A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites, Journal of the Mechanics and Physics of Solids, vol.44, issue.4, pp.497-524, 1996. ,
DOI : 10.1016/0022-5096(96)00007-5
Nonlocal Damage Theory, Journal of Engineering Mechanics, vol.113, issue.10, pp.1512-1533, 1987. ,
DOI : 10.1061/(ASCE)0733-9399(1987)113:10(1512)
Some observations on localisation in non-local and gradient damage models, Eur. J. Mech. A/Solids, vol.15, issue.6, pp.937-953, 1996. ,
Simulation of strain localization with gradient enhanced damage models, Computational Materials Science, vol.16, issue.1-4, pp.176-185, 2000. ,
DOI : 10.1016/S0927-0256(99)00060-9
Damage, gradient of damage and principle of virtual power, International Journal of Solids and Structures, vol.33, issue.8, pp.1083-1103, 1996. ,
DOI : 10.1016/0020-7683(95)00074-7
A variational formulation for nonlocal damage models, International Journal of Plasticity, vol.15, issue.2, pp.119-138, 1999. ,
DOI : 10.1016/S0749-6419(98)00057-6
From the onset of damage to rupture: construction of responses with damage localization for a general class of gradient damage models, Continuum Mechanics and Thermodynamics, vol.30, issue.6, pp.25-147, 2013. ,
DOI : 10.1007/s00161-011-0228-3
URL : https://hal.archives-ouvertes.fr/hal-00647860
Simulation of localization failure with strain-gradient-enhanced damage mechanics, International Journal for Numerical and Analytical Methods in Geomechanics, vol.38, issue.8, pp.26-793, 2002. ,
DOI : 10.1002/nag.225
Fracture of plain concrete, Indian Concrete Journal, vol.46, issue.11 ,
Nonlocal Integral Formulations of Plasticity and Damage: Survey of Progress, Journal of Engineering Mechanics, vol.128, issue.11, pp.1119-1149, 2002. ,
DOI : 10.1061/(ASCE)0733-9399(2002)128:11(1119)
Stress-based nonlocal damage model, International Journal of Solids and Structures, vol.48, issue.25-26, pp.25-26, 2011. ,
DOI : 10.1016/j.ijsolstr.2011.08.012
Nonlocal Damage Theory Based on Micromechanics of Crack Interactions, Journal of Engineering Mechanics, vol.120, issue.3, pp.593-617, 1994. ,
DOI : 10.1061/(ASCE)0733-9399(1994)120:3(593)
A class of constitutive relations with internal variable derivatives: derivation from homogenization, C.R. Acad. Sci, vol.323, pp.629-636, 1996. ,
URL : https://hal.archives-ouvertes.fr/jpa-00254479
A micromechanics-based strain gradient damage model for fracture prediction of brittle materials ??? Part I: Homogenization methodology and constitutive relations, International Journal of Solids and Structures, vol.48, issue.24, pp.3336-3345, 2011. ,
DOI : 10.1016/j.ijsolstr.2011.08.007
On the inversion of non symmetric sixth-order isotropic tensors and conditions of positiveness of third-order tensor valued quadratic functions, Mechanics Research Communications, vol.38, issue.4, pp.326-329, 2011. ,
DOI : 10.1016/j.mechrescom.2011.03.006
URL : https://hal.archives-ouvertes.fr/hal-00687815
Micromechanical Analysis of Anisotropic Damage in Brittle Materials, Journal of Engineering Mechanics, vol.128, issue.8, pp.889-897, 2002. ,
DOI : 10.1061/(ASCE)0733-9399(2002)128:8(889)
URL : https://hal.archives-ouvertes.fr/hal-00140762
Modelling of brittle and fatigue damage for elastic material by growth of microvoids, Engineering Fracture Mechanics, vol.21, issue.4, pp.861-874, 1985. ,
DOI : 10.1016/0013-7944(85)90093-1
Elasticity theory of composites Mechanics of Solids: The R. Hill 60th Anniversary, pp.653-686 ,
ON THE TWO-POINT CORRELATION FUNCTION FOR DISPERSIONS OF NONOVERLAPPING SPHERES, Mathematical Models and Methods in Applied Sciences, vol.08, issue.02, pp.359-377, 1998. ,
DOI : 10.1142/S0218202598000159
A micromechanics-based nonlocal constitutive equation for elastic composites containing randomly oriented spheroidal heterogeneities, Journal of the Mechanics and Physics of Solids, vol.52, issue.2, pp.359-393, 2004. ,
DOI : 10.1016/S0022-5096(03)00103-0
Stochastic Problems in Physics and Astronomy, Reviews of Modern Physics, vol.15, issue.1, pp.1-89, 1943. ,
DOI : 10.1103/RevModPhys.15.1
Second gradient of strain and surface-tension in linear elasticity, International Journal of Solids and Structures, vol.1, issue.4, pp.417-438, 1965. ,
DOI : 10.1016/0020-7683(65)90006-5
Strain Gradient Plasticity, Advances in Applied Mechanics, vol.33, pp.295-361, 1997. ,
DOI : 10.1016/S0065-2156(08)70388-0
Vector and tensor analysis, p.222, 1947. ,
Generalized hooke's law for isotropic second gradient materials, Proc. R. Soc. A, pp.2177-2196, 2009. ,
Experiments and theory in strain gradient elasticity, Journal of the Mechanics and Physics of Solids, vol.51, issue.8, pp.1477-1508, 2003. ,
DOI : 10.1016/S0022-5096(03)00053-X
Strain gradient plasticity: Theory and experiment, Acta Metallurgica et Materialia, vol.42, issue.2, pp.475-487, 1994. ,
DOI : 10.1016/0956-7151(94)90502-9
Size effects and micromechanics of a porous solid, Journal of Materials Science, vol.2, issue.9, pp.2572-2580, 1983. ,
DOI : 10.1007/BF00547573
Transient Study of Couple Stress Effects in Compact Bone: Torsion, Journal of Biomechanical Engineering, vol.103, issue.4, pp.275-279, 1981. ,
DOI : 10.1115/1.3138292
On the structure of the mode III crack-tip in gradient elasticity, Scripta Metallurgica et Materialia, vol.26, issue.2, pp.319-324, 1992. ,
DOI : 10.1016/0956-716X(92)90194-J
Numerical analysis of plane cracks in strain-gradient elastic materials, International Journal of Fracture, vol.92, issue.1, pp.403-430, 2006. ,
DOI : 10.1007/s10704-006-9004-y
A micromechanical-based damage analysis of a cylindrical bar under torsion: theoretical results and verification by finite elements computations, Theoretical and Applied Fracture Mechanics, pp.74-116, 2014. ,
Micro-structure in linear elasticity, Archive for Rational Mechanics and Analysis, vol.16, issue.1, pp.51-78, 1964. ,
DOI : 10.1007/BF00248490
Two exact micromechanics-based nonlocal constitutive equations for random linear elastic composite materials, Journal of the Mechanics and Physics of Solids, vol.51, issue.9, pp.1745-1772, 2003. ,
DOI : 10.1016/S0022-5096(03)00049-8
The issues of the uniqueness and the stability of the homogeneous response in uniaxial tests with gradient damage models, Journal of the Mechanics and Physics of Solids, vol.59, issue.6, pp.1163-1190, 2011. ,
DOI : 10.1016/j.jmps.2011.03.010
URL : https://hal.archives-ouvertes.fr/hal-00578995
Basis for isotropic sixth-order tensors The 6 th order tensors (6) K n for n = 1, ..., 6 and (6) J m for m = 1 ,