# Dynamics of Nonlocal Rod by Means of Fractional Laplacian

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Fractional Laplacian Model

#### 2.2. The Local/Nonlocal Differential Model

“Nevertheless, for a meaningful comparison between a nonlocal elasticity model and experimental size effect data, two important conditions must be satisfied: (i) the classical continuum theory is recovered for vanishing nonlocal length, and (ii) the nonlocal system is stiffer than the local one.”

#### 2.3. Nonlocal Rod Dynamics by Means of Fractional Laplacian Model

#### 2.4. Numerical Approximation of Fractional Laplacian Problems

## 3. Results

#### 3.1. The Local/Nonlocal Differential Model

#### 3.2. Fractional Laplacian Model and Comparison

#### 3.3. Response in Dynamics

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## List of Symbols

x | position |

u | displacement |

L | half-length |

$\sigma $ | stress |

$\u03f5$ | strain |

E | Young’s modulus |

$\rho $ | mass density |

${\beta}_{1}$ | local fraction for fractional Laplacian model |

${\beta}_{2}$ | nonlocal fraction for fractional Laplacian model |

${\xi}_{1}$ | local fraction for local/nonlocal differential model |

${\xi}_{2}$ | nonlocal fraction for local/nonlocal differential model |

k | contant related to nonlocal behaviour in fractional Laplacian model |

g | attenuation function |

s | order of fractional Laplacian |

${(-\Delta )}^{s}$ | fractional Laplacian operator |

b | distributed load |

$\mathcal{F},{\mathcal{F}}^{-1}$ | direct and inverse Fourier transform |

${}_{-L}{D}_{x}^{2s}\phi (x),\phantom{\rule{0.166667em}{0ex}}{}_{x}{D}_{L}^{2s}\phi (x)$ | forward and backwards Riemann–Liouville fractional derivatives of order $2s$ |

$\Gamma $ | Gamma function |

c | parameter of fractional Laplacian model, $c={\beta}_{1}$ |

$\kappa $ | parameter of fractional Laplacian model, $\kappa =-2{\beta}_{2}kcoss\pi $ |

${\mathcal{K}}_{l}$ | kernel of local/nonlocal differential model |

l | nonlocal characteristic length for local/nonlocal differential model |

n | points used for discretisation in space |

${w}_{j}$ | weight for term in approximation of fractional Laplacian |

M | number of terms in approximation of fractional Laplacian |

m | points used for discretisation in time |

T | length of time history |

$\beta ,\gamma $ | parameters of Newmark’s method |

${T}_{0}$ | period of dynamic load |

## Appendix A

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**Figure 2.**Values of $b(x)=1+cos\left(\frac{2\pi}{L}\left(x-\frac{L}{2}\right)\right)$ for $L=200\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}.$

**Figure 6.**Contour levels for the values of $\tilde{u}$ for different combinations of ${\beta}_{1}$, k and s.

**Figure 7.**Response of the rod to load $b(x)$: (

**a**) displacements. (

**b**) detailed view for $-80\le x\le -60$. Parameters for local/nonlocal differential model: $l=0.65$ and ${\xi}_{1}=1.5$. Parameters for fractional Laplacian model: ${\beta}_{1}=0.5$, k and s shown in legend.

**Figure 8.**Dynamic displacement in (

**a**) time and (

**b**) frequency domain for local case and the local/nonlocal case with ${\beta}_{1}=0.75$ and two different combination of k and s, which gives the same midspan displacement ratio: $k=0.508$ and $s=0.930$; $k=0.300$ and $s=0.522$.

**Figure 9.**Dynamic displacement in (

**a**,

**b**) time and (

**b**) frequency domain local case and the local/nonlocal case with two combinations of $\beta $, k and s, which gives the same midspan displacement ratio: ${\beta}_{1}=0.5$, $k=0.508$ and $s=0.930$; ${\beta}_{1}=0.75$, $k=0.300$ and $s=0.522$.

**Figure 10.**Displacements in dynamics for local case and the local/nonlocal case with ${\beta}_{1}=0.5$, $k=0.508$ and $s=0.930$ at time instants $t=0\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$ and $t=2.08\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$.

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**MDPI and ACS Style**

Gusella, V.; Autuori, G.; Pucci, P.; Cluni, F.
Dynamics of Nonlocal Rod by Means of Fractional Laplacian. *Symmetry* **2020**, *12*, 1933.
https://doi.org/10.3390/sym12121933

**AMA Style**

Gusella V, Autuori G, Pucci P, Cluni F.
Dynamics of Nonlocal Rod by Means of Fractional Laplacian. *Symmetry*. 2020; 12(12):1933.
https://doi.org/10.3390/sym12121933

**Chicago/Turabian Style**

Gusella, Vittorio, Giuseppina Autuori, Patrizia Pucci, and Federico Cluni.
2020. "Dynamics of Nonlocal Rod by Means of Fractional Laplacian" *Symmetry* 12, no. 12: 1933.
https://doi.org/10.3390/sym12121933