Tilings induced by a class of cubic Rauzy fractals

We study aperiodic and periodic tilings induced by the Rauzy fractal and its subtiles associated with beta-substitutions related to the polynomial x3-ax2-bx-1 for a≥b≥1. In particular, we compute the corresponding boundary graphs, describing the adjacencies in the tilings. These graphs are a valuable tool for more advanced studies of the topological properties of the Rauzy fractals. As an example, we show that the Rauzy fractals are not homeomorphic to a closed disc as soon as a≤2b-4. The methods presented in this paper may be used to obtain similar results for other classes of substitutions.© 2012 Elsevier B.V. All rights reserved.

A class of cubic Rauzy fractals

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Fractals de Rauzy

We study arithmetical and topological properties of a class of Rauzy Fractals. In particular, we give a parametrization of the boundaries of these sets and show that they are quasi-disks.

Propriedades topológicas e aritméticas dos fractais de Rauzy

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Máquina de somar, conjuntos de Julia e fractais de Rauzy

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Fractais de Rauzy, autômatos e frações contínuas

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Fractals and Scaling in Fracture Induced by Microcrack Coalescence

A numerical model is proposed to simulate fracture induced by the coalescence of numerous microcracks, in which the condition for coalescence between two randomly nucleated microcracks is determined in terms of a load-sharing principle. The results of the simulation show that, as the number density of nucleated microcracks increases, stochastic coalescence first occurs followed by a small fluctuation, and finally a newly nucleated microcrack triggers a cascade coalescence of microcracks resulting in catastrophic failure. The fracture profiles exhibit self-affine fractal characteristics with a universal roughness exponent, but the critical damage threshold is sensitive to details of the model. The spatiotemporal distribution of nucleated microcracks in the vicinity of critical failure follows a power-law behaviour, which implies that the microcrack system may evolve to a critical state.

Fractals in fracture of bulk metallic glass

We investigate the nanoscale periodic corrugation (NPC) structures on the dynamic fracture surface of a typical tough bulk metallic glass, submitted to high-velocity plate impact and scanned by atomic force microscopy (AFM). The detrended fluctuation analysis (DFA) of the recorded AFM profiles reveals that the valley landscapes of the NPC are nearly memoryless, characterized by Hurst parameter of 0.52 and exhibiting a self-similar fractal character with the dimension of about 1.48. Our findings confirm the existence of the “quasi-cleavage” fracture underpinned by tension transformation zones (TTZs) in metallic glasses.

SURFACE FRACTALS IN BLOCK-COPOLYMERS

A surface fractal model was presented to describe the interface in block copolymers. It gives a simple power-law relationship between the scattering intensity I(q) and the wave vector q in a relatively wide range as qxi >> 1, I(q) is-proportional-to q(D-6