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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 |
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Vol. 15 (2020), No. 3, 405–434
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Abstract |
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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. |
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Keywords
FG nanobeam, Hermite–Ritz method, bi-Helmholtz function,
porosity, Winkler–Pasternak elastic foundation, vibration
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Milestones
Received: 17 March 2020
Revised: 10 April 2020
Accepted: 23 April 2020
Published: 12 July 2020
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Authors |
| Subrat Kumar Jena | |
| Department of Mathematics National Institute of Technology Rourkela Unit 1 Rourkela India |
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| Snehashish Chakraverty | |
| Department of Mathematics National Institute of Technology Rourkela Unit 1 Rourkela India |
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| Mohammad Malikan | |
| Department of Mechanics of Materials
and Structures Gdansk University of Technology ul. G. Narutowicza 11/12 Gdansk Poland |
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| 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 |
