Vol. 15, No. 3, 2020

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Implementation of Hermite–Ritz method and Navier's technique for vibration of functionally graded porous nanobeam embedded in Winkler–Pasternak elastic foundation using bi-Helmholtz nonlocal elasticity

Subrat Kumar Jena, Snehashish Chakraverty, Mohammad Malikan and Hamid Mohammad-Sedighi

Vol. 15 (2020), No. 3, 405–434
Abstract

The vibration characteristics of functionally graded porous nanobeam embedded in an elastic substrate of Winkler–Pasternak type are investigated. Classical beam theory or Euler–Bernoulli beam theory has been incorporated to address the displacement of the FG nanobeam. bi-Helmholtz type of nonlocal elasticity is being used to capture the small scale effect of the FG nanobeam. Further, the nanobeam is assumed to have porosity, distributed evenly along the thickness throughout the cross-section. Young’s modulus and mass density of the nanobeam are considered to vary along the thickness from ceramic to metal constituents in accordance with power-law exponent model. A numerically efficient method, namely the Hermite–Ritz method, is incorporated to compute the natural frequencies of hinged-hinged, clamped-hinged, and clamped-clamped boundary conditions. A closed-form solution is also obtained for hinged-hinged (HH) boundary condition by employing Navier’s technique. The advantages of using Hermite polynomials as shape functions are orthogonality, a large domain that makes the method more computationally efficient and avoids ill-conditioning for higher values of polynomials. Additionally, the present results are validated with other existing results in special cases demonstrating excellent agreement. A comprehensive study has been carried out to justify the effectiveness or convergence of the present model or method. Likewise, impacts of various scaling parameters such as Helmholtz and bi-Helmholtz types of nonlocal elasticity, porosity volume fraction index, power-law exponent, and elastic foundation on frequency parameters have been investigated.

Keywords
FG nanobeam, Hermite–Ritz method, bi-Helmholtz function, porosity, Winkler–Pasternak elastic foundation, vibration
Milestones
Received: 17 March 2020
Revised: 10 April 2020
Accepted: 23 April 2020
Published: 12 July 2020
Authors
Subrat Kumar Jena
Department of Mathematics
National Institute of Technology Rourkela
Unit 1
Rourkela
India
Snehashish Chakraverty
Department of Mathematics
National Institute of Technology Rourkela
Unit 1
Rourkela
India
Mohammad Malikan
Department of Mechanics of Materials and Structures
Gdansk University of Technology
ul. G. Narutowicza 11/12
Gdansk
Poland
Hamid Mohammad-Sedighi
Mechanical Engineering Department
Faculty of Engineering
Shahid Chamran University of Ahvaz
Ahvaz
Iran
Drilling Center of Excellence and Research Center
Shahid Chamran University of Ahvaz
Ahvaz
Iran