Recipes for selecting failure modes in 2-d lattices

We present an analytical model to investigate the mechanics of 2-dimensional lattices composed of elastic beams of non-uniform cross-section. Our approach is based on reducing a lattice to a single beam subject to the action of a set of linear and torsional springs, thus allowing the problem to be solved through a transfer matrix method. We show a non-trivial region of design space that yields materials with short wavelength modes (closely associated with auxetic behaviour) for strains greater than that required to trigger elastic instability. The critical loading required to make this transition from long to short wavelength buckling modes is calculated. Furthermore, we present lattice parameters that provide direction-dependent deformation modes offering great tailorability of the mechanical properties of finite size lattices. Not only is our analytical formulation in good agreement with the finite element simulation results, but it provides an insight into the role of the interplay between structure and elastic instability, and gives an efficient methodology to pursue questions of rational design in the field of mechanical metamaterials.