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 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.

15 pages

References in corpus (2)

• Onset of Non-Linearity in the Elastic Bending of Blocks
• Solitary and compact-like shear waves in the bulk of solids

Cited by in corpus (5)

• Methodical Fitting for Mathematical Models of Rubber-like Materials
• Generalization of the Zabolotskaya equation to all incompressible isotropic elastic solids
• Likely oscillatory motions of stochastic hyperelastic solids
• Topological charges and conservation laws involving an arbitrary function of time for dynamical PDEs
• Computation of viscoelastic shear shock waves using finite volume schemes with artificial compressibility