A continuum theory for one-dimensional self-similar elasticity and applications to wave propagation and diffusion - Sorbonne Université Accéder directement au contenu
Article Dans Une Revue European Journal of Applied Mathematics Année : 2012

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

Résumé

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.
Fichier non déposé

Dates et versions

hal-01587113 , version 1 (13-09-2017)

Identifiants

Citer

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, 2012, 23 (06), pp.709 - 735. ⟨10.1017/S095679251200023X⟩. ⟨hal-01587113⟩
57 Consultations
0 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More