RUI: Dispersive Shock Waves in Nonlinear Lattices: Theory to Application

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Understanding how systems respond to sudden changes is critical to science and engineering applications. Classic examples include explosions in confined areas, gases compressed in a piston chamber, or breaking of water reservoir dams. Important applications include design of impact absorbers, which are important in protecting civil infrastructure and passengers in automobile accidents. They also play a crucial role in the deployment and landing of spacecraft. This project studies a class of systems modeled by nonlinear lattices, mathematical models of chains of particles that interact with each other in a nonlinear fashion. Models of origami-based materials will be the primary nonlinear lattices considered in this project since such materials exhibit desirable properties for applications. For example, the harder an origami lattice is hit, the slower a wave will travel through it. The project aims to improve understanding of wave propagation in nonlinear lattices through mathematical modeling, computer simulation, and experimentation. Results are expected to aid in the design of impact-mitigating devices. Students who belong to groups underrepresented in the sciences will be trained and recruited to participate in summer research experiences via a work-study program. In some systems subjected to a sudden change in state, an oscillating wave is formed that connects (local) states of different amplitude. Such oscillating waves are called dispersive shock waves (DSWs). The standard approach to analyze a DSW is based on Whitham modulation theory. In the case of nonlinear lattices, the Whitham modulation equations are prohibitively complex. In this project, a low-dimensional differential equation that accurately describes the waves that make up a DSW in a lattice will be sought using data-driven methodologies. This low-dimensional differential equation will be exploited to obtain a simple analytical description of a DSW. A quasi-continuum approach will also be employed to analytically identify the underlying low-dimensional differential equation. A systematic study of two-dimensional DSWs will also be conducted. Numerical simulations, small-amplitude approximations, modulation theory, and experimentation will be employed.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.