Engineering Mechanics Institute Conference 2013

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Physical interpretation of the gradient elasticity parameters via the dispersion analysis of longitudinal waves

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Roman Tokmashev
University of Minnesota
United States

Egor Dontsov
University of Minnesota
United States

Bojan Guzina
University of Minnesota
United States

The primary goal of this study is to understand the physical meaning and evaluate the intrinsic length scale parameters, featured in the gradient elasticity models, by correlating the analytical, simulated, and experimentally measured dispersion relations in a 1D periodic medium. The developments are effected by scrutinizing the dispersion of longitudinal waves in an aluminum rod with periodically varying cross-section introduced by a series of rectangular notches.

First, the analytical solution for the dispersion of longitudinal waves, based on the periodic cell analysis of bi-layered laminate and Bloch theorem, is compared to two competing gradient elasticity (GE) models. It is shown that the customary GE model with two length scale parameters is able to accurately capture the observed dispersion curve up to the beginning of the first band gap. At the same time, the GE model with an additional fourth-order time derivative (and an extra length scale parameter) allows for a faithful description of the dispersion relationship past the first band gap and into the second branch of phase velocity-wavenumber relationship. Closed-form relations between the microstructure GE parameters and intrinsic length scales are obtained for both gradient elastic models.

Next, the analysis is verified through experimental measurements of the phase velocity and amplitude attenuation of ultrasonic longitudinal waves propagating through a damaged rod. The wave motion was generated by a P-wave transducer attached at one end of the rod. To facilitate the time-of-flight measurements, the notches were machined only over a limited section in the interior of a long (otherwise intact) rod. A Laser Doppler Vibrometer (LDV) was used to monitor the time histories of the longitudinal rod motion at different locations before and after the notched section. A cross-correlation technique and amplitude decay were utilized to determine respectively the phase velocity and attenuation coefficient in the damaged “micro-structured” section. The observed dispersion relations, both within and outside the first band gap, are found to be in good agreement with those predicted by periodic cell analysis and the GE model with three length-scale parameters.


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