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Fractal and long-memory effects in solid, fluid, and granular media mechanics
Martin Ostoja-Starzewski

Last modified: 2023-05-15


There is a need for realistic random field (RF) models when dealing with geophysical-type phenomena. Simple models do not grasp the fractal-and-Hurst characteristics.  Heuristically speaking, the fractal dimension is a measure of the roughness of a field’s graph, whereas the Hurst exponent describes its long-range character: above 0.5 (or below 0.5) it indicates a persistent (respectively, anti-persistent) character of the field.  Here we present three such cases.

(1) Lamb’s type problems (concentrated force or moment) on elastic half-spaces, under anti-plane and in-plane conditions, with fractal-and-Hurst effects. We use cellular automata as the solution method, in each case first verifying it on progressively finer meshes against the continuum elastodynamic solution in a homogeneous, isotropic continuum. Then, a sensitivity of wave propagation on random fields is assessed for a wide range of fractal and Hurst coefficients.  In all the cases, the mean response amplitude is lowered by the mass density field’s randomness, while the Hurst exponent is found to have a stronger influence than the fractal dimension on the response.  In the in-plane problems, the associated Rayleigh wave is modified more than the pressure wave for the same random field parameters.

(2) Turbulent velocity fields. Extensive atmospheric field data indicate velocity correlation structures with fractal-and-Hurst effects. We show that so-called Generalized Cauchy and Dagum models provide excellent fits and much better performance than the conventionally used models (von Kármán and Kaimal). The fractal dimension, D, of both models is consistent with the well-known Kolmogorov 5/3 power law in the inertial sub-range.

(3) Couette flows of granular media. Motivated by departures from the second law of thermodynamics in Couette flows of molecular fluids, we analyze analogous flows of granular media on macroscales. Planar systems of monosized circular disks with frictional-Hertzian contacts, under Lees-Edwards boundary conditions, are simulated using LAMMPS. It is found that, in steady-state flows involving sufficiently small disk numbers, the dissipation function of such a system always exhibits spontaneous negative entropy increments. These negative entropy increments indicate that the fluctuation theorem applies to violations of the Clausius-Duhem inequality in flows of granular media. The Clausius-Duhem inequality and the energy balances are calculated according to the granular flows treated as micropolar (Cosserat-type) media with disk translations and rotations taken as two indepdented degrees of freedom. Conducting extensive sampling of LAMMPS-generated realizations of Couette flows, we determine that the dissipation function is always a random process with fractal-and-Hurst properties.