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.
This project studied the nonlinear waveforms that arise in granular crystals, which are assemblied of particles in contact with each other, organized in ordered configurations. The properties of granular crystals (material types, sizes, shapes) are extremely tunable and their dynamic response may be modified to be weakly or strongly nonlinear, for example, by adding a small static precompression to the particles. This level of tunability is valuable not only for basic physics studies but also for several engineering applications (for example, shock and energy/sound absorbing layers, sound focusing devices, and sound scramblers). The project studied the behavior of traveling waves, defect modes, and energy localization. We approached these studies using experiments and informed our experimental tests and analysis with numerical and analytical approaches. As a result, we were able to build a roadmap to identify how to tailor stress wave propagation in granular crystals. This will enable, in the future, the design of novel composite materials and mechanical systems to absorb impact more efficiently, focus mechanical energy and potentially harvest it to power small electronics. Our work was fundamental in nature and focused initially on simpler geometries (for example, one dimensional arrays of solid particles), building complexity at later stages of the work. For example, in the final year of the project, we studied how hollow particles respond to acoustic excitations and how particles filled with fluids can absorb impact energy, or how particles with an internal resonator can trap vibrations. From a techical standpoint, the findings of our work provided significant new insights on the physics of complex and inherently discrete dynamical systems. The fundamental understanding obtained in wave localization and energy transport will be applicable to the development of novel engineering systems (for example, usable in energy harvesting, impact absorption and non destructive evaluation of materials). The project trained two undergraduate students, which joined the research team for summer research projects, and supported one graduate student towards obtaining a doctoral degree (PhD) at Caltech. It also partially supported a post doctoral scholar and supported the continuation of his career in academia. The research team also hosted high school students for visits of variable duration of time. The research for this project has been developed in collaboration with other research groups in the United States and abroad, exposing the students and researchers involved in the work to a diverse and international research environment. The findings of this work have been published in technical journal papers, presented at international conferences and also presented in broad audience lectures and departmental seminars. They were also included in a book chapter. Some of the findings related to this projects have been presented and highlighted in broad news media reports (CNN, Popular Science, CBS, Scientific American, etc.).
