Finite element simulation of pressure-loaded phase-field fractures. (English) Zbl 1390.74169
Summary: A non-standard aspect of phase-field fracture formulations for pressurized cracks is the application of the pressure loading, due to the fact that a direct notion of the fracture surfaces is absent. In this work we study the possibility to apply the pressure loading through a traction boundary condition on a contour of the phase field. Computationally this requires application of a surface-extraction algorithm to obtain a parametrization of the loading boundary. When the phase-field value of the loading contour is chosen adequately, the recovered loading contour resembles that of the sharp fracture problem. The computational scheme used to construct the immersed loading boundary is leveraged to propose a hybrid model. In this hybrid model the solid domain (outside the loading contour) is unaffected by the phase field, while a standard phase-field formulation is used in the fluid domain (inside the loading contour). We present a detailed study and comparison of the \(\varGamma\)-convergence
behavior and mesh convergence behavior of both models using a one-dimensional model problem. The extension of these results to multiple dimensions is also considered.
Keywords:
brittle fracture; phase-field modeling; pressurized cracks; immersed finite element method; finite cell methodSoftware:
ISOGATReferences:
| [1] | Bourdin B, Francfort GA, Marigo J (2008) The variational approach to fracture. J Elast 91:5-148 · Zbl 1176.74018 · doi:10.1007/s10659-007-9107-3 |
| [2] | 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(45-48):2765-2778 · Zbl 1231.74022 · doi:10.1016/j.cma.2010.04.011 |
| [3] | 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 · Zbl 1202.74014 · doi:10.1002/nme.2861 |
| [4] | Karma A (2001) Phase-field formulation for quantitative modeling of alloy solidification. Phys Rev Lett 87(11):115701 · doi:10.1103/PhysRevLett.87.115701 |
| [5] | de Borst R, Verhoosel CV (2016) Gradient damage vs phase-field approaches for fracture: similarities and differences. Comput Methods Appl Mech Eng 312:78-94 · Zbl 1439.74347 · doi:10.1016/j.cma.2016.05.015 |
| [6] | Borden MJ, Verhoosel CV, Scott MA, Hughes TJ, Landis CM (2012) A phase-field description of dynamic brittle fracture. Comput Methods Appl Mech Eng 217-220:77-95 · Zbl 1253.74089 · doi:10.1016/j.cma.2012.01.008 |
| [7] | Hofacker M, Miehe C (2012) Continuum phase field modeling of dynamic fracture: variational principles and staggered FE implementation. Int J Fract 178(1-2):113-129 · doi:10.1007/s10704-012-9753-8 |
| [8] | Schlüter A, Willenbücher A, Kuhn C, Müller R (2014) Phase field approximation of dynamic brittle fracture. Comput Mech 54(5):1141-1161 · Zbl 1311.74106 · doi:10.1007/s00466-014-1045-x |
| [9] | Karma A, Kessler DA, Levine H (2001) Phase-field model of mode III dynamic fracture. Phys Rev Lett 87(4):045501 · doi:10.1103/PhysRevLett.87.045501 |
| [10] | Hildebrand F, Miehe C (2012) A phase field model for the formation and evolution of martensitic laminate microstructure at finite strains. Philos Mag 92(34):4250-4290 · doi:10.1080/14786435.2012.705039 |
| [11] | Miehe C, Schänzel LM (2013) Phase field modeling of fracture in rubbery polymers. Part I: finite elasticity coupled with brittle failure. J Mech Phys Solids 65:93-113 · Zbl 1323.74012 · doi:10.1016/j.jmps.2013.06.007 |
| [12] | Miehe C, Welschinger F, Hofacker M (2010) A phase field model of electromechanical fracture. J Mech Phys Solids 58(10):1716-1740 · Zbl 1200.74123 · doi:10.1016/j.jmps.2010.06.013 |
| [13] | 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 · doi:10.1016/j.jmps.2012.06.014 |
| [14] | Verhoosel CV, de Borst R (2013) A phase-field model for cohesive fracture. Int J Numer Methods Eng 96(1):43-62 · Zbl 1352.74029 · doi:10.1002/nme.4553 |
| [15] | Vignollet J, May S, de Borst R, Verhoosel CV (2014) Phase-field models for brittle and cohesive fracture. Meccanica 49(11):2587-2601 · doi:10.1007/s11012-013-9862-0 |
| [16] | Miehe C, Schänzel LM, Ulmer H (2015) Phase field modeling of fracture in multi-physics problems. Part I. Balance of crack surface and failure criteria for brittle crack propagation in thermo-elastic solids. Comput Methods Appl Mech Eng 294:449-485 · Zbl 1423.74838 · doi:10.1016/j.cma.2014.11.016 |
| [17] | Miehe C, Hofacker M, Schänzel LM, Aldakheel F (2015) Phase field modeling of fracture in multi-physics problems. Part II. Coupled brittle-to-ductile failure criteria and crack propagation in thermo-elastic-plastic solids. Comput Methods Appl Mech Eng 294:486-522 · Zbl 1423.74837 · doi:10.1016/j.cma.2014.11.017 |
| [18] | Chukwudozie C, Bourdin B, Yoshioka K (eds) (2013) A variational approach to the modeling and numerical simulation of hydraulic fracturing under in-situ stresses. In: Thirty-Eighth Workshop on Geothermal Reservoir Engineering, Stanford University, Stanford, California |
| [19] | Miehe C, Mauthe S, Teichtmeister S (2015) Minimization principles for the coupled problem of Darcy-Biot-type fluid transport in porous media linked to phase field modeling of fracture. J Mech Phys Solids 82:186-217 · doi:10.1016/j.jmps.2015.04.006 |
| [20] | Miehe C, Mauthe S (2016) Phase field modeling of fracture in multi-physics problems. Part III. Crack driving forces in hydro-poro-elasticity and hydraulic fracturing of fluid-saturated porous media. Comput Methods Appl Mech Eng 304:619-655 · Zbl 1425.74423 · doi:10.1016/j.cma.2015.09.021 |
| [21] | Wheeler MF, Wick T, Wollner W (2014) An augmented-Lagrangian method for the phase-field approach for pressurized fractures. Comput Methods Appl Mech Eng 271:69-85 · Zbl 1296.65170 · doi:10.1016/j.cma.2013.12.005 |
| [22] | Mikelić A, Wheeler MF, Wick T (2015) Phase-field modeling of a fluid-driven fracture in a poroelastic medium. Comput Geosci 19(6):1171-1195 · Zbl 1390.86010 · doi:10.1007/s10596-015-9532-5 |
| [23] | Wilson ZA, Landis CM (2016) Phase-field modeling of hydraulic fracture. J Mech Phys Solids 96:264-290 · Zbl 1482.74020 · doi:10.1016/j.jmps.2016.07.019 |
| [24] | Verhoosel CV, van Zwieten GJ, van Rietbergen B, de Borst R (2015) Image-based goal-oriented adaptive isogeometric analysis with application to the micro-mechanical modeling of trabecular bone. Comput Methods Appl Mech Eng 284:138-164 · Zbl 1423.74929 · doi:10.1016/j.cma.2014.07.009 |
| [25] | Chambolle A (2004) An approximation result for special functions with bounded deformation. Journal de Mathématiques Pures et Appliquées 83(7):929-954 · Zbl 1084.49038 · doi:10.1016/j.matpur.2004.02.004 |
| [26] | Chambolle A (2005) Addendum to “An approximation result for special functions with bounded deformation” [J Math Pures Appl (9) 83 (7) (2004) 929-954]: the N-dimensional case. Journal de Mathématiques Pures et Appliquées 84(1):137-145 · Zbl 1253.74089 |
| [27] | 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 · Zbl 1296.74098 · doi:10.1016/j.cma.2014.01.016 |
| [28] | van Zwieten GJ, van Brummelen EH, van der Zee KG, Gutiérrez MA, Hanssen RF (2014) Discontinuities without discontinuity: the weakly-enforced slip method. Comput Methods Appl Mech Eng 271:144-166 · Zbl 1296.86008 · doi:10.1016/j.cma.2013.12.004 |
| [29] | Babuška I, Miller A (1984) The post-processing approach in the finite element method—part 1: calculation of displacements, stresses and other higher derivatives of the displacements. Int J Numer Methods Eng 20(6):1085-1109 · Zbl 0535.73052 · doi:10.1002/nme.1620200610 |
| [30] | Sneddon I, Lowengrub M (1969) Crack problems in the classical theory of elasticity. The SIAM series in applied mathematics. Wiley, Hoboken · Zbl 0201.26702 |
| [31] | Vuong AV, Giannelli C, Jüttler B, Simeon B (2011) A hierarchical approach to adaptive local refinement in isogeometric analysis. Comput Methods Appl Mech Eng 200(49-52):3554-3567 · Zbl 1239.65013 · doi:10.1016/j.cma.2011.09.004 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
