# Nonlocal Elasticity for Nanostructures: A Review of Recent Achievements

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Eringen’s Theory of Integral Elasticity

#### 2.1. Averaging Kernel and Green’s Function

- Symmetry and positivity$${\varphi}_{\lambda}(x-\overline{x})={\varphi}_{\lambda}(\overline{x}-x)\ge 0$$
- Normalization$${\int}_{-\infty}^{+\infty}{\varphi}_{\lambda}\left(x\right)\phantom{\rule{0.166667em}{0ex}}dx=1$$
- Limit impulsivity$$\underset{\lambda \to {0}^{+}}{lim}{\int}_{-\infty}^{+\infty}{\varphi}_{\lambda}(x-\overline{x})\phantom{\rule{0.166667em}{0ex}}f\left(\overline{x}\right)d\overline{x}=f\left(x\right)$$

**Proposition**

**1.**

**Proposition**

**2.**

#### 2.2. The Alleged Paradox of the Nonlocal Elastic Cantilever

## 3. Strain-Driven Nonlocal Methodologies

## 4. The Stress-Driven Nonlocal Model

#### Nonlinear Mechanics of Nonlocal Elastic Beams

## 5. A Nonlocal Methodology for Shear Deformation Beam Theories

## 6. Stress-Driven Two-Phase Elasticity

#### 6.1. Two-Phase Elasticity for Stubby Curved Beams

#### 6.2. Two-Phase Elasticity for Plates

## 7. Dynamics of Nanobeams

## 8. Nonlocal Elasticity for Structural Assemblages

**Proposition**

**3.**

## 9. Nanostructures on Nonlocal Foundations

**Proposition**

**4.**

## 10. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Current configuration $\phantom{\rule{0.166667em}{0ex}}y\phantom{\rule{0.166667em}{0ex}}\left[\mu m\right]$ of beam axis versus $\phantom{\rule{0.166667em}{0ex}}x\phantom{\rule{0.166667em}{0ex}}\left[\mu m\right]$ for $\phantom{\rule{0.166667em}{0ex}}\lambda =\{0.05,0.10,0.15,0.20\}$.

**Figure 6.**Simply supported beam under uniformly distributed loading: shape function $\phantom{\rule{0.166667em}{0ex}}w$ versus $\phantom{\rule{0.166667em}{0ex}}x$.

**Figure 7.**Simply supported beam under uniformly distributed loading: shape function $\phantom{\rule{0.166667em}{0ex}}\phi $ versus $\phantom{\rule{0.166667em}{0ex}}x$.

**Figure 8.**Nanoplate with clamped edges under uniformly distributed loading: transverse displacement fields $\phantom{\rule{0.166667em}{0ex}}u\phantom{\rule{0.166667em}{0ex}}\left[nm\right]$ for $\phantom{\rule{0.166667em}{0ex}}\alpha =0.2$.

**Figure 9.**First five eigenfunctions of nonlocal elastic cantilever for nonlocal parameter $\lambda =0.15$.

**Figure 10.**Beam with clamped and simply supported ends under concentrated couple $\phantom{\rule{0.166667em}{0ex}}\mathcal{M}$ at mid-span $\phantom{\rule{0.166667em}{0ex}}\overline{x}=1/2$: elastic curvature $\phantom{\rule{0.166667em}{0ex}}{\overline{\chi}}^{el}\phantom{\rule{0.166667em}{0ex}}$ versus $\phantom{\rule{0.166667em}{0ex}}\overline{x}\phantom{\rule{0.166667em}{0ex}}$ for increasing nonlocal parameter $\phantom{\rule{0.166667em}{0ex}}\lambda $.

**Figure 11.**Beam with clamped and simply supported ends under concentrated couple $\phantom{\rule{0.166667em}{0ex}}\mathcal{M}$ at mid-span $\phantom{\rule{0.166667em}{0ex}}\overline{x}=1/2$: transverse displacement $\phantom{\rule{0.166667em}{0ex}}\overline{v}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{10}^{-2}\phantom{\rule{0.166667em}{0ex}}$ versus $\phantom{\rule{0.166667em}{0ex}}\overline{x}\phantom{\rule{0.166667em}{0ex}}$ for increasing nonlocal parameter $\phantom{\rule{0.166667em}{0ex}}\lambda $.

**Figure 12.**Non-dimensional transverse displacement fields for ${k}^{\ast}=100$ and ${\lambda}_{b}=0.2$.

$\mathit{\lambda}$ | ${\mathit{v}}_{\mathit{\perp}}\left(\mathit{L}\right)\phantom{\rule{0.222222em}{0ex}}\left[{10}^{-2}\mathbf{nm}\right]$ | ${\mathit{v}}_{\mathit{t}}\left(\mathit{L}\right)\phantom{\rule{0.222222em}{0ex}}\left[{10}^{-2}\mathbf{nm}\right]$ | $\mathit{\phi}\left(\mathit{L}\right)\phantom{\rule{0.222222em}{0ex}}\left[{10}^{-3}\right]$ | ${\mathit{v}}_{\mathit{max}}^{\mathit{tot}}\left(\mathit{L}\right)\phantom{\rule{0.222222em}{0ex}}\left[{10}^{-2}\mathbf{nm}\right]$ | ||||
---|---|---|---|---|---|---|---|---|

$\mathbf{\alpha}=\mathbf{0}.\mathbf{2}$ | $\mathbf{\alpha}=\mathbf{0}.\mathbf{5}$ | $\mathbf{\alpha}=\mathbf{0}.\mathbf{2}$ | $\mathbf{\alpha}=\mathbf{0}.\mathbf{5}$ | $\mathbf{\alpha}=\mathbf{0}.\mathbf{2}$ | $\mathbf{\alpha}=\mathbf{0}.\mathbf{5}$ | $\mathbf{\alpha}=\mathbf{0}.\mathbf{2}$ | $\mathbf{\alpha}=\mathbf{0}.\mathbf{5}$ | |

0.1 | 2.039 | 2.118 | −1.300 | −1.349 | 0.9429 | 0.9699 | 2.418 | 2.511 |

0.2 | 1.818 | 1.980 | −1.170 | −1.268 | 0.8629 | 0.9199 | 2.162 | 2.351 |

0.3 | 1.629 | 1.862 | −1.058 | −1.198 | 0.7876 | 0.8728 | 1.943 | 2.214 |

0.4 | 1.478 | 1.768 | −0.9655 | −1.140 | 0.7227 | 0.8323 | 1.766 | 2.104 |

0.5 | 1.358 | 1.692 | −0.8905 | −1.094 | 0.6686 | 0.7985 | 1.624 | 2.015 |

0.6 | 1.261 | 1.632 | −0.8293 | −1.055 | 0.6235 | 0.7703 | 1.510 | 1.944 |

0.7 | 1.183 | 1.583 | −0.7787 | −1.024 | 0.5858 | 0.7468 | 1.416 | 1.885 |

0.8 | 1.117 | 1.542 | −0.7364 | −0.9973 | 0.554 | 0.7269 | 1.338 | 1.836 |

0.9 | 1.062 | 1.508 | −0.7006 | −0.9749 | 0.5269 | 0.7099 | 1.273 | 1.795 |

$\mathit{\lambda}$ | ${\mathit{v}}_{\mathit{\perp}}\left(0\right)\phantom{\rule{0.222222em}{0ex}}\left[{10}^{-2}\mathbf{nm}\right]$ | ${\mathit{v}}_{\mathit{\perp}}\left(\mathit{L}\right)\phantom{\rule{0.222222em}{0ex}}\left[{10}^{-2}\mathbf{nm}\right]$ | $\mathit{\phi}\left(\mathit{L}\right)\phantom{\rule{0.222222em}{0ex}}\left[{10}^{-3}\right]$ | ${\mathit{v}}_{\mathit{max}}^{\mathit{tot}}\left(0\right)\phantom{\rule{0.222222em}{0ex}}\left[{10}^{-2}\mathbf{nm}\right]$ | ||||
---|---|---|---|---|---|---|---|---|

$\mathbf{\alpha}=\mathbf{0}.\mathbf{2}$ | $\mathbf{\alpha}=\mathbf{0}.\mathbf{5}$ | $\mathbf{\alpha}=\mathbf{0}.\mathbf{2}$ | $\mathbf{\alpha}=\mathbf{0}.\mathbf{5}$ | $\mathbf{\alpha}=\mathbf{0}.\mathbf{2}$ | $\mathbf{\alpha}=\mathbf{0}.\mathbf{5}$ | $\mathbf{\alpha}=\mathbf{0}.\mathbf{2}$ | $\mathbf{\alpha}=\mathbf{0}.\mathbf{5}$ | |

0.1 | 7.233 | 7.522 | −6.970 | −7.221 | −3.903 | −4.03 | 7.233 | 7.522 |

0.2 | 6.447 | 7.031 | −6.214 | −6.748 | −3.568 | −3.821 | 6.447 | 7.031 |

0.3 | 5.783 | 6.616 | −5.563 | −6.341 | −3.261 | −3.629 | 5.783 | 6.616 |

0.4 | 5.252 | 6.284 | −5.042 | −6.016 | −2.996 | −3.464 | 5.252 | 6.284 |

0.5 | 4.830 | 6.020 | −4.628 | −5.757 | −2.775 | −3.325 | 4.830 | 6.020 |

0.6 | 4.489 | 5.807 | −4.296 | −5.550 | −2.591 | −3.210 | 4.489 | 5.807 |

0.7 | 4.211 | 5.633 | −4.026 | −5.380 | −2.436 | −3.113 | 4.211 | 5.633 |

0.8 | 3.980 | 5.489 | −3.802 | −5.241 | −2.305 | −3.032 | 3.980 | 5.489 |

0.9 | 3.786 | 5.367 | −3.614 | −5.123 | −2.193 | −2.962 | 3.786 | 5.367 |

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**MDPI and ACS Style**

Barretta, R.; Marotti de Sciarra, F.; Vaccaro, M.S.
Nonlocal Elasticity for Nanostructures: A Review of Recent Achievements. *Encyclopedia* **2023**, *3*, 279-310.
https://doi.org/10.3390/encyclopedia3010018

**AMA Style**

Barretta R, Marotti de Sciarra F, Vaccaro MS.
Nonlocal Elasticity for Nanostructures: A Review of Recent Achievements. *Encyclopedia*. 2023; 3(1):279-310.
https://doi.org/10.3390/encyclopedia3010018

**Chicago/Turabian Style**

Barretta, Raffaele, Francesco Marotti de Sciarra, and Marzia Sara Vaccaro.
2023. "Nonlocal Elasticity for Nanostructures: A Review of Recent Achievements" *Encyclopedia* 3, no. 1: 279-310.
https://doi.org/10.3390/encyclopedia3010018