Scale effect and higher-order boundary conditions for generalized lattices, with direct and indirect interactions

The effects of higher-order boundary conditions in the dynamics behaviour of some higher-order lattices are studied from exact and asymptotic solutions. Higher-order lattices considered herein are generalized axial lattices with direct and indirect symmetrical elastic interactions. More specifically, this two-neighbour interaction lattice is composed of different springs connected to adjacent nodes and to next to adjacent nodes, with possible different stiffness values for each interaction. The boundary nodes at each extremity of this generalized lattice are assumed to be fixed. The natural frequencies of such a fixed-fixed generalized lattice with both symmetrical and truncated higher-order boundary conditions are analytically calculated, from the resolution of a fourth-order boundary difference value problem. The physical meaning of both higher-order boundary conditions is discussed. Whereas the so-called symmetrical higher-order boundary condition is associated with a boundary spring twice the internal one, the truncated higher-order boundary condition preserves the stiffness value of the boundary spring to the internal one. For both higher-order boundary conditions, the vibration modes and dimensionless frequencies are exactly calculated. In both cases, the dimensionless frequency of the general lattice is shown to be lower than the asymptotic continuous one. However, an asymptotic analysis shows that the scaling law for such generalized lattice is strongly sensitive to each higher-order boundary condition. A power law of order 1 or order 2 is obtained for the scaling laws associated with each higher-order boundary condition. As generalized lattices can be also understood as the physical discrete support of some distributed nonlocal elastic models with continuous kernels, it is expected that the strong scale dependence observed in this paper also concerns nonlocal elastic problems.