Wave propagation and wrinkling localization in a prestressed periodic elastic half-space

Abstract

We investigate surface waves in a prestressed elastic half-space whose material properties are constant in the vertical direction but vary periodically along the direction of wave propagation. Using Floquet-Bloch theory, Fourier expansion, and the Stroh formalism, we reduce the dynamic problem to the solution of a matrix Riccati equation for a high-dimensional (truncated) surface impedance matrix. Illustrative results are presented for the case in which each unit cell consists of two homogeneous materials. A two-term asymptotic expression for the wave speed is derived for wavenumbers near the edges of the Brillouin zone and is used to validate the numerical solutions. By increasing the prestress until the wave speed vanishes, we show that surface wrinkling in such a periodic half-space always localizes in the material that is more susceptible to instability and is not governed by homogenized properties. Furthermore, the critical prestress is found to be independent of the wavenumber, as in the classical homogeneous case, although surface waves remain dispersive before the onset of instability.