On the vibration of nanobeams with consistent two-phase nonlocal strain gradient theory: exact solution and integral non
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ORIGINAL ARTICLE
On the vibration of nanobeams with consistent two‑phase nonlocal strain gradient theory: exact solution and integral nonlocal finite‑element model Mahmood Fakher1 · Shahrokh Hosseini‑Hashemi1,2 Received: 6 August 2020 / Accepted: 23 October 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020
Abstract Recently, it has been proved that the common nonlocal strain gradient theory has inconsistence behaviors. The order of the differential nonlocal strain gradient governing equations is less than the number of all mandatory boundary conditions, and therefore, there is no solution for these differential equations. Given these, for the first time, transverse vibrations of nanobeams are analyzed within the framework of the two-phase local/nonlocal strain gradient (LNSG) theory, and to this aim, the exact solution as well as finite-element model are presented. To achieve the exact solution, the governing differential equations of LNSG nanobeams are derived by transformation of the basic integral form of the LNSG to its equal differential form. Furthermore, on the basis of the integral LNSG, a shear-locking-free finite-element (FE) model of the LNSG Timoshenko beams is constructed by introducing a new efficient higher order beam element with simple shape functions which can consider the influence of strains gradient as well as maintain the shear-locking-free property. Agreement between the exact results obtained from the differential LNSG and those of the FE model, integral LNSG, reveals that the LNSG is consistent and can be utilized instead of the common nonlocal strain gradient elasticity theory. Keywords Two-phase local/nonlocal strain gradient · Exact solution · Finite-element method · Euler–Bernoulli · Timoshenko · Shear-locking · Vibration
1 Introduction Due to paradoxical behaviors [1–4] of the common differential nonlocal elasticity [5–8] which has been widely utilized by researchers to consider the size effects for studying the mechanics of nano structures [9–15], other size-dependent elasticity theories such as stress-driven integral nonlocal [16, 17] and two-phase local/nonlocal have been recently attracted the attentions of the nano-mechanic researchers. Although, some valuable efforts [18, 19] have been made to resolve the weakness of the differential nonlocal, it has been indicated [20, 21] that using the differential form of * Mahmood Fakher [email protected] 1
School of Mechanical Engineering, Iran University of Science and Technology, Narmak, 16846‑13114 Tehran, Iran
Center of Excellence in Railway Transportation, Iran University of Science and Technology, Narmak, 16842‑13114 Tehran, Iran
2
nonlocal elasticity instead of its integral form is allowable only in a few certain cases in which satisfying additional constitutive boundary conditions (CBCs), resulted from this transformation, is possible. It should be noted that the integral form of nonlocal elasticity has been seldom used in the past years [22–24], while after presentation of Ref.[20], a significan
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