Topological features in continuum mechanics: from simple to complex bodies

Topological features have a ubiquitous influence on the behaviour of condensed matter, from the qualitative analysis of fluids flows to the classification of defects in simple and complex solids. We review results concerning the incidence of topological features on foundational aspects of the mechanics of simple and complex bodies, with particular emphasis on the mechanics of defects. Pertinent to this last issue, unusual concepts such as the relative power or the structure invariance of the Clausius–Duhem inequality under diffeomorphism-based observer changes are discussed.