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On fractional and fractal formulations of gradient linear and nonlinear elasticity

Acta Mechanica volume 230, pages 2043–2070 (2019)Cite this article

Abstract

In this paper, we consider extensions of the gradient elasticity models proposed earlier by the second author to describe materials with fractional non-locality and fractality using the techniques developed recently by the first author. We derive a generalization of three-dimensional continuum gradient elasticity theory, starting from integral relations and assuming a weak non-locality of power-law type that gives constitutive relations with fractional Laplacian terms, by utilizing the fractional Taylor series in wave-vector space. In the sequel, we consider more general field equations with fractional derivatives of non-integer order to describe nonlinear elastic effects for gradient materials with power-law long-range interactions in the framework of weak non-locality approximation. The special constitutive relation that we elaborate upon can form the basis for developing a fractional extension of deformation theory of gradient plasticity. Using the perturbation method, we obtain corrections to the constitutive relations of linear fractional gradient elasticity, when the perturbations are caused by weak deviations from linear elasticity or by fractional gradient non-locality. Finally, we discuss fractal materials described by continuum models in non-integer dimensional spaces. Using a recently suggested vector calculus for non-integer dimensional spaces, we consider problems of fractal gradient elasticity.

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Authors and Affiliations

  1. Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow, Russia, 119991

    Vasily E. Tarasov

  2. Laboratory of Mechanics and Materials, Aristotle University of Thessaloniki, 54006, Thessaloníki, Greece

    Elias C. Aifantis

  3. Emeritus Professor of Engineering, Michigan Tech, Houghton, MI, 49931, USA

    Elias C. Aifantis

  4. Distinguished Adjunct Professor, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Elias C. Aifantis

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  1. Vasily E. Tarasov
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  2. Elias C. Aifantis
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Correspondence to Vasily E. Tarasov.

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Tarasov, V.E., Aifantis, E.C. On fractional and fractal formulations of gradient linear and nonlinear elasticity. Acta Mech 230, 2043–2070 (2019). https://doi.org/10.1007/s00707-019-2373-x

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  • DOI: https://doi.org/10.1007/s00707-019-2373-x

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