Transverse waves in nonlinearly elastic solids and the Milne-Pinney equation
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Destrade, M., Saccomandi, G. (2011) 'Transverse waves in nonlinearly elastic solids and the Milne-Pinney equation'. Mathematics and Mechanics of Solids, Special Issue in Honour of M.M. Carroll [invited contribution], 16 :586-593.
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.