Explicit dispersion relations for elastic waves in extremely deformed soft matter with application to nearly incompressible and auxetic materials

TL;DR

This paper derives explicit dispersion relations for elastic waves in highly deformed soft materials, revealing how deformation influences wave velocities in nearly incompressible and auxetic materials, with implications for biological tissues and engineered materials.

Contribution

It provides the first explicit dispersion relations for elastic waves in extremely deformed soft matter, linking deformation to wave velocity changes in various material types.

Findings

Transverse wave velocities depend strongly on direction and strain in nearly incompressible materials.

Longitudinal wave velocities are stable until extreme deformations occur.

Deformation affects wave velocities through changes in effective compressibility and stiffness.

Abstract

We analyze the propagation of elastic waves in soft materials subjected to finite deformations. We derive explicit dispersion relations, and apply these results to study elastic wave propagation in (i) nearly incompressible materials such as biological tissues and polymers, and (ii) negative Poisson's ratio or auxetic materials. We find that for nearly incompressible materials transverse wave velocities exhibit strong dependence on direction of propagation and initial strain state, whereas the longitudinal component is not affected significantly until extreme levels of deformations are attained. For highly compressible materials, we show that both pressure and shear wave velocities depend strongly on initial deformation and direction of propagation. When compression is applied, longitudinal wave velocity decreases in positive bulk modulus materials, and increases for negative bulk modulus materials; this is regardless the direction of wave prorogation. We demonstrate that finite deformations influence elastic wave propagation through combinations of induced effective compressibility and stiffness.