, ABAQUS Version 6.11, 2011. Documentation. Dassault Systémes Simulia Corp

A. Alderson, J. Rasburn, S. Ameer-beg, P. G. Mullarkey, W. Perrie et al., An auxetic filter: a tuneable filter displaying enhanced size selectivity or defouling properties, Industrial Engineering Chemistry Research, vol.39, pp.654-665, 2000.

G. Allaire, Shape optimization by the homogenization method, 2012.

K. Anoukou, R. Brenner, F. Hong, M. Pellerin, and K. Danas, Random distribution of polydisperse ellipsoidal inclusions and homogenization estimates for porous elastic material, 2018.

J. Banhart, J. Baumeister, and M. Weber, Damping properties of aluminium foams, Materials Science and Engineering: A, vol.205, pp.221-228, 1996.

M. W. Barclift and C. Williams, Examining variability in the mechanical properties of parts manufactured via polyjet direct 3D printing. 23rd Annual International Solid Freeform Fabrication Symposium-An Additive Manufacturing Conference, 2012.

M. P. Bendsoe, A. Ben-tal, and J. Zowe, Optimization methods for truss geometry and topology design, Structural optimization, vol.7, pp.141-159, 1994.

J. B. Berger, H. N. Wadley, and R. M. Mcmeeking, Mechanical metamaterials at the theoretical limit of isotropic elastic stiffness, Nature, vol.543, p.533, 2017.

H. Böhm, A. Eckschlager, and W. Han, Multi-inclusion unit cell models for metal matrix composites with randomly oriented discontinuous reinforcements, Computational Materials Science, vol.25, pp.42-53, 2002.

H. J. Böhm and W. Han, Comparisons between three-dimensional and two-dimensional multi-particle unit cell models for particle reinforced metal matrix composites. Modelling and Simulation in Materials Science and Engineering 9, p.47, 2001.

I. Bucataru and M. A. Slawinski, Invariant properties for finding distance in space of elasticity tensors, Journal of Elasticity, vol.94, pp.97-114, 2008.

O. Cansizoglu, O. Harrysson, D. Cormier, H. West, and T. Mahale, Properties of ti-6al-4v non-stochastic lattice structures fabricated via electron beam melting, Materials Science and Engineering: A, vol.492, pp.468-474, 2008.

K. Danas, Effective response of classical, auxetic and chiral magnetoelastic materials by use of a new variational principle, Journal of the Mechanics and Physics of Solids, vol.105, pp.25-53, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01627522

G. J. Davies and S. Zhen, Metallic foams: their production, properties and applications, Journal of Materials Science, vol.18, pp.1899-1911, 1983.
DOI : 10.1007/bf00554981

V. Deshpande and N. Fleck, Isotropic constitutive models for metallic foams, Journal of the Mechanics and Physics of Solids, vol.48, pp.1253-1283, 2000.
DOI : 10.1016/s0022-5096(99)00082-4
URL : http://www-mech.eng.cam.ac.uk/profiles/vsd/papers/jmps_foam_const.pdf

J. Eshelby, The determination of elastic field of an ellipsoidal inclusion and related problems, Proceedings of the Royal Society of London, pp.379-396, 1957.

N. Fleck, V. Deshpande, and M. Ashby, Micro-architectured materials: past, present and future, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol.466, pp.2495-2516, 2010.
DOI : 10.1098/rspa.2010.0215
URL : http://rspa.royalsocietypublishing.org/content/466/2121/2495.full.pdf

G. A. Francfort and F. Murat, Homogenization and optimal bounds in linear elasticity, Archive for Rational Mechanics and Analysis, vol.94, pp.307-334, 1986.

E. Ghossein and M. Lévesque, A comprehensive validation of analytical homogenization models: The case of ellipsoidal particles reinforced composites, Mechanics of Materials, vol.75, pp.135-150, 2014.

L. J. Gibson and M. F. Ashby, Cellular Solids: Structure and Properties. Cambridge Solid State Science Series, 1997.

P. Göransson, Acoustic and vibrational damping in porous solids, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol.364, pp.89-108, 2006.

B. Gorny, T. Niendorf, J. Lackmann, M. Thoene, T. Troester et al., In situ characterization of the deformation and failure behavior of non-stochastic porous structures processed by selective laser melting, Materials Science and Engineering: A, vol.528, pp.7962-7967, 2011.

Z. Hashin, The elastic moduli of heterogeneous materials, Journal of Applied Mechanics, pp.481-505, 1962.

Z. Hashin and S. Shtrikman, A variational approach to the theory of the elastic behaviour of multiphase materials, Journal of the Mechanics and Physics of Solids, pp.127-140, 1963.

P. Heinl, A. Rottmair, C. Körner, F. Singer, and R. , Cellular titanium by selective electron beam melting, Advanced Engineering Materials, vol.9, pp.360-364, 2007.
DOI : 10.1002/adem.200700025

S. Hengsbach and A. D. Lantada, Direct laser writing of auxetic structures: present capabilities and challenges, Smart Materials and Structures, vol.23, p.85033, 2014.
DOI : 10.1088/0964-1726/23/8/085033

R. Hill, The elastic behaviour of a crystalline aggregate, Proceedings of the Physical Society. Section A, vol.65, p.349, 1952.

R. Hill, Elastic properties of reinforced solids: Some theoretical principles, Journal of the Mechanics and Physics of Solids, vol.11, pp.357-372, 1963.
DOI : 10.1016/0022-5096(63)90036-x

M. Hossain, D. Vu, and P. Steinmann, Experimental study and numerical modelling of vhb 4910 polymer, Computational Materials Science, vol.59, pp.65-74, 2012.
DOI : 10.1016/j.commatsci.2012.02.027

C. Huet, Application of variational concepts to size effects in elastic heterogeneous bodies, Journal of the Mechanics and Physics of Solids, vol.38, pp.813-841, 1990.

M. I. Idiart, Modeling the macroscopic behavior of two-phase nonlinear composites by infinite-rank laminates, Journal of the Mechanics and Physics of Solids, vol.56, pp.2599-2617, 2008.
DOI : 10.1016/j.jmps.2008.03.004

D. Jeulin, Random structure models for homogenization and fracture statistics, Mechanics of Random and Multiscale Microstructures, 2001.
DOI : 10.1007/978-3-7091-2780-3_2

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, International Journal of Solids and Structures, vol.40, pp.3647-3679, 2003.

O. Lopez-pamies, T. Goudarzi, and K. Danas, The nonlinear elastic response of suspensions of rigid inclusions in rubber: Ii-a simple explicit approximation for finite-concentration suspensions, Journal of the Mechanics and Physics of Solids, vol.61, pp.19-37, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00870885

Y. Ma, F. Scarpa, D. Zhang, B. Zhu, L. Chen et al., A nonlinear auxetic structural vibration damper with metal rubber particles, Smart Materials and Structures, vol.22, p.84012, 2013.
DOI : 10.1088/0964-1726/22/8/084012

A. Mbiakop, A. Constantinescu, and K. Danas, An analytical model for porous single crystals with ellipsoidal voids, Journal of the Mechanics and Physics of Solids, vol.84, pp.436-467, 2015.
DOI : 10.1016/j.jmps.2015.07.011
URL : https://hal.archives-ouvertes.fr/hal-01219387

M. Messner, Optimal lattice-structured materials, Journal of Mechanics Physics of Solids, vol.96, pp.162-183, 2016.
DOI : 10.1016/j.jmps.2016.07.010

L. Meza, S. Das, and J. Greer, Strong, lightweight, and recoverable three-dimensional ceramic nanolattices, Science, vol.345, pp.1322-1326, 2014.
DOI : 10.1126/science.1255908
URL : https://authors.library.caltech.edu/49512/7/Meza.SM.pdf

L. R. Meza, G. P. Phlipot, C. M. Portela, A. Maggi, L. C. Montemayor et al., Reexamining the mechanical property space of three-dimensional lattice architectures, Acta Materialia, vol.140, pp.424-432, 2017.
DOI : 10.1016/j.actamat.2017.08.052
URL : https://authors.library.caltech.edu/80858/1/1-s2.0-S1359645417307073-main.pdf

M. Moakher and A. N. Norris, The closest elastic tensor of arbitrary symmetry to an elasticity tensor of lower symmetry, Journal of Elasticity, vol.85, pp.215-263, 2006.

H. Moussaddy, D. Therriault, and M. Lévesque, Assessment of existing and introduction of a new and robust efficient definition of the representative volume element, International Journal of Solids and Structures, vol.50, pp.3817-3828, 2013.

D. T. Queheillalt, Y. Murty, and H. N. Wadley, Mechanical properties of an extruded pyramidal lattice truss sandwich structure, Scripta Materialia, vol.58, pp.76-79, 2008.
DOI : 10.1016/j.scriptamat.2007.08.041

M. Rintoul and S. Torquato, Reconstruction of the structure of dispersions, Journal of Colloid and Interface Science, vol.186, pp.467-476, 1997.

J. Schöberl, Netgen an advancing front 2d/3d-mesh generator based on abstract rules, Computing and Visualization in Science, vol.1, pp.41-52, 1997.

J. Segurado and J. Llorca, A numerical approximation to the elastic properties of sphere-reinforced composites, Journal of the Mechanics and Physics of Solids, vol.50, 2002.

O. Sigmund, New class of extremal composites, Journal of the Mechanics and Physics of Solids, vol.48, pp.397-428, 2000.
DOI : 10.1016/s0022-5096(99)00034-4

P. S. Spoor, J. D. Maynard, and A. R. Kortan, Elastic isotropy and anisotropy in quasicrystalline and cubic alculi, Phys. Rev. Lett, vol.75, pp.3462-3465, 1995.
DOI : 10.1103/physrevlett.75.3462

P. Suquet, Homogenization Techniques for Composite Media: Lectures Delivered at the CISM International Center for Mechanical Sciences, pp.193-230, 1985.

D. J. Sypeck, Cellular truss core sandwich structures, Applied Composite Materials, vol.12, pp.229-246, 2005.
DOI : 10.1007/s10443-005-1129-z

T. Tancogne-dejean and D. Mohr, Elastically-isotropic truss lattice materials of reduced plastic anisotropy, International Journal of Solids and Structures, 2017.
DOI : 10.1016/j.ijsolstr.2017.12.025

S. Torquato, Random Heterogeneous Materials: Microstructure and Macroscopic Properties, 2002.
DOI : 10.1115/1.1483342

D. W. Wang, F. Li, M. Liu, G. Lu, and H. M. Cheng, 3d aperiodic hierarchical porous graphitic carbon material for high rate electrochemical capacitive energy storage, Angewandte Chemie International Edition, vol.47, pp.373-376, 2008.
DOI : 10.1002/anie.200702721

J. Willis, Bounds and self-consistent estimates for the overall properties of anisotropic composites, Journal of the Mechanics and Physics of Solids, vol.25, pp.185-202, 1977.
DOI : 10.1016/0022-5096(77)90022-9

M. Wissler and E. Mazza, Mechanical behavior of an acrylic elastomer used in dielectric elastomer actuators, Sensors and Actuators A: Physical, vol.134, pp.494-504, 2007.

C. M. Zener and S. Siegel, Elasticity and anelasticity of metals, The Journal of Physical and Colloid Chemistry, vol.53, pp.1468-1468, 1949.

F. Zok, S. Waltner, Z. Wei, H. Rathbun, R. Mcmeeking et al., A protocol for characterizing the structural performance of metallic sandwich panels: application to pyramidal truss cores, International Journal of Solids and Structures, vol.41, pp.6249-6271, 2004.