# Proper Generalized Decomposition for Parametric Study and Material Distribution Design of Multi-Directional Functionally Graded Plates Based on 3D Elasticity Solution

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## Abstract

**:**

## 1. Introduction

- FGM plates with gradation in directions other than the thickness direction are of current research interest, both from a theoretical and industrial point of view.
- In most cases, the Poisson’s ratios are considered constants, or a similar grading function is used for different orthotropic moduli components (e.g., [20]) or directly for the orthotropic stiffness coefficients (e.g., [17]). These non-physical assumptions are performed to simplify the solution procedure, but can represent crude approximations in many cases.
- Shear deformations are important in thick plates and, although high-order plate bending theories are adopted to somehow consider it, the use of the 3D elasticity approach is inevitable. There are only a few of this type of solution, and even fewer for cases of orthotropic FGM plates and rarely for parametric analyses and material distribution design problems.
- There are some analytical or semi-analytical solutions for three-dimensional elasticity plate problems. Although these methods are successful in decreasing computational costs, they are limited in special boundary conditions, loading and material characteristics, and their application in general plate problems is restricted.
- Parametric studies and material distribution design problems are very limited because they need many independent simulations, which leads to high computational costs using numerical solutions of 3D elasticity problems.
- To the best of the authors’ knowledge, PGD, as an advanced model order reduction technique, has not been applied in the elasticity solution of FGM composite plates thus far.

- Material grading is considered along three physical directions.
- Orthotropic FGM material constants are considered, consistent with real physics, based on the volume fraction of the constitutive materials and using established micromechanical models.
- An analysis is conducted based on the 3D theory of elasticity to consider shearing deformations perfectly.
- All types of boundary conditions are considered without any limitations.
- Parametric studies are performed in an efficient manner in a high-dimensional coordinate space.
- The application of the PGD technique in the parametric study of thick FGM plate problems is introduced for the first time. The material distribution design is then determined using the resulting parametric study.

## 2. Governing Equations and Weak Form

## 3. Separated Approximate Representation (SAR)

## 4. SAR of a Given Field Function

## 5. Proper Generalized Decomposition

## 6. Numerical Examples

#### 6.1. Example 1

#### 6.2. Example 2

#### 6.3. Example 3

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**The first six modes of displacement components (${u}_{x},{u}_{y},{u}_{z}$) with respect to space coordinates ($x,y,z,p$) for Example 1 for boundary conditions case SSSS.

**Figure 3.**The displacement components at two locations ($x,y,z$) = ($0.75,0.75,0$) and ($0.75,0.75,1$) versus material parameter p for Example 1 considering boundary conditions SSSS; the reference solution given in (Pan 2003) [24] is shown by solid dots.

**Figure 4.**The displacement components at two locations ($x,y,z$)=($0.75,0.75,0$) and ($0.75,0.75,1$) versus material parameter p for Example 1 considering boundary conditions CCCC.

**Figure 5.**The displacement components at two locations ($x,y,z$)=($0.75,0.75,0$) and ($0.75,0.75,1$) versus material parameter p for Example 1 considering boundary conditions CSFF.

**Figure 6.**Schematic representation of fiber orientations and volume fraction distributions for the FGM composite plate considered in Example 2; (

**a**) ${90}^{\circ}$, $p=0$; (

**b**) ${90}^{\circ}$, $p=1$; (

**c**) ${0}^{\circ}$, $p=0$; (

**d**) ${0}^{\circ}$, $p=1$.

**Figure 7.**The dimensionless displacement, ${\overline{u}}_{z}$, at locations ($x,y,z$) = (${L}_{x}/2,{L}_{y}/2,{L}_{z}/2$) versus material parameter p for Example 2 considering boundary conditions SSSS; the reference solution given in (Ravindran 2019) [19] is shown by solid dots.

**Figure 8.**The dimensionless displacement, ${\overline{u}}_{z}$, at locations ($x,y,z$) = (${L}_{x}/2,{L}_{y}/2,{L}_{z}/2$) versus material parameter p for Example 2 considering boundary conditions CCCC.

**Figure 9.**The dimensionless displacement, ${\overline{u}}_{z}$, components at locations ($x,y,z$) = (${L}_{x}/2,{L}_{y}/2,{L}_{z}/2$) versus material parameter p for Example 2 considering boundary conditions CSFF.

**Figure 11.**The first six modes of displacement components (${u}_{x},{u}_{y},{u}_{z}$) with respect to space coordinates ($x,y,z,{p}_{1},{p}_{2}$) for Example 3.

**Figure 12.**Contour plot of the field of minimum factor of safety, ${\eta}_{min}({p}_{1},{p}_{2})$, for the full range of material parameters ${p}_{1}$ and ${p}_{2}$ for Example 3.

**Figure 13.**Contour plot of volume fraction distribution ${V}^{c}(x,y,z)$ for the optimum material parameters $({p}_{1},{p}_{2})=(0.3,0.9)$ for Example 3.

**Figure 14.**Deformed configuration (not scaled) and contour plot of displacement ${u}_{z}$ for the optimum material parameters $({p}_{1},{p}_{2})=(0.3,0.9)$ for Example 3.

Equation | Term | a | b | c | d | e | f |
---|---|---|---|---|---|---|---|

Equation (4) | 1 | x | x | x | y | 1 | 1 |

2 | x | x | y | y | 1 | 2 | |

3 | x | x | z | z | 1 | 3 | |

4 | x | y | y | x | 4 | 4 | |

5 | x | y | x | y | 4 | 4 | |

6 | x | z | x | z | 6 | 6 | |

7 | x | z | z | x | 6 | 6 | |

Equation (5) | 1 | y | x | y | x | 4 | 4 |

2 | y | x | x | y | 4 | 4 | |

3 | y | y | x | x | 1 | 2 | |

4 | y | y | y | y | 2 | 2 | |

5 | y | y | z | z | 2 | 3 | |

6 | y | z | z | y | 5 | 5 | |

7 | y | z | y | z | 5 | 5 | |

Equation (6) | 1 | z | x | x | z | 6 | 6 |

2 | z | x | z | x | 6 | 6 | |

3 | z | y | z | y | 5 | 5 | |

4 | z | y | y | z | 5 | 5 | |

5 | z | z | x | x | 1 | 3 | |

6 | z | z | y | y | 2 | 3 | |

7 | z | z | z | z | 3 | 3 |

B.C. Type | Face ($\mathit{x}=0$) | Face ($\mathit{x}={\mathit{L}}_{\mathit{x}}$) | Face ($\mathit{y}=0$) | Face ($\mathit{y}={\mathit{L}}_{\mathit{y}}$) |
---|---|---|---|---|

SSSS | ${u}_{y}={u}_{z}=0$ | ${u}_{y}={u}_{z}=0$ | ${u}_{x}={u}_{z}=0$ | ${u}_{x}={u}_{z}=0$ |

CCCC | ${u}_{x}={u}_{y}={u}_{z}=0$ | ${u}_{x}={u}_{y}={u}_{z}=0$ | ${u}_{x}={u}_{y}={u}_{z}=0$ | ${u}_{x}={u}_{y}={u}_{z}=0$ |

CSFF | ${u}_{x}={u}_{y}={u}_{z}=0$ | — | ${u}_{y}={u}_{z}=0$ | — |

CFSF | ${u}_{x}={u}_{y}={u}_{z}=0$ | ${u}_{y}={u}_{z}=0$ | — | — |

${E}_{x}$ | 6.89476 | GPa |

${E}_{y}$ | 172.369 | GPa |

${E}_{z}$ | 6.89476 | GPa |

${G}_{xy}$ | 3.44738 | GPa |

${G}_{yz}$ | 3.44738 | GPa |

${G}_{zx}$ | 1.37895 | GPa |

${\nu}_{xy}$ | 0.01 | |

${\nu}_{yz}$ | 0.25 | |

${\nu}_{zx}$ | 0.25 |

**Table 4.**The displacement components for three values of material parameter p, for Example 1, for boundary conditions SSSS obtained using different grid sizes.

Dis. | Loc. | p | Ref. [24] | Present | ||
---|---|---|---|---|---|---|

{21 × 21 × 11 × 11} | {41 × 41 × 15 × 15} | {61 × 61 × 21 × 21} | ||||

${u}_{x}$ | z = 0 | −1 | 6.4876 | 6.3892 (1.5) | 6.4416 (0.7) | 6.4629 (0.4) |

0 | 4.5491 | 4.4825 (1.5) | 4.5182 (0.7) | 4.5324 (0.4) | ||

1 | 3.0359 | 2.9864 (1.6) | 3.0128 (0.8) | 3.0236 (0.4) | ||

z = 1 | −1 | −7.0921 | −6.9544 (1.9) | −7.0267 (0.9) | −7.0577 (0.5) | |

0 | −3.9492 | −3.8791 (1.8) | −3.9161 (0.8) | −3.9317 (0.4) | ||

1 | −2.0888 | −2.0502 (1.9) | −2.0705 (0.9) | −2.0792 (0.5) | ||

${u}_{y}$ | z = 0 | −1 | 2.6853 | 2.5854 (3.7) | 2.6357 (1.8) | 2.6599 (0.9) |

0 | 1.7737 | 1.7070 (3.8) | 1.7406 (1.9) | 1.7568 (1.0) | ||

1 | 1.1232 | 1.0779 (4.0) | 1.1007 (2.0) | 1.1117 (1.0) | ||

z = 1 | −1 | −3.643 | −3.4934 (4.1) | −3.5678 (2.1) | −3.6046 (1.1) | |

0 | −2.0733 | −1.9945 (3.8) | −2.0338 (1.9) | −2.0531 (1.0) | ||

1 | −1.1419 | −1.0992 (3.7) | −1.1206 (1.9) | −1.1310 (1.0) | ||

${u}_{z}$ | z = 0 | −1 | 21.134 | 20.823 (1.5) | 20.987 (0.7) | 21.055 (0.4) |

0 | 13.095 | 12.918 (1.4) | 13.012 (0.6) | 13.050 (0.3) | ||

1 | 7.7749 | 7.6602 (1.5) | 7.7207 (0.7) | 7.7457 (0.4) | ||

z = 1 | −1 | 28.412 | 28.076 (1.2) | 28.249 (0.6) | 28.322 (0.3) | |

0 | 16.568 | 16.384 (1.1) | 16.480 (0.5) | 16.519(0.3) | ||

1 | 9.4808 | 9.3593 (1.3) | 9.4229 (0.6) | 9.4492 (0.3) |

Fiber | ${E}_{L}^{f}$ | 388 | GPa |

${E}_{T}^{f}$ | 7.17 | GPa | |

${G}_{LT}^{f}$ | 6.79 | GPa | |

${G}_{TT}^{f}$ | 2.41 | GPa | |

${\nu}_{LT}^{f}$ | 0.230 | ||

${\nu}_{TT}^{f}$ | 0.486 | ||

Matrix | ${E}^{m}$ | 3.5 | GPa |

${G}^{m}$ | 1.3 | GPa | |

${\nu}^{m}$ | 0.35 |

**Table 6.**The vertical dimensionless displacement, ${\overline{u}}_{z}$, for material parameter $p=0$, for Example 2, for different thickness ratios, S, obtained using different grid sizes.

$\mathit{\theta}$ | S | Ref. [19] | Present | |||
---|---|---|---|---|---|---|

{21 × 21 × 11 × 11} | {41 × 41 × 11 × 11} | {61 × 61 × 11 × 11} | {61 × 61 × 21 × 21} | |||

${0}^{\circ}$ | 5 | 0.752 | 0.742 (1.3) | 0.746 (0.8) | 0.746 (0.7) | 0.750 (0.3) |

10 | 0.329 | 0.324 (1.5) | 0.327 (0.7) | 0.327 (0.6) | 0.328 (0.3) | |

20 | 0.209 | 0.203 (2.7) | 0.207 (0.8) | 0.208 (0.5) | 0.208 (0.3) | |

${90}^{\circ}$ | 5 | 0.576 | 0.570 (1.0) | 0.572 (0.7) | 0.572 (0.7) | 0.575 (0.2) |

10 | 0.27 | 0.267 (0.9) | 0.269 (0.5) | 0.269 (0.4) | 0.270 (0.1) | |

20 | 0.186 | 0.183 (1.6) | 0.185 (0.3) | 0.186 (0.1) | 0.186 (0.0) |

Ceramic | ${E}^{c}$ | 348.43 | GPa |

${\nu}^{c}$ | 0.24 | ||

${S}_{tns}^{c}$ | 60 | MPa | |

${S}_{cmp}^{c}$ | 344.5 | MPa | |

Metal | ${E}^{m}$ | 201.04 | GPa |

${\nu}^{m}$ | 0.3262 | ||

${S}_{tns}^{m}$ | 215 | MPa | |

${S}_{cmp}^{m}$ | 215 | MPa |

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**MDPI and ACS Style**

Kazemzadeh-Parsi, M.-J.; Chinesta, F.; Ammar, A. Proper Generalized Decomposition for Parametric Study and Material Distribution Design of Multi-Directional Functionally Graded Plates Based on 3D Elasticity Solution. *Materials* **2021**, *14*, 6660.
https://doi.org/10.3390/ma14216660

**AMA Style**

Kazemzadeh-Parsi M-J, Chinesta F, Ammar A. Proper Generalized Decomposition for Parametric Study and Material Distribution Design of Multi-Directional Functionally Graded Plates Based on 3D Elasticity Solution. *Materials*. 2021; 14(21):6660.
https://doi.org/10.3390/ma14216660

**Chicago/Turabian Style**

Kazemzadeh-Parsi, Mohammad-Javad, Francisco Chinesta, and Amine Ammar. 2021. "Proper Generalized Decomposition for Parametric Study and Material Distribution Design of Multi-Directional Functionally Graded Plates Based on 3D Elasticity Solution" *Materials* 14, no. 21: 6660.
https://doi.org/10.3390/ma14216660