# The Study of Buckling and Post-Buckling of a Step-Variable FGM Box

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Description

_{2}O

_{3}with 11 layers) and pure ceramics Al

_{2}O

_{3}(Figure 2). The material properties of the analysed box were based on two basic materials, Al and Al

_{2}O

_{3}, as given in Table 1. Material properties for the remaining layers in the FGM (as in Figure 2) were determined using the mixture law of two components.

#### 2.1. Finite Element Model (FEM)

^{®}software (2018) [36]. To create an appropriate numerical model a 281-shell element was assumed. This finite element possessed eight nodes with six degrees of freedom in each of them and was suitable to analyse moderately thick shell structures. With the use of that element one can perform linear and nonlinear simulation of multilayered structures including a sandwich structure. The accuracy of modelling composite materials by means of a 281-shell element was based on a first-order deformation theory (FODT) (it usually refers to Mindlin-Reissner’s shell theory). The plate box was divided into 40,000 finite elements (100 elements along the edge—Figure 3a).

#### 2.2. Koiter’s Asymptotic Approach

## 3. Results and Discussion

#### 3.1. Buckling Forces

#### 3.2. Post-Buckling State

## 4. Conclusions

- (1)
- For variant_3 (with a mixed arrangement of material) the first critical force was the lowest. For higher critical forces, differences were not noticeable and for FEM they did not exceed 1%. Furthermore, in this case deflections on all walls appeared the fastest in comparison to other considered options, but finally at greater overload curves were grew closer to each other.
- (2)
- The difference in the deflection of walls achieved by SAM amounted to 0.1% in the whole scope and this does not give substantial contrast in comparison to analysed variants as it was observed in FEM.
- (3)
- The SAM allows results of equilibrium paths to be achieved significantly faster than FEM. SAM is very useful for interpreting the phenomena accompanying the interaction of different modes of buckling in the full scope of the load. This method furthers our understanding of phenomena that occur during coupled buckling.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**The functionally graded material (FGM) box with its dimensions and coordinate system: Ceramic inside (

**a**: variant_1), ceramic outside (

**b**: variant_2) and “mixed” case (

**c**: variant_3).

Components | Young Modulus (GPa) | Poisson’s Ratio (-) |
---|---|---|

Al | 70 | 0.33 |

Al_{2}O_{3} | 393 | 0.25 |

**Table 2.**Assumed boundary conditions in finite element (FE) model according to Figure 3.

Number of Edges | u_{x} | u_{y} | u_{z} | rot_{x} | rot_{y} | rot_{z} | Couple Degree of Freedom in Nodes in the z-Direction | Load in the z-Direction |
---|---|---|---|---|---|---|---|---|

1 | ○ | ● | const | ○ | ○ | ○ | ● | ● |

2 | ● | ○ | const | ○ | ○ | ○ | ● | ● |

3 | ○ | ● | const | ○ | ○ | ○ | ● | ● |

4 | ● | ○ | const | ○ | ○ | ○ | ● | ● |

5 | ○ | ● | ● | ○ | ○ | ○ | ○ | ○ |

6 | ● | ○ | ● | ○ | ○ | ○ | ○ | ○ |

7 | ○ | ● | ● | ○ | ○ | ○ | ○ | ○ |

8 | ● | ○ | ● | ○ | ○ | ○ | ○ | ○ |

Method of Solution | Buckling Mode | Variant_1 | Variant_2 | Variant_3 |
---|---|---|---|---|

Finite element method (FEM) | First | 115.490 | 114.961 | 112.176 |

Second | 162.128 | 163.014 | 162.669 | |

Third | 177.761 | 177.344 | 177.056 | |

Semi-analytical method (SAM) | First | 113.564 | 112.076 | 112.787 |

Second | 162.331 | 161.046 | 162.452 | |

Third | 176.572 | 175.507 | 176.014 |

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**MDPI and ACS Style**

Czechowski, L.; Kołakowski, Z. The Study of Buckling and Post-Buckling of a Step-Variable FGM Box. *Materials* **2019**, *12*, 918.
https://doi.org/10.3390/ma12060918

**AMA Style**

Czechowski L, Kołakowski Z. The Study of Buckling and Post-Buckling of a Step-Variable FGM Box. *Materials*. 2019; 12(6):918.
https://doi.org/10.3390/ma12060918

**Chicago/Turabian Style**

Czechowski, Leszek, and Zbigniew Kołakowski. 2019. "The Study of Buckling and Post-Buckling of a Step-Variable FGM Box" *Materials* 12, no. 6: 918.
https://doi.org/10.3390/ma12060918