M. Abouaf, J. Chenot, G. Raisson, and P. Bauduin, Finite element simulation of hot isostatic pressing of metal powders, Int. J. Numer. Methods Eng, vol.25, pp.191-212, 1988.

A. Alderson, K. L. Alderson, D. Attard, K. E. Evans, R. Gatt et al., Elastic constants of 3-, 4-and 6-connected chiral and anti-chiral honeycombs subject to uniaxial in-plane loading, Compos. Sci. Technol, vol.70, issue.7, pp.1042-1048, 2010.

G. Allaire, Shape Optimization by the Homogenization Method, 2002.

R. F. Almgren, An isotropic three-dimensional structure with Poisson's ratio-1, J. Elast, vol.15, pp.427-430, 1985.

, Generalized Continua from the Theory to Engineering Applications, 2013.

H. Altendorf, D. Jeulin, and F. Willot, Influence of the fiber geometry on the macroscopic elastic and thermal properties, Int. J. Solids Struct, vol.51, pp.3807-3822, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01002835

J. C. Alvarez-elipe and A. Diaz-lantada, Comparative study of auxetic geometries by means of computer-aided design and engineering, Smart Mater. Struct, vol.21, p.105004, 2012.

E. Andreassen, B. Lazarov, and O. Sigmund, Design of manufacturable 3d extremal elastic microstructure, Mech. Mater, vol.69, issue.1, pp.1-10, 2014.

U. Andreaus, F. Isola, I. Giorgio, L. Placidi, T. Lekszycki et al., Numerical simulations of classical problems in two-dimensional (non) linear second gradient elasticity, Int. J. Eng. Sci, vol.108, pp.34-50, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01378498

A. Asadpoure, M. Tootkaboni, and L. Valdevit, Topology optimization of multiphase architected materials for energy dissipation, Comput. Methods Appl. Mech. Eng, vol.325, pp.314-329, 2017.

M. F. Ashby and Y. Bréchet, Designing hybrid materials, Acta Mater, vol.51, pp.5801-5821, 2003.

N. Auffray, Analytical expressions for odd-order anisotropic tensor dimension, C. R. Mécaniques, vol.342, issue.5, pp.284-291, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00922063

N. Auffray, J. Dirrenberger, and G. Rosi, A complete description of bi-dimensional anisotropic strain-gradient elasticity, Int. J. Solids Struct, pp.195-210, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01145736

J. Auriault, Heterogeneous medium is an equivalent macroscopic description possible?, Int. J. Eng. Sci, vol.29, issue.7, pp.785-795, 1991.

S. Babaee, J. Shim, J. Weaver, E. Chen, N. Patel et al., 3d soft metamaterials with negative poisson's ratio, Adv. Mater, vol.25, issue.36, pp.5044-5049, 2013.

A. Bacigalupo and L. Gambarotta, Homogenization of periodic hexa-and tetrachiral cellular solids, Compos. Struct, vol.116, pp.461-476, 2014.

X. Badiche, S. Forest, T. Guibert, Y. Bienvenu, J. Bartout et al., 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, vol.289, pp.276-288, 2000.

E. Bafekrpour, A. Dyskin, E. Pasternak, A. Molotnikov, and Y. Estrin, Internally architectured materials with directionally asymmetric friction, Sci. Rep, vol.5, p.10732, 2015.

E. Bafekrpour, A. Molotnikov, J. C. Weaver, Y. Brechet, and Y. Estrin, Responsive materials: a novel design for enhanced machine-augmented composites, Sci. Rep, vol.4, p.3783, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00805042

C. Barbier, R. Dendievel, and D. Rodney, Numerical study of 3D-compressions of entangled materials, Comput. Mater. Sci, vol.45, pp.593-596, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00388904

C. Barbier, R. Dendievel, and D. Rodney, Role of friction in the mechanics of nonbounded fibrous materials, Phys. Rev. E, vol.80, issue.1, p.16115, 2009.

S. Bargmann, B. Klusemann, J. Markmann, J. E. Schnabel, K. Schneider et al., Generation of 3d representative volume elements for heterogeneous materials: a review, Prog. Mater. Sci, vol.96, pp.322-384, 2018.

R. J. Bathurst and L. Rothenburg, Note on a random isotropic granular material with negative Poisson's ratio, Int. J. Eng. Sci, vol.26, issue.4, pp.373-383, 1988.

M. Bendsøe and O. Sigmund, Topology Optimization, 2004.

I. Benedetti and F. Barbe, Modelling polycrystalline materials: an overview of three-dimensional grain-scale mechanical models, J. Multiscale Model, vol.5, issue.1, p.1350002, 2013.

M. J. Beran, Statistical Continuum Theories, 1968.

C. Berdin, Z. Y. Yao, and S. Pascal, Internal stresses in polycrystalline zirconia: microstructure effects, Comput. Mater. Sci, vol.70, pp.140-144, 2013.

K. Bertoldi, P. Reis, S. Willshaw, and T. Mullin, Negative poisson's ratio behavior induced by an elastic instability, Adv. Mater, vol.22, issue.3, pp.361-366, 2010.

J. Besson, G. Cailletaud, J. Chaboche, S. Forest, and M. Blétry, Solid Mechanics and Its Applications, vol.167, 2010.

D. Bigoni and W. Drugan, Analytical derivation of cosserat moduli via homogenization of heterogeneous elastic materials, J. Appl. Mech, vol.74, pp.741-753, 2007.

A. Bironeau, J. Dirrenberger, C. Sollogoub, G. Miquelard-garnier, and S. Roland, Evaluation of morphological representative sample sizes for nanolayered polymer blends, J. Microsc, vol.264, issue.1, pp.48-58, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01383497

M. Bornert, T. Bretheau, and P. Gilormini, Homogénéisation en mécanique des matériaux, Matériaux aléatoires élastiques et milieux périodiques, vol.1, 2001.

O. Bouaziz, Y. Bréchet, and J. D. Embury, Heterogeneous and architectured materials: a possible strategy for design of structural materials, Adv. Eng. Mater, vol.10, issue.1-2, pp.24-36, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00358408

O. Bouaziz, J. P. Masse, S. Allain, L. Orgéas, and P. Latil, Compression of crumpled aluminum thin foils and comparison with other cellular materials, Mater. Sci. Eng. Struct. Mater. Prop. Microstruct. Process, vol.570, pp.1-7, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00821940

Y. Bréchet and J. D. Embury, Architectured materials: expanding materials space, Scripta Mater, vol.68, issue.1, pp.1-3, 2013.

H. J. Bunge, Texture Analysis in Materials Science, 1982.

B. D. Caddock and K. E. Evans, Microporous materials with negative Poisson's ratios: I. Microstructure and mechanical properties, J. Phys. D: Appl. Phys, vol.22, pp.1877-1882, 1989.

G. Cailletaud, S. Forest, D. Jeulin, F. Feyel, I. Galliet et al., Some elements of microstructural mechanics, Comput. Mater. Sci, vol.27, pp.351-374, 2003.

G. Cailletaud, D. Jeulin, and P. Rolland, Size effect on elastic properties of random composites, Eng. Comput, vol.11, issue.2, pp.99-110, 1994.

O. Caty, E. Maire, and R. Bouchet, Fatigue of metal hollow spheres structures, Adv. Eng. Mater, vol.10, issue.3, pp.179-184, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00434025

V. J. Challis, A. P. Roberts, and A. H. Wilkins, Design of three dimensional isotropic microstructures for maximized stiffness and conductivity, Int. J. Solids Struct, vol.45, pp.4130-4146, 2008.

C. Chateau, L. Gélébart, M. Bornert, J. Crépin, and D. Caldemaison, Multiscale approach of mechanical behaviour of sic/sic composites: elastic behaviour at the scale of the tow, Tech. Mechanik, vol.30, issue.1-3, pp.45-55, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00490495

I. Chekkal, M. Bianchi, C. Remillat, F. Becot, L. Jaouen et al., Vibro-acoustic properties of auxetic open cell foam: model and experimental results, Acta Acustica united Acustica, vol.96, issue.2, pp.266-274, 2010.

C. P. Chen and R. S. Lakes, Micromechanical analysis of dynamic behavior of conventional and negative Poisson's ratio foams, J. Eng. Mater. Technol, vol.118, issue.3, pp.285-288, 1996.

Y. Chen, X. N. Liu, G. K. Hu, Q. P. Sun, and Q. S. Zheng, Micropolar continuum modelling of bidimensional tetrachiral lattices, Proc. R. Soc. A Math. Phys. Eng. Sci, vol.470, p.20130734, 2014.

J. B. Choi and R. S. Lakes, Design of a fastener based on negative Poisson's ratio foam, Cell. Polym, vol.10, issue.3, pp.205-212, 1991.

L. Courtois, E. Maire, M. Perez, D. Rodney, O. Bouaziz et al., Mechanical properties of monofilament entangled materials, Adv. Eng. Mater, vol.14, issue.12, pp.1128-1133, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00781282

A. S. Dalaq, D. W. Abueidda, R. K. Al-rub, and I. M. Jasiuk, Finite element prediction of effective elastic properties of interpenetrating phase composites with architectured 3d sheet reinforcements, Int. J. Solids Struct, vol.83, pp.169-182, 2016.

L. Decker, D. Jeulin, and I. Tovena, 3d morphological analysis of the connectivity of a porous medium, Acta Stereologica, vol.17, issue.1, pp.107-112, 1998.

C. Delisée, D. Jeulin, and F. Michaud, Morphological characterization and porosity in 3D of cellulosic fibrous materials. C.-R. de l'Acad, des Sci. Serie IIb : Mécanique, vol.329, issue.3, pp.179-185, 2001.

F. Isola, A. Della-corte, and I. Giorgio, Higher-gradient continua: The legacy of piola, mindlin, sedov and toupin and some future research perspectives, Math. Mech. Solids, vol.22, issue.4, pp.852-872, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01256929

V. S. Deshpande and N. A. Fleck, Isotropic constitutive models for metallic foams, J. Mech. Phys. Solids, vol.48, pp.1253-1283, 2000.

F. Di-paola, Modélisation multi-échelles du comportement thermo-élastique de composites à particules sphériques, 2010.

J. Dirrenberger, Effective properties of architectured materials, 2012.
URL : https://hal.archives-ouvertes.fr/hal-01713384

J. Dirrenberger, S. Forest, and D. Jeulin, Elastoplasticity of auxetic materials, Comput. Mater. Sci, vol.64, pp.57-61, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00732417

J. Dirrenberger, S. Forest, and D. Jeulin, Effective elastic properties of auxetic microstructures: anisotropy and structural applications, Int. J. Mech. Mater. Des, vol.9, issue.1, pp.21-33, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00790180

J. Dirrenberger, S. Forest, and D. Jeulin, Towards gigantic RVE sizes for stochastic fibrous networks, Int. J. Solids Struct, vol.51, issue.2, pp.359-376, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00931281

J. Dirrenberger, S. Forest, D. Jeulin, and C. Colin, Homogenization of periodic auxetic materials, 11th International Conference on the Mechanical Behavior of Materials (ICM11) Procedia Engineering, vol.10, pp.1847-1852, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00607077

L. Djumas, A. Molotnikov, G. P. Simon, and Y. Estrin, Enhanced mechanical performance of bio-inspired hybrid structures utilising topological interlocking geometry, Sci. Rep, vol.6, p.26706, 2016.

L. Djumas, G. P. Simon, Y. Estrin, and A. Molotnikov, Deformation mechanics of non-planar topologically interlocked assemblies with structural hierarchy and varying geometry, Sci. Rep, vol.7, issue.1, p.11844, 2017.

W. J. Drugan and J. R. Willis, A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites, J. Mech. Phys. Solids, vol.44, issue.4, pp.497-524, 1996.

A. V. Dyskin, Y. Estrin, A. J. Kanel-belov, and E. Pasternak, Toughening by fragmentation-how topology helps, Adv. Eng. Mater, vol.3, pp.885-888, 2001.

D. Embury and O. Bouaziz, Steel-based composites: driving forces and classifications, Ann. Rev. Mater. Res, vol.40, pp.213-241, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00523239

J. Escoda, D. Jeulin, F. Willot, and C. Toulemonde, Three-dimensional morphological modelling of concrete using multiscale poisson polyhedra, J. Microsc, vol.258, issue.1, pp.31-48, 2015.

Y. Estrin, A. V. Dyskin, and E. Pasternak, Topological interlocking as a material design concept, Mater. Sci. Eng. C, vol.31, pp.1189-1194, 2011.

Y. Estrin, A. V. Dyskin, E. Pasternak, H. C. Khor, and A. J. , Kanel-Belov, Topological interlocking of protective tiles for the space shuttle, Philos. Mag. Lett, vol.83, pp.351-355, 2003.

K. E. Evans, The design of doubly curved sandwich panels with honeycomb cores, Compos. Struct, vol.17, issue.2, pp.95-111, 1991.

K. E. Evans and A. Alderson, Auxetic materials: Functional materials and structures from lateral thinking!, Adv. Mater, vol.12, issue.9, pp.617-628, 2000.

K. E. Evans, M. A. Nkansah, I. J. Hutchinson, and S. C. Rogers, Molecular network design, Nature, vol.353, p.124, 1991.

A. Fallet, P. Lhuissier, L. Salvo, and Y. Bréchet, Mechanical behaviour of metallic hollow spheres foam, Adv. Eng. Mater, vol.10, issue.9, pp.858-862, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00374960

A. Faure, G. Michailidis, G. Parry, N. Vermaak, and R. Estevez, Design of thermoelastic multimaterial structures with graded interfaces using topology optimization, Struct. Multi. Optim, vol.56, issue.4, pp.823-837, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01383393

Y. Feng, T. Siegmund, E. Habtour, and J. Riddick, Impact mechanics of topologically interlocked material assemblies, Int. J. Impact Eng, vol.75, pp.140-149, 2015.

J. Fish, V. Filonova, and D. Fafalis, Computational continua revisited, Int. J. Numer. Methods Eng, vol.102, issue.3-4, pp.332-378, 2015.

J. Fish and S. Kuznetsov, Computational continua, Int. J. Numer. Methods Eng, vol.84, pp.774-802, 2010.

S. Forest, Mechanics of Generalized Continua and Heterogeneous Materials (Les Presses de lEcole des Mines de, 2005.

S. Forest, J. Blazy, Y. Chastel, and F. Moussy, Continuum modeling of strain localization phenomena in metallic foams, J. Mater. Sci, vol.40, pp.5903-5910, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00153590

D. François, A. Pineau, and A. Zaoui, Micro-and Macroscopic Constitutive Behaviour, vol.1, 2012.

A. J. Freeman, Materials by design and the exciting role of quantum computation/simulation, J. Comput. Appl. Math, vol.149, issue.1, pp.27-56, 2002.

F. Fritzen, S. Forest, D. Kondo, and T. Böhlke, Computational homogenization of porous materials of green type, Comput. Mech, vol.52, issue.1, pp.121-134, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00874802

N. Gaspar, X. J. Ren, C. W. Smith, J. N. Grima, and K. E. Evans, Novel honeycombs with auxetic behaviour, Acta Mater, vol.53, pp.2439-2445, 2005.

M. G. Geers and J. Yvonnet, Multiscale modeling of microstructure property relations, MRS Bull, vol.41, issue.8, pp.610-616, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01354910

L. Gélébart, C. Chateau, and M. Bornert, Conditions aux limites mixtes normales, 19ème Congrès Français de Mécanique, pp.24-28, 2009.

A. Ghaedizadeh, J. Shen, X. Ren, and Y. M. Xie, Tuning the performance of metallic auxetic metamaterials by using buckling and plasticity, Materials, vol.9, issue.54, pp.1-17, 2016.

I. M. Gitman, H. Askes, and L. J. Sluys, Representative volume: Existence and size determination, Eng. Fract. Mech, vol.74, pp.2518-2534, 2007.

R. J. Green, A plasticity theory for porous solids, International J. Mech. Sci, vol.14, pp.215-224, 1972.

J. K. Guest and J. H. Prévost, Optimizing multifunctional materials: design of microstructures for maximized stiffness and fluid permeability, Int. J. Solids Struct, vol.43, pp.7028-7047, 2006.

L. Guiducci, P. Fratzl, Y. Bréchet, and J. Dunlop, Pressurized honeycombs as soft-actuators: a theoretical study, J. R. Soc. Interface, vol.11, p.20140458, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01138231

Z. Hashin, Analysis of composite materials-a survey, J. Appl. Mech, vol.50, pp.481-505, 1983.

H. Hatami-marbini and R. C. Picu, Heterogeneous long-range correlated deformation of semiflexible random fiber networks, Phys. Rev. E, vol.80, issue.4, 2009.

S. Hazanov, Hill condition and overall properties of composites, Arch. Appl. Mech, vol.68, pp.385-394, 1998.

S. Hazanov and C. Huet, Order relationships for boundary conditions effect in heterogeneous bodies smaller than the representative volume, J. Mech. Phys. Solids, vol.42, issue.12, pp.1995-2011, 1994.

J. Henry and S. Pimenta, Increasing damage tolerance in composites using hierarchical brickand-mortar microstructures, J. Mech. Phys. Solids, vol.118, pp.322-340, 2018.

C. T. Herakovich, Composite laminates with negative through-the-thickness poisson's ratios, J. Compos. Mater, vol.18, issue.5, pp.447-455, 1984.

R. Hill, Elastic properties of reinforced solids: Some theoretical principles, J. Mech. Phys. Solids, vol.11, pp.357-372, 1963.

R. Hill, The essential structure of constitutive laws for metal composites and polycrystals, J. Mech. Phys. Solids, vol.15, pp.79-95, 1967.

J. B. Hopkins, L. A. Shaw, T. H. Weisgraber, G. R. Farquar, C. D. Harvey et al., Design of nonperiodic microarchitectured materials that achieve graded thermal expansions, J. Mech. Rob, vol.8, issue.5, p.51010, 2016.

A. Hor, N. Saintier, C. Robert, T. Palin-luc, and F. Morel, Statistical assessment of multiaxial hcf criteria at the grain scale, Int. J. Fatigue, vol.67, pp.151-158, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01057879

C. Huet, Application of variational concepts to size effects in elastic heterogeneous bodies, J. Mech. Phys. Solids, vol.38, issue.6, pp.813-841, 1990.

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, vol.58, issue.5-6, pp.459-556, 1997.

T. P. Hughes, A. Marmier, and K. E. Evans, Auxetic frameworks inspired by cubic crystals, Int. J. Solids Struct, vol.47, pp.1469-1476, 2010.

A. Iltchev, V. Marcadon, S. Kruch, and S. Forest, Computational homogenisation of periodic cellular materials: application to structural modelling, Int. J. Mech. Sci, vol.93, pp.240-255, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01149848

R. Jänicke, H. Sehlhorst, A. Dster, and S. Diebels, Micromorphic two-scale modelling of periodic grid structures, Int. J. Multiscale Comput. Eng, vol.11, pp.161-176, 2013.

R. Jänicke and H. Steeb, Minimal loading conditions for higher-order numerical homogenisation schemes, Arch. Appl. Mech, vol.82, issue.8, pp.1075-1088, 2012.

A. Jean and G. C. Engelmayr, Finite element analysis of an accordion-like honeycomb scaffold for cardiac tissue engineering, J. Biomech, vol.43, pp.3035-3043, 2010.

A. Jean, D. Jeulin, S. Forest, S. Cantournet, and F. N'guyen, A multiscale microstructure model of carbon black distribution in rubber, J. Microsc, vol.241, issue.3, pp.243-260, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00585338

A. Jean, F. Willot, S. Cantournet, S. Forest, and D. Jeulin, Large-scale computations of effective elastic properties of rubber with carbon black fillers, Int. J. Multiscale Comput. Eng, vol.9, issue.3, pp.271-303, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00661612

D. Jeulin, Modèles de fonctions aléatoires multivariables. Sci. de la Terre, vol.30, pp.225-256, 1991.

D. Jeulin, Random texture models for material structures, Stat. Comput, vol.10, issue.2, pp.121-132, 2000.

D. Jeulin, Caractérisation Morphologique et Modèles de Structures Aléatoires, vol.1, pp.95-132, 2001.

D. Jeulin, Variance scaling of Boolean random varieties, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00618967

D. Jeulin, Power laws variance scaling of Boolean random varieties, Methodol. Comput. Appl. Probab, pp.1-15, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01207456

D. Jeulin and M. Ostoja-starzewski, Mechanics of Random and Multiscale Microstructures, 2001.

N. Kaminakis, G. Drosopoulos, and G. Stavroulakis, Design and verification of auxetic microstructures using topology optimization and homogenization, Arch. App. Mech, vol.85, issue.9, pp.1289-1306, 2015.

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, vol.40, pp.3647-3679, 2003.

T. Kanit, F. N'guyen, S. Forest, D. Jeulin, M. Reed et al., Apparent and effective physical properties of heterogeneous materials: representativity of samples of two materials from food industry, Comput. Methods Appl. Mech. Eng, vol.195, pp.3960-3982, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00139164

S. Khakalo and J. Niiranen, Isogeometric analysis of higher-order gradient elasticity by user elements of a commercial finite element software, Comput.-Aided Des, vol.82, pp.154-169, 2017.

S. Khandelwal, T. Siegmund, R. J. Cipra, and J. S. Bolton, Adaptive mechanical properties of topologically interlocking material systems, Smart Mater. Struct, vol.24, issue.4, p.45037, 2015.

A. Kolopp, S. Rivallant, and C. Bouvet, Experimental study of sandwich structures as armour against medium-velocity impacts, Int. J. Impact Eng, vol.61, pp.24-35, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00906305

C. Körner and Y. Liebold-ribeiro, A systematic approach to identify cellular auxetic materials, Smart Mater. Struct, vol.24, issue.2, p.25013, 2015.

M. Kotani and S. Ikeda, Materials inspired by mathematics. Sci. Technol. Adv. Mater, vol.17, issue.1, pp.253-259, 2016.

N. Kowalski, L. Delannay, P. Yan, and J. F. Remacle, Finite element modeling of periodic polycrystalline aggregates with intergranular cracks, Int. J. Solids Struct, vol.90, pp.60-68, 2016.

V. Krasavin and A. Krasavin, Auxetic properties of cubic metal single crystals, Phys. Status Solidi b, vol.251, issue.11, pp.2314-2320, 2014.

T. Krause, A. Molotnikov, M. Carlesso, J. Rente, K. Rezwan et al., Mechanical properties of topologically interlocked structures with elements produced by freeze gelation of ceramic slurries, Adv. Eng. Mater, vol.14, issue.5, pp.335-341, 2012.

R. S. Lakes, Foam structures with a negative Poisson's Ratio, Science, vol.235, pp.1038-1040, 1987.

R. S. Lakes, Deformation mechanisms in negative Poisson's ratio materials: structural aspects, J. Mater. Sci, vol.26, pp.2287-2292, 1991.

C. , Ergodicity and integral range, J. Microsc, vol.161, pp.387-403, 1991.

C. , Geostatistical Simulation: Models and Algorithms, 2002.

L. Laszczyk, R. Dendievel, O. Bouaziz, Y. Br'echet, and G. Parry, Design of architectured sandwich core materials using topological optimization methods, Symposium LL Architectured Multifunctional Materials, vol.1188, 2009.

A. Lebée and K. Sab, Homogenization of thick periodic plates: application of the bending-gradient plate theory to a folded core sandwich panel, Int. J. Solids Struct, vol.49, pp.2778-2792, 2012.

M. Lewandowski, M. Amiot, and A. Perwuelz, Development and characterization of 3D nonwoven composites, Mater. Sci. Forum, vol.714, pp.131-137, 2012.

A. W. Lipsett and A. I. Beltzer, Reexamination of dynamic problems of elasticity for negative poisson's ratio, J. Acoust. Soc. Am, vol.84, issue.6, pp.2179-2186, 1988.

J. Liu, T. Gu, S. Shan, S. H. Kang, J. C. Weaver et al., Harnessing buckling to design architected materials that exhibit effective negative swelling, Adv. Mater, vol.28, issue.31, pp.6619-6624, 2016.

K. Madi, S. Forest, M. Boussuge, S. Gailliègue, E. Lataste et al., Finite element simulations of the deformation of fused-cast refractories based on xray computed tomography, Comput. Mater. Sci, vol.39, pp.224-229, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00140558

K. Madi, S. Forest, P. Cordier, and M. Boussuge, Numerical study of creep in two-phase aggregates with a large rheology contrast: implications for the lower mantle, Earth Planet. Sci. Lett, vol.237, issue.1-2, pp.223-238, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00146056

J. Martin, J. Heyder-bruckner, C. Remillat, F. Scarpa, K. Potter et al., The hexachiral prismatic wingbox concept, Phys. Status Solidi (b), vol.245, issue.3, pp.570-577, 2008.

A. Mather, R. Cipra, and T. Siegmund, Structural integrity during remanufacture of a topologically interlocked material, Int. J. Struct. Integrity, vol.3, issue.1, pp.61-78, 2012.

G. Matheron, The Theory of Regionalized Variables and its Applications Les Cahiers du Centre de, Morphologie Mathématique de Fontainebleau, 1971.

G. Matheron, Random Sets and Integral Geometry, 1975.

G. Matheron, Estimating and Choosing, 1989.

K. Matou?, M. G. Geers, V. G. Kouznetsova, and A. Gillman, A review of predictive nonlinear theories for multiscale modeling of heterogeneous materials, J. Comput. Phys, vol.330, pp.192-220, 2017.

L. Mezeix, C. Bouvet, J. Huez, and D. Poquillon, Mechanical behavior of entangled fibers and entangled cross-linked fibers during compression, J. Mater. Sci, vol.44, issue.14, pp.3652-3661, 2009.
URL : https://hal.archives-ouvertes.fr/hal-01853233

J. Michel, H. Moulinec, and P. Suquet, Effective properties of composite materials with periodic microstructure: a computational approach, Comput. Methods Appl. Mech. Eng, vol.172, pp.109-143, 1999.

R. E. Miller, A continuum plasticity model for the constitutive and indentation behaviour of foamed metals, Int. J. Mech. Sci, vol.42, pp.729-754, 2000.

G. W. Milton, Composite materials with Poisson's Ratios close to -1, J. Mech. Phys. Solids, vol.40, issue.5, pp.1105-1137, 1992.

A. Molotnikov, Y. Estrin, A. V. Dyskin, E. Pasternak, and A. J. , Kanel-Belov, Percolation mechanism of failure of a planar assembly of interlocked osteomorphic elements, Eng. Fracture Mech, vol.74, pp.1222-1232, 2007.

A. Molotnikov, R. Gerbrand, O. Bouaziz, and Y. Estrin, Sandwich panels with a core segmented into topologically interlocked elements, Adv. Eng. Mater, vol.15, issue.8, pp.728-731, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00853692

A. Molotnikov, R. Gerbrand, Y. Qi, G. P. Simon, and Y. Estrin, Design of responsive materials using topologically interlocked elements, Smart Mater. Struct, vol.24, issue.2, p.25034, 2015.

G. B. Olson, Beyond discovery: design for a new material world, Calphad, vol.25, issue.2, pp.175-190, 2001.

M. Osanov and J. K. Guest, Topology optimization for architected materials design, Ann. Rev. Mater. Res, vol.46, pp.211-233, 2016.

M. Ostoja-starzewski, Microstructural randomness versus representative volume element in thermomechanics, J. Appl. Mech, vol.69, issue.1, pp.25-35, 2002.

M. Ostoja-starzewski, Microstructural Randomness and Scaling in Mechanics of Materials, 2008.

M. Oumarou, D. Jeulin, and J. Renard, Etude statistique multi-échelle du comportement élastique et thermique d'un composite thermoplastique, Rev. des Compos. et des matériaux avancés, vol.21, pp.221-254, 2011.

M. Oumarou, D. Jeulin, J. Renard, and P. Castaing, Multi-scale statistical approach of the elastic and thermal behavior of a thermoplastic polyamid-glass fiber composite, Tech. Mechanik, vol.32, issue.2-5, pp.484-506, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00722633

D. H. Pahr and P. K. Zysset, Influence of boundary conditions on computed apparent elastic properties of cancellous bone, Biomech. Model. Mechanobiol, vol.7, pp.463-476, 2008.

E. Pasternak and A. Dyskin, Materials and structures with macroscopic negative poissons ratio, Int. J. Eng. Sci, vol.52, pp.103-114, 2012.

C. Pelissou, J. Baccou, Y. Monerie, and F. Perales, Determination of the size of the representative volume element for random quasi-brittle composites, Int. J. Solids Struct, vol.46, pp.2842-2855, 2009.
URL : https://hal.archives-ouvertes.fr/hal-02095427

X. L. Peng, E. Husser, G. Y. Huang, and S. Bargmann, Modeling of surface effects in crystalline materials within the framework of gradient crystal plasticity, J. Mech. Phys. Solids, vol.112, pp.508-522, 2018.

C. Peyrega, D. Jeulin, C. Delisée, and J. Malvestio, 3D morphological modelling of a random fibrous network, Image Anal. Stereol, vol.28, pp.129-141, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00906502

C. Peyrega, D. Jeulin, C. Delisée, and J. Malvestio, 3D morphological characterization of phonic insulation fibrous media, Adv. Eng. Mater, vol.13, issue.3, pp.156-164, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00833504

T. T. Pham, Un modèle d'endommagement à gradient de déformation à partir de la méthode d'homogénéisation pour les matériaux fragiles, 2010.

R. C. Picu, Mechanics of random fiber networks-a review, Soft Matter, vol.7, pp.6768-6785, 2011.

R. C. Picu and H. Hatami-marbini, Long-range correlations of elastic fields in semi-flexible fiber networks, Comput. Mech, vol.46, pp.635-640, 2010.

E. Piollet, Amortissement non-linéaire des structures sandwichs à matériau d'âme en fibres enchevêtrées, ISAE, 2014.

E. Piollet, G. Michon, and D. Poquillon, Nonlinear vibration behavior of sandwich beams with entangled fiber core material, ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, pp.8-13, 2013.

E. Piollet, D. Poquillon, and G. Michon, Dynamic hysteresis modelling of entangled cross-linked fibres in shear, J. Sound Vibr, vol.383, pp.248-264, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01758580

L. Placidi, E. Barchiesi, and A. Della-corte, 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, Mathematical Modelling in Solid Mechanics, pp.211-232, 2017.

L. Placidi and A. R. El-dhaba, Semi-inverse method à la saint-venant for two-dimensional linear isotropic homogeneous second-gradient elasticity, Math. Mech. Solids, vol.22, issue.5, pp.919-937, 2015.

M. Poncelet, A. Somera, C. Morel, C. Jailin, and 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, vol.147, pp.223-237, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01704787

D. Prall and R. S. Lakes, Properties of a Chiral honeycomb with a Poisson's ratio of -1, Int. J. Mech. Sci, vol.39, issue.3, pp.305-314, 1997.

Y. Rahali, I. Goda, and J. Ganghoffer, Numerical identification of classical and nonclassical moduli of 3d woven textiles and analysis of scale effects, Compos. Struct, vol.135, pp.122-139, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01416410

X. Ren, J. Shen, A. Ghaedizadeh, H. Tian, and Y. M. Xie, A simple auxetic tubular structure with tuneable mechanical properties, Smart Mater. Struct, vol.5, issue.6, p.65012, 2016.

A. Reuss, Berechnung der fließgrenze von mischkristallen auf grund der plastizitätsbedingung für einkristalle, Zeitschrift für angewandte Mathematik und Mechanik, vol.9, issue.1, pp.49-58, 1929.

D. Rodney, B. Gadot, O. R. Martinez, S. R. Du-roscoat, and L. Orgéas, Reversible dilatancy in entangled single-wire materials, Nat. Mater, vol.15, issue.1, p.72, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01956941

G. Rosi and N. Auffray, Anisotropic and dispersive wave propagation within strain-gradient framework, Wave Motion, vol.63, pp.120-134, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01263683

G. Rosi, V. H. Nguyen, A. Loseille, and S. Naili, Ultrasonic characterization of porous gyroid scaffolds for bone tissue engineering: mechanical modelling and numerical validation, J. Acoust. Soc. Am, vol.144, issue.3, p.1854, 2018.

G. Rosi, L. Placidi, and N. Auffray, On the validity range of strain-gradient elasticity: a mixed static-dynamic identification procedure, Eur. J. Mech. A/Solids, vol.69, pp.179-191, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01674513

G. Rosi, I. Scala, V. Nguyen, and S. Naili, Wave propagation in strain gradient poroelastic medium with microinertia: closed-form and finite element solutions. Zeitschrift für angewandte Mathematik und, Physik, vol.63, issue.3, p.58, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01595385

L. Rothenburg, A. A. Berlin, and R. J. Bathurst, Microstructure of isotropic materials with negative Poisson's ratio, Nature, vol.354, pp.470-472, 1991.

K. Sab, On the homogenization and the simulation of random materials, Eur. J. Mech. A/Solids, vol.11, issue.5, pp.585-607, 1992.

K. Sab and A. Lebée, Homogenization of Heterogeneous Thin and Thick Plates, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01266702

K. Sab and B. Nedjar, Periodization of random media and representative volume element size for linear composites. C.-R. de l'Acad, des Sci. Serie IIb : Mécanique, vol.333, pp.187-195, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00121487

M. Salmi, F. Auslender, M. Bornert, and M. Fogli, Apparent and effective mechanical properties of linear matrix-inclusion random composites: Improved bounds for the effective behavior, Int. J. Solids Struct, vol.49, pp.1195-1211, 2012.
URL : https://hal.archives-ouvertes.fr/hal-01157362

M. Salmi, F. Auslender, M. Bornert, and 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, vol.340, pp.230-246, 2012.
URL : https://hal.archives-ouvertes.fr/hal-01157361

K. Salonitis, D. Chantzis, and 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, vol.90, pp.2689-2701, 2017.

E. Sanchez-palencia and A. Zaoui, Homogenization Techniques for Composite Media, vol.272, 1987.

V. Sansalone, P. Trovalusci, and F. Cleri, Multiscale modeling of composite materials by a multifield finite element approach, Int. J. Multiscale Comput. Eng, vol.3, pp.463-480, 2005.

B. Sarac, J. Wilmers, and S. Bargmann, Property optimization of porous metallic glasses via structural design, Mater. Lett, vol.134, pp.306-310, 2014.

F. Scarpa, J. R. Yates, L. G. Ciffo, and S. Patsias, Dynamic crushing of auxetic open-cell polyurethane foam, Proc. Inst. Mech. Eng. Part C-J. Mech. Eng. Sci, vol.216, issue.12, pp.1153-1156, 2002.

S. Schaare, W. Riehemann, and Y. Estrin, Damping properties of an assembly of topologically interlocked cubes, Mater. Sci. Eng. A Struct. Mater. Prop. Microstruct. Process, pp.380-383, 2009.

T. A. Schaedler and W. B. Carter, Architected cellular materials, Ann. Rev. Mater. Res, vol.46, pp.187-210, 2016.

T. A. Schaedler, A. J. Jacobsen, A. Torrents, A. E. Sorensen, J. Lian et al., Ultralight metallic microlattices, Science, vol.334, issue.6058, pp.962-965, 2011.

K. Schladitz, S. Peters, D. Reinel-bitzer, A. Wiegmann, and J. Ohser, Design of acoustic trim based on geometric modeling and flow simulation for non-woven, Comput. Mater. Sci, vol.38, pp.56-66, 2006.

I. Shufrin, E. Pasternak, and A. Dyskin, Planar isotropic structures with negative poissons ratio, Int. J. Solids Struct, vol.49, issue.17, pp.2239-2253, 2012.

I. Shufrin, E. Pasternak, and A. Dyskin, Negative poissons ratio in hollow sphere materials, Int. J. Solids Struct, vol.54, pp.192-214, 2015.

P. Smith and S. Torquato, Computer simulation results for the two-point probability function of composite media, J. Comput. Phys, vol.76, issue.1, pp.176-191, 1988.

M. A. Soare and R. C. Picu, An approach to solving mechanics problems for materials with multiscale self-similar microstructure, Int. J. Solids Struct, vol.44, pp.7877-7890, 2007.

C. Soyarslan, S. Bargmann, M. Pradas, and J. Weissmuller, 3D stochastic bicontinuous microstructures: generation, topology and elasticity, Acta Mater, vol.149, pp.326-340, 2018.
DOI : 10.1016/j.actamat.2018.01.005
URL : https://doi.org/10.1016/j.actamat.2018.01.005

A. Spadoni, M. Ruzzene, S. Gonella, and F. Scarpa, Phononic properties of hexagonal chiral lattices, Wave Motion, vol.46, issue.7, pp.435-450, 2009.

J. Teixeira-pinto, C. Nadot-martin, F. Touchard, M. Gueguen, and S. Castagnet, Towards the size estimation of a representative elementary domain in semi-crystalline polymers, Mechanics of Materials in press, 2016.

K. Terada, M. Hori, T. Kyoya, and N. Kikuchi, Simulation of the multi-scale convergence in computational homogenization approaches, Int. J. Solids Struct, vol.37, pp.2285-2311, 2000.

S. Torquato, Morphology and effective properties of disordered heterogeneous media, Int. J. Solids Struct, vol.35, pp.2385-2406, 1998.

S. Torquato, Random Heterogeneous Materials, 2001.

D. K. Trinh, R. Jänicke, N. Auffray, S. Diebels, and S. Forest, Evaluation of generalized continuum substitution models for heterogeneous materials, Int. J. Multiscale Comput. Eng, vol.10, issue.6, pp.527-549, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00718875

N. Vermaak, G. Michailidis, G. Parry, R. Estevez, G. Allaire et al., Material interface effects on the topology optimizationof multi-phase structures using a level set method, Struct. Mult. Optim, vol.50, issue.4, pp.623-644, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01120063

W. M. Vicente, Z. H. Zuo, R. Pavanello, T. K. Calixto, R. Picelli et al., Concurrent topology optimization for minimizing frequency responses of two-level hierarchical structures, Comput. Methods Appl. Mech. Eng, vol.301, pp.116-136, 2016.

W. Voigt, Ueber die beziehung zwischen den beiden elasticitätsconstanten isotroper körper, Ann. der Phys. und Chem, vol.38, pp.573-587, 1889.
DOI : 10.1002/andp.18892741206
URL : https://zenodo.org/record/1423864/files/article.pdf

Y. Wang, H. Xu, and D. Pasini, Multiscale isogeometric topology optimization for lattice materials, Comput. Methods Appl. Mech. Eng, vol.316, pp.568-585, 2017.
DOI : 10.1016/j.cma.2016.08.015
URL : http://digitool.Library.McGill.CA:80/webclient/DeliveryManager?pid=156444&custom_att_2=direct

Z. P. Wang, L. H. Poh, J. Dirrenberger, Y. Zhu, and S. Forest, Isogeometric shape optimization of smoothed petal auxetic structures via computational periodic homogenization, Comput. Methods Appl. Mech. Eng, vol.323, pp.250-271, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01540814

P. M. Weaver and M. F. Ashby, The optimal selection of material and section-shape, J. Eng. Des, vol.7, issue.2, pp.129-150, 1996.

B. Xu, X. Huang, S. W. Zhou, and Y. M. Xie, Concurrent topological design of composite thermoelastic macrostructure and microstructure with multi-phase material for maximum stiffness, Compos. Struct, vol.150, pp.84-102, 2016.

S. Xu, J. Shen, S. Zhou, X. Huang, and Y. M. Xie, Design of lattice structures with controlled anisotropy, Mater. Des, vol.93, pp.443-447, 2016.

W. Yang, Review on auxetic materials, J. Mater. Sci, vol.39, pp.3269-3279, 2004.

C. L. Yeong and S. Torquato, Reconstructing random media, Phys. Rev. E, vol.57, issue.1, p.495, 1998.