Size-dependent geometrically nonlinear free vibration analysis of fractional viscoelastic nanobeams based on the nonlocal elasticity theory |
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| Institution: | 1. Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran;2. Department of Mechanical Engineering, Lahijan Branch, Islamic Azad University, P.O. Box 1616, Lahijan, Iran;1. School of Mechanical Engineering, Iran University of Science and Technology, Narmak, 16846-13114, Tehran, Islamic Republic of Iran;2. School of Engineering, Damghan University, 36716-41167, Damghan, Islamic Republic of Iran;3. Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA;4. Center of Excellence in Railway Transportation, Iran University of Science and Technology, 16846-13114, Narmak, Tehran, Islamic Republic of Iran;1. Mathematical Institute of the SASA, Kneza Mihaila 36, 11001 Belgrade, Serbia;2. School of Engineering, University of the West of Scotland, Paisley PA12BE, UK;3. College of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, UK;1. School of Mechanical Engineering, University of Adelaide, South Australia, 5005, Australia;2. Department of Mechanical and Construction Engineering, Northumbria University, Newcastle upon Tyne, NE1 8ST, UK |
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| Abstract: | In recent decades, mathematical modeling and engineering applications of fractional-order calculus have been extensively utilized to provide ef?cient simulation tools in the field of solid mechanics. In this paper, a nonlinear fractional nonlocal Euler–Bernoulli beam model is established using the concept of fractional derivative and nonlocal elasticity theory to investigate the size-dependent geometrically nonlinear free vibration of fractional viscoelastic nanobeams. The non-classical fractional integro-differential Euler–Bernoulli beam model contains the nonlocal parameter, viscoelasticity coefficient and order of the fractional derivative to interpret the size effect, viscoelastic material and fractional behavior in the nanoscale fractional viscoelastic structures, respectively. In the solution procedure, the Galerkin method is employed to reduce the fractional integro-partial differential governing equation to a fractional ordinary differential equation in the time domain. Afterwards, the predictor–corrector method is used to solve the nonlinear fractional time-dependent equation. Finally, the influences of nonlocal parameter, order of fractional derivative and viscoelasticity coefficient on the nonlinear time response of fractional viscoelastic nanobeams are discussed in detail. Moreover, comparisons are made between the time responses of linear and nonlinear models. |
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| Keywords: | Fractional viscoelastic nanobeams Nonlocal elasticity theory Geometrically nonlinear free vibration Size effect Time response |
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