NOTICE: this is the author’s version of a work that was accepted for publication in International Journal of Solids and Structures. 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 International Journal of Solids and Structures, 51 (11-12), 2014 DOI: 10.1016/j.ijsolstr.2014.02.029
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[en] In this work we propose to study the behavior of cellular materials using a second–order multi–scale computational homogenization approach. During the macroscopic loading, micro-buckling of thin components, such as cell walls or cell struts, can occur. Even if the behavior of the materials of which the micro–structure is made remains elliptic, the homogenized behavior can lose its ellipticity. In that case, a localization band is formed and propagates at the macro–scale. When the localization occurs, the assumption of local action in the standard approach, for which the stress state on a material point depends only on the strain state at that point, is no–longer suitable, which motivates the use of the second-order multi–scale computational homogenization scheme. At the macro–scale of this scheme, the discontinuous Galerkin method is chosen to solve the Mindlin strain gradient continuum.
At the microscopic scale, the classical finite element resolutions of representative volume elements are considered. Since the meshes generated from cellular materials exhibit voids on the boundaries and are not conforming in general, the periodic boundary conditions are reformulated and are enforced by a polynomial interpolation method. With the presence of instability phenomena at both scales, the arc–length path following technique is adopted to solve both macroscopic and microscopic problems.
ARC 09/14-02 BRIDGING - From imaging to geometrical modelling of complex micro structured materials: Bridging computational engineering and material science
Funders :
Communauté française de Belgique : Direction Générale de l'Enseignement Non Obligatoire et de la Recherche Scientifique - DGENORS CÉCI - Consortium des Équipements de Calcul Intensif [BE]
M.F.A. Lorna, and J. Gibson Cellular Solids: Structure and Properties second ed. 1997 Cambridge University Press
E. Andrews, G. Gioux, P. Onck, and L. Gibson Size effects in ductile cellular solids. Part II: Experimental results Int. J. Mech. Sci. 43 3 2001 701 713 10.1016/S0020-7403(00)00043-6
S. Papka, and S. Kyriakides Experiments and full-scale numerical simulations of in-plane crushing of a honeycomb Acta Mater. 46 8 1998 2765 2776 10.1016/S1359-6454(97)00453-9
H. Zhu, and N. Mills The in-plane non-linear compression of regular honeycombs Int. J. Solids Struct. 37 13 2000 1931 1949 10.1016/S0020-7683(98) 00324-2
H. Bart-Smith, A.-F. Bastawros, D. Mumm, A. Evans, D. Sypeck, and H. Wadley Compressive deformation and yielding mechanisms in cellular al alloys determined using x-ray tomography and surface strain mapping Acta Mater. 46 10 1998 3583 3592 10.1016/S1359-6454(98)00025-1
W.-Y. Jang, and S. Kyriakides On the crushing of aluminum open-cell foams: Part I. Experiments Int. J. Solids Struct. 46 34 2009 617 634 10.1016/j.ijsolstr.2008.09.008
C. Tekoglu, L. Gibson, T. Pardoen, and P. Onck Size effects in foams: experiments and modeling Prog. Mater. Sci. 56 2 2011 109 138 10.1016/j.pmatsci.2010.06.001
Mangipudi, K.; Onck, P.; in press. Multiscale modelling of damage and failure in two-dimensional metallic foams. J. Mech. Phys. Solids. http://dx.doi.org/10.1016/j.jmps.2011.02.008. .
C. Chen, and N. Fleck Size effects in the constrained deformation of metallic foams J. Mech. Phys. Solids 50 5 2002 955 977 10.1016/S0022-5096(01) 00128-4
S. Forest, J.-S. Blazy, Y. Chastel, and F. Moussy Continuum modeling of strain localization phenomena in metallic foams J. Mater. Sci. 40 2005 5903 5910 10.1007/s10853-005-5041-6
A. Hanssen, O. Hopperstad, M. Langseth, and H. Ilstad Validation of constitutive models applicable to aluminium foams Int. J. Mech. Sci. 44 2 2002 359 406 10.1016/S0020-7403(01)00091-1
J. Yvonnet, H. Zahrouni, and M. Potier-Ferry A model reduction method for the post-buckling analysis of cellular microstructures Comput. Methods Appl. Mech. Eng. 197 1-4 2007 265 280 10.1016/j.cma.2007.07.026 http://www. sciencedirect.com/science/article/pii/S0045782507003271
M. Laroussi, K. Sab, and A. Alaoui Foam mechanics: nonlinear response of an elastic 3d-periodic microstructure Int. J. Solids Struct. 39 13-14 2002 3599 3623 10.1016/S0020-7683(02)00172-5
N. Ohno, D. Okumura, and H. Noguchi Microscopic symmetric bifurcation condition of cellular solids based on a homogenization theory of finite deformation J. Mech. Phys. Solids 50 5 2002 1125 1153 10.1016/S0022-5096(01) 00106-5 http://www.sciencedirect.com/science/article/pii/S0022509601001065
D. Okumura, N. Ohno, and H. Noguchi Elastoplastic microscopic bifurcation and post-bifurcation behavior of periodic cellular solids J. Mech. Phys. Solids 52 3 2004 641 666 10.1016/j.jmps.2003.07.002
D. Okumura, N. Ohno, and H. Noguchi Post-buckling analysis of elastic honeycombs subject to in-plane biaxial compression Int. J. Solids Struct. 39 13-14 2002 3487 3503 10.1016/S0020-7683(02)00165-8
H. Sehlhorst, R. Jnicke, A. Dster, E. Rank, H. Steeb, and S. Diebels Numerical investigations of foam-like materials by nested high-order finite element methods Comput. Mech. 45 2009 45 59 10.1007/s00466-009-0414-3
Kouznetsova, V.G.; 2002. Computational Homogenization for the Multi-Scale Analysis of Multi-Phase Materials (Ph.D. thesis). Technische Universiteit, Eindhoven.
V.G. Kouznetsova, M.G.D. Geers, and W.A.M. Brekelmans Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy Comput. Methods Appl. Mech. Eng. 193 48-51 2004 5525 5550 10.1016/j.cma.2003.12.073 (Advances in Computational Plasticity)
T. Ebinger, H. Steeb, and S. Diebels Modeling macroscopic extended continua with the aid of numerical homogenization schemes Comput. Mater. Sci. 32 34 2005 337 347 10.1016/j.commatsci.2004.09.034
M.G.D. Geers, V.G. Kouznetsova, and W.A.M. Brekelmans Multi-scale computational homogenization: trends and challenges J. Comput. Appl. Math. 234 7 2010 2175 2182 10.1016/j.cam.2009.08.077 (Fourth International Conference on Advanced COmputational Methods in Engineering (ACOMEN 2008))
V.P. Nguyen, O. Lloberas-Valls, M. Stroeven, and L.J. Sluys Computational homogenization for multiscale crack modeling Implementational and computational aspects Int. J. Numer. Methods Eng. 2011 10.1002/nme.3237
T. Massart, R. Peerlings, and M. Geers An enhanced multi-scale approach for masonry wall computations Int. J. Numer. Method Eng. 69 5 2007 1022 1059
E. Coenen, V. Kouznetsova, and M. Geers Multi-scale continuous discontinuous framework for computational-homogenization localization J. Mech. Phys. Solids 60 8 2012 1486 1507 10.1016/j.jmps.2012.04.002
L. Kaczmarczyk, C.J. Pearce, and N. Bićanić Scale transition and enforcement of RVE boundary conditions in second-order computational homogenization Int. J. Numer. Methods Eng. 74 3 2008 506 522
V.-D. Nguyen, G. Becker, and L. Noels Multiscale computational homogenization methods with a gradient enhanced scheme based on the discontinuous Galerkin formulation Comput. Methods Appl. Mech. Eng. 260 2013 63 77 10.1016/j.cma.2013.03.024
G.A. Wempner Discrete approximations related to nonlinear theories of solids Int. J. Solids Struct. 7 11 1971 1581 1599 10.1016/0020-7683(71)90038-2
E. Riks An incremental approach to the solution of snapping and buckling problems Int. J. Solids Struct. 15 7 1979 529 551 10.1016/0020-7683(79)90081-7
P. Bellini, and A. Chulya An improved automatic incremental algorithm for the efficient solution of nonlinear finite element equations Comput. Struct. 26 1-2 1987 99 110 10.1016/0045-7949(87)90240-9
Riks, E.; 1992. On formulations of path-following techniques for structural stability analysis, NASA STI/Recon Technical Report N 931, p. 16346.
M. Fafard, and B. Massicotte Geometrical interpretation of the arc-length method Comput. Struct. 46 4 1993 603 615 10.1016/0045-7949(93)90389-U
Z. Zhou, and D. Murray An incremental solution technique for unstable equilibrium paths of shell structures Comput. Struct. 55 5 1995 749 759 10.1016/0045-7949(94)00474-H
R. Kouhia, and M. Mikkola Some aspects on efficient path-following Comput. Struct. 72 45 1999 509 524 10.1016/S0045-7949(98)00336-8
P.L. Grognec, and A. Le van Elastoplastic bifurcation and collapse of axially loaded cylindrical shells Int. J. Solids Struct. 45 1 2008 64 86 10.1016/j.ijsolstr.2007.07.017
P.L. Grognec, P. Casari, and D. Choqueuse Influence of residual stresses and geometric imperfections on the elastoplastic collapse of cylindrical tubes under external pressure Mar. Struct. 22 4 2009 836 854 10.1016/j.marstruc.2009. 09.003
G. Becker, C. Geuzaine, and L. Noels A one field full discontinuous Galerkin method for Kirchhoff-Love shells applied to fracture mechanics Comput. Methods Appl. Mech. Eng. 200 4546 2011 3223 3241 10.1016/j.cma.2011.07.008
L. Wu, D. Tjahjanto, G. Becker, A. Makradi, A. Jrusalem, and L. Noels A micromeso-model of intra-laminar fracture in fiber-reinforced composites based on a discontinuous Galerkin/cohesive zone method Eng. Fract. Mech. 104 2013 162 183 10.1016/j.engfracmech.2013.03.018
T. Kanit, S. Forest, I. Galliet, V. Mounoury, and D. Jeulin Determination of the size of the representative volume element for random composites: statistical and numerical approach Int. J. Solids Struct. 40 13-14 2003 3647 3679 10.1016/S0020-7683(03)00143-4
K. Terada, M. Hori, T. Kyoya, and N. Kikuchi Simulation of the multi-scale convergence in computational homogenization approaches Int. J. Solids Struct. 37 16 2000 2285 2311 10.1016/S0020-7683(98)00341-2
F. Larsson, K. Runesson, S. Saroukhani, and R. Vafadari Computational homogenization based on a weak format of micro-periodicity for RVE-problems Comput. Methods Appl. Mech. Eng. 200 1-4 2011 11 26 10.1016/j.cma.2010.06.023
V.-D. Nguyen, E. Béchet, C. Geuzaine, and L. Noels Im posing periodic boundary condition on arbitrary meshes by polynomial interpolation Comput. Mater. Sci. 55 2012 390 406 10.1016/j.commatsci.2011.10.017
E. Coenen, V. Kouznetsova, and M. Geers Novel boundary conditions for strain localization analyses in microstructural volume elements Int. J. Numer. Methods Eng. 90 1 2012 1 21 10.1002/nme.3298
Z. Yuan, and J. Fish Toward realization of computational homogenization in practice Int. J. Numer. Methods Eng. 73 3 2008 361 380
J.M. Tyrus, M. Gosz, and E. DeSantiago A local finite element implementation for imposing periodic boundary conditions on composite micromechanical models Int. J. Solids Struct. 44 9 2007 2972 2989 10.1016/j.ijsolstr.2006.08.040
R. Mindlin Second gradient of strain and surface-tension in linear elasticity Int. J. Solids Struct. 1 1965 417 438
C. Miehe, and A. Koch Computational micro-to-macro transitions of discretized microstructures undergoing small strains Arch. Appl. Mech. 72 4 2002 300 317 10.1007/s00419-002-0212-2
Ainsworth Mark Essential boundary conditions and multi-point constraints in finite element analysis Comput. Methods Appl. Mech. Eng. 190 48 2001 6323 6339 10.1016/S0045-7825(01)00236-5
V. Kouznetsova, M. Geers, and W. Brekelmans Size of a representative volume element in a second-order computational homogenization framework Int. J. Multiscale Comput. Eng. 2 4 2004 575 598
R. Peerlings, L. Poh, and M. Geers An implicit gradient plasticity damage theory for predicting size effects in hardening and softening Eng. Fract. Mech. 95 2012 2 12 10.1016/j.engfracmech.2011.12.016 (Cracks in Microstructures and Engineering Components)
C. Tekoglu, and P.R. Onck Size effects in two-dimensional Voronoi foams: a comparison between generalized continua and discrete models J. Mech. Phys. Solids 56 12 2008 3541 3564 10.1016/j.jmps.2008.06.007
D. Lam, F. Yang, A. Chong, J. Wang, and P. Tong Experiments and theory in strain gradient elasticity J. Mech. Phys. Solids 51 8 2003 1477 1508 10.1016/S0022-5096(03)00053-X
N. Fleck, and J. Hutchinson Strain gradient plasticity Advances in Applied Mechanics vol. 33 1997 Elsevier pp. 295-361
A. Cuitiño, and M. Ortiz A material-independent method for extending stress update algorithms from small-strain plasticity to finite plasticity with multiplicative kinematics Eng. Comput. 9 1992 437 451 10.1108/eb023876
C. Geuzaine, and J.-F. Remacle Gmsh: a 3-d finite element mesh generator with built-in pre- and post-processing facilities Int. J. Numer. Methods Eng. 79 11 2009 1309 1331
