Nonlinear Waves in Lattices and Metamaterials

University Of Pittsburgh, Pittsburgh PA

Investigators

Abstract

The project investigates the dynamics of transition fronts and other nonlinear waves in spatially discrete systems. These waves play a major role in transporting energy in lattices and mechanical metamaterials, engineered structures that exploit instabilities of their components to yield a desired collective response. The project aims to advance the fundamental understanding of the energy transfer phenomena associated with the wave propagation. This information is important for designing novel mechanical metamaterials with desired characteristics that enable applications in soft robotics, morphing surfaces, reconfigurable devices, and mechanical logic, among others. Due to the ubiquity of nonlinear transition waves in physical and biological settings, understanding their properties, as well as the conditions for their existence and stability, is relevant in fields such as mechanical engineering, materials science, condensed matter physics, and biophysics. The project will provide research training opportunities for doctoral students. This project will study traveling transition waves in a Fermi-Pasta-Ulam lattice and its various extensions that include viscous dissipation, diatomic structure, and long-range interactions. The intent is to provide insights into the effects of nonlinearity, nonconvexity, dissipation, heterogeneity, and nonlocality on the existence and stability of different types of transition waves and the extent to which some of these effects can be captured by quasicontinuum approximations. Open fundamental questions in the dynamics of transition waves will be addressed, including the dispersive mechanism of energy transfer inside the transition front of a supersonic kink and the structure of the kinetic relations for subsonic kinks. The research program involves the development of analytical and numerical approaches, such as representing traveling waves as fixed points of a nonlinear map and constructing semi-analytical solutions of the problems with piecewise linear nearest-neighbor interactions, that will be useful in other studies of discrete nonlinear systems. The constructed solutions will serve as benchmark cases for testing quasicontinuum approximations and inform the investigation of the fully nonlinear problems. Numerical explorations of the consequences of instability will clarify the solution structure and possibly reveal bifurcations of other nonlinear waveforms. 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.