zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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 phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits. (English) Zbl 1231.74022
Summary: The computational modeling of failure mechanisms in solids due to fracture based on $sharp$ crack discontinuities suffers in situations with complex crack topologies. This can be overcome by a {\it diffusive} crack modeling based on the introduction of a crack phase field. Following our recent work [Int. J. Numer. Methods Eng. 83, No. 10, 1273--1311 (2010; Zbl 1202.74014)] on phase-field-type fracture, we propose in this paper a new variational framework for rate-independent diffusive fracture that bases on the introduction of a local history field. It contains a maximum reference energy obtained in the deformation history, which may be considered as a measure for the maximum tensile strain obtained in history. It is shown that this local variable drives the evolution of the crack phase field. The introduction of the history field provides a very transparent representation of the balance equation that governs the diffusive crack topology. In particular, it allows for the construction of a new algorithmic treatment of diffusive fracture. Here, we propose an extremely robust operator split scheme that successively updates in a typical time step the history field, the crack phase field and finally the displacement field. A regularization based on a viscous crack resistance that even enhances the robustness of the algorithm may easily be added. The proposed algorithm is considered to be the canonically simple scheme for the treatment of diffusive fracture in elastic solids. We demonstrate the performance of the phase field formulation of fracture by means of representative numerical examples.

74A45Theories of fracture and damage
74S05Finite element methods in solid mechanics
Full Text: DOI
[1] C. Miehe, F. Welschinger, M. Hofacker, Thermodynamically-consistent phase field models of fracture: Variational principles and multi-field fe implementations, International Journal for Numerical Methods in Engineering DOI: 10.1002/nme.2861. · Zbl 1202.74014
[2] Griffith, A. A.: The phenomena of rupture and flow in solids, Philosophical transactions of the royal society London A 221, 163-198 (1921)
[3] Irwin, G. R.: Fracture, Encyclopedia of physics 6, 551-590 (1958)
[4] Francfort, G. A.; Marigo, J. J.: Revisiting brittle fracture as an energy minimization problem, Journal of the mechanics and physics of solids 46, 1319-1342 (1998) · Zbl 0966.74060 · doi:10.1016/S0022-5096(98)00034-9
[5] Bourdin, B.; Francfort, G. A.; Marigo, J. J.: The variational approach to fracture, (2008) · Zbl 1176.74018
[6] Dal Maso, G.; Toader, R.: A model for the quasistatic growth of brittle fractures: existence and approximation results, Archive for rational mechanics and analysis 162, 101-135 (2002) · Zbl 1042.74002 · doi:10.1007/s002050100187
[7] Buliga, M.: Energy minimizing brittle crack propagation, Journal of elasticity 52, 201-238 (1999) · Zbl 0947.74055 · doi:10.1023/A:1007545213010
[8] Bourdin, B.; Francfort, G. A.; Marigo, J. J.: Numerical experiments in revisited brittle fracture, Journal of the mechanics and physics of solids 48, 797-826 (2000) · Zbl 0995.74057 · doi:10.1016/S0022-5096(99)00028-9
[9] Mumford, D.; Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems, Communications on pure and applied mathematics 42, 577-685 (1989) · Zbl 0691.49036 · doi:10.1002/cpa.3160420503
[10] Ambrosio, L.; Tortorelli, V. M.: Approximation of functionals depending on jumps by elliptic functionals via ${\gamma}$-convergence, Communications on pure and applied mathematics 43, 999-1036 (1990) · Zbl 0722.49020 · doi:10.1002/cpa.3160430805
[11] Dal Maso, G.: An introduction to ${\Gamma}$-convergence, (1993) · Zbl 0816.49001
[12] Braides, D. P.: Approximation of free discontinuity problems, (1998) · Zbl 0909.49001
[13] Braides, D. P.: {$\Gamma$}-convergence for beginners, (2002) · Zbl 1198.49001
[14] Hakim, V.; Karma, A.: Laws of crack motion and phase-field models of fracture, Journal of the mechanics and physics of solids 57, 342-368 (2009) · Zbl 05600179
[15] Karma, A.; Kessler, D. A.; Levine, H.: Phase-field model of mode iii dynamic fracture, Physical review letters 92 (2001)
[16] Eastgate, L. O.; Sethna, J. P.; Rauscher, M.; Cretegny, T.; Chen, C. -S.; Myers, C. R.: Fracture in mode i using a conserved phase-field model, Physical review E 65 (2002)
[17] Belytschko, T.; Chen, H.; Xu, J.; Zi, G.: Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment, International journal for numerical methods in engineering 58, 1873-1905 (2003) · Zbl 1032.74662 · doi:10.1002/nme.941
[18] Song, J. -H.; Belytschko, T.: Cracking node method for dynamic fracture with finite elements, International journal for numerical methods in engineering 77, 360-385 (2009) · Zbl 1155.74415 · doi:10.1002/nme.2415
[19] Gürses, E.; Miehe, C.: A computational framework of three-dimensional configurational-force-driven brittle crack propagation, Computer methods in applied mechanics and engineering 198, 1413-1428 (2009) · Zbl 1227.74070 · doi:10.1016/j.cma.2008.12.028
[20] Miehe, C.; Gürses, E.: A robust algorithm for configurational-force-driven brittle crack propagation with r-adaptive mesh alignment, International journal for numerical methods in engineering 72, 127-155 (2007) · Zbl 1194.74444 · doi:10.1002/nme.1999
[21] Miehe, C.; Gürses, E.; Birkle, M.: A computational framework of configurational-force-driven brittle fracture based on incremental energy minimization, International journal of fracture 145, 245-259 (2007) · Zbl 1198.74008 · doi:10.1007/s10704-007-9078-1
[22] Capriz, G.: Continua with microstructure, (1989) · Zbl 0676.73001
[23] Mariano, P. M.: Multifield theories in mechanics of solids, Advances in applied mechanics 38, 1-93 (2001)
[24] Frémond, M.: Non-smooth thermomechanics, (2002) · Zbl 0990.80001
[25] Miehe, C.: Comparison of two algorithms for the computation of fourth-order isotropic tensor functions, Computers & structures 66, 37-43 (1998) · Zbl 0929.74128 · doi:10.1016/S0045-7949(97)00073-4
[26] Miehe, C.; Lambrecht, M.: Algorithms for computation of stresses and elasticity moduli in terms of seth-Hill’s family of generalized strain tensors, Communications in numerical methods in engineering 17, 337-353 (2001) · Zbl 1049.74011 · doi:10.1002/cnm.404
[27] Bittencourt, T. N.; Wawrzynek, P. A.; Ingraffea, A. R.; Sousa, J. L.: Quasi-automatic simulation of crack propagation for 2d lefm problems, Engineering fracture mechanics 55, 321-334 (1996)
[28] Miehe, C.: Discontinuous and continuous damage evolution in ogden-type large-strain elastic materials, European journal of mechanics A / solids 14, 697-720 (1995) · Zbl 0837.73054