Transverse Waves in Nonlinearly Elastic Solids and the Milne-Pinney (or Ermakov) Equation
Michel Destrade, Giuseppe Saccomandi
TL;DR
This paper links nonlinear elastodynamics equations to the Milne-Pinney equation, enabling the derivation of new exact and approximate solutions for various incompressible solid models using traveling wave solutions.
Contribution
It establishes a novel connection between elastodynamics and the Milne-Pinney equation, facilitating new solution methods for complex nonlinear elastic models.
Findings
Derived new exact solutions for neo-Hookean and Mooney-Rivlin solids.
Developed approximate solutions for third- and fourth-order elasticity models.
Provided a systematic approach to solve nonlinear elastodynamics equations.
Abstract
We establish a connection between the general equations of nonlinear elastodynamics and the nonlinear ordinary differential equation of Pinney [Proc. Amer. Math. Soc. 1 (1950) 681]. As a starting point, we use the exact travelling wave solutions of nonlinear elasticity discovered by Carroll [Acta Mechanica 3 (1967) 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.
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