On causality of wave motion in nonlocal theories of elasticity: a Kramers–Kronig relations study

Abstract

In this paper, the subject of the principle of causality and the physical realizability in the wave motion characteristics within a linear nonlocal elastic medium is examined. The principle of primitive causality is examined via Kramers–Kronig (K–K) relations and the principle of relativistic causality or the Einstein causality is examined via wave motion responses. Gradient as well as integral type nonlocality has been considered. Methodology here involves a Fourier frequency domain based spectral analysis and the wave motion characteristics include: wave modes, group speeds and frequency response function. In general, due to existence of atleast one of the non-physical features in the characteristics, violation of the causality is observed. The non-physical features include: existence of infinitesimally small or zero speeds; existence of very large or infinite speeds; existence of negative speeds; and absence of attenuation of waves. Violation to the primitive causality takes place as a disagreement to the K–K relations, either due to existence of negative and/or zero group speeds or due to absence of wave attenuation in the possible wave modes. Violation to Einstein causality is observed due to existence of infinitely large group speeds. Agreement to the primitive causality is achieved due to the presence of both wave dispersion and wave attenuation in the wave modes. Although existence of infinitely large group speeds violates Einstein causality, however, violation of the primitive causality is not observed. Upon considering only the physically realizable wavemodes, it is observed that, a local Neumann type boundary condition may be sufficient to conduct a wave motion study in a class of nonlocal boundary value problems. As an application of the primitive causality to the Fourier domain analysis, the wavenumbers from the K–K relations are utilized to demonstrate a mitigation effect of certain non-physical features in the wave motion responses.