure 1 .
Figure 1.Shows average values for statistical parameters (a) R 2 , (b) δ a , and (c) δ r for all models, for two forms of flow stress viz.z ln s = and z = σ.
(a)) than z = σ (figure 6(b)).It is clear that the improvement in the prediction accuracy for F 1 over F 0 is more pronounced for z ln s = (figure 6(c)) as compared to z = σ (figure 6(d)).The accuracy of fitting for model G is almost equal for z ln s = (figure 6(e)) and z = σ (figure 6(f)).These observations are corroborated by statistical parameters corresponding to these models (table 1).It can be seen from figure 5 that for σ > 57 MPa, the variations in flow stress become nonlinear for z ln sinh( ) as = and z = σ.The value of α in equation (5) for D 1 is 0.023 which gives ασ ¬ 1.3.The non-linear

Figure 2 .
Figure 2. Shows histogram of statistical parameters (a) R 2 , (b) δ a , and (c) δ r corresponding to z ln s = for all models, for all sample data sets analysed in the present investigation.

Figure 3 .
Figure 3. Shows histogram of statistical parameters (a) R 2 , (b) δ a , and (c) δ r corresponding to z = σ for all models, for all sample data sets analysed in the present investigation.

Figure 4 .
Figure 4. Shows three-dimensional representations of results of multivariate fitting of sample data from D 1 , corresponding to F and G for variations in z with x and y, for all forms of z.Solid-symbols are the values of Z calculated from experimental data in all figures.

Figure 5 .
Figure 5. Shows the variations in flow stress, for D 1 , with strain rate x, temperature y and fitted curves for constitutive models corresponding to single-variable fitting.The experimental data are represented as symbols.The dashed lines represent linear (A), and the solid lines represent quadratic fitting (B) curves.

Figure 6 .
Figure 6.Shows three-dimensional representations of results of multivariate fitting of sample data from D 1 , corresponding to F and G for variations in z with x and y, for all forms of z.Solid-symbols are experimental data in all figures.

Figure 7 .
Figure 7. Showing results of single-variable constitutive models for experimental data (shown as open symbols) for D 2 .The dashed lines represent linear fitting (A models), and solid lines represent quadratic fitting (B models).

Figure 8 .
Figure 8. Shows three-dimensional representations of results of multivariate fitting of sample data from D 2 , corresponding to F and G for variations in z with x and y, for all forms of z.Solid-symbols are the values of Z calculated from experimental data in all figures.
(a)).It is observed that 87% of the sample datasets show the values of m s in the range of 0.11-0.23.According to equation (3), n m 1 ; s ¢ = therefore, the value of n¢ lies in the range of 4.44-9.52.The value of R 2 for fitting using equation (21) is 0.91.

Figure 9 .
Figure 9. Showing fitted three-dimensional curves (for multivariate models) for D 2 along with experimental data (as solid-symbols) corresponding to z ln s = for model (a) F 0 (c) F 1 , (e) G and z = σ for model (b) F 0 (d) F 1 , (f) G.

Figure 10 .
Figure 10.Normal distributions of slopes m s and b s , corresponding to model F 0 , for all sample datasets and all forms of flow stress.Two peaks represent bimodal distributions, and solid-lines represent curves fitted using equation(21).
Similar to m s for z ln s = , the distribution for m s for z ln sinh( ) as = is bimodal (figure 10(b)) with two expected values of μ = 0.1679 and μ = 0.2850 and the corresponding probabilities p = 0.26 and p = 0.18.It is also observed that 86% of the values of b s lie in the range of 0.15 to 0.31 with m s = 0.15 and m s = 0.19 have equal probability of p = 0.23 (figure 10(b)).Generally, the slope m s is represented as n defined in equation (5); thus, 86% of the values of n fall in the range of 3.23-6.67with the expected value of μ = 3.51 and μ = 5.96.The R 2 for the fitting using equation (21) is 0.97.
28, μ = − 1.46 × 10 −4 , p = 0.29, and μ = 0.42, p = 0.49 for z ln s = , z ln sinh( ) as = , and z = σ, respectively.The distribution of parameters for model G for three forms of flow stress are shown in figures 12 through 14 along with the expected values, probabilities, and R 2 values.The average value of R 2 for the fitting of all parameters for G using equation (21) are 0.94, 0.95, and 0.97 for z ln s = , z ln sinh( ) as =

Figure 11 .
Figure 11.Normal distributions of slopes m s and b s , corresponding to model F 1 , for all sample datasets and for all forms of flow stress.Two peaks represent bimodal distributions, and solid lines represent curves fitted using equation(21).

Figure 12 .
Figure 12.Normal distributions of material parameters of model G for all sample datasets corresponding to flow stress z ln s = .Solid-symbols represent parameters and corresponding probability while solid-lines represented curves fitted using equation(21).

Figure 13 .
Figure 13.Normal distributions of material parameters of model G for all sample datasets corresponding to flow stress z ln sinh ln ( ) a s = .Solid-symbols represent parameters and corresponding probability while solid-lines represented curves fitted using equation (21).

2 .
35%, and, F 1 has the largest difference between the respective values of b s and Q d , as in the case of m s .Further, the values of δ a for b s and Q d are comparable.

F 1 .
Parameters m s b s , and Q d Slopes m s , b s , and Q d are tabulated in tables 5 and 6 for D 1 and D 2 , respectively.The values of m s and b s are the average values of individual values at different data points.The lowest value of δ a for Q d , with respect to z ln s = , is observed for models B and G for z ln sinh( ) as = .For for z ln s = and z ln sinh( ) as =

Figure 14 .
Figure 14.Normal distributions of material parameters of model G for all sample datasets corresponding to flow stress z = σ.Solid symbols represent parameters and corresponding probability while solid lines represented curves fitted using equation (21).

Figure 15 .
Figure 15.Variations in slopes m s and b s with strain rate (x ln  e =) and temperature (y = 1/T) corresponding to multivariate models for z ln s = .
for D 1 (figures 15(a) and 15(b)) and D 1 (figures 15(c) and (d)).Clearly, for F 0 , the values of m s and b s are constants while for F 1 and G, the nature of variations in m s and b s are identical for D 1 and D 2 .This is evident from the expressions for m s and b s for F 1 and G models (equation (16) and equation (17)); the expressions for m s and b s represent a three-dimensional plane.

Figure 16 .
Figure 16.Variations in Q d with strain rate (x ln  e = ) and temperature (y = 1/T) corresponding to model G for D 1 and D 2 .

1 .
Nonlinear single-variable and multivariate constitutive equations proposed in the present investigation have improved flow stress prediction accuracy compared to the conventional Arrhenius-like model.Following second-order multivariate equation (z is a given form of flow stress, x ln  e = , y = 1/T) can be used to model variations in flow stress during hot deformation.

2 . 3 .
The average values of rate of change in flow stress with strain rate and temperature are almost identical for nearly linear variations in flow stress.In contrast, for nonlinear variations in flow stress, these values are comparable.As in the case of rate of change in flow stress, the average values of the apparent activation energy (Q d ) remain the same for different models for linear variations in flow stress.A simpler expression for variations in Q d with strain rate and temperature is derived.

Table 2 .
Constitutive equations for multivariate models for all forms of z.

Table 3 .
Statistical parameters for all constitutive models for D 2 .

Table 4 .
Constitutive equations for multivariate models for all forms of z.

Table 5 .
Slopes and statistical parameters for all constitutive models for different representations of flow stress.The activation energy Q d is in kJ mol −1 .

Table 6 .
Slopes and statistical parameters for all constitutive models for different representations of flow stress.The activation energy Q d is in kJ mol −1 .