  It is the aim of the paper to present a new point of view on rotational
elasticity in a nonlinear setting using orthogonal matrices. The proposed
model, in the linear approximation, can be compared to the well known
equilibrium equations of static linear elasticity. An appropriate kinetic
energy is identified and we present a dynamical model of rotational elasticity.
The propagation of elastic waves in such a medium is studied and we find two
classes of waves, transversal rotational waves and longitudinal rotational
waves, both of which are solutions of the nonlinear partial differential
equations. For certain parameter choices, the transversal wave velocity can be
greater than the longitudinal wave velocity. Moreover, parameter ranges are
identified where the model describes an auxetic material. However, in all cases
the potential energy functional is positive definite. Finally, we couple the
rotational waves to linear elastic waves to study the behaviour of the coupled
system. We find wave like solutions to the coupled equations and can visualise
our results with the help of suitable figures.
