[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
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
Lee, S.H., Park, C.M., Seo, Y.M., Wang, Z.G., Kim, C.K., Acoustic metamaterial with negative modulus. J. Phys.: Condens. Matter, 21(17), 2009, 175704, 10.1088/0953-8984/21/17/175704.
Eidini, M., Zigzag-base folded sheet cellular mechanical metamaterials. Extreme Mech. Lett. 6 (2016), 96–102, 10.1016/j.eml.2015.12.006 URL https://www.sciencedirect.com/science/article/pii/S2352431615300109.
Kadic, M., Bückmann, T., Stenger, N., Thiel, M., Wegener, M., On the practicability of pentamode mechanical metamaterials. Appl. Phys. Lett., 100(19), 2012, 191901, 10.1063/1.4709436.
Eringen, A., Suhubi, E., Nonlinear theory of simple micro-elastic solids—I. Internat. J. Engrg. Sci. 2:2 (1964), 189–203, 10.1016/0020-7225(64)90004-7 URL https://www.sciencedirect.com/science/article/pii/0020722564900047.
Eringen, A.C., Microcontinuum Field Theories: I. Foundations and Solids. 2012, Springer Science & Business Media.
Nowacki, W., Theory of Asymmetric Elasticity. 1986, Pergamon Press, Headington Hill Hall, Oxford OX 3 0 BW, UK.
Forest, S., Sab, K., Cosserat overall modeling of heterogeneous materials. Mech. Res. Commun. 25:4 (1998), 449–454, 10.1016/S0093-6413(98)00059-7 URL https://www.sciencedirect.com/science/article/pii/S0093641398000597.
Forest, S., Pradel, F., Sab, K., Asymptotic analysis of heterogeneous Cosserat media. Int. J. Solids Struct. 38:26 (2001), 4585–4608, 10.1016/S0020-7683(00)00295-X URL https://www.sciencedirect.com/science/article/pii/S002076830000295X.
Barboura, S., Li, J., Establishment of strain gradient constitutive relations by using asymptotic analysis and the finite element method for complex periodic microstructures. Int. J. Solids Struct. 136–137 (2018), 60–76, 10.1016/j.ijsolstr.2017.12.003 URL https://www.sciencedirect.com/science/article/pii/S0020768317305383.
Smyshlyaev, V., Cherednichenko, K., On rigorous derivation of strain gradient effects in the overall behaviour of periodic heterogeneous media. J. Mech. Phys. Solids 48:6 (2000), 1325–1357, 10.1016/S0022-5096(99)00090-3 URL https://www.sciencedirect.com/science/article/pii/S0022509699000903.
Tran, T.-H., Monchiet, V., Bonnet, G., A micromechanics-based approach for the derivation of constitutive elastic coefficients of strain-gradient media. Int. J. Solids Struct. 49:5 (2012), 783–792, 10.1016/j.ijsolstr.2011.11.017 URL https://www.sciencedirect.com/science/article/pii/S0020768311003994.
Goda, I., Ganghoffer, J.-F., Construction of first and second order grade anisotropic continuum media for 3D porous and textile composite structures. Compos. Struct. 141 (2016), 292–327, 10.1016/j.compstruct.2016.01.061 URL https://www.sciencedirect.com/science/article/pii/S026382231600074X.
Trinh, D.K., Janicke, R., Auffray, N., Diebels, S., Forest, S., Evaluation of generalized continuum substitution models for heterogeneous materials. Int. J. Multiscale Comput. Eng. 10:6 (2012), 527–549.
Bacigalupo, A., Gambarotta, L., Homogenization of periodic hexa- and tetrachiral cellular solids. Compos. Struct. 116 (2014), 461–476, 10.1016/j.compstruct.2014.05.033 URL https://www.sciencedirect.com/science/article/pii/S0263822314002438.
Trovalusci, P., Ostoja-Starzewski, M., De Bellis, M.L., Murrali, A., Scale-dependent homogenization of random composites as micropolar continua. Eur. J. Mech. A Solids 49 (2015), 396–407, 10.1016/j.euromechsol.2014.08.010 URL https://www.sciencedirect.com/science/article/pii/S0997753814001223.
Yuan, X., Tomita, Y., Andou, T., A micromechanical approach of nonlocal modeling for media with periodic microstructures. Mech. Res. Commun. 35:1 (2008), 126–133, 10.1016/j.mechrescom.2007.07.004 Multiscale modeling, URL https://www.sciencedirect.com/science/article/pii/S0093641307000419.
Berkache, K., Deogekar, S., Goda, I., Picu, R., Ganghoffer, J.-F., Construction of second gradient continuum models for random fibrous networks and analysis of size effects. Compos. Struct. 181 (2017), 347–357, 10.1016/j.compstruct.2017.08.078 URL https://www.sciencedirect.com/science/article/pii/S0263822317313508.
Yvonnet, J., Auffray, N., Monchiet, V., Computational second-order homogenization of materials with effective anisotropic strain-gradient behavior. Int. J. Solids Struct. 191–192 (2020), 434–448, 10.1016/j.ijsolstr.2020.01.006 URL https://www.sciencedirect.com/science/article/pii/S0020768320300123.
Feyel, F., A multilevel finite element method (FE2) to describe the response of highly non-linear structures using generalized continua. Comput. Methods Appl. Mech. Engrg. 192:28 (2003), 3233–3244, 10.1016/S0045-7825(03)00348-7 Multiscale computational mechanics for materials and structures, URL https://www.sciencedirect.com/science/article/pii/S0045782503003487.
Kouznetsova, V., Geers, M., Brekelmans, W., Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy. Comput. Methods Appl. Mech. Engrg. 193:48–51 (2004), 5525–5550, 10.1016/j.cma.2003.12.073 Advances in computational plasticity.
Blanco, P., Sánchez, P., de Souza Neto, E., Feijóo, R., The method of multiscale virtual power for the derivation of a second order mechanical model. Mech. Mater. 99 (2016), 53–67, 10.1016/j.mechmat.2016.05.003 URL https://www.sciencedirect.com/science/article/pii/S0167663616300400.
Rokoš, O., Ameen, M., Peerlings, R., Geers, M., Micromorphic computational homogenization for mechanical metamaterials with patterning fluctuation fields. J. Mech. Phys. Solids 123 (2019), 119–137, 10.1016/j.jmps.2018.08.019 The N.A. Fleck 60th anniversary volume, URL https://www.sciencedirect.com/science/article/pii/S0022509618306148.
van Bree, S., Rokoš, O., Peerlings, R., Doškář, M., Geers, M., A Newton solver for micromorphic computational homogenization enabling multiscale buckling analysis of pattern-transforming metamaterials. Comput. Methods Appl. Mech. Engrg., 372, 2020, 113333, 10.1016/j.cma.2020.113333 URL https://www.sciencedirect.com/science/article/pii/S0045782520305181.
Biswas, R., Poh, L.H., Shedbale, A.S., A micromorphic computational homogenization framework for auxetic tetra-chiral structures. J. Mech. Phys. Solids, 135, 2020, 103801, 10.1016/j.jmps.2019.103801 URL https://www.sciencedirect.com/science/article/pii/S0022509619307203.
Kaczmarczyk, L., Pearce, C.J., Bićanić, N., Scale transition and enforcement of RVE boundary conditions in second-order computational homogenization. Internat. J. Numer. Methods Engrg. 74:3 (2008), 506–522, 10.1002/nme.2188 URL https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.2188.
Bacigalupo, A., Gambarotta, L., Second-order computational homogenization of heterogeneous materials with periodic microstructure. ZAMM Z. Angew. Math. Mech. 90:10–11 (2010), 796–811, 10.1002/zamm.201000031 URL https://onlinelibrary.wiley.com/doi/abs/10.1002/zamm.201000031.
Huet, C., Application of variational concepts to size effects in elastic heterogeneous bodies. J. Mech. Phys. Solids 38:6 (1990), 813–841, 10.1016/0022-5096(90)90041-2 URL https://www.sciencedirect.com/science/article/pii/0022509690900412.
Ostoja-Starzewski, M., Boccara, S.D., Jasiuk, I., Couple-stress moduli and characteristics length of a two-phase composite. Mech. Res. Commun. 26:4 (1999), 387–396, 10.1016/S0093-6413(99)00039-7 URL https://www.sciencedirect.com/science/article/pii/S0093641399000397.
Li, J., Establishment of strain gradient constitutive relations by homogenization. C. R. Méc. 339:4 (2011), 235–244, 10.1016/j.crme.2011.02.002 URL https://www.sciencedirect.com/science/article/pii/S1631072111000271.
Monchiet, V., Auffray, N., Yvonnet, J., Strain-gradient homogenization: A bridge between the asymptotic expansion and quadratic boundary condition methods. Mech. Mater., 143, 2020, 103309, 10.1016/j.mechmat.2019.103309 URL https://www.sciencedirect.com/science/article/pii/S0167663619308385.
Bertoldi, K., Boyce, M.C., Wave propagation and instabilities in monolithic and periodically structured elastomeric materials undergoing large deformations. Phys. Rev. B, 78, 2008, 184107, 10.1103/PhysRevB.78.184107 URL https://link.aps.org/doi/10.1103/PhysRevB.78.184107.
Bertoldi, K., Boyce, M., Deschanel, S., Prange, S., Mullin, T., Mechanics of deformation-triggered pattern transformations and superelastic behavior in periodic elastomeric structures. J. Mech. Phys. Solids 56:8 (2008), 2642–2668, 10.1016/j.jmps.2008.03.006 URL https://www.sciencedirect.com/science/article/pii/S0022509608000434.
Kouznetsova, V., Computational Homogenization for the Multi-Scale Analysis of Multi-Phase Materials. (Ph.D. thesis), 2002, Technische Universiteit Eindhoven, 10.6100/IR560009 Mechanical Engineering.
Nguyen, V.-D., Noels, L., Computational homogenization of cellular materials. Int. J. Solids Struct. 51:11 (2014), 2183–2203, 10.1016/j.ijsolstr.2014.02.029 URL https://www.sciencedirect.com/science/article/pii/S0020768314000778.
Nguyen, V.-D., Becker, G., Noels, L., Multiscale computational homogenization methods with a gradient enhanced scheme based on the discontinuous Galerkin formulation. Comput. Methods Appl. Mech. Engrg. 260 (2013), 63–77, 10.1016/j.cma.2013.03.024 URL https://www.sciencedirect.com/science/article/pii/S0045782513000832.
Poncelet, M., Somera, A., Morel, C., Jailin, C., Auffray, N., An experimental evidence of the failure of Cauchy elasticity for the overall modeling of a non-centro-symmetric lattice under static loading. Int. J. Solids Struct. 147 (2018), 223–237, 10.1016/j.ijsolstr.2018.05.028 URL https://www.sciencedirect.com/science/article/pii/S0020768318302221.
Nguyen, V.-D., Wu, L., Noels, L., Unified treatment of microscopic boundary conditions and efficient algorithms for estimating tangent operators of the homogenized behavior in the computational homogenization method. Comput. Mech. 59:3 (2017), 483–505.
Ainsworth, M., Essential boundary conditions and multi-point constraints in finite element analysis. Comput. Methods Appl. Mech. Engrg. 190:48 (2001), 6323–6339, 10.1016/S0045-7825(01)00236-5 URL https://www.sciencedirect.com/science/article/pii/S0045782501002365.
Hii, A.K., El Said, B., A kinematically consistent second-order computational homogenisation framework for thick shell models. Comput. Methods Appl. Mech. Engrg., 398, 2022, 115136, 10.1016/j.cma.2022.115136 URL https://www.sciencedirect.com/science/article/pii/S0045782522003139.
Mindlin, R., Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1:4 (1965), 417–438, 10.1016/0020-7683(65)90006-5 URL https://www.sciencedirect.com/science/article/pii/0020768365900065.
Cuitino, A., Ortiz, M., A material-independent method for extending stress update algorithms from small-strain plasticity to finite plasticity with multiplicative kinematics. Eng. Comput., 9, 1992, 437.
