| Summary: | <p>Phenomena related to disorder and nonlinearity have been studied in many different context in physics, from condensed-matter physics to optics, with a substantial potential for applications. In particular, in mechanical systems, one can think of "smart" materials that adapt their properties, such as thermal conductivity or wave-transmission properties, depending on external stimuli, such as changes in pressure. It is the aim of this dissertation study transport and localization properties of one-dimensional disordered and quasiperiodic granular crystals as a function of an external precompression on the system. To archive this, we first study the scattering problem of a single impurity in a homogeneous granular chain and we demonstrate the existence of analogs to quantum resonances. </p>
<p>We then study spreading of initially localized excitations. We thereby investigate localization phenomena in strongly nonlinear systems, which are fundamentally different from such phenomena in linear and weakly nonlinear systems. We conduct a thorough comparison of wave dynamics in chains with three different types of disorder: an uncorrelated (Anderson-like) disorder and two types of correlated disorders (which are produced by random dimer arrangements). We find that, in most of the cases studied the behavior of the second moment and the inverse participation ratio for large times are given by power laws. However, for low levels of precompression and initial perturbations on the displacement of the particles, we find out that there is not a clear trend for the second moment. For an Anderson-like uncorrelated disorder, we find some regimes in the parameter space where a transition from subdiffusive to superdiffusive dynamics emerges depending on the amount of precompression in the chain. By contrast, for the correlated disorders, we find that the dynamics is superdiffusive for any precompression level. Additionally, for large precompression, the inverse participation ratio decreases slowly and the dynamics leads to partial localization around the initial wave. This localization phenomenon does not occur in the sonic vacuum regime, which yields that spontaneous localization is no longer possible. </p>
<p>We also study quasipediodic granular crystals, in particular, localization of waves in strongly precompressed granular chains induced by quasiperiodicity. We propose three different set-ups, inspired by the Aubry--André (AA) model, of quasiperiodic chains; and we use these models to compare the effects of on-site and off-site quasiperiodicity. When there is purely on-site quasiperiodicity, which we implement in two different ways, we show for a chain of spherical particles that there is a localization transition (as in the original AA model). However, we observe no localization transition in a chain of cylindrical particles in which we incorporate quasiperiodicity in the distribution of contact angles between adjacent cylinders by making the angle periodicity incommensurate with that of the chain. For each of our three models, we compute the Hofstadter spectrum and the associated Minkowski--Bouligand fractal dimension, and we demonstrate that the fractal dimension decreases as one approaches the localization transition (when it exists). We also show, using the chain of cylinders as an example, how to recover the Hofstadter spectrum from the system dynamics. Finally, in a suite of numerical computations, we demonstrate localization and also that there exist regimes of ballistic, superdiffusive, diffusive and subdiffusive transport.</p>
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