Abstract
Crack propagation involves the creation of new internal surfaces of a priori unknown paths. A first challenge for modeling and simulation of crack propagation is to identify the location of the crack initiation accurately, a second challenge is to follow the crack paths accurately. Phase-field models address both challenges in an elegant way, as they are able to represent arbitrary crack paths by means of a damage parameter. Moreover, they allow for the representation of complex crack patterns without changing the computational mesh via the damage parameter—which however comes at the cost of larger spatial systems to be solved. Phase-field methods have already been proven to predict complex fracture patterns in two and three dimensional numerical simulations for brittle fracture. In this paper, we consider phase-field models and their numerical simulation for conchoidal fracture. The main characteristic of conchoidal fracture is that the point of crack initiation is typically located inside of the body. We present phase-field approaches for conchoidal fracture for both, the linear-elastic case as well as the case of finite deformations. We moreover present and discuss efficient methods for the numerical simulation of the arising large scale non-linear systems. Here, we propose to use multigrid methods as solution technique, which leads to a solution method of optimal complexity. We demonstrate the accuracy and the robustness of our approach for two and three dimensional examples related to mussel shell like shape and faceted surfaces of fracture and show that our approach can accurately capture the specific details of cracked surfaces, such as the rippled breakages of conchoidal fracture. Moreover, we show that using our approach the arising systems can also be solved efficiently in parallel with excellent scaling behavior.
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References
en.wikipedia.org/wiki/Obsidian/media
Abdollahi A, Arias I (2012) Phase-field modeling of crack propagation in piezoelectric and ferroelectric materials with different electromechanical crack conditions. J Mech Phys Solids 60(12):2100–2126
Ambrosio L, Tortorelli VM (1990) Approximation of functionals depending on jumps by elliptic functionals via \(\varGamma\)-convergence. Commun Pure Appl Math 43:999–1036
Amestoy PR, Duff IS, L’Excellent J-Y, Koster J (2000) Mumps: a general purpose distributed memory sparse solver. In: International workshop on applied parallel computing. Springer, pp 121–130
Balay S, Brown J, Buschelman K, Eijkhout V, Gropp W, Kaushik D, Knepley M, McInnes LC, Smith B, Zhang H (2012) PETSc users manual revision 3.3. Computer Science Division, Argonne National Laboratory, Argonne, IL
Borden MJ, Verhoosel CV, Scott MA, Hughes TJR, Landis CM (2012) A phase-field description of dynamic brittle fracture. Comput Methods Appl Mech Eng 217–220:77–95
Borden MJ, Hughes TJR, Landis CM, Verhoosel CV (2014) A higher-order phase-field model for brittle fracture: formulation and analysis within the isogeometric analysis framework. Comput Methods Appl Mech Eng 273:100–118
Borden MJ, Hughes TJR, Landis CM, Anvari A, Lee IJ (2016) A phase-field formulation for fracture in ductile materials: finite deformation balance law derivation, plastic degradation, and stress triaxiality effects. Comput Methods Appl Mech Eng 312:130–166
Bourdin B (2007) The variational formulation of brittle fracture: numerical implementation and extensions. In: Volume 5 of IUTAM symposium on discretization methods for evolving discontinuities, IUTAM bookseries, chapter 22. Springer, Dordrecht, pp 381–393
Bourdin B, Francfort GA, Marigo J-J (2000) Numerical experiments in revisited brittle fracture. J Mech Phys Solids 45:797–826
Briggs WL, McCormick SF et al (2000) A multigrid tutorial. SIAM, Philadelphia
Francfort GA, Marigo J-J (1998) Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids 46:1319–1342
Gaston D, Newmann C, Hansen G, Lebrun-Grandie D (2009) MOOSE: a parallel computational framework for coupled systems of nonlinear equations. Nucl Eng Des 239:1768–1778
Geist GA, Romine CH (1988) Lu factorization algorithms on distributed-memory multiprocessor architectures. SIAM J Sci Stat Comput 9(4):639–649
Gerasimov T, De Lorenzis L (2016) A line search assisted monolithic approach for phase-field computing of brittle fracture. Comput Methods Appl Mech Eng. doi:10.1016/j.cma.2015.12.017
Guide MU (1998) The mathworks, vol 5. Inc, Natick, p 333
Henry H, Levine H (2004) Dynamic instabilities of fracture under biaxial strain using a phase field model. Phys Rev Lett 93:105505
Hesch C, Gil AJ, Ortigosa R, Dittmann M, Bilgen C, Betsch P, Franke M, Janz A, Weinberg K (2017) A framework for polyconvex large strain phase-field methods to fracture. Comput Methods Appl Mech Eng 317:649–683
Hesch C, Weinberg K (2014) Thermodynamically consistent algorithms for a finite-deformation phase-field approach to fracture. Int J Numer Methods Eng 99:906–924
Johnson KL (1987) Contact mechanics. Cambridge University Press, Cambridge
Karma A, Kessler DA, Levine H (2001) Phase-field model of mode III dynamic fracture. Phys Rev Lett 81:045501
Kuhn C, Müller R (2010) A continuum phase field model for fracture. Eng Fract Mech 77:3625–3634
Mariani S, Perego U (2003) Extended finite element method for quasi-brittle fracture. Int J Numer Methods Eng 58:103–126
Miehe C, Hofacker M, Welschinger F (2010) A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits. Comput Methods Appl Mech Eng 199:2765–2778
Miehe C, Welschinger F, Hofacker M (2010) Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations. Int J Numer Methods Eng 83(10):1273–1311
Nocedal J, Wright SJ (2006) Numerical optimization. Springer, New York
Müller R (2016) A benchmark problem for phase-field models of fracture. Presentation at the annual meeting of SPP 1748: reliable simulation techniques in solid mechanics. Development of non-standard discretisation methods, mechanical and mathematical analysis, Pavia
Ortiz M, Pandolfi A (1999) Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis. Int J Numer Methods Eng 44(9):1267–1282
Pandolfi A, Ortiz M (2012) An eigenerosion approach to brittle fracture. Int J Numer Methods Eng 92:694–714
Roe KL, Siegmund T (2003) An irreversible cohesive zone model for interface fatigue crack growth simulation. Eng Fract Mech 70(2):209–232
Saad Y (2003) Iterative methods for sparse linear systems. SIAM, Philadelphia
Schenk O, Gärtner K (2004) Solving unsymmetric sparse systems of linear equations with pardiso. Future Gener Comput Syst 20(3):475–487
Schmidt B, Leyendecker S (2009) \(\varGamma\)-convergence of variational integrators for constraint systems. J Nonlinear Sci 19:153–177
Sneddon Ian N (1965) The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile. Int J Eng Sci 3:47–57
Sukumar N, Srolovitz DJ, Baker TJ, Prevost J-H (2003) Brittle fracture in polycrystalline microstructures with the extended finite element method. Int J Numer Methods Eng 56:2015–2037
Verhoosel CV, de Borst R (2013) A phase-field model for cohesive fracture. Int J Numer Methods Eng 96:43–62
Wallner H (1939) Linienstrukturen an Bruchflächen. Zeitschrift für Physik 114:368–378
Weinberg K, Dally T, Schuss S, Werner M, Bilgen C (2016) Modeling and numerical simulation of crack growth and damage with a phase field approach. GAMM-Mitt 39:55–77
Weinberg K, Hesch C (2017) A high-order finite deformation phase-field approach to fracture. Contin Mech Thermodyn 29:935–945
Xu X-P, Needlemann A (1994) Numerical simulations of fast crack growth in brittle solids. J Mech Phys Solids 42(9):1397–1434
Zulian P, Kopaničáková A, Schneider T (2016) Utopia: A c++ embedded domain specific language for scientific computing. https://bitbucket.org/zulianp/utopia
Acknowledgements
The authors gratefully acknowledge the support of the Deutsche Forschungsgesellschaft (DFG) under the project “Large-scale simulation of pneumatic and hydraulic fracture with a phase-field approach” as part of the Priority Programme SPP1748 “Reliable Simulation Techniques in Solid Mechanics. Development of Non-standard Discretisation Methods, Mechanical and Mathematical Analysis”.
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This study was funded by the German Research Foundation (DFG) under Grant WE2525-4/1.
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Bilgen, C., Kopaničáková, A., Krause, R. et al. A phase-field approach to conchoidal fracture. Meccanica 53, 1203–1219 (2018). https://doi.org/10.1007/s11012-017-0740-z
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DOI: https://doi.org/10.1007/s11012-017-0740-z