Influence of stiffening on elastic wave propagation in extremely deformed soft matter: From nearly incompressible to auxetic materials

We analyse the propagation of elastic waves in soft materials subjected to finite deformations. We derive explicit phase velocity relations for matter with pronounced stiffening effect, namely Gent model, and apply these results to study elastic wave propagation in (a) nearly incompressible materials such as biological tissues and polymers, (b) highly compressible and (c) negative Poisson's ratio or auxetic materials. We find, that for nearly incompressible materials transverse wave velocities exhibit strong dependence on the direction of propagation and initial strain state, whereas the longitudinal wave velocity is not affected significantly until extreme levels of deformation are attained. For highly compressible materials, we show that both longitudinal and transversal wave velocities depend strongly on deformation and direction of propagation. Moreover, the dependence becomes stronger when stiffening effects increase. When compression is applied, the longitudinal wave velocity decreases regardless the direction of wave propagation in highly compressible materials, and increases for most of the directions in materials with negative Poisson's ratio behaviour. We demonstrate that finite deformations can influence elastic wave propagation through combinations of induced effective compressibility and stiffness.