^{1}

^{1}

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Plastic flow behavior of the SNCM8 steel was investigated by performing hot compression tests within the temperature range of 850
˚C to 1200
˚C and strain rates of 0.01 s
^{−1} to 10 s
^{−1}. Constitutive modeling based on dynamic recrystallization was established, in which Cingara equation was applied to represent work hardening up to peak stress and Avrami equation to describe dynamic softening beyond peak stress up to steady state. It was found that stress-strain responses predicted by the combined model fairly agreed with experimentally resulted curves for the particular conditions. The correlation coefficient (
*R*) of 0.9485 and average absolute relative error (
*AARE*) of 2.3614% was calculated for the modeled flow curves.

SNCM8 alloy is a commercial medium carbon Ni-Cr-Mo alloy steel. Due to its good balance of high strength, excellent toughness and wear resistance, this steel has been widely used for several general-purpose parts including automotive components, heavy-duty axles and gears, shaft and structural parts. Generally, the steel alloy is worked and fabricated at elevated temperatures [

Those developed stress-strain relationships for predicting flow stress curves using peak stress, peak strain and four constants that were independent from deformation conditions could be simply calculated by non-linear regression of one or more experimental stress-strain curves [^{−1} [^{−1} [

In this study, stress-strain curves of the SNCM8 steel were investigated using hot compression tests, which were carried out at the temperatures of 850˚C, 950˚C, 1050˚C, 1150˚C and 1200˚C with the strain rates of 0.01, 0.1, 1 and 10 s^{−1}. By the flow curve modeling, experimentally determined stress-strain responses were predicted using two different constitutive approaches. Firstly, the hyperbolic sine equation was used to describe relationships between flow stresses, plastic strains, strain rates and temperatures. Secondly, the model according to the Cingara equation was applied for representing the flow curves up to their peak points. Then, successive flow stresses were characterized by the Avrami equation, which was based on the DRX kinetics model [

In this work, a commercial steel grade SNCM8 was investigated, for which the chemical composition was given in ^{−1}. The temperature-time diagram of the hot compression tests was depicted in

C | Si | Mn | P | S | Cr | Mo | Ni | V | Al | W |
---|---|---|---|---|---|---|---|---|---|---|

0.387 | 0.273 | 0.695 | 0.025 | 0.170 | 0.768 | 0.156 | 1.871 | 0.010 | 0.015 | 0.080 |

in order to obtain homogeneous temperature distribution before compression. The specimens were subsequently upset to a height reduction of 60%. During the hot compression tests, glass powder as lubricant was applied at the contact surfaces between specimens and dies for minimizing friction. To reveal recrystallized grain structures, specimens were also rapidly cooled in nitrogen gas directly after various stages of compression from the true strain of 0.1 up to 0.9. Finally, the quenched specimens were sliced along the axial section. The sectioned samples were polished and etched with saturated picric acid and corresponding micrographs were taken by optical microscope.

Flow curves experimentally determined from hot compression tests were shown in

Strain rate (s^{−1}) | Temperature (˚C) | |||||
---|---|---|---|---|---|---|

850 | 950 | 1050 | 1150 | 1200 | ||

σ p | 0.01 | 120.02 | 78.96 | 58.93 | 37.38 | 27.55 |

0.1 | 214.17 | 146.03 | 89.41 | 63.39 | 51.11 | |

1.0 | 278.38 | 206.98 | 144.22 | 93.10 | 77.22 | |

10 | 303.78 | 227.11 | 192.85 | 103.74 | 84.72 | |

ε p | 0.01 | 0.1583 | 0.1628 | 0.1402 | 0.104 | 0.0905 |

0.1 | 0.2442 | 0.2352 | 0.1673 | 0.1312 | 0.1402 | |

1.0 | 0.4206 | 0.3799 | 0.2940 | 0.2216 | 0.1764 | |

10 | 0.2487 | 0.3256 | 0.3121 | 0.2487 | 0.2307 |

peak strain and steady state strain decreased with increasing deformation temperatures. At low strain rate conditions such as 0.01 s^{−1}, stress-strain curves with multiple peaks were observed. This was a typical phenomenon of DRX process at low deformation rate due to repeated recrystallization and grain growth cycles that took place before reaching the steady state [

In this work, constitutive modeling according to Arrhenius model was applied for describing relationship between flow stress, strain rate and temperature. Material constants of the examined steel for the constitutive equations were determined from experimental results of the hot compression tests. By this manner, effects of both temperature and strain rate on plastic flow behavior of the material could be described by the Zener-Hollomon parameter with an exponent-type equation. The Arrhenius model represented functions between the Zener-Hollomon parameters and material flow stress that could be expressed as [

Z = ε ˙ exp ( Q R T ) (1)

ε ˙ = A F ( σ p ) exp ( − Q R T ) (2)

where

F ( σ ) = sinh ( α σ ) n (3)

σ is the material flow stress in MPa for a given strain. R is the universal gas constant (8.31 J∙mol^{−1}∙K^{−1}). T is the absolute temperature in K. ε ˙ is the strain rate (s^{−1}). Q is the activation energy during hot deformation (kJ∙mol^{−1}). A, α and n are the material constants, and α = β/n. In this work, values of peak stress ( σ p ) from each condition were taken into account for the calculations. It was previously found that results obtained by using incremental flow stress values were similar to those only from the peak stress. Moreover, the hyperbolic-sine law, as provided in Equation (3) was applied, since it was more representative and commonly used. In the following, procedure for determining the material constants by considering the peak stresses was shown. Firstly, relationships between flow stresses and strain rates were described as:

ln σ = 1 n ln ε ˙ − 1 n ln B (4)

σ = 1 β ln ε ˙ − 1 β ln B ′ (5)

B and B' are materials constants that were independent from the forming temperatures. Then, substituting values of the flow stress at the peak points and corresponding strain rates into Equations (4) and (5). This resulted in relationships between flow stresses and strain rates, as illustrated in

For a given constant temperature, the values of n and β were then derived from slopes of the regression lines in the ln σ - ln ε ˙ and σ - ln ε ˙ diagrams, respectively. Here, flow stresses obtained from the hot compression tests could be represented by a group of parallel and straight lines. As seen, by using the linear regression method, the parameters n and β could be approximately determined for different deformation temperatures. Afterwards, the mean values of n and β of

7.073 and 0.071 MPa, were calculated for all examined temperatures, respectively. For the SNCM8 steel, the obtained parameter α = β/n was equal to 0.011 MPa^{−1}. For the peak stress, Equation (2) could be rewritten as following.

Taking natural logarithm to both side of Equation (2) gave

Q / R T = ln A − ln ε ˙ + n ln [ sinh ( α σ p ) ] . (6)

The apparent activation energy Q was calculated through differentiating Equation (6).

Q = R [ ∂ ln ε ˙ ∂ ln [ sinh ( α σ p ) ] ] T [ ∂ ln [ sinh ( α σ p ) ] ∂ ( 1 / T ) ] ε ˙ (7)

The values of forming temperatures, strain rates and corresponding stresses were substituted into Equation (6). Relationships of ln [ sinh ( α σ p ) ] - 1 / T and ln [ sinh ( α σ p ) ] - ln ε ˙ could be evaluated at constant strain rate and temperature and then plotted in ^{−1} was obtained. For the peak stresses, Equation (1) could be expressed now as following.

Z = ε ˙ exp ( Q R T ) = A [ sinh ( α σ p ) ] η (8)

Then, taking the logarithm to both sides of Equation (8) gave

ln Z = ln A + n ln [ sinh ( α σ p ) ] (9)

From the experimental results, relationship between ln [ sinh ( α σ p ) ] and ln Z could be determined and plotted in ^{14} s^{−1} could be directly calculated. Additionally,

the mean value of n from Equation (9) was computed to be 4.18.

Finally, relationships between Q, lnA, β, n, α and plastic strain for the investigated steel SNCM8 were fitted in a polynomial form. From Equation (8) the flow stresses σ were subsequently written as a function of the Zener-Hollomon parameter. Also, the proposed constitutive model could be summarized in the following equation.

ε ˙ = 1.019 × 10 14 [ sinh ( 0.011 σ ) ] 4.18 exp ( − 385.6 / R T ) (10)

Then, the flow stress of the examined steel could be now expressed as a function of the Z parameter under consideration of the hyperbolic law.

σ = 90.9 × { ( Z 1.019 × 10 14 ) 1 4.18 + [ ( Z 1.019 × 10 14 ) 2 4.18 + 1 ] 1 2 } (11)

Peak state and onset of steady state deformation were the important characteristic points of a typical flow curve with DRX. Using calculated value of the activation energy Q, values of these states could be determined regarding to the Z parameter.

ε p = 0.0140 ⋅ Z 0.77 (12)

ε s s = 0.2731 ⋅ Z 0.23 (13)

σ p = 0.7445 ⋅ Z 0.143 (14)

σ s s = 0.4343 ⋅ Z 0.154 (15)

According to Equations (14) and (15), a linear relationship between peak stress and steady state stress was obtained, as given in Equation (16) and depicted in

σ s s = 0.961 σ p (16)

In this work, experimentally determined flow curves were firstly described using the Cingara equation [

σ = σ p [ ( ε ε p exp ( 1 − ε ε p ) ) ] c (17)

Taking natural logarithm to this equation yielded the following equation.

ln ( σ σ p ) = C [ 1 − ε ε p + ln ( ε ε p ) ] (18)

From Equation (18), relationship between ln ( σ / σ p ) and 1 − ( ε / ε p ) + ln ( ε / ε p )

could be obtained and plotted for determining the constant C by a linear regression, as illustrated for the temperature of 1050˚C and strain rate of 0.1 s^{−1} in ^{−1} and different temperatures in

Strain rate (s^{−1}) | Temperature (˚C) | |||||
---|---|---|---|---|---|---|

850 | 950 | 1050 | 1150 | 1200 | ||

C | 0.01 | 0.440 | 0.585 | 0.583 | 0.592 | 0.389 |

0.1 | 0.417 | 0.514 | 0.551 | 0.671 | 0.582 | |

1.0 | 0.282 | 0.281 | 0.239 | 0.247 | 0.198 | |

10 | 0.273 | 0.239 | 0.271 | 0.171 | 0.125 |

As seen in

X d = σ p − σ σ p − σ s s (19)

σ p is the peak stress, σ s s is the steady state stress. The ( σ p − σ ) term indicated material flow softening from the peak point to any flow stress σ, whereas the ( σ p − σ s s ) term represented the maximum achievable softening. This flow softening was directly related to the DRX volume fraction X d , in which effects of DRV on flow softening were not considered here [

Thus, Equation (20) has been often used as the general form of the DRX kinetics model in alloy [

X d = 1 − exp [ ( − k ) ( ε − ε c ε p ) n ] (20)

where k, n are the material constants; ε is the true strain; ε c is the critical strain and ε p is the peak strain. However, ε c was known as the critical strain for the initiation of DRX, but the flow softening intrinsically started at peak strain ε p [

X d = 1 − exp [ ( − k ) ( ε − ε p ε p ) n ] (21)

ln ln ( 1 1 − X d ) = ln k + n ln ( ε − ε p ε p ) (22)

The relationship between ln ln ( 1 / ( 1 − X d ) ) and ln ( ( ε − ε p ) / ε p ) under various deformation conditions was almost linear, as depicted in

X d = 1 − exp [ − 2.065 ( ε − ε p ε p ) 1.865 ] (23)

Based on the calculated results of this model, effects of deformation temperature, strain rate and strain on volume fraction of the occurred DRX were shown in ^{−1}, a completely recrystallized structure was predicted for all tested temperatures beyond the true strain of 0.9. In contrast, at higher strain rates of 1 and 10 s^{−1}, incomplete recrystallization was shown for the lower temperature of 850˚C, 950˚C and 1050˚C.

According to Equation (19) to Equation (23) in combination with Equation (16), flow stresses after peak points could be expressed as:

σ = σ p − ( σ p − σ s s ) X d (24)

According to Equation (24), Equation (21) and Equation (16), flow stresses beyond peak points could be given then as a function of the peak stress and peak strain as following.

σ = 0.961 σ p + 0.039 σ p exp [ ( − 2.065 ) ( ε − ε p ε p ) 1.865 ] (25)

The values of σ p , ε p , k and n were determined for each deformation condition and subsequently stress-strain curves were calculated by using Equation (25).

In order to verify the introduced flow stress models, experimental and numerically predicted stress-strain results were compared.

deformation temperatures and strain rates. It could be observed that flow stresses predicted by the proposed models were in good agreement with the experimental data for all entire ranges of examined deformation temperature and strain rate. A scatter diagram of the predicted values against experimental results for all deformation conditions was provided in

R = ∑ i = 1 N ( P i − P ¯ ) ( E i − E ¯ ) ∑ i = 1 N ( P i − P ¯ ) 2 ∑ i = 1 n ( E i − E ¯ ) 2 (26)

A A R E = I N ∑ i = 1 N | E i − P i E i | × 100 (27)

E is the experimental flow stress and P is the predicted flow stress obtained from the introduced combined constitutive equations. E ¯ and P ¯ were the mean values of E and P, respectively. N is the total number of data used in this study. R is a commonly employed statistical parameter and provided information on the reliability of linear relationship between the experimental and predicted data. Sometimes, the higher value of R may not necessarily indicate a better performance, whereby tendency of the equation could be biased towards higher or lower values [

Hot compression tests of the steel alloy SNCM8 were conducted in the temperature range of 850˚C - 1200˚C and the strain rate range of 0.01 - 10 s^{−1}. It was found that the strain rate and temperature significantly affected flow stress-strain behavior of this steel. The flow stress decreased with increasing deformation temperature and decreasing strain rate.

1) Stress-strain curves of the SNCM8 steel exhibited typical DRX phenomenon with single peak stress followed by a gradual fall towards steady state stress.

2) Constitutive equations were developed using the hyperbolic-sine type of the Arrhenius model for describing the stress-strain responses independence on the deformation temperatures and strain rates and following relationship was obtained.

ε ˙ = 1.019 × 10 14 [ sinh ( 0.0110 σ ) ] 4.1832 exp ( − 385.584 / R T )

3) Relationships between the characteristic points of flow stress and the parameter Z were determined as ε p = 0.0140 ⋅ Z 0.77 , ε s s = 0.2731 ⋅ Z 0.23 , σ p = 0.7445 ⋅ Z 0.143 , σ s s = 0.4343 ⋅ Z 0.154 and the linear function between the peak and steady state stress was σ s s = 0.961 σ p .

4) The DRX flow curves obtained from experiments were successfully described by using a combination of Cingara and Avrami equations.

5) Predictability of the introduced combination approach of the Cingara and Avrami equations was evaluated in terms of the correlation coefficient (R) and average absolute relative error (AARE). The R and AARE were found to be 0.9486% and 2.3614% respectively, which indicated good prediction capabilities of the combined model.

The authors would like to acknowledge King Mongkut’s University of Technology Thonburi through the “KMUTT 55^{th} Anniversary Commemorative Fund” and National Research Council of Thailand (NRCT) for the financial support. Also, sincere gratitude must be given to “S.B. - CERA Co., Ltd.” for the material testing and the experiment part.

The authors declare no conflicts of interest regarding the publication of this paper.

Jantepa, N. and Suranuntchai, S. (2021) Investigation of Hot Deformation Behavior of SNCM8 Alloy Steel. World Journal of Mechanics, 11, 17-33. https://doi.org/10.4236/wjm.2021.113003