Variational method for non-conservative instability of a cantilever SWCNT in the presence of variable mass or crack
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O R I G I NA L
M. A. De Rosa · M. Lippiello · N. M. Auciello · H. D. Martin · M. T. Piovan
Variational method for non-conservative instability of a cantilever SWCNT in the presence of variable mass or crack
Received: 21 April 2020 / Accepted: 28 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In the present paper, the non-conservative instability of a cantilever single-walled carbon nanotube (SWCNT) through nonlocal theory is investigated. The nanotube is modeled as clamped-free beam carrying a concentrated mass, located at a generic position, or in the presence of crack, and subjected to a compressive axial load, at the free end. Nonlocal Euler–Bernoulli beam theory is used in the formulation and the governing equations of motion and the corresponding boundary conditions are derived using an extended Hamilton’s variational principle. The governing equations are solved analytically. In order to show the sensitivity of the SWCNT to the values of an added mass, or crack and the influence of the nonlocal parameter and nondimensional crack severity coefficient on the fundamental frequencies values, some numerical examples have been performed and discussed. Also, the validity and the accuracy of the proposed analysis have been confirmed by comparing the results with those obtained from the literature. Keywords Non-conservative instability · Nonlocal elasticity · Nanosensor · Crack · Variational method
1 Introduction Carbon nanotubes (CNTs) play a key role in a variety of engineering fields due to their superior mechanical, physical and electronic properties [1–3]. Owing to these properties, CNTs have met applications in the emerging M. A. De Rosa · N. M. Auciello School of Engineering, University of Basilicata, Viale dell’Ateneo Lucano 10, Potenza 85100, Italy E-mail: [email protected] N. M. Auciello E-mail: [email protected] M. Lippiello (B) Department of Structures for Engineering and Architecture, University of Naples ”Federico II”, Via Forno Vecchio 36, Napoli 80134, Italy E-mail: [email protected] H. D. Martin Facultad Regional Reconquista Universidad Tecnológica Nacional, Parque Industrial Reconquista, Calle 44 1000,, Reconquista Santa Fe S3560, Argentina E-mail: [email protected] M. T. Piovan Centro de investigaciones de Mecanica Teorica y Aplicada, Universidad Tecnologica Nacional FRBB, 11 de Abril 461, Bahia Blanca B8000LMI, Argentina E-mail: [email protected]
M. A. De Rosa et al..
field of nanoelectronics, nanosensors, nanocomposites, bio-nanocomposites and so on [4–6]. According to literature, the nanoscale of these structures suggests an atomistic model, but this approach turns out to be very expensive. On the other hand, although the classical continuum theories (Euler–Bernoulli, Timoshenko, or even higher-order theories) are able to predict the behaviors of nanostructures, it is found to be inadequate because of ignoring the small size effects. Thus adopting the nonlocal elasticity theory, as developed by Eringen in [7,8], is usual.
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