Modeling and designing micro- and nano-structured metamaterials: Towards the application of exotic behaviors

The lack of suitable computational methods has significantly restricted the creativity of engineers in designing the materials to be used in technological applications. When one wants exact analytical solutions for a given physical system, then usually drastic and restrictive simplifying assumptions are needed. In particular, homogeneity of physical and geometrical properties at lower length-scale is the standard assumption in continuum mechanics. On the other hand, it is well-known since the pioneering work of Gabrio Piola, and then re-established in the works by Mindlin, Toupin, Green, Adkins and Germain, that it is possible to synthetically describe microscopic inhomogeneity by means of field theories incorporating additional kinematical fields. The characteristic length-scale affecting macro-behavior can even be of the order of nanometers, in which case the intuition due to Richard Feynman about the importance of quantum effects at macro-scale could open the path to technological advancements. In the present paper we review some of the literature in the field and try to indicate some research perspectives that seem to us potentially ground-breaking. In particular, following the suggestion of Professor dell’Isola, we briefly describe his concept of pantographic lattices and sheets whose importance in nano-technology could be relevant.