Waves and complexity of microstructured solids

A complex system as it is understood nowadays is composed by its constituents that interact with each other resulting in emergent properties of the system as a whole. In mechanics the concepts of complexity has been analysed by Engelbrecht [7] with a focus on wave propagation. This theory is based on some cornerstones like the introduction of internal structures at different scales and the nonlinearity of the models which in other words means incorporating intrinsic microstructural and nonlinear effects. In this case complexity means that we have different scales, with several interaction processes which encompass many physically meaningful phenomena. Usually a microstructured body, as we shall see, is modeled as a solid with an internal structure at a different scale, which is apt to describe the mechanical behaviour of solids with dislocations, polycrystalline solids, ceramic composites, granular media, etc. One main aspect of such theories is that they always take into account the nonlinearity of the materials, the nonlocality and the interactions between micro- and macroscales. It is possible, and useful, to develop also models with a hierarchy of microstructures, i.e. a first level micro-structure which contains a second level micro-structure, and it is meaningful also the case of concurrent micro-structures (see Berezovski et al [1]). In this paper we want to analyze the subject, recalling some main results in the theory of complex microstructures, developing new results in the case of multiple microstructures, exploiting hierarchical governing equations and analyzing non-linear wave propagation, which is crucial to put in evidence the weight of the different scales and the interaction of micro- and macro-structures.