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An overview of the modelling of fracture by gradient damage models. (English) Zbl 1374.74109
Summary: The paper is devoted to gradient damage models which allow us to describe all the process of degradation of a body including the nucleation of cracks and their propagation. The construction of such model follows the variational approach to fracture and proceeds into two stages: (1) definition of the energy; (2) formulation of the damage evolution problem. The total energy of the body is defined in terms of the state variables which are the displacement field and the damage field in the case of quasi-brittle materials. That energy contains in particular gradient damage terms in order to avoid too strong damage localizations. The formulation of the damage evolution problem is then based on the concepts of irreversibility, stability and energy balance. That allows us to construct homogeneous as well as localized damage solutions in a closed form and to illustrate the concepts of loss of stability, of scale effects, of damage localization, and of structural failure. Moreover, the variational formulation leads to a natural numerical method based on an alternate minimization algorithm. Several numerical examples illustrate the ability of this approach to account for all the process of fracture including a 3D thermal shock problem where the crack evolution is very complex.

MSC:
74R10 Brittle fracture
Software:
FEniCS; PETSc
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[1] Alessi, R; Marigo, J-J; Vidoli, S, Nucleation of cohesive cracks in gradient damage models coupled with plasticity, Arch Ration Mech Anal, 214, 575-615, (2014) · Zbl 1321.74006
[2] Alessi, R; Marigo, J-J; Vidoli, S, Gradient damage models coupled with plasticity: variational formulation and main properties, Mech Mater, 80, 351-367, (2015)
[3] Ambrosio L, Fusco N, Pallara D (2000) Functions of bounded variation and free discontinuity problems. Oxford Science Publications, Oxford Mathematical Monographs, Oxford · Zbl 0957.49001
[4] Amor, H; Marigo, J-J; Maurini, C, Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments, J Mech Phys Solids, 57, 1209-1229, (2009) · Zbl 1426.74257
[5] Artina, M; Fornasier, M; Micheletti, S; Perotto, S, Anisotropic mesh adaptation for crack detection in brittle materials, SIAM J Sci Comput, 37, b633-b659, (2015) · Zbl 1325.74134
[6] Bahr, HA; Fischer, G; Weiss, HJ, Thermal-shock crack patterns explained by single and multiple crack propagation, J Mater Sci, 21, 2716-2720, (1986)
[7] 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 · Zbl 1343.74004
[8] 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) · Zbl 1352.74290
[9] Benallal, A; Marigo, J-J, Bifurcation and stability issues in gradient theories with softening, Model Simul Mater Sci Eng, 15, s283-s295, (2007)
[10] Borden, MJ; Verhoosel, CV; Scott, MA; Hughes, TJ; Landis, CM, A phase-field description of dynamic brittle fracture, Comput Methods Appl Mech Eng, 217-220, 77-95, (2012) · Zbl 1253.74089
[11] Bourdin, B, Numerical implementation of the variational formulation of quasi-static brittle fracture, Interfaces Free Bound, 9, 411-430, (2007) · Zbl 1130.74040
[12] Bourdin, B; Francfort, G; Marigo, J-J, Numerical experiments in revisited brittle fracture, J Mech Phys Solids, 48, 797-826, (2000) · Zbl 0995.74057
[13] Bourdin, B; Marigo, J-J; Maurini, C; Sicsic, P, Morphogenesis and propagation of complex cracks induced by thermal shocks, Phys Rev Lett, 112, 014301, (2014)
[14] Braides A (1998) Approximation of free-discontinuity problems. Lecture notes in Mathematics. Springer, Berlin · Zbl 0909.49001
[15] Comi, C, Computational modelling of gradient-enhanced damage in quasi-brittle materials, Mech Cohes Frict Mater, 4, 17-36, (1999)
[16] Comi, C; Perego, U, Fracture energy based bi-dissipative damage model for concrete, Int J Solids Struct, 38, 6427-6454, (2001) · Zbl 0996.74069
[17] Farrell, P; Maurini, C, Linear and nonlinear solvers for variational phase-field models of brittle fracture, Int J Numer Methods Eng, (2016)
[18] Farrell P, Maurini C (2016) Solvers for variational damage and fracture. https://bitbucket.org/pefarrell/varfrac-solvers · Zbl 1263.74046
[19] Francfort, G; Marigo, J-J, Revisiting brittle fracture as an energy minimization problem, J Mech Phys Solids, 46, 1319-1342, (1998) · Zbl 0966.74060
[20] Freddi, F; Royer-Carfagni, G, Regularized variational theories of fracture: a unified approach, J Mech Phys Solids, 58, 1154-1174, (2010) · Zbl 1244.74114
[21] Geyer, J; Nemat-Nasser, S, Experimental investigations of thermally induced interacting cracks in brittle solids, Int J Solids Struct, 18, 137-356, (1982)
[22] Giacomini, A, Ambrosio-tortorelli approximation of quasi-static evolution of brittle fractures, Calc Var Partial Differ Equ, 22, 129-172, (2005) · Zbl 1068.35189
[23] Hakim, V; Karma, A, Laws of crack motion and phase-field models of fracture, J Mech Phys Solids, 57, 342-368, (2009) · Zbl 1421.74089
[24] Hossain, M; Hsueh, C-J; Bourdin, B; Bhattacharya, K, Effective toughness of heterogeneous media, J Mech Phys Solids, 71, 15-32, (2014)
[25] Karma, A; Kessler, DA; Levine, H, Phase-field model of mode iii dynamic fracture, Phys Rev Lett, 87, 045501, (2001)
[26] Kuhn, C; Müller, R, A continuum phase field model for fracture, Eng Fract Mech, 77, 3625-3634, (2010) · Zbl 1003.83012
[27] Lancioni, G; Royer-Carfagni, G, The variational approach to fracture mechanics. A practical application to the French panthéon in Paris, J Elast, 95, 1-30, (2009) · Zbl 1166.74029
[28] Li, B; Peco, C; Millán, D; Arias, I; Arroyo, M, Phase-field modeling and simulation of fracture in brittle materials with strongly anisotropic surface energy, Int J Numer Methods Eng, 102, 711-727, (2014) · Zbl 1352.74290
[29] Logg A, Mardal K-A, Wells G N et al (2012) Automated solution of differential equations by the finite element method. Springer, Berlin · Zbl 1247.65105
[30] Lorentz, E; Andrieux, S, Analysis of non-local models through energetic formulations, Int J Solids Struct, 40, 2905-2936, (2003) · Zbl 1038.74506
[31] Marigo, J-J, Constitutive relations in plasticity, damage and fracture mechanics based on a work property, Nucl Eng Design, 114, 249-272, (1989)
[32] Maurini C (2013) Fenics codes for variational damage and fracture. https://bitbucket.org/cmaurini/varfrac_for_cism · Zbl 1130.74040
[33] Miehe, C; Hofacker, M; Welschinger, F, A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits, Comput Methods Appl Mech Eng, 199, 2765-2778, (2010) · Zbl 1231.74022
[34] Mumford, D; Shah, J, Optimal approximations by piecewise smooth functions and associated variational problems, Commun Pure Appl Math, 42, 577-685, (1989) · Zbl 0691.49036
[35] Peerlings, R; Borst, R; Brekelmans, W; Vree, J; Spee, I, Some observations on localisation in non-local and gradient damage models, Eur J Mech A Solids, 15, 937-953, (1996) · Zbl 0891.73055
[36] Peerlings, R; Borst, R; Brekelmans, W; Geers, M, Gradient-enhanced damage modelling of concrete fracture, Mech Cohes Frict Mater, 3, 323-342, (1998) · Zbl 0995.74056
[37] Pham, K; Amor, H; Marigo, J-J; Maurini, C, Gradient damage models and their use to approximate brittle fracture, Int J Damage Mech, 20, 618-652, (2011)
[38] Pham, K; Marigo, J-J, Approche variationnelle de l’endommagement: I. LES concepts fondamentaux, Comptes Rendus Mécanique, 338, 191-198, (2010) · Zbl 1300.74046
[39] Pham, K; Marigo, J-J, Approche variationnelle de l’endommagement: II. LES modèles à gradient, Comptes Rendus Mécanique, 338, 199-206, (2010) · Zbl 1300.74047
[40] Pham, K; Marigo, J-J, 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, 147-171, (2013) · Zbl 1343.74004
[41] Pham, K; Marigo, J-J, Stability of homogeneous states with gradient damage models: size effects and shape effects in the three-dimensional setting, J Elast, 110, 63-93, (2013) · Zbl 1263.74046
[42] Pham, K; Marigo, J-J; Maurini, C, The issues of the uniqueness and the stability of the homogeneous response in uniaxial tests with gradient damage models, J Mech Phys Solids, 59, 1163-1190, (2011) · Zbl 1270.74015
[43] Pons, A; Karma, A, Helical crack-front instability in mixed mode fracture, Nature, 464, 85-89, (2010)
[44] Schlüter, A; Willenbücher, A; Kuhn, C; Müller, R, Phase field approximation of dynamic brittle fracture, Comput Mech, 54, 1141, (2014) · Zbl 1311.74106
[45] Shao, Y; Xu, X; Meng, S; Bai, G; Jiang, C; Song, F, Crack patterns in ceramic plates after quenching, J Am Ceram Soc, 93, 3006-3008, (2010)
[46] Sicsic, P; Marigo, J-J, From gradient damage laws to griffith’s theory of crack propagation, J Elast, 113, 55-74, (2013) · Zbl 1274.74436
[47] Sicsic, P; Marigo, J-J; Maurini, C, Initiation of a periodic array of cracks in the thermal shock problem: a gradient damage modeling, J Mech Phys Solids, 63, 256-284, (2014) · Zbl 1303.74008
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