# Evaluation on Flexibility of Phenomenological Hardening Law for Automotive Sheet Metals

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Ordinary Differential Equation of Existing Hardening Laws

#### 2.1. Saturation Laws

_{4}for all examined materials in their study, which led to excellent estimation for all tested materials, particularly in large strain ranges. However, other values of parameter c

_{4}can be used for different materials (see the Supplementary Material).

#### 2.2. Power Laws

#### 2.3. Combination of Hardening Laws

#### 2.3.1. Additive Combination

_{1}denotes a power hardening law that could be the Hollomon, Ludwik, or Swift models, while H

_{2}denotes a saturation hardening law in form of Voce or Hockett–Sherby models; ${c}_{1}$ is a linear combined factor ($0<{c}_{1}<1$). Equation (7b) also includes an ODE formulation of the corresponding hardening rate. As an example, Figure 2 shows an application of a linear combination of the Swift and Voce models (LSV model) with ${c}_{1}=0.7$ on modeling the stress-strain data of AA6016-T4 sheets. It is worth noticing that H

_{1}and H

_{2}are both excellent approximations for the experimental data. As a result, the first term of Equation (7b) yields an average of the two functions over the entire strain range, as illustrated in Figure 2. The second term, on the other hand, involves their difference, which primarily contributes to the extrapolation ranges.

#### 2.3.2. Multiplicative Combination

## 3. New Strain Hardening Law

#### 3.1. Saturation Law

#### 3.2. Power Law

## 4. Application for Automotive Sheet Metals

#### 4.1. Investigated Materials

#### 4.2. Calibration Method

#### 4.2.1. Common Curve Fitting Method

#### 4.2.2. Constrained Curve Fitting Method

#### 4.3. Calibration Result

#### 4.4. Discussion

#### 4.4.1. Diffuse Neck Prediction

#### 4.4.2. Hardening Rate Curve Prediction

## 5. Validation

#### 5.1. Finite Element Model

#### 5.2. Effect of Calibration Method

#### 5.3. Selection of a Proper Hardening Law

_{2}of the proposed hardening laws for all tested materials. According to Figure 10a, using the common curve fitting method yields a significant variation in calculated δ

_{2}based on different hardening laws. Adopting the constrained curve fitting method reduces the variation where the δ

_{2}calculations of these hardening laws are closer together. However, the constraints lead to underestimations for the tensile force of many materials. Therefore, it is suggested that the constrained method is applicable for limited materials.

_{2}. However, there exist several materials of which all of these hardening laws give unsatisfied δ

_{2}calculation, for instance, DP780, SPCC, and TRIP980 sheets. For such kinds of materials, a different hardening law or a different calibration method should be adopted to identify the parameters of the hardening laws. Hence, the calculated δ

_{2}of all mentioned hardening laws for all tested materials are reported in Supplementary Material (Figure S1). From this perspective, if there are a lot of hardening laws available for a particular investigated material, which one is adaptable will need to be determined.

_{2}criteria could be considered to meet the demand. A small value of RMSE can be used as a necessary condition to ensure the accuracy of the selected hardening law in the pre-necking ranges. Whereas, the δ

_{2}can be used as a sufficient condition to qualify the goodness of the model’s predictions in the post-necking ranges. For example, the Proposed 1 provides good descriptions for several materials, such as TRIP1180, AA7075, and AA6021 sheets. In another hand, the Proposed 4 gives excellent results for DP590, AA6016, and AA6022 sheets. Figure 11 depicts their predictions of tensile forces for the mentioned materials. For other materials, one may choose a proper hardening law based on the results reported in Figure 5 and Figure S1.

## 6. Conclusions

_{2}criterion expressed in Equation (22) can be used as an indicator for choosing a proper hardening law of which the parameters were identified by curve fitting methods.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Applications of four hardening laws in capturing the experimental data obtained from the uniaxial tensile test of AA6016-T4 sheets. Experimental data after [11]. (

**a**) Stress-strain curves. (

**b**) Hardening rate—flow stress curves.

**Figure 2.**Application of a linear combined Swift and Voce models for AA6016-T4 sheets. Experimental data after [11].

**Figure 3.**Stress-strain data obtained from uniaxial tensile tests for six steel sheets. (

**a**) Engineering stress-strain curves. (

**b**) True stress-strain curves.

**Figure 4.**Stress-strain data obtained from uniaxial tensile tests for six aluminum alloy sheets. (

**a**) Engineering stress-strain curves. (

**b**) True stress-strain curves.

**Figure 5.**RMSE of the identified hardening laws for tested materials. Column charts indicate the results of the common curve fitting method; opened symbols indicate the results of the constrained curve fitting method.

**Figure 6.**Percentage errors of diffuse neck prediction of all examined hardening laws identified by the common curve fitting method.

**Figure 7.**Normalized RMSE of hardening rate predictions. (

**a**) Hardening laws identified by the common curve fitting method. (

**b**) Hardening laws identified by the constrained curve fitting method.

**Figure 9.**Comparison between the experimental and predicted tensile forces according to gauge length displacement of DP590 sheets. (

**a**) Hardening laws identified by the common curve fitting method. (

**b**) Tensile force predictions of the hardening laws reported in (

**a**). (

**c**) Hardening laws identified by the constrained curve fitting method. (

**d**) Tensile force predictions of the hardening laws reported in (

**c**).

**Figure 10.**Calculated δ

_{2}of the identified hardening laws for tested materials. (

**a**) Hardening laws identified by the common curve fitting method. (

**b**) Hardening laws identified by the constrained curve fitting method.

**Figure 11.**Tensile force predictions of (

**a**) Proposed 1 and (

**b**) Proposed 4 models for several materials.

**Table 1.**Material properties were obtained from the uniaxial tensile test of the investigated materials.

Material | Thickness (mm) | Young Modulus (GPa) | Initial Yield Stress (MPa) | Ultimate Tensile Strength (MPa) | Maximum Uniform Strain | Elongation (%) |
---|---|---|---|---|---|---|

DP590 | 1.4 | 205 | 401 | 603 | 0.156 | 25.7 |

DP780 | 1.2 | 206 | 489 | 822 | 0.123 | 20.5 |

DP980 | 1.6 | 200 | 800 | 1030 | 0.050 | 11.0 |

SPCC | 0.9 | 210 | 158 | 309 | 0.158 | 41.4 |

TRIP980 | 1.2 | 213 | 640 | 1026 | 0.120 | 19.5 |

TRIP1180 | 1.25 | 207 | 854 | 1117 | 0.229 | 40.9 |

AA6016 | 1.2 | 69 | 158 | 277 | 0.238 | 33.4 |

AA6022 | 1.1 | 67 | 123 | 238 | 0.209 | 30.1 |

AA7075 | 1.6 | 67 | 478 | 554 | 0.091 | 13.5 |

AA5052 | 0.8 | 73 | 173 | 229 | 0.090 | 12.2 |

AA6021 | 1.4 | 71 | 146 | 279 | 0.157 | 20.9 |

AA3004 | 0.51 | 62 | 73 | 156 | 0.171 | 28.7 |

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**MDPI and ACS Style**

Pham, Q.T.; Kim, Y.-S. Evaluation on Flexibility of Phenomenological Hardening Law for Automotive Sheet Metals. *Metals* **2022**, *12*, 578.
https://doi.org/10.3390/met12040578

**AMA Style**

Pham QT, Kim Y-S. Evaluation on Flexibility of Phenomenological Hardening Law for Automotive Sheet Metals. *Metals*. 2022; 12(4):578.
https://doi.org/10.3390/met12040578

**Chicago/Turabian Style**

Pham, Quoc Tuan, and Young-Suk Kim. 2022. "Evaluation on Flexibility of Phenomenological Hardening Law for Automotive Sheet Metals" *Metals* 12, no. 4: 578.
https://doi.org/10.3390/met12040578