Phase field modeling of brittle fracture based on the cell-based smooth FEM by considering spectral decomposition. (English) Zbl 07341991
Summary: The spectral decomposition of the strain tensor is an essential technique to deal with the fracture problems via phase field method, and some incorrect results may be obtained without it. A novel phase field model for brittle fracture is developed based on cell-based smooth finite element (CS-FEM) and the spectral decomposition is taken into account. In order to describe the nonlinearity behaviors which contain the varied stress and elastic constitutive response caused by spectral decomposition. A second-order stress tensor and a fourth-order constitutive tensor based on decomposition of strain tensor are derived. A fundamental framework of CS-FEM is established to solve the phase field fracture problems, implemented by user-defined element (UEL) subroutine of ABAQUS software. The proposed model is validated by a typical Mode II crack, and the results show that the derived tensors are effective. Phase field parameter, CS-FEM parameter and mesh inhomogeneity are investigated to provide some useful suggestion for further development. Some classical numerical examples are solved by using the present model. The studies demonstrate that the proposed method can successfully overcome mesh distortion; the number of smooth cell does not show influences on the accuracy. Moreover, some results show that this method has the advantage over the standard FEM in convergence and computing efficiency.
Keywords:
phase field; brittle fracture; cell-based smooth finite element; spectral decomposition; crack pathReferences:
| [1] | Abaqus, Abaqus Documentation (Dassault Systems Simulia Corp., Providence, RI, 2012). |
| [2] | Ambati, M., Gerasimov, T. and De Lorenzis, L. [2015] “ Phase-field modeling of ductile fracture,” Comput Mech.55(5), 1017-1040. · Zbl 1329.74018 |
| [3] | Ambati, M., Gerasimov, T. and De Lorenzis, L. [2014] “ A review on phase-field models of brittle fracture and a new fast hybrid formulation,” Comput Mech.55(2), 383-405. · Zbl 1398.74270 |
| [4] | Amestoy, M. [1987] Propagations de fissures en élasticité plane. thèse d’Etat, Paris. |
| [5] | Badnava, H., Msekh, M. A., Etemadi, E. and Rabczuk, T. [2017] “ An \(h\)-adaptive thermo-mechanical phase field model for fracture,” Finite Elem. Anal. Des.138, 31-47. |
| [6] | Bhowmick, S. and Liu, G. R. [2018a] “ Three dimensional CS-FEM phase-field modeling technique for brittle fracture in elastic solids,” Appl. Sci.8, 2488. |
| [7] | Bhowmick, S. and Liu, G. R. [2018b] “ A phase-field modeling for brittle fracture and crack propagation based on the cell-based smoothed finite element method,” Eng. Fract. Mech.204, 369-387. |
| [8] | Bittencourt, T. N., Wawrzynek, P. A., Ingraffea, A. R. and Sousa, J. L. [1996] “ Quasi-automatic simulation of crack propagation for 2d LEFM problems,” Eng. Fract. Mech.55, 321-334. |
| [9] | Borden, M. J., Hughes, T. J. R., Landis, C. M. and Verhoosel, C. V. [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 |
| [10] | Bourdin, B. [2007] “ Numerical implementation of the variational formulation for quasi-static brittle fracture,” Interfaces Free Bound.9, 411-430. · Zbl 1130.74040 |
| [11] | Bourdin, B., Francfort, G. A. and Marigo, J. J. [2008] “ The variational approach to fracture,” Int. J. Elast.91(1-3), 5-148. · Zbl 1176.74018 |
| [12] | Bourdin, B., Francfort, G. and Marigo, J. J. [2000] “ Numerical experiments in revisited brittle fracture,” J. Mech. Phys. Solids48(4), 797-826. · Zbl 0995.74057 |
| [13] | Chen, L., Liu, G. R. and Zeng, K. [2011] “ A combined extended and edge-based smoothed finite element method (ES-XFEM) for fracture analysis of 2D elasticity,” Int. J. Comput. Methods8(4), 773-786. · Zbl 1245.74081 |
| [14] | Cui, X. Y., Li, G. Y. and Liu, G. R. [2013] “ An explicit smoothed finite element method (SFEM) for elastic dynamic problems,” Int. J. Comput. Methods10(1), 1340002. · Zbl 1359.74398 |
| [15] | Cui, X., Han, X., Duan, S. Y. and Liu, G. R. [2018] “ An abaqus implementation of the cell-based smoothed finite element method (CS-FEM),” Int. J. Comput. Methods17(2), S021987621850127X. · Zbl 07124751 |
| [16] | Dai, K. Y. and Liu, G. R. [2007] “ Free and forced vibration analysis using the smoothed finite element method (SFEM),” J. Sound Vib.301(3-5), 803-820. |
| [17] | Francfort, G. A. and Marigo, J. J. [1998] “ Revisiting brittle fracture as an energy minimization problem,” J. Mech. Phys. Solid46(8), 1319-1342. · Zbl 0966.74060 |
| [18] | Giovanardi, B., Scotti, A. and Formaggia, L. [2017] “ A hybrid XFEM - Phase field (Xfield) method for crack propagation in brittle elastic materials,” Comput. Methods Appl. Mech. Eng.320, 396-420. · Zbl 1439.74350 |
| [19] | He, Z. C., Li, G. Y., Li, E., Zhong, Z. H. and Liu, G. R. [2014] “ Mid-frequency acoustic analysis using edge-based smoothed tetrahedron radialpoint interpolation methods,” Int. J. Comput. Methods11(5), 1350103. · Zbl 1359.76255 |
| [20] | Hirsikesh, S. N., Pramod, A., Annabattula, R., Ooi, E. and Song, C. [2019] “ Adaptive phase-field modeling of brittle fracture using the scaled boundary finite element method,” Comput. Methods Appl. Mech. Eng.355, 284-307. · Zbl 1441.74206 |
| [21] | Kumar, V. [2013] “ Smoothed finite element methods for thermomechanical impact problems,” Int. J. Comput. Methods10(1), 13400100. · Zbl 1359.74418 |
| [22] | Kumbhar, P. Y., Francis, A., Swaminathan, N., Annabattula, R. K. and Natarajan, S. [2020] “ Development of User Element Routine (UEL) for cell based smoothed finite element method (CSFEM) in Abaqus,” Int. J. Comput. Methods.17(2), 1850128. · Zbl 07124752 |
| [23] | Lee, K., Lim, J. H., Sohn, D. and Im, S. J. [2015] “ A three-dimensional cell-based smoothed finite element method for elasto-plasticity,” J. Mech. Sci. Technol.29(2), 611-623. |
| [24] | Li, E.et al. [2014] “ Hybrid smoothed finite element method for acoustic problems,” Comput. Methods Appl. Mech. Eng.283, 664-688. · Zbl 1423.74902 |
| [25] | Li, E.et al. [2017a] “ Volumetric locking issue with uncertainty in the design of locally resonant acoustic metamaterials,” Comput. Methods Appl. Mech. Eng.324, 128-148. · Zbl 1439.74137 |
| [26] | Li, E.et al. [2017b] “ An ultra-accurate numerical method in the design of liquid phononic crystals with hard inclusion,” Comput. Mech.60, 983-996. |
| [27] | Li, E. and He, Z. C. [2019] “ Optimal balance between mass and smoothed stiffness in simulation of acoustic problems,” Appl. Math. Model.75, 1-22. · Zbl 1481.76191 |
| [28] | Li, E., He, Z. C. and Wang, G. [2016] “ An exact solution to compute the band gap in phononic crystals,” Comput. Mater. Sci.122, 72-85. |
| [29] | Li, E.et al. [2018] “ An efficient algorithm to analyze wave propagation in fluid or solid and solid or fluid phononic crystals,” Comput. Methods Appl. Mech. Eng.333, 421-442. · Zbl 1440.74416 |
| [30] | Liu, G. R. [2008] “ A generalized gradient smoothing technique and the smoothed bilinear form for Galerkin formulation of a wide class of computational methods,” Int. J. Comput. Methods5(2), 199-236. · Zbl 1222.74044 |
| [31] | Liu, G. R., Chen, L. and Li, M. [2014] “ S-FEM for fracture problems, theory, formulation and application,” Int. J. Comput. Methods11(3), 1343003. · Zbl 1359.74025 |
| [32] | Liu, G. R., Dai, K. Y.Nguyen, T. T. [2006] “ A smoothed finite element method for mechanics problems,” Comput. Mech.39(6), 859-877. · Zbl 1169.74047 |
| [33] | Liu, G. R., Nguyen, T. T., Dai, K. Y. and Lam, K. Y. [2007] “ Theoretical aspects of the smoothed finite element method (SFEM),” Int. J. Numer. Methods Eng.71(8), 902-930. · Zbl 1194.74432 |
| [34] | Liu, G. R., Nguyen, T. T., Nguyen, X. H., Dai, K. Y. and Lam, K. Y. [2009] “ On the essence and the evaluation of the shape functions for the smoothed finite element method (SFEM),” Int. J. Numer. Methods Eng.77(13), 1863-1869. · Zbl 1181.74137 |
| [35] | Liu, G. R., Nourbakhshnia, N., Chen, L. and Zhang, Y. W. [2010b] “ A novel general formulation for singular stress field using the ES-FEM method for the analysis of mixed-mode cracks,” Int. J. Comput. Methods7(1), 191-214. · Zbl 1267.74112 |
| [36] | Liu, G. R., Zeng, W.Nguyen, X. H. [2013] “ Generalized stochastic cell-based smoothed finite element method(GS-CS-FEM) for solid mechanics,” Finite Elem. Anal. Des.63, 51-61. · Zbl 1282.74090 |
| [37] | Liu, G. R., Nguyen, X. H. and Nguyen, T. T. [2010a] “ A theoretical study on the smoothed FEM (S-FEM) models: Properties, accuracy and convergence rates,” Int. J. Numer. Methods Eng.84(10), 1222-1256. · Zbl 1202.74180 |
| [38] | Liu, G., Li, Q., Msekh, M. A. and Zuo, Z. [2016] “ Abaqus implementation of monolithic and staggered schemes for quasi-static and dynamic fracture phase-field model,” Comput. Mater. Sci.121, 35-47. |
| [39] | Liu, G. R. [2016] “ An overview on meshfree methods: For computational solid mechanics,” Int. J. Comput. Methods13(5), 1630001. · Zbl 1359.74388 |
| [40] | Miehe, C., Hofacker, M. and Welschinger, F. [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. · Zbl 1231.74022 |
| [41] | Miehe, C. and 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 |
| [42] | Miehe, C., Welschinger, F. and Hofacker, M. [2010b] “ 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 |
| [43] | Moës, N., Dolbow, J. and Belytschko, T. [1999] “ A finite element method for crack growth without remeshing,” Int. J. Numer. Methods Eng.46(1), 131150. · Zbl 0955.74066 |
| [44] | Molnár, G. and Gravouil, A. [2017] “ 2D and 3D Abaqus implementation of a robust staggered phase-field solution for modeling brittle fracture,” Finite Elem. Anal. Des.130, 27-38. |
| [45] | Msekh, M. A., Sargado, J. M., Jamshidian, M., Areias, P. M. and Rabczuk, T. [2015] “ Abaqus implementation of phase-field model for brittle fracture,” Comput. Mater. Sci.96, 472-484. |
| [46] | Nagaraja, S., Elhaddad, M., Ambati, M., Kollmannsberger, S., De Lorenzis, L. and Rank, E. [2018] “ Phase-field modeling of brittle fracture with multi-level hp-FEM and the finite cell method,” Comput. Mech.63, https://doi.org/10.1007/s00466-018-1649-7. · Zbl 1465.74152 |
| [47] | Nguyen, T. T., Liu, G. R. and Nguyen, X. H. [2009] “ Additional properties of the node-based smoothed finite element method (NS-FEM) for solid mechanics problems,” Int. J. Comput. Methods6(4), 633-666. · Zbl 1267.74115 |
| [48] | Nguyen, T. T., Phung, V. P., Rabczuk, T., Nguyen, X. H. and LeVan, C. [2013] “ Free and forced vibration analysis using the \(n\)-sided polygonal cell-based smoothed finite element method (nCS-FEM),” Int. J. Comput. Methods10(1), 1340008. · Zbl 1359.74433 |
| [49] | Nguyen, T. T., Réthoré, J. and Baietto, M. C. [2017] “ Phase field modeling of anisotropic crack propagation,” Eur. J. Mech. A Solid65, 279-288. · Zbl 1406.74602 |
| [50] | Nguyen, T. T., Liu, G. R., Dai, K. Y. and Lam, K. Y. [2007] “ Selective smoothed finite element method,” Tsing. Sci. Technol.12(5), 497-508. |
| [51] | Phan, D. H., Nguyen, X. H., Thai, H. C., Nguyen, T. T. and Rabczuk, T. [2013] “ An edge-based smoothed finite element method for analysis of laminated composite plates,” Int. J. Comput. Methods10(1), 1340005. · Zbl 1359.74434 |
| [52] | Roy, P., Pathrikar, A., Deepu, S. P. and Roy, D. [2017] “ Peridynamics damage model through phase field theory,” Int. J. Mech. Sci.128-129, 181-193. |
| [53] | Rybicki, E. F. and Kanninen, M. F. [1977] “ A finite element calculation of stress intensity factors by a modified crack closure integral,” Eng. Fract. Mech.9(4), 931-938. |
| [54] | Schlüter, A., Willenbücher, A., Kuhn, C. and Müller, R. [2014] “ Phase field approximation of dynamic brittle fracture,” Comput. Mech.54(5), 1141-1161. · Zbl 1311.74106 |
| [55] | Verhoosel, C. V. and De Borst, R. [2013] “ A phase field model for cohesive fracture,” Int. J. Numer. Methods Eng.96(1), 43-62. · Zbl 1352.74029 |
| [56] | Wang, S. [2014] An ABAQUS implementation of the cell-based smoothed finite element method using quadrilateral elements, Master thesis, University of Cincinnati. |
| [57] | Winkler, B. [2001] Traglastunter suchungen von unbewehrten undbewehrten Betonstrukturen auf der Grundlage eines objektiven Werkstoffgesetzes fürBeton, Dissertation, University of Innsbruck, Austria. |
| [58] | Wu, F., Liu, G. R., Li, G. Y., Cheng, A. G. and He, Z. C. [2014] “ A new hybrid smoothed FEM for static and free vibration analyses of Reissner-Mindlin Plates,” Comput. Mech.54(3), 865-890. · Zbl 1311.74131 |
| [59] | Xu, X. P. and Needleman, A. [1994] “ Numerical simulations of fast crack growth in brittle solids,” J. Mech. Phys. Solid42(9), 1397-1434. · Zbl 0825.73579 |
| [60] | Zeng, W., Liu, G. R., Jiang, C., Dong, X. W., Chen, H. D., Bao, Y. and Jiang, Y. [2016] “ An effective fracture analysis method based on the virtual crack closure-integral technique implemented in CS-FEM,” Appl. Math. Model.40(5-6), 3783-3800. · Zbl 1459.74168 |
| [61] | Zeng, W. and Liu, G. R. [2018] “ Smoothed finite element methods (S-FEM): An overview and recent developments,” Arch. Comput. Method E25(2), 397-435. · Zbl 1398.65312 |
| [62] | Zhang, Z. Q., Liu, G. R. and Khoo, B. C. [2012] “ Immersed smoothed finite element method for two dimensional fluid-structure interaction problems,” Int. J. Numer. Methods Eng.90(10), 1292-1320. · Zbl 1242.74178 |
| [63] | Zhang, Z. Q., Liu, G. R. and Khoo, B. C. [2013] “ A three dimensional immersed smoothed finite element method (3D IS-FEM) for fluid-structure interaction problems,” Comput. Mech.51(2), 129-150. · Zbl 1312.74049 |
| [64] | Zhang, G.et al. [2018] “ A combination of singular cell-based smoothed radial point interpolation method and FEM in solving fracture problem,” Int. J. Comput. Methods15(8), 1850079. · Zbl 1404.74179 |
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.
