# Thermal Buckling and Postbuckling Behaviors of Couple Stress and Surface Energy-Enriched FG-CNTR Nanobeams

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## Abstract

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## 1. Introduction

^{2}-type differential quadrature plate element. In another work, Zhang et al. [39] presented surface energy-enriched gradient elastic Euler–Bernoulli, Timoshenko, and Reddy beam models and gave a novel numerical solution method. Khabaz et al. [40] studied the combined effects of strain gradient and surface energy on the dynamic behavior of an advanced sandwich composite microbeam including piezoelectric layers. Dangi et al. [41] proposed a theoretical model for bidirectional FG Euler–Bernoulli nanobeams with nonlocal stress, strain gradient, and surface energy effects and applied the model developed to evaluate three types of size effects on the natural frequencies of nanobeams. Attia and Shanab [42] made a combination of the symmetrical CST and classical SET to study the size-dependent geometric nonlinear behavior of FG Euler–Bernoulli and Timoshenko nanobeams. Shaat et al. [43] employed Newton’s second law to establish the governing equation for surface energy and couple stress enriched Kirchhoff nanoplates. Lu et al. [44] developed three isotropic plate modes with nonlocal stress, strain gradient, and surface energy effects according to the Kirchhoff, Mindlin, and Reddy deformation assumptions, respectively. Duong et al. [45] explored the static responses and stress concentration phenomenon of FG-CNTR composite cylindrical shells with various boundary conditions using the higher-order, shear-normal deformation theory and Laplace transform. Doan et al. [46] examined the vibration response and static buckling of variable flexoelectric nanoplates using the FEM and novel hyperbolic sine shear deformation theory, in which the thickness is adjusted by linear and nonlinear rules. Thom et al. [47] proposed a phase-field model to investigate the thermal buckling of fractured FG plates based on the third-order shear deformation plate theory and demonstrated the difference between the plate’s static stability response to temperature-dependent and temperature-independent cases. Do et al. [48] used the phase field model and Mindlin plate theory to predict the thermal buckling of cracked FG plates and considered two cases with and without the difference between the neutral surface and middle surface.

## 2. Theoretical Formulation

#### 2.1. Geometrical and Material Description

#### 2.2. Governing Equations

## 3. Solution Methodology

#### 3.1. Immovable SS Ends

#### 3.2. Immovable CC Ends

## 4. Results and Discussion

#### 4.1. Verification Study

#### 4.2. Parametric Studies

#### 4.2.1. Effects of Surface Energy and Couple Stress

#### 4.2.2. Effects of Geometrical Property and Substrate Stiffness

#### 4.2.3. Effect of Material Properties

## 5. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Comparisons of the postbuckling equilibrium path for the metal/ceramic macrobeam: CC ends [55].

**Figure 4.**Thermal postbuckling response of UD nanobeams under different scale parameters: (

**a**) SE ($l=0$); (

**b**) CS (${\mu}_{S}^{(1)}={\mu}_{S}^{(2)}=0$).

**Figure 5.**Combined effects of SE and CS on the critical buckling temperature $\Delta {T}_{cr}$: (

**a**) SS ends; (

**b**) CC ends.

**Figure 6.**Combined effects of surface energy and CS on thermal postbuckling response: (

**a**) SS ends; (

**b**) CC ends.

**Figure 7.**Variation of critical buckling temperature with respect to $L/{H}_{b}$: (

**a**) SS ends; (

**b**) CC ends.

**Figure 9.**Effects of various elastic media on thermal postbuckling response: (

**a**) SS ends; (

**b**) CC ends.

**Figure 11.**Effects of CNT volume percent on thermal postbuckling response: (

**a**) SS ends; (

**b**) CC ends.

**Table 1.**Critical buckling temperature $\Delta {T}_{cr}{\alpha}_{0}{\left(L/{H}_{b}\right)}^{2}$ of SS beams at $L/{H}_{b}=20$.

Materials | TID | TD | ||
---|---|---|---|---|

Shen and Wang [54] | Present | Shen and Wang [54] | Present | |

SUS304 | 0.619 | 0.661 | 0.582 | 0.595 |

Si_{3}N_{4} | 1.350 | 1.355 | 1.185 | 1.189 |

ZrO_{2} | 0.518 | 0.544 | 0.416 | 0.416 |

Al_{2}O_{3} | 1.379 | 1.406 | 1.326 | 1.342 |

**Table 2.**Critical buckling temperature $\Delta {T}_{cr}{\alpha}_{0}{\left(L/{H}_{b}\right)}^{2}$ of CC beams at $L/{H}_{b}=25$.

Materials | TID | TD | ||
---|---|---|---|---|

Esfahani et al. [55] | Present | Esfahani et al. [55] | Present | |

Si_{3}N_{4} | 5.338 | 5.421 | 3.916 | 3.916 |

$n=2.0$ | 3.263 | 3.315 | 2.659 | 2.659 |

$n=5.0$ | 3.039 | 3.089 | 2.515 | 2.515 |

$n=10.0$ | 2.898 | 2.946 | 2.416 | 2.416 |

SUS304 | 2.604 | 2.644 | 2.196 | 2.196 |

BC | $\left({\mathit{\mu}}_{\mathit{S}}^{(1)},{\mathit{\mu}}_{\mathit{S}}^{(2)}\right)$ | ||||||
---|---|---|---|---|---|---|---|

(0.0,0.0) | (0.01,0.01) | (0.3,0.3) | (0.4,0.4) | (0.5,0.5) | (1.0,1.0) | ||

SS | TID | 99.666 | 102.403 | 181.784 | 209.157 | 318.649 | 373.396 |

TD | 84.673 | 86.757 | 144.498 | 163.580 | 238.170 | 275.293 | |

CC | TID | 398.665 | 401.411 | 481.068 | 508.536 | 536.004 | 673.345 |

TD | 286.336 | 288.179 | 342.325 | 361.386 | 380.695 | 481.673 |

BC | $\mathit{l}/{\mathit{H}}_{\mathit{b}}$ | ||||||
---|---|---|---|---|---|---|---|

0.0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | ||

SS | TID | 99.666 | 99.692 | 110.095 | 123.131 | 141.383 | 164.848 |

TD | 84.673 | 84.692 | 92.488 | 102.092 | 115.266 | 131.816 | |

CC | TID | 398.665 | 398.768 | 440.381 | 492.527 | 565.531 | 659.393 |

TD | 286.336 | 286.403 | 313.451 | 347.616 | 396.229 | 460.486 |

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**MDPI and ACS Style**

Kong, L.; Zhang, B.; Li, C.
Thermal Buckling and Postbuckling Behaviors of Couple Stress and Surface Energy-Enriched FG-CNTR Nanobeams. *Symmetry* **2022**, *14*, 2228.
https://doi.org/10.3390/sym14112228

**AMA Style**

Kong L, Zhang B, Li C.
Thermal Buckling and Postbuckling Behaviors of Couple Stress and Surface Energy-Enriched FG-CNTR Nanobeams. *Symmetry*. 2022; 14(11):2228.
https://doi.org/10.3390/sym14112228

**Chicago/Turabian Style**

Kong, Liulin, Bo Zhang, and Cheng Li.
2022. "Thermal Buckling and Postbuckling Behaviors of Couple Stress and Surface Energy-Enriched FG-CNTR Nanobeams" *Symmetry* 14, no. 11: 2228.
https://doi.org/10.3390/sym14112228