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

dc.contributor.author | Destrade, Michel | |

dc.contributor.author | Pucci, Edvige | |

dc.contributor.author | Saccomandi, Giuseppe | |

dc.date.accessioned | 2019-09-04T13:57:41Z | |

dc.date.available | 2019-09-04T13:57:41Z | |

dc.date.issued | 2019-07-03 | |

dc.identifier.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 | en_IE |

dc.identifier.issn | 1364-5021 | |

dc.identifier.uri | http://hdl.handle.net/10379/15394 | |

dc.description.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. | en_IE |

dc.description.sponsorship | EP and GS have been partially supported for this work by the Gruppo Nazionale per la Fisica Matematica (GNFM) of the Italian non-profit research institution Istituto Nazionale di Alta Matematica Francesco Severi (INdAM) and the PRIN2017 project “Mathematics of active materials: From mechanobiology to smart devices” funded by the Italian Ministry of Education, Universities and Research (MIUR). We are most grateful to Gianmarco Pinton and David Esp´ındola for sharing the experimental data used to generate Figure 1. | en_IE |

dc.format | application/pdf | en_IE |

dc.language.iso | en | en_IE |

dc.publisher | The Royal Society | en_IE |

dc.relation.ispartof | Proceedings Of The Royal Society Of London Series A-Mathematical Physical And Engineering Sciences | en |

dc.subject | ANTIPLANE SHEAR DEFORMATIONS | en_IE |

dc.subject | WAVES | en_IE |

dc.subject | NONLINEARITY | en_IE |

dc.subject | BURGERS | en_IE |

dc.subject | BEAMS | en_IE |

dc.subject | 3RD | en_IE |

dc.title | Generalization of the Zabolotskaya equation to all incompressible isotropic elastic solids | en_IE |

dc.type | Article | en_IE |

dc.date.updated | 2019-08-25T11:47:10Z | |

dc.identifier.doi | 10.1098/rspa.2019.0061 | |

dc.local.publishedsource | https://doi.org/10.1098/rspa.2019.0061 | en_IE |

dc.description.peer-reviewed | peer-reviewed | |

dc.internal.rssid | 17366705 | |

dc.local.contact | Michel Destrade, Room Adb-1002, Áras De Brun, School Of Mathematics, Nui Galway. 2344 Email: michel.destrade@nuigalway.ie | |

dc.local.copyrightchecked | Yes | |

dc.local.version | ACCEPTED | |

nui.item.downloads | 20 |

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