A deep understanding of wave motion through heterogeneous solids and structures is critical to predict their dynamic behavior, or to tailor them for the purposes of e.g. cloaking, vibration control, seismic protection, information transport, and sub-wavelength sensing. In this vein, we aim to provide a review of recent developments on the subject by elucidating:
1. Wave propagation through periodic and non-periodic media and structures governed by either linear or non-linear field equations;
2. Homogenization of wave motion through “micro-structured” media; importance and utilization of the spectral singularities characterizing the dispersion map;
3. Boundary effects that are critical for understanding the dynamics of finite structures and phenomena such as topologically protected states;
4. Design and optimization of the heterogeneous (micro-) structure to achieve the sought wave phenomena.

The subjects will be covered from diverse viewpoints, including continuum mechanics, experimental mechanics, physics, and applied mathematics. The course will be structured as follows.

Introductory lectures will focus on the fundamentals of elastic wave motion in periodic media such as Bloch-Floquet theory, band-gaps, and spectral degeneracies. Essential mathematical tools will be recalled, including the (classical) low-frequency homogenization, high frequency homogenization, and Willis’ approach. Higher-order extensions of low-frequency homogenization will be introduced to facilitate topological optimization. Analogies will be drawn between the homogenization results and enriched continuum theories. Generalizations catering for the asymptotic treatment of aperiodic systems and arbitrary spectral neighborhoods will be included.

The above fundamentals, introduced in the linear regime, will be partly extended to investigate the nonlinear dynamics of systems triggered e.g. by large deformations or multistable architectures. Effective description via non-local temporal operators will be included. Besides theoretical issues, experimental realizations will demonstrate the rich phenomenology of nonlinear metamaterials, including solitons and transition waves. Overall, it will be shown how a carefully designed architecture of engineering structures can be leveraged to control their dynamic performance.

Lastly the role of boundaries will be examined. The germane evanescent waves and underpinning eigenvalue problem will be highlighted. Boundary and interface correctors will be introduced to capture the effective transmission conditions. Recent developments on the asymmetric transport and topological insulators will be included. The robustness of interface states to disorder will be examined through the prism of topology. The analysis will focus on the class of partial differential Hamiltonians to which a topological charge can be assigned.

With this lecture series, we aim to attract doctoral students and researchers interested in the dynamics of micro-structured media and the design of next-generation of modern ("meta") materials. The objective is to provide the audience with a theoretical framework, computational tools, and experimental evidence to better understand the most recent developments on the subject and so facilitate the technology transfer from research to applications.