Wave propagation analysis of a beam according to the strain gradient elasticity theory with vertical inhomogeneity

In this paper, wave propagation (WP) in Euler–Bernoulli (EB) beams is investigated through the framework of strain gradient elasticity (GradEla) theory by introducing additional higher-order terms that account for the contribution of the beam cross section. These terms provide an effective representation of the vertical inhomogeneity associated with the cross-sectional deformation. The governing equation of motion is derived using Hamilton's principle, and the corresponding dispersion relation is obtained analytically. The influences of the internal length scale parameter (ILSP), the inertia time parameter (ITP) and the vertical inhomogeneity on the WP characteristics are examined. The results demonstrate that the proposed GradEla beam model exhibits pronounced size-dependent behavior and predicts a stiffer mechanical response than both the classical beam theory and the conventional GradEla formulation. The inclusion of these additional higher-order terms leads to higher angular frequencies and phase velocities, particularly at higher wave numbers. Furthermore, the incorporation of the ITP ensures physically admissible dispersion behavior by maintaining bounded phase velocities at large wave numbers.