# Review of the Upper Bound Method for Application to Metal Forming Processes

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## Abstract

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## 1. Introduction

## 2. Upper Bound Theorem

#### 2.1. Work Function

**s**,

**u**, and $\mathbf{\xi}$ denote the actual deviatoric stress, velocity, and strain rate. The kinematically admissible velocity and strain rate are denoted as

**u***and ${\mathbf{\xi}}^{*}$, respectively.

#### 2.2. Specific Materials

#### 2.3. Discontinuous Velocity Fields

## 3. Solution Methods

#### 3.1. UBET and TEUBA

#### 3.2. Stream Functions

#### 3.3. Singular Velocity Fields

## 4. Restrictions of the Method

#### 4.1. Stress Fields and Combined Loading

#### 4.2. Stationary Processes

#### 4.2.1. Geometry

#### 4.2.2. Strain Hardening

#### 4.3. Non-Stationary Processes

#### 4.4. Friction

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

k | shear yield stress |

$m$ | friction factor |

${s}_{ij}$ | deviatoric stress tensor |

${s}_{ij}^{*}$ | deviatoric stress tensor calculated using kinematically admissible velocity fields |

${s}_{1}$, ${s}_{2}$, ${s}_{3}$ | principal components of the deviatoric stress tensor |

${t}_{i}$ | surface tractions |

${u}_{i}$ | velocity components |

${u}_{x}$, ${u}_{y}$ | velocity components referred to Cartesian coordinates $\left(x,\hspace{0.17em}y\right)$ |

${u}_{r}$ | radial velocity |

${u}_{z}$ | axial velocity |

${u}_{\tau}$ | velocity components tangent to the surface |

z | coordinate normal to the maximum friction surface |

E | work function |

S | surface of plastic mass |

${S}_{d}$ | velocity discontinuity surface |

${S}_{f}$ | friction surface |

${S}_{F}$ | surface over which the traction is given |

${S}_{u}$ | surface over which the velocity is given |

$\left[{u}_{\tau}\right]$ | amount of velocity discontinuity |

$\gamma $ | angle between the friction traction and the velocity vector tangent to the friction surface |

$\mu $ | friction coefficient |

${\xi}_{ij}$ | strain rate tensor |

${\xi}_{ij}^{*}$ | strain rate tensor calculated using kinematically admissible velocity fields |

ξ_{eq} | equivalent strain rate |

${\xi}_{1}$, ${\xi}_{2}$, ${\xi}_{3}$ | principal strain rates |

${\xi}_{\mathrm{max}}$ | maximum principal strain rate |

${\sigma}_{ij}$ | stress tensor |

${\sigma}_{0}$ | tensile yield stress |

${\sigma}_{1}$,${\sigma}_{2}$,${\sigma}_{3}$ | principal stresses |

${\sigma}_{n}$ | traction component normal to the friction surface |

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

${\tau}_{Y}$ | shear yield stress |

$\psi \left(x,\hspace{0.17em}y\right)$ | stream function |

$\left(r,\hspace{0.17em}\theta ,z\right)$ | cylindrical coordinate system |

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**Figure 5.**Orientation of actual and kinematically admissible velocities at a generic point of the friction surface.

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**MDPI and ACS Style**

Alexandrov, S.; Rynkovskaya, M.
Review of the Upper Bound Method for Application to Metal Forming Processes. *Metals* **2022**, *12*, 1962.
https://doi.org/10.3390/met12111962

**AMA Style**

Alexandrov S, Rynkovskaya M.
Review of the Upper Bound Method for Application to Metal Forming Processes. *Metals*. 2022; 12(11):1962.
https://doi.org/10.3390/met12111962

**Chicago/Turabian Style**

Alexandrov, Sergei, and Marina Rynkovskaya.
2022. "Review of the Upper Bound Method for Application to Metal Forming Processes" *Metals* 12, no. 11: 1962.
https://doi.org/10.3390/met12111962