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摘要
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 %26apos;self-similar%26apos; 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%26apos;s function due to a unit delta-force. In the dynamic framework we derive the solution of the Cauchy problem and the retarded Green%26apos;s function. We deduce the distributions of a self-similar variant of diffusion problem with Levi-stable distributions as solutions with infinite mean fluctuations. In both dynamic cases we obtain a hierarchy of solutions for the self-similar Poisson%26apos;s equation, which we call %26apos;self-similar potentials%26apos;. These non-local singular potentials are in a sense self-similar analogues to Newtonian potentials and to the 1D Dirac%26apos;s delta-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.
- 出版日期2012-12
