A class of periodic lattices for tuning elastic instabilities

Abstract

With in mind microstructures exhibiting unconventional macroscopic mechanical behaviors, characterized by overall auxetic responses and strain localization due to local elastic instabilities, in this work we conceived a simple two-dimensional non-chiral architecture in the form of a periodic lattice, whose drawing is decided by varying the thickness ratios of the cells’ walls and in turn their slenderness. Inspired by nature, which often uses supposedly naive geometries that conceal complex functions, and focusing the attention on two limit geometrical configurations of main interest, we show how these non-chiral settings, as a function of the prescribed boundary conditions, might lead to symmetry breaking associated to a variety of non-trivial deformation modes, which in some cases also retrace and generalize chiral as well as auxetic responses already observed in literature in simpler microstructures. Interestingly, for both the above mentioned limit cases, we were able to idealize the mechanical response of the system through geometrically nonlinear beam-based models, in this way obtaining helpful analytical solutions and explicit formulas to estimate the tangent effective stiffness of the lattice as well as the critical load at the onset of instability. The post-buckling was instead analyzed and discussed in detail through parametric numerical finite element simulations and ad hoc laboratory experiments, performed by faithfully realizing 3D printed prototypes, constructed via additive manufacturing technologies and made of rubber-like material, to follow extreme deformation patterns, including multi-stable states, localization and compaction with possible self-contact/touching of the elements. At the end, we exploited the obtained closed-form solutions and some associated inequalities to derive the transition from the discrete lattice to its continuum limit, which – together with the multiple equilibrium bifurcation points exhibited by the proposed microstructure – could be used to broaden the spectrum of metamaterial geometries designed to exhibit complex tunable properties.