Scalar evolution equations for shear waves in incompressible solids: a simple derivation of the Z, ZK, KZK and KP equations
Abstract
We study the propagation of twodimensional finiteamplitude shear waves in a nonlinear prestrained incompressible solid, and derive several asymptotic amplitude equations in a simple, consistent, and rigorous manner. The scalar Zabolotskaya (Z) equation is shown to be the asymptotic limit of the equations of motion for all elastic generalized neoHookean solids (with strain energy depending only on the first principal invariant of CauchyGreen strain). However, we show that the Z equation cannot be a scalar equation for the propagation of twodimensional shear waves in general elastic materials (with strain energy depending on the first and second principal invariants of strain). Then we introduce dispersive and dissipative terms to deduce the scalar KadomtsevPetviashvili (KP), ZabolotskayaKhokhlov (ZK) and KhokhlovZabolotskayaKuznetsov (KZK) equations of incompressible solid mechanics.
 Publication:

Proceedings of the Royal Society of London Series A
 Pub Date:
 July 2011
 DOI:
 10.1098/rspa.2010.0508
 arXiv:
 arXiv:1302.0109
 Bibcode:
 2011RSPSA.467.1823D
 Keywords:

 Condensed Matter  Soft Condensed Matter;
 Mathematical Physics
 EPrint:
 15 pages