A FEniCS implementation of the phase field method for quasi-static brittle fracture

Abstract

In the recent years, the phase field method for simulating fracture problems has received considerable attention. This is due to the salient features of the method: 1) it can be incorporated into any conventional finite element software; 2) has a scalar damage variable is used to represent the discontinuous surface implicitly and 3) the crack initiation and subsequent propagation and branching are treated with less complexity. Within this framework, the linear momentum equations are coupled with the diffusion type equation, which describes the evolution of the damage variable. The coupled nonlinear system of partial differential equations are solved in a ‘staggered’ approach. The present work discusses the implementation of the phase field method for brittle fracture within the open-source finite element software, FEniCS. The FEniCS provides a framework for the automated solutions of the partial differential equations. The details of the implementation which forms the core of the analysis are presented. The implementation is validated by solving a few benchmark problems and comparing the results with the open literature.

Keywords

phase field method / FEniCS / brittle fracture / crack propagation / variational theory of fracture

References

[1] Trevelyan J. Boundary Elements for Engineers: Theory and Applications, vol. 1. Southampton: Computational Mechanics, 1994

[2] Aliabadi M H. A new generation of boundary element methods in fracture mechanics. International Journal of Fracture, 1997, 86(1–2): 91–125

[3] Ooi E T, Song C, Tin-Loi F, Yang Z. Polygon scaled boundary finite elements for crack propagation modelling. International Journal for Numerical Methods in Engineering, 2012, 91(3): 319–342

[4] Rabczuk T, Bordas S, Zi G. A three-dimensional meshfree method for continuous multiple-crack initiation, propagation and junction in statics and dynamics. Computational Mechanics, 2007, 40(3): 473–495

[5] Nguyen V P, Rabczuk T, Bordas S, Duflot M. Meshless methods: a review and computer implementation aspects. Mathematics and Computers in Simulation, 2008, 79(3): 763–813

[6] Dechaumphai P, Phongthanapanich S, Sricharoenchai T. Combined Delaunay triangulation and adaptive finite element method for crack growth analysis. Acta Mechanica Sinica, 2003, 19(2): 162–171

[7] Moës N, Dolbow J, Belytschko T. A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering, 1999, 46(1): 131–150

[8] Wang C, Xu X. An extended phantom node method study of crack propagation of composites under fatigue loading. Composite Structures, 2016, 154: 410–418

[9] Areias P, Msekh M A, Rabczuk T. Damage and fracture algorithm using the screened Poisson equation and local remeshing. Engineering Fracture Mechanics, 2016, 158: 116–143

[10] Aranson I S, Kalatsky V A, Vinokur V M. Continuum field description of crack propagation. Physical Review Letters, 2000, 85(1): 118–121

[11] Marigo J J, Maurini C, Pham K. An overview of the modelling of fracture by gradient damage models. Meccanica, 2016, 51(12): 3107–3128

[12] de Borst R, Verhoosel C V. Gradient damage vs phase-field approaches for fracture: similarities and differences. Computer Methods in Applied Mechanics and Engineering, 2016, 312: 78–94

[13] Bordas S P A, Rabczuk T, Hung N X, Nguyen V P, Natarajan S, Bog T, Quan D M, Hiep N V. Strain smoothing in FEM and XFEM. Computers & Structures, 2010, 88(23‒24): 1419–1443

[14] Nguyen-Xuan H, Liu G R, Nourbakhshnia N, Chen L. A novel singular ES-FEM for crack growth simulation. Engineering Fracture Mechanics, 2012, 84: 41–66

[15] Nguyen-Xuan H, Liu G R, Bordas S, Natarajan S, Rabczuk T.An adaptive singular ES-FEM for mechanics problems with singular field of arbitrary order. Computer Methods Applied Mechanics and Engineering, 2013, 253: 252–273

[16] Natarajan S, Bordas S P A, Ooi E T. Virtual and smoothed finite elements: a connection and its application to polygonal/polyhedral finite element methods. International Journal for Numerical Methods in Engineering, 2015, 104(13): 1173–1199

[17] Francis A, Ortiz-Bernardin A, Bordas S P A, Natarajan S. Linear smoothed polygonal and polyhedral finite elements. International Journal for Numerical Methods in Engineering, 2017, 109(9): 1263–1288

[18] Surendran M, Natarajan S, Bordas S P A, Palani G S. Linear smoothed extended finite element method. International Journal for Numerical Methods in Engineering, 2017, 112(12): 1733–1749

[19] Abaqus. Abaqus documentation. Dassault Systmes Simulia Corp. , Provid. RI, USA, 2012

[20] Giner E, Sukumar N, Tarancón J E, Fuenmayor F J. An Abaqus implementation of the extended finite element method. Engineering Fracture Mechanics, 2009, 76(3): 347–368

[21] Shi J, Chopp D, Lua J, Sukumar N, Belytschko T. Abaqus implementation of extended finite element method using a level set representation for three-dimensional fatigue crack growth and life predictions. Engineering Fracture Mechanics, 2010, 77(14): 2840–2863

[22] Rabczuk T, Zi G, Bordas S, Nguyen-Xuan H. A geometrically non-linear three-dimensional cohesive crack method for reinforced concrete structures. Engineering Fracture Mechanics, 2008, 75(16): 4740–4758

[23] Bourdin B, Francfort G A, Marigo J J. The Variational Approach to Fracture. Springer Netherlands, 2008

[24] Mesgarnejad A, Bourdin B, Khonsari M M. Validation simulations for the variational approach to fracture. Computer Methods in Applied Mechanics and Engineering, 2015, 290: 420–437

[25] Areias P, Rabczuk T, Msekh M A. Phase-field analysis of finite-strain plates and shells including element subdivision. Computer Methods in Applied Mechanics and Engineering, 2016, 312: 322–350

[26] Bourdin B, Francfort G A, Marigo J J. Numerical experiments in revisited brittle fracture. Journal of the Mechanics and Physics of Solids, 2000, 48(4): 797–826

[27] Cahn J W, Hilliard J E. Free energy of a nonuniform system. I. Interfacial free energy. Journal of Chemical Physics, 1958, 28(2): 258–267

[28] Chen L Q, Khachaturyan A G. Computer simulation of structural transformations during precipitation of an ordered intermetallic phase. Acta Metallurgica et Materialia, 1991, 39(11): 2533–2551

[29] Galenko P K, Herlach D M, Funke O, Gandham P. Phase-field modeling of dendritic solidification: verification for the model predictions with latest experimental data. In: Herlach D M, ed. Solidification and Crystallization. Wiley-VCH, 2005, 52–60

[30] Moelans N, Blanpain B, Wollants P. An introduction to phase-field modeling of microstructure evolution. Calphad, 2008, 32(2): 268–294

[31] Francfort G A, Marigo J J. Revisiting brittle fracture as an energy minimization problem. Journal of the Mechanics and Physics of Solids, 1998, 46(8): 1319–1342

[32] Borden M J, Verhoosel C V, Scott M A, Hughes T J R, Landis C M. A phase-field description of dynamic brittle fracture. Computer Methods in Applied Mechanics and Engineering, 2012, 217: 77–95

[33] Schlüter A, Willenbücher A, Kuhn C, Müller R. Phase field approximation of dynamic brittle fracture. Computational Mechanics, 2014, 54(5): 1141–1161

[34] Raina A, Miehe C. A phase-field model for fracture in biological tissues. Biomechanics and Modeling in Mechanobiology, 2016, 15(3): 479–496

[35] Miehe C, Aldakheel F, Raina A. Phase field modeling of ductile fracture at finite strains: a variational gradient-extended plasticity-damage theory. International Journal of Plasticity, 2016, 84: 1–32

[36] Ambati M, Gerasimov T, De Lorenzis L. Phase-field modeling of ductile fracture. Computational Mechanics, 2015, 55(5): 1017–1040

[37] Ambati M, Kruse R, De Lorenzis L. A phase-field model for ductile fracture at finite strains and its experimental verification. Computational Mechanics, 2016, 57(1): 149–167

[38] Kuhn C, Müller R. Phase field simulation of thermomechanical fracture. Proceedings in Applied Mathematics and Mechanics, 2009, 9(1): 191–192

[39] Schlüter A, Kuhn C, Müller R, Tomut M, Trautmann C, Weick H, Plate C. Phase field modelling of dynamic thermal fracture in the context of irradiation damage. Continuum Mechanics and Thermodynamics, 2017, 29(4): 977–988

[40] Msekh M A, Sargado J M, Jamshidian M, Areias P M, Rabczuk T. Abaqus implementation of phase-field model for brittle fracture. Computational Materials Science, 2015, 96: 472–484

[41] Liu G, Li Q, Msekh M A, Zuo Z. Abaqus implementation of monolithic and staggered schemes for quasi-static and dynamic fracture phase-field mode. Computational Materials Science, 2016, 121: 35–47

[42] Molnár G, Gravouil A. 2D and 3D Abaqus implementation of a robust staggered phase-field solution for modeling brittle fracture. Finite Elements in Analysis and Design, 2017, 130: 27–38

[43] Nguyen T T, Yvonnet J, Bornert M, Chateau C, Sab K, Romani R, Le Roy R. On the choice of parameters in the phase field method for simulating crack initiation with experimental validation. International Journal of Fracture, 2016, 197(2): 213–226

[44] Ambati M, Gerasimov T, De Lorenzis L. A review on phase-field models of brittle fracture and a new fast hybrid formulation. Computational Mechanics, 2015, 55(2): 383–405

[45] Alnæs M S, Blechta J, Hake H, Hoansson A, Kehlet B, Logg A, Richardson C, Ring J, Rognes M E, Wells G N. The FEniCS Project Version 1.5. Archive of Numerical Software, 2015, 3(100): 9–23

[46] Logg A, Mardal K A, Wells G N. Automated Solution of Differential Equations by the Finite Element Method. Berlin: Springer, 2012

[47] Miehe C, Welschinger F, Hofacker M. Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations. International Journal for Numerical Methods in Engineering, 2010, 83(10): 1273–1311

[48] Amor H, Marigo J J, Maurini C. Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments. Journal of the Mechanics and Physics of Solids, 2009, 57(8): 1209–1229

[49] Moës N, Stolz C, Bernard P E, Chevaugeon N. A level set based model for damage growth: the thick level set approach. International Journal for Numerical Methods in Engineering, 2011, 86(3): 358–380

[50] Miehe C, Hofacker M, Welschinger F. A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits. Computer Methods in Applied Mechanics and Engineering, 2010, 199(45): 2765–2778

[51] Prasad N N V, Aliabadi M H, Rooke D P. Incremental crack growth in thermoelastic problems. International Journal of Fracture, 1994, 66(3): R45–R50

[52] Duflot M. The extended finite element method in thermoelastic fracture mechanics. International Journal for Numerical Methods in Engineering, 2008, 74(5): 827–847

[53] Bouchard P O, Bay F, Chastel Y. Numerical modelling of crack propagation: automatic remeshing and comparison of different criteria. Computer Methods in Applied Mechanics and Engineering, 2003, 192(35–36): 3887–3908

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