Transverse waves in nonlinearly elastic solids and the Milne

Abstract

We establish a connection between the general equations of nonlinear elastodynamics and the nonlinear ordinary differential equation of Pinney [Proc Amer Math Soc 1950; 1: 681]. As a starting point, we use the exact travelling wave solutions of nonlinear elasticity discovered by Carroll [Acta Mechanica 1967; 3: 167]. The connection provides a method for finding new exact and approximate dynamic solutions for neo-Hookean and Mooney—Rivlin solids, and for the general third- and fourth-order elasticity models of incompressible solids.

Keywords

nonlinear elastodynamics exact solutions Milne-Pinney equation Ermakov equation

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Transverse waves in nonlinearly elastic solids and the Milne—Pinney (or Ermakov) equation

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