Strongly Nonlinear Elastic Wave Propagation and the Essence of Spatial Invariance

Abstract:
Wave motion lies at the heart of many disciplines in the physical sciences and engineering. For example, problems and applications involving light, sound, heat or fluid flow are all likely to involve wave dynamics at some level. In this seminar, I will present our recent work on a class of problems involving intriguing nonlinear wave phenomena‒large-deformation elastic waves in solids; that is, the “large-on-small” problem.
Specifically, we examine the propagation of a large-amplitude wave in an elastic one-dimensional medium that is undeformed at its nominal state. In this context, our focus is on the effects of inherent nonlinearities on the dispersion relation. Considering a thin rod, where the thickness is small compared to the wavelength, I will present an exact formulation for the treatment of a nonlinearity in the strain-displacement gradient relation. As examples, we consider Green Lagrange strain and Hencky strain. The ideas presented, however, apply generally to other types of nonlinearities The derivation starts with an implementation of Hamilton’s principle and terminates with an expression for the finite-strain dispersion relation in closed form. The derived relation is then verified by direct time-domain simulations, examining both instantaneous dispersion (by direct observation) and short-term, pre-breaking dispersion (by Fourier transformations), as well as by perturbation theory. The results establish a perfect match between theory and simulation and reveal that an otherwise linearly nondispersive elastic solid may exhibit dispersion solely due to the presence of a nonlinearity. The same approach is also applied to flexural waves in an Euler Bernoulli beam, demonstrating qualitatively different nonlinear dispersive effects compared to longitudinal waves. Finally, I will present a method for extending this analysis to a continuous thin rod with a periodic arrangement of material properties. The method, which is based on a standard transfer matrix augmented with a nonlinear enrichment at the constitutive material level, yields an approximate band structure that accounts for the finite wave amplitude. Using this method, I will present an analysis on the condition required for the existence of spatial invariance in the wave profile.
This work provides insights into the fundamentals of nonlinear wave propagation in solids, both natural and engineered-a problem relevant to a range of disciplines including dislocation and crack dynamics, geophysical and seismic waves, material nondestructive evaluation, biomedical imaging, elastic metamaterial engineering, among others.