[en] Although "classical" multi-scale methods can capture the behaviour of cellular, including lattice, materials, when considering lattices or metamaterial local instabilities, corresponding to a change of the micro-structure morphology, classical computational homogenisation methods fail. On the one hand, first order computational homogenisation, which considers a classical continuum at the macro-scale cannot capture localisation bands inherent to cell buckling propagation. On the other hand, second-order computational homogenisation, which considers a higher order continuum at the macro-scale, introduces a size effect with respect to the Representative Volume Element (RVE) size, which is problematic when the RVE has to consider several cells to recover periodicity during local instability. In this paper we reformulate in a finite-strain setting the second-order computational homogenisation using the idea of equivalent homogenised volume. From this equivalence, arises at the micro-scale a non-uniform body force that acts as a supplementary volume term over the RVE. In the presented method, this non-uniform body-force term arises from the equivalence of energy, i.e. the Hill-Mandel condition, between the micro- and macroscopic volumes and depends mainly on the relation between the micro-scale and macro-scale deformation gradient. We show by considering elastic and elasto-plastic metamaterials and cellular materials that this approach reduces the RVE size dependency on the homogenised response.
Wu, Ling ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > Computational & Multiscale Mechanics of Materials (CM3)
Mustafa, Syed Mohib ; Université de Liège - ULiège > Aérospatiale et Mécanique (A&M) ; IMDEA Materials Institute
Segurado, Javier; IMDEA Materials Institute
Noels, Ludovic ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > Computational & Multiscale Mechanics of Materials (CM3)
Language :
English
Title :
Second-order computational homogenisation enhanced with non-uniform body forces for non-linear cellular materials and metamaterials
Publication date :
15 March 2023
Journal title :
Computer Methods in Applied Mechanics and Engineering
ISSN :
0045-7825
eISSN :
1879-2138
Publisher :
Elsevier, Amsterdam, Netherlands
Volume :
407
Pages :
115931
Peer reviewed :
Peer Reviewed verified by ORBi
European Projects :
H2020 - 862015 - MOAMMM - Multi-scale Optimisation for Additive Manufacturing of fatigue resistant shock-absorbing MetaMaterials
Name of the research project :
Multiscale Optimisation for Additive Manufacturing of fatigue resistant shock-absorbing MetaMaterials (MOAMMM)
Funders :
EU - European Union [BE]
Funding number :
862015
Funding text :
This project has received funding from the European Unions Horizon 2020 research and innovation
programme under grant agreement No 862015 for the project \Multi-scale Optimisation for Additive Manufacturing
of fatigue resistant shock-absorbing MetaMaterials (MOAMMM) of the H2020-EU.1.2.1. - FET
Open Programme.
Commentary :
NOTICE: this is the author’s version of a work that was accepted for publication in Computer Methods in Applied Mechanics and Engineering. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Computer Methods in Applied Mechanics and Engineering 407 (2023) 115931, DOI: 10.1016/j.cma.2023.115931
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