Vibration analysis of stress-driven nonlocal integral model of viscoelastic axially FG nanobeams
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Vibration analysis of stress-driven nonlocal integral model of viscoelastic axially FG nanobeams Mahmood Fakher1,a , Shahin Behdad1, Shahrokh Hosseini-Hashemi1,2 1 School of Mechanical Engineering, Iran University of Science and Technology, 16846-13114 Narmak,
Tehran, Iran
2 Center of Excellence in Railway Transportation, Iran University of Science and Technology, 16846-
13114 Narmak, Tehran, Iran Received: 13 July 2020 / Accepted: 5 November 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Finite element method (FEM) and generalized differential quadrature method (GDQM) are developed for damping vibration analysis of viscoelastic axially functionally graded (VAFG) nanobeams within the framework of stress-driven nonlocal integral model (SDM). The equivalent differential law of SDM and two constitutive non-classic boundary conditions (CBCs) are utilized to construct FE and GDQ models. In FEM, three different types of beam elements are derived: two different types for the both ends of the nanobeam and another type for elements located in the middle of nanobeam. Convergence and accuracy of the present FEM and GDQM are evaluated, and it is shown that the present results and formulations are efficient and reliable. Also, in the damping vibration analysis of VAFG nanobeams by SDM, variations of the mechanical properties are considered as a power-law function. After validation of present results and mathematical modeling, various benchmark results are presented to determine the influences of several parameters, such as nonlocal SDM parameter, FG index and damping factor on both parts of size-dependent complex frequencies of VAFG nanobeams with different boundary conditions.
List of symbols El , Er ρl , ρr η l , ηr E(x) ρ(x) η(x) c L b H x, y, z eff
Young’s modulus in left and right side of nanobeam Mass density in left and right side of nanobeam Damping factor in left and right side of nanobeam Variable Young’s modulus Variable mass density Variable damping factor Non-dimensional damping parameter Length of nanobeam Width of nanobeam Thickness of nanobeam Cartesian coordinate system Effective material properties of nanobeam
a e-mail: [email protected] (corresponding author)
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l , r Vl (x) np εx x σx x k μ I M A w(x, t) T (t) t m1 ω [M] [K s] [D] ng xi (r ) Hi j Ub , Ud Z [S] Z¯ [I ] ζ
Eur. Phys. J. Plus
(2020) 135:905
Material properties in left and right side of nanobeam Volume fraction of material in left side of nanobeam FG index Axial strain Normal stress Nonlocal parameter Non-dimensional nonlocal parameter Moment of inertia of the cross section Bending moment Area of the cross section Transverse displacement Time response function Time Mass distribution per length Vibration frequency Mass matrix Strain stiffness matrix Damping matrix Number of grid points Location of grid points Hermite shape functions Boundary and domain grid points State vector State matrix Eigenvectors Identity matrix Ratio
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