Size effects of mechanical metamaterials: a computational study based on a second-order asymptotic homogenization method
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O R I G I NA L
Hua Yang
· Wolfgang H. Müller
Size effects of mechanical metamaterials: a computational study based on a second-order asymptotic homogenization method
Received: 17 April 2020 / Accepted: 2 October 2020 © The Author(s) 2020
Abstract In this paper, size effects exhibited by mechanical metamaterials have been studied. When the sizescale of the metamaterials is reduced, stiffening or softening responses are observed in experiments. In order to capture both the stiffening and softening size effects fully, a second-order asymptotic homogenization method based on strain gradient theory is used. By this method, the metamaterials are homogenized and become effective strain gradient continua. The effective metamaterial parameters including the classical and strain gradient stiffness tensors are calculated. Comparisons between a detailed finite element analysis and the effective strain gradient continua model have been made for metamaterials under different boundary conditions, different aspect ratios, different unit cells (closed or open cells) and different topologies. It shows that both stiffening and softening size effects can be captured by using the effective strain gradient continua models. Keywords Size effects · Mechanical metamaterials · Asymptotic homogenization method · Strain gradient elasticity · Finite element method
1 Introduction Mechanical metamaterials are of growing interest due to their extraordinary mechanical properties for engineering applications [13,16,30,51]. Mechanical metamaterials are designed with well-defined microstructures in order to achieve controllable properties. Consequently, it is not only the composition of materials but also the microstructures that govern the behavior of metamaterial [39]. Most of the existing designs of mechanical metamaterials are achieved either by utilizing origami and kirigami deformation mechanisms [17,52] or by tailoring the topology of the lattice cells and by arranging them periodically at the microscale [64]. For example, so-called pantographic microstructures [25–27,53,56,65] are constituted by two parallel arrays of beams connected by small internal cylinders (pivots) and display novel functionalities. Their mechanical properties are reported to be usually size dependent [4]. In other words, remarkable size effects are observed when metamaterials are tested. For example, in the experiments of [4,45] the samples become stiffer with decreasing sample size. But smaller is not always stiffer. For example, a softening effect is observed in the studies of [59]. In order to capture the size-dependent behaviors completely, a direct finite element computation with a detailed mesh and an appropriate constitutive relation connecting stress and strains is always possible [62] and usually provides good accuracy. However, due to the complexity of the microstructures, it carries too much computational burden even for modern computer technology. The homogenization method [5,12,20– 22,32,34,36,37,42,63] is thus used as an alternative way to model meta
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