title: Imperfections in a two-dimensional hierarchical structure.
abstract: Hierarchical and fractal designs have been shown to yield high mechanical efficiency under a variety of loading conditions. Here a fractal frame is optimized for compressive loading in a two-dimensional space. We obtain the dependence of volume required for stability against loading for which the structure is optimized and a set of scaling relationships is found. We evaluate the dependence of the Hausdorff dimension of the optimal structure on the applied loading and establish the limit to which it tends under gentle loading. We then investigate the effect of a single imperfection in the structure through both analytical and simulational techniques. We find that a single asymmetric perturbation of beam thickness, increasing or decreasing the failure load of the individual beam, causes the same decrease in overall stability of the structure. A scaling relationship between imperfection magnitude and decrease in failure loading is obtained. We calculate theoretically the limit to which the single perturbation can effect the overall stability of higher generation frames.
authors.affiliation: Aalto Science Institute, School of Science, Aalto University, 02150 Espoo, Finland.
authors.forename: Daniel
authors.initials: D
authors.lastname: Rayneau-Kirkhope
authors.affiliation: School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom.
authors.forename: Yong
authors.initials: Y
authors.lastname: Mao
authors.affiliation: Unilever R&D, Colworth House, Sharnbrook, Bedford MK44 1LQ, United Kingdom and London Institute for Mathematical Sciences, 35a South Street, Mayfair, London, United Kingdom.
authors.forename: Robert
authors.initials: R
authors.lastname: Farr
mesh_terms: D019245:Compressive Strength; D003198:Computer Simulation; D055119:Elastic Modulus; D008420:Manufactured Materials; D008962:Models, Theoretical; D013314:Stress, Mechanical; D013718:Tensile Strength