A geometrically nonlinear size-dependent hypothesis for porous functionally graded micro-plate
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ORIGINAL ARTICLE
A geometrically nonlinear size‑dependent hypothesis for porous functionally graded micro‑plate Cuong Le Thanh1 · Trong Nghia Nguyen1 · Truong Huu Vu2 · Samir Khatir2 · Magd Abdel Wahab3,4 Received: 3 July 2020 / Accepted: 19 August 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020
Abstract The static bending behavior of porous functionally graded (PFG) micro-plate under the geometrically nonlinear analysis is studied in this article. A small-scale nonlinear solution is established using the Von-Kármán hypothesis and the modified couple stress theory (MCST). To obtain the deflection of the plate, the Reddy higher-order plate theory coupled with isogeometric analysis (IGA) is utilized. The distribution of porosities is assumed to be even and uneven across the plate’s thickness and the effective material properties of porous functionally graded micro-plate are calculated using the refined rule-of-mixture hypothesis. The influence of power index, porosity parameter and material length scale parameter on the nonlinear behaviors of static bending of porous FGM micro-plates are also investigated using several numerical examples. Keywords Porous functionally graded · Modified couple stress theory · Isogeometric analysis · Length scale parameter · Nonlinear analysis
1 Introduction When the size of structures is down to micro order, the conventional mechanics theories cannot accurately predict their responses under thermal and mechanical loads [1, 2]. To fill in this gap, various higher-order continuum theories were constructed by Toupin [3], Mindlin [4], Koiter [5] and Fleck et al. [1]. Recently, many micro/nano solution was established based on higher-order continuum theories including nonlocal strain gradient solution for vibration response of post buckling of laminated FG-GPRLC micro/nanobeam [6]. Sahnami et al. [7] also extended their solution for large amplitude vibration of micro/nano porous-beams, and nonlocal size effect post-buckling solution for FGM micro-sized * Magd Abdel Wahab [email protected] 1
Faculty of Civil Engineering, Ho Chi Minh City Open University, Ho Chi Minh City, Vietnam
2
Department of Electrical Energy, Metals, Mechanical Constructions, and Systems, Faculty of Engineering and Architecture, Ghent University, 9000 Gent, Belgium
3
Division of Computational Mechanics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
4
Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam
shell. Yang et al. [8] and Wu et al. [9] developed a sizedependent FG-GPRLC nano-beam using nonlocal strain gradient theory. The size-dependent solutions on the basic of strain gradient theory can be found in Refs. [10–12]. Recently, Yang et al. [13] proposed a higher-order elastic theory named the modified couple stress theory (MCST) that could predict the behavior of micro-structures using one material length scale parameter. Therefore, the MCST is simpler than other theories in predicting the small size effect. Based on MCST, a number of
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