# Effect of Frictional Slipping on the Strength of Ribbon-Reinforced Composite

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## Abstract

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## 1. Introduction

## 2. Formulation of the Problem

## 3. Materials and Methods

## 4. Numerical Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

SSS | stress–strain state; |

SSIE | system of singular integral equations; |

SLAE | system of linear algebraic equations; |

$x,\text{\hspace{0.17em}}y,\text{\hspace{0.17em}}z$ | Cartesian coordinates; |

${f}_{r}$ | jump functions; |

${E}_{k},\text{}{\nu}_{k},\text{\hspace{0.17em}}{G}_{k}$ | elastic modulus of the material; |

${S}_{k}$ | half-planes (sections of the body); |

$a,\text{}h,\text{}{b}^{\pm},\text{}{c}^{\pm}$ | dimensions of the inclusion and slip zones; |

$w,\text{}{\sigma}_{xz},\text{}{\sigma}_{yz},\text{}{\sigma}_{xx},\text{}{\sigma}_{yy}$ | displacement, stresses (components of SSS); |

${L}^{\prime}=[-a;\text{\hspace{0.17em}}a]$ | line, modelling the presence of thin inclusion; |

${Q}_{k},\text{}{b}_{k}$ | magnitudes of concentrated forces and screw dislocations; |

${\sigma}_{yz}^{\infty}$, ${\sigma}_{xzk}^{\infty}$, ${\sigma}_{yy}^{\infty}$,${\sigma}_{xxk}^{\infty}$ | uniformly distributed in infinity shear stresses; |

$\alpha $ | coefficient of the sliding friction; |

Special denotations | |

$\text{}{\left[\phi \right]}_{h}=\phi \left(x,-h\right)-\phi \left(x,+h\right)$, $\text{}{\langle \phi \rangle}_{h}=\phi \left(x,-h\right)+\phi \left(x,+h\right)$; | |

superscripts “+” and “−” | denotes boundary values of functions on the upper and the lower with respect to width inclusion borders accordingly; |

superscript “in” | marks the values corresponding to inclusion; |

superscript “°” | marks the values in the corresponding problem without any inclusion; |

superscript “~” | marks the terms that become dimensionless; |

subscript “k” | denotes the terms corresponding to half-plains. |

## References

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**Figure 2.**The boundary of the “safety zone” when loading the structure with a softer than the matrix inclusion by a concentrated force: 1—${\tilde{Q}}^{*}=0.5$; 2—${\tilde{Q}}^{*}=0.75$; 3—${\tilde{Q}}^{*}=0.9$.

**Figure 3.**The boundary of the “safety zone” when loading with the concentrated force ${\tilde{Q}}^{*}$ of the structure without (1—${\tilde{G}}_{y}^{in}=1$) and with harder than matrix inclusion (2—${\tilde{G}}_{y}^{in}=10$).

**Figure 4.**Stress distribution along with the inclusion–matrix boundary, (

**a**) and the size of the slip zone (

**b**) depending on the ratio ${\tilde{G}}_{y}^{in}/{\tilde{G}}_{k}$

**Figure 5.**Stress distribution along with the inclusion–matrix boundary (

**a**) and the value of the slip zone (

**b**) for inclusion harder than matrix, depending on the distance of the force application point from its axis.

**Figure 6.**Stress distribution along with the inclusion–matrix boundary (

**a**) and the size of the slip zone (

**b**) for a softer than matrix inclusion as a function of force intensity growth.

**Figure 7.**Stress distribution along with the inclusion–matrix boundary (

**a**) and the value of the slip zone (

**b**) for a harder than matrix inclusion as a function of force intensity growth.

**Figure 8.**Stress distribution along with the inclusion–matrix boundary (

**a**) and the value of the slip zone (

**b**) for a softer inclusion than matrix depending on the change in the coordinates of the force application points along the inclusion axis.

**Figure 9.**Stress distribution along with the inclusion–matrix boundary (

**a**) and the value of the slip zone (

**b**) for a softer than matrix inclusion under screw dislocation loading.

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**MDPI and ACS Style**

Piskozub, Y.; Sulym, H.
Effect of Frictional Slipping on the Strength of Ribbon-Reinforced Composite. *Materials* **2021**, *14*, 4928.
https://doi.org/10.3390/ma14174928

**AMA Style**

Piskozub Y, Sulym H.
Effect of Frictional Slipping on the Strength of Ribbon-Reinforced Composite. *Materials*. 2021; 14(17):4928.
https://doi.org/10.3390/ma14174928

**Chicago/Turabian Style**

Piskozub, Yosyf, and Heorhiy Sulym.
2021. "Effect of Frictional Slipping on the Strength of Ribbon-Reinforced Composite" *Materials* 14, no. 17: 4928.
https://doi.org/10.3390/ma14174928