Discrete and continuum modelling of size effects in architectured unstable metamaterials

Abstract

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.

Selecting Generalized Continuum Theories for Nonlinear Periodic Solids Based on the Instabilities of the Underlying Microstructure

“…Author would like to emphasize that the more springs are present in the system, the smaller are the amplitudes of the observed oscillations but this amplitude also depends on the system's stiffness parameters (K i , M i , N i i = 1, 2). This oscillating feature has been observed also experimentally and numerically in the literature [27,28].The localization starts at one end of the system due to imperfect boundary conditions that break the periodicity. When using the micromorphic equivalent medium, the principal path is appropriately captured and the limit point corresponds to point (∆ E , F ) = (0.525731, 0.9711376579) with relative error of the order of magnitude of the computer's accuracy on both displacement and force values.…”