NUMERICAL METHODS FOR MODELING HETEROGENEOUS MATERIALS TRAN THI QUYNH NHU (B.Eng., HCMC University of Technology (2003)) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2009 Acknowledgments Foremost, I would like to express my deep and sincere gratitude to my supervisors, Prof. Lee Heow Pueh and Prof. Lim Siak Piang, for their patient guidance, encouragement and advice during my time in NUS. I would like to thank National University of Singapore for offering me the opportunity and the financial support to pursue the PhD study. I am grateful to the staffs of Dynamics Lab for their help during my time there. I would like to mention all of my Vietnamese friends in Singapore for our unforgettable friendship. Hoang Quang Hung, Dau Van Huan, Le Ngoc Thuy, Nguyen Hoang Huy . (the list would be very long) and the rest of Hoi Coc Oi. I will always remember the fun and the encouragement I have got from them. And finally, I owe my loving gratitude to my parents Tran Du Sinh and Ho Thi Van Nga, and my husband Huynh Dinh Bao Phuong, who have suffered very much along with me during my PhD study. Without their love and support, I would not be able to fulfil my dream. To them I dedicate this thesis. Summary In this thesis, the applications of the conventional finite element method (FEM) and its variations in modeling heterogeneous materials are presented. At first the preliminary work on functionally graded materials (FGM) is presented in Chapter and Chapter 3. In these chapters, the FGM plates under thermal load are investigated using the conventional FEM with the aid of the FEM package ABAQUS. The Voronoi cell finite element method (VCFEM) is studied in Chapter for analyzing the heterogeneous materials. In this chapter, various numerical examples from simple to complicated compositions of heterogeneous materials containing inclusions are studied. In some examples, the quadratic quadrilateral elements are introduced as the 8-node elements for the VCFEM instead of the Voronoi cells. Chapter shows the application of the extended finite element method (XFEM) for heterogeneous materials, including porous structures. Instead of using the Heaviside enrichment function for the strong discontinuity of the holes’ interfaces, the penalty method is introduced to simulate the porous parts. Finally, the thesis is concluded in Chapter with the summary and the suggestions for future work. Contents Introduction 12 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2 Literature Reviews . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3 Review of Functionally graded materials . . . . . . . . . . . . . . . . 14 1.3.1 Review of micromechanical modeling for functionally graded materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3.2 Review of the Voronoi Cell Finite Element Method . . . . . . 17 1.3.3 Review of the Extended Finite Element Method . . . . . . . . 20 1.4 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Thermal induced vibration of functionally graded thin plate 24 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2.1 Material properties . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2.2 The finite element model . . . . . . . . . . . . . . . . . . . . . 27 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3.1 Benchmarking of simulation results for vibration of FG plates 28 2.3.2 Thermal induced vibration of FG plates . . . . . . . . . . . . 31 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.3 2.4 Transient thermal mechanical response of functionally graded thick plates 40 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2.1 Material properties . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2.2 The finite element model . . . . . . . . . . . . . . . . . . . . . 43 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3.1 Time-dependent prescribed temperature at the top surface . . 45 3.3.2 Time-dependent prescribed heat flux at the top surface . . . . 48 3.3.3 Time-dependent prescribe heat flux at partial top surface . . . 52 3.3.4 Comparison between continuous model and layered model . . 54 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.3 3.4 Voronoi Cell Finite Element Method 59 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.3 Voronoi Cell Finite Element Method . . . . . . . . . . . . . . . . . . 62 4.3.1 Element formulation for homogeneous materials . . . . . . . . 62 4.3.2 Element formulation for heterogeneous materials . . . . . . . . 66 4.3.3 Interpolation stress function and the shape of heterogeneities . 69 4.3.4 Numerical implementation . . . . . . . . . . . . . . . . . . . . 72 4.3.5 Mesh generation . . . . . . . . . . . . . . . . . . . . . . . . . 76 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.4.1 A Cantilever beam . . . . . . . . . . . . . . . . . . . . . . . . 77 4.4.2 A unit cell containing a circular fiber . . . . . . . . . . . . . . 79 4.4.3 A unit cell containing elliptic inclusions . . . . . . . . . . . . . 81 4.4.4 A unit cell containing more circular inclusions . . . . . . . . . 82 4.4.5 A unit cell of a composite model with more elliptic inclusions 86 4.4 4.5 4.4.6 A composite plate containing a hole . . . . . . . . . . . . . . . 87 4.4.7 A FGM specimen . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.4.8 A FG cantilever beam . . . . . . . . . . . . . . . . . . . . . . 93 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Extended Finite Element Method 99 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Element formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.3 5.4 99 5.2.1 Level set method . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.2.2 General formulation 5.2.3 Choice of enriched nodes . . . . . . . . . . . . . . . . . . . . . 104 5.2.4 Enrichment function . . . . . . . . . . . . . . . . . . . . . . . 105 5.2.5 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 106 . . . . . . . . . . . . . . . . . . . . . . . 103 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.3.1 A unit cell containing a circular inclusion . . . . . . . . . . . . 106 5.3.2 A FGM specimen . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.3.3 A unit cell containing holes . . . . . . . . . . . . . . . . . . . 110 5.3.4 A specimen made of porous material . . . . . . . . . . . . . . 112 5.3.5 A unit cell of a bone model . . . . . . . . . . . . . . . . . . . 115 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Conclusions 122 6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.2 Suggestions for future work . . . . . . . . . . . . . . . . . . . . . . . 125 A List of Publications 127 B Material properties implementation into ABAQUS 128 C Interpolation Polynomial Matrices 130 List of Figures 2-1 FGM thin plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2-2 Temperature distribution through the thickness of square FG plates subjected to suddenly applied heat flux q = 106 W/m2 . . . . . . . . . . . . . 33 2-2 Temperature distribution through the thickness of square FG plates subjected to suddenly applied heat flux q = 106 W/m2 . . . . . . . . . . . . . 34 2-3 Historical central displacement of simply supported FG square plates subjected to a suddenly applied heat flux q = 106 W/m2 . . . . . . . . . . . . 35 2-4 Historical central displacement of simply supported FG square plates subjected to a suddenly temperature rise Ttop = 200o K . . . . . . . . . . . 36 2-5 Historical central displacement of simply supported FG square plates (n = 2) under different temperature rises. . . . . . . . . . . . . . . . . . . . . 37 2-6 Temperature distribution through the thickness of square FG plate: a comparison between the dynamic solution and the quasi-static solution. . . . . 38 3-1 Geometry of the plate . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3-2 A 10 × 10 × 20 finite element mesh of the plate . . . . . . . . . . . . . . 44 3-3 Historical response of the FGM Al/SiC plate with the sinusoidal temperature distribution over the top surface . . . . . . . . . . . . . . . . . . . 47 3-4 Historical response of the FGM Al/SiC plate with the uniform temperature distribution over the top surface . . . . . . . . . . . . . . . . . . . . . . 49 3-5 Historical response of an FGM Al/SiC plate with the sinusoidal heat flux distribution over the top surface . . . . . . . . . . . . . . . . . . . . . . 51 3-6 Historical response of an FGM Al/SiC plate with the uniform heat flux distribution over the top surface . . . . . . . . . . . . . . . . . . . . . . 52 3-7 Historical response of the FGM Al/SiC plate with the uniform heat flux distribution on partial area of the top surface . . . . . . . . . . . . . . . 53 3-8 Historical response of the FGM Al/SiC plate with the uniform temperature distribution over the top surface . . . . . . . . . . . . . . . . . . . . . . 55 3-9 Through the thickness profile of a FGM Al/SiC plate with the uniform temperature distribution over the top surface . . . . . . . . . . . . . . . 56 4-1 A Voronoi cell finite element with an inclusion . . . . . . . . . . . . . . . 60 4-2 Element sub-division and Gaussian integration points: (a) no additional ring (b) one additional ring with smaller Gaussian integration points in each triangle. Here (+) are Gauss points for domain integration; (◦) are Gauss points for line integration . . . . . . . . . . . . . . . . . . . . . . 74 4-3 Meshes of the cantilever beam . . . . . . . . . . . . . . . . . . . . . . . 78 4-4 Displacement of the cantilever beam: VCFEM (thick line) and FEM (thin line). Displacement solution is scaled by a factor of 1000 . . . . . . . . . 78 4-5 Von Mises stress distribution: VCFEM (top . . . . . . . . . . . . . . . . 79 4-6 Meshes of the unit cell containing an inclusion . . . . . . . . . . . . . . . 80 4-7 Displacement of the unit cell containing an inclusion: VCFEM (thick line) and FEM (thin line) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4-8 Von Mises stress of the unit cell containing an inclusion: FEM (left) and VCFEM (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4-9 Meshes of a bi-unit cell containing inclusions . . . . . . . . . . . . . . . 82 4-10 Displacement solution: VCFEM (thick line) and FEM (thin line) . . . . . 83 4-11 Von Mises stress comparison: FEM (left) and VCFEM (right) . . . . . . . 83 4-12 Meshes of the composite unit cell containing 29 circular inclusions . . . . 84 4-13 Displacement of the composite unit cell containing 29 circular inclusions: VCFEM (thickline) and FEM (thin line) . . . . . . . . . . . . . . . . . . 85 4-14 Von Mises stress distribution of the composite unit cell containing 29 circular inclusions: FEM (left) and VCFEM (right) . . . . . . . . . . . . . . . . 85 4-15 Meshes of the composite unit cell containing 36 elliptic inclusions . . . . . 87 4-16 Displacement of the composite unit cell containing 36 elliptic inclusions: VCFEM (thick line) and FEM (thin line) . . . . . . . . . . . . . . . . . 87 4-17 Von Mises stress of the composite unit cell containing 36 elliptic inclusions: FEM (left) and VCFEM (right) . . . . . . . . . . . . . . . . . . . . . . 88 4-18 Meshes of a plate containing a hole . . . . . . . . . . . . . . . . . . . . 89 4-19 Displacement of a composite plate containing a hole: VCFEM (thick line) and FEM (thin line). Displacement is scaled by a factor of 1/5 . . . . . . 89 4-20 Von Mises stress of a composite plate containing a hole: FEM (left) and VCFEM (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4-21 Meshes of the FGM specimen . . . . . . . . . . . . . . . . . . . . . . . 91 4-22 Displacement of the FGM specimen: VCFEM (thick line) and FEM (thin . . . . . . . . . . . . . . 92 4-23 Von Mises stress of the FGM specimen: VCFEM (top . . . . . . . . . . . 92 line). Displacement is scaled by a factor of 1/5 4-24 Effect of the Young’s modulus on the maximum displacement of the FGM specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4-25 Meshes of the FG cantilever beam . . . . . . . . . . . . . . . . . . . . . 95 4-26 Displacement of the FG cantilever beam: VCFEM (thick line) and FEM . . . . . . 96 4-27 Von Mises stress distribution of the FG cantilever beam: VCFEM (top . . 96 (thin line). Displacement solution is scaled by a factor of 1000 4-28 Effect of the Young’s modulus on the tip displacement of the FG cantilever beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5-1 Level set (a) zero level set (b) level set contour (c) level set function . . . 101 5-2 Level set (a) zero level set (b) level set contour (c) level set function . . . 102 5-3 Domain subdivision and integration points . . . . . . . . . . . . . . . . . 106 5-4 Meshes (a) FEM (b) XFEM: ◦ indicates enriched nodes, and enriched elements are those with thick line . . . . . . . . . . . . . . . . . . . . . . . 107 5-5 Total displacement of the unit cell containing one inclusion . . . . . . . . 108 5-6 Von-mises stress distribution of the unit cell containing one inclusion: FEM (left) and XFEM (right) . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5-7 XFEM mesh of the FGM specimen: ◦ indicates enriched nodes, and enriched elements are those with thick line . . . . . . . . . . . . . . . . . . . . . 109 5-8 Displacement of the FGM specimen: FEM solution(left) and XFEM solution(right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5-9 Von-mises stress distribution of the FGM specimen: FEM (left) and XFEM (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5-10 Meshes of the porous unit cell(a) FEM (b) VCFEM . . . . . . . . . . . . 111 5-11 XFEM mesh of the porous unit cell: ◦ indicates enriched nodes, and enriched elements are those with thick line . . . . . . . . . . . . . . . . . . . . . 112 5-12 Displacement of the porous unit cell: VCFEM solution . . . . . . . . . . 113 5-13 Displacement of the porous unit cell: FEM solution(left) and XFEM solution(right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5-14 Von-mises stress distribution of the porous unit cell: FEM (left) and XFEM (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5-15 Meshes of the specimen made of porous material (a) FEM (b) XFEM: ◦ indicates enriched nodes, and enriched elements are those with thick line . 115 5-16 Displacement solution of the specimen made of porous material: FEM (left) and XFEM (right). Displacement is scaled by a factor of 1/5 . . . . . . . 116 5-17 Von-mises stress comparison: FEM (left) and XFEM (right) . . . . . . . 116 [...]... 7.8902 7 .14 01 7.2544 6.6047 6.62 81 3 7.6724 7.8902 7 .14 01 7.2544 6.6047 6.62 81 4 10 .7287 11 .18 34 10 .0472 10 .3924 9.3246 9.5990 5 12 .6073 12 .58 81 11. 83 51 11. 7054 10 .9 913 10 .8285 6 12 .7254 13 .18 67 11 .9554 12 . 317 5 11 .11 13 11 .4350 7 15 .2304 15 .4530 14 .3329 14 .4520 13 .3245 13 .4 412 8 15 .2304 16 .0 017 14 .3329 15 .0 019 13 .3245 13 .9756 free temperature Is and Ds were determined as follows: Is = ρh Ds = Eh3 12 (1 −... 7 31. 12 933.30 925.45 935.88 925.45 12 11. 19 11 80.93 12 11. 19 11 80.93 15 82.63 15 76. 91 1582.63 15 76.94 n = 10 0 Present Ref [12 ] 255.54 259.35 639.50 645.55 639.50 645.55 10 20.24 10 14.94 12 79.30 12 90.78 12 79.38 12 90.78 16 57.04 16 34.65 16 57.04 16 34.65 217 2.77 219 9.46 217 2.77 219 9.47 n =1 Present Ref [12 ] 204.97 19 8.92 502.62 495.62 502.62 495.62 803.00 778.94 10 02.09 993 .11 10 05 .18 993 .11 13 01. 08 12 55.98 13 01. 08... 13 .723 555 .11 2 01. 04 × 10 9 0.3262 12 .330 × 10 −6 15 .379 496.56 P 1 0 0 0 0 0 0 0 0 0 0 P1 −3.070 × 10 −4 0 9.095 × 10 −4 1. 032 × 10 −3 1. 016 × 10 −5 3.079 × 10 −4 −2.002 × 10 −4 8.086 × 10 −4 1. 264 × 10 −3 1. 1 51 × 10 −3 P2 2 .10 6 × 10 −7 0 0 5.466 × 10 −7 2.920 × 10 −7 −6.534 × 10 −7 3.797 × 10 −7 0 2.092 × 10 −6 1. 636 × 10 −6 P3 −8.946 × 10 11 0 0 −7.876 × 10 11 1. 670 × 10 10 0 0 0 −7.223 × 10 10 −5.863 × 10 10 Example... Present Ref [12 ] 17 3.55 16 8.74 429.87 420.66 429.87 420.66 686.09 665. 01 858. 41 8 41. 26 859.78 8 41. 26 11 12.82 10 73.70 11 12.82 10 73.70 14 56.39 14 32 .16 14 56.39 14 32 .16 n = 15 Present Ref [12 ] 247.95 247.30 619 .89 615 .58 619 .89 615 .58 989 .12 967.78 12 39.88 12 31. 00 12 40 .13 12 31. 00 16 06.36 15 58.77 16 06.36 15 58.77 210 5.77 2097.97 210 5.77 2097.97 (Hz) of FG plates n = 0.5 Present Ref [12 ] 19 0.42 18 5.45 467.96... 3 4 5 6 7 8 9 10 Mode 1 2 3 4 5 6 7 8 9 10 Table 2 .1: n=0 Present Ref [12 ] 14 4.54 14 4.25 3 61. 75 359.00 3 61. 75 359.00 577.09 564 .10 723.60 717 .80 723.62 717 .80 937.20 908.25 937.20 908.25 12 28.77 12 23 .14 12 28.77 12 23 .14 n=5 Present Ref [12 ] 233 .10 230.46 579.70 573.82 579.70 573.82 925.47 902.04 11 58. 51 114 8 .12 11 59.62 11 48 .12 15 02 .14 14 53.32 15 02 .14 14 53.32 19 66.94 19 58 .17 19 66.94 19 58 .17 Natural frequencies... 13 01. 08 12 55.98 16 99 .18 16 97 .15 16 99 .18 16 97 .15 n = 10 00 Present Ref [12 ] 262.43 2 61. 73 656.90 6 51. 49 656.90 6 51. 49 10 47.90 10 24.28 13 14 .15 13 02.64 13 14.20 13 02.64 17 01. 98 16 49.70 17 01. 98 16 49.70 22 31. 94 2 219 .67 22 31. 94 2 219 .68 29 Table 2.2: Material properties Material Properties E (Pa) ν Si3 N4 α(/K) κ (W/mK) c (J/kgK) E(P a) ν SUS304 α (/K) κ (W/mK) c (J/kgK) Po 348.43 × 10 9 0.2400 5.8723 × 10 −6 13 .723... in thermal environment 31 3 .1 Material properties of the constituents of the FGM 44 11 Chapter 1 Introduction 1. 1 Motivation Heterogeneous materials are very popular in nature The constituents of heterogeneous materials can be two or more different material phases, compositions or states in one length scale or multiple length scales Common examples of heterogeneous materials include bones,... n =1 n=0.5 n=0.2 −0.4 −0.5 300 305 310 Temperature 315 320 (a) t = 0.004s 0.5 0.4 0.3 0.2 z/h 0 .1 0 −0 .1 −0.2 −0.3 n =1 n=0.5 n=0.2 −0.4 −0.5 300 305 310 315 Temperature 320 325 (b) t = 0.006s Figure 2-2: Temperature distribution through the thickness of square FG plates subjected to suddenly applied heat flux q = 10 6 W/m2 33 0.5 0.4 0.3 0.2 z/h 0 .1 0 −0 .1 −0.2 −0.3 n =1 n=0.5 n=0.2 −0.4 −0.5 300 305 310 ... enriched nodes, and enriched elements are those with thick line 11 9 5-22 Displacement and von-mises stress solutions for 12 8 12 8 XFEM mesh (left) and 256 × 256 XFEM mesh (right) Displacement is scaled by a factor of 1/ 100 12 0 10 List of Tables 2 .1 Natural frequencies (Hz) of FG plates 29 2.2 Material properties ... for analyzing the heterogeneous materials In this chapter, various numerical examples from simple to complicated compositions of heterogeneous materials containing inclusions are studied In some examples, the quadratic quadrilateral elements are introduced as the 8-node elements for the VCFEM instead of the Voronoi cells Chapter 5 shows the application of the XFEM for heterogeneous materials, including . . . . . . . . . . . . 14 1. 3 Review of Functionally graded materials . . . . . . . . . . . . . . . . 14 1. 3 .1 Review of micromechanical modeling for functionally graded materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1 5 -10 Meshes of the porous unit cell(a) FEM (b) VCFEM . . . . . . . . . . . . 11 1 5 -11 XFEM mesh of the porous unit cell: ◦ indicates enriched. in thermal environment . 31 3 .1 Material properties of the constituents of the FGM . . . . . . . . . . . . 44 11 Chapter 1 Introduction 1. 1 Motivation Heterogeneous materials are very popular