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Vol. 8. Issue 3.
Pages 2481-3388 (May - June 2019)
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Vol. 8. Issue 3.
Pages 2481-3388 (May - June 2019)
Original Article
DOI: 10.1016/j.jmrt.2019.04.007
Open Access
General modeling of flow stress curves of alloys at elevated temperatures using bi-linearly interpolated or closed-form functions for material parameters
Mohd Kaswandee Razali, Missam Irani, ManSoo Joun
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Corresponding author.
Gyeongsang National University, Gyeongsangnam-do, Jinju-si, Gajwa-do, Republic of Korea
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Figures (13)
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Tables (8)
Table 1. Summary of two models.
Table 2. Flow stress information matrix for AHS-2.
Table 3. Experimental data constants of a CFF model for AHS-2.
Table 4. Maximum flow stress errors when only Ch and Cs are expressed as closed-form functions.
Table 5. Errors in peak strain, peak stress and steady-state stress fitted by Eqs. (12)–(14).
Table 6. Maximum and average errors and their standard deviations of different models used to evaluate 20MoCrS4 at a strain rate of 0.1/s.
Table 7. Experimental data constants of the CFF model for AISI 1020.
Table 8. Experimental data constants of the CFF model for AZ80.
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In this study, a promising model is presented to describe the flow stress curves with acceptable accuracy as well as generality. In this model, peak strain, peak stress, steady-state stress and hardening and softening constants are uncoupled and expressed as either bi-linearly interpolated or closed-form functions of temperature and strain rate. A practical method to obtain material constants is developed to render the model applicable in practice, which is based on optimization scheme. The flow stress curves of four materials are obtained using the new model and compared with experiments or the other models at some specific temperatures and strain rates. The comparison has revealed that the new model is feasible and general to describe the flow stress curves of various metals and alloys at different temperatures and strain rates with acceptable accuracy.

Flow stress model
Closed-form function
Bi-linear interpolation
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In recent years, metal researchers have sought high-strength materials to optimize the safety and lifespan of futuristic vehicles. Accurate prediction of phenomena occurring during plastic deformation of such materials is thus essential. Various finite element methods predict metallurgical phenomena during plastic deformation with acceptable accuracy [1–4] when material and friction models are reasonable.

Flow stresses of metals and alloys [5] and friction along mechanically complicated interfaces [6,7] are important in this context. It has been known that the flow stress is sensitive especially to temperature and strain rate [8]. Compared to elevated temperature, flow stresses at room temperature are simple and weakly dependent on temperature and strain rate. To the contrary, flow stresses at the elevated temperature are very complicated, which are necessary to predict the material properties of metal products, including grain size, hardness and strength [9]. Owing to the importance, the material identification has attracted significant interest of material and/or mechanical researchers for more than five decades. Several practical approaches [10–13] have thus been developed to describe the flow stress behavior of metals and alloys.

However, flow stresses at elevated temperatures are complicated such that simple modeling is inadequate. For example, hot materials harden at small strains (below the peak strain) and soften at higher strains because of recovery and recrystallization, which are the dominant restoration processes during microstructural evolution [14–16]. Accuracy of flow stress data critically influence not only on predicted plastic deformation but also on predicted metallurgical phenomena to be considered in numerically modeling metal-forming processes [17,18]. Generally, an ideal flow stress model for metals and alloys should be able to describe strain hardening and softening behaviors with acceptable accuracy.

Prior to 1960, many pioneers including Ludwik [14], Hollomon [19], Swift [20], Johnson and Cook [21], Hensel-Spittel [22], Voce [12] and Misaka and Yoshimoto [23] developed mathematical flow stress models of selected materials over certain ranges of temperature and strain rate. Their studies were based on either phenomenological constitutive model or physics-based constitutive model. All phenomenological models use limited numbers of material constants [12] while physics-based models employ more constants which should be obtained via time-consuming mathematical procedures using large experimental data sets [24].

The work of Voce attracted considerable attention in the 1970s when large-strain flow stress curves were analyzed. The Voce model features two equations describing strain hardening and softening associated with dynamic recrystallization. The volume fraction of dynamic recrystallization plays a key role [12,18,25]. To generalize the equations, Voce expressed material constants as functions of Zener–Hollomon parameters. Thus, the number of constants required to define flow behavior increased enormously. Although the work contributed greatly to the understanding of plastic deformation, practical applications have been scarce. However, recent advances in applied computational material science have triggered a revival of the approach. Cingara and McQueen [26] proposed a well-known equation for a flow stress curve up to the peak stress point. Peak stress values were derived using the Sinh-Arrhenius constitutive equation.

In a recent study, the modified power-law model [5], relating stress (σ) and (ε˙); (σ=Cε˙m), well-represented the complicated flow stress curves of hot metals, in which C and m values at any strain and temperature, were interpolated using constants defined at particular strains and temperatures. However, this does not afford insight into the metallurgical characteristics of the material. Moreover, it is computationally inefficient and it does not encourage process design engineers to be creative. Lin et al. [13] developed another constitutive model based on dislocation density. Many material constants are also required in this physics-based constitutive model. Twelve years ago, Ebrahimi et al. [17] presented a flow stress model based on both Cingara and McQueen model and Voce model, taking a mathematical and graphical approach toward determination of material constants from peak stress, peak strain and steady-state stress. By reference to metallurgical concepts, the constants were specified as power-law functions of temperature and strain rate and generalized flow stress equations were then derived. The material constants (coefficients and exponents of power-law functions) were obtained by observing the behaviors of lines or curves derived from graphical experimental data. This approach was used to study flow stress behavior of a Ti-IF steel, yielding generally acceptable results. However, the maximum error reached almost 10%, which cannot be neglected. Also, graphical approaches are computer-unfriendly and difficult to be applied in practice.

In this study, a new model is presented, which expresses the flow stress as either bi-linearly interpolated function (PLF) or closed-form function (CFF) for applicability and accuracy. The model is based on the works of Ebrahimi et al. [17,27] and Cingara and McQueen [26]. The appropriate number of material constants are used to improve generality and accuracy of the model and a systematic approach of identifying them are also suggested for practical application. The model is applied to analyze various metals and alloys, including the steels 20MoCrS4 [12] and S20C [3], the aluminum alloy AHS-2 [3] and the magnesium alloy AZ80 [27]. The predicted flow stresses are compared with experiments found in the literature and/or those calculated using earlier models.

2The variants and limitations of Voce flow stress model

Fig. 1 shows a schematic diagram of a typical flow stress curve at an elevated temperature; restoration occurs during deformation. The curve is characterized by three major sets of experimental data: peak strain, peak stress and steady-state stress [28].

Fig. 1.

Schematic diagram of a flow stress curve.


Voce suggested the use of a metallurgical approach to represent flow stress as a function of strain, strain rate and temperature with an emphasis on microstructural evolution as follows [12]:

where σp, σo, σs, Z, Xdyn, ε, εp and εs are the peak stress, initial stress, steady-state stress, a Zener–Hollomon parameter, the volume fraction of dynamic recrystallization, strain, peak strain and steady-state strain, respectively. b,co, do, d1, d2, c1, c2, eo, e1, e2, fo, f1, f2, go, g1 and g2 in the above equations are material constants to be obtained for the specific material. Unfortunately, no well-established method for identification of such constants is available and applications of the model are few in number, despite the model being based on metallurgical theory. Considering this difficulty, certain practical approaches based on phenomenological models with fewer material constants have been developed. Cingara and McQueen [26] suggested use of the following equation for flow stress curves prior to development of peak stress, thus in the strain-hardening region:

This equation has been employed by several researchers, including Ebrahimi et al. [17], Fereshteh-Sanaiee et al. [27] and Meyer et al. [12]. Note that the above function passes through the peak stress point (εp, σp) and the slope vanishes at this point, regardless of the Ch value, implying that the peak stress point is an extremum. The material constant Ch can thus be used to control the curve pattern in the strain-hardening region.

The flow stress in the softening region where the flow stress decreases prior to attainment of steady-state stress (attributable to dynamic recrystallization and recovery) was expressed by Ebrahimi et al. [17] as:

where Cs is a material constant that must be determined. Noted that their theoretical background is different even though Eqs. (1) and (9) look similar. Based on both phenomenological and physics-based approaches, Ebrahimi et al. [17] derived a general systematic scheme using defined experimental plots showing relationships between variables and/or parameters, not only to calculate Ch and Cs but also to formulate εp, σp and σs as functions of temperature and strain rate. The scheme assumes that the experimental plots are linear. The fitted linear functions were used to interrelate variables and/or parameters to identify flow stresses at particular strains, strain rates and temperatures.

Notably, Eq. (9) also passes through the peak stress point of the curve, i.e., (εp, σp), regardless of the Cs value and the slope at the peak stress point vanishes for all Cs. Thus, the flow stresses formulated by Eqs. (8) and (9) are naturally C1-continuity functions regardless of the Ch and Cs values, implying that the stresses can be calculated by focusing only on reductions in stress differences between the model functions and experiments over the strain ranges of interest.

Fig. 2 applies the Ebrahimi's model to identify the steel AISI 1020 and the fitted curves agree rather well with the experiments. The maximum error is 10% at temperature of 950°C and strain rate of 10.0/s and the average errors, the mean errors of all sampled points at strain rates of 0.1 and 10.0/s are 4.56 and 5.93%, respectively. All variables and plot parameters exhibit good linearity [3,17], as shown in Fig. 3. Fig. 4 shows purposely organized characteristic plots for AZ80, which exhibit high-level non-linearity compared to those of AISI 1020. As shown in Fig. 5, the error between the Ebrahimi's model and experiments is not negligible. The maximum error reaches 18.0% at temperature of 220°C and strain rate of 0.10/s. Moreover, the average errors are 7.12 and 9.63% at strain rates of 0.01 and 0.10/s, respectively. The Ebrahimi's model is applicable only to the materials for which the characteristic plots are linear.

Fig. 2.

Comparison of experiments and Ebrahimi's model for AISI 1020 at two strain rates. (a) 0.1/s and (b) 10.0/s.

Fig. 3.

Characteristic plots of AISI 1020 for Ebrahimi's model. (a) Peak stress (MPa) and (b) steady-state stress (MPa).

Fig. 4.

Characteristic plots of AZ80 for Ebrahimi's model. (a) Peak stress (MPa) and (b) steady-state stress (MPa).

Fig. 5.

Comparison of experiments and Ebrahimi's model for AZ80 at two strain rates. (a) 0.01/s and (b) 0.10/s.

3Validation of the new approach

Based on the pioneering studies summarized above and on typical experimental flow stress curves, it is evident that Eqs. (8) and (9) can be used to express most flow stress curves (i.e., strain-flow stress curves) if the coefficients and experimental data are handled properly. Thus, these equations form the core of the new approach in this study and new equations and schemes were developed for improving the accuracy and generality of flow stress expression assuming that the effects of temperature and strain rate on material constants including Ch in Eq. (8) and Cs in Eq. (9) cannot be neglected. Therefore, the constants are considered as bi-linear functions of temperature and strain rate. Of course, experimental peak strain, peak stress and steady-state stress are also expressed as either closed-form functions (CFFs) or piecewise bi-linear functions (PLFs) of temperature and strain rate.

Fig. 6 describes the concept of expressing ϕ as piecewise bi-linear functions of temperature T and strain rate ε˙. Eq. (10) is a formula to calculate the function values of εp, σp, σS, Ch and Cs at the specific point (T,ε˙) marked A in the figure using the experimental data or sampled function values, denoted as ϕ1, ϕ2, ϕ3 and ϕ4 at the associated sample points of the rectangular patch where the point A belongs, i.e., ε˙i≤ε˙<ε˙i+1 and Tj≤T<Tj+1. Note that ϕ in Eq. (10) represents the material functions or experiments.

Fig. 6.

Schematic drawing to describe bi-linear functions.


The material functions including Ch and Cs and experimental data including peak strain, peak stress and steady-state stress yield a matrix at sampled temperatures and strain rates, termed a flow stress information matrix. The material functions should be calculated to minimize error between experimental data and fitted curves. Of course, the experimental data can be directly measured using experimental strain-flow stress curves.

Unlike in previous works, the material functions Ch and Cs for each temperature/strain rate pair in the flow stress information matrix are calculated independently using an optimization technique. Data are derived directly from experimental strain-flow stress curves. For example, Ebrahimi et al. [17] defined the unique material constants Ch and Cs over the entire range of temperature and strain rate. Note that decoupling Ch and Cs calculations from those of other variables or parameters is very important because the identification can be simplified and generalized and more accurate mathematical treatment becomes possible.

Two application-oriented models, i.e., PLF model and CFF model, using different methods to express material functions and experimental data functions in the flow stress information matrix are summarized in Table 1 and the equations of the CFF model are given.

Table 1.

Summary of two models.

Parameter  PLF model  CFF model 
Ch  Piecewise bi-linear function  Ch=h1T+h2ε˙+h3 
Cs  Piecewise bi-linear function  Cs=s1T+s2ε˙+s3 
Peak strain, εp  Piecewise bi-linear function  εp=a1ε˙(a3T+a4)e(a2/Ta6)+a5 
Peak stress, σp (MPa)  Piecewise bi-linear function  σp=b5+b1ε˙(b2+b3Tb6)εpb7(eb4/T+b8T) 
Steady-state stress, σs (MPa)  Piecewise bi-linear function  σs=c5+c1ε˙(c2+c3Tc6)σpc7(ec4/T+c8T) 

In the model on the left (Table 1), all functions are PLFs and thus all experimental flow stress information in the matrix directly expresses the flow stress. The PLF model was used to express the flow stress of the aluminum alloy AHS-2 (chemical composition in wt%: Si 11.5; Cu 1.2; Mg 3.0; Cr 0.3; Ni 0.2; Zn 0.1; Fe 0.3; Mn 0.3; Ti 0.05; Al to 100wt%) at strain rates of 0.1, 1.0 and 10.0/s and at temperatures of 300, 350, 400 and 450°C, using material constants Ch and Cs optimally determined by the least-squares method. The experimental flow stresses (solid lines in Fig. 7) were derived via hot compression tests of solid cylinders at the aforementioned temperatures. The strain rates and their corresponding flow stress information matrix are shown in Table 2.

Fig. 7.

Comparison of experiments with piecewise bi-linear function (PLF) model for AHS-2 at three strain rates. (a) 0.1/s, (b) 1.0/s, and (c) 10.0/s.

Table 2.

Flow stress information matrix for AHS-2.

ε˙ (/s)  T (°C)  εp  σp (MPa)  σs (MPa)  Ch  Cs 
0.1300  0.1231  140.4600  114.4300  0.1249  3.6073 
350  0.1095  93.8420  71.4870  0.0965  4.0670 
400  0.0996  61.9180  48.0240  0.1030  4.9431 
450  0.0966  46.9670  34.3780  0.1094  5.8193 
1.0300  0.1300  155.6100  138.5100  0.1296  3.5605 
350  0.1230  102.0000  93.3900  0.1146  4.4555 
400  0.1119  74.8960  69.8010  0.0996  5.3505 
450  0.1050  58.0000  54.0420  0.0836  6.2455 
10.0300  0.1437  196.9100  155.8600  0.1764  3.6964 
350  0.1409  131.3900  104.4800  0.1614  4.5914 
400  0.1385  105.1600  78.9700  0.1464  5.4864 
450  0.1317  85.6660  61.1940  0.1314  6.3814 

Fig. 7 compares the flow stress functions derived using the PLF model with the experiments, revealing that the agreement is excellent. The average and maximum errors are less than 1.69 and 5.45%, respectively. The non-negligible error region exists between strains of 0.40–0.45 only at a strain rate of 10.0/s (Fig. 7(c)). Notably, all strain-flow stress curves at a strain rate of 10.0/s differ slightly from the typical flow stress curve of Fig. 1, especially just before the steady-state is attained. All other errors are negligible, implying that Eqs. (8) and (9) well-describe the flow stress of materials exhibiting typical strain-flow stress curves. Of course, the fitted curves of Fig. 7 refer only to the sample pairs of temperatures/strain rates. Interpolation or curve-fitting is required to calculate flow stresses at any other temperature/strain rate pair.

As explained above, when drawing fitted flow stress curves for only a few sampled temperature/strain rate pairs, the PLF model is optimal for all sampled couples. The model guarantees generality and is relatively reliable, simple and easy to use. However, during numerical simulation or real-time process control, the flow stress at any temperature/strain rate pair is calculated after finding the patch and local coordinates to interpolate the sampled values of Ch and Cs and experimental data, which is costly. That is, interpolation after identification of relevant element in a flow stress information matrix requires excessive computation, compromising real-time process control on the shop floor.

To the contrary, the CFF model is computationally efficient, concise and metallurgical in nature. Using the Ch and Cs values obtained from the PLF model, the material functions of the CFF model are assumed linear functions of temperature and strain rate as follows:

which cover the entire range of interest. The material constants h1, h2 and h3 for Ch and s1, s2 and s3 for Cs are determined by minimizing the objective squared errors between the experimental and modeled strain-flow stress curves. In the CFF model, experimental peak strain, peak stress and steady-state stress values are fitted and expressed as experimental data functions, for example, as follows:
as indicated after a detailed parametric study on functional optimization for the four different materials (see below). In the above equations, the ai's, bi's and ci's are called “experimental data constants”. In Eqs. (13)–(15), T and ε˙ are normalized by 100 and 10, respectively.

The CFF model was also applied to identification of the aluminum alloy AHS-2. All constants obtained via optimization are shown in Table 3 and graphs of strain rate versus temperature are provided in Figs. 8 and 9. The prominent slope of the surface of Fig. 8 indicates that Ch and Cs are highly dependent on the independent variables (temperature and strain rate) over the ranges 300–450°C and 0.1–10.0/s, respectively. Such high-level dependence emphasizes the validity of the present approach. To explore errors imparted by the CFFs of Ch and Cs, experimental flow stress curves for sampled temperature/strain rate pairs were compared with the flow stress curves fitted by Ch and Cs in terms of the experimental peak strains, peak stresses and steady-state stresses summarized in Table 2. From Table 4, the average and maximum errors were 2.4 and 5.5%, respectively, implying that the fitted functions of Ch and Cs well-reflected real phenomena.

Table 3.

Experimental data constants of a CFF model for AHS-2.

ChCsεpσp (MPa)σs (MPa)
h1  −0.0003  s1  0.0179  a1  0.1521  b1  4.3589  c1  1.3094 
h2  0.0052  s2  −0.0413  a2  −0.2047  b2  0.1771  c2  1.0947 
h3  0.2963  s3  −5.8596  a3  0.1862  b3  −1.1790  c3  −2.4246 
–  –  –  –  a4  −0.9527  b4  29.9981  c4  52.5035 
–  –  –  –  a5  0.0881  b5  5.0000  c5  3.0000 
–  –  –  –  a6  −0.9014  b6  −1.2129  c6  −0.5218 
–  –  –  –  –  –  b7  0.8339  c7  −0.8488 
–  –  –  –  –  –  b8  4.5903  c8  78.0020 
Fig. 8.

Variations in material functions. (a) Ch and (b) Cs.

Fig. 9.

Fitted functions of peak strain, peak stress and steady-state stress for AHS-2. (a) Peak strain, (b) peak stress (MPa) and (c) steady-state stress (MPa).

Table 4.

Maximum flow stress errors when only Ch and Cs are expressed as closed-form functions.

ε˙ (/s)T (°C)ChCsMax. flow stress error (%) 
Optimized  Eq. (11)  Optimized  Eq. (12)   
0.1300  0.1268  0.1249  3.6073  4.3929  1.4672 
350  0.1387  0.1099  4.0670  5.2879  2.0500 
400  0.1321  0.0949  4.9431  6.1829  1.2439 
450  0.1026  0.0811  5.8193  7.0780  1.2869 
1.0300  0.1312  0.1296  3.5605  4.3558  2.3001 
350  0.0808  0.1146  4.4555  5.2508  2.4255 
400  0.0811  0.0996  5.3505  6.1458  1.8935 
450  0.0801  0.0836  6.2455  7.0408  1.3912 
10.0300  0.1777  0.1764  3.6964  3.9841  5.5344 
350  0.1657  0.1614  4.5914  4.8791  3.2315 
400  0.1536  0.1464  5.4864  5.7741  2.8720 
450  0.1299  0.1314  6.3814  6.6691  3.1123 

Fig. 9 shows the fitted peak strain, peak stress and steady-state stress functions. As shown in Table 5, the maximum and average errors of peak strain, peak stress and steady-state stress were 1.87 and 1.05%, 4.64 and 2.74%, and 3.55 and 2.09%, respectively. The results indicate that the fitted curves were all acceptable and the functions of Eqs. (13)–(15) well-matched physical phenomena. Note that the sharp changes in functional values at higher temperatures and lower strain rates (Fig. 9) demand the use of highly non-linear functions in Eqs. (13)–(15). Fig. 9(a) shows that peak strain increases with a decrease in temperature and an increase in strain rate. In other words, the peak strain is related to the Zener–Hollomon parameter, as stated in Ref. [19]. Fig. 9(b) and (c) depicts variations in peak and steady-state stresses, respectively, by temperature and strain rate. The peak stress increases when the strain rate increases or the temperature decreases, in good agreement with literature data [3].

Table 5.

Errors in peak strain, peak stress and steady-state stress fitted by Eqs. (12)–(14).

ε˙ (/s)  T (°C)  Peak strainPeak stress (MPa)Steady-state stress (MPa)
    Exp.  CFF  Error (%)  Exp.  CFF  Error (%)  Exp.  CFF  Error (%) 
0.1300  0.1231  0.1217  1.1373  140.4600  146.3020  4.1592  114.4300  116.5053  1.8136 
350  0.1095  0.1084  1.0046  93.4820  89.1464  4.6375  71.4870  69.7577  2.4190 
400  0.0996  0.1004  0.8032  61.9180  61.1578  1.2278  48.0240  49.6235  3.3306 
450  0.0966  0.0955  1.1387  46.9670  48.3112  2.8620  34.3780  33.6478  2.1240 
1.0300  0.1300  0.1317  1.3077  155.6100  161.6338  3.8711  138.5100  139.7302  0.8809 
350  0.1230  0.1207  1.8699  102.0000  106.6581  4.5668  93.3900  93.6843  0.3151 
400  0.1119  0.1125  0.5362  74.8960  74.6780  0.2911  69.8010  67.3230  3.5501 
450  0.1050  0.1064  1.3333  58.0000  57.8102  0.3272  54.0420  54.2510  0.3867 
10.0300  0.1437  0.1448  0.7655  196.9100  189.5624  3.7315  155.8600  152.1056  2.4088 
350  0.1409  0.1406  0.2129  131.3900  133.7247  1.7769  104.4800  107.3452  2.7423 
400  0.1385  0.1367  1.2996  105.1600  103.5292  1.5508  78.9700  76.8599  2.6720 
450  0.1317  0.1332  1.1390  85.6660  82.3276  3.8970  61.1940  62.6927  2.4491 

The complete flow stress curves of the CFF model for the commercial aluminum alloy AHS-2 were compared with those of the PLF model and experiments in Fig. 10. It can be seen that the flow stresses yielded by the CFF model are in good agreement with the experiments with the average and maximum errors less than 2.51 and 4.64%, respectively (see Table 5). Notably, the average errors of the CFF model are greater than those of the PLF model, but the maximum error of the CFF model is slightly less than that of the PLF model because of unusual flow behavior between the strains of 0.40 and 0.45 at a strain rate of 10.0/s. When this region is excluded, the maximum error of the PLF model falls to 2.43%.

Fig. 10.

Comparison of the experiments with Ebrahimi's model, PLF model and CFF model for AHS-2. (a) 0.1/s, (b) 1.0/s, and (c) 10.0/s.

4Applications and discussion

Many researchers have modeled the flow stress behaviors of various materials. Fig. 11 compares experimental data of hot compression test on the steel 20MoCrS4 (chemical composition in wt%: C 0.20; Si 0.25; Mn 0.75; Cr 0.40; Mo 0.45; S 0.02; Pb to 100wt%) with curves fitted by Hensel-Spittel [22], Voce [18] and Ebrahimi et al. [17] and by CFF and PLF models. The Hensel-Spittel and Voce data were reported by Meyer et al. [12]. Comparison reveals that the CFF model optimally expresses flow stress behavior (Fig. 11), as is also apparent in Table 4, in which the maximum and average errors are compared. The maximum error of the CFF model is less than those of other models at all temperatures. The lower CFF deviation also emphasizes the reliability and superiority of the model.

Fig. 11.

Comparisons of models and experiments.


Table 6 shows that the maximum and average CFF errors at all temperatures (4.55 and 0.95%) are less than those of the Hensel-Spittel model [12] (15.57 and 7.18%), the Voce model [12] (7.51 and 3.85%) and the improved Voce model (14.60 and 3.24%), respectively. However, the PLF model error should be less than that of the CFF model because all PLF variables (peak strain, peak stress, steady-state stress, Ch and Cs) are calculated by piecewise bi-linear functions.

Table 6.

Maximum and average errors and their standard deviations of different models used to evaluate 20MoCrS4 at a strain rate of 0.1/s.

Temperature (°C)  Index  CFF  PLF  Ebrahimi's model  Voce  Hensel-Spittel 
900Max. error (%)  4.55  3.47  14.60  7.51  15.57 
Average error (%)  0.95  0.92  4.62  2.45  7.65 
Standard deviation  1.74  1.04  3.16  1.57  3.50 
1,050Max. error (%)  2.88  1.83  5.50  6.02  11.69 
Average error (%)  0.99  0.80  2.85  3.88  6.83 
Standard deviation  0.63  0.92  2.63  1.10  3.66 
1,200Max. error (%)  1.44  1.45  2.77  4.68  7.13 
Average error (%)  0.95  0.83  2.25  5.21  7.05 
Standard deviation  0.86  0.90  2.62  0.91  1.83 

The material constants of the CFF model for the steel AISI 1020 (chemical composition in wt%: Mn 0.4; C 0.20; S 0.03; P 0.02; Fe to 100wt%) and the magnesium alloy AZ80 (chemical composition in wt%: Al 7.83; Zn 0.46; Mn 0.25; Si 0.03; 0.001; Mg to 100wt%) are listed in Tables 7 and 8, respectively. Employing the constants of Tables 7 and 8, the flow stresses of each material can be calculated at specified temperature/strain rate pairs. The experimental flow stress curves on hot compression testing of AISI 1020 and AZ80 are shown in Figs. 12 and 13, respectively, with the fitted strain-flow stress curves. The CFF curves lie close to the experimental curves at all sampled temperatures and strain rates for both materials. The errors between the fitted curves and experiments of AZ80 are higher than those for AISI 1020, but are acceptable.

Table 7.

Experimental data constants of the CFF model for AISI 1020.

ChCsεpσp (MPa)σs (MPa)
h1  0.0009  s1  −0.0990  a1  2.7920  b1  6.4983  c1  0.4163 
h2  −0.0210  s2  3.1916  a2  −10.9966  b2  −0.5489  c2  −0.7145 
h3  −0.5586  s3  142.7361  a3  0.1636  b3  0.0819  c3  1.4621 
–  –  –  –  a4  −1.9300  b4  35.2092  c4  −4.2347 
–  –  –  –  a5  0.1059  b5  3.0000  c5  5.4623 
–  –  –  –  a6  0.5571  b6  0.8349  c6  −0.2766 
–  –  –  –  –  –  b7  −0.3232  c7  1.3740 
–  –  –  –  –  –  b8  −0.0762  c8  −0.0309 
Table 8.

Experimental data constants of the CFF model for AZ80.

ChCsεpσp (MPa)σs (MPa)
h1  0.0025  s1  0.0066  a1  53.0000  b1  −8.7979  c1  0.7072 
h2  −0.7367  s2  −24.9890  a2  −0.0853  b2  2.6787  c2  0.0457 
h3  −0.9100  s3  4.2262  a3  −0.6229  b3  −6.4970  c3  −8.0846 
–  –  –  –  a4  4.1396  b4  8.7274  c4  7.6908 
–  –  –  –  a5  0.2163  b5  6.4326  c5  2.0000 
–  –  –  –  a6  −2.1826  b6  −0.549  c6  −3.9162 
–  –  –  –  –  –  b7  −2.1850  c7  0.7208 
–  –  –  –  –  –  b8  −1.5024  c8  0.2159 
Fig. 12.

Comparison of the experiments with Ebrahimi's model, PLF model and CFF model for AISI 1020. (a) 0.1/s and (b) 10.0/s.

Fig. 13.

Comparison of the experiments with Ebrahimi's model, PLF model and CFF model for AZ80. (a) 0.01/s and (b) 0.10/s.


The new models were presented to express flow stress as a set of piecewise bi-linear functions (PLF) and closed-form functions (CFF) of temperature and strain rate. The accuracy, generality and applicability were evaluated.

In PLF model all material parameters or functions and experimental data are expressed and/or calculated using bi-linear interpolation. The PLF model is thus characterized by its flexibility and accuracy at the expense of computational efficiency in numerical calculation or real-time process control. To improve the practicability and feasibility of the model, peak strain, peak stress and steady-state stress as well as two major parameters, Ch and Cs were defined as closed-form functions of temperature and strain rate in the CFF model. For simplicity, each function was dealt with independently of all other functions.

In the two present material models, the number of material constants thus increased in order to improve the model generality and accuracy, compared to the previous research works. However, the constants or coefficients to identify the material functions expressed by the complicated PLFs or CFFs are all systematically and automatically obtained through minimizing error between experiments and the model, which is one of the essences of the present approach in terms of practicability. Note that the new models identify the flow stresses of metals and alloys at the elevated temperature in an automatic way via optimization of a set of variables for material functions or experimental data in terms of error, i.e., the difference in flow stress between experiments and the mathematical models.

The new models were applied to two steels, an aluminum alloy and a magnesium alloy, revealing that they are all in good agreement with the experiments or given flow stress information in all cases. It has been concluded that the PLF model is most accurate and the CFF model well-balances the accuracy and computational efficiency in numerical simulation.

Conflicts of interest

The authors declare no conflicts of interest..


This work is financially supported by Ministry of Trade, Industry and Energy (MOTIE) as a part of the project, “Development of Commercialization Technology for High Strength Ti Alloy Aircraft Fasteners using Warm Forming (10081334)”, and by The Ministry of Small and Medium-sized Enterprises and Startups (MSS) as a part of WC 300 project (S2415560).

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