An elastica theory for compressible imperfect beams with application to mechanical metamaterials

Abstract

Beam models are fundamental to structural engineering and crucial to the field of mechanical metamaterials. When modeling the failure of lattice-based mechanical metamaterials, we need formulations able to capture not only their linear response but also the post-buckling strength and deformation of its beam members, including how they are affected by the presence of geometric imperfections conferred to the structure by the manufacturing process. Any significant geometric imperfection can affect the maximum load a beam can withstand as well as its effective stiffness over the deformation regime. Unfortunately, current analytical models for beams that account for geometric imperfections focus on the small displacements regime, and models that capture large displacements focus on perfect beams. Consequently, current approaches to model failure of lightweight mechanical metamaterials rely on computational techniques, and generally employ a Lagrangian finite element formulation that explicitly models imperfections. While effective, this approach becomes computationally intractable for lattice problems where the number of degrees of freedoms is inherently large. To close this gap, we introduce a new analytical model that unifies the small and large post-buckling deflection behavior of simply supported imperfect beams. Our model is based on an imperfect Elastica theory, and exhibits great accuracy at very low computational cost.