Fractals and Randomness in Mechanics of Materials

These notes provide an introduction to two aspects of mechanics of materials: (i) effects of randomness on scaling to effective constitutive responses and (ii) fractal geometries. The first aspect is relevant when the separation of scales does not hold [i.e. when dominant (macroscopic) length scales are large relative to microscale ones]. Then, various concepts of continuum solid mechanics need to be re-examined and new methods developed. Thus, 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. While the linear conductive and elastic microstructures were treated in (Ostoja-Starzewski, 2001), here we review this scaling in (non)linear (thermo)elasticity, elasto-plasticity, and viscoelasticity. We also signal the new concept of a scaling function as well as touch on scale effects in stochastic damage mechanics. The above approach also allows one to ask the question: Why are fractal patterns observed in inelastic materials? This issue is addressed in the setting of microheterogeneous elastic-plastic materials, whose grain-level properties are weak noise-to-signal random fields lacking any spatial correlation structure. We find that, under monotonic loadings of Dirichlet or Neumann type, the RVE-level response involves plasticized grains forming fractal patterns and gradually filling the entire material domain. Simultaneously, 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 metal or soil type, made of isotropic or anisotropic grains, possibly with thermal stress effects, and irrespective of which material property is a random field.