ANALISI DI FOURIER E LE SUE APPLICAZIONI ALLA PERIDINAMICA

In this thesis we evaluate the importance of the Fourier analysis in real phenomena such as fracture mechanics. It is divided in two main sections: the first one is based on the theory of Fourier analysis; the second, instead, aim to describe the foundations of peridynamic model and provides an important numerical result in this topic. In the first section we quickly introduce the genesis of Fourier theory through the definition of heat equation and wave equation. We continue giving some properties of the Fourier series and Fourier transform on R. Some fundamental concepts of the peridynamic theory, such as non-locality, linear kernels and non-linear kernels, are going to be described in fourth chapter. These provided informations will be central in the main studies about the resolution of peridynamic equation of motion, that will be described, with the associated innovative linear periodic peridynamic model, in the fifth chapter. The formal investigation that is preponderant in most of the thesis, will be numerically plotted in the sixth chapter. It will provide some vital results; in fact, we will observe that the nucleation of a crack in the domain is intrinsic of the solution and not a preset condition. It shows that, due to the presence of multiple high-frequencies armonic contributes, the jump discontinuity will be dissipated for positive times and the same discontinuity can appear at a certain time. The last observation, will be about the multiple discontinuities in a dense set of times. This model is particularly useful in specific situations such as the dynamic damage of hyperelastic materials.

La tesi si propone di spiegare i principali oggetti matematici dell'analisi di Fourier con una buona parte dimostrativa. Successivamente si applicano le nozioni introdotte nello studio dei modelli peridinamici al fine di creare delle simulazioni numeriche di funzioni. Si commentano, infine, le conclusioni ottenute nello studio dei materiali iperelastici e il loro comportamento.