Novel dynamic-stiffness approaches to the dynamics of metamaterial structures

Metamaterial structures are structural systems that are tailored in such a way to possess periodicity in conjunction with local resonance effects. To the first class belong all the structures that exhibit a structural periodicity whose scale is comparable to the wave length of the propagating wave; these structures go by the name of phononic crystals. In the second class are included all the structural systems, known as locally-resonant acoustic metamaterials, that are endowed with periodic arrays of suitably tuned resonant subsystems. The purpose of this thesis is to propose new exact and analytical methods for the computation of the dynamic response and the wave propagation analysis of metamaterial structures by means of dynamic-stiffness based techniques. The dynamic-stiffness method presents itself as a very powerful method since it allows to exactly relate the response variables of continuous system, i.e. containing an infinite number of degrees of freedom, in the frequency domain through a finite set of degrees of freedom, namely the response variables sampled at the boundary nodes of the system. It is demonstrated that the dynamic-stiffness method is particularly suitable for computing the dynamic response and performing the wave propagation analysis of metamaterial structures very efficiently, as it allows implementing an ad hoc exact dynamic condensation of the degrees of freedom within the resonators with a significant model order reduction, not obtainable by approximate finite element methods. Furthermore, in this thesis it is shown that the dynamic-stiffness method can be profitably employed to compute the dynamic response in the time domain by deriving appropriate orthogonality conditions for the modes of the structure under the assumption of both classical and non-classical damping and by employing the modal superposition principle. This is an important result, because the dynamic-stiffness method is typically implemented in the frequency domain and no study in existing literature has ever demonstrated its applicability in the time domain as well, for any structure including metamaterial ones.