# Quadratic Solid–Shell Finite Elements for Geometrically Nonlinear Analysis of Functionally Graded Material Plates

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## Abstract

**:**

## 1. Introduction

## 2. SHB15 and SHB20 Solid‒Shell Elements

#### 2.1. Element Reference Geometries

#### 2.2. Quadratic Approximation for the SHB Elements

#### 2.3. Strain Field and Gradient Operator

#### 2.4. Hu–Washizu Variational Principle

#### 2.5. Description of Functionally Graded Elastic Behavior

## 3. Nonlinear Benchmark Problems

_{1}× N

_{2}) × N

_{3}for the hexahedral SHB20 element, where N

_{1}is the number of elements along the length, N

_{2}is the number of elements along the width, and N

_{3}is the number of elements along the thickness direction. As to the prismatic SHB15 element, the mesh strategy consists of (N

_{1}× N

_{2}× 2) × N

_{3}, due to the in-plane subdivision of a rectangular element into two triangles.

#### 3.1. Cantilever Beam Sujected to End Shear Force

#### 3.2. Slit Annular Plate

#### 3.3. Clamped Square Plate under Pressure

#### 3.4. Hinged Cylindrical Roof

#### 3.5. Pull-Out of an Open-Ended Cylinder

#### 3.6. Pinched Hemispherical Shell

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Reference geometry of (

**a**) quadratic prismatic SHB15 element and (

**b**) quadratic hexahedral SHB20 element, and position of the associated integration points.

**Figure 4.**Volume fraction distribution of the ceramic phase as function of the power-law exponent n.

**Figure 6.**Load–deflection curves for the cantilever beam. (

**a**) Prismatic SHB15 element; (

**b**) hexahedral SHB20 element.

**Figure 8.**Load–deflection curves at the outer point A for the slit annular plate. (

**a**) Prismatic SHB15 element; (

**b**) hexahedral SHB20 element.

**Figure 10.**Load–deflection curves at the center point for the square plate. (

**a**) Prismatic SHB15 element; (

**b**) hexahedral SHB20 element.

**Figure 12.**Deflection at the central point A under concentrated force for the thick hinged roof. (

**a**) prismatic SHB15 element; (

**b**) hexahedral SHB20 element.

**Figure 13.**Deflection at the central point A under concentrated force for the thin hinged roof. (

**a**) prismatic SHB15 element; (

**b**) hexahedral SHB20 element.

**Figure 14.**Pull-out of an open-ended cylinder: (

**a**) geometry and (

**b**) undeformed and deformed configurations.

**Figure 15.**Radial displacement at point A under concentrated force for the open-ended cylinder. (

**a**) Prismatic SHB15 element; (

**b**) hexahedral SHB20 element.

**Figure 16.**Radial displacement at point B under concentrated force for the open-ended cylinder. (

**a**) prismatic SHB15 element; (

**b**) hexahedral SHB20 element.

**Figure 17.**Radial displacement at point C under concentrated force for the open-ended cylinder. (

**a**) prismatic SHB15 element; (

**b**) hexahedral SHB20 element.

**Figure 19.**Load–displacement curves at point A for the pinched hemispherical shell, obtained with a mixture of prismatic SHB15 and hexahedral SHB20 elements.

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**MDPI and ACS Style**

Chalal, H.; Abed-Meraim, F.
Quadratic Solid–Shell Finite Elements for Geometrically Nonlinear Analysis of Functionally Graded Material Plates. *Materials* **2018**, *11*, 1046.
https://doi.org/10.3390/ma11061046

**AMA Style**

Chalal H, Abed-Meraim F.
Quadratic Solid–Shell Finite Elements for Geometrically Nonlinear Analysis of Functionally Graded Material Plates. *Materials*. 2018; 11(6):1046.
https://doi.org/10.3390/ma11061046

**Chicago/Turabian Style**

Chalal, Hocine, and Farid Abed-Meraim.
2018. "Quadratic Solid–Shell Finite Elements for Geometrically Nonlinear Analysis of Functionally Graded Material Plates" *Materials* 11, no. 6: 1046.
https://doi.org/10.3390/ma11061046