Improved Continuous Models for Discrete Media.pdf.pdf

Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2010, Article ID 986242, 35 pages doi: Review Article Improved Continuous Models for Discrete Media I. V. Andrianov, 1J. Awrejcewicz, 2and D. Weichert 1 1Institute of General Mechanic, RWTH Aachen University, Templergraben 64, 52056 Aachen, Germany 2Department of Automatics and Biomechanics, Technical University of ? ′ od ′ z, 1/15 Stefanowski St., 90-924 ? ′ od ′z, Poland Correspondence should be addressed to I. V. Andrianov, ******@- Received 9 June 2009; Accepted 23 September 2009 Academic Editor: Yuri Vladimirovich Mikhlin Copyright q2010 I. V. Andrianov et al. This is an open access article distributed under the mons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The paper focuses on continuous models derived from a discrete microstructure. Various continualization procedures that take into account the nonlocal interaction between variables of the discrete media are analysed. 1. Introduction In the recent years new classes of ultra-dispersive and nanocrystalline materials ? 1 – 3 ?, which require a di ?erent modern approach than that of a classical continuous media, have been proposed. For example, the nanocrystalline material is represented by a regular or quasiregular lattice with small size bodies ?domains, granules, fullerenes, nanotubes, or clusters nanoparticles ?possessing internal degrees of freedom occupying the lattice sites ? 4 ?. The situation mentioned is also a characteristic for various problems of nanomechanics ? 5 , 6 ?, because the transition of a material to the nanostructural state is panied by dimensional e ?ects in its mechanical properties. Models established on a classical continuous media cannot govern high frequency vibrations, material behavior in the vicinity of cracks and on the fronts of destruction waves, and during phas