Programmable higher-order Euler buckling modes in hierarchical beams

Abstract

We present a numerical-aided experimental study on the buckling of hierarchical beams comprising multiple self-similar modules. Each module consists of multiple elemental beams and is arranged in series to form the hierarchical beam. We show, through a combination of experiments and computations, that these beams exhibit stable and realizable higher-order buckling modes. By contrast to the canonical Euler buckling problem, such modes emerge naturally in the proposed self-similar beams since they correspond to almost identical critical loads. By harnessing the imperfection sensitivity of the hierarchical structures, we 3D-print weakly imperfect polymer samples with a small geometric imperfection corresponding to the desired eigenmode. We subsequently carry out uniaxial compression experiments and show in practice that higher-order patterns can be triggered selectively upon buckling. Moreover, these patterns are preserved in the post-bifurcation regime in many cases and are reversible upon load release. The ability to trigger higher-order buckling modes is found to depend on two main geometrical parameters which lead to scale coupling. Those are the slenderness of the macroscopic hierarchical beam and the slenderness of the lower-scale elemental beam. With increasing slenderness of the hierarchical beam, we observe a significant softening in the overall stress-strain response and patterns exhibiting curvature localization in the post-bifurcation regime. The numerical finite-strain simulations carried out in the present study are found to be in very good agreement with the experiments and are used to quantify further the observed curvature localization in the hierarchical beams. The present study and the obtained results are geometric in nature and thus can be extended to different scales and hierarchies ad infinitum.

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