Cnoidal waves in solids Dedicated to Professor Ioannis Vardoulakis

Cnoidal waves are nonlinear and exact periodic stationary waves, well known in the shallow water theory of fluid mechanics. In this study we retrieve such periodic stationary wave solutions as singularities of the problem of homogeneous volumetric deformation of a rate-dependent, heterogeneous solid material. In accordance to the classical Hill stationary wave localization instability, which provides velocity gradient discontinuities in shear failure, cnoidal waves are dilational and compactional manifestations of volumetric localization along lines of stress discontinuities. They therefore emerge along the volumetric component of the classical slip line field theory, with their regular distance being a tell tale indication of rate-dependent volumetric deformation. We discuss applications for the dominant mode of I1 compaction in geomaterials where distinct cnoidal wave instabilities appear as localisation features in compaction. We also discuss the case of localisation features in a classical (J2 plastic) material where a small but important cnoidal contribution may trigger equidistant bands of localisation known as Lüders lines. We therefore postulate that cnoidal waves constitute fundamental material instabilities stemming from the propagation of elasto-plastic P-waves.

Duke Scholars

Author Manolis Veveakis Civil and Environmental Engineering