Nonlocal equivalent continua for buckling and vibration analyses of microstructured beams

This paper is focused on the buckling and the vibration analyses of microstructured structural elements, i.e., elements composed of repetitive structural cells. The relationship between the discrete and the equivalent nonlocal continuum is specifically addressed from a numerical and a theoretical point of view. The microstructured beam considered herein is modeled by some repetitive cells composed of finite rigid segments and elastic rotational springs. The microstructure may come from the discreteness of the matter for small-scale structures (such as for nanotechnology applications), but it can also be related to some larger scales as for civil engineering applications. The buckling and vibration results of the discrete system are numerically obtained from a discrete-element code, whereas the nonlocal-based results for the equivalent continuum can be analytically performed. It is shown that Eringen's nonlocal elasticity coupled to the Euler-Bernoulli beam theory is relevant to capture the main-scale phenomena of such a microstructured continuum. The small-scale coefficient of the equivalent nonlocal continuum is identified from the specific microstructure features, namely, the length of each cell. However, the length scale calibration depends on the type of analysis, namely, static versus dynamic analysis. A perfect agreement is found for the microstructured beam with simply supported boundary conditions. The specific identification of the equivalent stiffness for modeling the equivalent clamped continuum is also discussed. The equivalent stiffness of the discrete system appears to be dependent on the static-dynamic analyses, but also on the boundary conditions applied to the overall system. Satisfactory results are also obtained for the comparison between the discrete and the equivalent continuum for other type of boundary conditions.