Higher order and scale-dependent micro-inertia effect on the longitudinal dispersion based on the modified couple stress theory

The conventional modified couple stress theory cannot model the correct behavior of the longitudinal dispersion and acts the same as the classical theory in the face of such problems. In this paper, the micro-inertia-based couple stress theory is used to triumph over this deficiency. The developed theory is imposed to tackle the longitudinal dispersion of aluminum beams in two distinct scales. Convenient available experimental data obtained for a macro-scale aluminum rod and aluminum crystals are utilized to determine the corresponding micro-inertia length scale parameters and show the scale-dependent nature of this parameter for the first time. In addition, a higher order micro-rotation relation is employed to describe the higher order micro-inertia effects. This relation leads to a developed equation of motion containing an additional term compared with the first-order relation. The obtained results indicate that only higher order micro-inertia effect that is proposed in this study for the first time is able to capture the highly nonlinear behavior of dispersion curves (including an extremum/inflection point), which has experimentally been observed for phonons propagating in the longitudinal direction in an aluminum crystal.