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A higher-order Eringen model for Bernoulli–Euler nanobeams

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Abstract

Small-scale effects in carbon nanotubes are effectively assessed by resorting to the methods of nonlocal continuum mechanics. The crucial point of this approach consists in defining suitable constitutive laws which lead to reliable results. A nonlocal elastic law, diffusely adopted in literature, is that proposed by Eringen. According to this theory, the elastic equilibrium problem of a nonlocal nanostructure is equivalent to that of a corresponding local nanostructure subjected to suitable distortions simulating the nonlocality effect. Accordingly, transverse displacements and bending moments of a Bernoulli–Euler nonlocal nanobeam can be obtained by solving a corresponding linearly elastic (local) nanobeam, subjected to the same loading and kinematic constraint conditions of the nonlocal nanobeam, but with the prescription of suitable inelastic bending curvature fields. This observation leads naturally to the definition of a higher-order Eringen version for Bernoulli–Euler nanobeams, in which the elastic energy is assumed to be dependent on the total and inelastic bending curvatures and on their derivatives. Weak and strong formulations of elastic equilibrium of first-order gradient nanobeams are provided by a consistent thermodynamic approach. Exact solutions of fully clamped and cantilever nanobeams are given and compared with those of literature.

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Notes

  1. The higher-order boundary condition Eq. (22)\(_3\) turns out to be identically fulfilled for the Eringen model, formulated in Sect. 4, which is got by setting \(c_1=0\).

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Acknowledgments

The support of “Polo delle Scienze e delle Tecnologie”—University of Naples Federico II—through the research project FARO is gratefully acknowledged.

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Authors and Affiliations

  1. Department of Structures for Engineering and Architecture, University of Naples Federico II, Via Claudio 21, 80125, Naples, Italy

    Raffaele Barretta & Francesco Marotti de Sciarra

  2. Department of Engineering Mechanics, Faculty of Engineering, University of Rijeka, Vukovarska 58, 51000, Rijeka, Croatia

    Marko Čanadija

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  1. Raffaele Barretta
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  2. Marko Čanadija
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  3. Francesco Marotti de Sciarra
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Correspondence to Raffaele Barretta.

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Barretta, R., Čanadija, M. & Marotti de Sciarra, F. A higher-order Eringen model for Bernoulli–Euler nanobeams. Arch Appl Mech 86, 483–495 (2016). https://doi.org/10.1007/s00419-015-1037-0

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