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.

Generalized continua represent a class of models whose potential applicability seems to have been underestimated. The mathematical structure of these models is discussed and the reasons why it has been underestimated are made clear. Their importance in the theory of metamaterials is highlighted and their potential impact on future technological applications is carefully argued. It is shown how the original ideas of Lagrange and Piola can be developed by using the modern tools of differential geometry, as formulated by Ricci and Levi-Civita. It has to be concluded that variational principles are the most powerful tool also in the mathematical modeling of metamaterials.

A most crucial aspect of the intellectual activity needed to comprehend the theory of metamaterials consists in the capacity to distinguish between the physical object which is studied and the possibly different models used to describe it, in different situations. A metamaterial is a material whose behavior is chosen "a priori" by fixing the mathematical model to be used to describe it, in a specific set of conditions. In a sense, ontologically, in the theory of metamaterials the models are used to define "a posteriori" some physical objects. In this context the "feasibility" or "possibility of existence" of a certain material represents a major conceptual problem. For these reasons a scientist working in this field must be aware of some basic concepts of Model Theory.

To get predictions from theoretical models of complex mechanical systems, the numerical tools are essential, as very few results can be obtained using analytical methods, especially when large deformations are involved. Variational methods are the preferred (or probably the most powerful) tool to formulate the numerical codes to be used, also in the study of metamaterials. A presentation focused on some aspects of numerical techniques, relevant to the considered class of problems, is presented.