A high-order FEM formulation for free and forced vibration analysis of a nonlocal nonlinear graded Timoshenko nanobeam b
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O R I G I NA L
M. Trabelssi · S. El-Borgi · M. I. Friswell
A high-order FEM formulation for free and forced vibration analysis of a nonlocal nonlinear graded Timoshenko nanobeam based on the weak form quadrature element method Received: 13 November 2019 / Accepted: 28 April 2020 © The Author(s) 2020
Abstract The purpose of this paper is to provide a high-order finite element method (FEM) formulation of nonlocal nonlinear nonlocal graded Timoshenko based on the weak form quadrature element method (WQEM). This formulation offers the advantages and flexibility of the FEM without its limiting low-order accuracy. The nanobeam theory accounts for the von Kármán geometric nonlinearity in addition to Eringen’s nonlocal constitutive models. For the sake of generality, a nonlinear foundation is included in the formulation. The proposed formulation generates high-order derivative terms that cannot be accounted for using regular first- or second-order interpolation functions. Hamilton’s principle is used to derive the variational statement which is discretized using WQEM. The results of a WQEM free vibration study are assessed using data obtained from a similar problem solved by the differential quadrature method (DQM). The study shows that WQEM can offer the same accuracy as DQM with a reduced computational cost. Currently the literature describes a small number of high-order numerical forced vibration problems, the majority of which are limited to DQM. To obtain forced vibration solutions using WQEM, the authors propose two different methods to obtain frequency response curves. The obtained results indicate that the frequency response curves generated by either method closely match their DQM counterparts obtained from the literature, and this is despite the low mesh density used for the WQEM systems. Keywords Functionally graded nanobeam · Nonlocal theory · Weak form quadrature element method (WQEM) · Free and forced vibration · Nonlinear von’Kármán strain · Frequency response curve 1 Introduction Nanobeams, nanoplates, nanoshells and other small-scale structural elements constitute the building blocks of micro- and nanoelectromechanical systems (MEMS and NEMS), actuators, sensors and atomic force microM. Trabelssi Applied Mechanics and Systems Research Laboratory, Tunisia Polytechnic School, University of Carthage, B.P. 743, La Marsa 2078, Tunisia M. Trabelssi Department of Mechanical Engineering, Tunis Higher National Engineering School, University of Tunis, 1008 Tunis, Tunisia S. El-Borgi (B) Mechanical Engineering Program, Texas A and M University at Qatar, Engineering Building, Education City, P.O. Box 23874, Doha, Qatar E-mail: [email protected] M. I. Friswell Swansea University, Bay Campus, Fabian Way, Swansea SA1 8EN, UK
M. Trabelssi et al.
scopes [1–3]. The choice of integrating small-scale components is related to exotic mechanical properties and size effects experimentally observed [4–8] at the nanoscale. While these size effects can be accurately captured and studied using molecular dynamics (MD)
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