R. Alessi, J. Marigo, and S. Vidoli, Nucleation of cohesive cracks in gradient damage models coupled with plasticity. Archives for Rational Mechanics and Analysis, pp.575-615, 2014.

R. Alessi, J. Marigo, and S. Vidoli, Gradient damage models coupled with plasticity: Variational formulation and main properties, Mechanics of Materials, vol.80, pp.351-367, 2015.
DOI : 10.1016/j.mechmat.2013.12.005

L. Ambrosio, N. Fusco, and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs, 2000.

H. Amor, J. Marigo, and C. Maurini, Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments, Journal of the Mechanics and Physics of Solids, vol.57, issue.8, pp.1209-1229, 2009.
DOI : 10.1016/j.jmps.2009.04.011

M. Artina, M. Fornasier, S. Micheletti, and S. Perotto, Anisotropic Mesh Adaptation for Crack Detection In Brittle Materials, SIAM Journal on Scientific Computing, vol.37, issue.4, pp.633-659, 2015.
DOI : 10.1137/140970495

H. A. Bahr, G. Fischer, and H. J. Weiss, Thermal-shock crack patterns explained by single and multiple crack propagation, Journal of Materials Science, vol.52, issue.8, pp.2716-2720, 1986.
DOI : 10.1007/BF00551478

S. Balay, S. Abhyankar, M. F. Adams, J. Brown, P. Brune et al., PETSc users manual, 2014.

A. Benallal, R. Billardon, and G. Geymonat, Bifurcation and localization in rate independent materials, C.I.S.M Lecture Notes on Bifurcation and Stability of Dissipative Systems International Centre for Mechanical Sciences, vol.327, pp.1-44, 1993.

A. Benallal and J. Marigo, Bifurcation and stability issues in gradient theories with softening, Modelling and Simulation in Materials Science and Engineering, vol.15, issue.1, pp.283-295, 2007.
DOI : 10.1088/0965-0393/15/1/S22

URL : https://hal.archives-ouvertes.fr/hal-00551073

M. J. Borden, C. V. Verhoosel, M. A. Scott, T. J. Hughes, and C. M. Landis, A phasefield description of dynamic brittle fracture, Computer Methods in Applied Mechanics and Engineering, issue.0, pp.217-22077, 2012.

B. Bourdin, Numerical implementation of the variational formulation of 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.797-826, 2000.
DOI : 10.1016/S0022-5096(99)00028-9

B. Bourdin, J. Marigo, C. Maurini, and P. Sicsic, Morphogenesis and Propagation of Complex Cracks Induced by Thermal Shocks, Physical Review Letters, vol.112, issue.1, p.14301, 2014.
DOI : 10.1103/PhysRevLett.112.014301

URL : https://hal.archives-ouvertes.fr/hal-00911118

A. Braides, Approximation of Free-Discontinuity Problems, Lecture notes in Mathematics, vol.1694, 1998.

C. Comi, Computational modelling of gradient???enhanced damage in quasi???brittle materials, Mechanics of Cohesive-Frictional Materials, pp.17-36, 1999.
DOI : 10.1002/(SICI)1099-1484(199901)4:1<17::AID-CFM55>3.0.CO;2-6

C. Comi and U. Perego, Fracture energy based bi-dissipative damage model for concrete, International Journal of Solids and Structures, vol.38, issue.36-37, pp.36-376427, 2001.
DOI : 10.1016/S0020-7683(01)00066-X

P. Farrell and C. Maurini, Linear and nonlinear solvers for variational phase-field models of brittle fracture, International Journal for Numerical Methods in Engineering, vol.18, issue.1, p.2016
DOI : 10.1016/j.cma.2015.12.017

URL : https://hal.archives-ouvertes.fr/hal-01326326

P. Farrell and C. Maurini, Solvers for variational damage and fracture. https:// bitbucket.org/pefarrell/varfrac-solvers, 2016.

G. 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

F. Freddi and G. Royer-carfagni, Regularized variational theories of fracture: A unified approach, Journal of the Mechanics and Physics of Solids, vol.58, issue.8, pp.1154-1174, 2010.
DOI : 10.1016/j.jmps.2010.02.010

J. Geyer and S. Nemat-nasser, Experimental investigation of thermally induced interacting cracks in brittle solids, International Journal of Solids and Structures, vol.18, issue.4, pp.137-356, 1982.
DOI : 10.1016/0020-7683(82)90059-2

A. Giacomini, Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fractures, Calculus of Variations, vol.12, issue.2, pp.129-172, 2005.
DOI : 10.1007/s00526-004-0269-6

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

M. Hossain, C. Hsueh, B. Bourdin, and K. Bhattacharya, Effective toughness of heterogeneous media, Journal of the Mechanics and Physics of Solids, vol.71, issue.0, pp.7115-7147, 2014.
DOI : 10.1016/j.jmps.2014.06.002

A. Karma, D. A. Kessler, and H. Levine, Phase-Field Model of Mode III Dynamic Fracture, Physical Review Letters, vol.87, issue.4, p.45501, 2001.
DOI : 10.1103/PhysRevLett.87.045501

C. Kuhn and R. Müller, A continuum phase field model for fracture, Engineering Fracture Mechanics, vol.77, issue.18, pp.3625-3634, 2010.
DOI : 10.1016/j.engfracmech.2010.08.009

G. Lancioni and G. Royer-carfagni, The Variational Approach to Fracture Mechanics. A??Practical Application to the French Panth??on in Paris, Journal of Elasticity, vol.36, issue.1-2, pp.1-30, 2009.
DOI : 10.1007/s10659-009-9189-1

B. Li, C. Peco, D. Millán, I. Arias, and M. Arroyo, Phase-field modeling and simulation of fracture in brittle materials with strongly anisotropic surface energy, International Journal for Numerical Methods in Engineering, vol.14, issue.7, pp.3-4711, 2014.
DOI : 10.1002/nme.4726

A. Logg, K. Mardal, and G. N. Wells, Automated Solution of Differential Equations by the Finite Element Method, 2012.
DOI : 10.1007/978-3-642-23099-8

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, 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

C. Maurini, Fenics codes for variational damage and fracture, 2013.

C. Miehe, M. Hofacker, and F. Welschinger, A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits, Computer Methods in Applied Mechanics and Engineering, vol.199, issue.45-48, pp.45-482765, 2010.
DOI : 10.1016/j.cma.2010.04.011

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 : 10.1002/cpa.3160420503

R. Peerlings, R. De-borst, W. Brekelmans, J. De-vree, and I. Spee, Some observations on localisation in non-local and gradient damage models, European Journal of Mechanics A/Solids, vol.15, pp.937-953, 1996.

R. Peerlings, R. De-borst, W. Brekelmans, and M. Geers, Gradient-enhanced damage modelling of concrete fracture, Mechanics of Cohesive-frictional Materialsf, pp.323-342, 1998.
DOI : 10.1002/(SICI)1099-1484(1998100)3:4<323::AID-CFM51>3.0.CO;2-Z

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 until the rupture: construction of the responses with damage localization for a general class of gradient damage models, Continuum Mech. Thermodyn, vol.25, pp.2-4147, 2013.

K. Pham and J. Marigo, Stability of Homogeneous States with Gradient Damage Models: Size Effects and Shape Effects in the Three-Dimensional Setting, Journal of Elasticity, vol.30, issue.13, pp.63-93, 2013.
DOI : 10.1007/s10659-012-9382-5

URL : https://hal.archives-ouvertes.fr/hal-00655488

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

A. Pons and A. Karma, Helical crack-front instability in mixed-mode fracture, Nature, vol.362, issue.7285, pp.85-89, 2010.
DOI : 10.1038/nature08862

A. Schlüter, A. Willenbücher, C. Kuhn, and R. Müller, Phase Field Approximation of Dynamic Brittle Fracture, PAMM, vol.14, issue.1, pp.1-21, 2014.
DOI : 10.1002/pamm.201410059

Y. Shao, X. Xu, S. Meng, G. Bai, C. Jiang et al., Crack Patterns in Ceramic Plates after Quenching, Journal of the American Ceramic Society, vol.15, issue.[5], pp.933006-3008, 2010.
DOI : 10.1111/j.1551-2916.2010.03971.x

P. Sicsic and J. Marigo, From Gradient Damage Laws to Griffith???s Theory of Crack Propagation, Journal of Elasticity, vol.59, issue.6, pp.55-74, 2013.
DOI : 10.1007/s10659-012-9410-5

P. Sicsic, J. Marigo, and C. Maurini, Initiation of a periodic array of cracks in the thermal shock problem: A gradient damage modeling, Journal of the Mechanics and Physics of Solids, vol.63, pp.63256-284
DOI : 10.1016/j.jmps.2013.09.003

URL : https://hal.archives-ouvertes.fr/hal-00843625