# Analysis of the Functionally Step-Variable Graded Plate Under In-Plane Compression

^{*}

## Abstract

**:**

## 1. Introduction

^{®}[39]. The full Green’s strain tensor, the second Piola-Kirchhoff’s stress tensor, and the transition matrix using Godunov’s orthogonalization were used in the description of the problem. In reference [17], on the basis of Koiter’s theory, FGM plates have non-symmetric stable post-buckling equilibrium paths. This feature explains differences in the plate response dependence on the imperfection sign. An FGM plate has a non-trivial coupling matrix B and the coupling between extensional and bending deformations exists as is in the case of non-symmetric laminated plates. On the basis of both of the above-mentioned methods, the present paper reveals the results of influence on the critical and post-critical state of plates made of an alumina-FGM with a finite number of layers (a total of five to 15 layers were assumed). Furthermore, several boundary conditions of the edge support and different contents of alumina in relation to the FGM were taken into account. Considered variants of material distributions were assumed to reflect real FGM.

## 2. Problem Description

_{2}0

_{3}) and the pure ceramics Al

_{2}0

_{3}(See Figure 1b).

#### 2.1. FE Model

^{®}software [39]. To generate an adequate numerical model, an 8-node 281 shell element was assumed. The plate was divided into 10 k finite elements (100 elements along the edge—Figure 3). A nonlinear analysis for large deflections was performed on the basis of Green-Lagrangian equations. During computations, nonlinear calculations were conducted in accordance with the Newton-Raphson algorithm. The critical- and post-critical state of the plate with different boundary conditions was analyzed (See Table 3). The initial deflection of the plate in all cases was assumed to be 0.01t

_{t}, which referred to the first buckling mode.

#### 2.2. Koiter’s Asymptotic Approach

## 3. Results and Discussion

#### 3.1. Buckling Forces

#### 3.2. Post-Buckling Behavior of the Plate

## 4. Summary

- The gradation of layers in the number of plates from five to 15 revealed a slight influence on the plate stability because the obtained curves in both the methods ran almost identically. It indicates that the overall bending stiffness remains on a comparable level, though a non-symmetrical distribution of normal forces with respect to the neutral axis of the plate exists;
- in a comparison of critical forces based on the two applied methods, a sufficiently good agreement was achieved (a few percent difference, at most). The SAM gave slightly lower values;
- in the cases under consideration, the growth in ceramics thickness (from 0.2 mm to 1 mm) played an insignificant role in post-buckling paths. Indeed, the differences in curves are visible but they differ only slightly from one another;
- a higher discrepancy could be seen by comparing the behavior of the plate obtained using the two methods. Firstly, in the FEM analysis, the plate deflects earlier than in the SAM analysis, but after exceeding the critical load, the SAM indicates a larger deflection. In addition, there is a change in the buckling mode during the plate compression. In contrast to the SAM, the FEM reveals a transformation of the defection function from one half-wave to two or three half-waves;
- when analyzing the curves obtained for three different boundary conditions, a similarity in the plate deflection up to three-fold overloads was noticed if static loads were referred to their critical buckling loads (see Figure 7b).

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**A functionally graded material (FGM) plate with its dimensions and a coordinate system (

**a**) and a schematic view of the material distribution (

**b**).

**Figure 4.**Static load vs. the normalized deflection in the middle of the plate for the plate SSSS: (

**a**) a different number of layers (

**b**) different variants.

**Figure 5.**Static load vs. the normalized deflection in the middle of the plate for the plate SCSC: (

**a**) a different number of layers (

**b**) different variants.

**Figure 6.**Static load vs. the normalized deflection (FEM) in the middle of the plate for the plate CCCC: (

**a**) a different number of layers (

**b**) different variants.

**Figure 7.**Comparison of the plots (FEM) for Var_1: (

**a**) static load vs. the normalized deflection and (

**b**) the critical load of static load vs. the normalized deflection.

Description of Variant | t_{t}[mm] | t_{c}[mm] | t_{FGM}[mm] |
---|---|---|---|

Var_1 | 2 | 0.2 | 1.8 |

Var_2 | 2 | 0.4 | 1.6 |

Var_3 | 2 | 0.6 | 1.4 |

Var_4 | 2 | 0.8 | 1.2 |

Var_5 | 2 | 1 | 1 |

Components | Young’s Modulus [GPa] | Poisson’s Ratio [-] |
---|---|---|

Al | 70 | 0.33 |

Al_{2}0_{3} | 393 | 0.25 |

Type of BC | Edge 1 | Edge 2 | Edge 3 | Edge 4 |
---|---|---|---|---|

SSSS | u_{z} = 0u _{x} = moveableapplied load Couple degree of freedom for nodes in x-directions | u_{y},u_{z} = 0 | u_{x},u_{z} = 0 | u_{z} = 0u _{y} = moveableCouple degree of freedom for nodes in y-directions |

SCSC | u_{z} = 0u _{x} = moveableapplied load Couple degree of freedom for nodes in x-directions | u_{y},u_{z} = 0rot _{x} = 0 | u_{x},u_{z} = 0 | u_{z} = 0u _{y} = moveablerot _{x} = 0Couple degree of freedom for nodes in y-directions |

CCCC | u_{z} = 0u _{x} = moveablerot _{y} = 0applied load Couple degree of freedom for nodes in x-directions | u_{y},u_{z} = 0rot _{x} = 0 | u_{x},u_{z} = 0rot _{y} = 0 | u_{z} = 0u _{y} = moveablerot _{x} = 0Couple degree of freedom for nodes in y-directions |

Number of Layers in the FGM (Var_1) | Type of BC | FEM [N] | SAM [N] |
---|---|---|---|

5 | SSSS | 28,994 | 27,876 |

7 | SSSS | 29,179 | 28,776 |

11 | SSSS | 29,663 | 29,392 |

15 | SSSS | 29,830 | 29,672 |

5 | SCSC | 53,491 | 50,888 |

7 | SCSC | 53,895 | 52,820 |

11 | SCSC | 54,970 | 54,160 |

15 | SCSC | 55,339 | 54,772 |

5 | CCCC | 68,584 | ------ |

7 | CCCC | 69,152 | ------ |

11 | CCCC | 70,667 | ------ |

15 | CCCC | 71,186 | ------ |

Variant | Type of BC | FEM [N] | SAM [N] |
---|---|---|---|

Var_1 | SSSS | 29,663 | 29,392 |

Var_2 | SSSS | 30,777 | 30,524 |

Var_3 | SSSS | 31,910 | 31,716 |

Var_4 | SSSS | 33,244 | 33,140 |

Var_5 | SSSS | 34,954 | 34,956 |

Var_1 | SCSC | 54,970 | 54,160 |

Var_2 | SCSC | 57,161 | 56,384 |

Var_3 | SCSC | 59,519 | 58,836 |

Var_4 | SCSC | 62,374 | 61,840 |

Var_5 | SCSC | 66,030 | 65,688 |

Var_1 | CCCC | 70,667 | ------ |

Var_2 | CCCC | 73,571 | ------ |

Var_3 | CCCC | 76,776 | ------ |

Var_4 | CCCC | 80,704 | ------ |

Var_5 | CCCC | 85,734 | ------ |

**Table 6.**Maps of displacements in direction of Z-axis (Var_1) for the boundary conditions under consideration.

SSSS | SCSC | CCC | |||
---|---|---|---|---|---|

F_{st} = 16.5 kN | F_{st} = 16.5 kN | F_{st} = 16.5 kN | |||

F_{st} = 88.5 kN | F_{st} = 88.5 kN | F_{st} = 88.5 kN | |||

F_{st} = 154.5 kN | F_{st} = 154.5 kN | F_{st} = 154.5 kN | |||

F_{st} = 190.5 kN | F_{st} = 183 kN | F_{st} = 190.5 kN | |||

F_{st} = 300 kN | F_{st} = 300 kN | F_{st} = 300 kN |

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**MDPI and ACS Style**

Czechowski, L.; Kołakowski, Z.
Analysis of the Functionally Step-Variable Graded Plate Under In-Plane Compression. *Materials* **2019**, *12*, 4090.
https://doi.org/10.3390/ma12244090

**AMA Style**

Czechowski L, Kołakowski Z.
Analysis of the Functionally Step-Variable Graded Plate Under In-Plane Compression. *Materials*. 2019; 12(24):4090.
https://doi.org/10.3390/ma12244090

**Chicago/Turabian Style**

Czechowski, Leszek, and Zbigniew Kołakowski.
2019. "Analysis of the Functionally Step-Variable Graded Plate Under In-Plane Compression" *Materials* 12, no. 24: 4090.
https://doi.org/10.3390/ma12244090