2013/02/01 by Michel Destrade, Alain Goriely, Giuseppe Saccomandi · 1 citation
Physics and Astronomy · Mathematics · #cond-mat.soft #math-ph #math.MP
paper · pdf · doi:10.1098/rspa.2010.0508
published as Proceedings of the Royal Society A 467 (2011) 1823-1834 · 15 pages
arxiv created 2013/02/01 · arxiv updated 2013/02/04
We study the propagation of two-dimensional finite-amplitude shear waves in a nonlinear pre-strained 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 neo-Hookean solids (with strain energy depending only on the first principal invariant of Cauchy-Green strain). However, we show that the Z equation cannot be a scalar equation for the propagation of two-dimensional 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 Kadomtsev-Petviashvili (KP), Zabolotskaya-Khokhlov (ZK) and Khokhlov-Zabolotskaya-Kuznetsov (KZK) equations of incompressible solid mechanics.