Phase-field modeling of crack propagation in piezoelectric and ferroelectric materials with different electromechanical crack conditions, Journal of the Mechanics and Physics of Solids, vol.60, issue.12, pp.602100-2126, 2012. ,
DOI : 10.1016/j.jmps.2012.06.014
Gradient Damage Models Coupled with Plasticity and Nucleation of Cohesive Cracks, Archive for Rational Mechanics and Analysis, vol.110, issue.1, pp.575-615, 2014. ,
DOI : 10.1007/s10659-012-9410-5
Phase-field modeling of ductile fracture, Computational Mechanics, vol.92, issue.3???4, pp.1017-1040, 2015. ,
DOI : 10.1017/CBO9780511762956
A phase-field model for ductile fracture at finite strains and its experimental verification, Computational Mechanics, vol.186, issue.4, pp.149-167, 2015. ,
DOI : 10.1007/s10704-013-9904-6
Approximation of functional depending on jumps by elliptic functional via t-convergence, Communications on Pure and Applied Mathematics, vol.17, issue.8, pp.999-1036, 1990. ,
DOI : 10.1080/01621459.1987.10478393
On the approximation of free discontinuity problems, Boll. Un. Mat. Ital. B, vol.6, issue.71, pp.105-123, 1992. ,
PETSc users manual, 2016. ,
Efficient Management of Parallelism in Object-Oriented Numerical Software Libraries, Modern Software Tools in Scientific Computing, pp.163-202, 1997. ,
DOI : 10.1007/978-1-4612-1986-6_8
Scaling of quasibrittle fracture: asymptotic analysis, International Journal of Fracture, vol.83, issue.1, pp.19-40, 1997. ,
DOI : 10.1023/A:1007387823522
Scaling of Structural Strength, 2005. ,
Discrete approximation of a free discontinuity problem, Numerical Functional Analysis and Optimization, vol.1, issue.3-4, pp.201-224, 1994. ,
DOI : 10.1007/978-1-4612-1015-3
Bifurcation and stability issues in gradient theories with softening. Modelling and Simulation in, Materials Science and Engineering, vol.15, issue.1, p.283, 2007. ,
DOI : 10.1088/0965-0393/15/1/s22
URL : https://hal.archives-ouvertes.fr/hal-00551073
A phase-field formulation for fracture in ductile materials: Finite deformation balance law derivation, plastic degradation, and stress triaxiality effects, Computer Methods in Applied Mechanics and Engineering, vol.312, pp.130-166, 2016. ,
DOI : 10.1016/j.cma.2016.09.005
A higher-order phase-field model for brittle fracture: Formulation and analysis within the isogeometric analysis framework, Computer Methods in Applied Mechanics and Engineering, vol.273, issue.0, pp.273100-118, 2014. ,
DOI : 10.1016/j.cma.2014.01.016
Image segmentation with a finite element method, ESAIM: Mathematical Modelling and Numerical Analysis, vol.33, issue.2, pp.229-244, 1999. ,
DOI : 10.1051/m2an:1999114
Numerical implementation of the variational formulation for quasi-static brittle fracture, Interfaces and Free Boundaries, vol.9, pp.411-430, 2007. ,
DOI : 10.4171/IFB/171
Implementation of an adaptive finite-element approximation of the Mumford-Shah functional, Numerische Mathematik, vol.85, issue.4, pp.609-646, 2000. ,
DOI : 10.1007/PL00005394
A Variational Approach to the Numerical Simulation of Hydraulic Fracturing, SPE Annual Technical Conference and Exhibition, 2012. ,
DOI : 10.2118/159154-MS
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
The Variational Approach to Fracture, Journal of Elasticity, vol.125, issue.8, pp.5-148, 2008. ,
DOI : 10.1016/S1874-5717(06)80009-5
URL : https://hal.archives-ouvertes.fr/hal-00551079
Morphogenesis and Propagation of Complex Cracks Induced by Thermal Shocks, Physical Review Letters, vol.112, issue.1, p.14301, 2014. ,
DOI : 10.1016/0377-0273(94)90092-2
URL : https://hal.archives-ouvertes.fr/hal-00911118
Approximation of Free-Discontinuity Problems. Number 1694 in Lecture Notes in Mathematics, 1998. ,
DOI : 10.1007/bfb0097344
?-convergence for beginners, of Oxford Lecture Series in Mathematics and its Applications, 2002. ,
DOI : 10.1093/acprof:oso/9780198507840.001.0001
An Adaptive Finite Element Approximation of a Variational Model of Brittle Fracture, SIAM Journal on Numerical Analysis, vol.48, issue.3, pp.980-1012, 2010. ,
DOI : 10.1137/080741033
AN ADAPTIVE FINITE ELEMENT APPROXIMATION OF A GENERALIZED AMBROSIO???TORTORELLI FUNCTIONAL, Mathematical Models and Methods in Applied Sciences, vol.19, issue.09, pp.1663-1697, 2013. ,
DOI : 10.1007/b98874
An approximation result for special functions with bounded deformation, Journal de Math??matiques Pures et Appliqu??es, vol.83, issue.7, pp.929-954, 2004. ,
DOI : 10.1016/j.matpur.2004.02.004
URL : http://doi.org/10.1016/j.matpur.2004.02.004
An approximation result for special functions with bounded deformation, Journal de Math??matiques Pures et Appliqu??es, vol.83, issue.7, pp.929-954137, 2004. ,
DOI : 10.1016/j.matpur.2004.02.004
URL : http://doi.org/10.1016/j.matpur.2004.02.004
Initiation of cracks with cohesive force models: a variational approach, European Journal of Mechanics - A/Solids, vol.25, issue.4, pp.649-669, 2006. ,
DOI : 10.1016/j.euromechsol.2006.05.002
URL : https://hal.archives-ouvertes.fr/hal-00551071
Slow crack growth, its modeling and crack-layer approach: A review, International Journal of Engineering Science, vol.83, pp.6-41, 2014. ,
DOI : 10.1016/j.ijengsci.2014.05.015
Application of the Variational Fracture Model to Hydraulic Fracturing in Poroelastic Media, 2016. ,
Phase field approximation of cohesive fracture models, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol.33, issue.4, pp.1033-1067, 2016. ,
DOI : 10.1016/j.anihpc.2015.02.001
URL : http://arxiv.org/abs/1405.6883
Viscous approximation of quasistatic evolutions for a coupled elastoplastic-damage model, Calculus of Variations and Partial Differential Equations, vol.90, issue.2, pp.1-54, 2016. ,
DOI : 10.1002/zamm.200900243
An introduction to ?-convergence, 1993. ,
Cohesive-zone models, higher-order continuum theories and reliability methods for computational failure analysis, International Journal for Numerical Methods in Engineering, vol.60, issue.1, pp.289-315, 2004. ,
DOI : 10.1002/nme.963
URL : https://hal.archives-ouvertes.fr/hal-00450612
A diffuse cohesive energy approach to fracture and plasticity: the one-dimensional case, J. Mech. Phys. Solids, vol.8, pp.2-4109, 2013. ,
Sur une Interpr??tation Math??matique de l'Int??grale de Rice en Th??orie de la Rupture Fragile, Mathematical Methods in the Applied Sciences, vol.III, issue.1, pp.70-87, 1981. ,
DOI : 10.1215/S0012-7094-63-03008-4
Fracture initiation at sharp notches: Correlation using critical stress intensities, International Journal of Solids and Structures, vol.34, issue.29, pp.3873-3883, 1997. ,
DOI : 10.1016/S0020-7683(96)00236-3
A critical evaluation of cohesive zone models of dynamic fracture, J. Phys IV, vol.11, issue.PR5, pp.43-50, 2001. ,
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
Free discontinuity finite element models in two-dimensions for in-plane crack problems, Theoretical and Applied Fracture Mechanics, vol.47, issue.3, pp.274-282, 2007. ,
DOI : 10.1016/j.tafmec.2007.01.006
Numerical insight of a variational smeared approach to cohesive fracture, Journal of the Mechanics and Physics of Solids, vol.98, pp.156-171, 2017. ,
DOI : 10.1016/j.jmps.2016.09.003
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 general solution of an infinite elastic plate with an elliptic hole under biaxial loading, International Journal of Pressure Vessels and Piping, vol.67, issue.1, pp.95-104, 1996. ,
DOI : 10.1016/0308-0161(94)00173-1
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
A Discontinuous Finite Element Approximation of Quasi-static Growth of Brittle Fractures, Numerical Functional Analysis and Optimization, vol.43, issue.7-8, pp.813-850, 2003. ,
DOI : 10.1016/S0022-5096(98)00034-9
A fracture criterion for sharp V-notched samples, International Journal of Fracture, vol.123, issue.3/4, pp.163-175, 2003. ,
DOI : 10.1023/B:FRAC.0000007374.80996.a2
Failure criteria for linear elastic materials with U-notches, International Journal of Fracture, vol.125, issue.2, pp.99-113, 2006. ,
DOI : 10.1007/s10704-006-0066-7
Modeling fracture in the context of a strain-limiting theory of elasticity: A single plane-strain crack, International Journal of Engineering Science, vol.88, pp.73-82, 2015. ,
DOI : 10.1016/j.ijengsci.2014.04.018
On cracks emanating from wedges in expanded PVC foam, Engineering Fracture Mechanics, vol.54, issue.4, pp.445-456, 1996. ,
DOI : 10.1016/0013-7944(95)00231-6
The Phenomena of Rupture and Flow in Solids, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.221, issue.582-593, pp.163-198, 1921. ,
DOI : 10.1098/rsta.1921.0006
Sur les matériaux standards généralisés, J. Mécanique, vol.14, issue.1, pp.39-63, 1975. ,
Effective toughness of heterogeneous media, Journal of the Mechanics and Physics of Solids, vol.71, pp.320-348, 2014. ,
DOI : 10.1016/j.jmps.2014.06.002
Stresses in plates due to the presence of cracks and sharp corners. Transactions of the Institute of Naval Architects, pp.219-241, 1913. ,
Size effects in concrete fracture: Part I, experimental setup and observations, International Journal of Fracture, vol.102, issue.1, pp.1-24, 2000. ,
DOI : 10.1023/A:1007533218153
A density result for GSBD and its application to the approximation of brittle fracture energies, Calculus of Variations and Partial Differential Equations, vol.6, issue.1-2, pp.315-342, 2014. ,
DOI : 10.1007/BF00284617
Phase-Field Model of Mode III Dynamic Fracture, Physical Review Letters, vol.71, issue.4, 2001. ,
DOI : 10.1103/PhysRevLett.71.2417
An assessment of the phase field formulation for crack growth, Computer Methods in Applied Mechanics and Engineering, vol.294, pp.313-330, 2015. ,
DOI : 10.1016/j.cma.2015.06.009
On degradation functions in phase field fracture models, Computational Materials Science, vol.108, issue.10, pp.374-384 ,
DOI : 10.1016/j.commatsci.2015.05.034
A generalized stress intensity factor to be applied to rounded V-shaped notches, International Journal of Solids and Structures, vol.43, issue.9, pp.2461-2478, 2006. ,
DOI : 10.1016/j.ijsolstr.2005.03.007
URL : http://doi.org/10.1016/j.ijsolstr.2005.03.007
Strength or toughness? A criterion for crack onset at a notch, European Journal of Mechanics - A/Solids, vol.21, issue.1, pp.61-72, 2002. ,
DOI : 10.1016/S0997-7538(01)01184-6
Computation of Singular Solutions in Elliptic Problems and Elasticity, 1987. ,
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 : 10.1007/BF00665906
Gradient Damage Modeling of Dynamic Brittle Fracture, 2016. ,
URL : https://hal.archives-ouvertes.fr/tel-01487449
Gradient damage modeling of brittle fracture in an explicit dynamics context, International Journal for Numerical Methods in Engineering, vol.44, issue.20, p.2016 ,
DOI : 10.1016/j.ijsolstr.2007.02.044
URL : https://hal.archives-ouvertes.fr/hal-01248263
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
An overview of the modelling of fracture by gradient damage models, Meccanica, vol.63, issue.1, pp.513107-3128, 2016. ,
DOI : 10.1016/j.jmps.2013.09.003
URL : https://hal.archives-ouvertes.fr/hal-01374814
Crack patterns obtained by unidirectional drying of a colloidal suspension in a capillary tube: experiments and numerical simulations using a two-dimensional variational approach, International Journal of Fracture, vol.362, issue.6418, pp.75-91, 2013. ,
DOI : 10.1038/362329a0
Validation simulations for the variational approach to fracture, Computer Methods in Applied Mechanics and Engineering, vol.290, pp.420-437, 2015. ,
DOI : 10.1016/j.cma.2014.10.052
Phase field modeling of ductile fracture at finite strains: A variational gradient-extended plasticity-damage theory, International Journal of Plasticity, vol.84, 2016. ,
DOI : 10.1016/j.ijplas.2016.04.011
Phase field modeling of fracture in multi-physics problems. Part II. Coupled brittle-to-ductile failure criteria and crack propagation in thermo-elastic???plastic solids, Computer Methods in Applied Mechanics and Engineering, vol.294, pp.486-522, 2015. ,
DOI : 10.1016/j.cma.2014.11.017
Evolution of rate-indipendent system, Handbook of Differential Equations, Evolutionary Equations, pp.461-559, 2005. ,
Some Basic Problems of the Mathematical Theory of Elasticity: Fundamental Equations, Plane Theory of Elasticity, Torsion, and Bending (translated from Russian), 1977. ,
DOI : 10.1007/978-94-017-3034-1
Numerical minimization??of the Mumford-Shah functional, Calcolo, vol.38, issue.2, pp.67-84, 2001. ,
DOI : 10.1007/s100920170004
On the choice of parameters in the phase field method for simulating crack initiation with experimental validation, International Journal of Fracture, vol.59, issue.9, pp.213-226, 2016. ,
DOI : 10.1002/nme.857
URL : https://hal.archives-ouvertes.fr/hal-01258035
Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis, International Journal for Numerical Methods in Engineering, vol.142, issue.9, pp.1267-1282, 1999. ,
DOI : 10.1007/BF00033830
URL : http://aero.caltech.edu/~ortiz/Pubs/1999/OrtizPandolfi1999.pdf
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
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
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
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
Experimental validation of a phase-field model for fracture, International Journal of Fracture, vol.27, issue.6, pp.83-101, 2017. ,
DOI : 10.1016/j.jmps.2011.03.010
The mechanics of earthquake rupture, Proc. Int. School of Physics " Enrico Fermi " Physics of the Earth's Interior, pp.555-649, 1980. ,
High-accuracy phase-field models for brittle fracture based on a new family of degradation functions, 2017. ,
Eigenfracture: An Eigendeformation Approach to Variational Fracture, Multiscale Modeling & Simulation, vol.7, issue.3, pp.1237-1266, 2009. ,
DOI : 10.1137/080712568
Brittle fracture criterion for structures with sharp notches, Engineering Fracture Mechanics, vol.47, issue.5, pp.673-681, 1994. ,
DOI : 10.1016/0013-7944(94)90158-9
From Gradient Damage Laws to Griffith???s Theory of Crack Propagation, Journal of Elasticity, vol.59, issue.6, pp.55-74, 2013. ,
DOI : 10.1016/j.jmps.2011.03.010
Reformulation of elasticity theory for discontinuities and long-range forces, Journal of the Mechanics and Physics of Solids, vol.48, issue.1, pp.175-209, 2000. ,
DOI : 10.1016/S0022-5096(99)00029-0
Fracture at V-notches with contained plasticity, Engineering Fracture Mechanics, vol.69, issue.3, pp.403-415, 2002. ,
DOI : 10.1016/S0013-7944(01)00079-0
An augmented-Lagrangian method for the phase-field approach for pressurized fractures, Computer Methods in Applied Mechanics and Engineering, vol.271, issue.4, pp.69-85 ,
DOI : 10.1016/j.cma.2013.12.005
A phase-field model for fracture in piezoelectric ceramics, International Journal of Fracture, vol.58, issue.10, pp.135-153, 2013. ,
DOI : 10.1016/S0020-7683(97)00168-6
Phase-field modeling of hydraulic fracture, Journal of the Mechanics and Physics of Solids, vol.96, pp.264-290, 2016. ,
DOI : 10.1016/j.jmps.2016.07.019
Failure criteria for brittle elastic materials, International Journal of Fracture, vol.125, issue.3/4, pp.307-333, 2004. ,
DOI : 10.1023/B:FRAC.0000022244.31825.3b
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.132.7500
Numerical evaluation of the phase-field model for brittle fracture with emphasis on the length scale, Computational Mechanics, vol.133, issue.5, pp.1-16, 2017. ,
DOI : 10.1016/j.compstruct.2015.08.051