Analytical length scale calibration of nonlocal continuum from a microstructured buckling model

This paper deals with the buckling of a column which is modeled by some finite rigid segments and elastic rotational springs and relating its solution to continuum nonlocal elasticity. This problem, which can be referred to Hencky's chain, can serve as a basic model to rigorously investigate the effect of the microstructure on the buckling behaviour of a simple equivalent continuum structural model. The buckling problem of the pinned-pinned discretized column is analytically investigated by introducing a Lagrange multiplier. Such a buckling problem is mathematically treated as an iterative eigenvalue problem. It is shown that the buckling load of this finite degree-of-freedom system is exactly obtained by a recursive formula involving Chebyschev polynomials. Euler's buckling load is asymptotically obtained at larger scales. However, at smaller scales, the buckling model highlights some scale effect that can be only captured by nonlocal elasticity for the equivalent continuum. We show that Eringen's nonlocal continuum is well suited to capture this scale effect. The small scale coefficient of the equivalent nonlocal continuum is then identified from the specific microstructure features, namely the length of each cell. It is shown that the small length scale coefficient valid for this buckling problem is very close to the one already identified from a comparison with the Born-Karman model of lattice dynamics using dispersive wave properties.