Exact and nonlocal solutions for vibration of axial lattice with direct and indirect neighboring interactions

This paper investigates the effect of long-range interaction forces in the longitudinal vibration of an axial linear elastic lattice. The eigenfrequencies of this axial lattice with direct and indirect linear elastic interactions are expressed in closed-form solutions for fixed-fixed and fixed-free boundary conditions. The paper starts with the two-neighbor interaction problem, and then generalizes the result to the more general N-neighbor interaction problem. Starting from a continualization of the higher-order difference equations, a nonlocal elastic continuum is constructed to approximate the behavior of the axial lattice with generalized short and long-range interactions. It is shown, while preserving the definite positiveness of the generalized lattice strain energy, that the associated nonlocal continuum is equivalent to a stress gradient type of Eringen's nonlocal model, which is able to capture the main scale phenomena of the generalized lattice. The length scale factor of the nonlocal model is calibrated to the size of the influence domain for the long-range interaction. The paper ends with a discussion about the effect of the discrete interaction kernel on the sensitivity of the eigenfrequencies with respect to scale effects. Exact analytical natural frequencies are calculated for discrete power law or exponential-based kernels. The stress gradient nonlocal theory offers an efficient engineering continuum framework, which accurately fits the response of generalized lattice with N-neighbor interaction.