Surrogate model for a mechanical metamaterial undergoing microstructure instabilities and phase transformations

Abstract

This paper addresses the modeling of periodic metamaterials with microstructure instabilities. Such instabilities are produced by bistable elastic mechanisms, i.e., snap-through deformation mechanisms, inducing phase transformations. Metamaterials with these features can be used for energy dissipation in elastic regimes between other applications. A new surrogate model of the bistable mechanism to replace High-Fidelity Finite Element simulations is developed in two sequential stages. In the first stage, a semi-analytical model of the bistable structural element is presented. Following an approach taken from the literature, the analysis of the bistable element is generalized to include load cases that had not been considered in the original study. These loading conditions naturally happen in functionally 2D and 3D metamaterials with complex microarchitectures. In a second stage, a frame element with linear kinematics and small deformations is described. The constitutive relations of the frame element, i.e., axial stress and moment in terms of axial strain and curvature, follow the same equations of the semi-analytical model derived in the first stage. A Variational Principle of Energetic Equivalence supplies the link between the semi-analytical model of the curved beam and the frame element. In this way, the response of this frame element, of only six degrees of freedom, reproduces the non-linear geometric behavior of the bistable element. The surrogate model is particularly efficient for simulating a large number of unit cells of the metamaterial and with the capacity to reproduce its behavior under general loading conditions. Thus, it is appropriate to assessing the metamaterial limit behavior, typically concerning phenomena such as energy dissipation and hysteresis under closed load cycles.