Discrete systems behave as nonlocal structural elements: Bending, buckling and vibration analysis

It is shown herein that the bending, buckling and vibration problems of a microstructured beam can be modeled by Eringenʹs nonlocal elasticity model. The microstructured model is composed of rigid periodic elements elastically connected by rotational springs. It is shown that this discrete system is the finite difference formulation of a continuous problem, i.e. the Euler–Bernoulli beam problem. Starting from the discrete equations, a continualization method leads to the formulation of an Eringenʹs type nonlocal equivalent continuum. The sensitivity phenomenon of the apparent nonlocal length scale with respect to the bending, the vibrations and the buckling analyses is investigated in more detail. A unified length scale can be used for the microstructured-based model with both nonlocal constitutive law and nonlocal governing equations. The Finite Difference Method is used for studying the exact discrete problem and leads to tractable engineering formula. The bending behaviour of the microstructured cantilever beam does not reveal any scale effect in the presence of concentrated loads. This scale invariance is not a deficiency of Eringenʹs nonlocality because it is in fact supported by the exact discreteness of the microstructured beam. A comparison of the discrete and the continuous problems (for both static and dynamics analyses) show the efficiency of the nonlocal-based modelling for capturing scale effects. As it has already been shown for buckling or vibrations studies, small scale effects tend to soften the material in this case.