Seminarium Instytutowego Seminarium Mechaniki im. W. Olszaka i A. Sawczuka

Randomness and Fractals in Mechanics of Materials

M. Ostoja-Starzewski

poniedziałek, 28 maja 2012, godz. 10:00, sala Aula (II p.)

Microstructural randomness is present in just about all solid materials. When macroscopic length scales are very large relative to microscale ones, one can quite safely work within deterministic solid mechanics. However, when the separation of scales does not hold, various concepts of continuum mechanics need to be re-examined and new methods developed. We first show the approach to a representative volume element (RVE) in terms of two hierarchies of bounds stemming, respectively, from uniform kinematic and traction boundary value problems set up on a statistical volume element (SVE). This finite-size scaling is illustrated in the settings of conductivity, linear/finite (thermo)elasticity, plasticity, and Darcy permeability. This methodology then forms a physical basis for developing locally anisotropic, mesoscale continuum random fields and stochastic finite element methods.

The above approach also allows one to ask the question: Does the elastic-plastic transition display a fractal character? We address this issue in 2d and 3d settings, where the grain-level properties are weak- and white-noise random fields. We find that, under monotonic loadings of kinematic and traction type applied to the SVE, plasticized grains form evolving 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 metal or soil type, made of isotropic or anisotropic grains, possibly with thermal stress effects. By analogy to scaling analyses of phase transitions, we recognize the fully plastic state as a critical point and find the critical exponents to be universal in 2d and 3d, regardless of the randomness in various constitutive properties and their random noise levels.