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A phase-field model for cohesive fracture. (English) Zbl 1352.74029
Summary: In this paper, a phase-field model for cohesive fracture is developed. After casting the cohesive zone approach in an energetic framework, which is suitable for incorporation in phase-field approaches, the phase-field approach to brittle fracture is recapitulated. The approximation to the Dirac function is discussed with particular emphasis on the Dirichlet boundary conditions that arise in the phase-field approximation. The accuracy of the discretisation of the phase field, including the sensitivity to the parameter that balances the field and the boundary contributions, is assessed at the hand of a simple example. The relation to gradient-enhanced damage models is highlighted, and some comments on the similarities and the differences between phase-field approaches to fracture and gradient-damage models are made. A phase-field representation for cohesive fracture is elaborated, starting from the aforementioned energetic framework. The strong as well as the weak formats are presented, the latter being the starting point for the ensuing finite element discretisation, which involves three fields: the displacement field, an auxiliary field that represents the jump in the displacement across the crack, and the phase field. Compared to phase-field approaches for brittle fracture, the modelling of the jump of the displacement across the crack is a complication, and the current work provides evidence that an additional constraint has to be provided in the sense that the auxiliary field must be constant in the direction orthogonal to the crack. The sensitivity of the results with respect to the numerical parameter needed to enforce this constraint is investigated, as well as how the results depend on the orders of the discretisation of the three fields. Finally, examples are given that demonstrate grid insensitivity for adhesive and for cohesive failure, the latter example being somewhat limited because only straight crack propagation is considered.

74A45 Theories of fracture and damage
74R10 Brittle fracture
74S05 Finite element methods applied to problems in solid mechanics
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[1] Ngo, Finite element analysis of reinforced concrete beams, Journal of the American Concrete Institute 64 pp 152– (1967)
[2] Rashid, Analysis of reinforced concrete pressure vessels, Nuclear Engineering and Design 7 pp 334– (1968) · doi:10.1016/0029-5493(68)90066-6
[3] Ingraffea, Fracture Mechanics of Concrete pp 171– (1985)
[4] Camacho, Computational modelling of impact damage in brittle materials, International Journal of Solids and Structures 33 pp 2899– (1996) · Zbl 0929.74101 · doi:10.1016/0020-7683(95)00255-3
[5] Fleming, Enriched element-free Galerkin methods for crack tip fields, International Journal for Numerical Methods in Engineering 40 pp 1483– (1997) · doi:10.1002/(SICI)1097-0207(19970430)40:8<1483::AID-NME123>3.0.CO;2-6
[6] Belytschko, Elastic crack growth in finite elements with minimal remeshing, International Journal for Numerical Methods in Engineering 45 pp 601– (1999) · Zbl 0943.74061 · doi:10.1002/(SICI)1097-0207(19990620)45:5<601::AID-NME598>3.0.CO;2-S
[7] Moës, A finite element method for crack growth without remeshing, International Journal for Numerical Methods in Engineering 46 pp 131– (1999) · Zbl 0955.74066 · doi:10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J
[8] Wells, A new method for modelling cohesive cracks using finite elements, International Journal for Numerical Methods in Engineering 50 pp 2667– (2001) · Zbl 1013.74074 · doi:10.1002/nme.143
[9] Remmers, A cohesive segments method for the simulation of crack growth, Computational Mechanics 31 pp 69– (2003) · Zbl 1038.74679 · doi:10.1007/s00466-002-0394-z
[10] Borst, Mesh-independent discrete numerical representations of cohesive-zone models, Engineering Fracture Mechanics 73 pp 160– (2006) · doi:10.1016/j.engfracmech.2005.05.007
[11] Moës, Non-planar 3D crack growth by the extended finite element and level sets - part I: mechanical model, International Journal for Numerical Methods in Engineering 53 pp 2549– (2002) · Zbl 1169.74621 · doi:10.1002/nme.429
[12] Gravouil, Non-planar 3D crack growth by the extended finite element and level sets - part II: level set update, International Journal for Numerical Methods in Engineering 53 pp 2569– (2002) · Zbl 1169.74621 · doi:10.1002/nme.430
[13] Borst, A unified framework for concrete damage and fracture models including size effects, International Journal of Fracture 95 pp 261– (1999) · doi:10.1023/A:1018664705895
[14] Borst, Non-linear Finite Element Analysis of Solids and Structures, 2. ed. (2012)
[15] Pijaudier-Cabot, Nonlocal damage theory, ASCE Journal of Engineering Mechanics 113 pp 1512– (1987) · Zbl 0788.73012 · doi:10.1061/(ASCE)0733-9399(1987)113:10(1512)
[16] Peerlings, Gradient-enhanced damage for quasi-brittle materials, International Journal for Numerical Methods in Engineering 39 pp 3391– (1996) · Zbl 0882.73057 · doi:10.1002/(SICI)1097-0207(19961015)39:19<3391::AID-NME7>3.0.CO;2-D
[17] Francfort, Revisiting brittle fracture as an energy minimization problem, Journal of the Mechanics and Physics of Solids 46 pp 1319– (1998) · Zbl 0966.74060 · doi:10.1016/S0022-5096(98)00034-9
[18] Bourdin, Numerical experiments in revisited brittle fracture, Journal of the Mechanics and Physics of Solids 48 pp 797– (2000) · Zbl 0995.74057 · doi:10.1016/S0022-5096(99)00028-9
[19] Bourdin, The variational approach to fracture, Journal of Elasticity 91 pp 5– (2008) · Zbl 1176.74018 · doi:10.1007/s10659-007-9107-3
[20] Mumford, Optimal approximations by piecewise smooth functions and associated variational problems, Communications in Pure and Applied Mathematics 42 pp 577– (1989) · Zbl 0691.49036 · doi:10.1002/cpa.3160420503
[21] Miehe, Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations, International Journal for Numerical Methods in Engineering 83 pp 1273– (2010) · Zbl 1202.74014 · doi:10.1002/nme.2861
[22] Miehe, A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits, Computer Methods in Applied Mechanics and Engineering 199 pp 2765– (2010) · Zbl 1231.74022 · doi:10.1016/j.cma.2010.04.011
[23] Hofacker, A phase field model of dynamic fracture: robust field updates for the analysis of complex crack patterns, International Journal for Numerical Methods in Engineering 93 pp 276– (2012) · Zbl 1352.74022 · doi:10.1002/nme.4387
[24] Bourdin, A time-discrete model for dynamic fracture based on crack regularization, International Journal of Fracture 168 pp 133– (2011) · Zbl 1283.74055 · doi:10.1007/s10704-010-9562-x
[25] Borden, A phase-field description of dynamic brittle fracture, Computer Methods in Applied Mechanics and Engineering 217-220 pp 77– (2012) · Zbl 1253.74089 · doi:10.1016/j.cma.2012.01.008
[26] Borst, A gradient-enhanced damage approach to fracture, Journal de Physique IV C6 pp 491– (1999)
[27] Comi, Computational modelling of gradient-enhanced damage in quasi-brittle materials, Mechanics of Cohesive-frictional Materials 4 pp 17– (1999) · doi:10.1002/(SICI)1099-1484(199901)4:1<17::AID-CFM55>3.0.CO;2-6
[28] Geers, Strain-based transient-gradient damage model for failure analyses, Computer Methods in Applied Mechanics and Engineering 160 pp 133– (1998) · Zbl 0938.74006 · doi:10.1016/S0045-7825(98)80011-X
[29] Auricchio, Encyclopedia of Computational Mechanics pp 237– (2004)
[30] Xu, Void nucleation by inclusion debonding in a crystal matrix, Modelling and Simulation in Materials Science and Engineering 1 pp 111– (1993) · doi:10.1088/0965-0393/1/2/001
[31] Remmers JJC Discontinuities in materials and structures: a unifying computational approach PhD thesis 2006
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