A beam that can only bend on the Cantor set

Abstract

In this work, we address the following question: Is it possible for a one‐dimensional, linearly elastic beam to only bend on the Cantor set and, if so, what would the bending energy of such a beam look like? We answer this question by considering a sequence of beams, indexed by , each one only able to bend on the set associated with the ‐th step in the construction of the Cantor set and compute the ‐limit of the bending energies. The resulting energy in the limit has a structure similar to the traditional bending energy, a key difference being that the measure used for the integration is the Hausdorff measure of dimension , which is the dimension of the Cantor set.