The in-plane mechanics of a family of curved 2D lattices

Abstract

We propose an analytical framework to understand the mechanics and quantify the essential elastic properties of two-dimensional hexagonal lattices with curved elements. Generalised closed-form expressions for the in-plane Young's moduli and Poisson's ratios are obtained. It is of utmost importance to develop physics-based efficient computational models for the design and analysis of cellular metamaterials. This paper develops fundamental analytical approaches for obtaining generalised expressions to capture a large class of geometry. The closed-form expressions are obtained utilising the stiffness coefficients of the constituent structural members of the unit cell with curved beams. The new expressions for the equivalent in-plane properties are then explored to investigate seven other unique unit cell geometries including two auxetic configurations. Curved beam element as a constituent member of the unit cell has a significant effect in increasing the flexibility of the lattice and it also expands the design space for lattice materials. The Poisson's ratios also vary in a controlled way and this favourable feature can be exploited for obtaining designer values for both the regular and auxetic cases. The proposed analytical approach and the new closed-form expressions provide a computationally efficient and physically intuitive framework for the analysis and parametric design of curved lattice materials. The equivalent in-plane properties can be utilised as per the design requirements and the expressions can be considered as benchmark results for future numerical and experimental investigations.