On the convergence of 3D free discontinuity models in variational fracture
Source abstract
Abstract
Free discontinuity problems arising in the variational theory for fracture mechanics are considered. A Γ -convergence proof for an r-adaptive 3D finite element discretization is given in the case of a brittle material. The optimal displacement field, crack pattern and mesh geometry are obtained through a variational procedure that encompasses both mechanical and configurational forces. Possible extensions to cohesive fracture and quasi-static evolutions are discussed.
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Research context
OpenAlex topics: Numerical methods in engineering · Rock Mechanics and Modeling · Composite Material Mechanics
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References (42)
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1. Ambrosio L, Braides A (1990) Functionals defined on partitions in sets of finite perimeter. II.Semicontinuity, relaxation and homogenization. J Math Pures Appl 69(3): 307–333
2. Ambrosio L, Coscia A, Dal Maso G (1997) Fine properties of functions with bounded deformation. Arch Rational Mech Anal 139(3): 201–23810.1007/s002050050051 ↗
3. Ambrosio L, Fusco N, Pallara D (2000) Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs, Oxford University Press, New York10.1093/oso/9780198502456.001.0001 ↗
4. Angelillo M, Babilio E, Fortunato A (2005) A computational approach to quasi-static propagation of brittle fracture. In: Fremond M, Maceri F (eds) Lecture Notes in Applied and Computational Mechanics. Springer-Verlag, Berlin, 23:277–288
5. Bellettini G, Coscia A, Dal Maso G (1998) Compactness and lower semicontinuity properties in SBD(Ω). Math Z 228(2): 337–35110.1007/PL00004617 ↗
6. Belytschko T, Moes M, Usui S, Parimi C (2001) Arbitrary discontinuities in finite elements. Int J Numer Meth Eng 50(4): 993–101310.1002/1097-0207(20010210)50:4<993::AID-NME164>3.0.CO;2-M ↗
7. Bourdin B (2007) Numerical implementation of the variational formulation of brittle fracture. Interfaces Free Bound 9: 411–43010.4171/IFB/171 ↗
8. Bourdin B, Francfort GA, Marigo JJ (2000) Numerical experiments in revisited brittle fracture. J Mech Phys Solids 48: 797–82610.1016/S0022-5096(99)00028-9 ↗
9. Braides A, Coscia A (1994) The interaction between bulk energy and surface energy in multiple integrals. Proc Roy Soc Edinb Sect A 124(4): 737–75610.1017/S0308210500028638 ↗
10. Carpinteri A, Chiaia B, Cornetti P (2003) On the mechanics of quasi-brittle materials with a fractal microstructure. Engrg Fract Mech 70: 2321–234910.1016/S0013-7944(02)00220-5 ↗
11. Chambolle A (2003) A density result in two-dimensional linearized elasticity, and applications. Arch Ration Mech Anal 167(3): 211–23310.1007/s00205-002-0240-7 ↗
12. Chambolle A (2005) Addendum to: ”An approximation result for special functions with bounded deformation. J Math Pures Appl 84(1): 137–14510.1016/j.matpur.2004.11.001 ↗
13. Ciarlet PG (1978) The finite element method for elliptic problems. Studies in mathematics and its applications, Vol. 4. North-Holland Publishing Co., Amsterdam. Arch Rational Mech Anal 139(3):201–238, 199710.1115/1.3424474 ↗
14. Cortesani G, Toader R (1999) A density result in SBV with respect to non-isotropic energies. Nonlinear Anal 38(5): 585–60410.1016/S0362-546X(98)00132-1 ↗
15. DalMaso G (1993) An introduction to Γ -convergence. Progress in nonlinear differential equations and their applications, 8. Birkhäuser, Boston
16. Dal Maso G, Francfort GA, Toader R (2005) Quasistatic crack growth in nonlinear elasticity. Arch Rat Mech Anal 176(2): 165–22510.1007/s00205-004-0351-4 ↗
17. Dal Maso G, Garroni A (2008) Gradient bounds for minimizers of free discontinuity problems related to cohesive zone models in fracture mechanics. Calc Var Partial Differ Equ 31(2): 137–14510.1007/s00526-006-0084-3 ↗
18. Dal Maso G, Toader R (2002) A model for the quasi-static growth of brittle fractures based on local minimization. Math Models Methods Appl Sci 12(12): 1773–179910.1142/S0218202502002331 ↗
19. Dolbow J, Moes N, Belytschko T (2000) Discontinuous enrichment in finite elements with a partition of unity method. Finite Elem Anal Des 36(34): 235–26010.1016/S0168-874X(00)00035-4 ↗
20. Dvorkin EN, Cuitino AM, Gioia G (1990) Finite-elements with displacement interpolated embedded localization lines insensitive to meshsize and distortions. Int J Numer Meth Eng 30(3): 541–56410.1002/nme.1620300311 ↗
21. Francfort GA, Marigo JJ (1998) Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids 46(8): 1319–134210.1016/S0022-5096(98)00034-9 ↗
22. Fraternali F (2007) Free discontinuity finite element models in two-dimensions for in-plane crack problems. Theor Appl Fract Mech 47: 274–28210.1016/j.tafmec.2007.01.006 ↗
23. Gasser TC, Holzapfel GA (2005) Modeling 3D crack propagation in unreinforced concrete using PUFEM. Comp Method App M 194(2526): 2859–289610.1016/j.cma.2004.07.025 ↗
24. Gasser TC, Holzapfel GA (2006) 3D Crack propagation in unreinforced concrete. A two-step algorithm for tracking 3D crack paths. Comp Method App M 195(3740): 5198–521910.1016/j.cma.2005.10.023 ↗
25. Griffith AA (1920) The phenomena of rupture and flow in solids. Phil Trans Roy Soc Lond 18: 16–98
26. Gurtin ME (2000) Configurational forces as basic concepts of continuum physics. Series of applied mathematical sciences, vol. 137. Springer, New York
27. Hansbo A, Hansbo P (2004) A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Comp Method App M 193(3335): 3532–3540
28. Jäger P, Steinmann P, Kuhl E (2008) Modeling three-dimensional crack propagation–a comparison of crack path tracking strategies. Int J Numer Meth Eng 76: 1328–135210.1002/nme.2353 ↗
29. Lussardi L, Negri M (2007) Convergence of non-local finite element energies for fracture mechanics. Numer Funct Anal Optim 28: 83–10910.1080/01630560701189994 ↗
30. Mergheim J, Kuhl E, Steinmann P (2005) A finite element method for the computational modelling of cohesive cracks. Int J Numer Meth Eng 63(2): 276–28910.1002/nme.1286 ↗
31. Negri M (1999) The anisotropy introduced by the mesh in the finite element approximation of the Mumford-Shah functional. Numer Funct Anal Optim 20: 957–98210.1080/01630569908816934 ↗
32. Negri M (2005) A discontinuous finite element approximation of free discontinuity problems. Adv Math Sci Appl 15: 283–306
33. Negri M (2007) Convergence analysis for a smeared crack approach in brittle fracture. Interfaces Free Bound 9: 307–33010.4171/IFB/166 ↗
34. Negri M (2010) A comparative analysis on variational models for quasi-static brittle crack propagation. Adv Calc Var (to appear)10.1515/acv.2010.008 ↗
35. Negri M, Ortner C (2008) Quasi-static crack propagation by Griffith’s criterion. Math Models Methods Appl Sci 18(11): 1895–192510.1142/S0218202508003236 ↗
36. Oliver J (1996a) Modelling strong discontinuities in solid mechanics via strain softening constitutive equations. 1. Fundamentals. Int J Numer Meth Eng 39(21): 3575–360010.1002/(SICI)1097-0207(19961115)39:21<3575::AID-NME65>3.0.CO;2-E ↗
37. Oliver J (1996b) Modelling strong discontinuities in solid mechanics via strain softening constitutive equations. 2. Numerical simulation. Int J Numer Meth Eng 39(21): 3601–362310.1002/(SICI)1097-0207(19961115)39:21<3601::AID-NME64>3.0.CO;2-4 ↗
38. Ortiz M, Pandolfi A (1999) A class of cohesive elements for the simulation of three-dimensional crack propagation. Int J Numer Methods Eng 44: 1267–128210.1002/(SICI)1097-0207(19990330)44:9<1267::AID-NME486>3.0.CO;2-7 ↗
39. Ortiz M, Pandolfi A (2002) An efficient adaptive procedure for three-dimensional fragmentation simulations. Eng Comput 18(2): 148–15910.1007/s003660200013 ↗
40. Ortiz M, Stainier L (1999) The variational formulation of viscoplastic constitutive updates. Comput Method Appl Mech 171(3–4): 419–44410.1016/S0045-7825(98)00219-9 ↗
41. Schmidt B, Fraternali F, Ortiz M (2009) Eigenfracture: an eigendeformation approach to variational fracture. Multiscale Model Simul 7(3): 1237–126610.1137/080712568 ↗
42. Thoutireddy P, Ortiz M (2004) A variational r-adaption and shape-optimization method for finite deformation elasticity. Int J Numer Meth Eng 61: 1–2110.1002/nme.1052 ↗