Field-Enriched Finite-Element Method for Simulating Crack Propagation and Coalescence in Geomaterials
Publication: Journal of Engineering Mechanics
Volume 147, Issue 10
Abstract
A field-enriched finite-element method for simulating crack propagation and coalescence in geomaterials is proposed in this work. In the novel numerical method, the physical position of the crack is characterized by the field variables, and the propagation evolution of the crack is controlled by the fracture criterion. The field-enriched finite-element method is first verified by a benchmark example. Then, it is used to simulate crack initiation, propagation, and coalescence in the different configuration specimens under various loads. The numerical results show that the field-enriched finite-element method has the capability to efficiently simulate crack propagation and coalescence in geomaterials.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
All codes used in this study are available from the corresponding author upon reasonable request.
Acknowledgments
The work is supported by the National Natural Science Foundation of China (Nos. 52027814 and 51839009) and by Graduate Research and Innovation Foundation of Chongqing, China (Grant No. CYB20030), which are gratefully acknowledged.
References
Al-Shayea, N. A. 2005. “Crack propagation trajectories for rocks under mixed mode I-II fracture.” Eng. Geol. 81 (1): 84–97. https://doi.org/10.1016/j.enggeo.2005.07.013.
Ambati, M., T. Gerasimov, and L. De Lorenzis. 2014. “A review on phase-field models of brittle fracture and a new fast hybrid formulation.” Comput. Mech. 55 (2): 383–405. https://doi.org/10.1007/s00466-014-1109-y.
Amor, H., J. J. Marigo, and C. Maurini. 2009. “Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments.” J. Mech. Phys. Solids 57 (8): 1209–1229. https://doi.org/10.1016/j.jmps.2009.04.011.
Belytschko, T., and T. Black. 1999. “Elastic crack growth in finite elements with minimal remeshing.” Int. J. Numer. Methods Eng. 45 (5): 601–620. https://doi.org/10.1002/(SICI)1097-0207(19990620)45:5%3C601::AID-NME598%3E3.0.CO;2-S.
Bernard, P. E., N. Moës, and N. Chevaugeon. 2012. “Damage growth modeling using the thick level set (TLS) approach: Efficient discretization for quasi-static loadings.” Comput. Methods Appl. Mech. Eng. 233–236 (Aug): 11–27. https://doi.org/10.1016/j.cma.2012.02.020.
Borden, M. J., C. V. Verhoosel, M. A. Scott, T. J. Hughes, and C. M. Landis. 2012. “A phase-field description of dynamic brittle fracture.” Comput. Methods Appl. Mech. Eng. 217–220 (Apr): 77–95. https://doi.org/10.1016/j.cma.2012.01.008.
Bourdin, B., G. A. Francfort, and J. J. Marigo. 2008. “The variational approach to fracture.” J. Elast. 91 (1): 5–148. https://doi.org/10.1007/s10659-007-9107-3.
Bryant, E. C., and W. C. Sun. 2018. “A mixed-mode phase field fracture model in anisotropic rocks with consistent kinematics.” Comput. Methods Appl. Mech. Eng. 342 (Dec): 561–584. https://doi.org/10.1016/j.cma.2018.08.008.
Bryant, E. C., and W. C. Sun. 2021. “Phase field modeling of frictional slip with slip weakening/strengthening under non-isothermal conditions.” Comput. Methods Appl. Mech. Eng. 375 (Mar): 113557. https://doi.org/10.1016/j.cma.2020.113557.
Dolbow, J., N. Moës, and T. Belytschko. 2000. “Discontinuous enrichment in finite elements with a partition of unity method.” Finite Elem. Anal. Des. 36 (3): 235–260. https://doi.org/10.1016/S0168-874X(00)00035-4.
Engelen, R. A. B., M. G. D. Geers, and F. P. T. Baaijens. 2003. “Nonlocal implicit gradient-enhanced elasto-plasticity for the modelling of softening behavior.” Int. J. Plast. 19 (4): 403–433. https://doi.org/10.1016/S0749-6419(01)00042-0.
Erdogan, F., and G. C. Sih. 1997. “On the crack extension in plates under plane loading and transverse shear.” J. Fluids Eng. Trans. ASME 85 (4): 519–525. https://doi.org/10.1115/1.3656897.
Fei, F., and J. Choo. 2020. “A phase-field model of frictional shear fracture in geologic materials.” Comput. Methods Appl. Mech. Eng. 369 (Sep): 113265. https://doi.org/10.1016/j.cma.2020.113265.
Francfort, G. A., and J. J. Marigo. 1998. “Revisiting brittle fracture as an energy minimization problem.” J. Mech. Phys. Solids 46 (8): 1319–1342. https://doi.org/10.1016/S0022-5096(98)00034-9.
Gergely, M., et al. 2020. “An open-source Abaqus implementation of the phase-field method to study the effect of plasticity on the instantaneous fracture toughness in dynamic crack propagation.” Comput. Methods Appl. Mech. Eng. 365 (Jun): 113004. https://doi.org/10.1016/j.cma.2020.113004.
Giner, E., N. Sukumar, J. E. Tarancón, and F. J. Fuenmayor. 2009. “An Abaqus implementation of the extended finite element method.” Eng. Fract. Mech. 76 (3): 347–368. https://doi.org/10.1016/j.engfracmech.2008.10.015.
Griffith, A. 1920. “The phenomena of rupture and flow in solids.” Philos. Trans. R. Soc. London, Ser. A 221 (582–593): 163–198. https://doi.org/10.1098/rsta.1921.0006.
Hou, C., Z. Y. Wang, W. G. Liang, H. Yu, and Z. Wang. 2017. “Investigation of the effects of confining pressure on SIFs and T-stress for CCBD specimens using the XFEM and the interaction integral method.” Eng. Fract. Mech. 178 (Jun): 279–300. https://doi.org/10.1016/j.engfracmech.2017.03.049.
Kim, J. H., and G. H. Paulino. 2003. “T-stress, mixed-mode stress intensity factors, and crack initiation angles in functionally graded materials: A unified approach using the interaction integral method.” Comput. Methods Appl. Mech. Eng. 192 (11–12): 1463–1494. https://doi.org/10.1016/S0045-7825(02)00652-7.
Kumar, S., I. V. Singh, B. K. Mishra, and T. Rabczuk. 2015. “Modeling and simulation of kinked cracks by virtual node XFEM.” Comput. Methods Appl. Mech. Eng. 283 (Jan): 1425–1466. https://doi.org/10.1016/j.cma.2014.10.019.
Lan, M., H. Waisman, and I. Harari. 2013a. “A direct analytical method to extract mixed-mode components of strain energy release rates from Irwin’s integral using extended finite element method.” Int. J. Numer. Methods Eng. 95 (12): 1033–1052. https://doi.org/10.1002/nme.4542.
Lan, M., H. Waisman, and I. Harari. 2013b. “A high-order extended finite element method for extraction of mixed-mode strain energy release rates in arbitrary crack settings based on Irwin’s integral.” Int. J. Numer. Methods Eng. 96 (12): 787–812. https://doi.org/10.1002/nme.4584.
Lee, H., and S. Jeon. 2011. “An experimental and numerical study of fracture coalescence in pre-cracked specimens under uniaxial compression.” Int. J. Solids Struct. 48 (6): 979–999. https://doi.org/10.1016/j.ijsolstr.2010.12.001.
Liao, M. M., X. Deng, and Z. Y. Guo. 2018. “Crack propagation modelling using the weak form quadrature element method with minimal remeshing.” Theor. Appl. Fract. Mech. 93 (Feb): 293–301. https://doi.org/10.1016/j.tafmec.2017.09.012.
Liu, X. Y., Q. Z. Xiao, and B. L. Karihaloo. 2004. “XFEM for direct evaluation of mixed mode SIFs in homogeneous and bi-materials.” Int. J. Numer. Methods Eng. 59 (8): 1103–1118. https://doi.org/10.1002/nme.906.
Londono, J. G., R. Shen, and H. Waisman. 2020. “Temperature-dependent viscoelastic model for asphalt-concrete implemented within a novel nonlocal damage framework.” J. Eng. Mech. 146 (2): 04019119. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001702.
Meer, V. D., F. P. Sluys, and L. J. Sluys. 2015. “The thick level set method: Sliding deformations and damage initiation.” Comput. Methods Appl. Mech. Eng. 285 (Mar): 64–82. https://doi.org/10.1016/j.cma.2014.10.020.
Mesgarnejad, A., B. Bourdin, and M. M. Khonsari. 2014. “Validation simulations for the variational approach to fracture.” Comput. Methods Appl. Mech. Eng. 290 (Jun): 420–437. https://doi.org/10.1016/j.cma.2014.10.052.
Miehe, C., M. Hofacker, L. M. Schänzel, and F. Aldakheel. 2015a. “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 (Sep): 486–522. https://doi.org/10.1016/j.cma.2014.11.017.
Miehe, C., M. Hofacker, and F. Welschinger. 2010a. “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. https://doi.org/10.1016/j.cma.2010.04.011.
Miehe, C., L. M. Schänzel, and H. Ulmer. 2015b. “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 (Sep): 449–485. https://doi.org/10.1016/j.cma.2014.11.016.
Miehe, C., F. Welschinger, and M. Hofacker. 2010b. “Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementations.” Int. J. Numer. Methods Eng. 83 (10): 1273–1311. https://doi.org/10.1002/nme.2861.
Mobasher, M. E., H. Waisman, and L. Berger-Vergiat. 2018. “Thermodynamic framework for non-local transport-damage modeling of fluid driven fracture in porous media.” Int. J. Rock Mech. Min. Sci. 111 (Nov): 64–83. https://doi.org/10.1016/j.ijrmms.2018.08.006.
Moës, N., J. Dolbow, and T. Belytschko. 1999. “A finite element method for crack growth without remeshing.” Int. J. Numer. Methods Eng. 46 (1): 131–150. https://doi.org/10.1002/(SICI)1097-0207(19990910)46:1%3C131::AID-NME726%3E3.0.CO;2-J.
Moës, N., A. Gravouil, and T. Belytschko. 2002. “Non-planar 3D crack growth by the extended finite element and level sets—Part I: Mechanical model.” Int. J. Numer. Methods Eng. 53 (11): 2549–2568. https://doi.org/10.1002/nme.429.
Moës, N., C. Stolz, P.-E. Bernard, and N. Chevaugeon. 2011. “A level set based model for damage growth: The thick level set approach.” Int. J. Numer. Methods Eng. 86 (3): 358–380. https://doi.org/10.1002/nme.3069.
Nguyen, V. P., and J. Y. Wu. 2018. “Modeling dynamic fracture of solids with a phase-field regularized cohesive zone model.” Comput. Methods Appl. Mech. Eng. 340 (Oct): 1000–1022. https://doi.org/10.1016/j.cma.2018.06.015.
Nooru-Mohamed, M. B. 1992. “Mixed-mode fracture of concrete: An experimental approach.” Ph.D. thesis, Dept. of Civil Engineering, Delft Univ. of Technology.
Patil, R. U., B. K. Mishra, and I. V. Singh. 2018. “An adaptive multiscale phase field method for brittle fracture.” Comput. Methods Appl. Mech. Eng. 329 (Feb): 254–288. https://doi.org/10.1016/j.cma.2017.09.021.
Philip, K. K., and M. P. Emilio. 2020. “Phase field fracture modelling using quasi-Newton methods and a new adaptive step scheme.” Theor. Appl. Fract. Mech. 107 (Jun): 102446. https://doi.org/10.1016/j.tafmec.2019.102446.
Rice, J. R. 1967. “A path independent integral and the approximate analysis of strain concentration by notches and cracks.” J Appl. Mech. 35 (2): 379–386. https://doi.org/10.1115/1.3601206.
Shahani, A. R., and M. R. Amini Fasakhodi. 2009. “Finite element analysis of dynamic crack propagation using remeshing technique.” Mater. Des. 30 (4): 1032–1041. https://doi.org/10.1016/j.matdes.2008.06.049.
Shen, R. L., H. Waisman, and L. C. Guo. 2019. “Fracture of viscoelastic solids modeled with a modified phase field method.” Comput. Methods Appl. Mech. Eng. 346 (Apr): 862–890. https://doi.org/10.1016/j.cma.2018.09.018.
Shephard, M. S., N. A. B. Yehia, G. S. Burd, and T. J. Weidner. 1985. “Computational strategies for nonlinear and fracture mechanics problem: Automatic crack propagation tracking.” Comput. Struct. 20 (1–3): 211–223. https://doi.org/10.1016/0045-7949(85)90070-7.
Song, G., H. Waisman, M. Lan, and I. Harari. 2015. “Extraction of stress intensity factors from Irwin’s integral using high-order XFEM on triangular meshes.” Int. J. Numer. Methods Eng. 102 (3–4): 528–550. https://doi.org/10.1002/nme.4698.
Sukumar, N., N. Moës, B. Moran, and T. Belytschko. 2000. “Extended finite element method for three-dimensional crack modeling.” Int. J. Numer. Methods Eng. 48 (11): 1549–1570. https://doi.org/10.1002/1097-0207(20000820)48:11%3C1549::AID-NME955%3E3.0.CO;2-A.
Wang, L. F., and X. P. Zhou. 2020. “Phase field model for simulating the fracture behaviors of some disc-type specimens.” Eng. Fract. Mech. 226 (Mar): 106870. https://doi.org/10.1016/j.engfracmech.2020.106870.
Wang, Q., Y. T. Feng, W. Zhou, Y. Cheng, and G. Ma. 2020. “A phase-field model for mixed-mode fracture based on a unified tensile fracture criterion.” Comput. Methods Appl. Mech. Eng. 370 (Oct): 113270. https://doi.org/10.1016/j.cma.2020.113270.
Wang, T., X. Ye, Z. L. Liu, D. Chu, and Z. Zhuang. 2019. “Modeling the dynamic and quasi-static compression-shear failure of brittle materials by explicit phase field method.” Comput. Mech. 64 (6): 1537–1556. https://doi.org/10.1007/s00466-019-01733-z.
Wang, Y. T., X. P. Zhou, Y. Wang, and Y. Shou. 2018. “A 3-D conjugated bond-pair-based peridynamic formulation for initiation and propagation of cracks in brittle solids.” Int. J. Solids Struct. 134 (Mar): 89–115. https://doi.org/10.1016/j.ijsolstr.2017.10.022.
Wang, Y. X., and H. Waisman. 2017. “Material-dependent crack-tip enrichment functions in XFEM for modeling interfacial cracks in biomaterials.” Int. J. Numer. Methods Eng. 112 (11): 1495–1518. https://doi.org/10.1002/nme.5566.
Wang, Y. X., and H. Waisman. 2018. “An arc-length method for controlled cohesive crack propagation using high-order XFEM and Irwin’s crack closure integral.” Eng. Fract. Mech. 199 (Aug): 235–256. https://doi.org/10.1016/j.engfracmech.2018.05.018.
Wang, Y. X., H. Waisman, and I. Harari. 2017. “Direct evaluation of stress intensity factors for curved cracks using Irwin’s integral and XFEM with high-order enrichment functions.” Int. J. Numer. Methods Eng. 112 (7): 629–654. https://doi.org/10.1002/nme.5517.
Williams, M. 1957. “On the stress distribution at the base of a stationary crack.” J. Appl. Mech. Trans., ASME 24 (1): 109–114.
Wowk, D., L. Alousis, and P. R. Underhill. 2019. “An adaptive remeshing technique for predicting the growth of irregular crack fronts using p-version finite element analysis.” Eng. Fract. Mech. 207 (Feb): 36–47. https://doi.org/10.1016/j.engfracmech.2018.12.002.
Wu, J. Y., and V. P. Nguyen. 2018. “A length scale insensitive phase-field damage model for brittle fracture.” J. Mech. Phys. Solids 119 (Oct): 20–42. https://doi.org/10.1016/j.jmps.2018.06.006.
Xing, C., Y. X. Wang, and H. Waisman. 2019. “Fracture analysis of cracked thin-walled structures using a high-order XFEM and Irwin’s integral.” Comput. Struct. 212 (Feb): 1–19. https://doi.org/10.1016/j.compstruc.2018.10.010.
Xu, Y. J., and H. Yuan. 2011. “Applications of normal stress dominated cohesive zone models for mixed-mode crack simulation based on extended finite element methods.” Eng. Fract. Mech. 78 (3): 544–558. https://doi.org/10.1016/j.engfracmech.2010.03.029.
Yi, L. P., H. Waisman, Z. Z. Yang, and X. G. Li. 2020. “A consistent phase field model for hydraulic fracture propagation in poroelastic media.” Comput. Methods Appl. Mech. Eng. 372 (Dec): 113396. https://doi.org/10.1016/j.cma.2020.113396.
You, T., H. Waisman, and Q. Z. Zhu. 2021. “Brittle-ductile failure transition in geomaterials modeled by a modified phase-field method with a varying damage-driving energy coefficient.” Int. J. Plast. 136 (Jan): 102836. https://doi.org/10.1016/j.ijplas.2020.102836.
You, T., Q. Z. Zhu, P. F. Li, and J. F. Shao. 2020. “Incorporation of tension-compression asymmetry into plastic damage phase-field modeling of quasi brittle geomaterials.” Int. J. Plast. 124 (Jan): 71–95. https://doi.org/10.1016/j.ijplas.2019.08.003.
Zhang, J. Z., X. P. Zhou, J. Y. Zhu, C. Xian, and Y. T. Wang. 2018. “Quasi-static fracturing in double-flawed specimens under uniaxial loading: The role of strain rate.” Int. J. Fract. 211 (1–2): 75–102. https://doi.org/10.1007/s10704-018-0277-8.
Zhang, P., X. F. Hu, S. Yang, and W. Yao. 2019. “Modelling progressive failure in multi-phase materials using a phase field method.” Eng. Fract. Mech. 209 (Mar): 105–124. https://doi.org/10.1016/j.engfracmech.2019.01.021.
Zhang, S. F., W. Jiang, and M. R. Tonks. 2020. “A new phase field fracture model for brittle materials that accounts for elastic anisotropy.” Comput. Methods Appl. Mech. Eng. 358 (Jan): 112643. https://doi.org/10.1016/j.cma.2019.112643.
Zhang, X., S. W. Sloan, C. Vignes, and D. C. Sheng. 2017. “A modification of the phase-field model for mixed mode crack propagation in rock-like materials.” Comput. Methods Appl. Mech. Eng. 322 (Aug): 123–136. https://doi.org/10.1016/j.cma.2017.04.028.
Zhou, S. W., X. Y. Zhuang, and T. Rabczuk. 2019a. “Phase field modeling of brittle compressive-shear fractures in rock-like materials: A new driving force and a hybrid formulation.” Comput. Methods Appl. Mech. Eng. 355 (Oct): 729–752. https://doi.org/10.1016/j.cma.2019.06.021.
Zhou, S. W., X. Y. Zhuang, H. H. Zhu, and T. Rabczuk. 2018. “Phase field modelling of crack propagation, branching and coalescence in rocks.” Theor. Appl. Fract. Mech. 96 (Aug): 174–192. https://doi.org/10.1016/j.tafmec.2018.04.011.
Zhou, X. P., J. Bi, R. S. Deng, and B. Li. 2020a. “Effects of brittleness on crack behaviors in rock-like materials.” J. Test. Eval. 48 (4): 2829–2851. https://doi.org/10.1520/JTE20170595.
Zhou, X. P., and J. W. Chen. 2019. “Extended finite element simulation of step-path brittle failure in rock slopes with non-persistent en-echelon joints.” Eng. Geol. 250 (Feb): 65–88. https://doi.org/10.1016/j.enggeo.2019.01.012.
Zhou, X. P., Z. M. Jia, and F. Berto. 2019b. “Simulation of cracking behaviours in interlayered rocks with flaws subjected to tension using a phase-field method.” Fatigue Fract. Eng. Mater. Struct. 42 (8): 1679–1698. https://doi.org/10.1111/ffe.13009.
Zhou, X. P., L. F. Wang, and Y. D. Shou. 2020b. “Understanding the fracture mechanism of ring Brazilian disc specimens by the phase field method.” Int. J. Fract. 226 (1): 17–43. https://doi.org/10.1007/s10704-020-00476-w.
Zhou, X. P., and H. Q. Yang. 2012. “Multiscale numerical modeling of propagation and coalescence of multiple cracks in rock masses.” Int. J. Rock Mech. Min. Sci. 55 (Oct): 15–27. https://doi.org/10.1016/j.ijrmms.2012.06.001.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 147 • Issue 10 • October 2021
Copyright
© 2021 American Society of Civil Engineers.
History
Received: Aug 11, 2020
Accepted: Apr 9, 2021
Published online: Jul 23, 2021
Published in print: Oct 1, 2021
Discussion open until: Dec 23, 2021
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited by
- Hassan Ziari, Mohammad Reza Mohammad Aliha, Ehsan Sobhani Fard, Majid Jebalbarezi Sarbijan, Mixed mode I+II fracture parameters and cracking trajectory of heterogeneous multilayer pavement structure containing reflective crack, Fatigue & Fracture of Engineering Materials & Structures, 10.1111/ffe.13796, 45, 10, (2958-2977), (2022).