#
Numerical Fatigue Analysis of Induction-Hardened and Mechanically Post-Treated Steel Components^{ †}

^{*}

^{†}

## Abstract

**:**

^{®}. Secondly, the thermo-metallurgical-mechanical analysis of the hardening process is conducted by means of a user-defined interface, utilizing the software Sysweld

^{®}. Thirdly, mechanical post-treatment is numerically simulated by Abaqus

^{®}. Finally, a strain-based approach considering the evaluated local material properties is applied, which reveals sound accordance to the fatigue tests results, exhibiting a minor conservative deviation of only up to two per cent, which validates the applicability of the presented numerical fatigue approach.

## 1. Introduction

## 2. Materials and Methods

^{®}[24]. Thereby, a transient simulation model was built up, invoking moving mesh approach, such that the translation of the induction coils was reproduced correctly. The numerically computed position and time dependent temperature field was then transferred to Sysweld

^{®}[25], wherewith the simulation of the heat-treatment process was examined. The presented IH process simulation methodology was based on a preliminary work, which is given in [21,26]. Beginning with the electro-magnetic-thermal simulation part, selected application studies for the utilized software are presented in [27,28]. Further validations of numerical and experimental results are provided in [29]. The main principle of the incorporated electromagnetic analysis was based on Maxwell’s equations considering the magnetic vector potential by Equation (1) [30]:

_{0}is the permittivity of vacuum, ε

_{r}is the relative permittivity, A is the magnetic vector potential, µ

_{0}is the permeability in vacuum, µ

_{r}is the relative permeability, B is the magnetic flux density, and J

_{e}is external current density. The resulting induced heat energy is subsequently coupled with a thermal calculation of the heat transfer according to Equation (2):

_{ind}is the induced heat energy. In addition, the heat loss due to convection and radiation was considered in the course of the numerical analysis, details see [30,31].

^{®}is depicted in Figure 2. Due to the axial-symmetric specimen geometry, a two dimensional model was set up. Thereby, the surface layer is modeled with a fine mesh as the current density was mainly localized on the surface region based on the skin effect, which skin depth δ can be evaluated by Equation (3):

^{®}, the time-dependent temperature data of each node was transferred utilizing a self developed routine [26] to the software package Sysweld

^{®}, in order to conduct the numerical analysis of the heat-treatment process. This self-written interface converted the temperature-time distribution for each node of the numerical model, which is the result of the simulation in Comsol

^{®}, to Sysweld

^{®}. Herein, this input data acted as the initial heating step for the thermo-metallurgic-mechanical heat-treatment analysis. As the electro-magnetic properties majorly depended on the actual temperature (see Appendix A), and not on the actual metallurgical phase of the investigated steel material, this methodology was applicable. The thermo-metallurgic-mechanical simulation in Sysweld

^{®}was performed in accordance to the models provided in the reference manual, see [34]. Further details on the modeling of the quenching process, in order to evaluate the local phase proportions, as well as hardness, distortion, and residual stress conditions are given in [34,35,36,37,38]. Utilizing these models, the residual stress state during, and more importantly, at the end of the IH process was numerically computed, which acts, in addition to the hardness condition, as a significant input for the local fatigue analysis at the final stage of the numerical assessment methodology.

^{®}to Abaqus

^{®}, in order to perform the mechanical simulation. The numerical model was set up in agreement to the experimental investigations [16] incorporating a pin, which iteratively impacted the surface at the notched area of the specimen, as seen in Figure 3. The radius of the pin was comparable to the radius of the notch. The experimental results in [16] highlighted that the superimposed StrP process did not majorly affect the local microstructure and hardness condition. Nevertheless, the subsequent StrP post-treatment did significantly change the residual stress state in depth, which was also the aim of the numerical analysis of the StrP simulation within this work.

^{®}, the local node-dependent residual stress state, including the local IH-affected material behavior in terms of stress-strain data, acts as initial condition for the simulation of the mechanical post-treatment in Abaqus

^{®}. To properly cover hardening effects, a combined isotropic-kinematic hardening model [39,40] was invoked for the analysis, as this model has been shown to be applicable for the simulation of similar mechanical post-treatment processes, such as shot peening, for comparable steel materials (see [41]).

_{a}was calculated based on the cyclic stress-strain relationship by Ramberg-Osgood [42], considering the linear-elastic stress amplitude σ

_{a}, the Young’s modulus E, the cyclic strength coefficient K′, and the cyclic strain hardening exponent n′, as seen in Equation (4):

_{m}on the basis of the damage parameter P

_{SWT}by Smith, Watson, and Topper [43], as seen in Equation (5):

_{f}, the fatigue strength exponent b, the fatigue ductility coefficient ε‘

_{f}, and the fatigue ductility exponent c, as seen in Equation (6):

_{f}, ε′

_{f}, b, and c, which were mandatory to perform the fatigue assessment, could be either determined based on low-cycle fatigue tests or by an estimation based on the unified material law (UML), introduced by Bäumel and Seeger [47]. As the original UML was focused on mild steels, an extension for high-strength steels is given in [48]. In both cases, the parameter evaluation mostly depends on the ultimate tensile strength (UTS). As the local Vicker’s hardness value (HV) was numerically computed by the manufacturing process simulation, the UTS could be estimated based on a suggestion in [49] for steels, as seen in Equation (7):

## 3. Results

#### 3.1. Fatigue Tests

_{S}= 50%. It was shown that IH elevated the mean fatigue resistance at 5e6 load-cycles by a factor of 1.46, and IH + StrP by a factor of 1.75 compared to the BM. These results proved the beneficial effect of the post-treatment processes and the experimental values acted as basis to validate the numerical fatigue analysis. Further details of the fatigue test procedure, statistical evaluation of the fatigue data, as well as extensive fracture surface analysis, is provided in [16].

#### 3.2. Numerical Simulation

#### 3.2.1. Electro-Magnetic-Thermal Simulation

^{®}are illustrated in Figure 5. At first, Figure 5a depicts the increment at which induction coil #1 started at the top end of the specimen. It is shown that a peak temperature of about 330 °C occurred at this time step. Further on, Figure 5b presents the time increment at which induction coil #2 additionally began at the top end of the specimen. Thereby, a maximum temperature of around 900 °C was computed within the model. The final condition directly after finishing the inductive heating utilizing the two coils, is depicted in Figure 5c. In this case, the peak temperature was about 840 °C and the localization of the heat input in the surface layer due to skin-effect was clearly observable. Subsequently, these time-dependent nodal temperature data were transferred to Sysweld

^{®}, utilizing a self-written script.

#### 3.2.2. Thermo-Metallurgical-Mechanical Simulation

^{®}to Sysweld

^{®}, as an initial step, the heating process was computed in Sysweld

^{®}. The results are shown Figure 6. At first, Figure 6a displays the increment at which induction coil #1 started at the top end of the specimen. It was observed that a similar peak temperature of about 330 °C occurred at this time step. Further on, Figure 6b presents the time increment at which coil #2 began, where again a similar maximum temperature of around 900 °C was computed. The final condition after finishing the heating process is depicted in Figure 6c revealing a peak temperature of about 850 °C, which was in sound accordance to the previous simulation in Comsol

^{®}. To sum up, the numerically computed temperature distribution in Sysweld

^{®}was similar to the simulation results given by Comsol

^{®}, which validated the applicability of the self-written interface.

^{®}. Firstly, water spray cooling, and then air cooling was modeled. These cooling steps were defined in accordance to the experimental induction hardening process of the round specimen. Figure 7a demonstrates the temperature distribution during water spray cooling at the top end and Figure 7b at the notch area of the specimen. Figure 7c depicts the time increment after finishing the water spray cooling, which shows that the IH heat-treatment was effective especially at the surface-layer of the specimen. After cooling down to room temperature, the numerically computed metallurgical and mechanical results were analyzed. Figure 8a illustrates the total amount of the martensitic and Figure 8b of the ferritic/pearlitic phase. The simulated induction hardening depth fit well to a micrographical analysis of the one real specimen, see Figure 8c.

^{®}to Abaqus

^{®}.

#### 3.2.3. Simulation of Mechanical Stroke-Peening Process

^{®}to Abaqus

^{®}, the superimposed StrP process as mechanical post-treatment was simulated. The axial stress results before, see Figure 11a, and during the peak compressive load, see Figure 11b, are shown as follows.

#### 3.3. Local Fatigue Strength Assessment

_{S}= 90%, as presented within the subsequent discussion. As a conservative fatigue assessment with values of P

_{S}≥ 90% is preferentially applied, the presented fatigue model was capable of estimating the fatigue strength in this region.

## 4. Discussion

_{S}= 50%. It is shown that the normalized fatigue strength amplitude at 5e6 load-cycles was majorly elevated, by 46%, due to the IH, and by as much as 75% due to IH+StrP compared to the BM state. In addition, the number of load-cycles at the transition knee point N

_{T}were reduced, which contributed to the beneficial effect of the post-treatment processes. No significant change of the slope within the finite life region was evaluated for IH+StrP compared to IH, concluding that the StrP was most effective within the long life fatigue region at 5e6 load-cycles.

_{S}= 90%, considering a conservative fatigue assessment. It is shown that in both cases IH, as well as IH+StrP condition, the fatigue model is capable of estimating the experimental results. For both conditions, the deviation of the model to the experiments was only between one and two percent, which validates the applicability of the presented numerical fatigue assessment.

## 5. Conclusions

- The fatigue test results reveal that induction hardening (IH) leads to an increase of 46% of the long life fatigue strength amplitude at 5e6 load-cycles compared to the base material (BM). The superimposed stroke peening (StrP) process even elevates this value again by about 20%, leading to a benefit of 75% compared to the BM.
- The numerically computed hardness state, as well as residual stress condition utilizing an electro-magnetic-thermal and thermo-metallurgical-mechanical simulation for the IH process, shows a sound agreement to the measurements. Residual stress values at the surface in an axial direction reveal a difference of only about two percent when comparing simulation and X-ray residual stress measurements.
- The simulation of the superimposed StrP process leads to similar results, whereby a minor difference of the von Mises equivalent residual stress state at the surface of only one percent between the numerical analysis and measurements is also observed.
- The final fatigue assessment highlights a sound agreement of the fatigue model, which incorporates the local hardness and residual stress state, to the experiments. Focusing on the long life fatigue strength, a minor conservative estimation with a difference of only one to two percent is shown for both IH as well as IH+StrP condition.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Table A1.**Temperature dependence of selected material parameters for 50CrMo4 steel [32]

T [°C] | κ [MS m^{−1}] | λ [W m^{−1} K^{−1}] | ρC [J kg^{−1} K^{−1}] |
---|---|---|---|

0 | 4.91 | 43.1 | 451 |

100 | 3.90 | 42.7 | 488 |

200 | 3.02 | 41.7 | 528 |

300 | 2.40 | 40.6 | 552 |

400 | 1.95 | 38.9 | 619 |

500 | 1.60 | 37.3 | 731 |

600 | 1.25 | 33.7 | 762 |

700 | 1.05 | 31.0 | 828 |

800 | 1.00 | 29.5 | 836 |

900 | 0.97 | 28.4 | 830 |

1000 | 0.94 | 28.1 | 790 |

1100 | 0.92 | 28.8 | 732 |

1200 | 0.90 | 30.1 | 672 |

**Table A2.**Magnetization characteristic at room temperature for 50CrMo4 steel [32].

B [T] | H [A/m] |
---|---|

0.08 | 500 |

0.32 | 1000 |

0.88 | 1500 |

1.28 | 2000 |

1.40 | 2500 |

1.48 | 3000 |

1.52 | 3500 |

1.56 | 4000 |

1.60 | 4500 |

1.64 | 5000 |

1.66 | 5500 |

1.68 | 6000 |

1.70 | 6500 |

1.72 | 7000 |

## Appendix B

**Table A3.**Extended unified material law (UML) [48].

Parameter | Estimation |
---|---|

K′ | σ′_{f}/(ε′_{f})^{n′} |

n′ | b/c |

σ′_{f} | UTS(1 + ψ) |

ε′_{f} | 0.58ψ + 0.01 |

b | −log(σ′_{f}/σ_{E})/6 |

σ_{E} | UTS(0.32 + ψ/6) |

c | −0.58 |

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**Figure 1.**Geometry of the investigated notched round specimen [21].

**Figure 2.**Numerical model of the inductive heating process (based on [26]).

**Figure 3.**Experimental StrP process of specimen [16] and according to the numerical model.

**Figure 4.**Fatigue test results for induction hardening (IH) and IH + superimposed stroke peening (StrP) − condition (according to [16]).

**Figure 11.**Axial residual stress state before (

**a**) and during the peak compressive load (

**b**) of the mechanical StrP process.

**Table 1.**Comparison of the fatigue test results (evaluated at P

_{S}= 50%) for base material (BM), IH, and IH+StrP-condition [16].

Condition | σ_{a} at 5e6 | Slope k | Knee Point N_{T} |
---|---|---|---|

BM | 1.00 | 15.8 | 1.4e6 |

IH | 1.46 | 9.8 | 7.9e5 |

IH+StrP | 1.75 | 9.7 | 2.7e5 |

**Table 2.**Comparison of axial residual stresses for IH-condition by X-ray measurements and numerical simulation.

Evaluation | At Surface | In Depth |
---|---|---|

X-ray measurement | −0.54 | Not evaluable ^{1} |

Numerical simulation | −0.55 | +0.20 |

^{1}Electro-chemical polishing not applicable up to this depth.

**Table 3.**Comparison of von Mises residual stresses for IH+StrP-condition by X-ray measurements and numerical simulation.

Evaluation | At-Surface | In Depth |
---|---|---|

X-ray measurement | −1.58 | Not evaluable ^{1} |

Numerical simulation | −1.59 | −0.24 |

^{1}Electro-chemical polishing not applicable up to this depth

**Table 4.**Comparison of fatigue strength amplitude σ

_{a}at 5e6 load-cycles evaluated by model and experiments (P

_{S}= 90%), for IH and IH+StrP-condition.

Condition | Model | Experiment | Difference |
---|---|---|---|

IH | 1.37 | 1.39 | −1.4% |

IH+StrP | 1.70 | 1.72 | −1.2% |

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**MDPI and ACS Style**

Leitner, M.; Aigner, R.; Grün, F.
Numerical Fatigue Analysis of Induction-Hardened and Mechanically Post-Treated Steel Components. *Machines* **2019**, *7*, 1.
https://doi.org/10.3390/machines7010001

**AMA Style**

Leitner M, Aigner R, Grün F.
Numerical Fatigue Analysis of Induction-Hardened and Mechanically Post-Treated Steel Components. *Machines*. 2019; 7(1):1.
https://doi.org/10.3390/machines7010001

**Chicago/Turabian Style**

Leitner, Martin, Roman Aigner, and Florian Grün.
2019. "Numerical Fatigue Analysis of Induction-Hardened and Mechanically Post-Treated Steel Components" *Machines* 7, no. 1: 1.
https://doi.org/10.3390/machines7010001