# Numerical Study on the Buckling Behavior of FG Porous Spherical Caps Reinforced by Graphene Platelets

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Problem

#### 2.1. Description of Geometry and Mechanical Properties

- -
- Porosity distribution 1$$\left\{\begin{array}{c}E\left(r\right)={E}^{\ast}\left[1-{e}_{0}cos\left(\pi (\frac{r-{r}_{in}}{h}-\frac{1}{2})\right)\right]\hfill \\ G\left(r\right)={G}^{\ast}\left[1-{e}_{0}cos\left(\pi (\frac{r-{r}_{in}}{h}-\frac{1}{2})\right)\right]\hfill \\ \rho \left(r\right)={\rho}^{\ast}\left[1-{e}_{m}cos\left(\pi (\frac{r-{r}_{in}}{h}-\frac{1}{2})\right)\right]\hfill \end{array}\right.$$

- -
- Porosity distribution 2$$\left\{\begin{array}{c}E\left(r\right)={E}^{\ast}\left[\left(1-{e}_{0}^{\ast}\left(1-cos\left(\pi (\frac{r-{r}_{\mathrm{in}}}{h}-\frac{1}{2})\right)\right)\right]\right.\hfill \\ G\left(r\right)={G}^{\ast}\left[\left(1-{e}_{0}^{\ast}\left(1-cos\left(\pi (\frac{r-{r}_{\mathrm{in}}}{h}-\frac{1}{2})\right)\right)\right]\right.\hfill \\ \rho \left(r\right)={\rho}^{\ast}\left[\left(1-{e}_{m}^{\ast}\left(1-cos\left(\pi (\frac{r-{r}_{\mathrm{in}}}{h}-\frac{1}{2})\right)\right)\right]\right.\hfill \end{array}\right.$$

- -
- Uniform porosity distribution

#### 2.2. Governing Equations of the Problem

**Q**is the displacements vector and $\mathcal{L}$ is an operator matrix involving the partial derivatives of a function

## 3. Finite Element Modeling

- -
- For a spherical cap with θ = 180°, ϕ = 180°, u, v, w (r, θ and ϕ = 0, 180°), $\overline{{\sigma}_{\mathit{rr}}}$ = 1 at r = b.
- -
- For a spherical cap with θ = 180°, ϕ = 90°, u, v, w (r, θ and ϕ = 0, 90°), $\overline{{\sigma}_{\mathit{rr}}}$ = 1 at r = b.

## 4. Numerical Results and Discussion

#### 4.1. Validation

_{0}= 0, γ

_{GPL}= 0. As far as the mechanical properties are concerned, we assumed E

_{m}= 130 GPa, ρ

_{m}= 8960 kg/m

^{3}, ν

_{m}= 0.34 for the copper material. As geometrical dimensions, we assumed: a = 0.225 m, b = 0.25 m, θ = 180°, ϕ = 180°, 90°. In this way, the FG-GPL porous structure changes to an isotropic homogenous structure. The comparison between our results and predictions from Ansys Workbench is shown in Table 2, with an excellent agreement among them.

#### 4.2. Parametric Analysis of the Buckling Load

_{m}= 130 GPa, ρ

_{m}= 8960 kg/m

^{3}and ν

_{m}= 0.34 for the copper material [28]; and E

_{GPL}= 1.01 TPa, ρ

_{GPL}= 1062.5 kg/m

^{3}, ν

_{GPL}= 0.186, w

_{GPL}= 1.5 μm, l

_{GPL}= 2.5 μm and t

_{GPL}= 1.5 nm for GPLs.

_{0}= 0.2 and γ = 0.01 wt%. The extreme values of buckling load are related to GPLX and GPL-O distributions, respectively, which means that, when GPLs accumulate around the inner and outer surfaces of the shell, the stiffness reaches its highest value. Moreover, when GPLs are sparser around the outer and inner surfaces of the shell, the minimum buckling load is obtained. Note also that the critical buckling loads for a GPLA and GPL-V distributions are approximately the same. The results also show that the surface area of the shell increases by increasing the polar angle, and there are consecutive increases in the structural stiffness and buckling load.

_{0}= 0.4, γ = 0.01 wt%), which shows that the maximum and minimum buckling loads belong to PD1 and PD2 distributions, respectively. PD1 provides higher structural stiffness, and PD2 gives the minimum stiffness of the spherical cap shell. The main difference between the extreme values of critical buckling load is approximately 90% for different porosity patterns. This means that the porosity distribution has a considerable effect on the buckling loads of FG-GPL, porous, spherical cap shells.

_{0}= 0.5, GPLX) are given in Table 5. Note that, by increasing the weight fraction of GPLs, the critical buckling loads of shell increases significantly (approximately 100%), along with a small variation of the structural mass. This issue can be useful for aerospace structures where the high stiffness and low density are extremely important.

## 5. Concluding Remarks

- (a)
- The maximum and minimum buckling loads seem to be reached for GPL-X and GPL-O distributions, respectively.
- (b)
- The maximum and minimum buckling loads belong to the PD1 and PD2 cases, respectively.
- (c)
- The difference between the maximum and minimum critical buckling loads for different porosity distributions is approximately equal to 90%, and the buckling loads of the selected structure increase considerably (approximately of 100%) with an increase in the weight fraction of GPLs.
- (d)
- The effect of the porosity coefficient on the critical buckling load for porous spherical cap shells made of FG-GPLs is lower than the weight fraction of the nanofillers, being approximately equal to 12.5%.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**GPL distributions: (

**A**) symmetric distribution I, (

**B**) symmetric distribution II and (

**C**) uniform distribution and patterns of porosities: (

**a**) GPL-X, (

**b**) GPL-O, (

**c**) GPL-UD, (

**d**) GPL-A, (

**e**) GPL-V.

**Figure 3.**The first six buckling mode shapes of an FG-GPL, porous spherical cap ($\vartheta $ = 180°, $\varphi $ = 180°, GPL-X, PD1, e

_{0}= 0.4, γ = 0.01 wt%).

**Figure 4.**The first six buckling mode shapes of FG-GPL, porous spherical cap ($\vartheta $ = 180°, ϕ = 90°, GPL-X, PD1, e

_{0}= 0.4, γ = 0.01 wt%).

${\mathit{e}}_{0}$ | ${\mathit{e}}_{0}^{\ast}$ | $\mathit{\alpha}$ |
---|---|---|

0.1 | 0.1738 | 0.9361 |

0.2 | 0.3442 | 0.8716 |

0.3 | 0.5103 | 0.8064 |

0.4 | 0.6708 | 0.7404 |

0.5 | 0.8231 | 0.6733 |

0.6 | 0.9612 | 0.6047 |

Polar Angle | ${\mathit{\omega}}_{1}$ | ${\mathit{\omega}}_{2}$ | ${\mathit{\omega}}_{3}$ | ${\mathit{\omega}}_{4}$ | ${\mathit{\omega}}_{5}$ | ${\mathit{\omega}}_{6}$ | |
---|---|---|---|---|---|---|---|

(Ansys Workbench) | 2.890 | 2.901 | 2.990 | 3.001 | 3.012 | 3.078 | |

180° | (Present) | 2.908 | 2.924 | 3.008 | 3.021 | 3.033 | 3.132 |

(Ansys Workbench) | 1.861 | 1.867 | 2.110 | 2.101 | 2.159 | 2.299 | |

90° | (Present) | 1.873 | 1.875 | 2.180 | 2.211 | 2.222 | 2.320 |

**Table 3.**Buckling loads (GPa) of FG-GPL, porous spherical caps for various polar angles and GPL patterns (PD3, e

_{0}= 0.2, γ = 0.01 wt%).

GPL Pattern | $\mathit{\varphi}$ | λ_{1} | λ_{2} | λ_{3} | λ_{4} | λ_{5} | λ_{6} |
---|---|---|---|---|---|---|---|

90° | 3.174 | 3.174 | 3.670 | 3.775 | 3.778 | 3.939 | |

GPL-X | 180° | 4.450 | 4.462 | 4.572 | 4.600 | 4.617 | 4.798 |

90° | 1.898 | 1.901 | 2.227 | 2.254 | 2.264 | 2.354 | |

GPL-A | 180° | 2.914 | 2.925 | 2.997 | 3.009 | 3.018 | 3.109 |

90° | 1.885 | 1.888 | 2.212 | 2.238 | 2.248 | 2.336 | |

GPL-V | 180° | 2.906 | 2.917 | 2.988 | 3.001 | 3.009 | 3.098 |

90° | 1.653 | 1.657 | 1.949 | 1.950 | 1.958 | 2.018 | |

GPL-O | 180° | 2.731 | 2.747 | 2.801 | 2.821 | 2.824 | 2.894 |

90° | 1.903 | 1.906 | 2.233 | 2.259 | 2.270 | 2.359 | |

GPL-UD | 180° | 2.926 | 2.937 | 3.009 | 3.022 | 3.030 | 3.121 |

**Table 4.**Buckling loads (GPa) of FG-GPL, porous spherical caps for various polar angles and porosity distributions (GPLX, e

_{0}= 0.4, γ = 0.01wt%).

Porosity Distribution | $\mathit{\varphi}$ | λ_{1} | λ_{2} | λ_{3} | λ_{4} | λ_{5} | λ_{6} |
---|---|---|---|---|---|---|---|

90° | 2.842 | 2.842 | 3.268 | 3.346 | 3.359 | 3.473 | |

PD1 | 180° | 3.964 | 3.974 | 4.076 | 4.092 | 4.111 | 4.259 |

90° | 1.581 | 1.584 | 1.862 | 1.863 | 1.870 | 1.926 | |

PD2 | 180° | 2.613 | 2.632 | 2.684 | 2.703 | 2.707 | 2.774 |

90° | 2.249 | 2.249 | 2.607 | 2.677 | 2.683 | 2.799 | |

PD3 | 180° | 3.191 | 3.199 | 3.280 | 3.297 | 3.310 | 3.434 |

**Table 5.**Buckling loads (GPa) of an FG-GPL, porous spherical cap for various polar angle and weight fractions of GPL nano-fillers (PD1, e

_{0}= 0.5, GPLX).

Weight Fraction of Nano-Fillers (%wt) | $\mathit{\varphi}$ | λ_{1} | λ_{2} | λ_{3} | λ_{4} | λ_{5} | λ_{6} |
---|---|---|---|---|---|---|---|

90° | 1.458 | 1.459 | 1.696 | 1.737 | 1.744 | 1.819 | |

0% | 180° | 2.102 | 2.107 | 2.162 | 2.171 | 2.180 | 2.258 |

90° | 2.191 | 2.191 | 2.528 | 2.601 | 2.602 | 2.710 | |

0.5% | 180° | 3.048 | 3.056 | 3.130 | 3.151 | 3.163 | 3.290 |

90° | 2.744 | 2.746 | 3.187 | 3.261 | 3.268 | 3.403 | |

1% | 180° | 3.927 | 3.940 | 4.032 | 4.062 | 4.076 | 4.246 |

**Table 6.**Buckling loads (GPa) of FG-GPL, porous spherical caps for various polar angles and porosity coefficients (PD1, γ = 0.01 wt%, GPLX).

${\mathit{e}}_{0}$ | $\mathit{\varphi}$ | λ_{1} | λ_{2} | λ_{3} | λ_{4} | λ_{5} | λ_{6} |
---|---|---|---|---|---|---|---|

90° | 3.262 | 3.262 | 3.780 | 3.881 | 3.889 | 4.055 | |

0.2 | 180° | 4.626 | 4.635 | 4.752 | 4.777 | 4.794 | 4.975 |

90° | 2.842 | 2.842 | 3.268 | 3.346 | 3.359 | 3.473 | |

0.4 | 180° | 3.964 | 3.974 | 4.076 | 4.092 | 4.111 | 4.259 |

90° | 2.744 | 2.746 | 3.187 | 3.261 | 3.268 | 3.403 | |

0.5 | 180° | 3.927 | 3.940 | 4.032 | 4.062 | 4.076 | 4.246 |

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**MDPI and ACS Style**

Zhou, Z.; Wang, Y.; Zhang, S.; Dimitri, R.; Tornabene, F.; Asemi, K. Numerical Study on the Buckling Behavior of FG Porous Spherical Caps Reinforced by Graphene Platelets. *Nanomaterials* **2023**, *13*, 1205.
https://doi.org/10.3390/nano13071205

**AMA Style**

Zhou Z, Wang Y, Zhang S, Dimitri R, Tornabene F, Asemi K. Numerical Study on the Buckling Behavior of FG Porous Spherical Caps Reinforced by Graphene Platelets. *Nanomaterials*. 2023; 13(7):1205.
https://doi.org/10.3390/nano13071205

**Chicago/Turabian Style**

Zhou, Zhimin, Yun Wang, Suying Zhang, Rossana Dimitri, Francesco Tornabene, and Kamran Asemi. 2023. "Numerical Study on the Buckling Behavior of FG Porous Spherical Caps Reinforced by Graphene Platelets" *Nanomaterials* 13, no. 7: 1205.
https://doi.org/10.3390/nano13071205