Multifractal properties of self-similar stress distributions

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    Abstract

    We consider highly heterogeneous materials with multiscale microstructures, such as, for instance, the Earth's crust, and focus on the case when the microstructure could be considered self-similar, at least in a range of scales. The paper develops a conceptual framework to treat such materials within the paradigm of continuum mechanics. We represent the material, within its range of self-similarity, by a self-similar sequence of what we call H-continua. Each H-continuum replaces the original materials with all structural elements of sizes smaller than H, such that the average (over the volume elements of size H) response of the continuum is similar to the original material. In each continuum, stress, strain and displacement fields are defined as usual. We then continue this sequence in a self-similar manner to the limits H -> 0 and H -> infinity, such that a truly self-similar system is obtained. The intersection of these continua (generally a fractal F) consists of points where all fields scale according to power laws. In this system, the equations of equilibrium are determined using the H-derivative, which is a difference quotient with the increment equal to H. Multifractal formalism is developed for non-uniform stress scaling and applied to the growth of a crack in a continuous material with fractal distribution of body forces. Fractal dimension of the future fracture surface is determined and the conditions of stable and unstable crack growth are formulated based on the multifractal spectrum.
    Original languageEnglish
    Pages (from-to)3117-3136
    JournalPhilosophical Magazine
    Volume86
    Issue number21/22
    DOIs
    Publication statusPublished - 2006

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