# Effect of Plastic Anisotropy on the Distribution of Residual Stresses and Strains in Rotating Annular Disks

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## Abstract

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## 1. Introduction

## 2. Statement of the Problem

_{r}is the radial strain, ε

_{θ}is the circumferential strain, and ε

_{z}is the axial strain. The superscript e denotes the elastic part of the strain and will denote the elastic part of the strain rate. The orthotropic yield criterion proposed in reference [14] and its associated flow rule are adopted in the plastic region. It is assumed that the principal axes of anisotropy coincide with coordinate curves of the cylindrical coordinate system. Under plane stress conditions, the yield criterion adopted reads $\left(G+H\right){\sigma}_{r}^{2}+\left(F+H\right){\sigma}_{\theta}^{2}-2H{\sigma}_{r}{\sigma}_{\theta}=1$ where G, H, and F are anisotropic constants. It is convenient to rewrite this criterion as:

## 3. Solution at Loading

#### 3.1. Purely Elastic Solution

#### 3.2. Elastic/Plastic Stress Solution

#### 3.3. Elastic/Plastic Strain Solution

## 4. Distribution of Residual Stresses and Strains

## 5. Illustrative Example

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Effect of anisotropic properties on the distribution of the radial stress in an $a=0.5$ disk.

**Figure 3.**Effect of anisotropic properties on the distribution of the circumferential stress in an $a=0.5$ disk.

**Figure 4.**Effect of anisotropic properties on the distribution of the residual radial stress in an $a=0.5$ disk.

**Figure 5.**Effect of anisotropic properties on the distribution of the residual circumferential stress in an $a=0.5$ disk.

**Figure 6.**Effect of anisotropic properties on the distribution of the radial strain in an $a=0.5$ disk.

**Figure 7.**Effect of anisotropic properties on the distribution of the circumferential strain in an $a=0.5$ disk.

**Figure 8.**Effect of anisotropic properties on the distribution of the axial strain in an $a=0.5$ disk.

**Figure 9.**Effect of anisotropic properties on the distribution of the radial plastic strain in an $a=0.5$ disk.

**Figure 10.**Effect of anisotropic properties on the distribution of the circumferential plastic strain in an $a=0.5$ disk.

**Figure 11.**Effect of anisotropic properties on the distribution of the axial plastic strain in an $a=0.5$ disk.

**Figure 12.**Effect of anisotropic properties on the distribution of the residual radial strain in an $a=0.5$ disk.

**Figure 13.**Effect of anisotropic properties on the distribution of the residual circumferential strain in an $a=0.5$ disk.

**Figure 14.**Effect of anisotropic properties on the distribution of the residual axial strain in an $a=0.5$ disk.

Material | F/(G + H) | H/(G + H) |
---|---|---|

DC06 | 0.243 | 0.703 |

AA6016 | 0.587 | 0.410 |

AA5182 | 0.498 | 0.419 |

AA3014 | 0.239 | 0.301 |

Isotropic | 0.5 | 0.5 |

**Table 2.**Comparison of the total circumferential strain at $\rho =a$ and ${\rho}_{f}=5/7$ found by the two methods.

F/(G + H) | H/(G + H) | ${\mathit{\epsilon}}_{\mathit{\theta},\mathit{F}\mathit{D}\mathit{M}}\text{}$ | ${\mathit{\epsilon}}_{\mathit{\theta},\mathit{N}}\text{}$ | $\mathbf{\Delta}$ (%) |
---|---|---|---|---|

0.452 | 0.681 | 0.00088 | 0.00134 | 26.3 |

0.421 | 0.615 | 0.0014 | 0.0019 | 34.4 |

0.283 | 0.634 | 0.00178 | 0.00212 | 16.0 |

0.811 | 0.454 | 0.0025 | 0.0036 | 30.6 |

0.5 | 0.5 | 0.0019 | 0.0022 | 13.6 |

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**MDPI and ACS Style**

Jeong, W.; Alexandrov, S.; Lang, L.
Effect of Plastic Anisotropy on the Distribution of Residual Stresses and Strains in Rotating Annular Disks. *Symmetry* **2018**, *10*, 420.
https://doi.org/10.3390/sym10090420

**AMA Style**

Jeong W, Alexandrov S, Lang L.
Effect of Plastic Anisotropy on the Distribution of Residual Stresses and Strains in Rotating Annular Disks. *Symmetry*. 2018; 10(9):420.
https://doi.org/10.3390/sym10090420

**Chicago/Turabian Style**

Jeong, Woncheol, Sergei Alexandrov, and Lihui Lang.
2018. "Effect of Plastic Anisotropy on the Distribution of Residual Stresses and Strains in Rotating Annular Disks" *Symmetry* 10, no. 9: 420.
https://doi.org/10.3390/sym10090420