A dimensão fractal de fenômenos físicos dos sistemas geométricos fractais

• Ano de defesa: 2011
• Autor(a) principal: Barros, Marcelo Miranda
• Orientador(a): Bevilácqua, Luiz , Galeão, Augusto Cesar Noronha Rodrigues
• Banca de defesa: Loula, Abimael Fernando Dourado , Karam Filho, José , Battista, Ronaldo Carvalho , Barra, Luis Paulo da Silva , Gonçalves, Marcelo Albano Moret Simões
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: por
• Instituição de defesa: Laboratório Nacional de Computação Científica
• Programa de Pós-Graduação: Programa de Pós-Graduação em Modelagem Computacional
• Departamento: Coordenação de Pós-Graduação e Aperfeiçoamento (COPGA)
• País: BR
• Palavras-chave em Português: Fractais
• Palavras-chave em Inglês: Fractals
• Área do conhecimento CNPq: CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA::GEOMETRIA E TOPOLOGIA::TOPOLOGIA ALGEBRICA
• Link de acesso: https://tede.lncc.br/handle/tede/158

The physics associated to geometric fractal systems is investigated. Discrete and continuous models, from statics and dynamics as well as computational and physical experiments help defining and evaluating dimensions associated to the physics of the systems. It is shown the relation between the mechanical dimensions (flexibility and dynamical) and the geometric fractal dimension. Moments of order 2 are shown to be useful in identifying randomness in the generation process of geometry. Mixed fractals are defined by more than one law of formation or organization: the case of alternating laws is studied. Weierstrass-Mandelbrot systems (SWM) are defined through a properly summation of senoidal functions, each with amplitude proportional to the associated period squared. A dimension for SWM is defined. An origin for 1/f noises from SWM is proposed. A new method to determine fractal dimensions is proposed. It consists in taking successive samples from the object and relating a given property with the size of the sample, called sampling method. It is tested with Koch, mixed and Weierstrass systems. Branched systems (fractal trees) in 2D are studied under the solid mechanics approach. It is shown that Murray's law corresponds to the state of constant normal stress in solids. A mechanical efficiency (stiffness x weight) of beams with cross sections given by a Sierpinski system is studied. Defined by the proportion between geometric mechanical stiffness (moment of inertia) and the cross section area squared, the efficiency is shown to grow with the advance of orders. In this way, the more porous the more efficient is the beam.