# zbMATH — the first resource for mathematics

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 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.

##### MSC:
 74A45 Theories of fracture and damage 74S05 Finite element methods applied to problems in solid mechanics
Full Text:
##### References:
 [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, (), 551-590 [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 [5] Bourdin, B.; Francfort, G.A.; Marigo, J.J., The variational approach to fracture, (2008), Springer Verlag Berlin · 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 [7] Buliga, M., Energy minimizing brittle crack propagation, Journal of elasticity, 52, 201-238, (1999) · Zbl 0947.74055 [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 [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 [10] Ambrosio, L.; Tortorelli, V.M., Approximation of functionals depending on jumps by elliptic functionals via γ-convergence, Communications on pure and applied mathematics, 43, 999-1036, (1990) · Zbl 0722.49020 [11] Dal Maso, G., An introduction to γ-convergence, (1993), Birkhäuser Boston · Zbl 0816.49001 [12] Braides, D.P., Approximation of free discontinuity problems, (1998), Springer Verlag Berlin · Zbl 0909.49001 [13] Braides, D.P., Γ-convergence for beginners, (2002), Oxford University Press New York · 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 1421.74089 [15] Karma, A.; Kessler, D.A.; Levine, H., Phase-field model of mode iii dynamic fracture, Physical review letters, 92, (2001), 8704.045501 [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), 036117-1-10 [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 [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 [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 [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 [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 [22] Capriz, G., Continua with microstructure, (1989), Springer Verlag · 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), Springer Verlag · 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 [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 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.