An overview of the modelling of fracture by gradient damage models

Alessi R, Marigo J-J, Vidoli S (2014) Nucleation of cohesive cracks in gradient damage models coupled with plasticity. Arch Ration Mech Anal 214:575–615

Alessi R, Marigo J-J, Vidoli S (2015) Gradient damage models coupled with plasticity: variational formulation and main properties. Mech Mater 80:351–367

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

Amor H, Marigo J-J, Maurini C (2009) Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments. J Mech Phys Solids 57(8):1209–1229

Artina M, Fornasier M, Micheletti S, Perotto S (2015) Anisotropic mesh adaptation for crack detection in brittle materials. SIAM J Sci Comput 37(4):B633–B659

Bahr HA, Fischer G, Weiss HJ (1986) Thermal-shock crack patterns explained by single and multiple crack propagation. J Mater Sci 21:2716–2720

Balay S, Abhyankar S, Adams MF, Brown J, Brune P, Buschelman K, Eijkhout V, Gropp WD, Kaushik D, Knepley MG, McInnes LC, Rupp K, Smith BF, Zhang H (2014) PETSc users manual. Technical Report ANL-95/11—Revision 3.5, Argonne National Laboratory

Benallal A, Billardon R, Geymonat G (1993) Bifurcation and localization in rate independent materials. In: Nguyen Q (ed), C.I.S.M lecture notes on bifurcation and stability of dissipative systems, vol 327 of international centre for mechanical sciences, pp 1–44. Springer (1993)

Benallal A, Marigo J-J (2007) Bifurcation and stability issues in gradient theories with softening. Model Simul Mater Sci Eng 15:S283–S295

Borden MJ, Verhoosel CV, Scott MA, Hughes TJ, Landis CM (2012) A phase-field description of dynamic brittle fracture. Comput Methods Appl Mech Eng 217–220:77–95

Bourdin B (2007) Numerical implementation of the variational formulation of quasi-static brittle fracture. Interfaces Free Bound 9:411–430

Bourdin B, Francfort G, Marigo J-J (2000) Numerical experiments in revisited brittle fracture. J Mech Phys Solids 48:797–826

Bourdin B, Marigo J-J, Maurini C, Sicsic P (2014) Morphogenesis and propagation of complex cracks induced by thermal shocks. Phys Rev Lett 112(1):014301

Braides A (1998) Approximation of free-discontinuity problems. Lecture notes in Mathematics. Springer, Berlin

Comi C (1999) Computational modelling of gradient-enhanced damage in quasi-brittle materials. Mech Cohes Frict Mater 4(1):17–36

Comi C, Perego U (2001) Fracture energy based bi-dissipative damage model for concrete. Int J Solids Struct 38(36–37):6427–6454

Farrell P, Maurini C (2016) Linear and nonlinear solvers for variational phase-field models of brittle fracture. Int J Numer Methods Eng. doi: 10.1002/nme.5300

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

Francfort G, Marigo J-J (1998) Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids 46(8):1319–1342

Freddi F, Royer-Carfagni G (2010) Regularized variational theories of fracture: a unified approach. J Mech Phys Solids 58:1154–1174

Geyer J, Nemat-Nasser S (1982) Experimental investigations of thermally induced interacting cracks in brittle solids. Int J Solids Struct 18(4):137–356

Giacomini A (2005) Ambrosio–Tortorelli approximation of quasi-static evolution of brittle fractures. Calc Var Partial Differ Equ 22:129–172

Hakim V, Karma A (2009) Laws of crack motion and phase-field models of fracture. J Mech Phys Solids 57:342–368

Hossain M, Hsueh C-J, Bourdin B, Bhattacharya K (2014) Effective toughness of heterogeneous media. J Mech Phys Solids 71:15–32

Karma A, Kessler DA, Levine H (2001) Phase-field model of mode iii dynamic fracture. Phys Rev Lett 87:045501

Kuhn C, Müller R (2010) A continuum phase field model for fracture. Eng Fract Mech 77(18):3625–3634 (Computational mechanics in fracture and damage: a special issue in Honor of Prof. Gross)

Lancioni G, Royer-Carfagni G (2009) The variational approach to fracture mechanics. A practical application to the French Panthéon in Paris. J Elast 95:1–30

Li B, Peco C, Millán D, Arias I, Arroyo M (2014) Phase-field modeling and simulation of fracture in brittle materials with strongly anisotropic surface energy. Int J Numer Methods Eng 102(3–4):711–727

Logg A, Mardal K-A, Wells G N et al (2012) Automated solution of differential equations by the finite element method. Springer, Berlin

Lorentz E, Andrieux S (2003) Analysis of non-local models through energetic formulations. Int J Solids Struct 40:2905–2936

Marigo J-J (1989) Constitutive relations in plasticity, damage and fracture mechanics based on a work property. Nucl Eng Design 114:249–272

Maurini C (2013) Fenics codes for variational damage and fracture. https://bitbucket.org/cmaurini/varfrac_for_cism

Miehe C, Hofacker M, Welschinger F (2010) A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits. Comput Methods Appl Mech Eng 199(45–48):2765–2778

Mumford D, Shah J (1989) Optimal approximations by piecewise smooth functions and associated variational problems. Commun Pure Appl Math 42:577–685

Peerlings R, de Borst R, Brekelmans W, de Vree J, Spee I (1996) Some observations on localisation in non-local and gradient damage models. Eur J Mech A Solids 15:937–953

Peerlings R, de Borst R, Brekelmans W, Geers M (1998) Gradient-enhanced damage modelling of concrete fracture. Mech Cohes Frict Mater 3:323–342

Pham K, Amor H, Marigo J-J, Maurini C (2011) Gradient damage models and their use to approximate brittle fracture. Int J Damage Mech 20(4):618–652

Pham K, Marigo J-J (2010) Approche variationnelle de l’endommagement: I. les concepts fondamentaux. Comptes Rendus Mécanique 338(4):191–198

Pham K, Marigo J-J (2010) Approche variationnelle de l’endommagement: II. Les modèles à gradient. Comptes Rendus Mécanique 338(4):199–206

Pham K, Marigo J-J (2013) 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 25(2–4):147–171

Pham K, Marigo J-J (2013) Stability of homogeneous states with gradient damage models: size effects and shape effects in the three-dimensional setting. J Elast 110(1):63–93 (Isiweb)

Pham K, Marigo J-J, Maurini C (2011) The issues of the uniqueness and the stability of the homogeneous response in uniaxial tests with gradient damage models. J Mech Phys Solids 59(6):1163–1190 (Isiweb)

Pons A, Karma A (2010) Helical crack-front instability in mixed mode fracture. Nature 464:85–89

Schlüter A, Willenbücher A, Kuhn C, Müller R (2014) Phase field approximation of dynamic brittle fracture. Comput Mech 54:1141

Shao Y, Xu X, Meng S, Bai G, Jiang C, Song F (2010) Crack patterns in ceramic plates after quenching. J Am Ceram Soc 93(10):3006–3008

Sicsic P, Marigo J-J (2013) From gradient damage laws to Griffith’s theory of crack propagation. J Elast 113(1):55–74

Sicsic P, Marigo J-J, Maurini C (2014) Initiation of a periodic array of cracks in the thermal shock problem: a gradient damage modeling. J Mech Phys Solids 63:256–284