# Physical and Numerical Simulations of Closed Die Hot Forging and Heat Treatment of Forged Parts

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model

#### 2.1. FE Model

^{®}package were used in mechanical (flow stress) and thermal (conductivity, specific heat, density, expansion coefficient) parts. The flow stress, which is the material parameter in the mechanical part, was calculated based on the Hansel-Spittel equation with coefficients, which were identified using inverse analysis for the hot compression tests results. The inverse algorithm described in [11] was used. Equations describing phase transformations during cooling were implemented in the user’s procedures in the FE program.

#### 2.2. Microstructure Evolution Model

_{0}—grain size prior to deformation, Z—Zener-Holomon parameter, Q

_{SRX}, Q

_{DRX}, Q

_{DSRX}, Q

_{GROWTH}—activation energy for the relevant process, n, a

_{0}−a

_{3}, b

_{0}−b

_{3}, p

_{1}−p

_{10}, q, K—coefficients determined by the inverse analysis of the experimental tests.

#### 2.3. Phase Transformation Model

_{γ}) was included as a variable in the model. The main equations in the model are given in Table 2. The following upgrades were introduced in the JMAK equation [16]:

- (1)
- Coefficient n is constant for each transformation and in the identification it is referred to as a
_{4}, a_{16}and a_{24}for ferrite, pearlite and bainite transformations, respectively. - (2)
- (3)
- Calculations of carbon concentration in the austenite during both ferrite and bainite transformations were added.
- (4)
- The T
_{0}temperature concept proposed in [18] was added. It allowed prediction of the return of the pearlitic transformation after bainitic has started during holding at the constant temperature. Beyond this, prediction of the occurrence of the retained austenite became possible.

_{γ}), temperatures of bainite start (B

_{s}) and martensite start (M

_{s}), as well as martensite volume fraction (F

_{m}) are shown in Table 2, where F

_{f}, F

_{p}and F

_{b}represent volume fractions of ferrite, pearlite and bainite, respectively. Parameter p in this table is explained in [19] and the numerical solution of the present model is described in [20].

- (1)
- A routine which prepares storage for the model’s intermediate variables, initializes or obtains simulation parameters passed to the model and calls the model main routine from a library file. The results are similarly stored in mesh fields and can be used in further simulation steps.
- (2)
- A routine which converts model results, especially transformation energy, to values accepted by the solver’s FE thermal coupling (in the analysed case, thermal power). This routine allows a full thermal coupling of transformation on an energy level to be obtained without using additional models to compensate the transformation with modifying material parameters and can be turned off to make computations faster, but ignores thermal coupling.

## 3. Experiments

#### 3.1. Methodology

#### 3.2. Materials

#### 3.3. Deformation and Thermal Cycles

#### 3.4. Physical Simulations

^{−1}.

- (1)
- Variant 0—cooling to the temperature above A
_{c}_{3}, heating to the holding temperature (all steels). - (2)
- Variant 1—cooling until the austenite decomposition is completed, heating to the holding temperature (steels B and C).
- (3)
- Variant 2—cooling until the ferritic transformation is completed and pearlitic transformation did not begin, heating to the holding temperature (steels B and C).

#### 3.5. Results for Steel A

_{0}line.

_{0}line. The possibility of modelling of this phenomenon is described in [19], and this approach was applied in the present work. Microstructures show also that even for very long holding times the austenite decomposition is not completed and the remaining austenite is transferred into bainite or martensite during subsequent cooling. Volume fractions of phases were determined for each sample and these results are presented in Section 5.1, where they are compared with the model predictions.

#### 3.6. Results for Steel B

#### 3.7. Results for Steel C

## 4. Identification and Verification of the Austenite Microstructure Evolution Model

_{T}, w

_{F}, w

_{t}—weights, T

_{m}, T

_{c}—measured and calculated start and end temperatures of phase transformations in the CCT tests, F

_{m}, F

_{c}—measured and calculated volume fractions of phases, t

_{m}, t

_{c}—measured and calculated times to the start and end of phase transformations in the TTT tests, k, l, m—number of temperatures, volume fractions and times, respectively.

## 5. Numerical Simulation of Forging-Cooling Sequences

#### 5.1. Steel A

#### 5.2. Steel B

#### 5.3. Steel C

#### Calculated Distribution of the Temperature

## 6. Conclusions

- (1)
- Introduction of the modified Gauss function to describe the relation of the Avrami coefficient k to the temperature.
- (2)
- Introduction of the austenite grain size prior to transformation as an independent variable in the model.
- (3)
- Prediction of the carbon concentration in the austenite during bainitic transformation and using the T
_{0}line concept to control the progress of the isothermal bainitic transformation.

- (1)
- The deformation process affects substantially the developed austenite microstructure and affects the kinetics of the decomposition of this phase to ferrite, pearlite and to a lesser extent to bainite and martensite. Therefore, it is important that this effect should be accounted for in the phase transformation model.
- (2)
- The model predicts the start and end temperatures of transformations with good accuracy.
- (3)
- The model predicts very well volume fractions of structural components for ferritic/pearlitic microstructures.
- (4)
- Despite the limitations, the methodology of the model validation and tuning adapted in this investigation using the results of the physical simulation of the forging process with Gleeble 3800 proved to be a very effective tool for improving the predictive capabilities of the mathematical models.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schematic illustration of the investigated forgings with the locations of sensors: (

**a**) adapter, (

**b**) flange, (

**c**) fork.

**Figure 2.**Thermal cycles in points P during cooling of the investigated forgings: (

**a**) annealing of the adapter, (

**b**) normalization of the flange, (

**c**) quenching and tempering of the fork.

**Figure 3.**(

**a**,

**b**) Selected microstructures of the samples after holding for 1 h at the temperature of 620 °C, various magnifications. P—pearlite; F—ferrite.

**Figure 4.**Microstructures of the samples after holding at the temperature of 560 °C for 600 s (

**a**), 1200 s (

**b**), 2400 s (

**c**) and 4000 s (

**d**) F—ferrite, P—pearlite, B—bainite, M—martensite.

**Figure 5.**Microstructures of the samples after holding at the temperature of 520 °C for 600 s (

**a**), 1000 s (

**b**), 2000 s (

**c**) and 4000 s (

**d**). F—ferrite, P—pearlite, B—bainite, M—martensite.

**Figure 6.**Microstructures of steel B before the first deformation (

**a**) and after the last deformation (

**b**).

**Figure 7.**Microstructures of steel B in point P after cooling variant 1 (

**a**) and variant 0 (

**b**). F—ferrite, P—pearlite.

**Figure 9.**Point P variant 1—microstructures of the samples after first cooling to 540 °C (

**a**), after heating to the holding temperature (

**b**) and at the end of holding (

**c**).

**Figure 10.**Point P variant 2—microstructures of the samples after first cooling to 670 °C (

**a**), after heating to the holding temperature (

**b**) and at the end of holding (

**c**).

**Figure 11.**Point Q variant 1—microstructures of the samples after first cooling to 500 °C (

**a**), after heating to the holding temperature (

**b**) and at the end of holding (

**c**).

**Figure 12.**Point Q variant 2—microstructures of the samples after first cooling to 609 °C (

**a**), after heating to the holding temperature (

**b**) and at the end of holding (

**c**).

**Figure 13.**Measured and calculated austenite grain size at various stages of the process for steel B, EoF—end of forging.

**Figure 14.**Measured and calculated austenite grain size at various stages of the process for steel C.

**Figure 15.**Measured (full symbols) and calculated (open symbols with lines) start and end temperatures of phase transformations in the CCT tests for steel B, grain size 30 ± 3 μm, (

**a**) and steel C, grain size 37 ± 4 μm (

**a**) and 51 ± 3 μm (

**c**).

**Figure 16.**Measured and calculated volume fraction of phases in steel A after physical simulations described in Section 3.3, holding temperature 560 °C (

**a**,

**b**) and 520 °C (

**c**,

**d**) in the massive part of the forging (

**a**,

**c**) and in the thinner part (

**b**,

**d**).

**Figure 17.**Measured and calculated volume fraction of ferrite in steel B after physical simulations described in Section 3.3.

**Figure 18.**Calculated distributions of the temperature (°C) (

**a**) and austenite grain size (μm) (

**b**) after forging of the adapter from steel A.

**Figure 19.**Cooling of the adapter from steel A—calculated distributions of the pearlite volume fraction after holding at 560 °C for 1 h (

**a**) and of the bainite volume fraction after holding at 520 °C for 1 h (

**b**).

**Figure 20.**Calculated distributions of the temperature (

**a**) and austenite grain size (

**b**) after forging of the flange from steel B.

**Figure 21.**Calculated distributions of the ferrite volume fraction after cooling of the flange from steel B according to the schedule in Figure 2b.

**Figure 22.**Calculated distributions of the temperature (

**a**) and austenite grain size (

**b**) after forging of the fork from steel C.

**Figure 23.**First stage of the cooling of the fork from steel C—calculated distribution of the austenite volume fraction after cooling to the temperature at which ferritic transformation is completed and pearlitic transformation has not begun yet.

Parameter | Equation | |
---|---|---|

Kinetics of SRX | ${X}_{SRX}=1-\mathrm{exp}\left[-0.693{\left(\frac{t}{{t}_{0.5}}\right)}^{n}\right]$ ${t}_{0.5}={a}_{0}{\epsilon}^{-{a}_{1}}{\dot{\epsilon}}^{{a}_{2}}{D}_{0}^{{a}_{3}}\mathrm{exp}\left(\frac{{Q}_{SRX}}{RT}\right)$ | (1) |

Grain size after SRX | ${D}_{SRX}={b}_{0}{\epsilon}^{-{b}_{1}}{\dot{\epsilon}}^{-{b}_{2}}{D}_{0}^{{b}_{3}}\mathrm{exp}\left(\frac{-{Q}_{DSRX}}{RT}\right)$ | (2) |

Critical strain for DRX | ${\epsilon}_{cr\_DRX}={p}_{1}{D}_{0}^{{p}_{2}}{Z}^{{p}_{3}}$ | (3) |

DRX volume fraction | ${X}_{DRX}=1-\mathrm{exp}\left[-{p}_{7}{\left(\frac{\epsilon -{\epsilon}_{cr\_DRX}}{{\epsilon}_{s}-{\epsilon}_{cr\_DRX}}\right)}^{{p}_{8}}\right]$ | (4) |

Saturation strain | ${\epsilon}_{s}={p}_{4}{D}_{0}^{{p}_{5}}{Z}^{{p}_{6}}$ | (5) |

Grain size after DRX | ${D}_{DRX}={p}_{9}{Z}^{-{p}_{10}}$ | (6) |

Grain growth | ${D}_{t+\mathsf{\Delta}t}^{q}={D}_{t}^{q}+K\text{\hspace{0.17em}}t\text{\hspace{0.17em}}\mathrm{exp}\left(-\frac{{Q}_{GROWTH}}{RT}\right)\text{\hspace{0.17em}}$ | (7) |

Parameter | Equation | |
---|---|---|

Incubation time of perlite for the remaining transformation | $\begin{array}{l}{\tau}_{P}=\frac{{a}_{9}{D}_{\gamma}^{{a}_{28}}}{{\left(\mathsf{\Delta}T\right)}^{{a}_{11}}}\mathrm{exp}\left(\frac{{a}_{10}}{RT}\right)\\ \mathsf{\Delta}T={A}_{e1}-T\end{array}$ | (9) |

Incubation time of bainite for the remaining transformation | $\begin{array}{l}{\tau}_{b}=\frac{{a}_{17}{D}_{\gamma}^{{a}_{30}}}{{\left(\mathsf{\Delta}T\right)}^{{a}_{19}}}\mathrm{exp}\left(\frac{{a}_{18}}{RT}\right)\\ \mathsf{\Delta}T={B}_{s}-T\end{array}$ | (10) |

Coefficients k for ferrite | $\begin{array}{l}{k}_{f}=\frac{{a}_{5}}{{D}_{\gamma}}\mathrm{exp}\left[-{\left(\frac{\left|\mathsf{\Delta}T\right|}{{a}_{7}}\right)}^{{a}_{8}}\right]\\ \mathsf{\Delta}T=T-{T}_{nose}\\ {T}_{nose}={A}_{e3}-\frac{400}{{D}_{\gamma}}+{a}_{6}\end{array}$ | (11) |

Coefficients k for perlite | $\begin{array}{l}{k}_{p}=\frac{{a}_{15}}{{D}_{\gamma}^{{a}_{14}}}\mathrm{exp}\left[-{\left(\frac{\left|\mathsf{\Delta}T\right|}{{a}_{13}}\right)}^{{a}_{8}}\right]\\ \mathsf{\Delta}T=T-{a}_{12}\end{array}$ | (12) |

Coefficients k for bainite | $\begin{array}{l}{k}_{b}=\frac{{a}_{23}}{{D}_{\gamma}^{{a}_{29}}}\mathrm{exp}\left[-{\left(\frac{\left|\mathsf{\Delta}T\right|}{{a}_{22}}\right)}^{{a}_{24}}\right]\\ \mathsf{\Delta}T=T-{a}_{21}\end{array}$ | (13) |

Average carbon content in austenite | ${c}_{\gamma}=\frac{\left[{c}_{0}-\left[{F}_{f}+{F}_{b}\left(1-p\right)\right]{c}_{\alpha}\right]}{1-{F}_{f}-{F}_{b}\left(1-p\right)}$ | (14) |

Bainite start temperatures | ${B}_{s}={a}_{20}$ | (15) |

Martensite start temperatures | ${M}_{s}={a}_{25}-{a}_{26}{c}_{\gamma}$ | (16) |

Martensite volume fraction | ${F}_{m}=\left(1-{F}_{f}-{F}_{p}-{F}_{b}\right)\left\{1-\mathrm{exp}\left[-{a}_{27}\left({M}_{s}-T\right)\right]\right\}$ | (17) |

Steel | C | Mn | Si | Cr | Ni | Cu | Mo |
---|---|---|---|---|---|---|---|

A | 0.42 | 0.744 | 0.237 | 1.049 | 0.122 | 0.209 | 0.171 |

B | 0.18 | 0.58 | 0.17 | 0.39 | 0.1 | 0.24 | 0.02 |

C | 0.48 | 0.58 | 0.23 | 0.12 | 0.16 | - | 0.04 |

Part | Point | Pass 1 | Pass 2 | Pass 3 | Pass 4 | ||||
---|---|---|---|---|---|---|---|---|---|

ε | T, °C | ε | T, °C | ε | T, °C | ε | T, °C | ||

Adapter | P | 0.48 | 1229 | 0.52 | 1235 | 0.1 | 1232 | - | - |

Q | 0.59 | 1228 | 0.89 | 1221 | 0.23 | 1211 | - | - | |

Flange | P | 0.4 | 1100 | 0.99 | 1103 | 0.07 | 1115 | - | - |

Q | 0.64 | 1099 | 0.87 | 1106 | 0.06 | 1103 | - | - | |

Fork | P | 0.11 | 1125 | 0.95 | 1120 | 0.39 | 1101 | 0.3 | 1093 |

Q | 0.12 | 1148 | 0.99 | 1147 | 1.69 | 1150 | 0.17 | 1120 |

Coefficient | Steel A | Steel B | Steel C |
---|---|---|---|

n | 1.3025 | 1.4919 | 0.86 |

a_{0} | 2.814 × 10^{−15} | 9.9684 × 10^{−13} | 2.199 × 10^{−16} |

a_{1} | −1.4016 | −0.73206 | −1.0784 |

a_{2} | −0.1195 | −0.15703 | −0.42 |

a_{3} | 2.1793 | 3.9289 | 3.597 |

Q_{SRX} | 244080 | 92147 | 186,000 |

b_{0} | 0.6534 | 0.6143 | 28.713 |

b_{1} | −0.2661 | −0.1017 | −0.3 |

b_{2} | −0.06558 | −0.013 | −0.0796 |

b_{3} | 1.1471 | 1.1683 | 0.1746 |

Q_{DSRX} | 10,002 | 5008 | 6446 |

p_{1} | 0.61928 × 10^{−3} | 1.9337 × 10^{−3} | 0.60634 × 10^{−4} |

p_{2} | 0.2871 | 0.092 | 0.79995 |

p_{3} | 0.1906 | 0.1814 | 0.18753 |

p_{4} | 2.433 × 10^{−3} | 0.51143 × 10^{−3} | 0.5267 × 10^{−5} |

p_{5} | 0.1481 | 0.5252 | 1.476 |

p_{6} | 0.1951 | 0.1865 | 0.19823 |

p_{7} | −1.4869 | −1.159 | −1.66863 |

p_{8} | 1.8849 | 1.5158 | 1.62773 |

p_{9} | 6822.71 | 3553 | 7676 |

p_{10} | −0.1934 | −0.1837 | −0.21461 |

Q_{DEF} | 284,798 | 278,878 | 280,204 |

q | 8.34 | 7 | 5 |

K | 4.3322 × 10^{30} | 4 × 10^{34} | 2.837 × 10^{16} |

Q_{GROWTH} | 412,430 | 580,000 | 302,086 |

a_{4} | a_{5} | a_{6} | a_{7} | a_{8} | a_{9} | a_{10} | a_{11} | a_{12} |

0.5417 | 0.74798 | 124.39 | 111.87 | 2.8367 | 6151.89 | 13.7849 | 1.118 | 666.76 |

a_{13} | a_{14} | a_{15} | a_{16} | a_{17} | a_{18} | a_{19} | a_{20} | a_{21} |

350 | 1.072 | 2.1496 | 0.05 | 7.185 | 76.43 | 3.4748 | 574.998 | 300 |

a_{22} | a_{23} | a_{24} | a_{25} | a_{26} | a_{27} | a_{28} | a_{29} | a_{30} |

236.55 | 10.2419 | 0.5 | 817.32 | 1299.8 | 0.011 | 0.0016 | 1.06559 | 1.2618 |

a_{4} | a_{5} | a_{6} | a_{7} | a_{8} | a_{9} | a_{10} | a_{11} | a_{12} |

3.0 | 9.676 | 205.7 | 22.98 | 1.326 | 16.000 | 35.06 | 3.5 | 517 |

a_{13} | a_{14} | a_{15} | a_{16} | a_{17} | a_{18} | a_{19} | a_{20} | a_{21} |

119 | 0.364 | 8.4 | 0.291 | 0.0167 | 9.804 | 0.168 | 569.8 | 378.7 |

a_{22} | a_{23} | a_{24} | a_{25} | a_{26} | a_{27} | a_{28} | a_{29} | a_{30} |

61.66 | 10.86 | 0.5 | 362.1 | 0.047 | 0.011 | 0.516 | 0.636 | 0.927 |

a_{4} | a_{5} | a_{6} | a_{7} | a_{8} | a_{9} | a_{10} | a_{11} | a_{12} |

1.33 | 5.896 | 185.7 | 75.85 | 2.713 | 16.000 | 12.54 | 3.5 | 594.9 |

a_{13} | a_{14} | a_{15} | a_{16} | a_{17} | a_{18} | a_{19} | a_{20} | a_{21} |

47.29 | 0.664 | 18.19 | 0.487 | 0.309 | 0.012 | 0.1 | 535.4 | 498.13 |

a_{22} | a_{23} | a_{24} | a_{25} | a_{26} | a_{27} | a_{28} | a_{29} | a_{30} |

15.63 | 13.2 | 3.7 | 656.7 | 696.4 | 0.011 | 1.3975 | 0.549 | 0.045 |

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**MDPI and ACS Style**

Poloczek, Ł.; Rauch, Ł.; Wilkus, M.; Bachniak, D.; Zalecki, W.; Pidvysotsk’yy, V.; Kuziak, R.; Pietrzyk, M.
Physical and Numerical Simulations of Closed Die Hot Forging and Heat Treatment of Forged Parts. *Materials* **2021**, *14*, 15.
https://doi.org/10.3390/ma14010015

**AMA Style**

Poloczek Ł, Rauch Ł, Wilkus M, Bachniak D, Zalecki W, Pidvysotsk’yy V, Kuziak R, Pietrzyk M.
Physical and Numerical Simulations of Closed Die Hot Forging and Heat Treatment of Forged Parts. *Materials*. 2021; 14(1):15.
https://doi.org/10.3390/ma14010015

**Chicago/Turabian Style**

Poloczek, Łukasz, Łukasz Rauch, Marek Wilkus, Daniel Bachniak, Władysław Zalecki, Valeriy Pidvysotsk’yy, Roman Kuziak, and Maciej Pietrzyk.
2021. "Physical and Numerical Simulations of Closed Die Hot Forging and Heat Treatment of Forged Parts" *Materials* 14, no. 1: 15.
https://doi.org/10.3390/ma14010015