  The paper studies the interaction of a longitudinal wave with transverse
waves in general isotropic and unconstrained hyperelastic materials, including
the possibility of dissipation. The dissipative term chosen is similar to the
classical stress tensor describing a Stokesian fluid and is commonly used in
nonlinear acoustics. The aim of this research is to derive the corresponding
general equations of motion, valid for any possible form of the strain energy
function and to investigate the possibility of obtaining some general and exact
solutions to these equations by reducing them to a set of ordinary differential
equations. Then the reductions can lead to some exact closed-form solutions for
special classes of materials (here the examples of the Hadamard, Blatz-Ko, and
power-law strain energy densities are considered, as well as fourth-order
elasticity). The solutions derived are in a time/space separable form and may
be interpreted as generalized oscillatory shearing motions and generalized
sinusoidal standing waves. By means of standard methods of dynamical systems
theory, some peculiar properties of waves propagating in compressible materials
are uncovered, such as for example, the emergence of destabilizing effects.
These latter features exist for highly nonlinear strain energy functions such
as the relatively simple power-law strain energy, but they cannot exist in the
framework of fourth-order elasticity.
