A weakly nonlinear Love hypothesis for longitudinal waves in elastic rods

When investigating nonlinear wave propagation in slender hyperelastic rods, the usual stance is to construct a reduced kinematics and then derive a system of coupled nonlinear PDEs for the unknown functions. To make further analytical progress, the linear Love hypothesis, that connects longitudinal and transversal strain, is often reverted to. The viability of this assumption, that was originally proposed within the framework of linear elasticity, remains uncertain. In this paper, a refined Love hypothesis is derived in the weakly nonlinear regime by slow-time perturbation of the motion equations. For the sake of illustration, the simplest two-modal setting is adopted. This refined Love assumption is not equivalent, not even in principle, to that derived by Porubov and Samsonov (1993) by accommodating for the free boundary conditions at the rod mantle. Besides, the perturbation process lends a uni-dimensional model equation which parallels that obtained by Ostrovskii and Sutin (1977) with the help of the linear Love hypothesis, with yet different coefficients in the dispersive term. The corresponding longitudinal motion is compared numerically against the solution of the bimodal nonlinear system and the transversal motion is contrasted with the linear Love hypothesis. For both motions, excellent agreement is found and the quality of the approximation extends to a wide range of values for the small parameter. Finally, within this setting, the corresponding unimodal Lagrangian is also derived, and it remains accurate regardless of the first correction terms to the linear Love hypothesis.