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

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.
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Article dans une revue
European Journal of Applied Mathematics, Cambridge University Press (CUP), 2012, 23 (06), pp.709 - 735. 〈10.1017/S095679251200023X〉
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Soumis le : mercredi 13 septembre 2017 - 16:36:16
Dernière modification le : lundi 9 avril 2018 - 12:20:07

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Thomas M. Michelitsch, GÉrard A. Maugin, Mujibur Rahman, Shahram Derogar, Andrzej F. Nowakowski, et al.. A continuum theory for one-dimensional self-similar elasticity and applications to wave propagation and diffusion. European Journal of Applied Mathematics, Cambridge University Press (CUP), 2012, 23 (06), pp.709 - 735. 〈10.1017/S095679251200023X〉. 〈hal-01587113〉

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