A micromorphic phase-field model for brittle and quasi-brittle fracture. (English) Zbl 07850336
Summary: In this manuscript, a robust and variationally consistent technique is proposed for local treatment of the phase-field fracture irreversibility. This technique involves an extension of the phase-field fracture energy functional through a micromorphic approach. Consequently, the phase-field is transformed into a local variable, while a micromorphic variable regularizes the problem. The local nature of the phase-field variable enables an easier implementation of its irreversibility using a pointwise ‘max’ with system level precision. Unlike the popular history variable approach, which also enforces local fracture irreversibility, the micromorphic approach yields a variationally consistent framework. The efficacy of the micromorphic approach in phase-field fracture modelling is demonstrated in this work with numerical experiments on benchmark brittle and quasi-brittle fracture problems in linear elastic media. Furthermore, the extensibility of the micromorphic phase-field fracture model towards multiphysics problems is demonstrated. To that end, a theoretical extension is carried out for modelling hydraulic fracture, and relevant numerical experiments exhibiting crack merging are presented. The source code as well as the data set accompanying this work would be made available on GitHub (https://github.com/ritukeshbharali/falcon).
MSC:
| 74R10 | Brittle fracture |
| 74B05 | Classical linear elasticity |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
| 74S05 | Finite element methods applied to problems in solid mechanics |
Keywords:
energy functional; Euler-Lagrange equations; fracture irreversibility; linear elastic medium; porous medium; hydraulic fracture; finite element methodReferences:
| [1] | Francfort, GA; Marigo, J-J, Revisiting brittle fracture as an energy minimization problem, J Mech Phys Solids, 46, 8, 1319-1342, 1998 · Zbl 0966.74060 · doi:10.1016/S0022-5096(98)00034-9 |
| [2] | Bourdin, B.; Francfort, GA; Marigo, J-J, Numerical experiments in revisited brittle fracture, J Mech Phys Solids, 48, 4, 797-826, 2000 · Zbl 0995.74057 · doi:10.1016/S0022-5096(99)00028-9 |
| [3] | Bourdin, B., Numerical implementation of the variational formulation for quasi-static brittle fracture, Interfaces Free Bound, 9, 411-430, 2007 · Zbl 1130.74040 · doi:10.4171/IFB/171 |
| [4] | Sukumar, N., Extended finite element method for three-dimensional crack modelling, Int J Numer Methods Eng, 48, 11, 1549-1570, 2000 · Zbl 0963.74067 · doi:10.1002/1097-0207(20000820)48:11<1549::AID-NME955>3.0.CO;2-A |
| [5] | Moës, N.; Belytschko, T., Extended finite element method for cohesive crack growth, Eng Fract Mech, 69, 7, 813-833, 2002 · doi:10.1016/S0013-7944(01)00128-X |
| [6] | Dugdale, DS, Yielding of steel sheets containing slits, J Mech Phys Solids, 8, 2, 100-104, 1960 · doi:10.1016/0022-5096(60)90013-2 |
| [7] | Barenblatt, GI; Dryden, HL, The mathematical theory of equilibrium cracks in brittle fracture, Advances in applied mechanics, 55-129, 1962, Amsterdam: Elsevier, Amsterdam · doi:10.1016/S0065-2156(08)70121-2 |
| [8] | Elices, MGGV, The cohesive zone model: advantages, limitations and challenges, Eng Fract Mech, 69, 2, 137-163, 2002 · doi:10.1016/S0013-7944(01)00083-2 |
| [9] | Miehe, C.; Welschinger, F.; Hofacker, M., Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations, Int J Numer Methods Eng, 83, 10, 1273-1311, 2010 · Zbl 1202.74014 · doi:10.1002/nme.2861 |
| [10] | Miehe, C., 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, 2015 · Zbl 1423.74837 · doi:10.1016/j.cma.2014.11.017 |
| [11] | Ambati, M.; Gerasimov, T.; De Lorenzis, L., Phase-field modeling of ductile fracture, Comput Mech, 55, 5, 1017-1040, 2015 · Zbl 1329.74018 · doi:10.1007/s00466-015-1151-4 |
| [12] | Teichtmeister, S., Phase field modeling of fracture in anisotropic brittle solids, Int J Non-Linear Mech, 97, 1-21, 2017 · doi:10.1016/j.ijnonlinmec.2017.06.018 |
| [13] | Bleyer, J.; Alessi, R., Phase-field modeling of anisotropic brittle fracture including several damage mechanisms, Comput Methods Appl Mech Eng, 336, 213-236, 2018 · Zbl 1440.74354 · doi:10.1016/j.cma.2018.03.012 |
| [14] | Wilson, ZA; Landis, CM, Phase-field modeling of hydraulic fracture, J Mech Phys Solids, 96, 264-290, 2016 · Zbl 1482.74020 · doi:10.1016/j.jmps.2016.07.019 |
| [15] | Heider, Y.; Markert, B., A phase-field modeling approach of hydraulic fracture in saturated porous media, Mech Res Commun, 80, 38-46, 2017 · doi:10.1016/j.mechrescom.2016.07.002 |
| [16] | Cajuhi, T.; Sanavia, L.; De Lorenzis, L., Phase-field modeling of fracture in variably saturated porous media, Comput Mech, 61, 3, 299-318, 2018 · Zbl 1458.74125 · doi:10.1007/s00466-017-1459-3 |
| [17] | Hu, T.; Guilleminot, J.; Dolbow, JE, A phase-field model of fracture with frictionless contact and random fracture properties: application to thin-film fracture and soil desiccation, Comput Methods Appl Mech Eng, 368, 2020 · Zbl 1506.74354 · doi:10.1016/j.cma.2020.113106 |
| [18] | Martínez-Pañeda, E.; Golahmar, A.; Niordson, CF, A phase field formulation for hydrogen assisted cracking, Comput Methods Appl Mech Eng, 342, 742-761, 2018 · Zbl 1440.82005 · doi:10.1016/j.cma.2018.07.021 |
| [19] | Kristensen, PK; Niordson, CF; Martínez-Pañeda, E., A phase field model for elastic-gradient-plastic solids undergoing hydrogen embrittlement, J Mech Phys Solids, 143, 2020 · doi:10.1016/j.jmps.2020.104093 |
| [20] | Mesgarnejad, A.; Bourdin, B.; Khonsari, MM, A variational approach to the fracture of brittle thin films subject to out-of-plane loading, J Mech Phys Solids, 61, 11, 2360-2379, 2013 · doi:10.1016/j.jmps.2013.05.001 |
| [21] | Jian-Ying, Wu, A unified phase-field theory for the mechanics of damage and quasi-brittle failure, J Mech Phys Solids, 103, 72-99, 2017 · doi:10.1016/j.jmps.2017.03.015 |
| [22] | Feng, D-C; Wu, J-Y, Phase-field regularized cohesive zone model (CZM) and size effect of concrete, Eng Fract Mech, 197, 66-79, 2018 · doi:10.1016/j.engfracmech.2018.04.038 |
| [23] | Wu, J-Y; Mandal, TK; Nguyen, VP, A phase-field regularized cohesive zone model for hydrogen assisted cracking, Comput Methods Appl Mech Eng, 358, 112614, 2020 · Zbl 1441.74219 · doi:10.1016/j.cma.2019.112614 |
| [24] | Wu, J-Y; Chen, W-X, Phase-field modeling of electromechanical fracture in piezoelectric solids: analytical results and numerical simulations, Comput Methods Appl Mech Eng, 387, 114125, 2021 · Zbl 1507.74428 · doi:10.1016/j.cma.2021.114125 |
| [25] | Mandal, TK, Fracture of thermo-elastic solids: phase-field modeling and new results with an efficient monolithic solver, Comput Methods Appl Mech Eng, 376, 2021 · Zbl 1506.74362 · doi:10.1016/j.cma.2020.113648 |
| [26] | Patil, RU; Mishra, BK; Singh, IV, A multiscale framework based on phase field method and XFEM to simulate fracture in highly heterogeneous materials, Theor Appl Fract Mech, 100, 390-415, 2019 · doi:10.1016/j.tafmec.2019.02.002 |
| [27] | Gerasimov, T., A non-intrusive global/local approach applied to phase-field modeling of brittle fracture, Adv Model Simul Eng Sci, 5, 1, 1-30, 2018 · doi:10.1186/s40323-018-0105-8 |
| [28] | Nguyen, LH; Schillinger, D., The multiscale finite element method for nonlinear continuum localization problems at full fine-scale fidelity, illustrated through phase-field fracture and plasticity, J Comput Phys, 396, 129-160, 2019 · Zbl 1452.74110 · doi:10.1016/j.jcp.2019.06.058 |
| [29] | Triantafyllou, SP; Kakouris, EG, A generalized phase field multiscale finite element method for brittle fracture, Int J Numer Methods Eng, 121, 9, 1915-1945, 2020 · Zbl 07843276 · doi:10.1002/nme.6293 |
| [30] | He, B.; Schuler, L.; Newell, P., A numerical-homogenization based phase-field fracture modeling of linear elastic heterogeneous porous media, Comput Mater Sci, 176, 2020 · doi:10.1016/j.commatsci.2020.109519 |
| [31] | Bharali, R.; Larsson, F.; Jänicke, R., Computational homogenisation of phase-field fracture, Eur J Mech A Solids, 88, 2021 · Zbl 1475.74105 · doi:10.1016/j.euromechsol.2021.104247 |
| [32] | Gerasimov, T.; De Lorenzis, L., A line search assisted monolithic approach for phase-field computing of brittle fracture, Comput Methods Appl Mech Eng, 312, 276-303, 2016 · Zbl 1439.74349 · doi:10.1016/j.cma.2015.12.017 |
| [33] | Kopaničáková, A.; Krause, R., A recursive multilevel trust region method with application to fully monolithic phase-field models of brittle fracture, Comput Methods Appl Mech Eng, 360, 112720, 2020 · Zbl 1441.74208 · doi:10.1016/j.cma.2019.112720 |
| [34] | Wick, T., Modified Newton methods for solving fully monolithic phase-field quasi-static brittle fracture propagation, Comput Methods Appl Mech Eng, 325, 577-611, 2017 · Zbl 1439.74375 · doi:10.1016/j.cma.2017.07.026 |
| [35] | Vignollet, J., Phase-field models for brittle and cohesive fracture, Meccanica, 49, 11, 2587-2601, 2014 · doi:10.1007/s11012-013-9862-0 |
| [36] | May, S.; Vignollet, J.; De Borst, R., A numerical assessment of phase-field models for brittle and cohesive fracture: r-convergence and stress oscillations, Eur J Mech A Solids, 52, 72-84, 2015 · Zbl 1406.74599 · doi:10.1016/j.euromechsol.2015.02.002 |
| [37] | Singh, N., A fracture-controlled path-following technique for phase-field modeling of brittle fracture, Finite Elem Anal Des, 113, 14-29, 2016 · doi:10.1016/j.finel.2015.12.005 |
| [38] | Bharali, R., A robust monolithic solver for phase-field fracture integrated with fracture energy based arc-length method and under-relaxation, Comput Methods Appl Mech Eng, 394, 2022 · Zbl 1507.74374 · doi:10.1016/j.cma.2022.114927 |
| [39] | Wu, J-Y; Huang, Y.; Nguyen, VP, On the BFGS monolithic algorithm for the unified phase field damage theory, Comput Methods Appl Mech Eng, 360, 112704, 2020 · Zbl 1441.74196 · doi:10.1016/j.cma.2019.112704 |
| [40] | Kristensen, PK; Martínez-Pañeda, E., Phase field fracture modelling using quasi-Newton methods and a new adaptive step scheme, Theor Appl Fract Mech, 107, 2020 · doi:10.1016/j.tafmec.2019.102446 |
| [41] | Heister, T.; Wheeler, MF; Wick, T., A primal-dual active set method and predictor-corrector mesh adaptivity for computing fracture propagation using a phase- field approach, Comput Methods Appl Mech Eng, 2015 · Zbl 1423.76239 · doi:10.1016/j.cma.2015.03.009 |
| [42] | De Lorenzis L, Gerasimov T (2020) Numerical implementation of phase-field models of brittle fracture. In: Modeling in engineering using innovative numerical methods for solids and fluids. Springer, pp 75-101 · Zbl 1481.74694 |
| [43] | Gerasimov, T.; De Lorenzis, L., On penalization in variational phase-field models of brittle fracture, Comput Methods Appl Mech Eng, 354, 990-1026, 2019 · Zbl 1441.74203 · doi:10.1016/j.cma.2019.05.038 |
| [44] | Wick, T., An error-oriented Newton/inexact augmented Lagrangian approach for fully monolithic phase-field fracture propagation, SIAM J Sci Comput, 39, 4, B589-B617, 2017 · Zbl 1403.74131 · doi:10.1137/16m1063873 |
| [45] | Miehe, C.; Hofacker, M.; Welschinger, F., 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, 2010 · Zbl 1231.74022 · doi:10.1016/j.cma.2010.04.011 |
| [46] | Samuel, Forest, Micromorphic approach for gradient elasticity, viscoplasticity, and damage, J Eng Mech, 135, 3, 117-131, 2009 |
| [47] | Forest, S.; Schröder, J.; Hackl, K., Micromorphic approach to crystal plasticity and phase transformation, Plasticity and beyond: microstructures, crystal-plasticity and phase transitions, 131-198, 2014, Vienna: Springer, Vienna · Zbl 1291.74046 · doi:10.1007/978-3-7091-1625-8_3 |
| [48] | Aslan, O., Micromorphic approach to single crystal plasticity and damage, Int J Eng Scie, 49, 12, 1311-1325, 2011 · Zbl 1423.74195 · doi:10.1016/j.ijengsci.2011.03.008 |
| [49] | Lindroos, M., Micromorphic crystal plasticity approach to damage regularization and size effects in martensitic steels, Int J Plast, 151, 2022 · doi:10.1016/j.ijplas.2021.103187 |
| [50] | Grammenoudis, P.; Tsakmakis, C.; Hofer, D., Micromorphic continuum. Part III: small deformation plasticity coupled with damage, Int J Non-Linear Mech, 45, 2, 140-148, 2010 · doi:10.1016/j.ijnonlinmec.2009.10.003 |
| [51] | Grammenoudis, P.; Tsakmakis, C.; Hofer, D., Micromorphic continuum. Part II: finite deformation plasticity coupled with damage, Int J Non-Linear Mech, 44, 9, 957-974, 2009 · doi:10.1016/j.ijnonlinmec.2009.05.004 |
| [52] | Miehe, C.; Teichtmeister, S.; Aldakheel, F., Phase-field modelling of ductile fracture: a variational gradient-extended plasticity-damage theory and its micromorphic regularization, Philos Trans R Soc A Math Phys Eng Sci, 374, 2066, 20150170, 2016 · Zbl 1353.74065 · doi:10.1098/rsta.2015.0170 |
| [53] | Gerke, HH; van Genuchten, MT, A dual-porosity model for simulating the preferential movement of water and solutes in structured porous media, Water Resour Res, 29, 2, 305-319, 1993 · doi:10.1029/92WR02339 |
| [54] | Lee, J.; Choi, S-U; Cho, W., A comparative study of dual-porosity model and discrete fracture network model, KSCE J Civ Eng, 3, 2, 171-180, 1999 · doi:10.1007/BF02829057 |
| [55] | Witherspoon, PA, Validity of Cubic Law for fluid flow in a deformable rock fracture, Water Resour Res, 16, 6, 1016-1024, 1980 · doi:10.1029/WR016i006p01016 |
| [56] | Miehe, C.; Mauthe, S.; Teichtmeister, S., 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, 2015 · doi:10.1016/j.jmps.2015.04.006 |
| [57] | Winkler, BJ, Traglastuntersuchungen von unbewehrten und bewehrten Betonstruk-turen auf der Grundlage eines objektiven Werkstoffgesetzes fur Beton, 2001, Innsbruck: Innsbruck University Press, Innsbruck |
| [58] | Rots J (1988) Computational modeling of concrete fracture. PhD thesis. Technische Universiteit Delft, Delft |
| [59] | Mikelić, A.; Wheeler, MF; Wick, T., A phase-field method for propagating fluid-filled fractures coupled to a surrounding porous medium, Multiscale Model Simul, 13, 1, 367-398, 2015 · Zbl 1317.74028 · doi:10.1137/140967118 |
| [60] | Pham, K., Gradient damage models and their use to approximate brittle fracture, Int J Damage Mech, 20, 4, 618-652, 2011 · doi:10.1177/1056789510386852 |
| [61] | Cornelissen, H.; Hordijk, D.; Reinhardt, H., Experimental determination of crack softening characteristics of normalweight and lightweight, Heron, 31, 2, 45-46, 1986 |
| [62] | Amor, H.; Marigo, J-J; Maurini, C., Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments, J Mech Phys Solids, 57, 8, 1209-1229, 2009 · Zbl 1426.74257 · doi:10.1016/j.jmps.2009.04.011 |
| [63] | Hmtermiiller, M.; Ito, K.; Kunisch, K., The primal-dual active set strategy as a semis-mooth Newton method, SIAM J Optim, 13, 3, 865-888, 2002 · Zbl 1080.90074 · doi:10.1137/S1052623401383558 |
| [64] | Zienkiewicz, OC, The finite element method, 1977, London: McGraw-Hill, London · Zbl 0435.73072 |
| [65] | Hughes, TJR, The finite element method: linear static and dynamic finite element analysis, 2012, Chelmsford: Courier Corporation, Chelmsford |
| [66] | De Borst, R., Nonlinear finite element analysis of solids and structures, 2012, Hoboken: Wiley, Hoboken · Zbl 1300.74002 · doi:10.1002/9781118375938 |
| [67] | Zienkiewicz, OC, Computational geomechanics, 1999, Citeseer: Princeton, Citeseer |
| [68] | Ambati M, Gerasimov T, De Lorenzis L (2015) A review on phase-field models of brittle fracture and a new fast hybrid formulation. Comput Mech 55(2):383-405. doi:10.1007/s00466-014-1109-y · Zbl 1398.74270 |
| [69] | Unger, JF; Eckardt, S.; Könke, C., Modelling of cohesive crack growth in concrete structures with the extended finite element method, Comput Methods Appl Mech Eng, 196, 41-44, 4087-4100, 2007 · Zbl 1173.74387 · doi:10.1016/j.cma.2007.03.023 |
| [70] | Bui, TQ; Hu, X., A review of phase-field models, fundamentals and their applications to composite laminates, Eng Fract Mech, 248, 107705, 2021 · doi:10.1016/j.engfracmech.2021.107705 |
| [71] | Nguyen-Thanh, C., Jive: An open source, research-oriented C++ library for solving partial differential equations, Adv Eng Softw, 150, 2020 · doi:10.1016/j.advengsoft.2020.102925 |
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.
