Free Vibration Analysis of Functionally Graded FG Nano-Plates with Porosities MERDACI Slimane 1,a* , HADJ MOSTEFA Adda 2,b , BOUTALEB Sabrina 3 , HELLAL Hadjira 3 1 Structures and Advanced Materials in Civil Engineering and Public Works Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering and Public Works Department, Algeria. 2 Industrial Engineering and Sustainable Development Laboratory, University Center of Rélizane, Institute of Science & Technology, Civil Engineering Department, Algeria. 3 Laboratory of Materials and Hydrology, University of Sidi Bel Abbes, Faculty of Technology, Department of Civil Engineering and Public Works, BP 89 Cité Ben M’hidi, 22000 Sidi Bel Abbes, Algeria. a *slimanem2016@gmail.com, b hadj9510@gmail.com Keywords: nanoplates FG; shear deformation theory; non-local elasticity; free vibration; porosity. Abstract. This study presents the analytical solutions of free vibration analysis of simply supported nano-plate FG porous using nonlocal high order shear deformation plate theory. This theory contains four unknowns without the use of shear correction factors unlike the others. The objective of this article is to develop a model to use the function f (z) on vibration and the natural frequencies of functionally graded nano-plates nonlocal to study the effect of the various parameters. The validity of the theory is shown by comparing the present results with obtained with those reported in the literature. The effects of various parameters are all discussed. Introduction Functionally graduated materials (FGM) are a new class of composite materials whose microstructure and composition vary gradually and continuously with position so as to optimize the mechanical and thermal performance of the structure they constitute. They are considered as intelligent materials whose desired functions are integrated, from the design, at the very heart of the material. At each interface, the material is chosen according to specific applications and environmental loads. These materials have multiple advantages which can make them attractive from the point of view of their application potential. It can be an improvement in rigidity, resistance to fatigue, resistance to corrosion or thermal conductivity in addition to having a gradation of properties thus making it possible to increase or modulate performances. Such as reducing local constraints or even improving heat transfer [1-3]. The concept of functionally graded material was first considered in Japan in 1984 during a space plane project [4]. The manufacture of FGMs, micro-porosities or voids can occur in the materials during the sintering process. This is due to the large difference in solidification temperature between the material constituents [5]. The porosities that occur within FGM specimens made by a sequential multi-step infiltration technique [6]. Therefore, it is important to take into account the effect of porosity in the design of the static and bending plates made of functionally graded materials FGM [7-12] and sandwich plates FGM using of high order shear-deformation theory presented by [13- 15]. Consequently, studies devoted to understanding the dynamic and free vibrations of the plate made of functionally graded materials FGM have been given more and more attention in recent years [16-22]. The investigated of the nano-plate made of functionally graded materials behavior dependent with small-scale effects by [23-26]. Analytical solutions for the analysis of free vibrations, which are made of materials and supported by elastic Winkler or Pasternak foundations, are presented in Ref. [27]. One of the well-known theories of continuum mechanics that includes small-scale effects with good precision is the non-local theory [28-29]. The Various non-local theories, the differential Journal of Nano Research Submitted: 2020-04-06 ISSN: 1661-9897, Vol. 64, pp 61-74 Revised: 2020-08-18 doi:10.4028/www.scientific.net/JNanoR.64.61 Accepted: 2020-09-04 © 2020 Trans Tech Publications Ltd, Switzerland Online: 2020-11-30 All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications Ltd, www.scientific.net. (#547830056, University of Sidi-Bel-Abbes, Algeria, Algeria-04/12/20,16:50:57)