V. Arsigny, P. Fillard, X. Pennec, and N. Ayache, Log-Euclidean metrics for fast and simple calculus on diffusion tensors, Magnetic Resonance in Medicine, vol.52, issue.2, pp.411-421, 2006.
DOI : 10.1002/mrm.20965

URL : https://hal.archives-ouvertes.fr/inria-00502678

I. Babu?ka, F. Nobile, and R. Tempone, A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data, SIAM Review, vol.52, issue.2, pp.317-355, 2010.
DOI : 10.1137/100786356

A. Bóna, I. Bucataru, and M. A. Slawinski, Material symmetries of elasticity tensors, The Quarterly Journal of Mechanics and Applied Mathematics, vol.57, issue.4, pp.584-598, 2004.
DOI : 10.1093/qjmam/57.4.583

A. Bóna, I. Bucataru, and M. A. Slawinski, Coordinate-free Characterization of the Symmetry Classes of Elasticity Tensors, Journal of Elasticity, vol.25, issue.4,5, pp.109-132, 2007.
DOI : 10.1007/s10659-007-9099-z

J. T. Browaeys and S. Chevrot, Decomposition of the elastic tensor and geophysical applications, Geophysical Journal International, vol.159, issue.2, pp.667-678, 2004.
DOI : 10.1111/j.1365-246X.2004.02415.x

I. Bucataru and M. A. Slawinski, Invariant Properties for Finding Distance in Space of??Elasticity Tensors, Journal of Elasticity, vol.391, issue.2, pp.97-114, 2009.
DOI : 10.1007/s10659-008-9186-9

P. Chadwick, M. Vianello, and S. C. Cowin, A new proof that the number of linear elastic symmetries is eight, Journal of the Mechanics and Physics of Solids, vol.49, issue.11, pp.2471-2492, 2001.
DOI : 10.1016/S0022-5096(01)00064-3

D. H. Chung and W. R. Buessem, The Elastic Anisotropy of Crystals, Journal of Applied Physics, vol.38, issue.5, pp.2010-2012, 1967.
DOI : 10.1063/1.1709819

T. M. Cover and J. A. Thomas, Elements of Information Theory, 2006.

S. C. Cowin and M. M. Mehrabadi, ON THE IDENTIFICATION OF MATERIAL SYMMETRY FOR ANISOTROPIC ELASTIC MATERIALS, The Quarterly Journal of Mechanics and Applied Mathematics, vol.40, issue.4, pp.451-476, 1987.
DOI : 10.1093/qjmam/40.4.451

S. Das and R. Ghanem, A Bounded Random Matrix Approach for Stochastic Upscaling, Multiscale Modeling & Simulation, vol.8, issue.1, pp.296-325, 2009.
DOI : 10.1137/090747713

F. I. Fedorov, Theory of Elastic Waves in Crystals, 1968.
DOI : 10.1007/978-1-4757-1275-9

S. Forte and M. Vianello, Symmetry classes for elasticity tensors, Journal of Elasticity, vol.31, issue.2, pp.81-108, 1996.
DOI : 10.1007/BF00042505

D. C. Gazis, I. Tadjbakhsh, and R. A. Toupin, The elastic tensor of given symmetry nearest to an anisotropic elastic tensor, Acta Crystallographica, vol.16, issue.9, pp.917-922, 1963.
DOI : 10.1107/S0365110X63002449

URL : https://hal.archives-ouvertes.fr/hal-00852597

R. G. Ghanem and P. D. Spanos, Stochastic Finite Elements: A Spectral Approach (rev, 2003.
DOI : 10.1007/978-1-4612-3094-6

J. Guilleminot, A. Noshadravan, C. Soize, and R. G. Ghanem, A probabilistic model for bounded elasticity tensor random fields with application to polycrystalline microstructures, Computer Methods in Applied Mechanics and Engineering, vol.200, issue.17-20, 2011.
DOI : 10.1016/j.cma.2011.01.016

URL : https://hal.archives-ouvertes.fr/hal-00684305

J. Guilleminot and C. Soize, Non-gaussian positive-definite matrixvalued random fields with constrained eigenvalues: application to random elasticity tensors with uncertain material symmetries, 2010.
DOI : 10.1002/nme.3212

URL : https://hal.archives-ouvertes.fr/hal-00684290

J. Guilleminot and C. Soize, A stochastic model for elasticity tensors with uncertain material symmetries, International Journal of Solids and Structures, vol.47, issue.22-23, pp.3121-3130, 2010.
DOI : 10.1016/j.ijsolstr.2010.07.013

URL : https://hal.archives-ouvertes.fr/hal-00684310

W. K. Hastings, Monte Carlo sampling methods using Markov chains and their applications, Biometrika, vol.57, issue.1, pp.57-97, 1970.
DOI : 10.1093/biomet/57.1.97

S. Hazanov and C. Huet, Order relationships for boundary conditions effect in heterogeneous bodies smaller than the representative volume, Journal of the Mechanics and Physics of Solids, vol.42, issue.12, pp.1995-2011, 1994.
DOI : 10.1016/0022-5096(94)90022-1

C. Huet, Application of variational concepts to size effects in elastic heterogeneous bodies, Journal of the Mechanics and Physics of Solids, vol.38, issue.6, pp.813-841, 1990.
DOI : 10.1016/0022-5096(90)90041-2

Y. Z. Huo and G. D. Piero, On the completeness of the crystallographic symmetries in the description of the symmetries of the elasticity tensor, Journal of Elasticity, vol.25, pp.203-246, 1991.

E. T. Jaynes, Information Theory and Statistical Mechanics, Physical Review, vol.106, issue.4, pp.620-630, 1957.
DOI : 10.1103/PhysRev.106.620

E. T. Jaynes, Information Theory and Statistical Mechanics, Physical Review, vol.106, issue.4, pp.171-190, 1957.
DOI : 10.1103/PhysRev.106.620

T. Kanit, Notion of Representative Volume Element for Heterogeneous Materials: Statistical and Numerical Approach, 2003.
URL : https://hal.archives-ouvertes.fr/tel-00005751

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, Computer Methods in Applied Mechanics and Engineering, vol.195, issue.33-36, pp.3960-3982, 2006.
DOI : 10.1016/j.cma.2005.07.022

URL : https://hal.archives-ouvertes.fr/hal-00139164

H. K. Kapur and J. N. Kesavan, Entropy Optimization Principles and Their Applications, 1992.
DOI : 10.1007/978-94-011-2430-0_1

M. Kochetov and M. A. Slawinski, Estimating effective elasticity tensors from Christoffel equations, GEOPHYSICS, vol.74, issue.5, pp.67-73, 2009.
DOI : 10.1190/1.3155163

M. Kochetov and M. A. Slawinski, On obtaining effective orthotropic elasticity tensors, The Quarterly Journal of Mechanics and Applied Mathematics, vol.62, issue.2, pp.149-166, 2009.
DOI : 10.1093/qjmam/hbp001

M. Kochetov and M. A. Slawinski, On Obtaining Effective Transversely Isotropic Elasticity??Tensors, Journal of Elasticity, vol.40, issue.4, pp.1-13, 2009.
DOI : 10.1007/s10659-008-9180-2

H. Ledbetter and A. Migliori, A general elastic-anisotropy measure, Journal of Applied Physics, vol.100, issue.6, pp.63516-63517, 2006.
DOI : 10.1063/1.2338835

J. Löfberg, Yalmip : A toolbox for modeling and optimization in MAT- LAB, Proceedings of the CACSD Conference, 2004.

M. M. Mehrabadi and S. C. Cowin, EIGENTENSORS OF LINEAR ANISOTROPIC ELASTIC MATERIALS, The Quarterly Journal of Mechanics and Applied Mathematics, vol.43, issue.1, pp.15-41, 1990.
DOI : 10.1093/qjmam/43.1.15

M. P. Mignolet and C. Soize, Nonparametric stochastic modeling of linear systems with prescribed variance of several natural frequencies, Probabilistic Engineering Mechanics, vol.23, issue.2-3, pp.267-278, 2008.
DOI : 10.1016/j.probengmech.2007.12.027

URL : https://hal.archives-ouvertes.fr/hal-00685147

M. Moakher, On the Averaging of Symmetric Positive-Definite Tensors, Journal of Elasticity, vol.38, issue.1, pp.273-296, 2006.
DOI : 10.1007/s10659-005-9035-z

M. Moakher and A. N. Norris, The Closest Elastic Tensor of Arbitrary Symmetry to an Elasticity Tensor of Lower Symmetry, Journal of Elasticity, vol.40, issue.31???32, pp.215-263, 2006.
DOI : 10.1007/s10659-006-9082-0

A. N. Norris, Elastic moduli approximation of higher symmetry for the acoustical properties of an anisotropic material, The Journal of the Acoustical Society of America, vol.119, issue.4, pp.2114-2121, 2006.
DOI : 10.1121/1.2173525

M. Ostoja-starzewski, Microstructural Randomness and Scaling in Mechanics of Materials, 2008.
DOI : 10.1201/9781420010275

S. I. Ranganathan and M. Ostoja-starzewski, Universal Elastic Anisotropy Index, Physical Review Letters, vol.101, issue.5, pp.55504-55505, 2008.
DOI : 10.1103/PhysRevLett.101.055504

J. Rychlewski, On Hooke's law, Journal of Applied Mathematics and Mechanics, vol.48, issue.3, pp.303-314, 1984.
DOI : 10.1016/0021-8928(84)90137-0

C. E. Shannon, A Mathematical Theory of Communication, Bell System Technical Journal, vol.27, issue.3, pp.379-423, 1948.
DOI : 10.1002/j.1538-7305.1948.tb01338.x

C. Soize, A nonparametric model of random uncertainties for reduced matrix models in structural dynamics, Probabilistic Engineering Mechanics, vol.15, issue.3, pp.277-294, 2000.
DOI : 10.1016/S0266-8920(99)00028-4

URL : https://hal.archives-ouvertes.fr/hal-00686293

C. Soize, Maximum entropy approach for modeling random uncertainties in transient elastodynamics, The Journal of the Acoustical Society of America, vol.109, issue.5, pp.1979-1996, 2001.
DOI : 10.1121/1.1360716

URL : https://hal.archives-ouvertes.fr/hal-00686287

C. Soize, Non-Gaussian positive-definite matrix-valued random fields for elliptic stochastic partial differential operators, Computer Methods in Applied Mechanics and Engineering, vol.195, issue.1-3, pp.26-64, 2006.
DOI : 10.1016/j.cma.2004.12.014

URL : https://hal.archives-ouvertes.fr/hal-00686157

C. Soize and R. G. Ghanem, Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary Probability Measure, SIAM Journal on Scientific Computing, vol.26, issue.2, pp.395-410, 2004.
DOI : 10.1137/S1064827503424505

URL : https://hal.archives-ouvertes.fr/hal-00686211

Q. A. Ta, D. Clouteau, and R. Cottereau, Modeling of random anisotropic elastic media and impact on wave propagation, Revue europ??enne de m??canique num??rique, vol.19, issue.1-3, pp.241-253, 2010.
DOI : 10.3166/ejcm.19.241-253

URL : https://hal.archives-ouvertes.fr/hal-00709537

T. C. Ting, Generalized Cowin???Mehrabadi theorems and a direct proof that the number of linear elastic symmetries is eight, International Journal of Solids and Structures, vol.40, issue.25, pp.7129-7142, 2003.
DOI : 10.1016/S0020-7683(03)00358-5

L. Vandenberghe and S. Boyd, Semidefinite Programming, SIAM Review, vol.38, issue.1, pp.49-95, 1996.
DOI : 10.1137/1038003

X. L. Wan and G. E. Karniadakis, Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures, SIAM Journal on Scientific Computing, vol.28, issue.3, pp.901-928, 2006.
DOI : 10.1137/050627630

N. Wiener, The Homogeneous Chaos, American Journal of Mathematics, vol.60, issue.4, pp.897-936, 1938.
DOI : 10.2307/2371268

C. Zener, Elasticity and Anelasticity of Metals., The Journal of Physical and Colloid Chemistry, vol.53, issue.9, 1948.
DOI : 10.1021/j150474a017