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Bending analysis of two different types of functionally graded material porous sandwich plates

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Abstract

In this paper, a new functionally graded material (FGM) porous sandwich plate analysis model considering pores with only four variables is proposed. It is applied to the nonlinear bending of two different types of FGM porous sandwich plates under mechanical loading, and the bending of two different types of FGM porous sandwich plates is compared. The governing equations are obtained through the principle of virtual work. The simple supported FGM porous sandwich plates are solved using the Navier method. The obtained numerical results are compared with the results in the literature to verify the accuracy of the theory in this paper. Finally, the effects of volume fraction index, porosity, layer thickness ratio, side-thickness ratio and symmetry on the mechanical bending properties of FGM porous sandwich plates are studied in detail. By comparing the bending characteristics of two types of FGM porous sandwich plates under different influencing factors, it is found that the bending characteristics of different FGM porous sandwich plate types are also very different. This has great guiding significance for the design of FGM porous sandwich plates.

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Acknowledgments

This research is supported by Natural Science Foundation of China (11862007, 52265020), Science and Technology Projects of Jiangxi Education Department of China (GJJ211322).

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Authors and Affiliations

  1. College of Mechanical and Electronic Engineering, Jingdezhen Ceramic University, Jingdezhen, 333001, Jiangxi, China

    Zhicheng Huang, Mengna Han & Xingguo Wang

  2. Department of Mechanical Engineering, Tsinghua University, Beijing, 100084, China

    Fulei Chu

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  1. Zhicheng Huang
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Appendices

Appendix 1

The displacement (δu1, δv1, δw1, δw2) can be given as:

$$\begin{gathered} A_{11} \frac{{\partial^{2} u_{1} }}{{\partial x^{2} }} + A_{12} \frac{{\partial^{2} v_{1} }}{\partial x\partial y} - A_{11}^{1} \frac{{\partial^{3} w_{1} }}{{\partial x^{3} }} - A_{12}^{1} \frac{{\partial^{3} w_{1} }}{{\partial x\partial y^{2} }} - B_{11}^{1} \frac{{\partial^{3} w_{2} }}{{\partial x^{3} }} - B_{11}^{1} \frac{{\partial^{3} w_{2} }}{{\partial x\partial y^{2} }} \hfill \\ + A_{66} \frac{{\partial^{2} u_{1} }}{{\partial y^{2} }} + A_{66} \frac{{\partial^{2} v_{1} }}{\partial x\partial y} - 2A_{66}^{1} \frac{{\partial^{3} w_{1} }}{{\partial x\partial y^{2} }} - 2B_{66}^{1} \frac{{\partial^{3} w_{2} }}{{\partial x\partial y^{2} }} = 0 \hfill \\ A_{12} \frac{{\partial^{2} u_{1} }}{\partial x\partial y} + A_{22} \frac{{\partial^{2} v_{1} }}{{\partial y^{2} }} - A_{12}^{1} \frac{{\partial^{3} w_{1} }}{{\partial y\partial x^{2} }} - A_{33}^{1} \frac{{\partial^{3} w_{1} }}{{\partial y^{3} }} - B_{12}^{1} \frac{{\partial^{3} w_{2} }}{{\partial y\partial x^{2} }} - B_{22}^{1} \frac{{\partial^{3} w_{2} }}{{\partial y^{3} }} \hfill \\ + A_{66} \frac{{\partial^{2} u_{1} }}{\partial x\partial y} + A_{66} \frac{{\partial^{2} v_{1} }}{{\partial x^{2} }} - 2A_{66}^{1} \frac{{\partial^{3} w_{1} }}{{\partial y\partial x^{2} }} - 2B_{66}^{1} \frac{{\partial^{3} w_{2} }}{{\partial y\partial x^{2} }} = 0 \hfill \\ A_{11}^{1} \frac{{\partial^{3} u_{1} }}{{\partial x^{3} }} + A_{12}^{1} \frac{{\partial^{3} v_{1} }}{{\partial y\partial x^{2} }} - B_{12} \frac{{\partial^{4} w_{1} }}{{\partial x^{4} }} - B_{22} \frac{{\partial^{4} w_{1} }}{{\partial x^{2} \partial y^{2} }} - C_{11}^{1} \frac{{\partial^{4} w_{2} }}{{\partial x^{4} }} - C_{11}^{1} \frac{{\partial^{4} w_{2} }}{{\partial x^{2} \partial y^{2} }} + A_{12}^{1} \frac{{\partial^{3} u_{1} }}{{\partial x\partial y^{2} }} + A_{22}^{1} \frac{{\partial^{3} v_{1} }}{{\partial y^{3} }} \hfill \\ - B_{12} \frac{{\partial^{4} w_{1} }}{{\partial x^{2} \partial y^{2} }} - B_{22} \frac{{\partial^{4} w_{1} }}{{\partial y^{4} }} - C_{11}^{1} \frac{{\partial^{4} w_{2} }}{{\partial x^{2} \partial y^{2} }} - C_{12}^{1} \frac{{\partial^{4} w_{2} }}{{\partial y^{4} }} + 2A_{66}^{1} \frac{{\partial^{3} u_{1} }}{{\partial x\partial y^{2} }} + 2A_{66}^{1} \frac{{\partial^{3} v_{1} }}{{\partial y\partial x^{2} }} \hfill \\ - 4B_{66} \frac{{\partial^{4} w_{1} }}{{\partial x^{2} \partial y^{2} }} - 4C_{66}^{1} \frac{{\partial^{4} w_{2} }}{{\partial x^{2} \partial y^{2} }} = - q \hfill \\ B_{11}^{1} \frac{{\partial^{3} u_{1} }}{{\partial x^{3} }} + B_{12}^{1} \frac{{\partial^{3} v_{1} }}{{\partial y\partial x^{2} }} - C_{11}^{1} \frac{{\partial^{4} w_{1} }}{{\partial x^{4} }} - C_{12}^{1} \frac{{\partial^{4} w_{1} }}{{\partial x^{2} \partial y^{2} }} - C_{11} \frac{{\partial^{4} w_{2} }}{{\partial x^{4} }} - C_{12} \frac{{\partial^{4} w_{2} }}{{\partial x^{2} \partial y^{2} }} + B_{12}^{1} \frac{{\partial^{3} u_{1} }}{{\partial x\partial y^{2} }} + B_{22}^{1} \frac{{\partial^{3} v_{1} }}{{\partial y^{3} }} \hfill \\ - C_{12}^{1} \frac{{\partial^{4} w_{1} }}{{\partial x^{2} \partial y^{2} }} - C_{22}^{1} \frac{{\partial^{4} w_{1} }}{{\partial y^{4} }} - C_{12} \frac{{\partial^{4} w_{2} }}{{\partial x^{2} \partial y^{2} }} - C_{22} \frac{{\partial^{4} w_{2} }}{{\partial y^{4} }} + 2B_{66}^{1} \frac{{\partial^{3} u_{1} }}{{\partial x\partial y^{2} }} + 2B_{66}^{1} \frac{{\partial^{3} v_{1} }}{{\partial y\partial x^{2} }} \hfill \\ - 4C_{66}^{1} \frac{{\partial^{4} w_{1} }}{{\partial x^{2} \partial y^{2} }} - 4C_{66} \frac{{\partial^{4} w_{2} }}{{\partial x^{2} \partial y^{2} }} + E_{44} \frac{{\partial^{3} w_{2} }}{{\partial y^{3} }} + E_{55} \frac{{\partial^{3} w_{2} }}{{\partial x^{3} }} = - q \hfill \\ \end{gathered}$$

Appendix 2

The coefficient matrix \(\Gamma\) is given by:

$$\begin{gathered} \Gamma_{11} = A_{11} mm^{2} + A_{66} nn^{2} \hfill \\ \Gamma_{12} = mm \times nn\left( {A_{12} + A_{66} } \right) \hfill \\ \Gamma_{13} = - mm\left[ {A_{11}^{1} mm^{2} + \left( {A_{12}^{1} + 2A_{66}^{1} } \right)nn^{2} } \right] \hfill \\ \Gamma_{14} = - mm\left[ {B_{11}^{1} mm^{2} + \left( {B_{12}^{1} + 2B_{66}^{1} } \right)nn^{2} } \right] \hfill \\ \Gamma_{22} = A_{66} mm^{2} + A_{66} nn^{2} \hfill \\ \Gamma_{23} = - nn\left[ {A_{22}^{1} nn^{2} + \left( {A_{12}^{1} + 2A_{66}^{1} } \right)mm^{2} } \right] \hfill \\ \Gamma_{24} = - nn\left[ {B_{22}^{1} nn^{2} + \left( {B_{12}^{1} + 2B_{66}^{1} } \right)mm^{2} } \right] \hfill \\ \Gamma_{33} = B_{11} mm^{4} + 2\left( {B_{12} + 2B_{66} } \right)mm^{2} nn^{2} + B_{22} nn^{4} \hfill \\ \Gamma_{34} = C_{11}^{1} mm^{4} + 2\left( {C_{12}^{1} + 2C_{66}^{1} } \right)mm^{2} nn^{2} + C_{22}^{1} nn^{4} \hfill \\ \Gamma_{44} = C_{11} mm^{4} + 2\left( {C_{12} + 2C_{66} } \right)mm^{2} nn^{2} + C_{22} nn^{4} + E_{44} nn^{4} + E_{55} mm^{4} \hfill \\ \end{gathered}$$

The generalized force vector \(\left[ F \right] = \left[ {\begin{array}{*{20}c} {F_{1} } & {F_{2} } & {F_{3} } & {F_{4} } \\ \end{array} } \right]^{T}\) can be written as:

$$\begin{array}{*{20}l} {F_{1} = 0} \hfill \\ {F_{2} = 0} \hfill \\ {F_{3} = q_{0} } \hfill \\ {F_{4} = q_{0} } \hfill \\ \end{array}$$

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Huang, Z., Han, M., Wang, X. et al. Bending analysis of two different types of functionally graded material porous sandwich plates. Arch Appl Mech 93, 3071–3091 (2023). https://doi.org/10.1007/s00419-023-02425-0

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