L. Ambrosio and V. M. Tortorelli, On the approximation of Free Discontinuity Problems, Bollettino dell'Unione Matematica Italiana, pp.105-123, 1992.

Z. P. Bazant, T. B. Belytschko, and T. P. Chang, Continuum Theory for Strain???Softening, Journal of Engineering Mechanics, vol.110, issue.12, pp.1666-1692, 1984.
DOI : 10.1061/(ASCE)0733-9399(1984)110:12(1666)

Z. P. Bazant and G. Pijaudier-cabot, Nonlocal Continuum Damage, Localization Instability and Convergence, Journal of Applied Mechanics, vol.55, issue.2, pp.287-293, 1988.
DOI : 10.1115/1.3173674

B. Bourdin, Numerical implementation of the variational formulation for quasi-static brittle fracture. Interfaces and Free Boundaries, pp.411-430, 2007.

B. Bourdin, G. Francfort, and J. Marigo, Numerical experiments in revisited brittle fracture, Journal of the Mechanics and Physics of Solids, vol.48, issue.4, pp.787-826, 2000.
DOI : 10.1016/S0022-5096(99)00028-9

A. Braides, Approximation of free-discontinuity problems, 1998.
DOI : 10.1007/bfb0097344

C. Comi, A non-local model with tension and compression damage mechanisms, European Journal of Mechanics - A/Solids, vol.20, issue.1, pp.1-22, 2001.
DOI : 10.1016/S0997-7538(00)01111-6

S. Forest, Nonlinear regularization operators as derived from the micromorphic approach to gradient elasticity, viscoplasticity and damage, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, vol.57, issue.31, pp.472-2016, 2188.
DOI : 10.1177/1056789510386852

URL : https://hal.archives-ouvertes.fr/hal-01337863

G. A. Francfort, B. Bourdin, and J. Marigo, The variational approach to fracture, Journal of Elasticity, vol.91, issue.1-3, pp.5-148, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00551079

G. A. Francfort and J. Marigo, Revisiting brittle fracture as an energy minimization problem, Journal of the Mechanics and Physics of Solids, vol.46, issue.8, pp.1319-1342, 1998.
DOI : 10.1016/S0022-5096(98)00034-9

M. Frémond and B. Nedjar, Damage, gradient of damage and principle of virtual power. International of Solids and Structures, pp.1083-1103, 1996.

V. Hakim and A. Karma, Laws of crack motion and phase-field models of fracture, Journal of the Mechanics and Physics of Solids, vol.57, issue.2, pp.342-368, 2009.
DOI : 10.1016/j.jmps.2008.10.012

URL : http://arxiv.org/pdf/0806.0593

B. Halphen and Q. S. Nguyen, Sur les matériaux standards généralisés, Journal de Mécanique, vol.14, pp.39-63, 1975.

A. Karma, H. Da-kessler, and . Levine, Phase-Field Model of Mode III Dynamic Fracture, Physical Review Letters, vol.71, issue.4, 2001.
DOI : 10.1103/PhysRevLett.71.2417

E. Lorentz and S. Andrieux, Analysis of non-local models through energetic formulations, International Journal of Solids and Structures, vol.40, issue.12, pp.2905-2936, 2003.
DOI : 10.1016/S0020-7683(03)00110-0

J. Marigo, Formulation d'une loi d'endommagement d'un matériaú elastique, C. R. Acad. Sci. Paris Sér. II, pp.2921309-1312, 1981.

J. J. Marigo, Constitutive relations in plasticity, damage and fracture mechanics based on a work property, Nuclear Engineering and Design, vol.114, issue.3, pp.249-272, 1989.
DOI : 10.1016/0029-5493(89)90105-2

D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Communications on Pure and Applied Mathematics, vol.3, issue.5, pp.577-685, 1989.
DOI : 10.1109/TPAMI.1984.4767596

URL : https://dash.harvard.edu/bitstream/handle/1/3637121/Mumford_OptimalApproxPiece.pdf?sequence=1

R. H. Peerlings, R. De-borst, W. A. Brekelmans, and J. H. De-vree, GRADIENT ENHANCED DAMAGE FOR QUASI-BRITTLE MATERIALS, International Journal for Numerical Methods in Engineering, vol.23, issue.19, pp.393391-3403, 1996.
DOI : 10.1016/0020-7683(87)90083-7

R. H. Peerlings, R. De-borst, W. A. Brekelmans, and M. G. Geers, Gradient-enhanced damage modelling of concrete fracture, Mechanics of Cohesive-frictional Materials, vol.48, issue.4, pp.323-342, 1998.
DOI : 10.1115/1.3157740

R. H. Peerlings, R. De-borst, W. A. Brekelmans, and M. G. Geers, Localisation issues in local and nonlocal continuum approaches to fracture, European Journal of Mechanics - A/Solids, vol.21, issue.2, pp.175-189, 2002.
DOI : 10.1016/S0997-7538(02)01211-1

URL : http://www.mate.tue.nl/mate/pdfs/1735.pdf

R. H. Peerlings, M. G. Geers, R. De-borst, and W. A. Brekelmans, A critical comparison of nonlocal and gradient-enhanced softening continua, International Journal of Solids and Structures, vol.38, issue.44-45, pp.44-457723, 2001.
DOI : 10.1016/S0020-7683(01)00087-7

R. H. Peerlings, T. J. Massart, and M. G. Geers, A thermodynamically motivated implicit gradient damage framework and its application to brick masonry cracking, Computer Methods in Applied Mechanics and Engineering, vol.193, issue.30-32, pp.3403-3417, 2004.
DOI : 10.1016/j.cma.2003.10.021

K. Pham, H. Amor, J. Marigo, and C. Maurini, Gradient Damage Models and Their Use to Approximate Brittle Fracture, International Journal of Damage Mechanics, vol.30, issue.4, pp.618-652, 2011.
DOI : 10.1016/0029-5493(92)90094-C

URL : https://hal.archives-ouvertes.fr/hal-00549530

K. Pham and J. Marigo, Approche variationnelle de l'endommagement : I. Les concepts fondamentaux, Comptes Rendus M??canique, vol.338, issue.4, pp.191-198, 2010.
DOI : 10.1016/j.crme.2010.03.009

URL : https://hal.archives-ouvertes.fr/hal-00490518

K. Pham and J. Marigo, Approche variationnelle de l'endommagement : II. Les mod??les ?? gradient, Comptes Rendus M??canique, vol.338, issue.4, pp.199-206, 2010.
DOI : 10.1016/j.crme.2010.03.012

URL : https://hal.archives-ouvertes.fr/hal-00490520

K. Pham and J. Marigo, 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.2-4147, 2013.
DOI : 10.1016/0020-7683(93)90232-V

URL : https://hal.archives-ouvertes.fr/hal-00647860

K. Pham, J. Marigo, and C. Maurini, 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

L. Placidi, A variational approach for a nonlinear 1-dimensional second gradient continuum damage model, Continuum Mechanics and Thermodynamics, vol.1, issue.2, pp.623-638, 2015.
DOI : 10.1061/(ASCE)NM.2153-5477.0000030

E. Tanné, T. Li, B. Bourdin, J. Marigo, and C. Maurini, Crack nucleation in variational phase-field models of brittle fracture, Journal of the Mechanics and Physics of Solids, vol.110, 2017.
DOI : 10.1016/j.jmps.2017.09.006