Non-existence of propagating Rayleigh waves in extremal materials

Extremal materials are a class of Cauchy materials with rank-deficient elastic matrix, i.e. exhibiting one or multiple zero eigenvalues and allowing energy-free deformation modes. In a previous study, we demonstrated that, in the Cauchy framework, no propagating Rayleigh wave exists when the extremal materials' principal axis is parallel to the free surface (Wei et al. 2024 J. Mech. Phys. Solids 193, 105842. (doi:10.1016/j.jmps.2024.105842)). However, a question is raised naturally: can any extremal material support propagating Rayleigh wave? In this paper, we theoretically investigate the propagation of Rayleigh waves in any extremal materials based on the Cauchy framework. Dispersion relations and polarizations of Rayleigh waves in extremal materials are derived analytically. Designing a conservative function in the weak form, we prove the non-existence of propagating Rayleigh waves in any two-dimensional extremal materials, and calculate the corresponding Rayleigh modes analytically. Moreover, we illustrate the existence condition for propagating Rayleigh waves in special three-dimensional extremal materials analytically in a similar way. A Rayleigh wave isolator is proposed and demonstrated by using a piece of extremal material. This study provides a continuum model for exploring surface waves in any extremal materials and paves the way to stimulate applications of extremal materials for controlling surface waves.