A fractional equation to approximate wave dispersion relation in elastic rods

Abstract

AbstractThe Kolsky‐Hopkinson bar technique is widely used to test materials at high strain rates. When large‐diameter bars have to be used, the processing of experimental signals needs to take into account wave dispersion effects. Such effects are either characterised experimentally, mostly using spectral techniques, or modelled analytically, using the Pochhammer‐Chree equation. In this paper, a phenomenological fractional approximation is proposed to model the wave dispersion relation in elastic bars. It involves the Poisson's ratio as a parameter. The proposed new equation approximates well the wave dispersion relation obtained by the Pochhammer‐Chree equation for Poisson's ratio ranging from 0.25 to 0.4 and for wavelengths 4 times higher than the radius of the bar. This roughly corresponds to frequencies up to 75 kHz. Thus, the proposed fractional equation is useful for processing signals obtained in Kolsky‐Hopkinson bar experiments. The approximate fractional equation is as simple as Love's equation; however, the former outperforms the latter equation.