On interfacial waves in pre-stressed layered incompressible elastic solids

Abstract

Interfacial waves along the plane boundary between two pre-stressed incompressible elastic solids are considered. One of the solids is a half-space while the other has arbitrary uniform thickness. The principal axes of the underlying pure homogeneous deformation in the two solids are aligned, with one axis normal to the interface. For propagation along an in-plane principal axis, the dispersion equation is derived in respect of a general strain-energy function. Conditions on the pre-strain, pre-stress and material parameters that ensure the existence of a unique interfacial wavespeed at low frequencies are obtained, and it is shown that, in special circumstances, non-dispersive waves can exist at the low-frequency limit. Asymptotic results at the high-frequency limit are also obtained. For the case of equibiaxial pre-strain, more specific conditions are derived for the existence of interfacial waves at the low- and high-frequency asymptotes, and these provide information on the existence of waves for the whole frequency range. A particular feature of the structure considered is that it may act as a mechanical filter in different frequency regimes depending on the pre-strain, pre-stress and material parameters. When the wavespeed vanishes, the dispersion equation reduces to a bifurcation equation, solutions of which define states of stress and deformation which form boundaries of the region of stability of the underlying state of stress and deformation in the two materials for given material properties. The bifurcation equation is examined separately and an explicit bifurcation criterion is given for equibiaxial deformations. The results are illustrated graphically by considering several numerical examples based on a certain class of strain-energy functions, which includes the neo-Hookean strain-energy function. The results highlight low- and high-frequency features and demonstrate the influence of pre-stress and deformation on the multiplicity of propagating modes.