Effective strain gradient continuum model of metamaterials and size effects analysis
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O R I G I NA L A RT I C L E
Hua Yang · Dmitry Timofeev · Ivan Giorgio · Wolfgang H. Müller
Effective strain gradient continuum model of metamaterials and size effects analysis
Received: 23 January 2020 / Accepted: 31 July 2020 © The Author(s) 2020
Abstract In this paper, a strain gradient continuum model for a metamaterial with a periodic lattice substructure is considered. A second gradient constitutive law is postulated at the macroscopic level. The effective classical and strain gradient stiffness tensors are obtained based on asymptotic homogenization techniques using the equivalence of energy at the macro- and microscales within a so-called representative volume element. Numerical studies by means of finite element analysis were performed to investigate the effects of changing volume ratio and characteristic length for a single unit cell of the metamaterial as well as changing properties of the underlying material. It is also shown that the size effects occurring in a cantilever beam made of a periodic metamaterial can be captured with appropriate accuracy by using the identified effective stiffness tensors. Keywords Effective continuum · Strain gradient elasticity · Asymptotic homogenization method · Finite element method 1 Introduction The modeling of solids and structures in the framework of classical Cauchy mechanics has been widely used in diverse fields. However, experimental evidence increasingly shows limitations of this approach to predict the behavior of some materials at small scales (micro- or nanometer scales), where the intrinsic microstructure of the materials becomes crucial [11,26,54,61]. Heterogeneity inherited by microstructure leads to the so-called size effects, which cannot be captured by strain-based Cauchy mechanics. Particularly this is relevant to metamaterials whose effective properties are mainly determined by their microstructure [18,37,80,85]. When metamaterials are tested, small samples may show different deformation behaviors than larger ones [86]. In order to evaluate metamaterials with appropriate accuracy, a qualitative but also quantitative understanding of size effects needs to be included either in a physically feasible way or, as an alternative method, by Communicated by Luca Placidi. H. Yang (B)· W. H. Müller Chair of Continuum Mechanics and Constitutive Theory, Institute of Mechanics, Technische Universität Berlin, Einsteinufer 5, 10587 Berlin, Germany E-mail: [email protected] D. Timofeev · I. Giorgio International Research Center for the Mathematics and Mechanics of Complex Systems, Università degli studi dell’Aquila, L’Aquila, Italy I. Giorgio Department of Civil, Construction-Architectural and Environmental Engineering, Università degli studi dell’Aquila, Via Giovanni Gronchi 18, 67100 L’Aquila, Italy
H. Yang et al.
homogenization or identification of effective material properties. Indeed, size effects can be captured by using FEM and detailed modeling on a microstructural level with a relatively simple constitutive law [83,84], which means that th
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