# Effect of Axial Porosities on Flexomagnetic Response of In-Plane Compressed Piezomagnetic Nanobeams

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## Abstract

**:**

## 1. Introduction

## 2. Formulation of the Problem

#### 2.1. Constitutive Relations for Piezo-Flexomagnetic Solids

#### 2.2. The Piezo-Flexomagnetic Beam Model

_{i}(i = 1,3) represent the points’ displacements in direction of x and z, u and w are the axial and transverse displacements of the mid-plan, respectively, see Figure 1. To show the thickness coordinate, the z parameter is used.

## 3. Solution of the Problem

## 4. Numerical Results

#### 4.1. Validation of Results

#### 4.2. Stability Analysis

_{0}a < 0.8 nm [69], and 0 < e

_{0}a ≤ 2 nm [70,71], are utilized.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Geometry and description of a continuum nanobeam as a square actuator installed on simple end conditions.

**Figure 2.**(

**a**) Nonlocal parameter vs. four cases of non-porous nanobeams (l = 0.5 nm, L = 10 h, ψ = 1 mA). (

**b**) The length scale strain gradient parameter vs. different cases of non-porous nanobeams (e

_{0}a = 0.5 nm, L = 10 h, ψ = 1 mA).

**Figure 3.**Types of porosity vs. two cases of nanobeams (e

_{0}a = 0.5 nm, l = 1 nm, L = 20 h, ψ = 1 mA).

**Figure 4.**Magnetic potential vs. two cases of non-porous nanobeams (e

_{0}a = 0.5 nm, l = 1 nm, L = 10 h).

**Figure 5.**Thickness vs. two cases of non-porous nanobeams (e

_{0}a = 0.5 nm, l = 1 nm, ψ = 1 mA, L = 20 h).

Porosity Type | $\mathit{\lambda}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ | $\mathbf{Ranges}\text{}\mathbf{of}\text{}\mathit{\alpha}$ |
---|---|---|

“O” type distribution | ${\eta}_{1}^{2}\mathrm{sin}\left(\frac{\pi}{L}x\right)$ | $0\le \alpha <0.344$ |

“$\overline{\mathrm{O}}$” type distribution | ${\eta}_{2}^{2}\left[1-\mathrm{sin}\left(\frac{\pi}{L}x\right)\right]$ | $0\le \alpha <0.112$ |

“X” type distribution | ${\eta}_{1}{\eta}_{2}\mathrm{sin}\left(\frac{\pi}{L}x\right)$ | $0\le \alpha <0.197$ |

“$\overline{\mathrm{X}}$” type distribution | ${\eta}_{1}{\eta}_{2}\left[1-\mathrm{sin}\left(\frac{\pi}{L}x\right)\right]$ | $0\le \alpha <0.197$ |

Uniform type distribution | 1 | $0\le \alpha <0.85$ |

${\eta}_{1}=\frac{\pi}{2}$, ${\eta}_{2}=\frac{\pi}{\pi -2}$ |

Nonlocal Strain Gradient Conditions at (0, L) | Local Conditions $\mathbf{(}\mathit{l}\mathbf{=}\mathit{\mu}\mathbf{=}\mathbf{0}\mathbf{)}\text{}\mathbf{at}\text{}\mathbf{(}\mathbf{0}\mathbf{,}\text{}\mathit{L}\mathbf{)}$ |
---|---|

w = 0 ${M}_{nl}=\left(1+{l}^{2}\frac{{d}^{2}}{d{x}^{2}}\right){M}_{l}+\mu \frac{d}{dx}\left(\frac{{d}^{2}{T}_{xxz}}{d{x}^{2}}+{N}_{x}^{0}\frac{{d}^{2}w}{d{x}^{2}}\right)=0$ *${T}_{xxz}=B\frac{{d}^{2}w}{d{x}^{2}}+{f}_{31}\psi =0$ | w = 0 ${M}_{l}=-D\frac{{d}^{2}w}{d{x}^{2}}=0$ *${T}_{xxz}=B\frac{{d}^{2}w}{d{x}^{2}}+{f}_{31}\psi =0$ |

P_{Cr} (nN) | |||||||||
---|---|---|---|---|---|---|---|---|---|

L (nm) | µ = 0 nm^{2} | µ = 1 nm^{2} | µ = 4 nm^{2} | ||||||

[67] | [68] | Present | [67] | [68] | Present | [67] | [68] | Present | |

10 | 4.8447 | 4.8447 | 4.8447 | 4.4095 | 4.4095 | 4.4095 | 3.4735 | 3.4735 | 3.4735 |

12 | 3.3644 | 3.3644 | 3.3644 | 3.1486 | 3.1486 | 3.1486 | 2.6405 | 2.6405 | 2.6405 |

14 | 2.4718 | 2.4718 | 2.4718 | 2.3533 | 2.3533 | 2.3533 | 2.0574 | 2.0574 | 2.0574 |

16 | 1.8925 | 1.8925 | 1.8925 | 1.8222 | 1.8222 | 1.8222 | 1.6396 | 1.6396 | 1.6396 |

18 | 1.4953 | 1.4953 | 1.4953 | 1.4511 | 1.4511 | 1.4511 | 1.3329 | 1.3329 | 1.3329 |

20 | 1.2112 | 1.2112 | 1.2112 | 1.182 | 1.182 | 1.182 | 1.1024 | 1.1024 | 1.1024 |

CoFe_{2}O_{4} |
---|

C_{11} = 286 GPaq _{31} = 580.3 N/A.ma _{33} = 1.57 × 10^{−4} N/A^{2}(A = Ampere) |

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**MDPI and ACS Style**

Malikan, M.; Eremeyev, V.A.; Żur, K.K.
Effect of Axial Porosities on Flexomagnetic Response of In-Plane Compressed Piezomagnetic Nanobeams. *Symmetry* **2020**, *12*, 1935.
https://doi.org/10.3390/sym12121935

**AMA Style**

Malikan M, Eremeyev VA, Żur KK.
Effect of Axial Porosities on Flexomagnetic Response of In-Plane Compressed Piezomagnetic Nanobeams. *Symmetry*. 2020; 12(12):1935.
https://doi.org/10.3390/sym12121935

**Chicago/Turabian Style**

Malikan, Mohammad, Victor A. Eremeyev, and Krzysztof Kamil Żur.
2020. "Effect of Axial Porosities on Flexomagnetic Response of In-Plane Compressed Piezomagnetic Nanobeams" *Symmetry* 12, no. 12: 1935.
https://doi.org/10.3390/sym12121935