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Nonlocal elasticity: an approach based on fractional calculus

  • Multi-Scale and Multi-Physics Modelling for Complex Materials
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Abstract

Fractional calculus is the mathematical subject dealing with integrals and derivatives of non-integer order. Although its age approaches that of classical calculus, its applications in mechanics are relatively recent and mainly related to fractional damping. Investigations using fractional spatial derivatives are even newer. In the present paper spatial fractional calculus is exploited to investigate a material whose nonlocal stress is defined as the fractional integral of the strain field. The developed fractional nonlocal elastic model is compared with standard integral nonlocal elasticity, which dates back to Eringen’s works. Analogies and differences are highlighted. The long tails of the power law kernel of fractional integrals make the mechanical behaviour of fractional nonlocal elastic materials peculiar. Peculiar are also the power law size effects yielded by the anomalous physical dimension of fractional operators. Furthermore we prove that the fractional nonlocal elastic medium can be seen as the continuum limit of a lattice model whose points are connected by three levels of springs with stiffness decaying with the power law of the distance between the connected points. Interestingly, interactions between bulk and surface material points are taken distinctly into account by the fractional model. Finally, the fractional differential equation in terms of the displacement function along with the proper static and kinematic boundary conditions are derived and solved implementing a suitable numerical algorithm. Applications to some example problems conclude the paper.

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Authors and Affiliations

  1. Department of Structural, Building and Geotechnical Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Turin, Italy

    Alberto Carpinteri, Pietro Cornetti & Alberto Sapora

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  1. Alberto Carpinteri
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  2. Pietro Cornetti
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  3. Alberto Sapora
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Correspondence to Pietro Cornetti.

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Carpinteri, A., Cornetti, P. & Sapora, A. Nonlocal elasticity: an approach based on fractional calculus. Meccanica 49, 2551–2569 (2014). https://doi.org/10.1007/s11012-014-0044-5

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  • DOI: https://doi.org/10.1007/s11012-014-0044-5

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