Discrete and continuum modelling of size effects in architectured unstable metamaterials

Metamaterials made of bi-stable building blocks gain their promising effective properties from a micromechanical mechanism, namely buckling, rather than by the chemical composition of its constituent. Both discrete and continuum modelling of unstable metamaterials is a challenging task. It requires great care in the stability analysis and a kinematic enhanced continuum theory to adequately describe the softening behaviour and related size effects. This paper presents a detailed analytical and numerical investigation of the modelling capabilities of a gradient enhanced continuum model compared to a discrete modelling approach which has been proven to be qualitatively consistent with experimental results on small structures. It is demonstrated that the gradient model is capable of describing size effects with respect to stability of small structures; however, the limit of very large structures is found to be inconsistent with both the discrete model and the Maxwell rule of a classical continuum. Based on an analytical investigation of the size of the energy barrier between local minima, it is discussed how the consistency of the limit case of both discrete and continuum models can be restored by redefining the classical meaning of stability.