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 and the key concept of continuum mechanics, Representative Volume Element (RVE), has to be generalized to a Statistical Volume Element (SVE). 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, impact waves, stochastic finite elements. See Book 1.
- Continuum thermomechanics in the presence of non-Fourier heat conduction: thermo-elastodynamics as a hyperbolic theory free of the paradox of infinitely fast heat propagation in 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 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. See papers [103, 108, 115, 116, 117, 120, 130, 131, 132, 142, 144, …].
- Morphogenesis of fractals at elastic-inelastic transitions. See papers [12, 118, 122, 127, 140, …].
- Modeling the spatio-temporal mutliscale dynamics of head trauma. This research is based on MRI of human brain, conducted jointly with Brad Sutton. See papers [123, 137, 151].
- Saturn’s rings have a fractal structure [See arxiv] – this has been speculated/proposed 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.
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