Department of Mechanical Science and Engineering,
also Institute for Condensed Matter Theory, and Beckman Institute
University of Illinois at Urbana-Champaign, USA
Microstructural randomness is present in just about all solid materials. When dominant (macroscopic) length scales are large relative to microscales, one can safely work within classical, deterministic solid mechanics. However, when the separation of scales does not hold, various concepts of continuum mechanics need to be re-examined and new methods developed. In this talk we focus on scaling from a Statistical Volume Element (SVE) to a Representative Volume Element (RVE). Starting from the Hill-Mandel homogeneity condition, without assuming any spatial periodicity, the RVE is approached in terms of two hierarchies of bounds stemming, respectively, from uniform kinematic and static boundary value problems set up on the SVE. This is illustrated in various settings: conductivity, linear or finite (thermo)elasticity, plasticity, viscoelasticity, and Darcy permeability. The entire methodology can also be extended to homogenization of random media by micropolar (Cosserat) rather than by classical (Cauchy) continua. This methodology also forms a systematic basis for setting up of random fields with continuous realizations and stochastic finite element methods.
Martin Ostoja-Starzewski holds a Ph.D. (1983) in mechanical engineering from McGill University, Canada. His research interests are primarily in (thermo)mechanics of random and fractal media, advanced continuum theories, as well as aerospace, bio- and geo-physical applications. He wrote 190+ journal papers as well as two books: 1. Microstructural Randomness and Scaling in Mechanics of Materials, CRC Press (2007); 2. Thermoelasticity with Finite Wave Speeds, Oxford University Press (2009). He also (co-)edited 15 books/journal special issues and co-organized various meetings. He is Editor of Acta Mechanica, Chair Managing Editor of Mathematics and Mechanics of Complex Systems, Associate Editor of Journal of Thermal Stresses, and sits on editorial boards of many journals, including Probabilistic Engineering Mechanics, Int. J. Damage Mech., and Mechanics Research Communications. He is also co-Editor of the CRC Modern Mechanics and Mathematics Series and Fellow of ASME, American Academy of Mechanics, Society of Engineering Science, and Assoc. Fellow of AIAA. In 2012 he was Timoshenko Distinguished Visitor at Stanford. Since 2014, he has been Site co-Director of NSF Industry/University Cooperative Research Center for Novel High Voltage/Temperature Materials and Structures.