Microstructured continua and scaling for wave motion

Abstract

This paper deals with wave motion in microstructured solids. A short introduction explains how the basic mathematical models for description of microstructure(s) of solids are derived. Based on the Mindlin-type micromorphic theory, the governing equations for wave motion in such solids are presented in one-dimensional setting. The focus of the paper is in explaining the importance of internal scales in microstructured solids. It is shown that the proper scaling permits to construct the mathematical models which involve hierarchies of wave operators. Depending on the scale parameter (the ratio of an internal scale over the wave length), the various operators govern the wave propagation. The main case analysed here consists of the second-order operators but the first-order operators which are characteristic to evolution equations, are also briefly explained.