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.
Intellectual Merit The general objectives of this project have been: (i) to accurately assess states of stress, displacement, and strain in fractal porous media through development of new analytical techniques; and (ii) to investigate the morphogenesis of fractal patterns in heterogeneous solids as related to growth, inelasticity and damage in materials and structures. The theoretical approach developed in area (i) has been based on dimensional regularization, leading to balance laws expressed in terms of fractional integrals whose order is directly tied to fractal dimensions of volumes and surfaces. As a result, highly complex media which have so far been the domain of condensed matter physics, geophysics and biophysics (polymer clusters, rocks, percolating networks, nervous and pulmonary systems...) are now becoming open to solution methods for initial/boundary value problems conventionally reserved for smooth materials. In the area (ii) it was found that elastic-plastic-brittle transitions lead to avalanches in response curves and fractal patterns of plasticized grains in a wide range of different elastic-plastic materials of metal or soil type, made of isotropic or anisotropic grains with random fluctuations in material properties (plastic limits, elastic and plastic moduli,…). Simultaneously, the sharp kink in the stress-strain curve is replaced by a smooth change. The grant has also provided support for closely related research on (iii) explicit representations of tensor-valued random fields with possibly fractal covariance structure; (iv) dynamical systems under random forcings with spatially/temporally fractal and Hurst characteristics; and (v) multiscale mechanics problems (homogenization, electromagnetism, tensor random fields, and MRI-based modeling of traumatic brain injury with brain imaged as a fractal-type structure). Broader Impacts and Outreach 1. Human Resource Development: Graduated students (including one female): S. Kale (M.Sc. 2012); M. Sena (2012); Y. Chen (Ph.D. 2011), H. Joumaa (Ph.D. 2012), J. Li (Ph.D. 2012), A. Saharan (Ph.D. 2014), Sohan Kale (Ph.D. 2015). 2. Gave 4-hour AGORA courses "Fractals" at University Laboratory High School, U-Illinois (Feb. 2013 & 2014). 3. Organizer of Symposia (i) "Randomness and Fractals in Mechanics," at 12th PACAM (2012); and (ii) "Randomness, Fractals, and Computational Mechanics" at WCCM-2012, USNCCM-2013. 4. Gave short courses "Mechanics of Random and Fractal Materials and Structures" at USNCCM-2009, ASCE-EMI-2013. 5. Chair the PACAM Committee at the American Academy of Mechanics (until Dec. 2012). 6. Lecturer on "Fractals and Randomness in Mechanics of Materials" in Course "Multiscale Modelling of Complex Materials," at CISM, Udine, Italy, 2012.
