Discrete and continuum modelling of size effects in architectured unstable metamaterials
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O R I G I NA L A RT I C L E
Claudio Findeisen Peter Gumbsch
· Samuel Forest · Jörg Hohe ·
Discrete and continuum modelling of size effects in architectured unstable metamaterials
Received: 24 June 2019 / Accepted: 28 January 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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. Keywords Gradient continuum · Stability size effects · Unstable metamaterials · Softening and localization 1 Introduction One of the fundamental assumptions of classical continuum mechanics is the assumption of a continuous distribution of particles [1]. If this assumption fails, the kinematic description can be enhanced in the framework of higher-order or higher-grade continua. Well-known examples of such extended theories are the Cosserat continuum [2], the micromorphic- [1] or second gradient continua [3], and its various special cases (e.g. [4] or [5]). Extended continuum theories become necessary if microstructural detail is relevant for the effective material response. This is for example the case in the continuum modelling of (boundary) size effects [6], or wave dispersion [7]. Moreover, the microstructural aspect is also relevant in the description of localization and material failure phenomena. Both material failure and localization are driven by the microstructure, and so it appears natural that their accurate description also requires a kinematic enhancement. A kinematic enhancement Communicated by Andreas Öchsner. C. Findeisen · P. Gumbsch Institute for Applied Materials, Karlsruhe Institute of Technology (KIT), 76128 Karlsruhe, Germany C. Findeisen (B) · J. Hohe · P. Gumbsch Fraunhofer Institute for Mechanics of Materials IWM, 79108 Freiburg, Germany E-mail: [email protected] S. Forest Centre des Matériaux, Mines-ParisTech, CNRS UMR 7633, PSL Researc
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