Stress distribution for a rigid fractal embedded in a two-dimensional elastic medium

Abstract

The stress distribution on the surface of a fractal embedded in an elastic medium has been explored with use of diffusion-limited-aggregation (DLA) clusters fixed in a network of bonds and nodes which form a triangular lattice. The distribution of forces on the bonds in the network which are connected to the fixed nodes in the cluster has been obtained for clusters containing 100–3200 sites. Our results indicate that in the asymptotic limit of large cluster sizes the distribution of forces on the surface of the fractal can be described by using the scaling form N(ln(F))=ln(S)h(ln(F)/ln(S)), where S is the size (number of nodes or number of surface bonds) of the DLA cluster and N(ln(F))δ ln(F) is the number of surface bonds for which the natural log of the normalized forces lies in the range ln(F) to ln(F)+δ ln(F). These results indicate that the stress distribution in the surface of a fractal can be described in terms of a fractal measure with an associated infinite hierarchy of scaling exponents and a continuous spectrum of singularities given by f(α)=Dh(-$D^{\mathrm{-}1}$α). .AE