Dynamic Poynting coupling

Abstract

Plane wave propagation in a compressible isotropic hyperelastic material is treated. Emphasis is on the Poynting coupling, which is analysed in a linear and a weakly nonlinear approximation for a statically predeformed material. Unidirectional and resonating waves in the material with and without predeformation are described. NotationThe symbols used in this paper are collected here for convenient reference. They are listed roughly in the order of the appearence in the text. uY vY w dimensional/dimensionless displacement components in longitudinal andspace-®xed, and particle-®xed coordinates, fast time, slow times .mass density of undeformed material R Helmholtz free energy density, internal energy per unit volume of undeformed material c L Y c T propagation velocities of in®nitesimal longitudinal and transversal deformation waves of linear elasticity c AE Y c LYT propagation velocities of in®nitesimal deformation waves in a homogeneously pre-strained hyperelastic material c propagation velocity of statically precompressed or pre-expanded hyperelastic material (without preshear) AY BY CY DY E second AY B and third order CY DY E elastic constants aY bY bY k nondimensional elastic constants associated with AY BY DY respectively 0 Y c 0 Y v 0 homogeneous prestrains wave amplitude-ordering parameter m c L ac T n X À t f Y gY hY FY G strain or particle velocity components aY bY c acceleration wave amplitudes a 0 Y b 0 Y c 0 initial (ballistic) acceleration wave amplitudes K a Y KY k bifurcation parameters in acceleration wave problems m aa4 b q nonlinearity parameter VY m transversal and longitudinal excitation displacement amplitude