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A continuum theory for one-dimensional self-similar elasticity and applications to wave propagation and diffusion

Published online by Cambridge University Press:  16 August 2012

THOMAS M. MICHELITSCH ,
GÉRARD A. MAUGIN ,
MUJIBUR RAHMAN ,
SHAHRAM DEROGAR ,
ANDRZEJ F. NOWAKOWSKI  and
FRANCK C. G. A. NICOLLEAU
THOMAS M. MICHELITSCH
Affiliation:
Université Pierre et Marie Curie, Paris 6, Institut Jean le Rond d'Alembert, CNRS UMR 7190, France email: [email protected], [email protected]
GÉRARD A. MAUGIN
Affiliation:
Université Pierre et Marie Curie, Paris 6, Institut Jean le Rond d'Alembert, CNRS UMR 7190, France email: [email protected], [email protected]
MUJIBUR RAHMAN
Affiliation:
General Electric Energy, Greenville, SC 29615, USA email: [email protected]
SHAHRAM DEROGAR
Affiliation:
Department of Architecture, Yeditepe University, Istanbul, Turkey email: [email protected]
ANDRZEJ F. NOWAKOWSKI
Affiliation:
Sheffield Fluid Mechanics Group, Department of Mechanical Engineering, University of Sheffield, South Yorkshire, UK email: [email protected], [email protected]
FRANCK C. G. A. NICOLLEAU
Affiliation:
Sheffield Fluid Mechanics Group, Department of Mechanical Engineering, University of Sheffield, South Yorkshire, UK email: [email protected], [email protected]
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Abstract

We analyse some fundamental problems of linear elasticity in one-dimensional (1D) continua where the material points of the medium interact in a self-similar manner. This continuum with ‘self-similar’ elastic properties is obtained as the continuum limit of a linear chain with self-similar harmonic interactions (harmonic springs) which was introduced in [19] and (Michelitsch T.M. (2011) The self-similar field and its application to a diffusion problem. J. Phys. A Math. Theor.44, 465206). We deduce a continuous field approach where the self-similar elasticity is reflected by self-similar Laplacian-generating equations of motion which are spatially non-local convolutions with power-function kernels (fractional integrals). We obtain closed-form expressions for the static displacement Green's function due to a unit δ-force. In the dynamic framework we derive the solution of the Cauchy problem and the retarded Green's function. We deduce the distributions of a self-similar variant of diffusion problem with Lévi-stable distributions as solutions with infinite mean fluctuations. In both dynamic cases we obtain a hierarchy of solutions for the self-similar Poisson's equation, which we call ‘self-similar potentials’. These non-local singular potentials are in a sense self-similar analogues to Newtonian potentials and to the 1D Dirac's δ-function. The approach can be a point of departure for a theory of self-similar elasticity in 2D and 3D and for other field theories (e.g. in electrodynamics) of systems with scale invariant interactions.

Keywords

Mechanics of fractal structuresSelf-similar elasticityFractional calculus
Type
Papers
Information
European Journal of Applied Mathematics , Volume 23 , Issue 6 , December 2012 , pp. 709 - 735
Copyright
Copyright © Cambridge University Press 2012

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