# An Upper Bound Solution for the Compression of an Orthotropic Cylinder

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## Abstract

**:**

## 1. Introduction

## 2. Statement of the Problem

## 3. Kinematically Admissible Velocity Field

## 4. Upper Bound Solution

## 5. Numerical Examples

## 6. Conclusions

- Plastic anisotropy affects the limit load required to deform the specimen. It may increase or decrease the limit load as compared to the isotropic case (Figure 5).
- The upsetting of a cylinder is often used as a friction test. Plastic anisotropy significantly affects the region of sticking friction (Figure 6). Since Equation (8) is not valid in this region, this effect of plastic anisotropy should be taken into account in the interpretation of the friction test results.
- Five dimensionless parameters classify the boundary value problem. Therefore, its systematic parametric analysis is invisible. An advantage of the proposed solution is that it quickly estimates the upper bound limit load for a given set of parameters.
- The real velocity field is singular near the friction surface if $m=1$ in Equation (8). The solution proposed accounts for this singularity, which is impossible when using ordinary finite element solutions.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$\mathbf{i}$ and $\mathbf{j}$ | unit base vectors |

m | friction factor |

$\mathbf{n}$ | unit vector normal to the velocity discontinuity line |

${p}_{u}$ | dimensionless upper bound limit load |

$(t,\theta ,z)$ | cylindrical coordinate system |

${u}_{r}$ and ${u}_{z}$ | radial and axial velocities |

F, G, H and M | material parameters introduced in Equation (2) |

${H}_{0}$ | half-height of the cylinder |

P | force |

${R}_{0}$ | radius of the cylinder |

S | shear yield stress in the $rz$-plane |

U | velocity of the plate |

${W}_{d}$ | plastic work rate at the velocity discontinuity line |

${W}_{f}$ | plastic work rate at the friction surface |

${W}_{v}$ | plastic work rate in the plastic region |

$\mathrm{Y}$, $\mathsf{\Theta}$ and Z | tensile yield stresses in the radial, circumferential, and axial directions |

${\xi}_{eq}$ | equivalent strain rate |

${\xi}_{rr}$, ${\xi}_{\theta \theta}$, ${\xi}_{zz}$ and ${\xi}_{rz}$ | strain rate components referred to the cylindrical coordinate system |

${\overline{\xi}}_{rr}$, ${\overline{\xi}}_{\theta \theta}$, ${\overline{\xi}}_{zz}$ and ${\overline{\xi}}_{rz}$ | dimensionless strain rate components introduced in Equation (11) |

$\rho $, $\zeta $ and h | dimensionless quantities introduced after Equation (9) |

${\sigma}_{rr}$, ${\sigma}_{\theta \theta}$, ${\sigma}_{zz}$ and ${\sigma}_{rz}$ | stress components referred to the cylindrical coordinate system |

${\tau}_{f}$ | friction stress |

$\phi $ | orientation of the tangent to the velocity discontinuity line |

$\omega $ | plastic work rate per unit volume |

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**Figure 3.**Existence of a critical value of h at which solutions (42) and (51) provide the same limit load for the isotropic material at $m=0.7$.

**Figure 4.**Effect of plastic anisotropy on the dependence of ${h}_{c}$ on m. No ${h}_{c}$ is found for Case 2 since ${p}_{u}^{\left(1\right)}<{p}_{u}^{\left(2\right)}$ in the m range.

**Table 1.**Experimental data from Ref. [37].

FZ^{2} | HZ^{2} | GZ^{2} | MZ^{2} | |
---|---|---|---|---|

Exp. Value | 0.378 | 0.1 | 0.623 | 2.558 |

**Table 2.**Yield stresses chosen to illustrate the effect of plastic anisotropy on the interpretation of the friction test.

Y/Z | Θ/Z | S/Z | |
---|---|---|---|

Isotropic Case | 1 | 1 | $1/\sqrt{3}$ |

Case 1 | 1.2 | 1.5 | $1.2/\sqrt{3}$ |

Case 2 | 0.8 | 0.7 | $0.9/\sqrt{3}$ |

Case 3 | 0.8 | 0.9 | $0.75/\sqrt{3}$ |

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Lang, L.; Alexandrov, S.; Wang, Y.-C.
An Upper Bound Solution for the Compression of an Orthotropic Cylinder. *Materials* **2021**, *14*, 5253.
https://doi.org/10.3390/ma14185253

**AMA Style**

Lang L, Alexandrov S, Wang Y-C.
An Upper Bound Solution for the Compression of an Orthotropic Cylinder. *Materials*. 2021; 14(18):5253.
https://doi.org/10.3390/ma14185253

**Chicago/Turabian Style**

Lang, Lihui, Sergei Alexandrov, and Yun-Che Wang.
2021. "An Upper Bound Solution for the Compression of an Orthotropic Cylinder" *Materials* 14, no. 18: 5253.
https://doi.org/10.3390/ma14185253