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Computational Homogenization of Architectured Materials

  • Justin DirrenbergerEmail author
  • Samuel Forest
  • Dominique Jeulin
Chapter
Part of the Springer Series in Materials Science book series (SSMATERIALS, volume 282)

Abstract

Architectured materials involve geometrically engineered distributions of microstructural phases at a scale comparable to the scale of the component, thus calling for new models in order to determine the effective properties of materials. The present chapter aims at providing such models, in the case of mechanical properties. As a matter of fact, one engineering challenge is to predict the effective properties of such materials; computational homogenization using finite element analysis is a powerful tool to do so. Homogenized behavior of architectured materials can thus be used in large structural computations, hence enabling the dissemination of architectured materials in the industry. Furthermore, computational homogenization is the basis for computational topology optimization which will give rise to the next generation of architectured materials. This chapter covers the computational homogenization of periodic architectured materials in elasticity and plasticity, as well as the homogenization and representativity of random architectured materials.

References

  1. 1.
    M. Abouaf, J.-L. Chenot, G. Raisson, P. Bauduin, Finite element simulation of hot isostatic pressing of metal powders. Int. J. Numer. Methods Eng. 25, 191–212 (1988)CrossRefGoogle Scholar
  2. 2.
    A. Alderson, K.L. Alderson, D. Attard, K.E. Evans, R. Gatt, J.N. Grima, W. Miller, N. Ravirala, C.W. Smith, K. Zied, Elastic constants of 3-, 4- and 6-connected chiral and anti-chiral honeycombs subject to uniaxial in-plane loading. Compos. Sci. Technol. 70(7), 1042–1048 (2010)CrossRefGoogle Scholar
  3. 3.
    G. Allaire, Shape Optimization by the Homogenization Method (Springer, Berlin, 2002)CrossRefGoogle Scholar
  4. 4.
    R.F. Almgren, An isotropic three-dimensional structure with Poisson’s ratio-1. J. Elast. 15, 427–430 (1985)CrossRefGoogle Scholar
  5. 5.
    H. Altenbach, V. Eremeyev (eds.), Generalized Continua from the Theory to Engineering Applications (Springer, Berlin, 2013)Google Scholar
  6. 6.
    H. Altendorf, D. Jeulin, F. Willot, Influence of the fiber geometry on the macroscopic elastic and thermal properties. Int. J. Solids Struct. 51(23–24), 3807–3822 (2014)CrossRefGoogle Scholar
  7. 7.
    J.C. Alvarez Elipe, A. Diaz Lantada, Comparative study of auxetic geometries by means of computer-aided design and engineering. Smart Mater. Struct. 21, 105004 (2012)Google Scholar
  8. 8.
    E. Andreassen, B. Lazarov, O. Sigmund, Design of manufacturable 3d extremal elastic microstructure. Mech. Mater. 69(1), 1–10 (2014)CrossRefGoogle Scholar
  9. 9.
    U. Andreaus, F. dell’Isola, I. Giorgio, L. Placidi, T. Lekszycki, N.L. Rizzi, Numerical simulations of classical problems in two-dimensional (non) linear second gradient elasticity. Int. J. Eng. Sci. 108, 34–50 (2016)CrossRefGoogle Scholar
  10. 10.
    A. Asadpoure, M. Tootkaboni, L. Valdevit, Topology optimization of multiphase architected materials for energy dissipation. Comput. Methods Appl. Mech. Eng. 325, 314–329 (2017)CrossRefGoogle Scholar
  11. 11.
    M.F. Ashby, Y. Bréchet, Designing hybrid materials. Acta Mater. 51, 5801–5821 (2003)CrossRefGoogle Scholar
  12. 12.
    N. Auffray, Analytical expressions for odd-order anisotropic tensor dimension. C. R. Mécaniques 342(5), 284–291 (2014)CrossRefGoogle Scholar
  13. 13.
    N. Auffray, J. Dirrenberger, G. Rosi, A complete description of bi-dimensional anisotropic strain-gradient elasticity. Int. J. Solids Struct. 69–70, 195–210 (2015)CrossRefGoogle Scholar
  14. 14.
    J.-L. Auriault, Heterogeneous medium is an equivalent macroscopic description possible? Int. J. Eng. Sci. 29(7), 785–795 (1991)CrossRefGoogle Scholar
  15. 15.
    S. Babaee, J. Shim, J. Weaver, E. Chen, N. Patel, K. Bertoldi, 3d soft metamaterials with negative poisson’s ratio. Adv. Mater. 25(36), 5044–5049 (2013)CrossRefGoogle Scholar
  16. 16.
    A. Bacigalupo, L. Gambarotta, Homogenization of periodic hexa- and tetrachiral cellular solids. Compos. Struct. 116, 461–476 (2014)CrossRefGoogle Scholar
  17. 17.
    X. Badiche, S. Forest, T. Guibert, Y. Bienvenu, J.-D. Bartout, P. Ienny, M. Croset, H. Bernet, Mechanical properties and non-homogeneous deformation of open-cell nickel foams: application of the mechanics of cellular solids and of porous materials. Mater. Sci. Eng. A 289, 276–288 (2000)CrossRefGoogle Scholar
  18. 18.
    E. Bafekrpour, A. Dyskin, E. Pasternak, A. Molotnikov, Y. Estrin, Internally architectured materials with directionally asymmetric friction. Sci. Rep. 5, 10732 (2015)CrossRefGoogle Scholar
  19. 19.
    E. Bafekrpour, A. Molotnikov, J.C. Weaver, Y. Brechet, Y. Estrin, Responsive materials: a novel design for enhanced machine-augmented composites. Sci. Rep. 4, 3783 (2014)CrossRefGoogle Scholar
  20. 20.
    C. Barbier, R. Dendievel, D. Rodney, Numerical study of 3D-compressions of entangled materials. Comput. Mater. Sci. 45, 593–596 (2009a)CrossRefGoogle Scholar
  21. 21.
    C. Barbier, R. Dendievel, D. Rodney, Role of friction in the mechanics of nonbounded fibrous materials. Phys. Rev. E 80(1), 016115 (2009b)CrossRefGoogle Scholar
  22. 22.
    S. Bargmann, B. Klusemann, J. Markmann, J.E. Schnabel, K. Schneider, C. Soyarslan, J. Wilmers, Generation of 3d representative volume elements for heterogeneous materials: a review. Prog. Mater. Sci. 96, 322–384 (2018)CrossRefGoogle Scholar
  23. 23.
    R.J. Bathurst, L. Rothenburg, Note on a random isotropic granular material with negative Poisson’s ratio. Int. J. Eng. Sci. 26(4), 373–383 (1988)CrossRefGoogle Scholar
  24. 24.
    M. Bendsøe, O. Sigmund, Topology Optimization (Springer, Berlin, 2004)CrossRefGoogle Scholar
  25. 25.
    I. Benedetti, F. Barbe, Modelling polycrystalline materials: an overview of three-dimensional grain-scale mechanical models. J. Multiscale Model. 5(1), 1350002 (2013)CrossRefGoogle Scholar
  26. 26.
    M.J. Beran, Statistical Continuum Theories (Wiley, Hoboken, 1968)CrossRefGoogle Scholar
  27. 27.
    C. Berdin, Z.Y. Yao, S. Pascal, Internal stresses in polycrystalline zirconia: microstructure effects. Comput. Mater. Sci. 70, 140–144 (2013)CrossRefGoogle Scholar
  28. 28.
    K. Bertoldi, P. Reis, S. Willshaw, T. Mullin, Negative poisson’s ratio behavior induced by an elastic instability. Adv. Mater. 22(3), 361–366 (2010)CrossRefGoogle Scholar
  29. 29.
    J. Besson, G. Cailletaud, J.-L. Chaboche, S. Forest, M. Blétry, Non-Linear Mechanics of Materials, Volume 167 of Solid Mechanics and Its Applications (Springer, Berlin, 2010)Google Scholar
  30. 30.
    D. Bigoni, W. Drugan, Analytical derivation of cosserat moduli via homogenization of heterogeneous elastic materials. J. Appl. Mech. 74, 741–753 (2007)CrossRefGoogle Scholar
  31. 31.
    A. Bironeau, J. Dirrenberger, C. Sollogoub, G. Miquelard-Garnier, S. Roland, Evaluation of morphological representative sample sizes for nanolayered polymer blends. J. Microsc. 264(1), 48–58 (2016)CrossRefGoogle Scholar
  32. 32.
    M. Bornert, T. Bretheau, P. Gilormini, Homogénéisation en mécanique des matériaux, Tome 1 : Matériaux aléatoires élastiques et milieux périodiques. (Hermès, 2001)Google Scholar
  33. 33.
    O. Bouaziz, Y. Bréchet, J.D. Embury, Heterogeneous and architectured materials: a possible strategy for design of structural materials. Adv. Eng. Mater. 10(1–2), 24–36 (2008)CrossRefGoogle Scholar
  34. 34.
    O. Bouaziz, J.P. Masse, S. Allain, L. Orgéas, P. Latil, Compression of crumpled aluminum thin foils and comparison with other cellular materials. Mater. Sci. Eng. Struct. Mater. Prop. Microstruct. Process. 570, 1–7 (2013)CrossRefGoogle Scholar
  35. 35.
    Y. Bréchet, J.D. Embury, Architectured materials: expanding materials space. Scripta Mater. 68(1), 1–3 (2013)CrossRefGoogle Scholar
  36. 36.
    H.J. Bunge, Texture Analysis in Materials Science (Butterworths, 1982)Google Scholar
  37. 37.
    B.D. Caddock, K.E. Evans, Microporous materials with negative Poisson’s ratios: I. Microstructure and mechanical properties. J. Phys. D: Appl. Phys. 22, 1877–1882 (1989)Google Scholar
  38. 38.
    G. Cailletaud, S. Forest, D. Jeulin, F. Feyel, I. Galliet, V. Mounoury, S. Quilici, Some elements of microstructural mechanics. Comput. Mater. Sci. 27, 351–374 (2003)CrossRefGoogle Scholar
  39. 39.
    G. Cailletaud, D. Jeulin, P. Rolland, Size effect on elastic properties of random composites. Eng. Comput. 11(2), 99–110 (1994)CrossRefGoogle Scholar
  40. 40.
    O. Caty, E. Maire, R. Bouchet, Fatigue of metal hollow spheres structures. Adv. Eng. Mater. 10(3), 179–184 (2008)CrossRefGoogle Scholar
  41. 41.
    V.J. Challis, A.P. Roberts, A.H. Wilkins, Design of three dimensional isotropic microstructures for maximized stiffness and conductivity. Int. J. Solids Struct. 45, 4130–4146 (2008)CrossRefGoogle Scholar
  42. 42.
    C. Chateau, L. Gélébart, M. Bornert, J. Crépin, D. Caldemaison, Multiscale approach of mechanical behaviour of sic/sic composites: elastic behaviour at the scale of the tow. Tech. Mechanik 30(1–3), 45–55 (2010)Google Scholar
  43. 43.
    I. Chekkal, M. Bianchi, C. Remillat, F.-X. Becot, L. Jaouen, F. Scarpa, Vibro-acoustic properties of auxetic open cell foam: model and experimental results. Acta Acustica united Acustica 96(2), 266–274 (2010)CrossRefGoogle Scholar
  44. 44.
    C.P. Chen, R.S. Lakes, Micromechanical analysis of dynamic behavior of conventional and negative Poisson’s ratio foams. J. Eng. Mater. Technol. 118(3), 285–288 (1996)CrossRefGoogle Scholar
  45. 45.
    Y. Chen, X.N. Liu, G.K. Hu, Q.P. Sun, Q.S. Zheng, Micropolar continuum modelling of bi-dimensional tetrachiral lattices. Proc. R. Soc. A Math. Phys. Eng. Sci. 470(2165), 20130734 (2014)Google Scholar
  46. 46.
    J.B. Choi, R.S. Lakes, Design of a fastener based on negative Poisson’s ratio foam. Cell. Polym. 10(3), 205–212 (1991)Google Scholar
  47. 47.
    L. Courtois, E. Maire, M. Perez, D. Rodney, O. Bouaziz, Y. Bréchet, Mechanical properties of monofilament entangled materials. Adv. Eng. Mater. 14(12), 1128–1133 (2012)CrossRefGoogle Scholar
  48. 48.
    A.S. Dalaq, D.W. Abueidda, R.K.A. Al-Rub, I.M. Jasiuk, Finite element prediction of effective elastic properties of interpenetrating phase composites with architectured 3d sheet reinforcements. Int. J. Solids Struct. 83, 169–182 (2016)CrossRefGoogle Scholar
  49. 49.
    L. Decker, D. Jeulin, I. Tovena, 3d morphological analysis of the connectivity of a porous medium. Acta Stereologica 17(1), 107–112 (1998)Google Scholar
  50. 50.
    C. Delisée, D. Jeulin, F. Michaud, Morphological characterization and porosity in 3D of cellulosic fibrous materials. C.-R. de l’Acad. des Sci. Serie IIb : Mécanique 329(3), 179–185 (2001)Google Scholar
  51. 51.
    F. dell’Isola, A. Della Corte, I. Giorgio, Higher-gradient continua: The legacy of piola, mindlin, sedov and toupin and some future research perspectives. Math. Mech. Solids 22(4), 852–872 (2017)Google Scholar
  52. 52.
    V.S. Deshpande, N.A. Fleck, Isotropic constitutive models for metallic foams. J. Mech. Phys. Solids 48, 1253–1283 (2000)CrossRefGoogle Scholar
  53. 53.
    F. di Paola, Modélisation multi-échelles du comportement thermo-élastique de composites à particules sphériques. Ph.D. thesis, Ecole Centrale Paris, 2010Google Scholar
  54. 54.
    J. Dirrenberger, Effective properties of architectured materials. Ph.D. thesis, MINES-ParisTech, Paris, 2012Google Scholar
  55. 55.
    J. Dirrenberger, S. Forest, D. Jeulin, Elastoplasticity of auxetic materials. Comput. Mater. Sci. 64, 57–61 (2012)CrossRefGoogle Scholar
  56. 56.
    J. Dirrenberger, S. Forest, D. Jeulin, Effective elastic properties of auxetic microstructures: anisotropy and structural applications. Int. J. Mech. Mater. Des. 9(1), 21–33 (2013)CrossRefGoogle Scholar
  57. 57.
    J. Dirrenberger, S. Forest, D. Jeulin, Towards gigantic RVE sizes for stochastic fibrous networks. Int. J. Solids Struct. 51(2), 359–376 (2014)CrossRefGoogle Scholar
  58. 58.
    J. Dirrenberger, S. Forest, D. Jeulin, C. Colin, Homogenization of periodic auxetic materials, in 11th International Conference on the Mechanical Behavior of Materials (ICM11) Procedia Engineering, vol. 10, pp. 1847–1852 (2011)Google Scholar
  59. 59.
    L. Djumas, A. Molotnikov, G.P. Simon, Y. Estrin, Enhanced mechanical performance of bio-inspired hybrid structures utilising topological interlocking geometry. Sci. Rep. 6, 26706 (2016)CrossRefGoogle Scholar
  60. 60.
    L. Djumas, G.P. Simon, Y. Estrin, A. Molotnikov, Deformation mechanics of non-planar topologically interlocked assemblies with structural hierarchy and varying geometry. Sci. Rep. 7(1), 11844 (2017)CrossRefGoogle Scholar
  61. 61.
    W.J. Drugan, J.R. Willis, A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites. J. Mech. Phys. Solids 44(4), 497–524 (1996)CrossRefGoogle Scholar
  62. 62.
    A.V. Dyskin, Y. Estrin, A.J. Kanel-Belov, E. Pasternak, Toughening by fragmentation–how topology helps. Adv. Eng. Mater. 3, 885–888 (2001)CrossRefGoogle Scholar
  63. 63.
    D. Embury, O. Bouaziz, Steel-based composites: driving forces and classifications. Ann. Rev. Mater. Res. 40, 213–241 (2010)CrossRefGoogle Scholar
  64. 64.
    J. Escoda, D. Jeulin, F. Willot, C. Toulemonde, Three-dimensional morphological modelling of concrete using multiscale poisson polyhedra. J. Microsc. 258(1), 31–48 (2015)CrossRefGoogle Scholar
  65. 65.
    Y. Estrin, A.V. Dyskin, E. Pasternak, Topological interlocking as a material design concept. Mater. Sci. Eng. C 31, 1189–1194 (2011)CrossRefGoogle Scholar
  66. 66.
    Y. Estrin, A.V. Dyskin, E. Pasternak, H.C. Khor, A.J. Kanel-Belov, Topological interlocking of protective tiles for the space shuttle. Philos. Mag. Lett. 83, 351–355 (2003)CrossRefGoogle Scholar
  67. 67.
    K.E. Evans, The design of doubly curved sandwich panels with honeycomb cores. Compos. Struct. 17(2), 95–111 (1991)CrossRefGoogle Scholar
  68. 68.
    K.E. Evans, A. Alderson, Auxetic materials: Functional materials and structures from lateral thinking!. Adv. Mater. 12(9), 617–628 (2000)CrossRefGoogle Scholar
  69. 69.
    K.E. Evans, M.A. Nkansah, I.J. Hutchinson, S.C. Rogers, Molecular network design. Nature 353, 124 (1991)CrossRefGoogle Scholar
  70. 70.
    A. Fallet, P. Lhuissier, L. Salvo, Y. Bréchet, Mechanical behaviour of metallic hollow spheres foam. Adv. Eng. Mater. 10(9), 858–862 (2008)CrossRefGoogle Scholar
  71. 71.
    A. Faure, G. Michailidis, G. Parry, N. Vermaak, R. Estevez, Design of thermoelastic multi-material structures with graded interfaces using topology optimization. Struct. Multi. Optim. 56(4), 823–837 (2017)CrossRefGoogle Scholar
  72. 72.
    Y. Feng, T. Siegmund, E. Habtour, J. Riddick, Impact mechanics of topologically interlocked material assemblies. Int. J. Impact Eng. 75, 140–149 (2015)CrossRefGoogle Scholar
  73. 73.
    J. Fish, V. Filonova, D. Fafalis, Computational continua revisited. Int. J. Numer. Methods Eng. 102(3–4), 332–378 (2015)CrossRefGoogle Scholar
  74. 74.
    J. Fish, S. Kuznetsov, Computational continua. Int. J. Numer. Methods Eng. 84, 774–802 (2010)CrossRefGoogle Scholar
  75. 75.
    S. Forest, Mechanics of Generalized Continua and Heterogeneous Materials (Les Presses de lEcole des Mines de Paris, 2005)Google Scholar
  76. 76.
    S. Forest, J.-S. Blazy, Y. Chastel, F. Moussy, Continuum modeling of strain localization phenomena in metallic foams. J. Mater. Sci. 40, 5903–5910 (2005)CrossRefGoogle Scholar
  77. 77.
    D. François, A. Pineau, A. Zaoui, Mechanical Behaviour of Materials, Volume 1: Micro- and Macroscopic Constitutive Behaviour, volume 180 of Solid Mechanics and Its Applications (Springer, Berlin, 2012)Google Scholar
  78. 78.
    A.J. Freeman, Materials by design and the exciting role of quantum computation/simulation. J. Comput. Appl. Math. 149(1), 27–56 (2002)CrossRefGoogle Scholar
  79. 79.
    F. Fritzen, S. Forest, D. Kondo, T. Böhlke, Computational homogenization of porous materials of green type. Comput. Mech. 52(1), 121–134 (2013)CrossRefGoogle Scholar
  80. 80.
    N. Gaspar, X.J. Ren, C.W. Smith, J.N. Grima, K.E. Evans, Novel honeycombs with auxetic behaviour. Acta Mater. 53, 2439–2445 (2005)CrossRefGoogle Scholar
  81. 81.
    M.G.D. Geers, J. Yvonnet, Multiscale modeling of microstructure property relations. MRS Bull. 41(8), 610–616 (2016)CrossRefGoogle Scholar
  82. 82.
    L. Gélébart, C. Chateau, M. Bornert, Conditions aux limites mixtes normales. In 19ème Congrès Français de Mécanique, 24–28 August 2009, Marseille (2009)Google Scholar
  83. 83.
    A. Ghaedizadeh, J. Shen, X. Ren, Y.M. Xie, Tuning the performance of metallic auxetic metamaterials by using buckling and plasticity. Materials 9(54), 1–17 (2016)Google Scholar
  84. 84.
    I.M. Gitman, H. Askes, L.J. Sluys, Representative volume: Existence and size determination. Eng. Fract. Mech. 74, 2518–2534 (2007)CrossRefGoogle Scholar
  85. 85.
    R.J. Green, A plasticity theory for porous solids. International J. Mech. Sci. 14, 215–224 (1972)CrossRefGoogle Scholar
  86. 86.
    J.K. Guest, J.H. Prévost, Optimizing multifunctional materials: design of microstructures for maximized stiffness and fluid permeability. Int. J. Solids Struct. 43(22–23), 7028–7047 (2006)CrossRefGoogle Scholar
  87. 87.
    L. Guiducci, P. Fratzl, Y. Bréchet, J. Dunlop, Pressurized honeycombs as soft-actuators: a theoretical study. J. R. Soc. Interface 11, 20140458 (2014)CrossRefGoogle Scholar
  88. 88.
    Z. Hashin, Analysis of composite materials–a survey. J. Appl. Mech. 50, 481–505 (1983)CrossRefGoogle Scholar
  89. 89.
    H. Hatami-Marbini, R.C. Picu, Heterogeneous long-range correlated deformation of semiflexible random fiber networks. Phys. Rev. E 80(4) (2009)Google Scholar
  90. 90.
    S. Hazanov, Hill condition and overall properties of composites. Arch. Appl. Mech. 68, 385–394 (1998)CrossRefGoogle Scholar
  91. 91.
    S. Hazanov, C. Huet, Order relationships for boundary conditions effect in heterogeneous bodies smaller than the representative volume. J. Mech. Phys. Solids 42(12), 1995–2011 (1994)CrossRefGoogle Scholar
  92. 92.
    J. Henry, S. Pimenta, Increasing damage tolerance in composites using hierarchical brick-and-mortar microstructures. J. Mech. Phys. Solids 118, 322–340 (2018)CrossRefGoogle Scholar
  93. 93.
    C.T. Herakovich, Composite laminates with negative through-the-thickness poisson’s ratios. J. Compos. Mater. 18(5), 447–455 (1984)CrossRefGoogle Scholar
  94. 94.
    R. Hill, Elastic properties of reinforced solids: Some theoretical principles. J. Mech. Phys. Solids 11, 357–372 (1963)CrossRefGoogle Scholar
  95. 95.
    R. Hill, The essential structure of constitutive laws for metal composites and polycrystals. J. Mech. Phys. Solids 15, 79–95 (1967)CrossRefGoogle Scholar
  96. 96.
    J.B. Hopkins, L.A. Shaw, T.H. Weisgraber, G.R. Farquar, C.D. Harvey, C.M. Spadaccini, Design of nonperiodic microarchitectured materials that achieve graded thermal expansions. J. Mech. Rob. 8(5), 051010 (2016)CrossRefGoogle Scholar
  97. 97.
    A. Hor, N. Saintier, C. Robert, T. Palin-Luc, F. Morel, Statistical assessment of multiaxial hcf criteria at the grain scale. Int. J. Fatigue 67, 151–158 (2014)CrossRefGoogle Scholar
  98. 98.
    C. Huet, Application of variational concepts to size effects in elastic heterogeneous bodies. J. Mech. Phys. Solids 38(6), 813–841 (1990)CrossRefGoogle Scholar
  99. 99.
    C. Huet, An integrated micromechanics and statistical continuum thermodynamics approach for studying the fracture behaviour of microcracked heterogeneous materials with delayed response. Eng. Fracture Mech. 58(5–6), 459–556 (1997)CrossRefGoogle Scholar
  100. 100.
    T.P. Hughes, A. Marmier, K.E. Evans, Auxetic frameworks inspired by cubic crystals. Int. J. Solids Struct. 47, 1469–1476 (2010)CrossRefGoogle Scholar
  101. 101.
    A. Iltchev, V. Marcadon, S. Kruch, S. Forest, Computational homogenisation of periodic cellular materials: application to structural modelling. Int. J. Mech. Sci. 93, 240–255 (2015)CrossRefGoogle Scholar
  102. 102.
    R. Jänicke, H. Sehlhorst, A. Dster, S. Diebels, Micromorphic two-scale modelling of periodic grid structures. Int. J. Multiscale Comput. Eng. 11, 161–176 (2013)CrossRefGoogle Scholar
  103. 103.
    R. Jänicke, H. Steeb, Minimal loading conditions for higher-order numerical homogenisation schemes. Arch. Appl. Mech. 82(8), 1075–1088 (2012)CrossRefGoogle Scholar
  104. 104.
    A. Jean, G.C. Engelmayr, Finite element analysis of an accordion-like honeycomb scaffold for cardiac tissue engineering. J. Biomech. 43, 3035–3043 (2010)CrossRefGoogle Scholar
  105. 105.
    A. Jean, D. Jeulin, S. Forest, S. Cantournet, F. N’Guyen, A multiscale microstructure model of carbon black distribution in rubber. J. Microsc. 241(3), 243–260 (2011a)CrossRefGoogle Scholar
  106. 106.
    A. Jean, F. Willot, S. Cantournet, S. Forest, D. Jeulin, Large-scale computations of effective elastic properties of rubber with carbon black fillers. Int. J. Multiscale Comput. Eng. 9(3), 271–303 (2011b)CrossRefGoogle Scholar
  107. 107.
    D. Jeulin, Modèles de fonctions aléatoires multivariables. Sci. de la Terre 30, 225–256 (1991)Google Scholar
  108. 108.
    D. Jeulin, Random texture models for material structures. Stat. Comput. 10(2), 121–132 (2000)CrossRefGoogle Scholar
  109. 109.
    D. Jeulin, Caractérisation Morphologique et Modèles de Structures Aléatoires, volume 1 of Homogénéisation en Mécanique des Matériaux, chapter 4, pp. 95–132. Hermès (2001)Google Scholar
  110. 110.
    D. Jeulin, Variance scaling of Boolean random varieties. Technical report, Centre de Morphologie Mathématique. N/10/11/MM (2011), hal-00618967, version 1 (2011)Google Scholar
  111. 111.
    D. Jeulin, Power laws variance scaling of Boolean random varieties. Methodol. Comput. Appl. Probab., pp. 1–15 (2015)Google Scholar
  112. 112.
    D. Jeulin, M. Ostoja-Starzewski, Mechanics of Random and Multiscale Microstructures (Springer, CISM Courses, 2001)CrossRefGoogle Scholar
  113. 113.
    N. Kaminakis, G. Drosopoulos, G. Stavroulakis, Design and verification of auxetic microstructures using topology optimization and homogenization. Arch. App. Mech. 85(9), 1289–1306 (2015)CrossRefGoogle Scholar
  114. 114.
    T. Kanit, S. Forest, I. Galliet, V. Mounoury, D. Jeulin, Determination of the size of the representative volume element for random composites: Statistical and numerical approach. Int. J. Solids Struct. 40, 3647–3679 (2003)CrossRefGoogle Scholar
  115. 115.
    T. Kanit, F. N’Guyen, S. Forest, D. Jeulin, M. Reed, S. Singleton, Apparent and effective physical properties of heterogeneous materials: representativity of samples of two materials from food industry. Comput. Methods Appl. Mech. Eng. 195, 3960–3982 (2006)CrossRefGoogle Scholar
  116. 116.
    S. Khakalo, J. Niiranen, Isogeometric analysis of higher-order gradient elasticity by user elements of a commercial finite element software. Comput.-Aided Des. 82, 154–169 (2017)Google Scholar
  117. 117.
    S. Khandelwal, T. Siegmund, R.J. Cipra, J.S. Bolton, Adaptive mechanical properties of topologically interlocking material systems. Smart Mater. Struct. 24(4), 045037 (2015)CrossRefGoogle Scholar
  118. 118.
    A. Kolopp, S. Rivallant, C. Bouvet, Experimental study of sandwich structures as armour against medium-velocity impacts. Int. J. Impact Eng. 61, 24–35 (2013)CrossRefGoogle Scholar
  119. 119.
    C. Körner, Y. Liebold-Ribeiro, A systematic approach to identify cellular auxetic materials. Smart Mater. Struct. 24(2), 025013 (2015)CrossRefGoogle Scholar
  120. 120.
    M. Kotani, S. Ikeda, Materials inspired by mathematics. Sci. Technol. Adv. Mater. 17(1), 253–259 (2016)CrossRefGoogle Scholar
  121. 121.
    N. Kowalski, L. Delannay, P. Yan, J.F. Remacle, Finite element modeling of periodic polycrystalline aggregates with intergranular cracks. Int. J. Solids Struct. 90, 60–68 (2016)CrossRefGoogle Scholar
  122. 122.
    V. Krasavin, A. Krasavin, Auxetic properties of cubic metal single crystals. Phys. Status Solidi b 251(11), 2314–2320 (2014)CrossRefGoogle Scholar
  123. 123.
    T. Krause, A. Molotnikov, M. Carlesso, J. Rente, K. Rezwan, Y. Estrin, D. Koch, Mechanical properties of topologically interlocked structures with elements produced by freeze gelation of ceramic slurries. Adv. Eng. Mater. 14(5), 335–341 (2012)CrossRefGoogle Scholar
  124. 124.
    R.S. Lakes, Foam structures with a negative Poisson’s Ratio. Science 235, 1038–1040 (1987)CrossRefGoogle Scholar
  125. 125.
    R.S. Lakes, Deformation mechanisms in negative Poisson’s ratio materials: structural aspects. J. Mater. Sci. 26, 2287–2292 (1991)CrossRefGoogle Scholar
  126. 126.
    C. Lantuéjoul, Ergodicity and integral range. J. Microsc. 161, 387–403 (1991)CrossRefGoogle Scholar
  127. 127.
    C. Lantuéjoul, Geostatistical Simulation: Models and Algorithms (Springer, Berlin, 2002)CrossRefGoogle Scholar
  128. 128.
    L. Laszczyk, R. Dendievel, O. Bouaziz, Y. Br’echet, G. Parry, Design of architectured sandwich core materials using topological optimization methods. In Symposium LL Architectured Multifunctional Materials, volume 1188 of MRS Proceedings (2009)Google Scholar
  129. 129.
    A. Lebée, K. Sab, Homogenization of thick periodic plates: application of the bending-gradient plate theory to a folded core sandwich panel. Int. J. Solids Struct. 49(19–20), 2778–2792 (2012)CrossRefGoogle Scholar
  130. 130.
    M. Lewandowski, M. Amiot, A. Perwuelz, Development and characterization of 3D nonwoven composites. Mater. Sci. Forum 714, 131–137 (2012)CrossRefGoogle Scholar
  131. 131.
    A.W. Lipsett, A.I. Beltzer, Reexamination of dynamic problems of elasticity for negative poisson’s ratio. J. Acoust. Soc. Am. 84(6), 2179–2186 (1988)CrossRefGoogle Scholar
  132. 132.
    J. Liu, T. Gu, S. Shan, S.H. Kang, J.C. Weaver, K. Bertoldi, Harnessing buckling to design architected materials that exhibit effective negative swelling. Adv. Mater. 28(31), 6619–6624 (2016)CrossRefGoogle Scholar
  133. 133.
    K. Madi, S. Forest, M. Boussuge, S. Gailliègue, E. Lataste, J.-Y. Buffière, D. Bernard, D. Jeulin, Finite element simulations of the deformation of fused-cast refractories based on x-ray computed tomography. Comput. Mater. Sci. 39, 224–229 (2007)CrossRefGoogle Scholar
  134. 134.
    K. Madi, S. Forest, P. Cordier, M. Boussuge, Numerical study of creep in two-phase aggregates with a large rheology contrast: implications for the lower mantle. Earth Planet. Sci. Lett. 237(1–2), 223–238 (2005)CrossRefGoogle Scholar
  135. 135.
    J. Martin, J.-J. Heyder-Bruckner, C. Remillat, F. Scarpa, K. Potter, M. Ruzzene, The hexachiral prismatic wingbox concept. Phys. Status Solidi (b) 245(3), 570–577 (2008)CrossRefGoogle Scholar
  136. 136.
    A. Mather, R. Cipra, T. Siegmund, Structural integrity during remanufacture of a topologically interlocked material. Int. J. Struct. Integrity 3(1), 61–78 (2012)CrossRefGoogle Scholar
  137. 137.
    G. Matheron, The Theory of Regionalized Variables and its Applications Les Cahiers du Centre de Morphologie Mathématique de Fontainebleau. (Ecole des Mines de Paris, 1971)Google Scholar
  138. 138.
    G. Matheron, Random Sets and Integral Geometry (Wiley, 1975)Google Scholar
  139. 139.
    G. Matheron, Estimating and Choosing (Springer, Berlin, 1989)CrossRefGoogle Scholar
  140. 140.
    K. Matouš, M.G.D. Geers, V.G. Kouznetsova, A. Gillman, A review of predictive nonlinear theories for multiscale modeling of heterogeneous materials. J. Comput. Phys. 330, 192–220 (2017)CrossRefGoogle Scholar
  141. 141.
    L. Mezeix, C. Bouvet, J. Huez, D. Poquillon, Mechanical behavior of entangled fibers and entangled cross-linked fibers during compression. J. Mater. Sci. 44(14), 3652–3661 (2009)CrossRefGoogle Scholar
  142. 142.
    J.-C. Michel, H. Moulinec, P. Suquet, Effective properties of composite materials with periodic microstructure: a computational approach. Comput. Methods Appl. Mech. Eng. 172, 109–143 (1999)CrossRefGoogle Scholar
  143. 143.
    R.E. Miller, A continuum plasticity model for the constitutive and indentation behaviour of foamed metals. Int. J. Mech. Sci. 42, 729–754 (2000)CrossRefGoogle Scholar
  144. 144.
    G.W. Milton, Composite materials with Poisson’s Ratios close to -1. J. Mech. Phys. Solids 40(5), 1105–1137 (1992)CrossRefGoogle Scholar
  145. 145.
    A. Molotnikov, Y. Estrin, A.V. Dyskin, E. Pasternak, A.J. Kanel-Belov, Percolation mechanism of failure of a planar assembly of interlocked osteomorphic elements. Eng. Fracture Mech. 74, 1222–1232 (2007)CrossRefGoogle Scholar
  146. 146.
    A. Molotnikov, R. Gerbrand, O. Bouaziz, Y. Estrin, Sandwich panels with a core segmented into topologically interlocked elements. Adv. Eng. Mater. 15(8), 728–731 (2013)CrossRefGoogle Scholar
  147. 147.
    A. Molotnikov, R. Gerbrand, Y. Qi, G.P. Simon, Y. Estrin, Design of responsive materials using topologically interlocked elements. Smart Mater. Struct. 24(2), 025034 (2015)CrossRefGoogle Scholar
  148. 148.
    G.B. Olson, Beyond discovery: design for a new material world. Calphad 25(2), 175–190 (2001)CrossRefGoogle Scholar
  149. 149.
    M. Osanov, J.K. Guest, Topology optimization for architected materials design. Ann. Rev. Mater. Res. 46, 211–233 (2016)CrossRefGoogle Scholar
  150. 150.
    M. Ostoja-Starzewski, Microstructural randomness versus representative volume element in thermomechanics. J. Appl. Mech. 69(1), 25–35 (2002)CrossRefGoogle Scholar
  151. 151.
    M. Ostoja-Starzewski, Microstructural Randomness and Scaling in Mechanics of Materials (Chapman & Hall/CRC, Mordern Mechanics and Mathematics, 2008)Google Scholar
  152. 152.
    M. Oumarou, D. Jeulin, J. Renard, Etude statistique multi-échelle du comportement élastique et thermique d’un composite thermoplastique. Rev. des Compos. et des matériaux avancés 21, 221–254 (2011)Google Scholar
  153. 153.
    M. Oumarou, D. Jeulin, J. Renard, P. Castaing, Multi-scale statistical approach of the elastic and thermal behavior of a thermoplastic polyamid-glass fiber composite. Tech. Mechanik 32(2–5), 484–506 (2012)Google Scholar
  154. 154.
    D.H. Pahr, P.K. Zysset, Influence of boundary conditions on computed apparent elastic properties of cancellous bone. Biomech. Model. Mechanobiol. 7, 463–476 (2008)CrossRefGoogle Scholar
  155. 155.
    E. Pasternak, A. Dyskin, Materials and structures with macroscopic negative poissons ratio. Int. J. Eng. Sci. 52, 103–114 (2012)CrossRefGoogle Scholar
  156. 156.
    C. Pelissou, J. Baccou, Y. Monerie, F. Perales, Determination of the size of the representative volume element for random quasi-brittle composites. Int. J. Solids Struct. 46, 2842–2855 (2009)CrossRefGoogle Scholar
  157. 157.
    X.L. Peng, E. Husser, G.Y. Huang, S. Bargmann, Modeling of surface effects in crystalline materials within the framework of gradient crystal plasticity. J. Mech. Phys. Solids 112, 508–522 (2018)CrossRefGoogle Scholar
  158. 158.
    C. Peyrega, D. Jeulin, C. Delisée, J. Malvestio, 3D morphological modelling of a random fibrous network. Image Anal. Stereol. 28, 129–141 (2009)CrossRefGoogle Scholar
  159. 159.
    C. Peyrega, D. Jeulin, C. Delisée, J. Malvestio, 3D morphological characterization of phonic insulation fibrous media. Adv. Eng. Mater. 13(3), 156–164 (2011)CrossRefGoogle Scholar
  160. 160.
    Pham, T. T.T, Un modèle d’endommagement à gradient de déformation à partir de la méthode d’homogénéisation pour les matériaux fragiles. Ph.D. thesis, Université Paris XIII, 2010Google Scholar
  161. 161.
    R.C. Picu, Mechanics of random fiber networks–a review. Soft Matter 7, 6768–6785 (2011)CrossRefGoogle Scholar
  162. 162.
    R.C. Picu, H. Hatami-Marbini, Long-range correlations of elastic fields in semi-flexible fiber networks. Comput. Mech. 46, 635–640 (2010)CrossRefGoogle Scholar
  163. 163.
    E. Piollet, Amortissement non-linéaire des structures sandwichs à matériau d’âme en fibres enchevêtrées. Ph.D. thesis, ISAE, Toulouse, 2014Google Scholar
  164. 164.
    E. Piollet, G. Michon, D. Poquillon, Nonlinear vibration behavior of sandwich beams with entangled fiber core material. In ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (American Society of Mechanical Engineers, 2013), pp. V008T13A013–V008T13A013Google Scholar
  165. 165.
    E. Piollet, D. Poquillon, G. Michon, Dynamic hysteresis modelling of entangled cross-linked fibres in shear. J. Sound Vibr. 383, 248–264 (2016)CrossRefGoogle Scholar
  166. 166.
    L. Placidi, E. Barchiesi, A. Della Corte, Mathematical Modelling in Solid Mechanics, volume 69 of Advanced Structured Materials, chapter Identification of Two-Dimensional Pantographic Structures with a Linear D4 Orthotropic Second Gradient Elastic Model Accounting for External Bulk Double Forces (Springer, Singapore, 2017), pp. 211–232Google Scholar
  167. 167.
    L. Placidi, A.R. El Dhaba, Semi-inverse method à la saint-venant for two-dimensional linear isotropic homogeneous second-gradient elasticity. Math. Mech. Solids 22(5), 919–937 (2015)CrossRefGoogle Scholar
  168. 168.
    M. Poncelet, A. Somera, C. Morel, C. Jailin, N. Auffray, 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, 223–237 (2018)CrossRefGoogle Scholar
  169. 169.
    D. Prall, R.S. Lakes, Properties of a Chiral honeycomb with a Poisson’s ratio of -1. Int. J. Mech. Sci. 39(3), 305–314 (1997)CrossRefGoogle Scholar
  170. 170.
    Y. Rahali, I. Goda, J.-F. Ganghoffer, Numerical identification of classical and nonclassical moduli of 3d woven textiles and analysis of scale effects. Compos. Struct. 135, 122–139 (2016)CrossRefGoogle Scholar
  171. 171.
    X. Ren, J. Shen, A. Ghaedizadeh, H. Tian, Y.M. Xie, A simple auxetic tubular structure with tuneable mechanical properties. Smart Mater. Struct. 5(6), 065012 (2016)CrossRefGoogle Scholar
  172. 172.
    A. Reuss, Berechnung der fließgrenze von mischkristallen auf grund der plastizitätsbedingung für einkristalle. Zeitschrift für angewandte Mathematik und Mechanik 9(1), 49–58 (1929)CrossRefGoogle Scholar
  173. 173.
    D. Rodney, B. Gadot, O.R. Martinez, S.R. Du Roscoat, L. Orgéas, Reversible dilatancy in entangled single-wire materials. Nat. Mater. 15(1), 72 (2016)CrossRefGoogle Scholar
  174. 174.
    G. Rosi, N. Auffray, Anisotropic and dispersive wave propagation within strain-gradient framework. Wave Motion 63, 120–134 (2016)CrossRefGoogle Scholar
  175. 175.
    G. Rosi, V.H. Nguyen, A. Loseille, S. Naili, Ultrasonic characterization of porous gyroid scaffolds for bone tissue engineering: mechanical modelling and numerical validation. J. Acoust. Soc. Am. 144(3), 1854 (2018a)CrossRefGoogle Scholar
  176. 176.
    G. Rosi, L. Placidi, N. Auffray, On the validity range of strain-gradient elasticity: a mixed static-dynamic identification procedure. Eur. J. Mech. A/Solids 69, 179–191 (2018b)CrossRefGoogle Scholar
  177. 177.
    G. Rosi, I. Scala, V. Nguyen, S. Naili, Wave propagation in strain gradient poroelastic medium with microinertia: closed-form and finite element solutions. Zeitschrift für angewandte Mathematik und Physik 63(3), 58 (2017)CrossRefGoogle Scholar
  178. 178.
    L. Rothenburg, A.A. Berlin, R.J. Bathurst, Microstructure of isotropic materials with negative Poisson’s ratio. Nature 354, 470–472 (1991)CrossRefGoogle Scholar
  179. 179.
    K. Sab, On the homogenization and the simulation of random materials. Eur. J. Mech. A/Solids 11(5), 585–607 (1992)Google Scholar
  180. 180.
    K. Sab, A. Lebée, Homogenization of Heterogeneous Thin and Thick Plates (Wiley, Hoboken, 2015)CrossRefGoogle Scholar
  181. 181.
    K. Sab, B. Nedjar, Periodization of random media and representative volume element size for linear composites. C.-R. de l’Acad. des Sci. Serie IIb : Mécanique 333, 187–195 (2005)Google Scholar
  182. 182.
    M. Salmi, F. Auslender, M. Bornert, M. Fogli, Apparent and effective mechanical properties of linear matrix-inclusion random composites: Improved bounds for the effective behavior. Int. J. Solids Struct. 49, 1195–1211 (2012a)CrossRefGoogle Scholar
  183. 183.
    M. Salmi, F. Auslender, M. Bornert, M. Fogli, Various estimates of representative volume element sizes based on a statistical analysis of the apparent behavior of random linear composites. C.-R. de l’Acad. des Sci. Serie IIb : Mécanique 340, 230–246 (2012b)Google Scholar
  184. 184.
    K. Salonitis, D. Chantzis, V. Kappatos, A hybrid finite element analysis and evolutionary computation method for the design of lightweight lattice components with optimized strut diameter. Int. J. Adv. Manufact. Technol. 90(9–12), 2689–2701 (2017)CrossRefGoogle Scholar
  185. 185.
    E. Sanchez-Palencia, A. Zaoui, Homogenization Techniques for Composite Media, vol. 272. Lecture Notes in Physics (Springer, Berlin, 1987)Google Scholar
  186. 186.
    V. Sansalone, P. Trovalusci, F. Cleri, Multiscale modeling of composite materials by a multifield finite element approach. Int. J. Multiscale Comput. Eng. 3, 463–480 (2005)CrossRefGoogle Scholar
  187. 187.
    B. Sarac, J. Wilmers, S. Bargmann, Property optimization of porous metallic glasses via structural design. Mater. Lett. 134, 306–310 (2014)CrossRefGoogle Scholar
  188. 188.
    F. Scarpa, J.R. Yates, L.G. Ciffo, S. Patsias, Dynamic crushing of auxetic open-cell polyurethane foam. Proc. Inst. Mech. Eng. Part C-J. Mech. Eng. Sci. 216(12), 1153–1156 (2002)Google Scholar
  189. 189.
    S. Schaare, W. Riehemann, Y. Estrin, Damping properties of an assembly of topologically interlocked cubes. Mater. Sci. Eng. A Struct. Mater. Prop. Microstruct. Process. 521–522, 380–383 (2009)Google Scholar
  190. 190.
    T.A. Schaedler, W.B. Carter, Architected cellular materials. Ann. Rev. Mater. Res. 46, 187–210 (2016)CrossRefGoogle Scholar
  191. 191.
    T.A. Schaedler, A.J. Jacobsen, A. Torrents, A.E. Sorensen, J. Lian, J.R. Greer, L. Valdevit, W.B. Carter, Ultralight metallic microlattices. Science 334(6058), 962–965 (2011)CrossRefGoogle Scholar
  192. 192.
    K. Schladitz, S. Peters, D. Reinel-Bitzer, A. Wiegmann, J. Ohser, Design of acoustic trim based on geometric modeling and flow simulation for non-woven. Comput. Mater. Sci. 38, 56–66 (2006)CrossRefGoogle Scholar
  193. 193.
    I. Shufrin, E. Pasternak, A. Dyskin, Planar isotropic structures with negative poissons ratio. Int. J. Solids Struct. 49(17), 2239–2253 (2012)CrossRefGoogle Scholar
  194. 194.
    I. Shufrin, E. Pasternak, A. Dyskin, Negative poissons ratio in hollow sphere materials. Int. J. Solids Struct. 54, 192–214 (2015)CrossRefGoogle Scholar
  195. 195.
    P. Smith, S. Torquato, Computer simulation results for the two-point probability function of composite media. J. Comput. Phys. 76(1), 176–191 (1988)CrossRefGoogle Scholar
  196. 196.
    M.A. Soare, R.C. Picu, An approach to solving mechanics problems for materials with multiscale self-similar microstructure. Int. J. Solids Struct. 44, 7877–7890 (2007)CrossRefGoogle Scholar
  197. 197.
    C. Soyarslan, S. Bargmann, M. Pradas, J. Weissmuller, 3D stochastic bicontinuous microstructures: generation, topology and elasticity. Acta Mater. 149, 326–340 (2018)CrossRefGoogle Scholar
  198. 198.
    A. Spadoni, M. Ruzzene, S. Gonella, F. Scarpa, Phononic properties of hexagonal chiral lattices. Wave Motion 46(7), 435–450 (2009)CrossRefGoogle Scholar
  199. 199.
    J. Teixeira-Pinto, C. Nadot-Martin, F. Touchard, M. Gueguen, S. Castagnet, Towards the size estimation of a representative elementary domain in semi-crystalline polymers. Mechanics of Materials in press (2016)Google Scholar
  200. 200.
    K. Terada, M. Hori, T. Kyoya, N. Kikuchi, Simulation of the multi-scale convergence in computational homogenization approaches. Int. J. Solids Struct. 37, 2285–2311 (2000)CrossRefGoogle Scholar
  201. 201.
    S. Torquato, Morphology and effective properties of disordered heterogeneous media. Int. J. Solids Struct. 35(19), 2385–2406 (1998)CrossRefGoogle Scholar
  202. 202.
    S. Torquato, Random Heterogeneous Materials (Springer, Berlin, 2001)Google Scholar
  203. 203.
    D.K. Trinh, R. Jänicke, N. Auffray, S. Diebels, S. Forest, Evaluation of generalized continuum substitution models for heterogeneous materials. Int. J. Multiscale Comput. Eng. 10(6), 527–549 (2012)CrossRefGoogle Scholar
  204. 204.
    N. Vermaak, G. Michailidis, G. Parry, R. Estevez, G. Allaire, Y. Brechet, Material interface effects on the topology optimizationof multi-phase structures using a level set method. Struct. Mult. Optim. 50(4), 623–644 (2014)CrossRefGoogle Scholar
  205. 205.
    W.M. Vicente, Z.H. Zuo, R. Pavanello, T.K.L. Calixto, R. Picelli, Y.M. Xie, Concurrent topology optimization for minimizing frequency responses of two-level hierarchical structures. Comput. Methods Appl. Mech. Eng. 301, 116–136 (2016)CrossRefGoogle Scholar
  206. 206.
    W. Voigt, Ueber die beziehung zwischen den beiden elasticitätsconstanten isotroper körper. Ann. der Phys. und Chem. 38, 573–587 (1889)CrossRefGoogle Scholar
  207. 207.
    Y. Wang, H. Xu, D. Pasini, Multiscale isogeometric topology optimization for lattice materials. Comput. Methods Appl. Mech. Eng. 316, 568–585 (2017a)CrossRefGoogle Scholar
  208. 208.
    Z.P. Wang, L.H. Poh, J. Dirrenberger, Y. Zhu, S. Forest, Isogeometric shape optimization of smoothed petal auxetic structures via computational periodic homogenization. Comput. Methods Appl. Mech. Eng. 323, 250–271 (2017b)CrossRefGoogle Scholar
  209. 209.
    P.M. Weaver, M.F. Ashby, The optimal selection of material and section-shape. J. Eng. Des. 7(2), 129–150 (1996)CrossRefGoogle Scholar
  210. 210.
    B. Xu, X. Huang, S.W. Zhou, Y.M. Xie, Concurrent topological design of composite thermoelastic macrostructure and microstructure with multi-phase material for maximum stiffness. Compos. Struct. 150, 84–102 (2016a)CrossRefGoogle Scholar
  211. 211.
    S. Xu, J. Shen, S. Zhou, X. Huang, Y.M. Xie, Design of lattice structures with controlled anisotropy. Mater. Des. 93, 443–447 (2016b)CrossRefGoogle Scholar
  212. 212.
    W. Yang et al., Review on auxetic materials. J. Mater. Sci. 39, 3269–3279 (2004)CrossRefGoogle Scholar
  213. 213.
    C.L.Y. Yeong, S. Torquato, Reconstructing random media. Phys. Rev. E 57(1), 495 (1998)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Justin Dirrenberger
    • 1
    Email author
  • Samuel Forest
    • 2
  • Dominique Jeulin
    • 3
  1. 1.Laboratoire PIMM, Arts et Métiers-ParisTech, CNRS, CnamParisFrance
  2. 2.Centre des Matériaux, MINES-ParisTech, CNRS UMR 7633Evry cedexFrance
  3. 3.Centre de Morphologie Mathématique, MINES-ParisTechFontainebleauFrance

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