  Extremal elastic materials here refer to a specific class of elastic
materials whose elastic matrices exhibit one or more zero eigenvalues,
resulting in soft deformation modes that, in principle, cost no energy. They
can be approximated through artificially designed solid microstructures.
Extremal elastic materials have exotic bulk wave properties unavailable with
conventional solids due to the soft modes, offering unprecedented opportunities
for manipulating bulk waves, e.g., acting as phonon polarizers for elastic
waves or invisibility cloaks for underwater acoustic waves. Despite their
potential, Rayleigh surface waves, crucially linked to bulk wave behaviors of
such extremal elastic materials, have largely remained unexplored so far. In
this paper, we theoretically investigate the propagation of Rayleigh waves in
extremal elastic materials based on continuum theory and verify our findings
with designed microstructure metamaterials based on pantographic structures.
Dispersion relations and polarizations of Rayleigh waves in extremal elastic
materials are derived, and the impact of higher order gradient effects is also
investigated by using strain gradient theory. This study provides a continuum
model for exploring surface waves in extremal elastic materials and may
stimulate applications of extremal elastic materials for controlling surface
waves.
