Dynamics of nonlocal thick nano-bars
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ORIGINAL ARTICLE
Dynamics of nonlocal thick nano‑bars S. Ali Faghidian1 · Hamid Mohammad‑Sedighi2,3 Received: 25 September 2020 / Accepted: 3 November 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020
Abstract The thick bar model, accounting for the lateral deformation, shear stiffness, and lateral inertia effect, is the most comprehensive structural theory to study the axial deformation of carbon nanotubes. Physically motivated definition of the axial force field and associated higher order boundary conditions are determined applying a consistent variational framework. The effects of long-range interactions are suitably realized in the framework of the nonlocal integral elasticity. The integral convolutions of the nonlocal constitutive law are determined and suitably resorted with the equivalent nonlocal differential model equipped with non-standard boundary conditions. Preceding contributions on the elastodynamic analysis of the elastic thick bar are, therefore, amended by properly taking into account the higher order and non-standard boundary conditions. The established size-dependent thick bar model is demonstrated to be exempt from the inherent drawbacks of the nonlocal differential formulation and leads to well-posed elastodynamic problems. The wave desperation response and free vibrational behavior of elastic thick bars with kinematic constraints of nano-mechanics interest are rigorously investigated by making recourse to a viable solution approach. New numerical benchmarks are detected for the elastodynamic response of nonlocal thick nano-bars. A consistent approach for nanoscopic study of the field quantities in the nonlocal mechanics is proposed that is capable of properly confirming the smaller-is-softer phenomenon. Keywords Thick nano-bar · Nonlocal integral elasticity · Wave dispersion · Vibrational behavior · Analytical modeling
1 Introduction Nano-structured materials, like carbon nanotubes (CNTs), have found various conceivable applications in the groundbreaking fields of nano-engineering in consequence of the significant material properties at micro- and nano-scales [1–4]. When the structural dimensions are comparable with the internal length-scales of the medium of interest, the validity of the classical theory of elasticity ceases to hold. To adequately describe the scale-effect phenomena, generalized elasticity theories equipped with intrinsic lengthscales are introduced in the literature. Implementation of the * S. Ali Faghidian [email protected]; [email protected] 1
Department of Mechanical Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran
2
Mechanical Engineering Department, Faculty of Engineering, Shahid Chamran University of Ahvaz, Ahvaz, Iran
3
Drilling Center of Excellence and Research Center, Shahid Chamran University of Ahvaz, Ahvaz, Iran
generalized elasticity theories can enhance the description of the physical behavior of media with nano-structural feature, and accordingly has stimulated a great deal of interest
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