Wave dispersion analysis in auxetic rods

Abstract

Very few works have dealt with the wave propagation in auxetic rods. They mainly considered the effect of density changes on the longitudinal wave velocity. To this goal, the elementary one-dimensional wave propagation equation has been modified to include the high variation in material density that occurs in auxetic materials. In this work, we are mainly interested in analysing wave dispersion in auxetic rods. To this purpose, the three-dimensional Pochhammer-Chree equation is solved, for a Poisson’s ratio ν ranging from −1 to 0.5, using the regula-falsi method. We mainly show that the Poisson’s ratio controls the wave dispersion in rods through two effects: radial inertia and shear stiffness. Increasing the module of the Poisson’s ratio leads to an increase in radial inertia thus reducing wave velocity. This happens at low frequencies or large wavelengths. Consequently, the wave velocity within this range is mainly governed by the Poisson’s ratio module. On the opposite, in the high frequencies range (short wavelengths range) the wave velocity is rather governed by the shear stiffness of the auxetic rod. The wave velocity at vanishing wavelengths converges to the velocity of Rayleigh waves, which increases as the Poisson’s ratio decreases. In the high frequencies range, wave velocity is sensitive to the algebraic value of the Poisson’s ratio. In this work, we have also dealt with higher modes (higher solutions). Namely, the Pochhammer-Chree equation have been solved for the second and third modes; and analytical approximations were proposed for both low and high wavenumber ranges.