# A Limit Load Solution for Anisotropic Welded Cracked Plates in Pure Bending

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## Abstract

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## 1. Introduction

## 2. Statement of the Problem

## 3. General Solution

#### 3.1. Kinematically Admissible Velocity Field

#### 3.2. Relations between Geometric Parameters

#### 3.3. Bending Moment

## 4. Numerical Example

## 5. Summary

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

a | crack length |

c | constitutive parameter introduced in Equation (1) |

c_{b} | value of c for the base material |

c_{w} | value of c for the weld material |

h | specimen width |

m_{u} | dimensionless upper bound on the bending moment |

(x, y) | Cartesian coordinates |

B | specimen thickness |

M | bending moment |

M_{u} | upper bound on the bending moment |

T | shear yield stress in the Cartesian coordinates |

T_{b} | value of T for the base material |

T_{w} | value of T for the weld material |

W | half-thickness of weld |

X_{0}, Y_{0} | parameters determining the position of point 0 relative to the crack tip (Figure 2). |

${\theta}_{1},\hspace{0.17em}{\theta}_{2},\hspace{0.17em}{\theta}_{3},\hspace{0.17em}{\theta}_{4}$ | angles introduced in Figure 2 |

$\rho $ | radius of velocity discontinuity lines |

${\sigma}_{xx},\hspace{0.17em}{\sigma}_{yy},\hspace{0.17em}{\sigma}_{xy}$ | stress components in the Cartesian coordinates |

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**Figure 2.**First pattern of the kinematically admissible velocity field on the right (

**a**) and left (

**b**) to the crack line.

**Figure 3.**Second pattern of the kinematically admissible velocity field on the right (

**a**) and left (

**b**) to the crack line.

**Figure 4.**Third pattern of the kinematically admissible velocity field on the right (

**a**) and left (

**b**) to the crack line.

**Figure 5.**Illustration of the geometric parameters involved in the description of the kinematically admissible velocity field.

**Figure 6.**Variation of ${m}_{u}$ with ${c}_{b}$ for several values of a at ${c}_{w}=-6,W/h=0.05,{T}_{w}/{T}_{b}=2$, and $t=W$ (symmetric case).

**Figure 7.**Variation of ${m}_{u}$ with ${c}_{b}$ for several values of a at ${c}_{w}=-6,W/h=0.05,{T}_{w}/{T}_{b}=2$, and $t=0$.

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**MDPI and ACS Style**

Alexandrov, S.; Lyamina, E.; Pirumov, A.; Nguyen, D.K.
A Limit Load Solution for Anisotropic Welded Cracked Plates in Pure Bending. *Symmetry* **2020**, *12*, 1764.
https://doi.org/10.3390/sym12111764

**AMA Style**

Alexandrov S, Lyamina E, Pirumov A, Nguyen DK.
A Limit Load Solution for Anisotropic Welded Cracked Plates in Pure Bending. *Symmetry*. 2020; 12(11):1764.
https://doi.org/10.3390/sym12111764

**Chicago/Turabian Style**

Alexandrov, Sergei, Elena Lyamina, Alexander Pirumov, and Dinh Kien Nguyen.
2020. "A Limit Load Solution for Anisotropic Welded Cracked Plates in Pure Bending" *Symmetry* 12, no. 11: 1764.
https://doi.org/10.3390/sym12111764