# Nonlocal Mechanical Behavior of Layered Nanobeams

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## Abstract

**:**

## 1. Introduction

## 2. Fundamental Concepts

#### 2.1. Geometry of the Beam

#### 2.2. Notation

#### 2.3. Kinematics of the Beam

## 3. Nonlocal Material Model

## 4. A Nonlocal Variational Setting for Beams

#### 4.1. Stationarity with Respect to Axial Displacement

#### 4.2. Stationarity with Respect to Transverse Displacement

#### 4.3. Neutral Surface Position

- all small-size parameters are equal, ${\lambda}_{i}=\lambda $, $i\in \{1,2,\dots ,n\}$; or
- symmetry about the $x-y$ plane in all material properties (Young’s modulus and small-size parameter) and geometry (width and height) of each layer exists.

## 5. Examples

#### 5.1. Cantilever Beam

#### 5.2. Doubly Clamped Beam

## 6. Conclusions

- The stress-driven integral approach based on Bernoulli–Euler kinematical hypotheses is extended to composite beams assembled of multiple layers, not necessarily of equal width. As demonstrated in the examples, the approach does not suffer from paradoxes present in some other formulations.
- The more standard approach that includes mixed boundary conditions, i.e., both stress resultants and prescribed displacements, is replaced by the purely kinematical framework. In this way, it is not necessary to explicitly determine support reactions in order to calculate displacements. Support reactions and stress resultant distributions are conveniently calculated in the post-processing phase.
- The example section demonstrates that in statically undetermined structural problems, reaction systems exhibit technically significant size effects which therefore have to be taken in due account in design and optimization of a wide variety of new-generation sensors and actuators.
- In the general case of layered beams, the resulting formulation exhibits coupling between axial and transverse displacements. This gives rise to unusual nonlocal phenomena, such as shortening of the nanobeam in the presence of tensile axial force. Coupling of axial and bending terms in the governing differential equations, as well as the neutral surface shift, give rise to such effects.
- Finally, as discussed in the Introduction, if the beams with larger length/thickness ratios are to be considered, one must be wary about the surface and interface effects. An extension with a specialized size-dependent model is recommended in such cases.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Axial displacement ${u}_{0}$ (nm) of the beam loaded with the longitudinal force at the free end, ${\lambda}_{1}=0.1,{\lambda}_{2}=0.2$ (continuous line) and local solution (dotted line), x (nm).

**Figure 3.**Three-dimensional representation of the free end’s longitudinal displacement ${u}_{0}\left(L\right)$ (nm) for different values of nonlocal parameters.

**Figure 4.**Distribution of the normal force $N\left(x\right)$ (nN) and bending moment $M\left(x\right)$ (nN nm) along the cantilever beam, x (nm).

**Figure 5.**Transverse displacement $w\left(L\right)$ (nm) of the beam loaded with the transverse force at the free end, ${\lambda}_{1}=0.1,{\lambda}_{2}=0.2$, x (nm).

**Figure 6.**Three-dimensional representation of the free end’s transverse displacement $w\left(L\right)$ (nm) for different values of nonlocal parameters.

**Figure 9.**Three-dimensional representation of the beam’s middle axial displacement ${u}_{0}(L/2)$ (nm) for different values of nonlocal parameters, ${\lambda}_{1}={\lambda}_{3},{\lambda}_{2}$.

**Figure 10.**Distribution of the normal force $N\left(x\right)$ (nN) along the doubly clamped beam, x (nm).

**Figure 12.**Three-dimensional representation of the beam’s middle transverse displacement $w(L/2)$ (nm) for different values of nonlocal parameters, ${\lambda}_{1}={\lambda}_{3},{\lambda}_{2}$.

**Figure 14.**Support reaction ${M}_{A}$ (nN nm) in the doubly clamped beam in terms of nonlocal parameters.

**Table 1.**Coupled nonlocal layered beams framework for beams with $x-z$ symmetry of the cross-section, ${k}_{ES}=0$.

Calculate stiffnesses: | ${k}_{EA}=(\mathbf{E}\odot \mathbf{A})\xb7\mathbf{1}$, | |

${k}_{EI}=(\mathbf{E}\odot \mathbf{I})\xb7\mathbf{1}$, | ||

${k}_{EA}^{NL}=(\mathbf{A}\odot \mathbf{E}\odot {\mathbf{L}}_{\lambda}\odot {\mathbf{L}}_{\lambda})\xb7\mathbf{1},$ | ||

${k}_{ES}^{NL}=(\mathbf{S}\odot \mathbf{E}\odot {\mathbf{L}}_{\lambda}\odot {\mathbf{L}}_{\lambda})\xb7\mathbf{1},$ | ||

${k}_{EI}^{NL}=(\mathbf{I}\odot \mathbf{E}\odot {\mathbf{L}}_{\lambda}\odot {\mathbf{L}}_{\lambda})\xb7\mathbf{1}.$ | ||

Coupled formulation: | $-{k}_{EA}^{NL}{u}_{0}^{\left(4\right)}+{k}_{ES}^{NL}{w}^{\left(5\right)}+{k}_{EA}{u}_{0}^{\left(2\right)}+{q}_{x}=0$, | |

$-{k}_{ES}^{NL}{u}_{0}^{\left(5\right)}+{k}_{EI}^{NL}{w}^{\left(6\right)}-{k}_{EI}{w}^{\left(4\right)}+{q}_{z}=0$. | ||

Boundary conditions: | ${\left.(-{k}_{EA}^{NL}{u}_{0}^{\left(3\right)}+{k}_{ES}^{NL}{w}^{\left(4\right)}+{k}_{EA}{u}_{0}^{\left(1\right)})\right|}_{x=0}=-{\mathcal{N}}_{0}$ | $\phantom{\rule{4.pt}{0ex}}\mathrm{or}\phantom{\rule{4.pt}{0ex}}\mathrm{prescribe}\phantom{\rule{4.pt}{0ex}}{u}_{0}\left(0\right),$ |

${\left.(-{k}_{EA}^{NL}{u}_{0}^{\left(3\right)}+{k}_{ES}^{NL}{w}^{\left(4\right)}+{k}_{EA}{u}_{0}^{\left(1\right)})\right|}_{x=L}={\mathcal{N}}_{L}$ | $\phantom{\rule{4.pt}{0ex}}\mathrm{or}\phantom{\rule{4.pt}{0ex}}\mathrm{prescribe}\phantom{\rule{4.pt}{0ex}}{u}_{0}\left(L\right),$ | |

${\left.(-{k}_{ES}^{NL}{u}_{0}^{\left(3\right)}+{k}_{EI}^{NL}{w}^{\left(4\right)}-{k}_{EI}{w}^{\left(2\right)})\right|}_{x=0}=-{\mathcal{M}}_{0}$ | $\phantom{\rule{4.pt}{0ex}}\mathrm{or}\phantom{\rule{4.pt}{0ex}}\mathrm{prescribe}\phantom{\rule{4.pt}{0ex}}{w}^{\left(1\right)}\left(0\right),$ | |

${\left.(-{k}_{ES}^{NL}{u}_{0}^{\left(3\right)}+{k}_{EI}^{NL}{w}^{\left(4\right)}-{k}_{EI}{w}^{\left(2\right)})\right|}_{x=L}={\mathcal{M}}_{L}$ | $\phantom{\rule{4.pt}{0ex}}\mathrm{or}\phantom{\rule{4.pt}{0ex}}\mathrm{prescribe}\phantom{\rule{4.pt}{0ex}}{w}^{\left(1\right)}\left(L\right),$ | |

${\left.(-{k}_{ES}^{NL}{u}_{0}^{\left(4\right)}+{k}_{EI}^{NL}{w}^{\left(5\right)}-{k}_{EI}{w}^{\left(3\right)})\right|}_{x=0}=-{\mathcal{T}}_{0}$ | $\phantom{\rule{4.pt}{0ex}}\mathrm{or}\phantom{\rule{4.pt}{0ex}}\mathrm{prescribe}\phantom{\rule{4.pt}{0ex}}w\left(0\right),$ | |

${\left.(-{k}_{ES}^{NL}{u}_{0}^{\left(4\right)}+{k}_{EI}^{NL}{w}^{\left(5\right)}-{k}_{EI}{w}^{\left(3\right)})\right|}_{x=L}={\mathcal{T}}_{L}$ | $\phantom{\rule{4.pt}{0ex}}\mathrm{or}\phantom{\rule{4.pt}{0ex}}\mathrm{prescribe}\phantom{\rule{4.pt}{0ex}}w\left(L\right).$ | |

Constitutive boundary | $({\mathbf{L}}_{\lambda}\odot \mathbf{E})\xb7\left({u}_{0}^{\left(2\right)}\mathbf{A}-{w}^{\left(3\right)}\mathbf{S}\right)-{u}_{0}^{\left(1\right)}\mathbf{E}\xb7\mathbf{A}={\left.0\right|}_{x=0},$ | |

conditions: | $({\mathbf{L}}_{\lambda}\odot \mathbf{E})\xb7\left({u}_{0}^{\left(2\right)}\mathbf{A}-{w}^{\left(3\right)}\mathbf{S}\right)+{u}_{0}^{\left(1\right)}\mathbf{E}\xb7\mathbf{A}={\left.0\right|}_{x=L},$ | |

$({\mathbf{L}}_{\lambda}\odot \mathbf{E})\xb7\left({u}_{0}^{\left(2\right)}\mathbf{S}-{w}^{\left(3\right)}\mathbf{I}\right)+{w}^{\left(2\right)}\mathbf{E}\xb7\mathbf{I}={\left.0\right|}_{x=0},$ | ||

$({\mathbf{L}}_{\lambda}\odot \mathbf{E})\xb7\left({u}_{0}^{\left(2\right)}\mathbf{S}-{w}^{\left(3\right)}\mathbf{I}\right)-{w}^{\left(2\right)}\mathbf{E}\xb7\mathbf{I}={\left.0\right|}_{x=L}.$ |

**Table 2.**Decoupled nonlocal layered beams framework for beams with material and geometric symmetry of the cross-section, ${k}_{ES}=0$ and ${k}_{ES}^{NL}=0$.

Calculate stiffnesses: | ${k}_{EA}=(\mathbf{E}\odot \mathbf{A})\xb7\mathbf{1}$, | |

${k}_{EI}=(\mathbf{E}\odot \mathbf{I})\xb7\mathbf{1}$, | ||

${k}_{EA}^{NL}=(\mathbf{A}\odot \mathbf{E}\odot {\mathbf{L}}_{\lambda}\odot {\mathbf{L}}_{\lambda})\xb7\mathbf{1},$ | ||

${k}_{EI}^{NL}=(\mathbf{I}\odot \mathbf{E}\odot {\mathbf{L}}_{\lambda}\odot {\mathbf{L}}_{\lambda})\xb7\mathbf{1}.$ | ||

Axial displacements: | $-{k}_{EA}^{NL}{u}_{0}^{\left(4\right)}+{k}_{EA}{u}_{0}^{\left(2\right)}+{q}_{x}=0$ | |

Constitutive boundary | $({\mathbf{L}}_{\lambda}\odot \mathbf{E})\xb7\mathbf{A}{u}_{0}^{\left(2\right)}-\mathbf{E}\xb7\mathbf{A}{u}_{0}^{\left(1\right)}={\left.0\right|}_{x=0},$ | |

conditions: | $({\mathbf{L}}_{\lambda}\odot \mathbf{E})\xb7\mathbf{A}{u}_{0}^{\left(2\right)}+\mathbf{E}\xb7\mathbf{A}{u}_{0}^{\left(1\right)}={\left.0\right|}_{x=L},$ | |

Boundary conditions: | ${\left.(-{k}_{EA}^{NL}{u}_{0}^{\left(3\right)}+{k}_{EA}{u}_{0}^{\left(1\right)})\right|}_{x=0}=-{\mathcal{N}}_{0}$ | $\phantom{\rule{4.pt}{0ex}}\mathrm{or}\phantom{\rule{4.pt}{0ex}}\mathrm{prescribe}\phantom{\rule{4.pt}{0ex}}{u}_{0}\left(0\right),$ |

${\left.(-{k}_{EA}^{NL}{u}_{0}^{\left(3\right)}+{k}_{EA}{u}_{0}^{\left(1\right)})\right|}_{x=L}={\mathcal{N}}_{L}$ | $\phantom{\rule{4.pt}{0ex}}\mathrm{or}\phantom{\rule{4.pt}{0ex}}\mathrm{prescribe}\phantom{\rule{4.pt}{0ex}}{u}_{0}\left(L\right).$ | |

Transverse displacements: | $-{k}_{EI}^{NL}{w}^{\left(6\right)}+{k}_{EI}{w}^{\left(4\right)}-{q}_{z}=0$ | |

Constitutive boundary | $-({\mathbf{L}}_{\lambda}\odot \mathbf{E})\xb7\mathbf{I}\phantom{\rule{0.277778em}{0ex}}{w}^{\left(3\right)}+\mathbf{E}\xb7\mathbf{I}\phantom{\rule{0.277778em}{0ex}}{w}^{\left(2\right)}={\left.0\right|}_{x=0},$ | |

conditions: | $-({\mathbf{L}}_{\lambda}\odot \mathbf{E})\xb7\mathbf{I}\phantom{\rule{0.277778em}{0ex}}{w}^{\left(3\right)}-\mathbf{E}\xb7\mathbf{I}\phantom{\rule{0.277778em}{0ex}}{w}^{\left(2\right)}={\left.0\right|}_{x=L},$ | |

Boundary conditions: | ${\left.({k}_{EI}^{NL}{w}^{\left(4\right)}-{k}_{EI}{w}^{\left(2\right)})\right|}_{x=0}=-{\mathcal{M}}_{0}$ | $\phantom{\rule{4.pt}{0ex}}\mathrm{or}\phantom{\rule{4.pt}{0ex}}\mathrm{prescribe}\phantom{\rule{4.pt}{0ex}}{w}^{\left(1\right)}\left(0\right),$ |

${\left.({k}_{EI}^{NL}{w}^{\left(4\right)}-{k}_{EI}{w}^{\left(2\right)})\right|}_{x=L}={\mathcal{M}}_{L}$ | $\phantom{\rule{4.pt}{0ex}}\mathrm{or}\phantom{\rule{4.pt}{0ex}}\mathrm{prescribe}\phantom{\rule{4.pt}{0ex}}{w}^{\left(1\right)}\left(L\right),$ | |

${\left.({k}_{EI}^{NL}{w}^{\left(5\right)}-{k}_{EI}{w}^{\left(3\right)})\right|}_{x=0}=-{\mathcal{T}}_{0}=0$ | $\phantom{\rule{4.pt}{0ex}}\mathrm{or}\phantom{\rule{4.pt}{0ex}}\mathrm{prescribe}\phantom{\rule{4.pt}{0ex}}w\left(0\right),$ | |

${\left.({k}_{EI}^{NL}{w}^{\left(5\right)}-{k}_{EI}{w}^{\left(3\right)})\right|}_{x=L}={\mathcal{T}}_{L}=0$ | $\phantom{\rule{4.pt}{0ex}}\mathrm{or}\phantom{\rule{4.pt}{0ex}}\mathrm{prescribe}\phantom{\rule{4.pt}{0ex}}w\left(L\right).$ |

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**MDPI and ACS Style**

Barretta, R.; Čanađija, M.; Marotti de Sciarra, F.
Nonlocal Mechanical Behavior of Layered Nanobeams. *Symmetry* **2020**, *12*, 717.
https://doi.org/10.3390/sym12050717

**AMA Style**

Barretta R, Čanađija M, Marotti de Sciarra F.
Nonlocal Mechanical Behavior of Layered Nanobeams. *Symmetry*. 2020; 12(5):717.
https://doi.org/10.3390/sym12050717

**Chicago/Turabian Style**

Barretta, Raffaele, Marko Čanađija, and Francesco Marotti de Sciarra.
2020. "Nonlocal Mechanical Behavior of Layered Nanobeams" *Symmetry* 12, no. 5: 717.
https://doi.org/10.3390/sym12050717