# Numerical Study of Dynamic Properties of Fractional Viscoplasticity Model

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fractional Viscoplasticity

#### 2.1. Remarks on Fractional Calculus

#### 2.2. Basic Concepts

^{e}denotes the Cauchy stress tensor and ${\mathcal{L}}^{e}$ denotes the elastic constitutive tensor. The rate of viscoplastic strain is analogous to the classical viscoplastic definition, namely

## 3. Implementation

## 4. Parametric Study: Uniaxial Tension

#### 4.1. Description of the Numerical Experiment

#### 4.2. Influence of the Order of FV and Non-Locality in a Stress State on Plastic Flow

#### 4.2.1. Study of Intensified Plastic Flow in Tension Direction for Different Orders of Flow

#### 4.2.2. Study of Intensified Plastic Flow Perpendicular to the Tension Direction for Different Orders of Flow

#### 4.3. Influence of the Relaxation Time and the Overstress Power

#### 4.3.1. Study of the Fractional Flow Under Different Dynamic Loading Rates for Intensified Plastic Flow in Tension Direction

#### 4.3.2. Study of the Fractional Flow Under Different Dynamic Loading for the Intensified Plastic Flow Perpendicular to the Tension Direction

#### 4.4. Study of the Disperse Character of the Fractional Viscoplastic Stress Waves

## 5. Conclusions

- Fractional viscoplasticity introduces an additional set of material parameters, namely flow order $\alpha $ and virtual stress state surrounding $\Delta $.
- Fractional parameters $\alpha $ and $\Delta $ control the dynamic properties of the fractional model, especially hardening, the character of the stress waves, and plastic anisotropy.
- The direction of the flow vector is controlled by $\Delta $, which in general leads to non-normality of plastic flow.
- As in the classical Perzyna model, the relaxation time ${T}_{m}$ and the overstress power m affect the strain rate hardening and the character of the stress waves.
- Induced plastic anisotropy of the fractional model should be regarded not only in the classical sense as directional deformation but also as directional viscosity, which results in directional dispersive character.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Influence of the order $\alpha $ and the value of material parameter ${\Delta}_{22}$ on the stress–strain relation, for: $v=1\phantom{\rule{0.277778em}{0ex}}\frac{m}{s},{T}_{m}=2.5e-6\phantom{\rule{0.277778em}{0ex}}s,m=1$.

**Figure 5.**Influence of the order $\alpha $ and the value of material parameter ${\Delta}_{22}$ on the stress–strain relation, for: $v=25\phantom{\rule{0.277778em}{0ex}}\frac{m}{s},{T}_{m}=2.5e-6\phantom{\rule{0.277778em}{0ex}}s,m=1$.

**Figure 6.**Influence of the order $\alpha $ and the value of material parameter ${\Delta}_{11}$ on the stress–strain relation, for: $v=1\phantom{\rule{0.277778em}{0ex}}\frac{m}{s},{T}_{m}=2.5e-6\phantom{\rule{0.277778em}{0ex}}s,m=1$.

**Figure 7.**Influence of the order $\alpha $ and the value of material parameter ${\Delta}_{11}$ on the relation between three normal stresses, for: $v=1\phantom{\rule{0.277778em}{0ex}}\frac{m}{s},{T}_{m}=2.5e-6\phantom{\rule{0.277778em}{0ex}}s,m=1$.

**Figure 8.**Influence of the order $\alpha $ and the value of material parameter ${\Delta}_{11}$ on the stress–strain relation, for: $v=25\phantom{\rule{0.277778em}{0ex}}\frac{m}{s},{T}_{m}=2.5e-6\phantom{\rule{0.277778em}{0ex}}s,m=1$.

**Figure 9.**Influence of the order $\alpha $ and the value of material parameter ${\Delta}_{11}$ on the relation between three normal stresses, for: $v=25\phantom{\rule{0.277778em}{0ex}}\frac{m}{s},{T}_{m}=2.5e-6\phantom{\rule{0.277778em}{0ex}}s,m=1$.

**Figure 10.**Influence of the relaxation parameter ${T}_{m}$ and the value of applied velocity field v on the stress–strain relation, for: $\alpha =0.75,m=1,{\Delta}_{22}=3.0$.

**Figure 11.**Influence of the material parameter m and the value of applied velocity field v on the stress–strain relation, for: $\alpha =0.75,{T}_{m}=2.5e-6\phantom{\rule{0.277778em}{0ex}}s,{\Delta}_{22}=3.0$.

**Figure 12.**Influence of the relaxation parameter ${T}_{m}$ and the value of applied velocity field v on the stress–strain relation, for: $\alpha =0.75,m=1,{\Delta}_{11}=3.0$.

**Figure 13.**Influence of the relaxation parameter ${T}_{m}$ and the value of applied velocity field v on the relation between three normal stresses, for: $\alpha =0.75,m=1,{\Delta}_{11}=3.0$.

**Figure 14.**Influence of the material parameter m and the value of applied velocity field v on the stress–strain relation, for: $\alpha =0.75,{T}_{m}=2.5e-6\phantom{\rule{0.277778em}{0ex}}s,{\Delta}_{11}=3.0$.

**Figure 15.**Influence of the material parameter m and the value of applied velocity field v on the relation between three normal stresses, for: $\alpha =0.75,{T}_{m}=2.5e-6\phantom{\rule{0.277778em}{0ex}}s,{\Delta}_{11}=3.0$.

${\mathbf{\Delta}}_{22}=3.0$ | ${\mathbf{\Delta}}_{11}=3.0$ | ||
---|---|---|---|

$v=25\phantom{\rule{0.277778em}{0ex}}\frac{m}{s}$ | $\alpha =1$ | 2.085 MHz | 2.085 MHz |

$\alpha =0.75$ | 2.108 MHz | 1.996 MHz | |

$v=50\phantom{\rule{0.277778em}{0ex}}\frac{m}{s}$ | $\alpha =1$ | 2.073 MHz | 2.073 MHz |

$\alpha =0.75$ | 2.157 MHz | 2.028 MHz |

**Table 2.**The stress wave frequencies for ${\Delta}_{22}=3.0\phantom{\rule{0.277778em}{0ex}}MPa$, $v=50\phantom{\rule{0.277778em}{0ex}}\frac{m}{s}$

${\mathit{T}}_{\mathit{m}}$ | ||||
---|---|---|---|---|

2.5e-7 | 2.5e-6 | 2.5e-5 | ||

$m=1$ | $\alpha =1$ | 2.073 MHz | 2.073MHz | 2.274 MHz |

$\alpha =0.75$ | 2.157 MHz | 2.157 MHz | 2.288 MHz | |

$m=2$ | $\alpha =1$ | 2.073 MHz | 2.085 MHz | 2.182 MHz |

$\alpha =0.75$ | 2.157 MHz | 2.157 MHz | 2.207 MHz | |

$m=3$ | $\alpha =1$ | 2.073 MHz | 2.085 MHz | 2.169 MHz |

$\alpha =0.75$ | 2.157 MHz | 2.157 MHz | 2.182 MHz |

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Szymczyk, M.; Nowak, M.; Sumelka, W.
Numerical Study of Dynamic Properties of Fractional Viscoplasticity Model. *Symmetry* **2018**, *10*, 282.
https://doi.org/10.3390/sym10070282

**AMA Style**

Szymczyk M, Nowak M, Sumelka W.
Numerical Study of Dynamic Properties of Fractional Viscoplasticity Model. *Symmetry*. 2018; 10(7):282.
https://doi.org/10.3390/sym10070282

**Chicago/Turabian Style**

Szymczyk, Michał, Marcin Nowak, and Wojciech Sumelka.
2018. "Numerical Study of Dynamic Properties of Fractional Viscoplasticity Model" *Symmetry* 10, no. 7: 282.
https://doi.org/10.3390/sym10070282