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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 slender rod, where the thickness is small compared to the wavelength, we present an exact formulation for the treatment of a nonlinearity in the strain-displacement gradient relation. As an example, we consider Green Lagrange 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. A method is then provided for extending this analysis to a continuous thin rod periodic layering (phononic crystal) or with periodically embedded local resonators (elastic metamaterial). 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. The effects of the nonlinearity on the subwavelength band gap, among other intriguing outcomes, are highlighted.
This work provides insights into the fundamentals of nonlinear wave propagation in solids, both natural and engineereda 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.
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