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.

The primary purpose of this book is to develop methods for the dynamic analysis of multibody systems (MBS) that consist of interconnected rigid and deformable components. In that sense, the objective may be considered as a generalization of methods of structural and rigid body analysis. Many mechanical and structural systems such as vehicles, space structures, robotics, mechanisms, and aircraft consist of interconnected components that undergo large translational and rotational displacements. Figure 1.1 shows examples of such systems that can be modeled as multibody systems. In general, a multibody system is defined to be a collection of subsystems called bodies, components, or substructures. The motion of the subsystems is kinematically constrained because of different types of joints, and each subsystem or component may undergo large translations and rotational displacements.

In this chapter, approximation methods are used to formulate a finite set of dynamic equations of motion of multibody systems that contain interconnected deformable bodies. As shown in Chapter 3, the dynamic equations of motion of the rigid bodies in the multibody system can be defined in terms of the mass of the body, the inertia tensor, and the generalized forces acting on the body. On the other hand, the dynamic formulation of the system equations of motion of linear structural systems requires the definition of the system mass and stiffness matrices as well as the vector of generalized forces. In this chapter, the formulation of the equations of motion of deformable bodies that undergo large translational and rotational displacements are developed using the floating frame of reference (FFR) formulation. It will be shown that the equations of motion of such systems can be written in terms of a set of inertia shape integrals in addition to the mass of the body, the inertia tensor, and the generalized forces that appear in the dynamic formulation of rigid body system equations of motion and the mass and stiffness matrices and the vector of generalized forces that appear in the dynamic equations of linear structural systems. These inertia shape integrals that depend on the assumed displacement field appear in the nonlinear terms that represent the inertia coupling between the reference motion and the elastic deformation of the body. It will be also shown that the deformable body inertia tensor depends on the elastic deformation of the body, and accordingly it is an implicit function of time.

This chapter provides explanations of some of the fundamental issues addressed in this book. It also provides detailed derivations of some of the important equations presented in previous chapters. The first two sections of this chapter show the detailed derivation of the quadratic velocity centrifugal and Coriolis force vector of the spatial flexible body presented in Chapter 5. The final expression of these forces is obtained using two different approaches; the kinetic energy and the virtual work. It is also shown in Section 3 of this chapter how a general expression of these forces that is applicable to any set of orientation parameters can be obtained. This is the expression used in the generalized Newton–Euler equations presented in Chapter 5 of the book. The generalized centrifugal and Coriolis inertia forces associated with any set of orientation parameters including Euler angles can be obtained from the forces that appear in the Newton–Euler equations using a simple velocity transformation.

Thus far, only the dynamics of multibody systems consisting of interconnected rigid bodies has been discussed. In Chapter 2, methods for the kinematic analysis of the rigid frames of reference were presented and many useful kinematic relationships and identities were developed. These kinematic equations were used in Chapter 3 to develop general formulations for the dynamic differential equations of motion of multi-rigid-body systems. In rigid body dynamics, it is assumed that the distance between two arbitrary points on the body remains constant. This implies that when a force is applied to any point on the rigid body, the resultant stresses set every other point in motion instantaneously, and as shown in the preceding chapter, the force can be considered as producing a linear acceleration for the whole body together with an angular acceleration about its center of mass. The dynamic motion of the body, in this case, can be described using Newton–Euler equations, developed in the preceding chapter.

In the preceding chapter, methods for the kinematic analysis of moving frames of reference were presented. The kinematic analysis presented in that chapter was of a preliminary nature and is fundamental for understanding the dynamic motion of moving rigid bodies or coordinate systems. In this chapter, techniques for developing the dynamic equations of motion of multibody systems (MBS) consisting of interconnected rigid bodies are introduced. The analysis of multibody systems consisting of deformable bodies that undergo large translational and rotational displacements will be deferred until we discuss in later chapters some concepts related to the body deformation. In the first three sections, a few basic concepts and definitions to be used repeatedly in this book are introduced. In these sections, the important concepts of the system generalized coordinates, holonomic and nonholonomic constraints, degrees of freedom, virtual work, and the system generalized forces are discussed. Although the reader previously may very well have met some, or even all, of these concepts and definitions, they are so fundamental for our purposes that it seems desirable to present them here in some detail. Since the direct application of Newton’s second law becomes difficult when large-scale multibody systems are considered, in Section 4, D’Alembert’s principle is used to derive Lagrange’s equation, which circumvents to some extent some of the difficulties found in applying Newton’s second law as demonstrated by the discussion and example presented in Sections 5 and 6. In contrast to Newton’s second law, the application of Lagrange’s equation requires scalar quantities such as the kinetic energy, potential energy, and virtual work. In Sections 7 and 8 the variational principles of dynamics, including Hamilton’s principle, are presented. Hamilton’s principle can also be used to derive the MBS dynamic equations of motion from scalar quantities. This chapter is concluded by discussing the numerical procedures and their relationship to the Lagrange–D’Alembert principle and by developing the equations of motion of multibody systems consisting of interconnected rigid components.