# Hot Deformation Constitutive Equation and Plastic Instability of 30Cr4MoNiV Ultra-High-Strength Steel

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

^{−1}. A constitutive equation with strain-dependent constants and processing maps suitable for 30Cr4MoNiV ultra-high-strength steel are established. The results show that the combination of the hyperbolic sine function and the Zener-Hollomon parameter can accurately represent the influences of deformation temperature, strain rate, and strain on the flow stress of the 30Cr4MoNiV ultra-high-strength steel. The applicability of plastic instability criteria such as m, $\dot{m}$, S, $\dot{S}$ and the instability parameter ξ are analyzed, the stability and instability regions are clarified accurately, and the optimized processing regions are given in the stability regions. The optimized regions are verified by the uniform equiaxed grains, and the plastic instability is validated by dynamic strain aging and the microstructure of the voids.

## 1. Introduction

^{−1}, and the strain of 0.05–0.6. A reliable constitutive equation is particularly important for describing the hot deformation behavior of the material. Rao and Prasad [7] established the constitutive equation of low carbon steel based on Arrhenius-type hyperbolic sine function, which could satisfactorily describe strain hardening, dynamic recovery and dynamic recrystallization, and temperature dependence in a wide range of strain rates. Jha et al. [8] pointed out that the power law equation was more suitable for describing the constitutive relationship between the flow strain and stress of Ti-6Al-4V alloys in both lamellar and equiaxed morphologies. So far, the constitutive equations of many types of metal materials, such as iron-based alloys [9,10], aluminum alloys [11], titanium alloys [8], copper alloys [12], magnesium alloys [13], and composite materials [14], had been constructed and discussed, which played an important guiding role in the hot processing of materials.

^{−1}strain rates. Zhai et al. [17] in his paper reported the processing performance of 40CrNiMo alloy steel could be improved by dynamic recovery (DRV) and dynamic recrystallization (DRX) at the optimum processing parameters. Liu et al. [18] pointed out that the adiabatic shear bands, voids, wedge cracks, and other defects were generated under non-optimal process parameters, especially in severe plastic instability areas. In the study of the hot deformation behavior of AZ31 alloy, Ding et al. [19] found that there are many voids and coarse grains in the stable region obtained by ξ, and which was usually regarded as a characteristic of instability. Therefore, it is necessary to pay attention to the microstructure and process parameters optimization of the material in the hot deformation stage, especially the suitable plastic instability criteria. Compared the processing maps with different plastic instability criteria such as m, ξ using DMM of Prasad [15], κ using modified DMM of Murty et al. [20], and ${\kappa}_{j}$ model of Poletti et al. [21], Rajput et al. [22] found that the instability areas obtained from different instability criteria were quite different, and the instability parameter ${\kappa}_{j}$ was most suitable for the instability areas division of AISI1010 low carbon steel. Besides, the plastic instability region could be divided by the following criteria based on the Lyapounov function, as presented in Prasad’s paper [23].

## 2. Materials and Methods

^{−1}, respectively, with a deformation of 60%. After compression, the specimens were quenched in water immediately to retain the deformed microstructure. The schematic diagram of the hot compression tests is shown in Figure 1a. Figure 1b illustrates the schematic diagram of the specimens before and after compression.

## 3. Results and Discussion

#### 3.1. Stress-Strain Curves

^{−1}have a zigzag or wavy characteristic, which is usually regarded as a sign of DSA, as reported in the paper by Ivanchenko et al. [26].

#### 3.2. Constitutive Equation

^{−1}), Q is the activation energy ($\mathrm{J}\xb7{\mathrm{mol}}^{-1}$), R is the gas constant (R = 8.314 $\mathrm{J}\xb7{\mathrm{mol}}^{-1}\xb7{\mathrm{K}}^{-1}$), T is the deformation temperature (K), $\sigma $ is the flow stress (MPa) and f (σ) is the flow stress expression.

_{1}, A

_{2}, A

_{3}, n

_{1}, n

_{2}, β and α (≈β/n

_{1}) are the material-dependent constants independent of the deformation temperature. The stress multiplier α is an adjustable constant that brings ασ into the correct range to make constant T curves of $\mathrm{ln}\dot{\epsilon}$ versus $\mathrm{ln}\left[\mathrm{sinh}\left(\alpha \sigma \right)\right]$ constant, as reported in the paper of Mirzadeh et al. [29] and McQueen [30].

_{P}) values in the temperature range of 1173–1373 K and the strain rate of 0.01–10 s

^{−1}. Since the strain for determination of flow stress is not specified, the description of the flow stress in Equation (1) is incomplete. Therefore, the steady-state stress, the peak stress, and the stress corresponding to a particular strain can be used for this purpose. However, the peak stress is usually used because the steady-state stress cannot be accurately obtained, or there has been some softening during the morphological evolution; similar findings were reported by Cai et al. [31] and Chen et al. [32].

_{1}value can be evaluated using the slope of curve of $\mathrm{ln}\dot{\epsilon}-{\mathrm{ln}\sigma}_{\mathrm{P}}$ (Figure 3a) and β is determined using the slope of the curve of $\mathrm{ln}\dot{\epsilon}-\sigma $ (Figure 3b). The values of n

_{1}and β are evaluated by taking an average of slopes at different temperatures. For the peak stresses, the n

_{1}, β and α are calculated as 11.278, 0.0655, and 0.00581 respectively. Furthermore, the slopes of curves $\mathrm{ln}\dot{\epsilon}-\mathrm{ln}\left[\mathrm{sinh}\left({\alpha \sigma}_{\mathrm{P}}\right)\right]$ (Figure 3c) and ${\mathrm{ln}\left[\mathrm{sinh}\right(\alpha \sigma}_{\mathrm{P}}\left)\right]-1/T$ (Figure 3d) are used to calculate n

_{2}and Q. The average values of n

_{2}and Q were found to be 8.343 and 520.815 $\mathrm{kJ}\xb7{\mathrm{mol}}^{-1}$. By comparing the similar steels with different compositions, such as 3Cr-1Si-1Ni UHS steel (431.2376 $\mathrm{kJ}\xb7{\mathrm{mol}}^{-1}$) in the paper of Lei et al. [6], 30Si2MnCrMoVE low-alloying UHS steel (362.48278 $\mathrm{kJ}\xb7{\mathrm{mol}}^{-1}$) in the paper of Wang et al. [33], and Nb–V–Ti micro-alloyed UHS steel (407.29 $\mathrm{kJ}\xb7{\mathrm{mol}}^{-1}$) in the paper of Dong et al. [34]. It has been found that the activation energy is significantly affected by the chemical composition. According to Equations (1) and (4), the A

_{3}can be determined by the average intercept of the curve of ${\mathrm{ln}Z-\mathrm{ln}\left[\mathrm{sinh}\right(\alpha \sigma}_{\mathrm{P}}\left)\right]$ (Figure 3e). The A

_{3}is 1.951 × 10

^{20}(lnA

_{3}= 46.72). The constitutive equation with optimized the material-dependent constants is as follows:

^{2}) are all greater than 0.985, and the highest is 0.9995, which indicates that the accuracy of the fitted curves is very high. The S in Equation (10) stands for any material-dependent constant (Q, α, n, and A) in Figure 4. The coefficients of the best fitting polynomial for any material constant are listed in Table 3.

^{−1}. The predicted values are obtained by Equation (11). The results show that the above-derived constitutive equation can accurately predict the flow stress of the material, especially at higher deformation temperatures, which can provide the main theoretical guidance for the actual production of UHS steel.

_{i}and P

_{i}are the experimental and predicted values at different strains, $\overline{E}$ and $\overline{P}$ are the corresponding average values, and N is the amount of data.

^{−1}. The R is 0.9906, very close to 1. The results show that the flow stress of 30Cr4MoNiV UHS steel can be accurately predicted by Equation (11) in the temperature range of 1173–1373 K and a strain rate of 0.01–10 s

^{−1}.

#### 3.3. Processing Maps and Plastic Instability

^{−1}), K is a constant, and m is the strain rate sensitivity, which is defined as follows:

^{−1}), and the region gradually moves towards high temperature with the increase of strain. When the strain increases from 0.3 to 0.9, the region of η > 0.2 is reduced from the deformation temperature of 1248–1373 K and the strain rate of 0.01–1 s

^{−1}to 1263–1373 K and 0.01–1 s

^{−1}, whereas the corresponding peak value of power dissipation changed from 29% to 24%. Meantime, the ξ value also changes greatly with the increase of strain. When the strain is 0.3, there are three plastic instability zones, but the total area is small, as shown in Figure 8a. In Figure 8b,c, with the increase of the strain, the area of plastic instability increases significantly and converges into one area, accounting for about one-third of the total area. The plastic instability zone is mainly concentrated under the conditions of a high strain rate (1–10 s

^{−1}), but does not include high-temperature conditions (1373 K). This is mainly because the higher strain rate is unfavorable to the occurrence of DRX, but the high-temperature environment is conducive, and the DRX can suppress the appearance of instability.

^{−1}, the values of m, $\dot{m}$, S and $\dot{S}$ vary irregularly with strain rate and deformation temperature. The dark cyan part in Figure 9a–d represent the area of $0<m<1$, $\dot{m}<0$, $S>1$, and $\dot{S}<0$, respectively. The dark cyan parts in Figure 9 are the stable areas of plastic deformation, and the shaded (gray) parts are the unstable areas. The hot deformation of the material accords with the m and S criteria within the range of deformation temperature 1173–1373 K and strain rate 0.01–10 s

^{−1}(Figure 9a,c). In Figure 9b, according to the instability criterion $\dot{m}$, the instability phenomenon mainly occurs at low strain rates (0.01–0.1 s

^{−1}), except the temperature in the range of 1258–1308 K. In Figure 9d, based on the instability criterion $\dot{S}$, in addition to deformation temperature 1173–1193 K, strain rate 0.01–0.1 s

^{−1}, and 1313–1373 K, 0.01–1 s

^{−1}, plastic instability also occurs in the range of 1198–1273 K and 1–10 s

^{−1}.

^{−1}, the value of η is higher in the range of 0.180–0.240.

^{−1}, the range of the η values in this area is 0.105–0.195.

^{−1}, the range of η in this area is 0.060–0.180.

^{−1}, the range of η in this area is 0.180–0.225.

#### 3.4. Microstructure

^{−1}, and initial casting structures have disappeared. It is shown that complete DRX takes place under this condition. The softening produced by DRX can make the material flow stable and has good processability. Therefore, area I can be regarded as a reasonable hot working region.

^{−1}. It can be found that the refined grains formed by DRX are clustered in a narrow region with all the boundaries almost in a straight line, which is characteristic of DSA, and it is the result of the unpinning and repeated pinning of interstitial atoms and dislocations. Another characteristic of DSA is the zigzag stress-strain curve or the Portevin-Le Chatelier effect, as reported in the paper of Ivanchenko et al. [26], which can be seen in Figure 2d. The area with DSA is usually the plastic instability area, because it will bring many adverse effects to the mechanical properties of the material, such as ductility loss, localized strain, the triple junction (wedge) crack, serrated flow, etc. Therefore, area II illustrates the unreasonable hot working region.

^{−1}, the new fine grains are also formed, but coarse casting grains still exist compared with Figure 11a,b. Simultaneously, the voids were found under this condition, which have a negative effect on the mechanical properties of the materials. Therefore, hot working of 30Cr4MoNiV UHS steel should be avoided in this deformation condition.

^{−1}. Initial casting structures have partly been replaced by recrystallized grains; however, the voids were also found in this area. Obviously, area IV is not the ideal region for hot working.

^{−1}.

## 4. Conclusions

^{−1}. The conclusions can be obtained.

^{20}, respectively, in addition, the constitutive equation with strain-dependent constants is established, which can accurately predict the flow stress of 30Cr4MoNiV UHS steel in the hot working process, and the correlation coefficient is 0.9906.

^{−1}. The voids and dynamic strain aging are found to be the cause of plastic instability by the analysis of microstructure.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Schematic diagram of the hot compression tests, (

**b**) Schematic diagram of the specimens before and after compression.

**Figure 2.**True stress–strain curves under the different deformation temperatures with strain rates. (

**a**) 0.01 s

^{−1}; (

**b**) 0.1 s

^{−1}; (

**c**) 1 s

^{−1}; (

**d**) 10 s

^{−1}.

**Figure 3.**Plots used for calculation of the material-dependent constants. (

**a**) $\mathrm{ln}\dot{\epsilon}-{\mathrm{ln}\sigma}_{\mathrm{P}}$, (

**b**) $\mathrm{ln}\dot{\epsilon}-{\sigma}_{\mathrm{P}}$, (

**c**) $\mathrm{ln}\dot{\epsilon}-\mathrm{ln}\left[\mathrm{sinh}\left({\alpha \sigma}_{\mathrm{P}}\right)\right]$, (

**d**) ${\mathrm{ln}\left[\mathrm{sinh}\right(\alpha \sigma}_{\mathrm{P}}\left)\right]-1/T$, (

**e**) ${\mathrm{ln}Z-\mathrm{ln}\left[\mathrm{sinh}\right(\alpha \sigma}_{\mathrm{P}}\left)\right]$ (the units of peak stress and strain rate are MPa and s

^{−1}, respectively).

**Figure 4.**The calculation flow chart of the material-dependent constants related to the constitutive equation.

**Figure 5.**Seventh-order fitting curves of each of the material-dependent constants: (

**a**) Q-ε; (

**b**) α-ε; (

**c**) n-ε; (

**d**) lnA-ε.

**Figure 6.**Comparison of predicted and experimental true stress-strain curves. (

**a**) 0.01 s

^{−1}; (

**b**) 0.1 s

^{−1}; (

**c**) 1 s

^{−1}; (

**d**) 10 s

^{−1}.

**Figure 7.**Correlation between predicted values and experimental values over a temperature range of 1173–1373 K with strain rate of 0.01–10 s

^{−1}.

**Figure 8.**Processing maps at different true strains: (

**a**) ε = 0.3; (

**b**) ε = 0.6; (

**c**) ε = 0.9. The shaded areas are the plastic instability areas, and the colored areas represent the different η values.

**Figure 9.**The response 3D surfaces of four criteria on deformation temperature and strain rate at the true strain of 0.9: (

**a**) m; (

**b**) $\dot{m}$; (

**c**) S; (

**d**) $\dot{S}$. The colored parts are the stable areas, and the shaded (gray) parts are the unstable areas.

**Figure 10.**Processing map with the true strain of 0.9. The purple dotted area is the instability region obtained by the instability parameter ξ, the cyan dotted areas are the instability regions obtained by the $\dot{m}$ criterion, the red dashed areas are the instability regions obtained by the $\dot{S}$ criterion, and the colored areas represent the η values.

**Figure 11.**Microstructures of 30Cr4MoNiV ultra-high-strength steel in I–IV areas: (

**a**) I (1273 K, 0.01 s

^{−1}); (

**b**) II (1323 K, 10 s

^{−1}); (

**c**) III (1223 K, 0.01 s

^{−1}); (

**d**) IV (1373 K, 0.1 s

^{−1}).

C | Si | Mn | Cr | Ni | Mo | V | Al | Fe |
---|---|---|---|---|---|---|---|---|

0.30 | 0.24 | 0.94 | 3.85 | 0.67 | 1.31 | 0.35 | 0.05 | Bal. |

Strain Rate [s^{−1}] | Deformation Temperature [K] | ||||
---|---|---|---|---|---|

1173 | 1223 | 1273 | 1323 | 1373 | |

0.01 | 193.24 | 158.85 | 115.62 | 93.17 | 87.91 |

0.1 | 221.00 | 182.26 | 152.81 | 121.79 | 109.83 |

1 | 270.99 | 228.80 | 187.61 | 171.35 | 143.43 |

10 | 311.71 | 263.33 | 215.62 | 194.19 | 177.01 |

Polynomial Coefficients | The Material-Dependent Constants | |||
---|---|---|---|---|

Q [kJ/mol] | α | n | ln A | |

B_{0} | 1219.831 | 0.00949 | 9.893 | 111.708 |

B_{1} | −8053.152 | −0.0543 | −49.28 | −722.781 |

B_{2} | 48,062.003 | 0.43802 | 350.185 | 4164.055 |

B_{3} | −168,839.713 | −1.90727 | −1248.333 | −14,174.679 |

B_{4} | 350,611.446 | 4.63567 | 2480.95 | 28,607.488 |

B_{5} | −424,298.811 | −6.31237 | −2796.554 | −33,725.332 |

B_{6} | 277,595.681 | 4.50402 | 1684.091 | 21,550.56 |

B_{7} | −76,115.557 | −1.31131 | −424.078 | −5790.606 |

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**MDPI and ACS Style**

Chen, G.; Yao, Y.; Jia, Y.; Su, B.; Liu, G.; Zeng, B.
Hot Deformation Constitutive Equation and Plastic Instability of 30Cr4MoNiV Ultra-High-Strength Steel. *Metals* **2021**, *11*, 769.
https://doi.org/10.3390/met11050769

**AMA Style**

Chen G, Yao Y, Jia Y, Su B, Liu G, Zeng B.
Hot Deformation Constitutive Equation and Plastic Instability of 30Cr4MoNiV Ultra-High-Strength Steel. *Metals*. 2021; 11(5):769.
https://doi.org/10.3390/met11050769

**Chicago/Turabian Style**

Chen, Gang, Yuanchao Yao, Yuzhen Jia, Bin Su, Guoyue Liu, and Bin Zeng.
2021. "Hot Deformation Constitutive Equation and Plastic Instability of 30Cr4MoNiV Ultra-High-Strength Steel" *Metals* 11, no. 5: 769.
https://doi.org/10.3390/met11050769