×

Brittle fracture in polycrystalline microstructures with the extended finite element method. (English) Zbl 1038.74652

Summary: A two-dimensional numerical model of microstructural effects in brittle fracture is presented, with an aim towards the understanding of toughening mechanisms in polycrystalline materials such as ceramics. Quasi-static crack propagation is modelled using the extended finite element method (X-FEM) and microstructures are simulated within the framework of the Potts model for grain growth. In the X-FEM, a discontinuous function and the two-dimensional asymptotic crack-tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity. This enables the domain to be modelled by finite elements with no explicit meshing of the crack surfaces. Hence, crack propagation can be simulated without any user-intervention or the need to remesh as the crack advances. The microstructural calculations are carried out on a regular lattice using a kinetic Monte Carlo algorithm for grain growth. We present a novel constrained Delaunay triangulation algorithm with grain boundary smoothing to create a finite element mesh of the microstructure. The fracture properties of the microstructure are characterized by assuming that the critical fracture energy of the grain boundary (\(G_{\text c}^{\text{gb}}\)) is different from that of the grain interior (\(G_{\text c}^{\text i}\)). Numerical crack propagation simulations for varying toughness ratios \(G_{\text c}^{\text{gb}}/G_{\text c}^{\text i}\) are presented, to study the transition from the intergranular to the transgranular mode of crack growth. This study has demonstrated the capability of modelling crack propagation through a material microstructure within a finite element framework, which opens-up exciting possibilities for the fracture analysis of functionally graded material systems.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74R10 Brittle fracture
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Watanabe, The impact of grain boundary character distribution on fracture in polycrystals, Materials Science and Engineering A 176 pp 39– (1994) · doi:10.1016/0921-5093(94)90957-1
[2] Aust, Interface control for resistance to intergrangular cracking, Materials Science and Engineering A 176 pp 329– (1994) · doi:10.1016/0921-5093(94)90995-4
[3] Kumar, Modifications to the microstructural topology in f.c.c. materials through thermomechanical processing, Acta Materialia 48 (9) pp 2081– (2000) · doi:10.1016/S1359-6454(00)00045-8
[4] Beale, Elastic fracture in random materials, Physics Review B 37 (10) pp 5500– (1998) · doi:10.1103/PhysRevB.37.5500
[5] Curtin, Brittle fracture of disordered materials, Journal of Materials Research 5 (3) pp 535– (1990)
[6] Curtin, Mechanics modeling using a spring network, Journal of Materials Research 5 (3) pp 554– (1990)
[7] Jagota, Nonlinearity and Breakdown in Soft Condensed Matter, Springer Lecture Notes in Physics 437 pp 186– (1994)
[8] Jagota, Element breaking rules in computational models for brittle fracture, Modelling and Simulation in Materials Science and Engineering 3 (4) pp 485– (1995) · doi:10.1088/0965-0393/3/4/005
[9] Schlangen, New method for simulating fracture using an elastically uniform random geometry, International Journal of Engineering Science 34 (10) pp 1131– (1996) · Zbl 0900.73598 · doi:10.1016/0020-7225(96)00019-5
[10] Bolander, Fracture analyses using spring networks with random geometry, Engineering Fracture Mechanics 61 pp 569– (1998) · doi:10.1016/S0013-7944(98)00069-1
[11] Yang, Simulation and Theory of Evolving Microstructures pp 277– (1990)
[12] Potts, Some generalized order-disorder transformations, Proceedings of the Cambridge Philosophical Society 48 pp 106– (1952) · Zbl 0048.45601
[13] Srolovitz, Grain growth in two dimensions, Scripta Metallurgica 17 pp 241– (1983) · doi:10.1016/0036-9748(83)90106-0
[14] Srolovitz, Computer simulation of grain growth: I. Kinetics, Acta Metallurgica 32 (5) pp 783– (1984) · doi:10.1016/0001-6160(84)90151-2
[15] Srolovitz, Computer simulation of grain growth: II Grain size distribution, topology and local dynamics, Acta Metallurgica 32 (5) pp 793– (1984) · doi:10.1016/0001-6160(84)90152-4
[16] Grest, Computer simulation of normal grain growth in three dimensions, Philosophical Magazine B59 (3) pp 293– (1988)
[17] Sridhar, Microstructural mechanics model of anisotropic-thermal-expansion-induced microcracking, Journal of the American Ceramic Society 77 (5) pp 1123– (1994)
[18] Zimmermann, Damage evolution during microcracking of brittle solids, Acta Materialia 49 pp 127– (2001) · doi:10.1016/S1359-6454(00)00294-9
[19] Holm, Surface formation energy for intergranular fracture in two-dimensional polycrystals, Journal of the American Ceramic Society 81 (3) pp 455– (1998)
[20] Kim, Characteristics of 2-dimensional crack propagation behavior by simulation and analysis, International Journal of Fracture 75 pp 247– (1996) · doi:10.1007/BF00037085
[21] Zhai, Finite element analysis of micromechanical failure modes in a heterogeneous ceramic material system, International Journal of Fracture 101 pp 161– (2000) · doi:10.1023/A:1007545105723
[22] Xu, Numerical simulations of fast crack growth in brittle solids, Journal of the Mechanics and Physics of Solids 42 pp 1397– (1994) · Zbl 0825.73579 · doi:10.1016/0022-5096(94)90003-5
[23] Zavattieri, A computational model of ceramic microstructures subjected to multi-axial dynamic loading, Journal of the Mechanics and Physics of Solids 49 pp 27– (2001) · Zbl 1013.74055 · doi:10.1016/S0022-5096(00)00028-4
[24] Klein, Advances in Computational Engineering and Sciences pp 1790– (2000)
[25] Ortiz, A finite element method for localized failure analysis, Computer Methods in Applied Mechanics and Engineering 61 pp 189– (1987) · Zbl 0597.73105 · doi:10.1016/0045-7825(87)90004-1
[26] Belytschko, A finite element with embedded localization zones, Computer Methods in Applied Mechanics and Engineering 70 pp 59– (1988) · Zbl 0653.73032 · doi:10.1016/0045-7825(88)90180-6
[27] Dvorkin, Finite elements with displacement interpolated embedded localization lines insensitive to mesh size and distortions, International Journal for Numerical Methods in Engineering 30 pp 541– (1990) · Zbl 0729.73209
[28] Simo, An analysis of strong discontinuities induced by strain softening in rate-independent inelastics solids, Computational Mechanics 12 pp 277– (1993) · Zbl 0783.73024 · doi:10.1007/BF00372173
[29] Simo, Computational Modelling of Concrete Structures pp 363– (1994)
[30] Armero, An analysis of strong discontinuities in multiplicative finite strain plasticity and their relation with the numerical simulation of strain localization in solids, International Journal of Solids and Structures 33 pp 2863– (1996) · Zbl 0924.73084 · doi:10.1016/0020-7683(95)00257-X
[31] Sluys, Discontinuous failure analysis for mode-I and mode-II localization problems, International Journal of Solids and Structures 35 pp 4257– (1998) · Zbl 0933.74060 · doi:10.1016/S0020-7683(97)00313-2
[32] Larsson, Finite element embedded localization band for finite strain plasticity based on a regularized strong discontinuity, Mechanics of Cohesive-frictional Materials 4 (2) pp 171– (1999) · doi:10.1002/(SICI)1099-1484(199903)4:2<171::AID-CFM81>3.0.CO;2-X
[33] Regueiro, Plane strain finite element analysis of pressure sensitive plasticity with strong discontinuity, International Journal of Solids and Structures 38 (21) pp 3647– (2001) · Zbl 1031.74013 · doi:10.1016/S0020-7683(00)00250-X
[34] Jirásek, Comparative study on finite elements with embedded discontinuities, Computer Methods in Applied Mechanics and Engineering 188 pp 307– (2000) · Zbl 1166.74427 · doi:10.1016/S0045-7825(99)00154-1
[35] Moës, A finite element method for crack growth without remeshing, International Journal for Numerical Methods in Engineering 46 (1) pp 131– (1999) · Zbl 0955.74066 · doi:10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J
[36] Daux, Arbitrary cracks and holes with the extended finite element method, International Journal for Numerical Methods in Engineering 48 (12) pp 1741– (2000) · Zbl 0989.74066 · doi:10.1002/1097-0207(20000830)48:12<1741::AID-NME956>3.0.CO;2-L
[37] Belytschko, Arbitrary discontinuities in finite elements, International Journal for Numerical Methods in Engineering 50 (4) pp 993– (2001) · Zbl 0981.74062 · doi:10.1002/1097-0207(20010210)50:4<993::AID-NME164>3.0.CO;2-M
[38] Melenk, The partition of unity finite element method: Basic theory and applications, Computer Methods in Applied Mechanics and Engineering 139 pp 289– (1996) · Zbl 0881.65099 · doi:10.1016/S0045-7825(96)01087-0
[39] Duarte, An H-p adaptive method using clouds, Computer Methods in Applied Mechanics and Engineering 139 pp 237– (1996) · Zbl 0918.73328 · doi:10.1016/S0045-7825(96)01085-7
[40] Sukumar, Extended finite element method for three-dimensional crack modeling, International Journal for Numerical Methods in Engineering 48 (11) pp 1549– (2000) · Zbl 0963.74067 · doi:10.1002/1097-0207(20000820)48:11<1549::AID-NME955>3.0.CO;2-A
[41] Oden, A new cloud-based hp finite element method, Computer Methods in Applied Mechanics and Engineering 153 (1-2) pp 117– (1998) · Zbl 0956.74062 · doi:10.1016/S0045-7825(97)00039-X
[42] Strouboulis, The generalized finite element method, Computer Methods in Applied Mechanics and Engineering 190 (32-33) pp 4081– (2001) · Zbl 0997.74069 · doi:10.1016/S0045-7825(01)00188-8
[43] Belytschko, Elastic crack growth in finite elements with minimal remeshing, International Journal for Numerical Methods in Engineering 45 (5) pp 601– (1999) · Zbl 0943.74061 · doi:10.1002/(SICI)1097-0207(19990620)45:5<601::AID-NME598>3.0.CO;2-S
[44] Sukumar, Extended finite element method and fast marching method for three-dimensional fatigue crack propagation, Engineering Fracture Mechanics 70 pp 29– (2003) · Zbl 1211.74199 · doi:10.1016/S0013-7944(02)00032-2
[45] Chopp DL Sukumar N Fatigue crack propagation of multiple coplanar cracks with the coupled extended finite element method/fast marching method 2003 · Zbl 1211.74199
[46] Moës, Non-planar 3D crack growth by the extended finite element and level sets. Part I: Mechanical model, International Journal for Numerical Methods in Engineering 53 (11) pp 2549– (2002) · Zbl 1169.74621 · doi:10.1002/nme.429
[47] Gravouil, Non-planar 3D crack growth by the extended finite element and the level sets-Part II: Level set update, International Journal for Numerical Methods in Engineering 53 (11) pp 2569– (2002) · Zbl 1169.74621 · doi:10.1002/nme.430
[48] Wells, A new method for modelling cohesive cracks using finite elements, International Journal for Numerical Methods in Engineering 50 (12) pp 2667– (2001) · Zbl 1013.74074 · doi:10.1002/nme.143
[49] Sukumar, Modeling holes and inclusions by level sets in the extended finite element method, Computer Methods in Applied Mechanics and Engineering 190 (46-47) pp 6183– (2001) · Zbl 1029.74049 · doi:10.1016/S0045-7825(01)00215-8
[50] Wu, The Potts model, Reviews of Modern Physics 54 pp 235– (1982) · doi:10.1103/RevModPhys.54.235
[51] Carter WC Langer SA Fuller Jr ER http://www.ctcms.nist.gov/oof 1998
[52] Langer, OOF: An image-based finite-element analysis of material microstructures, Computing in Science and Engineering 3 (3) pp 15– (2001) · Zbl 05092017 · doi:10.1109/5992.919261
[53] Ghosh, A material based finite-element analysis of heterogeneous media involving Dirichlet tessellations, Computer Methods in Applied Mechanics and Engineering 104 (2) pp 211– (1993) · Zbl 0775.73252 · doi:10.1016/0045-7825(93)90198-7
[54] Ghosh, Quantitative characterization and modeling of composite microstructures by Voronoi cells, Acta Materialia 45 (6) pp 2215– (1997) · doi:10.1016/S1359-6454(96)00365-5
[55] Ghosh, Elastic-plastic analysis of arbitrary heterogeneous materials with the Voronoi cell finite-element method, Computer Methods in Applied Mechanics and Engineering 121 (1-4) pp 373– (1995) · Zbl 0853.73065 · doi:10.1016/0045-7825(94)00687-I
[56] Gottstein, Grain Boundary Migration in Metals: Thermodynamics, Kinetics, Applications (1999)
[57] Lee, Generalized Delaunay triangulation for planar graphs, Discrete and Computational Geometry 1 pp 201– (1986) · Zbl 0596.52007
[58] Tanemura, A new algorithm for three-dimensional Voronoi tesselation, Journal of Computational Physics 51 pp 191– (1983) · Zbl 0529.05016 · doi:10.1016/0021-9991(83)90087-6
[59] Bowyer, Computing Dirichlet tessellations, The Computer Journal 24 pp 162– (1981)
[60] Watson, Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes, The Computer Journal 24 (2) pp 167– (1981)
[61] Hassan O Bayne LB Morgan K Weatherill NP An adaptive unstructured mesh method for transient flows involving moving boundaries 1998 668 674
[62] George, Delaunay Triangulation and Meshing (1998) · Zbl 0908.65143
[63] Holmes, Proceedings of the 2nd International Conference on Numerical Grid Generation pp 643– (1988)
[64] Rebay, Efficient unstructured mesh generation by means of Delaunay triangulation and Bowyer-Watson algorithm, Journal of Computational Physics 106 pp 125– (1993) · Zbl 0777.65064 · doi:10.1006/jcph.1993.1097
[65] Chew LP Guaranteed-quality mesh generation for curved surfaces 1993 274 280
[66] Baker, Frontiers of Computational Fluid Dynamics pp 101– (1994)
[67] Moran, Crack tip and associated domain integrals from momentum and energy balance, Engineering Fracture Mechanics 27 (6) pp 615– (1987) · doi:10.1016/0013-7944(87)90155-X
[68] Nikishkov, Calculation of fracture mechanics parameters for an arbitrary three-dimensional crack by the ’equivalent domain integral method’, International Journal for Numerical Methods in Engineering 24 pp 1801– (1987) · Zbl 0625.73118
[69] Yau, A mixed-mode crack analysis of isotropic solids using conservation laws of elasticity, Journal of Applied Mechanics 47 pp 335– (1980) · Zbl 0463.73103
[70] Erdogan, On the crack extension in plates under plane loading and transverse shear, Journal of Basic Engineering 85 pp 519– (1963)
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.