Published: April 3, 2015

Exact Dispersion Relations For Finite-Strain Elastic Waves In Solids  

Mahmoud Hussein

Department of Aerospace Engineering SciencesUniversity of Colorado Boulder

Date and time: 

Friday, April 3, 2015 - 3:00pm


ECCR 265


The introduction of nonlinearities to the dynamics of a homogeneous elastic medium alters the underlying wave dispersion characteristics.In this work, we present an exact formulation for the treatment of geometric nonlinearity in one-dimensional elastic wave propagation in a thin rod where the thickness is small compared to the wavelength. Our derivation starts with the implementation of Hamilton’s principle and terminates with an expression for the finite-strain dispersion relation in closed form. We explore the effect of wave amplitude on the derived dispersion relation and compare with results obtained by direct time-domain simulations followed by inverse Fourier transformations. Finally, we combine the standard transfer matrix method with our nonlinear formulation to obtain approximate band structures for a 1D elastic phononic crystal and a 1D elastic metamaterial. This work provides insights into the fundamentals of nonlinear wave propagation in solids. While often dispersion is attributed to only linear mechanisms, here we show that an otherwise linearly nondispersive elastic solid may exhibit dispersion solely due to the presence of a nonlinearity.