Axially symmetric nonlinear wave propagation in elastic rods made of strain-limiting materials

We study the propagation of nonlinear longitudinal waves in circular cylindrical elastic rods. We assume that the material is homogeneous, isotropic and is a strain-limiting elastic material with an asymptotic parameter that limits the overall range of strains. Starting from the three-dimensional equations of strain-limiting elasticity and using the Bernoulli–Navier hypothesis, we derive one-dimensional nonlinear equations governing the dynamics of the rod for two different material models proposed by Rajagopal and Rodriguez. Apart from this fully nonlinear case, the rod equations corresponding to linear and weakly nonlinear regimes are also derived. Two important differences between linear and weakly nonlinear cases are observed for the corresponding rod equations for two material models. The first one is that the equations describing the axial and radial dynamics are coupled for one material model and uncoupled for the other. The second one is that longitudinal wave propagation is dispersive for one of the material models but not for the other one. We also observe that when the asymptotic parameter is equal to 1, linear rod equations corresponding to one of these two models reduce to the well-known Mindlin–Herrmann equations.