Attempts at the formulation of response laws of many materials regularly encounter microstructural randomness and complexity of several scales. As a result, problems have to be cast in the framework of Stochastic (micro)Mechanics (e.g., 215), and the key concept of continuum mechanics, Representative Volume Element (RVE), has to be generalized to a Statistical Volume Element (SVE). SVE was first introduced here. This calls for a range of techniques lying at the intersection of mechanics, thermodynamics, materials science, and applied/stochastic mathematics. Applications include composite materials, polycrystals, granular media, functionally graded materials, and biomaterials. Methods involve classical and non-classical continuum mechanics, stochastic mechanics/dynamics, computational mechanics, random geometry (+ mathematical morphology), and experiments. See Book 1.
- Mechanics and physics of random media: stochastic models, constitutive responses including conductivity, elasticity, impact waves. See Books 1, 3, and 4,
- Tensor random fields: providing a basis for stochastic finite elements and stochastic PDEs. See Books 3, 4, and papers [154, 162, 180, 185, 237, 243].
- Generalization of continuum mechanics accounting for apparent/spontaneous violations of the Second Law of thermodynamics according to the fluctuation theorem and Jarzynski equality. See papers [158, 174, 189, 197, 210, 218, 221].
- Hyperbolic continuum thermomechanics: a hyperbolic theory free of the paradox of infinitely fast heat propagation in classical heat conduction and classical thermo-elasticity. See Book 2. Related issues are studied from the standpoint of thermodynamics and physics. See papers [118, 128, 134, 158].
- Thermo-mechanics and dynamics of fractal media. Fractal media are ubiquitous in nature, yet fall outside the realm of conventional continuum mechanics. However, they can be brought into the framework of continuum theories via dimensional regularization or random fields with fractal properties. See papers [103, 108, 115, 116, 117, 120, 130, 131, 132, 142, 144, 147, 150, 156, 164, 165, 169, 184, 194, 202, 214, 216, 217, 225, 229, 230, 240 …].
- Morphogenesis of fractals at elastic-inelastic transitions. See papers [12, 120, 121, 126, 139, …].
- Blunt head trauma. This research is based on MRI/MRE of the human brain, conducted jointly with Brad Sutton. See papers [123, 137, 151,190, 204, 219, 239].
- Convolution finite elements. A new method, based on alternative field equations in which the convolution product appears in place of dot products. In contradistinction to the Newmark method, it can adaptively conserve energy. See papers [224, 232, 237…].
- Fun project a few years ago: Saturn’s rings have a fractal structure [See arxiv] – this has been speculated for a long time in science and popular science literature, but ours is the first quantitative study based on the images from the Voyager and Cassini missions. See papers [140, 193].
Past/present funding sources
- National Science Foundation
- DTRA – Sandia National Labs
- US Army Corps of Engineers
- Air Force Office of Scientific Research
- San Diego Supercomputer Center
- Office of Naval Research
- American Forest and Paper Association
- US Department of Agriculture
- Canada Foundation for Innovation
- Atmospheric Environment Service Canada
- Canada Centre for Inland Lakes and Waters
- e-Xstream engineering and Ministry of Economy of the Wallonia Region, Belgium