Advertisement

Meccanica

, Volume 53, Issue 6, pp 1203–1219 | Cite as

A phase-field approach to conchoidal fracture

  • Carola Bilgen
  • Alena Kopaničáková
  • Rolf Krause
  • Kerstin Weinberg
Novel Computational Approaches to Old and New Problems in Mechanics

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.

Keywords

Phase field Multigrid method Brittle fracture Crack initiation Conchoidal fracture 

Notes

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”.

Funding

This study was funded by the German Research Foundation (DFG) under Grant WE2525-4/1.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    en.wikipedia.org/wiki/Obsidian/mediaGoogle Scholar
  2. 2.
    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):2100–2126ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Ambrosio L, Tortorelli VM (1990) Approximation of functionals depending on jumps by elliptic functionals via \(\varGamma\)-convergence. Commun Pure Appl Math 43:999–1036MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    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 121–130Google Scholar
  5. 5.
    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, ILGoogle Scholar
  6. 6.
    Borden MJ, Verhoosel CV, Scott MA, Hughes TJR, Landis CM (2012) A phase-field description of dynamic brittle fracture. Comput Methods Appl Mech Eng 217–220:77–95MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    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 273:100–118ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    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 312:130–166ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    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 381–393Google Scholar
  10. 10.
    Bourdin B, Francfort GA, Marigo J-J (2000) Numerical experiments in revisited brittle fracture. J Mech Phys Solids 45:797–826ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Briggs WL, McCormick SF et al (2000) A multigrid tutorial. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
  12. 12.
    Francfort GA, Marigo J-J (1998) Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids 46:1319–1342ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gaston D, Newmann C, Hansen G, Lebrun-Grandie D (2009) MOOSE: a parallel computational framework for coupled systems of nonlinear equations. Nucl Eng Des 239:1768–1778CrossRefGoogle Scholar
  14. 14.
    Geist GA, Romine CH (1988) Lu factorization algorithms on distributed-memory multiprocessor architectures. SIAM J Sci Stat Comput 9(4):639–649MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    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 MathSciNetGoogle Scholar
  16. 16.
    Guide MU (1998) The mathworks, vol 5. Inc, Natick, p 333Google Scholar
  17. 17.
    Henry H, Levine H (2004) Dynamic instabilities of fracture under biaxial strain using a phase field model. Phys Rev Lett 93:105505ADSCrossRefGoogle Scholar
  18. 18.
    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 317:649–683ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Hesch C, Weinberg K (2014) Thermodynamically consistent algorithms for a finite-deformation phase-field approach to fracture. Int J Numer Methods Eng 99:906–924MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Johnson KL (1987) Contact mechanics. Cambridge University Press, CambridgezbMATHGoogle Scholar
  21. 21.
    Karma A, Kessler DA, Levine H (2001) Phase-field model of mode III dynamic fracture. Phys Rev Lett 81:045501ADSCrossRefGoogle Scholar
  22. 22.
    Kuhn C, Müller R (2010) A continuum phase field model for fracture. Eng Fract Mech 77:3625–3634CrossRefGoogle Scholar
  23. 23.
    Mariani S, Perego U (2003) Extended finite element method for quasi-brittle fracture. Int J Numer Methods Eng 58:103–126MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    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 199:2765–2778ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    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):1273–1311MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Nocedal J, Wright SJ (2006) Numerical optimization. Springer, New YorkzbMATHGoogle Scholar
  27. 27.
    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, PaviaGoogle Scholar
  28. 28.
    Ortiz M, Pandolfi A (1999) Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis. Int J Numer Methods Eng 44(9):1267–1282CrossRefzbMATHGoogle Scholar
  29. 29.
    Pandolfi A, Ortiz M (2012) An eigenerosion approach to brittle fracture. Int J Numer Methods Eng 92:694–714MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Roe KL, Siegmund T (2003) An irreversible cohesive zone model for interface fatigue crack growth simulation. Eng Fract Mech 70(2):209–232CrossRefGoogle Scholar
  31. 31.
    Saad Y (2003) Iterative methods for sparse linear systems. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
  32. 32.
    Schenk O, Gärtner K (2004) Solving unsymmetric sparse systems of linear equations with pardiso. Future Gener Comput Syst 20(3):475–487CrossRefzbMATHGoogle Scholar
  33. 33.
    Schmidt B, Leyendecker S (2009) \(\varGamma\)-convergence of variational integrators for constraint systems. J Nonlinear Sci 19:153–177ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    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 3:47–57MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    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 56:2015–2037CrossRefzbMATHGoogle Scholar
  36. 36.
    Verhoosel CV, de Borst R (2013) A phase-field model for cohesive fracture. Int J Numer Methods Eng 96:43–62MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Wallner H (1939) Linienstrukturen an Bruchflächen. Zeitschrift für Physik 114:368–378ADSCrossRefGoogle Scholar
  38. 38.
    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 39:55–77MathSciNetCrossRefGoogle Scholar
  39. 39.
    Weinberg K, Hesch C (2017) A high-order finite deformation phase-field approach to fracture. Contin Mech Thermodyn 29:935–945ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Xu X-P, Needlemann A (1994) Numerical simulations of fast crack growth in brittle solids. J Mech Phys Solids 42(9):1397–1434ADSCrossRefzbMATHGoogle Scholar
  41. 41.
    Zulian P, Kopaničáková A, Schneider T (2016) Utopia: A c++ embedded domain specific language for scientific computing. https://bitbucket.org/zulianp/utopia

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  • Carola Bilgen
    • 1
  • Alena Kopaničáková
    • 2
  • Rolf Krause
    • 2
  • Kerstin Weinberg
    • 1
  1. 1.Lehrstuhl für Festkörpermechanik, Department MaschinenbauUniversität SiegenSiegenGermany
  2. 2.Chair for Advanced Scientific Computing, Institute of Computational ScienceUSI - Università della Svizzera italianaLuganoSwitzerland

Personalised recommendations