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Nonlocal dynamic response analysis of functionally graded porous L-shape nanoplates resting on elastic foundation using finite element formulation

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Abstract

This paper proposes a finite element method (FEM) based on a nonlocal theory for the dynamic response analysis of the functionally graded porous (FGP) L-shape nanoplate lying on the elastic foundation (EF). The FGP nanoplate with uneven porosity distribution includes two parameters as the power-law index (\(k\)) and the porosity factor (\(\xi\)). The EF is Pasternak’s model with two parameters: the spring stiffness (\(k_{1}\)) and the shear layer stiffness (\(k_{2}\)). For the first time, the motion equation of FGP nanoplates is established using an eight-node rectangular element (Q8). Some numerical results in our work are compared with other published to verify accuracy and reliability. Moreover, the influence of geometric parameters, material properties on the vibration response of the FGP nanoplates resting on the EF is comprehensively investigated.

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  1. Faculty of Mechanical Engineering, Le Quy Don Technical University, Hanoi, Vietnam

    Trung Thanh Tran & Pham Binh Le

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Appendix 1

Appendix 1

The Lagrange interpolation in Eq. (25):

$$\begin{array}{*{20}l} {\aleph_{1} = \frac{1}{4}\left( {1 - r} \right)\left( {1 - s} \right)\left( { - r - s - 1} \right);{ }\aleph_{2} = \frac{1}{2}\left( {1 - r^{2} } \right)\left( {1 - s} \right);} \hfill \\ {\aleph_{3} = \left( {1 + r} \right)\left( {1 - s} \right)\left( { - r - s - 1} \right);{ } \aleph_{4} = \frac{1}{2}\left( {1 + r} \right)\left( {1 - s^{2} } \right);} \hfill \\ {\aleph_{5} = \frac{1}{4}\left( {1 + r} \right)\left( {1 + s} \right)\left( {r + s - 1} \right);{ } \aleph_{6} = \frac{1}{2}\left( {1 - r^{2} } \right)\left( {1 + s} \right);} \hfill \\ {\aleph_{7} = \frac{1}{4}\left( {1 - r} \right)\left( {1 + s} \right)\left( { - r + s - 1} \right);{ }\aleph_{8} = \frac{1}{2}\left( {1 - r} \right)\left( {1 - s^{2} } \right),} \hfill \\ \end{array}$$

with \(r, s\) are natural coordinates.

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Tran, T.T., Le, P.B. Nonlocal dynamic response analysis of functionally graded porous L-shape nanoplates resting on elastic foundation using finite element formulation. Engineering with Computers 39, 809–825 (2023). https://doi.org/10.1007/s00366-022-01679-6

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