Hierarchies of waves in nonclassical materials

Wave propagation in microstructured materials is directly affected by the exis-tence of internal space scale(s) in the compound matter. In this case the classical continuum theory cannot be used. In this paper based on the Mindlin model, the balance laws for macro- and microstructure are formulated separately. Using the slaving principles relating macro- and microdisplacements, the governing equa-tions are derived for a single- and two-scale (scale within scale) cases. These equations exhibit hierarchical properties assigning the wave operators to inter-nal scales. In terms of macrodisplacements, higher-order dispersive terms appear having a clear physical background (microinertia, wave speed in microstructure) related to the scale of the microstructure. Full, approximated (corresponding to hierarchical models), and simplified dispersion relations are derived and analysed to demonstrate the validity of the hierarchical governing equations. Linear theory is based on the quadratic free energy function, in nonlinear theory the cubic terms should also be taken into account. The corresponding governing equation includes nonlinearities in both macro- and microscale. Such consistent modelling opens up new possibilities to Nondestructive Testing (NDT) of material properties.