Propagation of Elastic Waves in Nonlocal Bars and Beams

In this paper, wave dispersion properties in nonlocal theories of elasticity models are critically examined. Both gradient type as well as integral-type non-locality within a setting of the rod and beam are considered. The mathematical framework here involves the Fourier frequency analysis that leads to the frequency spectrum relation (FSR) and system transfer function. Utilizing the FSR, wave modes and group speeds are examined. One main difference that arises between the two nonlocal model types is in the number of wave modes that are possible from the FSR. In gradient-type non-locality, the number of modes is finite and equal to the order of the displacement gradient in the governing equation of motion. However, in an integral-type non-locality, wave modes are infinite in number. Further, in contrast to classical theories there exist nonclassical wave modes showing the existence of nonphysical features, such as either exponential instabilities or undefined wavenumbers, negative group speeds, infinitesimally small, and infinite group speed values. Existence of such features, then, naturally raises the aspect of physically realizable wave motion and causality in these models. In literature, there exist dispersion relations or Kramers–Kronig (K-K) relations as an aid to examine wave motion in a linear, passive, and causal system. In this paper, K-K relations are utilized in order to further examine the wave dispersion properties. Discrepancies are seen between FSR predictions and K-K predictions, especially at frequencies with nonphysical features. An example is also presented that contradicts the general concept that all time-domain formulations agree well with the K-K relations.