Designing recipes for auxetic behaviour of 2-d lattices

TL;DR

This paper develops an analytical model to design 2D elastic lattices with auxetic properties, enabling tailored mechanical responses and understanding instability-driven behavior in metamaterials.

Contribution

It introduces a transfer matrix-based analytical approach for designing auxetic 2D lattices with tunable, direction-dependent properties, validated by finite element simulations.

Findings

Identifies a design space for auxetic behavior beyond elastic instability thresholds

Provides lattice parameters for direction-dependent deformation modes

Shows good agreement between analytical model and finite element results

Abstract

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 auxetic properties for strains greater than that required to trigger elastic instability. The critical loading required to make this transition from positive to negative Poisson's ratio 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.