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Randomness and Fractals in Mechanics of Materials
=MARTIN OSTOJA-STARZEWSKI, Ph.D.= ===January 19, 2012=== RANDOMNESS AND FRACTALS IN MECHANICS OF MATERIALS Department of Mechanical Science and Engineering,<br> also Institute for Condensed Matter Theory, and Beckman Institute<br> University of Illinois at Urbana-Champaign Microstructural randomness is present in just about all materials. When dominant (macroscopic) length scales are large relative to the microscale, we can safely work within classical, deterministic solid mechanics. However, when the separation of scales does not hold (e.g. in FGM, geological, or biological materials) many concepts of continuum solid mechanics need to be re-examined and new methods developed. In this talk we focus on scaling from a Statistical Volume Element (SVE) to a Representative Volume Element (RVE). Using micromechanics, the RVE is approached in terms of two hierarchies of bounds stemming, respectively, from Dirichlet and Neumann boundary value problems set up on the SVE. This is illustrated in the settings of planar conductivity, (non)linear (thermo)elasticity, plasticity, and Darcy permeability. This methodology then forms a rational basis for setting up of mesoscale continuum random fields and stochastic finite element methods. The above approach also allows one to ask the question: Why are fractal patterns observed in inelastic materials? We address this issue in the setting of 2D or 3D elastic-plastic materials, whose grain-level properties are random fields lacking any spatial correlation structure. We find that, under monotonic loadings of Dirichlet or Neumann type, plasticized grains form fractal patterns and gradually fill the entire material domain, while the sharp kink in the stress-strain curve is replaced by a smooth transition. This is universally the case for a wide range of different elastic-plastic materials of solid or soil type, made of isotropic or anisotropic grains, possibly with thermal stress effects.
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