  This paper is devoted to the analysis of some fundamental problems of linear
elasticity in 1D continua with self-similar interparticle interactions. We
introduce a self-similar continuous field approach where the self-similarity is
reflected by 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
$\delta$-force. In the dynamic framework we derive the solution of the {\it
Cauchy problem} and the retarded Green's function. We deduce the distribution
of a self-similar variant of diffusion problem with L\'evi-stable distributions
as solutions with infinite mean fluctuations describing the statistics
L\'evi-flights. We deduce 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 the 1D-Dirac's
$\delta$-function. The approach can be the starting point to tackle a variety
of scale invariant interdisciplinary problems.
