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arXiv · 1906.08087

Generalization of the Zabolotskaya equation to all incompressible isotropic elastic solids

Michel Destrade·Edvige Pucci·Giuseppe Saccomandi
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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 non-linear 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 polarised) 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 polarisation.

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Michel Destrade, Edvige Pucci, Giuseppe Saccomandi. 2019-06-19. Generalization of the Zabolotskaya equation to all incompressible isotropic elastic solids. https://doi.org/10.1098/rspa.2019.0061

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