The research objective of the present proposal is to study nonlinear waveforms that may arise in granular crystals, including traveling waves, defect modes, and discrete breathers. It also intends to provide a roadmap for identifying when these different structures will arise, systematically building up in complexity, from monomer lattices, to dimmer, trimmer and heterogeneous ones, and from one- to multiple-defect chains and even random lattices. Granular crystals consist of chains of interacting particles that deform elastically when they collide. Their properties (material types, sizes, shapes) are extremely tunable and their dynamic response may be modified to be weakly or strongly nonlinear. The project will involve a fruitful cross-pollination of tools and methods from dynamical systems, nonlinear ordinary and partial differential equations, asymptotic analysis and numerical computation on the theoretical side. These will be interwoven with physical experiments that will attempt to excite and identify the relevant nonlinear waves and to investigate their properties (amplitude, width, energy, lifetime, interactions, higher-dimensional analogs etc.). A continuous feedback loop between mathematical theory, numerical simulation and physical experimentation will be the driving force of the study.
It is anticipated that this project will have a significant societal impact due to the potential discovery of mechanical systems that may affect aspects of everyday life. In particular, localized modes will be studied for potential use in energy harvesting systems, delay lines and shock protective or vibration absorption materials. Additionally, the relevant effort will involve student researchers with a vigorous educational component including both formal (associated with coursework and research conferences) and informal (inter-group exchanges and visits) educational opportunities. It will bring together students from fundamentally different backgrounds (mathematics and aeronautical/aerospace engineering) providing them with a rich interdisciplinary experience that will offer an excellent framework for learning and scientific exchange/collaboration. The program will also foster collaborations between the two institutions, but also with other Universities (Oxford University in the UK, Princeton University and the University of Athens). We expect that several conference presentations, journal articles and patent applications will emerge from this study.
Granular chains are chains of elements, most commonly of spherical shape which interact through elastic forces of compression. Despite its apparent simplicity in both physical visualization (a typical example is the well-known Newton's cradle toy) and in mathematical modeling (according to Newton's second law of dynamics accounting for the force on each bead from its neighboring ones), the system carries an extreme wealth of possibilities. These encompass both the potential heterogeneity in the chain (e.g. all beads the same, or one bead of different mass, or a chain of two types of alternating beads --a dimer--, or many other possible variations) and the role of the nonlinearity, ranging from linear, to weakly nonlinear to purely nonlinear. This diversity of setups produces a wide range of possibilities in terms of dynamical observations. Moreover, recent experimental techniques (including ones developed in collaboration with our group) permit the detailed visualization of the system's space-time evolution. The exploration of our group during the span of this award has had multi-faceted impacts on the mathematical and physical understanding of this system and a significant contribution even in its engineering applications. More specifically, we have systematically identified and classified the possible wave patterns that emerge in the system in its simplest homogeneous form, extending from shock waves, to traveling waves and finally to so-called dark breathers (exponentially localized, time periodic waveforms supported on top of a pedestal). Moreover, we have explored systematically how its dynamics substantially changes in the presence of heterogeneity (which generates the potential for trapping energy locally e.g. at a defect) and even creates the potential for fundamentally distinct types of excitations predicted theoretically/numerically and corroborated experimentally. We have also touched upon various frontiers consisting of significant departures from the main setup. As such we have considered: (a) chains with internal resonators which support nanoptera (localized waves but with nonvanishing tails) (b) disordered chains and how they transport the energy in a fundamentally different way than believed before and finally (c) two-dimensional realizations of self-similarly decaying (in amplitude) propagating waves in genuinely two-dimensional hexagonal crystals. In addition to these (and numerous other, such as the creation of granular switches and acoustic logic elements) fundamental discoveries, this grant has enabled the training of a wide range of young scientists and mathematicians on this theme of research. 4 undergraduate students have substantially benefited from Research Experiences for Undergraduates under the auspices of this award. Moreover, one M.Sc. student and 2 PhD students have also been partially supported by the award. The results of the research efforts have been disseminated in some of the best known/most highly reputed journals of the respective fields and in prestigious conferences (e.g. in Oberwolfach, AMS sectional meetings, the AIMS meeting in Madrid and most recently a Keynote speaker invitation at the IMACS Nonlinear Waves conference). In all of the above aspects, the award has proved catalytic for a wide range of lasting developments within the research, personnel, and curricular aspects of our work and a potential continuation thereof is expected to play a similarly critical contribution in the mission of the research team.
