Longitudinal Wave Propagation in Stretched Polymers

Abstract

The wave equation is derived for the propagation of longitudinal waves along a stretched filament of a highly elastic material. It is found that the tension in the filament reduces the effective modulus by twice the product of Poisson's ratio and the tensile traction, leaving the internal viscosity term unchanged. This result is illustrated by measurements of continuous 1 kcps wave propagation in natural rubber at 50°C, where the damping is small (tan δ<0.1). An alternative derivation is given for the purely elastic case of zero damping without restriction upon the amplitude. The ``equilibrium'' or ``static'' Young's modulus is obtained for extensions up to about 600% from the slope of the equilibrium stress-strain curve and used to predict the corresponding wave velocities from the wave equation for zero damping. The predicted velocities are slightly higher—by up to about 10%—than the measured velocities. It is shown that the deviations could arise from differences in rate of strain between the wave-propagation and the stress-strain measurements. At the higher extensions the rubber is very hysteretic for large deformations, and the Young's modulus governing the small-amplitude wave propagation is shown to relate substantially to the loading branch of the stress-strain curve.