Fractal and Hurst Effects in Stochastic Mechanics

Abstract Myriad random phenomena in nature possess fractal and Hurst characteristics. Random processes/fields, such as those with Cauchy or Dagum correlations, enable modeling such stochastic structures in time and space. In the first place, this paper provides a compact review of these models, including their spectral properties, for wide ranges of the fractal dimension and Hurst parameter. The Cauchy and Dagum models can be used to determine stochastic responses of dynamical systems and/or spatial problems in 1d, 2d, or 3d in the presence of fractal and Hurst characteristics. The paper surveys various examples ranging from vibration problems, rods and beams with random properties under random loadings, waves and wavefronts, fracture, homogenization of random media, and statistical turbulence, to stochastically evolving spontaneous violations of the entropy inequality in granular flows. The latter case shows the route to examine whether a mechanical system gives rise to stochastics with such intriguing features. Common features brought out in this survey show what can and how can be achieved with Cauchy and Dagum-type models and related constructs in mechanics.