zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A geometrically nonlinear FE approach for the simulation of strong and weak discontinuities. (English) Zbl 1126.74050
Summary: We introduce a discontinuous finite element method for computational modelling of strong and weak discontinuities in geometrically nonlinear elasticity. The location of the interface is independent of the mesh structure, and therefore discontinuous elements are introduced to capture the jump in the deformation map or its gradient, respectively. To model strong discontinuities, the cohesive crack concept is adopted. The inelastic material behaviour is covered by a cohesive constitutive law, which associates the cohesive tractions, acting on the crack surfaces, with the jump in the deformation map. In the case of weak discontinuities an extended Nitsche’s method is applied, which ensures the continuity of deformation map in a weak sense. The applicability of the proposed method is highlighted by means of numerical examples, dealing with both crack propagation and material interfaces.

MSC:
74S05Finite element methods in solid mechanics
74R10Brittle fracture
74B20Nonlinear elasticity
Software:
XFEM
WorldCat.org
Full Text: DOI
References:
[1] Hansbo, A.; Hansbo, P.: An unfitted finite element method, based on Nitsche’s method for elliptic interface problems. Comput. methods appl. Mech. engrg. 191, 5537-5552 (2002) · Zbl 1035.65125
[2] Hansbo, A.; Hansbo, P.: A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Comput. methods appl. Mech. engrg. 193, 3523-3540 (2004) · Zbl 1068.74076
[3] Nitsche, J.: Über ein variationsprinzip zur lösung von Dirichlet-problemen bei verwendung von teilräumen, die keinen randbedingungen unterworfen sind. Abh. math. Univ. Hamburg 36, 9-15 (1970) · Zbl 0229.65079
[4] Ortiz, M.; Leroy, Y.; Needleman, A.: A finite element method for localized failure analysis. Comput. methods appl. Mech. engrg. 61, 189-214 (1987) · Zbl 0597.73105
[5] Belytschko, T.; Fish, J.; Engelmann, B. E.: A finite element with embedded localization zones. Comput. methods appl. Mech. engrg. 70, 59-89 (1988) · Zbl 0653.73032
[6] Klisinski, M.; Runesson, K.; Sture, S.: Finite element with inner softening band. J. engrg. Mech., ASCE 117, 575-587 (1991)
[7] Jirásek, M.: Comparative study on finite elements with embedded cracks. Comput. methods appl. Mech. engrg. 188, 307-330 (2000) · Zbl 1166.74427
[8] Armero, F.; Garikipati, K.: An analysis of strong discontinuities in multiplicative finite strain plasticity and their relation with the numerical simulation of strain localization in solids. Int. J. Solids struct. 33, 2863-2885 (1996) · Zbl 0924.73084
[9] Oliver, J.; Huespe, A. E.; Pulido, M. D. G.; Samaniego, E.: On the strong discontinuity approach in finite deformation setting. Int. J. Numer. methods engrg. 56, 1051-1082 (2003) · Zbl 1031.74010
[10] Gasser, T. C.; Holzapfel, G. A.: Geometrically nonlinear and consistently linearized embedded strong discontinuity models for 3d problems with an application to the dissection analysis of soft biological tissues. Comput. methods appl. Mech. engrg. 192, 5059-5098 (2003) · Zbl 1088.74541
[11] Babuška, I.; Melenk, J. M.: The partition of unity method. Int. J. Numer. methods engrg. 40, 727-758 (1997) · Zbl 0949.65117
[12] Belytschko, T.; Black, T.: Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. methods engrg. 45, 601-620 (1999) · Zbl 0943.74061
[13] Moës, N.; Dolbow, J.; Belytschko, T.: A finite element method for crack growth without remeshing. Int. J. Numer. methods engrg. 46, 131-150 (1999) · Zbl 0955.74066
[14] Dolbow, J.; Möes, N.; Belytschko, T.: An extended finite element method for modeling crack growth with friction contact. Comput. methods appl. Mech. engrg. 190, 6825-6846 (2001) · Zbl 1033.74042
[15] Sukumar, N.; Möes, N.; Moran, B.; Belytschko, T.: Extended finite element method for three-dimensional crack modeling. Int. J. Numer. methods engrg. 48, No. 11, 1549-1570 (2000) · Zbl 0963.74067
[16] Daux, C.; Moës, N.; Dolbow, J.; Sukumar, N.; Belytschko, T.: Arbitrary branched and intersecting cracks with the extended finite element method. Int. J. Numer. methods engrg. 48, 1741-1760 (2000) · Zbl 0989.74066
[17] Mergheim, J.; Kuhl, E.; Steinmann, P.: A finite element method for the computational modelling of cohesive cracks. Int. J. Numer. methods engrg. 63, No. 2, 276-289 (2005) · Zbl 1118.74349
[18] Dugdale, D. S.: Yielding of steel sheets containing slits. J. mech. Solids 8, 100-108 (1960)
[19] Barenblatt, G. I.: The mathematical theory of equilibrium of cracks in brittle fracture. Adv. appl. Mech. 7, 55-129 (1962)
[20] Hillerborg, A.; Petersson, P. E.; Modéer, M.: Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement concr. Res. 6, 773-782 (1976)
[21] Ortiz, M.; Pandolfi, A.: Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis. Int. J. Numer. methods engrg. 44, 1267-1282 (1999) · Zbl 0932.74067
[22] De Borst, R.; Gutiérrez, M. A.; Wells, G. N.; Remmers, J. J. C.; Askes, H.: Cohesive-zone models, higher-order continuum theories and reliability methods for computational failure analysis. Int. J. Numer. methods engrg. 60, 289-315 (2004) · Zbl 1060.74620
[23] Wells, G. N.; Sluys, L. J.: A new method for modelling cohesive cracks using finite elements. Int. J. Numer. methods engrg. 50, 2667-2682 (2001) · Zbl 1013.74074
[24] Moës, N.; Belytschko, T.: Extended finite element method for cohesive crack growth. Engrg. fracture mech. 69, No. 7, 813-833 (2002)
[25] Zi, G.; Belytschko, T.: New crack-tip elements for XFEM and applications to cohesive cracks. Int. J. Numer. methods engrg. 57, 2221-2240 (2003) · Zbl 1062.74633
[26] Wells, G. N.; Sluys, L. J.; De Borst, R.: A consistent geometrically nonlinear approach for delamination. Int. J. Numer. methods engrg. 54, 1333-1355 (2002) · Zbl 1086.74043
[27] Gasser, T. C.; Holzapfel, G. A.: Modeling of 3d crack propagation in unreinforced concrete using PUFEM. Comput. methods appl. Mech. engrg. 194, No. 25-26, 2859-2896 (2005) · Zbl 1176.74180
[28] Sukumar, N.; Chopp, D. L.; Möes, N.; Belytschko, T.: Modelling holes and inclusions by level sets in the extended finite-element method. Comput. methods appl. Mech. engrg. 190, 6183-6200 (2001) · Zbl 1029.74049
[29] Ji, H.; Chopp, D.; Dolbow, J. E.: A hybrid extended finite element/level set method for modeling phase transformations. Int. J. Numer. methods engrg. 54, 1209-1233 (2002) · Zbl 1098.76572
[30] Chessa, J.; Smolinski, P.; Belytschko, T.: The extended finite element method (XFEM) for solidification problems. Int. J. Numer. methods engrg. 53, No. 8, 1959-1977 (2002) · Zbl 1003.80004
[31] Chessa, J.; Belytschko, T.: An extended finite element method for two-phase fluids. J. appl. Mech. 70, No. 1, 10-17 (2003) · Zbl 1110.74391
[32] Hansbo, P.; Larson, M. G.: Discontinuous Galerkin and the Crouzeix-Raviart element: application to elasticity. Math. modell. Numer. anal. 37, 63-72 (2003) · Zbl 1137.65431
[33] Mergheim, J.; Kuhl, E.; Steinmann, P.: A hybrid discontinuous Galerkin/interface method for the computational modelling of failure. Commun. numer. Methods engrg. 20, 511-519 (2004) · Zbl 1302.74166
[34] A. Fritz, S. Hüber, B.I. Wohlmuth, A comparison of mortar and Nitsche techniques for linear elasticity. Preprint, Universität Stuttgart, 008, 2003.
[35] Papoulia, K. D.; Sam, C. -H.; Vavasis, S. A.: Time continuity in cohesive finite element modeling. Int. J. Numer. methods engrg. 58, 679-701 (2003) · Zbl 1032.74676
[36] Osher, S.; Sethian, J. A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. comput. Phys. 79, No. 1, 12-49 (1988) · Zbl 0659.65132