## Generalization of the Zabolotskaya equation to all incompressible isotropic elastic solids

##### View/Open

##### Date

2019-07-03##### Author

Destrade, Michel

Pucci, Edvige

Saccomandi, Giuseppe

##### Metadata

Show full item record##### Usage

This item's downloads:

**202**(view details)Cited 2 times in Scopus (view citations)

##### Recommended Citation

Destrade, Michel, Pucci, Edvige, & Saccomandi, Giuseppe. (2019). Generalization of the Zabolotskaya equation to all incompressible isotropic elastic solids†. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 475(2227), 20190061. doi: 10.1098/rspa.2019.0061

##### Published Version

##### Abstract

We study elastic shear waves of small but finite amplitude, composed of an anti-plane shear motion and a general in-plane motion. We use a multiple scales expansion to derive an asymptotic system of coupled nonlinear equations describing their propagation in all isotropic incompressible nonlinear elastic solids, generalizing the scalar Zabolotskaya equation of compressible nonlinear elasticity. We show that for a general isotropic incompressible solid, the coupling between anti-plane and in-plane motions cannot be undone and thus conclude that linear polarization is impossible for general nonlinear two-dimensional shear waves. We then use the equations to study the evolution of a nonlinear Gaussian beam in a soft solid: we show that a pure (linearly polarized) shear beam source generates only odd harmonics, but that introducing a slight in-plane noise in the source signal leads to a second harmonic, of the same magnitude as the fifth harmonic, a phenomenon recently observed experimentally. Finally, we present examples of some special shear motions with linear polarization.