Nonlocal strain gradient finite element analysis of nanobeams using two-variable trigonometric shear deformation theory
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ORIGINAL ARTICLE
Nonlocal strain gradient finite element analysis of nanobeams using two‑variable trigonometric shear deformation theory Tarek Merzouki1 · Mohammed Sid Ahmed Houari2 · Mohamed Haboussi3 · Aicha Bessaim2 · Manickam Ganapathi4 Received: 12 May 2020 / Accepted: 23 August 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020
Abstract In the present paper, a new trigonometric two-variable shear deformation beam nonlocal strain gradient theory is developed and applied to investigate the combined effects of nonlocal stress and strain gradient on the bending, buckling and free vibration analysis of nanobeams. The model introduces a nonlocal stress field parameter and a length scale parameter to capture the size effect. The governing equations derived are solved employing finite element method using a 3-nodes beam element, developed for this purpose. The predictive capability of the proposed model is shown through illustrative examples for bending, buckling and free vibration of nanobeams. Comparisons with other higher-order shear deformation beam theory are also performed to validate its numerical implementation and assess its accuracy within the nonlocal context. Keywords Nonlocal strain gradient theory · Variational formulation · Finite element method · Static analysis · Free vibration · Elastic buckling
1 Introduction Nowadays, nanostructures such as nanorods, nanobeams and nanoplates are receiving a great attention in nanoscience and nanotechnology, due to their extraordinary mechanical, thermal, electrical, magnetic, and other properties [1–5]. Examples of applications and devices related to such nanostructures are oscillators [6], clocks [7], sensors [8–10], atomic force microscopy [11, 12], nano/micro electro-mechanical systems (NEMS/MEMS) [13, 14] and nano actuators [15, 16]. In nanostructures, the size effect is no longer negligible and becomes rather important. It is then necessary to take it account into the design of applications, such as * Tarek Merzouki [email protected] 1
LISV, University of Versailles Saint-Quentin, 10‑12 Avenue de l’Europe, 78140 Vélizy‑Villacoublay, France
2
Laboratoire d’Etude des Structures et de Mécanique des Matériaux, University Mustapha Stambouli of Mascara, Mascara, Algeria
3
Laboratoire des Sciences des Procédés et des Matériaux (LSPM), CNRS UPR 3407, Université Paris 13, Sorbonne-Paris-Cité, 93430 Villetaneuse, France
4
School of Mechanical Engineering, VIT University, Vellore 632 014, India
those mentioned above. There have been many theoretical and experimental investigations for better understanding and designing the mechanical and physical behavior of such small-scaled structures [17, 18]. It is known that classical continuum mechanics is a local theory that is sizeindependent. So, it is not really appropriate for small-scaled structures as it does not allow to capturing the size effect in such small structures. To overcome this limitation, nonclassical continuum theories are developed. Whether being of integral or gra
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