On the convergence of 3D free discontinuity models in variational fracture

International Journal of Fracture

Volumn 166, Issue 1-2, 2010, Pages 3-11

  Fraternali, Fernando ^c   Negri, Matteo ^a   Ortiz, Michael ^b  

^a UNIVERSITY OF PAVIA   (Italy)

^b California Institute of Technology   (United States)

^c NONE

Author keywords

Convergence; Configurational forces; Free discontinuity models; r Adaption; Variational fracture

Indexed keywords

3D FINITE ELEMENT; BRITTLE MATERIALS; COHESIVE FRACTURE; CONFIGURATIONAL FORCE; CRACK PATTERNS; DISPLACEMENT FIELD; FREE DISCONTINUITIES; FREE DISCONTINUITY PROBLEMS; MESH GEOMETRY; QUASI-STATIC EVOLUTION; R-ADAPTION; VARIATIONAL FRACTURE; VARIATIONAL THEORY;

FRACTURE; FRACTURE MECHANICS;

THREE DIMENSIONAL;

EID: 77956402251     PISSN: 03769429     EISSN: 15732673     Source Type: Journal    
DOI: 10.1007/s10704-010-9462-0     Document Type: Conference Paper

References (42)

1. L Ambrosio A Braides 1990 Functionals defined on partitions in sets of finite perimeter. II.Semicontinuity, relaxation and homogenization J Math Pures Appl 69 3 307 333 0676.49029 1070482
2. L Ambrosio A Coscia G Dal Maso 1997 Fine properties of functions with bounded deformation Arch Rational Mech Anal 139 3 201 238 0890.49019 10.1007/s002050050051 1480240 1997ArRMA.139..201A
3. Ambrosio L, Fusco N, Pallara D (2000) Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs, Oxford University Press, New York
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. G Bellettini A Coscia G Dal Maso 1998 Compactness and lower semicontinuity properties in SBD(Ω) Math Z 228 2 337 351 0914.46007 10.1007/PL00004617 1630504
6. T Belytschko M Moes S Usui C Parimi 2001 Arbitrary discontinuities in finite elements Int J Numer Meth Eng 50 4 993 1013 0981.74062 10.1002/1097-0207(20010210)50:4<993::AID-NME164>3.0.CO;2-M
7. B Bourdin 2007 Numerical implementation of the variational formulation of brittle fracture Interfaces Free Bound 9 411 430 1130.74040 10.4171/IFB/171 2341850
8. B Bourdin GA Francfort JJ Marigo 2000 Numerical experiments in revisited brittle fracture J Mech Phys Solids 48 797 826 0995.74057 10.1016/S0022-5096(99) 00028-9 1745759 2000JMPSo..48..797B
9. A Braides A Coscia 1994 The interaction between bulk energy and surface energy in multiple integrals Proc Roy Soc Edinb Sect A 124 4 737 756 0810.49015 1298590
10. A Carpinteri B Chiaia P Cornetti 2003 On the mechanics of quasi-brittle materials with a fractal microstructure Engrg Fract Mech 70 2321 2349 10.1016/S0013-7944(02)00220-5
11. A Chambolle 2003 A density result in two-dimensional linearized elasticity, and applications Arch Ration Mech Anal 167 3 211 233 1030.74007 10.1007/s00205-002-0240-7 1978582
12. A Chambolle 2005 Addendum to: "An approximation result for special functions with bounded deformation J Math Pures Appl 84 1 137 145 2112874
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, 1997
14. G Cortesani R Toader 1999 A density result in SBV with respect to non-isotropic energies Nonlinear Anal 38 5 585 604 0939.49024 10.1016/S0362-546X(98)00132-1 1709990
15. DalMaso G (1993) An introduction to Γ -convergence. Progress in nonlinear differential equations and their applications, 8. Birkhäuser, Boston
16. G Dal Maso GA Francfort R Toader 2005 Quasistatic crack growth in nonlinear elasticity Arch Rat Mech Anal 176 2 165 225 1064.74150 10.1007/s00205-004-0351-4 2186036
17. G Dal Maso A Garroni 2008 Gradient bounds for minimizers of free discontinuity problems related to cohesive zone models in fracture mechanics Calc Var Partial Differ Equ 31 2 137 145 1125.74006 10.1007/s00526-006-0084-3 2353867
18. G Dal Maso R Toader 2002 A model for the quasi-static growth of brittle fractures based on local minimization Math Models Methods Appl Sci 12 12 1773 1799 01882866 10.1142/S0218202502002331 1946723
19. J Dolbow N Moes T Belytschko 2000 Discontinuous enrichment in finite elements with a partition of unity method Finite Elem Anal Des 36 34 235 260 0981.74057 10.1016/S0168-874X(00)00035-4
20. EN Dvorkin AM Cuitino G Gioia 1990 Finite-elements with displacement interpolated embedded localization lines insensitive to meshsize and distortions Int J Numer Meth Eng 30 3 541 564 0729.73209 10.1002/nme.1620300311
21. GA Francfort JJ Marigo 1998 Revisiting brittle fracture as an energy minimization problem J Mech Phys Solids 46 8 1319 1342 0966.74060 10.1016/S0022-5096(98)00034-9 1633984 1998JMPSo..46.1319F
22. F Fraternali 2007 Free discontinuity finite element models in two-dimensions for in-plane crack problems Theor Appl Fract Mech 47 274 282 1:CAS:528:DC%2BD2sXjsVGjt7g%3D 10.1016/j.tafmec.2007.01.006
23. TC Gasser GA Holzapfel 2005 Modeling 3D crack propagation in unreinforced concrete using PUFEM Comp Method App M 194 2526 2859 2896 1176.74180 10.1016/j.cma.2004.07.025
24. TC Gasser GA Holzapfel 2006 3D Crack propagation in unreinforced concrete. A two-step algorithm for tracking 3D crack paths Comp Method App M 195 3740 5198 5219 1154.74376 10.1016/j.cma.2005.10.023 2241409
25. AA Griffith 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. A Hansbo P Hansbo 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. P Jäger P Steinmann E Kuhl 2008 Modeling three-dimensional crack propagation-a comparison of crack path tracking strategies Int J Numer Meth Eng 76 1328 1352 10.1002/nme.2353
29. L Lussardi M Negri 2007 Convergence of non-local finite element energies for fracture mechanics Numer Funct Anal Optim 28 83 109 1108.74059 10.1080/01630560701189994 2302706
30. J Mergheim E Kuhl P Steinmann 2005 A finite element method for the computational modelling of cohesive cracks Int J Numer Meth Eng 63 2 276 289 1118.74349 10.1002/nme.1286
31. M Negri 1999 The anisotropy introduced by the mesh in the finite element approximation of the Mumford-Shah functional Numer Funct Anal Optim 20 957 982 0953.49024 10.1080/01630569908816934 1728172
32. M Negri 2005 A discontinuous finite element approximation of free discontinuity problems Adv Math Sci Appl 15 283 306 1111.49025 2148285
33. M Negri 2007 Convergence analysis for a smeared crack approach in brittle fracture Interfaces Free Bound 9 307 330 05235721 10.4171/IFB/166 2341845
34. Negri M (2010) A comparative analysis on variational models for quasi-static brittle crack propagation. Adv Calc Var (to appear)
35. M Negri C Ortner 2008 Quasi-static crack propagation by Griffith's criterion Math Models Methods Appl Sci 18 11 1895 1925 1155.74035 10.1142/S0218202508003236 2472402
36. J Oliver 1996 Modelling strong discontinuities in solid mechanics via strain softening constitutive equations. 1. Fundamentals Int J Numer Meth Eng 39 21 3575 3600 0888.73018 10.1002/(SICI)1097-0207(19961115)39:21<3575::AID- NME65>3.0.CO;2-E
37. J Oliver 1996 Modelling strong discontinuities in solid mechanics via strain softening constitutive equations. 2. Numerical simulation Int J Numer Meth Eng 39 21 3601 3623 10.1002/(SICI)1097-0207(19961115)39:21<3601::AID- NME64>3.0.CO;2-4
38. M Ortiz A Pandolfi 1999 A class of cohesive elements for the simulation of three-dimensional crack propagation Int J Numer Methods Eng 44 1267 1282 0932.74067 10.1002/(SICI)1097-0207(19990330)44:9<1267::AID-NME486>3.0. CO;2-7
39. M Ortiz A Pandolfi 2002 An efficient adaptive procedure for three-dimensional fragmentation simulations Eng Comput 18 2 148 159 10.1007/s003660200013
40. M Ortiz L Stainier 1999 The variational formulation of viscoplastic constitutive updates Comput Method Appl Mech 171 3-4 419 444 0938.74016 10.1016/S0045-7825(98)00219-9 1685716
41. B Schmidt F Fraternali M Ortiz 2009 Eigenfracture: an eigendeformation approach to variational fracture Multiscale Model Simul 7 3 1237 1266 1173.74040 10.1137/080712568 2480119
42. P Thoutireddy M Ortiz 2004 A variational r-adaption and shape-optimization method for finite deformation elasticity Int J Numer Meth Eng 61 1 21 1079.74597 10.1002/nme.1052 2078677