# Stress-Invariants-Based Anisotropic Yield Functions and Its Application to Sheet Metal Plasticity

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## Abstract

**:**

## 1. Introduction

## 2. Proposed Model

#### 2.1. Proposed Anisotropic Yield Function

#### 2.2. The Convexity Condition of the Yield Function

#### 2.3. Effect of α and β on Yield Surface under Plane Stress

## 3. Application of the Proposed Yield Function Model

#### 3.1. AA6016-T4 Sheet

#### 3.2. AA2090-Ts3 Sheet

#### 3.3. Pure Titanium Sheet

## 4. Conclusions

- (1)
- The KV’12S yield function results in the Von Mises yield function when the two parameters α and β are zero.
- (2)
- KV’12S can also well represent various types of yield curves including strongly anisotropic materials. It was shown that the proposed yield function can implement various types of yield surface shapes by combining the two parameters α and β.
- (3)
- To verify the flexibility of the proposed model, the convexity of the yield surface was checked and compared with the Cazacu 2018 yield function. The convexity of KV’12S was satisfied in most stress ranges, but the Cazacu 2018 yield function for the material covered in this study did not satisfy the convexity in some stress ranges.
- (4)
- As a result of applying the KV’12S yield function proposed in this study to the AA2090-T4 material, which exhibited very strong anisotropy, the results deviated from some experimental data. Attention should be paid to the selection of experimental data used for the parameter optimization of the yield function proposed in this study. In this case, twelve experimental data (e.g., ten experimental data from the uniaxial tensile test and two experimental data from the equi-biaxial tension test) are recommended.
- (5)
- The KV’21A yield function was used to predict the yield surface of a commercial pure titanium sheet that exhibited an asymmetrical yield surface shape, and it was compared with the Yoon 2014 yield function and experimental data. Both yield functions were in good agreement with the experiment overall.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Yield locus according to proposed isotropic yield function for different values of parameters α and β: (

**a**) effect of parameter α, and (

**b**) effect of parameter β.

**Figure 2.**General yield loci shapes obtained using the isotropic yield: (

**a**) α = 3, β = 12; (

**b**) α = −6, β = −4; (

**c**) α = −2, β = −2; (

**d**) α = 20, β = −4.

**Figure 3.**General yield loci predicted by the proposed isotropic yield function for α = 0; β =−2.0~2.0.

**Figure 4.**Predicted anisotropy according to the Kim–Van and the Cazacu–Barlat yield functions (2001) for AA6016-T4 sheet with 16 equations for fitting (

**a**) yield locus, (

**b**) 3D yield surface, (

**c**) normalized uniaxial yield stresses, and (

**d**) R-values.

**Figure 5.**Predicted anisotropy according to the Kim–Van and the Cazacu–Barlat yield functions (2001) for AA6016-T4 sheet with 12 equations for fitting (

**a**) yield locus, (

**b**) 3D yield surface, (

**c**) normalized uniaxial yield stresses, and (

**d**) R-values.

**Figure 6.**Predicted anisotropy according to the Kim–Van and the Cazacu 2018 yield functions for AA2090-T3 sheet with 12 equations for fitting (

**a**) yield locus, (

**b**) 3D yield surface, (

**c**) normalized uniaxial yield stresses, and (

**d**) R-values.

**Figure 7.**Calculated second eigenvalue (λ

_{2}) of Hessian matrix at ${\sigma}_{12}=0$ for AA2090-T3 sheet: (

**a**) Kim–Van yield function; (

**b**) Cazacu 2018 yield function; (

**c**) contour lines of eigenvalue of Kim–Van model; (

**d**) contour lines of eigenvalue of Cazacu 2018 model.

**Figure 8.**Yield loci of high-purity α-titanium sheet predicted using the Kim–Van and the Yoon 2014 yield functions.

**Table 1.**Anisotropic coefficients of the Kim–Van and the Cazacu–Barlat yield functions (2001) for AA6016-T4 sheet (16 equations for fitting).

Parameters | a_{1} | a_{2} | a_{3} | a_{4} | b_{1} | b_{2} | b_{3} | b_{4} | b_{5} | b_{10} | α | β_{2} | c | Error |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Kim–Van | 1.039 | 1.018 | 0.729 | 0.720 | 0.722 | 2.006 | 2.825 | 0.251 | 1.574 | 1.633 | 1.546 | 2.004 | $\times $ | 0.0006 |

Cazacu–Barlat | 0.334 | 0.815 | 0.815 | 0.420 | 0.040 | −1.205 | −0.958 | 0.306 | 0.153 | −0.020 | $\times $ | $\times $ | 1.4 | 0.0007 |

**Table 2.**Anisotropic coefficients in the Kim–Van and the Cazacu–Barlat yield functions (2001) for AA6016-T4 sheet (12 equations for fitting).

Parameters | a_{1} | a_{2} | a_{3} | a_{4} | b_{1} | b_{2} | b_{3} | b_{4} | b_{5} | b_{10} | α | β_{2} | c | Error |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Kim–Van | 1.020 | 1.035 | 0.787 | 0.685 | 0.747 | 1.871 | 2.744 | 0.296 | 1.720 | 1.656 | 1.590 | 1.894 | $\times $ | 0.0013 |

Cazacu–Barlat | 0.334 | 0.815 | 0.815 | 0.420 | 0.040 | −1.205 | −0.958 | 0.306 | 0.153 | −0.020 | $\times $ | $\times $ | 1.4 | 0.0007 |

**Table 3.**Anisotropic coefficients of the Kim–Van and the Cazacu 2018 yield functions for AA2090-T3 sheet.

Parameters | a_{1} | a_{2} | a_{3} | a_{4} | b_{1} | b_{2} | b_{3} | b_{4} | b_{5} | b_{10} | α | β_{2} | c | Error |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Kim–Van | 0.989 | 1.430 | 0.252 | 1.663 | −0.544 | −3.571 | 0.065 | 0.337 | −2.049 | −2.148 | 2.368 | 0.090 | 0.0837 | 0.0013 |

Cazacu–Barlat | 0.779 | 1.596 | 1.427 | 1.813 | 2.009 | 0.255 | −0.990 | 0.646 | 1.589 | 2.735 | $-2.18$ | $\times $ | $\times $ | 0.0007 |

Parameters | a_{1} | a_{2} | a_{3} | a_{4} | b_{1} | b_{2} | b_{3} | b_{4} | b_{5} | b_{10} | α | β_{2} | c | Error |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Kim–Van | 1.014 | 0.673 | 0.660 | x | 1.159 | 1.765 | 1.063 | 1.089 | x | x | 1.124 | 1.654 | $\times $ | 0.0007 |

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**MDPI and ACS Style**

Kim, J.; Nguyen, P.V.; Hong, J.G.; Kim, Y.S.
Stress-Invariants-Based Anisotropic Yield Functions and Its Application to Sheet Metal Plasticity. *Metals* **2023**, *13*, 142.
https://doi.org/10.3390/met13010142

**AMA Style**

Kim J, Nguyen PV, Hong JG, Kim YS.
Stress-Invariants-Based Anisotropic Yield Functions and Its Application to Sheet Metal Plasticity. *Metals*. 2023; 13(1):142.
https://doi.org/10.3390/met13010142

**Chicago/Turabian Style**

Kim, Jinjae, Phu Van Nguyen, Jung Goo Hong, and Young Suk Kim.
2023. "Stress-Invariants-Based Anisotropic Yield Functions and Its Application to Sheet Metal Plasticity" *Metals* 13, no. 1: 142.
https://doi.org/10.3390/met13010142