Embedded Unit Cell Homogenization Approach for Fracture Analysis
Publication: Journal of Engineering Mechanics
Volume 150, Issue 8
Abstract
We extend the applicability of the embedded unit cell (EUC) method to three-dimensional (3D) fracture problems, which are modeled by the extended finite element method (XFEM). The EUC method is a concurrent multiscale method based on the computational homogenization theory for nonperiodic domains. Herein, we show that this method can accurately estimate fracture parameters and, in particular, stress intensity factors using the J-integral method. Additionally, the method is shown to capture crack propagation within the microscale domain, as well as cracks initiating at the microscale and propagating outwards onto the macroscale through the internal subdomain boundaries. To demonstrate the accuracy, robustness, and computational efficiency of the proposed method, several 3D numerical benchmark examples, including planar cracks with single and mixed-mode fractures, are considered. In particular, we analyze horizontal, inclined, square, and penny-shaped cracks embedded in a homogeneous material. The results are verified against full FEM models and known analytical solutions if available.
Practical Applications
The insights of this research offer practical application for engineers and scientists in designing more resilient and durable structures. By extending the EUC method to 3D fracture problems, the study addresses the ability to forecast and access fracture phenomena in materials. This method is shown to be an effective approach for exploring the interaction between local microscopic discontinuities and cross-scale crack propagation, crucial for evaluating the durability of engineering structures. The EUC approach, integrated with XFEM, provides a comprehensive methodology for analyzing different fracture scenarios, including stationary mixed-mode cracks and crack-propagation examples. Its ability to accurately transition from microscale to macroscale analysis without remeshing introduces a valuable computational advantage, making it a more cost-efficient solution in fracture analysis. This research’s application offers tangible benefits in industrial context, especially in aerospace, automotive, and construction industries, where precise evaluation of structure and material failure can lead to more cost-efficient and safer designs by reducing the maintenance costs and failure risks.
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Data Availability Statement
Some or all data, models, or code generated or used during the study are available in a repository online in accordance with funder data retention policies (Grigorovitch and Waisman 2024).
References
Abdel-Wahab, A. A., A. R. Maligno, and V. V. Silberschmidt. 2012. “Micro-scale modelling of bovine cortical bone fracture: Analysis of crack propagation and microstructure using X-FEM.” Comput. Mater. Sci. 52 (1): 128–135. https://doi.org/10.1016/j.commatsci.2011.01.021.
Akono, A. T., and F. J. Ulm. 2014. “An improved technique for characterizing the fracture toughness via scratch test experiments.” Wear 313 (1–2): 117–124. https://doi.org/10.1016/j.wear.2014.02.015.
Akono, A.-T., and F.-J. Ulm. 2011. “Scratch test model for the determination of fracture toughness.” Eng. Fract. Mech. 78 (2): 334–342. https://doi.org/10.1016/j.engfracmech.2010.09.017.
Alam, S. Y., and A. Loukili. 2020. “Effect of micro-macro crack interaction on softening behaviour of concrete fracture.” Int. J. Solids Struct. 182 (Jan): 34–45. https://doi.org/10.1016/j.ijsolstr.2019.08.003.
Amini, S., and R. S. Kumar. 2014. “A high-fidelity strain-mapping framework using digital image correlation.” Mater. Sci. Eng., A 594 (Jan): 394–403. https://doi.org/10.1016/j.msea.2013.11.020.
Andrew, A. M. 2000. “Level set methods and fast marching methods: Evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science (2nd edition).” Kybernetes 29 (2): 239–248. https://doi.org/10.1108/k.2000.06729bag.001.
Asareh, I., Y. C. Yoon, and J. H. Song. 2018. “A numerical method for dynamic fracture using the extended finite element method with non-nodal enrichment parameters.” Int. J. Impact Eng. 121 (Nov): 63–76. https://doi.org/10.1016/j.ijimpeng.2018.06.012.
Ayadi, W., L. Laiarinandrasana, and K. Saï. 2018. “Anisotropic (continuum damage mechanics)-based multi-mechanism model for semi-crystalline polymer.” Int. J. Damage Mech. 27 (3): 357–386. https://doi.org/10.1177/1056789516679494.
Azevedo, P. C. M. 2008. “Stress intensity factors determination for an inclined central crack on a plate subjected to uniform tensile loading using FE analysis.” Int. J. Fract. 150 (2): 193–209. https://doi.org/10.1007/s10704-008-9221-9.
Babaei, R., and A. Farrokhabadi. 2017. “A computational continuum damage mechanics model for predicting transverse cracking and splitting evolution in open hole cross-ply composite laminates.” Fatigue Fract. Eng. Mater. Struct. 40 (3): 375–390. https://doi.org/10.1111/ffe.12502.
Bansal, M., I. V. Singh, B. K. Mishra, and S. P. A. Bordas. 2019. “A parallel and efficient multi-split XFEM for 3-D analysis of heterogeneous materials.” Comput. Methods Appl. Mech. Eng. 347 (Apr): 365–401. https://doi.org/10.1016/j.cma.2018.12.023.
Barenblatt, G. I. 1962. “The mathematical theory of equilibrium cracks in brittle fracture.” Adv. Appl. Mech. 7 (C): 55–129. https://doi.org/10.1016/S0065-2156(08)70121-2.
Belytschko, T. 2008. “The finite element method: Linear static and dynamic finite element analysis: Thomas J. R. Hughes.” Comput.-Aided Civ. Infrastruct. Eng. 4 (3): 45–48. https://doi.org/10.1111/j.1467-8667.1989.tb00025.x.
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.
Bordas, S., and B. Moran. 2006. “Enriched finite elements and level sets for damage tolerance assessment of complex structures.” Eng. Fract. Mech. 73 (9): 1176–1201. https://doi.org/10.1016/j.engfracmech.2006.01.006.
Chaboche, J. L. 1988. “Continuum damage mechanics: Part I-general concepts.” J. Appl. Mech. Trans. ASME 55 (1): 59–64. https://doi.org/10.1115/1.3173661.
Clément, A., C. Soize, and J. Yvonnet. 2013. “Uncertainty quantification in computational stochastic multiscale analysis of nonlinear elastic materials.” Comput. Methods Appl. Mech. Eng. 254 (Feb): 61–82. https://doi.org/10.1016/j.cma.2012.10.016.
Dugdale, D. S. 1960. “Yielding of steel sheets containing slits.” J. Mech. Phys. Solids 8 (2): 100–104. https://doi.org/10.1016/0022-5096(60)90013-2.
Elliott, J. A. 2011. “Novel approaches to multiscale modelling in materials science.” Int. Mater. Rev. 56 (4): 207–225. https://doi.org/10.1179/1743280410Y.0000000002.
EMERSON. 2023. “National instruments Labview software.” Accessed October 12, 2023. https://www.ni.com/en/shop/labview.html.
Fries, T., and T. Belytschko. 2010. “The extended/generalized finite element method: An overview of the method and its applications.” Int. J. Numer. Methods Eng. 84 (3): 253–304. https://doi.org/10.1002/nme.2914.
Garrett, G. G. 1977. “Introduction to fracture mechanics.” Civ. Eng. South Afr. 19 (10): 78–82. https://doi.org/10.2472/jsms.32.935.
Garzon, J., C. A. Duarte, and J. P. Pereira. 2013. “Extraction of stress intensity factors for the simulation of 3-D crack growth with the generalized finite element method.” Key Eng. Mater. 560 (Aug): 1–36. https://doi.org/10.4028/www.scientific.net/KEM.560.1.
Ghosh, S., J. Bai, and P. Raghavan. 2007. “Concurrent multi-level model for damage evolution in microstructurally debonding composites.” Mech. Mater. 39 (3): 241–266. https://doi.org/10.1016/j.mechmat.2006.05.004.
González, C., and J. LLorca. 2006. “Multiscale modeling of fracture in fiber-reinforced composites.” Acta Mater. 54 (16): 4171–4181. https://doi.org/10.1016/j.actamat.2006.05.007.
Gravouil, A., N. Moës, and T. Belytschko. 2002. “Non-planar 3D crack growth by the extended finite element and level sets-Part II: Level set update.” Int. J. Numer. Methods Eng. 53 (11): 2570–2575. https://doi.org/10.1002/nme.430.
Griffith, A. A. 1921. “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.
Grigorovitch, M., and E. Gal. 2015. “The local response in structures using the embedded unit cell approach.” Comput. Struct. 157 (Sep): 189–200. https://doi.org/10.1016/j.compstruc.2015.05.006.
Grigorovitch, M., and E. Gal. 2017. “Homogenization of non-periodic zones in periodic domains using the embedded unit cell approach.” Comput. Struct. 179 (Jan): 95–108. https://doi.org/10.1016/j.compstruc.2016.11.001.
Grigorovitch, M., E. Gal, and H. Waisman. 2021. “Embedded unit cell homogenization model for localized non-periodic elasto-plastic zones.” Comput. Mech. 68 (6): 1437–1456. https://doi.org/10.1007/s00466-021-02077-3.
Grigorovitch, M., and H. Waisman. 2024. “The source materials code and FE models for EUC crack analysis.” Accessed January 11, 2024. https://github.com/marinagrig/EUCcrack/tree/code.
Guedes, J., and N. Kikuchi. 1990. “Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods.” Comput. Methods Appl. Mech. Eng. 83 (2): 143–198. https://doi.org/10.1016/0045-7825(90)90148-F.
Guilleminot, J., C. Soize, and D. Kondo. 2009. “Mesoscale probabilistic models for the elasticity tensor of fiber reinforced composites: Experimental identification and numerical aspects.” Mech. Mater. 41 (12): 1309–1322. https://doi.org/10.1016/j.mechmat.2009.08.004.
Han, Z., D. Li, and X. Li. 2022. “Experimental study on the dynamic behavior of sandstone with coplanar elliptical flaws from macro, meso, and micro viewpoints.” Theor. Appl. Fract. Mech. 120 (Jun): 103400. https://doi.org/10.1016/j.tafmec.2022.103400.
Hillerborg, A., M. Modéer, and P. E. Petersson. 1976. “Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements.” Cem. Concr. Res. 6 (6): 773–781. https://doi.org/10.1016/0008-8846(76)90007-7.
Hiriyur, B., H. Waisman, and G. Deodatis. 2011. “Uncertainty quantification in homogenization of heterogeneous microstructures modeled by XFEM.” Int. J. Numer. Methods Eng. 88 (3): 257–278. https://doi.org/10.1002/nme.3174.
Janson, J., and J. Hult. 1977. “Fracture mechanics and damage mechanics—A combined approach.” In Proc., 14th Int. Congress of Theoretical and Applied Mechanics, 69–84. Delft, Netherlands: North-Holland Publishing Company.
Jiang, R., F. Dai, Y. Liu, A. Li, and P. Feng. 2021. “Frequency characteristics of acoustic emissions induced by crack propagation in rock tensile fracture.” Rock Mech. Rock Eng. 54 (4): 2053–2065. https://doi.org/10.1007/s00603-020-02351-5.
Kaczmarek, J. 2015. “A method of multiscale modelling considered as a way leading to unified mechanics of materials.” Acta Mech. 226 (5): 1419–1443. https://doi.org/10.1007/s00707-014-1261-7.
Kouznetsova, V. G., M. G. D. Geers, and W. A. M. Brekelmans. 2004. “Multi-scale second-order computational homogenization of multi-phase materials: A nested finite element solution strategy.” Comput. Methods Appl. Mech. Eng. 193 (48–51): 5525–5550. https://doi.org/10.1016/j.cma.2003.12.073.
Kumar, M., S. Ahmad, I. V. Singh, A. V. Rao, J. Kumar, and V. Kumar. 2018. “Experimental and numerical studies to estimate fatigue crack growth behavior of Ni-based super alloy.” Theor. Appl. Fract. Mech. 96 (Aug): 604–616. https://doi.org/10.1016/j.tafmec.2018.07.002.
Kumar, P. 2009. Elements of fracture mechanics. New York: McGraw-Hill Education LLC.
Kumar, S., I. V. Singh, and B. K. Mishra. 2015. “A homogenized XFEM approach to simulate fatigue crack growth problems.” Comput. Struct. 150 (Apr): 1–22. https://doi.org/10.1016/j.compstruc.2014.12.008.
Landau, L. D., E. M. Lifshitz, J. B. Sykes, W. H. Reid, and E. H. Dill. 1960. “Theory of elasticity: Vol. 7 of course of theoretical physics.” Phys. Today 13 (7): 44–46. https://doi.org/10.1063/1.3057037.
Levén, M., and R. Daniel. 2012. “Stationary 3D crack analysis with Abaqus XFEM for integrity assessment of subsea equipment.” Master’s thesis, Dept. of Applied Mechanics, Chalmers Univ. of Technology.
Li, H., P. O’Hara, and C. A. Duarte. 2021. “Non-intrusive coupling of a 3-D Generalized Finite Element Method and Abaqus for the multiscale analysis of localized defects and structural features.” Finite Elem. Anal. Des. 193 (Oct): 103554. https://doi.org/10.1016/j.finel.2021.103554.
Li, X., W. Gao, and W. Liu. 2019. “A mesh objective continuum damage model for quasi-brittle crack modelling and finite element implementation.” Int. J. Damage Mech. 28 (9): 1299–1322. https://doi.org/10.1177/1056789518823876.
Li, Y., and M. Zhou. 2013. “Prediction of fracturess toughness of ceramic composites as function of microstructure: II. Analytical model.” J. Mech. Phys. Solids 61 (2): 489–503. https://doi.org/10.1016/j.jmps.2012.09.011.
Lin, M., Y. Liu, C. Oskay, and X. Zhang. 2022. “Informed reduced-order modeling of fatigue initiation in a titanium skin panel subjected to thermo-mechanical loading.” In Proc., AIAA Science and Technology Forum and Exposition, AIAA SciTech Forum 2022. Reston, VA: American Institute of Aeronautics and Astronautics.
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.
McDowell, D. L. 2010. “A perspective on trends in multiscale plasticity.” Int. J. Plast. 26 (9): 1280–1309. https://doi.org/10.1016/j.ijplas.2010.02.008.
Meguid, S. A., and X. D. Wang. 1995. “On the dynamic interaction between a microdefect and a main crack.” Proc. R. Soc. London A Math Phys. Sci. 448 (1934): 449–464. https://doi.org/10.1098/rspa.1995.0049.
Menouillard, T., and T. Belytschko. 2010. “Dynamic fracture with meshfree enriched XFEM.” Acta Mech. 213 (1): 53–69. https://doi.org/10.1007/s00707-009-0275-z.
Moës, N., and T. Belytschko. 2002. “Extended finite element method for cohesive crack growth.” Eng. Fract. Mech. 69 (7): 813–833. https://doi.org/10.1016/S0013-7944(01)00128-X.
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.
Mohammadnejad, M., H. Liu, A. Chan, S. Dehkhoda, and D. Fukuda. 2021. “An overview on advances in computational fracture mechanics of rock.” Geosyst. Eng. 24 (4): 206–229. https://doi.org/10.1080/12269328.2018.1448006.
Mota, A., I. Tezaur, and C. Alleman. 2017. “The Schwarz alternating method in solid mechanics.” Comput. Methods Appl. Mech. Eng. 319 (Jun): 19–51. https://doi.org/10.1016/j.cma.2017.02.006.
Mota, A., I. Tezaur, and G. Phlipot. 2022. “The Schwarz alternating method for transient solid dynamics.” Int. J. Numer. Methods Eng. 123 (21): 5036–5071. https://doi.org/10.1002/nme.6982.
Nguyen, V. P., M. Stroeven, and L. J. Sluys. 2011. “Multiscale continuous and discontinuous modeling of heterogeneous materials: A review on recent developments.” J. Multiscale Modell. 3 (4): 229–270. https://doi.org/10.1142/S1756973711000509.
Osher, S., and J. A. Sethian. 1988. “Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations.” J. Comput. Phys. 79 (1): 12–49. https://doi.org/10.1016/0021-9991(88)90002-2.
Panasenko, G. 2005. Multi-scale modelling for structures and composites, 65–72. Dordrecht, Netherlands: Springer. https://doi.org/10.1007/1-4020-2982-9.
Panchal, J. H., S. R. Kalidindi, and D. L. McDowell. 2013. “Key computational modeling issues in integrated computational materials engineering.” CAD Comput. Aided Des. 45 (1): 4–25. https://doi.org/10.1016/j.cad.2012.06.006.
Pandey, V. B., I. V. Singh, B. K. Mishra, S. Ahmad, A. V. Rao, and V. Kumar. 2019. “A new framework based on continuum damage mechanics and XFEM for high cycle fatigue crack growth simulations.” Eng. Fract. Mech. 206 (Apr): 172–200. https://doi.org/10.1016/j.engfracmech.2018.11.021.
Pathak, H., A. Singh, and I. V. Singh. 2012. “Numerical simulation of bi-material interfacial cracks using EFGM and XFEM.” Int. J. Mech. Mater. Des. 8 (1): 9–36. https://doi.org/10.1007/s10999-011-9173-3.
Richard, H. A., M. Fulland, and M. Sander. 2005. “Theoretical crack path prediction.” Fatigue Fract. Eng. Mater. Struct. 28 (1–2): 3–12. https://doi.org/10.1111/j.1460-2695.2004.00855.x.
Richardson, C. L., J. Hegemann, E. Sifakis, J. Hellrung, and J. M. Teran. 2011. “An XFEM method for modeling geometrically elaborate crack propagation in brittle materials.” Int. J. Numer. Methods Eng. 88 (10): 1042–1065. https://doi.org/10.1002/nme.3211.
Russ, J., V. Slesarenko, S. Rudykh, and H. Waisman. 2020. “Rupture of 3D-printed hyperelastic composites: Experiments and phase field fracture modeling.” J. Mech. Phys Solids 140 (Jul): 103941. https://doi.org/10.1016/j.jmps.2020.103941.
Shen, B., and G. H. Paulino. 2011. “Identification of cohesive zone model and elastic parameters of fiber-reinforced cementitious composites using digital image correlation and a hybrid inverse technique.” Cem. Concr. Compos. 33 (5): 572–585. https://doi.org/10.1016/j.cemconcomp.2011.01.005.
Singh, I. V., B. K. Mishra, S. Bhattacharya, and R. U. Patil. 2012. “The numerical simulation of fatigue crack growth using extended finite element method.” Int. J. Fatigue 36 (1): 109–119. https://doi.org/10.1016/j.ijfatigue.2011.08.010.
Su, Z., and C. Oskay. 2022. “Modeling arbitrarily oriented and reorienting multiscale cracks in composite materials with adaptive multiscale discrete damage theory.” Comput. Mech. 70 (5): 1041–1058. https://doi.org/10.1007/s00466-022-02205-7.
Sukumar, N., D. L. Chopp, N. Moës, and T. Belytschko. 2001. “Modeling holes and inclusions by level sets in the extended finite-element method.” Comput. Methods Appl. Mech. Eng. 190 (46–47): 6183–6200. https://doi.org/10.1016/S0045-7825(01)00215-8.
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.
Szabó, B. A., and I. Babuška. 2009. “Computation of the amplitude of stress singular terms for cracks and reentrant corners.” In Proc., Fracture Mechanics: 19th Symp., edited by T. A. Cruse, ASTM STP 969, 101–124. West Conshohocken, PA: ASTM.
Tabarraei, A., J. H. Song, and H. Waisman. 2013. “A two-scale strong discontinuity approach for evolution of shear bands under dynamic impact loads.” Int. J. Multiscale Comput. Eng. 11 (6): 543–563. https://doi.org/10.1615/IntJMultCompEng.2013005506.
Tada, H., P. C. Paris, and G. R. Irwin. 2010. The stress analysis of cracks handbook. 3rd ed. New York: ASME Press. https://doi.org/10.1115/1.801535.
Trias, D., J. Costa, B. Fiedler, T. Hobbiebrunken, and J. E. Hurtado. 2006. “A two-scale method for matrix cracking probability in fibre-reinforced composites based on a statistical representative volume element.” Compos. Sci. Technol. 66 (11–12): 1766–1777. https://doi.org/10.1016/j.compscitech.2005.10.030.
Waisman, H. 2010. “An analytical stiffness derivative extended finite element technique for extraction of crack tip strain energy release rates.” Eng. Fract. Mech. 77 (16): 3204–3215. https://doi.org/10.1016/j.engfracmech.2010.08.015.
Waisman, H., and L. Berger-Vergiat. 2013. “An adaptive domain decomposition preconditioner for crack propagation problems modeled by XFEM.” Int. J. Multiscale Comput. Eng. 11 (6): 633–654. https://doi.org/10.1615/IntJMultCompEng.2013006012.
Waisman, H., E. Chatzi, and A. W. Smyth. 2010. “Detection and quantification of flaws in structures by the extended finite element method and genetic algorithms.” Int. J. Numer. Methods Eng. 82 (3): 303–328. https://doi.org/10.1002/nme.2766.
Williams, M. L. 1960. “The bending stress distribution at the base of a stationary crack.” J. Appl. Mech. Trans. ASME 28 (1): 109–114. https://doi.org/10.1115/1.3640470.
Yasbolaghi, R., and A. R. Khoei. 2020. “Micro-structural aspects of fatigue crack propagation in atomistic-scale via the molecular dynamics analysis.” Eng. Fract. Mech. 226 (Mar): 106848. https://doi.org/10.1016/j.engfracmech.2019.106848.
Yosibash, Z. 2012. “Singularities in elliptic boundary value problems and elasticity and their connection with failure initiation.” In Interdisciplinary applied mathematics. Berlin: Springer.
Yuan, Z., and J. Fish. 2009. “Multiple scale eigendeformation-based reduced order homogenization.” Comput. Methods Appl. Mech. Eng. 198 (21–26): 2016–2038. https://doi.org/10.1016/j.cma.2008.12.038.
Zhu, C. L., J. B. Li, G. Lin, and H. Zhong. 2014. “Study on the relationship between stress intensity factor and J integral for mixed mode crack with arbitrary inclination based on SBFEM.” IOP Conf. Ser. Mater. Sci. Eng. 10 (1): 012066. https://doi.org/10.1088/1757-899X/10/1/012066.
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Journal of Engineering Mechanics
Volume 150 • Issue 8 • August 2024
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© 2024 American Society of Civil Engineers.
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Received: Nov 30, 2023
Accepted: Mar 21, 2024
Published online: Jun 11, 2024
Published in print: Aug 1, 2024
Discussion open until: Nov 11, 2024
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