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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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References

[1]Abramowitz, M. & Stegun, I. A. (editors) (1972) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Chapter 6), Dover, New York.Google Scholar
[2]Atanackovic, T. M. & Stankovic, B. (2009), Generalized wave equation in nonlocal elasticity. Acta. Mech. 208, 1–10.Google Scholar
[3]Bondarenko, A. N. & Levin, V. A. (2005) Self-similar spectrum of fractal lattice. Proceedings of Science and Technology KORUS 2005, Novosibirsk, Russia.Google Scholar
[4]Borodich, F. M. (1997) Some fractal models for fracture. J. Mech. Phys. Sol. 46 (2), 239–259.Google Scholar
[5]Carpinteri, A.Cornetti, P. & Sapora, A. (2011) A fractional calculus approach to non-local elasticity. Eur. Phys. J. Special Topics 193, 193–204.Google Scholar
[6]Epstein, M. & Adeebb, S. M. (2007) The stiffness of self-similar fractals. Int. J. Solids Struct. 45, 3238–3254.Google Scholar
[7]Gel'fand, I. M. & Shilov, G. E. (1964) Generalized Functions, Vol. 1: Properties and Operations, Academic Press, New York.Google Scholar
[8]Jackson, J. D. (1998) Classical Electrodynamics, 3rd ed., John Wiley, New York, ISBN: 10-047130932X, 13-978-0471309321.Google Scholar
[9]Jumarie, J. (2008) From Self-similarity to fractional derivative of non-differentiable functions via Mittag–Leffler Function. Appl. Math. Sci. 2 (40), 1949–1962.Google Scholar
[10]Kigami, J. (1989) A harmonic calculus on the Sierpinski spaces. Japan J. Appl. Math. 8, 259–290.Google Scholar
[11]Lévi, P. (1965) Processus Stochastiques et Mouvement Brownien (Jacques Gabay Editions), Gauthier-Villars, Paris, France.Google Scholar
[12]Li, X., Davison, M. & Essex, C. (2003) Fractional Differential Equations and Stable Distributions, Applied Probability Trust, University of Sheffield, UK.Google Scholar
[13]Majumdar, A. & Bushan, B. (1992) Elastic-plastic contact of bifractal surfaces. Wear 153, 53–64.Google Scholar
[14]Mandelbrot, B. (1992) The Fractal Geometry of Nature, W. Freeman, Longton, UK, ISBN: 0-716-71186-9.Google Scholar
[15]Mandelbrot, B. (1995) Les Objets Fractals – Form Hasard et Dimension, Mandelbrot, Flammarion, Paris, France, ISBN: 978-2-0812-4617-1.Google Scholar
[16]Mandelbrot, B. (1997) Fractales, Hasard et Finance, Flammarion, Paris, France, pp. 106ff, 156, ISBN: 978-2-0812-2510-7.Google Scholar
[17]Michelitsch, T. M. (2011) The self-similar field and its application to a diffusion problem. J. Phys. A Math. Theor. 44, 465206.Google Scholar
[18]Michelitsch, T. M., Gao, H. & Levin, V. M. (2003) Dynamic Eshelby tensor and potentials for ellipsoidal inclusions. Proc. R. Soc. 459 (2032), 863–890.Google Scholar
[19]Michelitsch, T. M., Maugin, G. A., Nicolleau, F. C. G. A., Nowakowski, A. F. & Derogar, S. (2009) Dispersion relations and wave operators in self-similar quasicontinuous linear chains. Phys. Rev. E 80, 011135.CrossRefGoogle ScholarPubMed
[20]Ostoja-Starzewski, M. (2007) Towards a thermomechanics of fractal media. ZAMP 58, 1085–1096.Google Scholar
[21]Peitgen, H.-O., Jürgens, H. & Saupe, D. (1991) Fractals for the Classroom: Part 1: Introduction to Fractals and Chaos, Springer, New York, ISBN: 10: 9780387970417.Google Scholar
[22]Sapoval, B. (1997) Universalités et Fractales, Flammarion, Paris, France, ISBN: 2-08-081466-4.Google Scholar
[23]Zähle, M. & Ziezold, H. (1996) Fractional derivatives of Weierstrass-type functions. J. Comput. Appl. Math. 76, 265–275.Google Scholar
[24]Yavari, A., Sarkani, S. & Moyer, E. T. (2002) The mechanics of self-similar and self-affine cracks. Int. J. Fracture 114 (1), 1–27.CrossRefGoogle Scholar