Theoretical estimates of the parameters of longitudinal undular bores in polymethylmethacrylate bars based on their measured initial speeds

Abstract

We study the evolution of the longitudinal release wave that is generated by induced tensile fracture as it propagates through solid rectangular polymethylmethacrylate (PMMA) bars of different constant cross-section. High-speed multi-point photoelasticity is used to register the strain wave at three distances from the fracture site in each experiment. In all cases, oscillations develop at the bottom of the release wave that exhibit the qualitative features of an undular bore. The pre-strain, post-strain, strain rate of the release wave and the cross-section dimensions determine the evolution of the oscillations. From the wave speed and strain rate close to the fracture site, we estimate the strain rate of the release wave as well as the growth of the amplitude and duration of the leading oscillation away from the fracture site by using formulae derived from the simple analytical solution of the linearized Gardner equation (linearized near the pre-strain level at fracture). Our estimates are then compared to experimental data, where good agreements of these three parameters are found between the predictions of the model and the experimental observations. Thus, we developed an approach to estimating the key characteristics of the developing unsteady undular bore based on the measured initial speeds of the longitudinal and shear waves. This does not require a prior knowledge of the elastic moduli for the conditions of the experiments, which in PMMA are known to be strain rate dependent.