On fractional and fractal formulations of gradient linear and nonlinear elasticity

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 gradi...

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Bibliographic Details
Published in:Acta mechanica 2019-06, Vol.230 (6), p.2043-2070
Main Authors: Tarasov, Vasily E., Aifantis, Elias C.
Format: Article
Language:English
Subjects:
Calculus
Classical and Continuum Physics
Constitutive relationships
Continuum modeling
Control
Deformation mechanisms
Dynamical Systems
Elasticity
Engineering
Engineering Thermodynamics
Formulations
Fractal models
Fractals
Heat and Mass Transfer
Original Paper
Perturbation methods
Power law
Solid Mechanics
Taylor series
Theoretical and Applied Mechanics
Vibration
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Summary: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.
ISSN:0001-5970
1619-6937
DOI:10.1007/s00707-019-2373-x