On the nano-structural dependence of nonlocal dynamics and its relationship to the upper limit of nonlocal scale parameter

Nonlocal elasticity theory is one of the most popular theoretical approaches to investigate the intrinsic scale effect of nano-materials/structures. The coupling of an internal characteristic length and a material parameter can be regarded as a nonlocal scale parameter in nano-meters. The range of this non-dimensional scale parameter is from zero up to different values previously. There is no doubt that the zero nonlocal scale parameter corresponds to a situation without any nonlocal effect. However, the determination of a peak value for the scale parameter is still uncertain. In fact, we frequently ask a simple but unresolved question, i.e., how strong is the nonlocal scale effect? This question is equivalent to what the maximum value of the nonlocal scale parameter is, since it was introduced to characterize the scale effect theoretically. Until now, various maximum values have been selected without rigorous verifications. In this paper, the nano-structural dependence of nonlocal dynamical behavior is investigated to present the existence of an upper limit for the scale parameter. Through three typical examples, the size-dependent behavior of nonlocal dynamics for various nano-structures is analyzed. The upper limit of the scale parameter can be determined accordingly. It is shown that an interval for the scale parameter in the illustrative examples can be found on the basis of the nonlocal softening physical mechanism, in which the equivalent stiffness of nano-structures is weakened than that predicted by the classical continuum theory. The present study contributes to a fuzzy zone in nonlocal elasticity where people are puzzled over the question how to select the upper limit of the nonlocal scale parameter. It is not only beneficial to the refinement of the nonlocal theory of elasticity, and also useful for the exploration of similar theories in nano-mechanics.