Mechanics of Fractal Materials

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

The objectives of this proposed research is to provide funding for (i) the analysis of formation of fractal patterns at elastic-inelastic transitions and (ii) the development of continuum poromechanics of fractal porous materials saturated with fluids. The first task will involve parallel computation, stochastic mathematics and physics of critical phenomena, so as to enable simulation of growth of fractal sets of plastic grains in 2D and 3D. Also, random field models will be developed in order to provide a theoretical framework for understanding elastic-inelastic transitions in spatially random solid-like and soil-like media, including functionally-graded materials. The second task will entail a generalization of continuum thermomechanics through dimensional regularization to materials with fractal geometries and arbitrary anisotropies, aiming at formulation of extremum and variational principles, governing equations and solution of initial-boundary value problems in static as well as dynamic settings. All the governing equations will explicitly depend on the fractal dimensions of mass distribution and bounding surfaces, so that a reduction to the well-known classical strong forms of partial differential equations can always be accomplished. While the subject matter of fractals in mechanics of materials offers several basic challenges, the pay-off will be significant for natural (geological, biological, etc.) as well as man-made materials. Thus, (i) the formation of fractal patterns at elastic-inelastic transitions in composites, solids, and soils will be better understood; and (ii) the highly complex, fractal-type media (polymer clusters, gels, fluid-saturated rock systems, percolating networks, neural and pulmonary systems, etc.) will become open to studies conventionally reserved for smooth materials. Overall, the results of this research will lead to a broadening of applicability of continuum mechanics and physics.