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Strain-based finite element formulation for the analysis of functionally graded plates

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Abstract

This work introduces a novel four-node quadrilateral finite element based on the strain approach and the first-order shear deformation theory for static and free vibration responses of functionally graded (FG) material plates. Material properties of the plate are assumed to be graded across the thickness direction by using a simple power law distribution of the volume fractions constituents. The developed element possesses five essential degrees of freedom per node. This element is obtained by the superposition of two strain-based elements where the first is a membrane with two degrees of freedom per node and the second is a Reissner–Mindlin plate that has three degrees of freedom per node. The displacements field of the proposed element which contains higher-order terms is based on assumed strain functions satisfying compatibility equations. The performance of the suggested element is evaluated through several tests and the obtained results are compared with available solutions from the literature. The results of the present element have proved excellent accuracy and efficiency in predicting bending and free vibration of FG plates.

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Authors and Affiliations

  1. Department of Civil Engineering, University Center of Tipaza, Tipaza, Algeria

    Abderahim Belounar

  2. NMISSI Laboratory, Faculty of Science and Technology, Biskra University, Biskra, Algeria

    Abderahim Belounar, Mohamed Nabil Houhou & Lahcene Fortas

  3. Department of Hydrocarbons and Renewable Energies, Adrar University, Adrar, Algeria

    Faiçal Boussem

  4. Laboratoire de Génie Energétique Et Matériaux, Biskra University, Biskra, Algeria

    Abdelouahab Tati

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  1. Abderahim Belounar
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  2. Faiçal Boussem
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Appendix

Appendix

For membrane behaviour, the three strains (εx, εy and γxy) given by Eq. (30) satisfy the following compatibility equation

$$\frac{{\partial^{2} \varepsilon_{x} }}{{\partial y^{2} }} + \frac{{\partial^{2} \varepsilon_{y} }}{{\partial x^{2} }} - \frac{{\partial^{2} \gamma_{xy} }}{\partial x\partial y} = 0.$$
(1a)

For Reissner–Mindlin plate theory, the curvatures (κx, κy and κxy) and the transverse shear strains (γxz and γyz) given in Eqs. (34)–(35) satisfy the following compatibility equations:

$$\frac{{\partial^{2} \kappa_{x} }}{{\partial y^{2} }} + \frac{{\partial^{2} \kappa_{y} }}{{\partial x^{2} }} = \frac{{\partial^{2} \kappa_{xy} }}{\partial x\partial y};$$
$$\frac{{\partial^{2} \gamma_{xz} }}{\partial x\partial y} - \frac{{\partial^{2} \gamma_{yz} }}{{\partial x^{2} }} + \frac{{\partial \kappa_{xy} }}{\partial x} = 2\frac{{\partial \kappa_{x} }}{\partial y};$$
$$\frac{{\partial^{2} \gamma_{yz} }}{\partial x\partial y} - \frac{{\partial^{2} \gamma_{xz} }}{{\partial y^{2} }} + \frac{{\partial \kappa_{xy} }}{\partial y} = 2\frac{{\partial \kappa_{y} }}{\partial x}.$$
(2a)

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Belounar, A., Boussem, F., Houhou, M.N. et al. Strain-based finite element formulation for the analysis of functionally graded plates. Arch Appl Mech 92, 2061–2079 (2022). https://doi.org/10.1007/s00419-022-02160-y

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