A new method for the generation of arbitrarily shaped 3D random polycrystalline domains. (English) Zbl 1309.74058

Summary: In this paper a new method for the generation and meshing of arbitrarily shaped three-dimensional polycrystalline models is presented. The discretization is based on Voronoi tessellation, which is shown to be statistically representative of the microstructure of polycrystalline materials. An original approach is introduced to define any possible (concave or convex) shape of the final domain, independently from the initial configuration of the aggregate. Firstly the Voronoi cells are cropped along arbitrarily oriented planes to generate a convex domain, and then an arbitrary number of cuts are performed along planar surfaces to generate the final concave domain. Finally the grains are discretised separately and assembled together to create a finite element model. Several examples are presented to show the capability of generated virtual samples to simulate the behaviour of real polycrystalline materials. The macroscopic elastic properties of polycrystals consisting of anisotropic (trigonal) grains and the stress intensity factor at the tip of a sharp notch are evaluated and compared both with analytical calculations and experimental evidences, showing excellent agreement.


74N15 Analysis of microstructure in solids
05C90 Applications of graph theory
74S05 Finite element methods applied to problems in solid mechanics


Gmsh; geom3d; Neper
Full Text: DOI


[1] Burke, S; Cousland, S; Scala, C, Nondestructive characterization of advanced composite materials, Met Forum, 18, 85-109, (1994)
[2] King, R; Delaney, P, Confocal microscopy, Met Forum, 18, 21-29, (1994)
[3] Marrow, TJ; Briggs, GAD; Roberts, SG, In-situ scanning acoustic microscopy of crack bridging in alumina, J Eur Ceram Soc, 14, 111-116, (1994)
[4] Wilkinson, AJ; Hirsch, PB, Electron diffraction based techniques in scanning electron microscopy of bulk materials, Micron, 28, 279-308, (1997)
[5] Bhandari, Y; Sarkar, S; Groeber, M; Uchic, MD; Dimiduk, DM; Ghosh, S, 3D polycrystalline microstructure reconstruction from FIB generated serial sections for FE analysis, Comput Mater Sci, 41, 222-235, (2007)
[6] Groeber, MA; Haley, BK; Uchic, MD; Dimiduk, DM; Ghosh, S, 3D reconstruction and characterization of polycrystalline microstructures using a FIB-SEM system, Mater Charact, 57, 259-273, (2006)
[7] Groeber, M, A framework for automated analysis and simulation of 3d polycrystalline microstructures. part 2: synthetic structure generation, Acta Mater, 56, 1274-1287, (2008)
[8] Groeber, M, A framework for automated analysis and simulation of 3d polycrystalline microstructures. part 1: statistical characterization, Acta Mater, 56, 1257-1273, (2008)
[9] Baczmanski, A; Wierzbanowski, K; Lipinski, P; Helmholdt, RB; Ekambaranathan, G; Pathiraj, B, Examination of the residual stress field in plastically deformed polycrystalline material, Philos Mag A, 69, 437-449, (1994)
[10] Ortiz, M; Suresh, S, Statistical properties of residual stresses and intergranular fracture in ceramic materials, J Appl Mech, 60, 77-84, (1993)
[11] Sukumar, N; Srolovitz, DJ; Baker, TJ; Prevost, JH, Brittle fracture in polycrystalline microstructures with the extended finite element method, Int J Numer Methods Eng, 56, 2015-2037, (2003) · Zbl 1038.74652
[12] Kumar, S; Kurtz, SK, Simulation of material microstructure using a 3d Voronoi tessellation: calculation of effective thermal expansion coefficient of polycrystalline materials, Acta Met Mater, 42, 3917-3927, (1994)
[13] Kumar, S; Kurtz, SK; Agarwala, V, Micro-stress distribution within polycrystalline aggregate, Acta Mech, 114, 203-216, (1996) · Zbl 0857.73068
[14] Ghosh, S; Lee, K; Moorthy, S, Multiple scale analysis of heterogeneous elastic structures using homogenization theory and Voronoi cell finite element method, Int J Solids Struct, 32, 27-62, (1994) · Zbl 0865.73060
[15] Wu, MS; Niu, J, A theoretical analysis of crack nucleation due to grain boundary dislocation pile-ups in a random ice microstructure, Philos Mag A, 71, 831-854, (1995)
[16] Wu, MS; He, MD, Prediction of crack statistics in a random polycrystal damaged by the pile-ups of extrinsic grain-boundary dislocations, Philos Mag A, 79, 271-292, (1999)
[17] Raabe, D; Zhao, Z; Mao, W, On the dependence of in-grain subdivision and deformation texture of aluminium on grain interaction, Acta Mater, 50, 4379-4394, (2002)
[18] Zhao, Z; Kuchnicki, S; Radovitzky, R; Cuitino, A, Influence of in-grain mesh resolution on the prediction of deformation textures in FCC polycrystals by crystal plasticity FEM, Acta Mater, 55, 2361-2373, (2007)
[19] Jivkov AP, Marrow TJ (2007) Rates of intergranular environment assisted cracking in three-dimensional model microstructures. Theor Appl Fract Mech 48(3):187-202 · Zbl 1228.74093
[20] Wakai F, Enomoto N, Ogawa H (2000) Three dimensional microstructural evolution in ideal grain growth—general statistics. Acta Mater 48:1297-1311
[21] Aurenhammer, F, Voronoi diagrams—a survey of fundamental geometric data structure, ACM Comput Surv, 23, 345-405, (1991)
[22] Barbe, F; Decker, L; Jeulin, D; Cailletaud, G, Intergranular and intragranular behaviour of polycrystalline aggregates. part 1: FE model, Int J Plast, 17, 513-536, (2001) · Zbl 1067.74011
[23] Barbe, F; Decker, L; Jeulin, D; Cailletaud, G, Intergranular and intragranular behaviour of polycrystalline aggregates. part 2: results, Int J Plast, 17, 537-563, (2001) · Zbl 1067.74012
[24] Fritzen, F; Böhlke, T; Schnack, E, Periodic three-dimensional mesh generation for crystalline aggregates based on Voronoi tessellations, Comput Mech, 43, 701-713, (2008)
[25] Fritzen, F; Böhlke, T, Periodic three-dimensional mesh generation for particle reinforced composites with application to metal matrix composites, Int J Solids Struct, 48, 706-718, (2011) · Zbl 1236.74058
[26] Quey, R; Dawson, P; Barbe, F, Large-scale 3d random polycrystals for the finite element method: generation, meshing and remeshing, Comput Methods Appl Mech Eng, 200, 1729-1745, (2011) · Zbl 1228.74093
[27] Qian, J; Zhang, Y; Wang, W; Lewis, AC; Qidwai, M; Geltmacher, AB, Quality improvement of non-manifold hexahedral meshes for critical feature determination of microstructure materials, Int J Numer Methods Eng, 82, 1406-1423, (2010) · Zbl 1188.74094
[28] Yoon, S; Akatsu, T; Yasuda, E, Anisotropy of creep deformation rate in hot-pressed si\(_3\)N\(_4\) with preferred orientation of the elongated grains, J Mater Sci, 32, 3813-3819, (1997)
[29] Liu, Q; Juul Jensen, D; Hansen, N, Effect of grain orientation on deformation structure in cold-rolled polycrystalline aluminium, Acta Mater, 46, 5819-5838, (1998)
[30] Ramella, M; Boschin, W; Fadda, D; Nonino, M, Finding galaxy clusters using Voronoi tessellations, Astron Astrophys, 368, 776-786, (2001)
[31] Poupon, A, Voronoi and Voronoi-related tessellations in studies of protein structure and interaction, Curr Opin Struct Biol, 14, 233-241, (2004)
[32] Boots, B, The arrangment of cells in random network, Metallography, 15, 53-62, (1982)
[33] Du, Q; Faber, V; Gunzburger, M, Centroidal Voronoi tessellations: applications and algorithms, SIAM Rev, 41, 637-676, (1999) · Zbl 0983.65021
[34] Ohser J, Schladitz K (2009) 3D images of materials structures: processing and analysis. Wiley, New York · Zbl 1165.94306
[35] Dobrich, K; Rau, M; Krill III, C, Quantitative characterization of the three-dimensional microstructure of polycrystalline al-sn using X-ray microtomography, Metall Mater Trans A, 35, 1953-1961, (2004)
[36] Rowenhorst DJ, Lewis AC, Spanos G (2010) Three-dimensional analysis of grain topology and interface curvature in a \(β \)-titanium alloy. Acta Mater 58(16):5511-5519
[37] Legland D (2009) Graphics library geom3d · Zbl 1062.74629
[38] Fritzen, F; Böhlke, T, Nonuniform transformation field analysis of materials with morphological anisotropy, Compos Sci Technol, 71, 433-442, (2011)
[39] Geuzaine, C; Remacle, J-F, Gmsh: a 3-d finite element mesh generator with built-in pre- and post-processing facilities, Int J Numer Methods Eng, 79, 1309-1331, (2009) · Zbl 1176.74181
[40] Vodenitcharova, T; Zhang, LC; Zarudi, I; Yin, Y; Domyo, H; Ho, T; Sato, M, The effect of anisotropy on the deformation and fracture of sapphire wafers subjected to thermal shocks, J Mater Process Technol, 194, 52-62, (2007)
[41] Nye J (1957) Physical properties of crystals: their representation by tensor and matrices. Clarendon Press, Oxford · Zbl 0079.22601
[42] Gladden, JR; So, JH; Maynard, JD; Saxe, PW; Page, Y, Reconciliation of ab initio theory and experimental elastic properties of al\(_2\)O\(_3\), Appl Phys Lett, 85, 392, (2004)
[43] Hovis, DB; Reddy, A; Heuer, AH, X-ray elastic constants for \(α \)-al\(_2\)O\(_3\), Appl Phys Lett, 88, 131910, (2006)
[44] Winey, JM; Gupta, YM; Hare, DE, R-axis sound speed and elastic properties of sapphire single crystals, J Appl Phys, 90, 3109, (2001)
[45] Ostoja-Starzewski, M, Material spatial randomness: from statistical to representative volume element, Probab Eng Mech, 21, 112-132, (2006)
[46] Benedetti, I; Aliabadi, MH, A three-dimensional grain boundary formulation for microstructural modeling of polycrystalline materials, Comput Mater Sci, 67, 249-260, (2013)
[47] Ren, ZY; Zheng, QS, Effects of grain sizes, shapes, and distribution on minimum sizes of representative volume elements of cubic polycrystals, Mech Mater, 36, 1217-1229, (2004)
[48] Nygards, M, Number of grains necessary to homogenize elastic materials with cubic symmetry, Mech Mater, 35, 1049-1057, (2003)
[49] Zener C (1948) Elasticity and anelasticity of metals. University of Chicago Press, Chicago · Zbl 0032.22202
[50] Chung, DH; Simmons, G, Pressure and temperature dependences of the isotropic elastic moduli of polycrystalline alumina, J Appl Phys, 39, 5316-5326, (1968)
[51] Kanit, T; Forest, S; Galliet, I; Mounoury, V; Jeulin, D, Determination of the size of the representative volume element for random composites: statistical and numerical approach, Int J Solids Struct, 40, 3647-3679, (2003) · Zbl 1038.74605
[52] Lemaitre J, Chaboche J (1990) Mechanics of soilid materials. Cambridge University Press, Cambridge · Zbl 0743.73002
[53] Gross, B; Mendelson, A, Plane elastostatic analysis of v-notched plates, Int J Fract Mech, 8, 267-276, (1972)
[54] Tabiei, A; Wu, J, Development od the DYNA3D simulation code with automated fracture procedure for brick elements, Int J Numer Methods Eng, 57, 1979-2006, (2002) · Zbl 1062.74629
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