Lagrangian discrete models are the most ancient and still most effective models used in mechanics to predict the behavior of complex systems. Their structure is presented here, in a very classical way, which follows the classical presentation given by Levi-Civita. Their flexibility is highlighted, together with their capacity to be used in very efficient numerical codes. They must be regarded as a preferred tool also in the formulation of problems in the theory of metamaterials.

A targeted discussion of the state of the art in the field of metamaterials' design, modeling and construction is presented. Only some of the most interesting aspects of the theoretical and experimental investigations available in the literature are described, by selecting the most innovative or methodologically interesting ones. After a preliminary analysis of these aspects, those which seem to be the most promising future research directions are sketched. The most important challenges in the field are delineated, in order to motivate the reader who wants to become acquainted with presented subject.

A long debate in the mechanicians' community was started by the seminal works by Piola, Mindlin, Rivlin, Toupin, Sedov and Germain. Higher gradient or microstructured continuum models have been questioned in several aspects. Sometimes they have been regarded as an empty mathematical "game" devoid of any physical application or, worse, they were considered to be inconsistent with the second principle of thermodynamics. Pantographic metamaterials, i.e. metamaterials having a multiscale pantographic microstructure, have been initially introduced in order to give an example of materials whose macroscopic continuous description must necessarily be given by a second gradient continuum model. Once 3D printing technology allowed for the realization of these microstructures it has been discovered that this class of metamaterials exhibits very interesting features, which may possibly lead to interesting technological applications.

The scope of this volume is limited to metamaterials based on microstructural phenomena involving purely mechanical interactions. In general the exotic behavior of metamaterials is obtained by using multiscale architectured internal structures: it is assumed here that at the lowest considered scale a mechanical description is sufficient. The literature in the field being enormous, only a targeted selection of mechanical metamaterials has been considered, aiming to give an analysis of the literature relevant to the specific application developed in Chapter 3.

Once a metamaterial has been conceived, designed and built, its expected properties must be experimentally verified, in order to validate the conceptual analysis leading to it and the construction process used to realize it. Using 3D printing technology is not always a trivial task, especially if the designed microstructures are complex and show large differences in their geometrical and mechanical properties, at lower scales. Moreover, once some specimens are built, some specific experimental apparatuses have to be designed that are able to manifest the specific desired exotic mechanical features which are the target of the whole research effort. Therefore it is not a simple task to prove that the pantographic microstructured metamaterials do really exhibit the behavior which is expected. The gathered evidence which shows the validity of the concept of pantographic metamaterial is carefully presented here.