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MINISTRY OF EDUCATION ANDTRAINING VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY SCIENCE AND TECHNOLOGY - VUONG THI MY HANH ESTIMATES AND SIMULATIONS FOR THE ELASTIC MODULI OF RANDOM POLYCRYSTALS Major: Mechanics of Solid Code: 9440107 SUMARY OF DOCTORAL THESIS IN MECHANICS HA NOI - 2020 The thesis has been completed at: Graduate University Science and Technology -Vietnam Academy of Science and Technology Supervisor 1: Prof DrSc Pham Duc Chinh Supervisor : Dr Le Hoai Chau Reviewer 1: Prof Dr Pham Chi Vinh Reviewer 2: Assoc Prof Dr La Duc Viet Reviewer 3: Assoc Prof Dr Tran Bao Viet Thesis is defended at Graduate University Scienc and TechnologyVietnam Academy of Science and Technology at : , on / / 2020 Hardcopy of the thesis be found at: - Library of Graduate University Science and Technology - Vietnam national library PREFACE Reason of choosing the thesis a Objective reason Polycrystalline materials are being used extensively in all areas of human life The study of elastic coefficients for this material yields many analytical results: Voigt, Ruess, Hill, Hashin-Strikman, Pham Duc Chinh However, the finite element method (FEM) results are not surffice for comparison The question is: are these estimates the best, how to calculate by the FEM, how the FEM results compared to these estimates b Subjective reason Homogenization of materials is a long-term research field of supervisor Pham Duc Chinh and Material Mechanics team with many published results The PhD candidate completed the master's thesis on homogenization of thermal conductivity for isotropic composite materials Therefore, author has selected the topic "Estimates and simulation for the elastic moduli of random polycrystals " as the research thesis Aim, research method of the thesis a Aim: to find better estimates, compare results of analytic method and FEM b Method: using energy principles and applying analytical and numerical methods simultaneously Research subject and scope of the thesis a Subject: macroscopic elastic moduli of random polycrystals b Scope: For estimates, thesis considers d- dimensional polyscrystals; For simulation, thesis only considers 2D polyscrystals with hexagonal shape of New contributions of the thesis a Theory: Generalized polarization fields, estimates and specific calculation results for elastic moduli of d-dimensional polyscrystals are new and better than the previous estimates b Numerical simulation: Large-scale FEM results for elastic moduli of 2D square, orthorhombic and tetragonal polycrystals for comparisons with the bounds are new Thesis layout Chapter presents the development history and research methods of the previous authors Chapter constructs general estimates for macroscopic elastic moduli Chapter applies Chapter results to 2D and 3D polycrystals; calculates and compares thesis estimates with V-R, HS, PĐC, SC estimates Chapter applies FEM to simulate values of 2D polycrystal macroscopic elastic moduli, compares with analytical results CHAPTER 1: OVERVIEW 1.1 Overview of polycrystaline materials Polycrystalline materials are aggregates of large numbers of individual crystals bonded perfectly together Figure 1.2: Random polycrystalline materials model 1.2 Research history of macroscopic elastic moduli 1.2.1 Outline the process of developing research field Common approach is using energy priciples, statistical isotropy and symmetric cell hypotheses have been applied to narrow the bounds of estimates from the first order to the second order and the third order ones Experimental data shows that the values of macroscopic properties concentrate within higher order bounds Therefore, third-order estimates are the best ones for the macroscopic properties of polycrystals as well as composites 1.2.2 Typical estimates a Voigt- Ruess- Hill estimate (first order) k eff ,  eff : macroscopic bulk and shear elastic moduli; kV , V , kR , R : Voigt, Reuss estimates; Cijkl , Sijkl (i, j, k , l  1, d ) are the stiffness and compliant elastic tensors of α- orientation crystal, respectively:  kV        Ciijj ; V   Cijij  Ciijj  d2 d d 2 d     kR  Siijj 1 ; R  (1.1)      Sijij  Siijj  d d d 2  1 kV  k eff  kR ; V   eff  R (1.2) (1.3) b Hashin- Strikman estimate (second order) HS used new variatinonal principle and polar field to buils new estimates better than the Hill ones In cubic case, HS L U estimates for bulk uper k HS and lower bound k HS : U L kHS  kHS  kV  kR   2C11  2C11  C33  L HS estimates for shear uper  UHS and lower bounds  HS : L UHS  P (C,k eff , 0UHS ) , HS  P (C,k eff , 0LHS ) , P (C, k , 0 )  C 11   C12  2* (C44  * ) 3(C11  C12 )  4C44  10*  * , (1.5) *  0   9k0  80  C  C12  , 0UHS  max  11 , C44  , 6k0  120        C11  C12  , C44      0LHS   (1.8) c Pham Duc Chinh estimate (third order) Using HS-type polarization trial fields, but coming derectly from classical minimum energy and complementary energy principles, PDC added three-point correlation parameters A , B and succeded in constructing tighter bounds PDC estimates have short forms for spherical cell polycrystals: k0   (1.10) Cij*kl  Tijkl (k* , * );k*  0 ; *  0 6k0  120  ε0 : Ceff : ε  ε : C * )1 1   σ : (Ceff )1 : σ  σ : C * )1   C* : ε , C0  C 1   C*  1 1 : σ0 , C  C (1.24) 1 (1.26) d Self- consistent value(SC) SC value is the solution C0 of the equation: C0  1  * 1 C   C* (1.27) Advantages: SC values are calculated simply and quickly; Disadvantages: they are valid only for perfect material model and has many deviations, so thesis only uses it for reference 1.3 General research method 1.3.1 Analytical method The problem is solved by finding extremums of energy functions on RVE domain Specifically: we choose one or more possible test fields for deformations and stresses, put in mechanical equations with constraints, and transform them to get evaluations This method is the traditional variational one that V-R, HS, PDC used 1.3.2 Numerical method FEM is commonly used, the basic steps are: random crystal orientation gereration, meshing RVE, setting stiffness matrix, equations describing the material balance, applying conditions, solving systems of equations to get the node displacements, deformation, stress, caculating effective elastic coefficients 1.4 Conclusion of chapter Studying elastic moduli of polycrystalline materials has high scientific and practical significance The analytical results have been developed well, but the FEM results are few Therefore, in this thesis PhD will use both analytical and numerical methods in solving this problem, compare them with each other and give specific conclusions CHAPTER 2: ESTIMATES FOR ELASTIC MODULI OF D- DIMENSIONAL RANDOM POLYCRYSTALS This chapter uses analytic methods to construct general upper and lower bounds for the bulk and shear elastic moduli of d-dimensional polycrystals Conclusions for these estimates are presented at the end of this chapter 2.1 Scientific basis 2.1.1 Elastic coefficients of single crystal Elastic properties of single crystals are anisotropic and often   used by the index-Voigt notation C  Cmn  , S  Smn  ,    m, n  1,6 or index C  Cijkl  , S  Sijkl , i, j, k , l  1, d 2.1.2 Elastic coeficicents of polycrystals Elastic moduli are determined by the folowing fomulae: a Hooke's law Average stress field and strain field are related: σ  Ceff : ε (2.22) b Minimum energy princilple ( ε is compatible) Wε  ε0 : Ceff : ε0  inf  ε : C : εdx ε ε (2.29) V c Minimum complementary energy princilple( σ is balanced)  Wσ  σ : Ceff  1 : σ  inf  σ : C1 : σdx σ σ (2.34) V 2.2 Bulk elastic modulus of d-dimensional polycrystals 2.2.1 Begining equations We consider RVE has volume V=1, v is corresponding volume ratio of V  V Three-point correlation parameters: A   ij ij dx , ij  ,ij  V v      ,ijkl  B    ijkl  ijkl dx ,  ijkl V   ,ij dx , V v   ,ijkl dx (2.50) V   ,  are harmonic and biharmonic functions Geometric parameters f1, f3, g1, g3 are restricted by: f1  f3  d 1 (d  1)(d  3) , g1  g3  d(d  2) d (2.52) d2 1 d 1 f1   g1  f1  f1  , d4 (d  2)(d  4) d4 d (2.54) 2.2.2 Upper bound of bulk elastic modulus HS polarization trial field has form:  ij  3k0  0 0 (3k0  40 )  p  kl ,ijkl  0  p (i ) m ,j m (2.55) This field has only free coefficients k0 , 0 Refering to HS field, PDC ’s thesis, PhD selects diffirent general polar fields for upper and lower bounds, specifically with the upper bound:  ij  n 1   ij ε0   aik ,kj  ajk,ki  bakl ,ijkl  d   1     (2.56) ε is volumetric strain field; a  aij  are free scalar constants n restricted by v a  ;    b  2 is free parameter 1 After putting trial strain field in to minimum energy expression, transforming it, we have:   W  kV ε0  2ε0 n   v CK : a  1 n v a : A : a   (2.60) 1  bkV   2b  A CK  CijK  Cijkk 1   , A  C pq    ij d  d  d         A C pq  C 'pqA  Dpq ,     1 Cikjl  C jkil B2  Ciipp kl  Cklppij B3 2 1  Cipjp kl  Ckplpij B4  Cipkp jl  C jpkpil  C jplpik  Ciplp jk B5  Cikpp jl  C jlpp ik  C jkpp il  Cilpp jk B6 , Dijkl   ij kl D1   ik jl   il jk D2 , 2   D1   kV  V   f3 F1  g3G1   V  f3 F2  g3G2  d   Cij' Akl  Cijkl B1           d2  d    dkV  f3 F3  g3G3    kV  V   f3 F4  g3G4  , d     d  1 d  3 d 1 b2 F7  G7  kV , d d  d  2  d  2 d 2   D2  2V  f3 F1  g3G1    kV  V   f3 F2  g3G2   d    d2  d   d 1   kV  V   f3 F5  g3G5   dkV  f3 F6  g3G6   F8 , d d    d  1 d  3 ,  2b   G8 B1  1    f1 F1  g1 G1 , d  d  2 d  d 2 B2  f1 F2  g1 G2 , B3  4b2 2b   f1 F3  g1 G3 d  d   d  d  2 B4  f1 F4  g1 G4 , B5  f1 F5  g1 G5 , B6  f1 F6  g1 G6 (2.61) Optimizing (2.60) over the free variables aij restricted by (2.59), using Lagrange multiplier method, we recive: k eff  k Ud  C, f1 , g1 , b  , k Ud  kV  CK : A-1 A-1 1  : A-1 : CK   CK : A1 : CK   : (2.63) Now optimizing (2.63) over the remaining parameter b, shape parameters f1, g1 restricted by (2.52), (2.54), we obtain the upper estimate: k eff  max K Ud  C, f1 , g1 , b  f1 , g1 b (2.64)  Here we choose minimum over b because: trial strain field admissible at all the values of b, so we choose the b in order to ensure the smallest bulk modulus  Choose maximum over f1, g1: these are two parameters representing the geometry of polycrystals, so select the biggest values to ensure the upper bound 11 k U , k L , U ,  L are upper and lower bounds of bulk and shear moduli respectively These measure parameters characterize the relative difference between upper and lower bounds, if they are smaller then the estimates are better 3.1 2D polycrystals 3.1.1 2D Orthorhombic a Upper bound of area elastic modulus Calcultating the terms in (2.64) for 2D orthorhombic, we obtain  AC K U  KV  CKAC 11  CK 22    A K R S pq  CKCAC (3.11) b Lower bound of area elastic modulus Similarly, from (2.73) we receive:  K Lfgb   K R1  CKAC11  CKAC22     A KV1 C pq  CKCAC     1 (3.15) c Result of estimates and comparison For numerical illustrations, we take some 2D orthorhombic crystals, their elastic constants are tabultated in Table 3.1 (all in GPa) Results in Table 3.2, K U , K L are thesis’ estimates; bU , f1U , g1U and b L , f1L , g1L are values of b and f1, g1, at which U L the respective extrema in the thesis’ bounds; Kcir are , Kcir estimates for circle cell crystals; S kLA , S kcir , S kVR are scatter measure parameters of thesis, circle cell and V-R respectively Table 3.1: Elastic constants of some 2D orthorhombic crystals Crytal C11 C22 C12 C33 S(1) 2.05 4.83 1.59 0.43 S(2) 2.40 2.05 1.33 0.76 U(1) 19.86 26.71 10.76 12.44 U(2) 21.47 19.86 4.65 7.43 12 Table 3.2: Estimates for area elastic modulus of orthorhombic 2D KR KL L K cir U K cir KU KV S(1) 1.9928 2.1365 2.1365 2.1612 2.1612 2.5150 S(2) 1.7604 1.7678 1.7678 1.7680 1.7774 1.7775 U(1) 16.554 16.739 16.7399 16.7489 16.7489 17.022 U(2) 12.637 12.643 12.6434 12.64341 12.64341 12.657 bL bU f1L f1U g1L -1.40 0.06 0.51 -0.52 0.41 -1.02 0.16 0.51 -0.05 0.31 g1U -0.67 0.20 -0.88 0.01 0.04 -0.97 0.31 0.41 -1.25 0.16 0.14 SkLA (%) Skcir (%) SkVR (%) 0.57 0.57 11.5 0.27 0.01 0.48 0.03 0.03 1.39 4.105 4.105 0.08 Comments of Table 3.2: The new estimates of the thesis are always in the range of V-R, proving that our results are better; The values S kLA are almost equal S kcir and much smaller the S kVR , proving that the thesis evaluation is close to the circle cell and much better than V-R 13 3.1.2 Square a Estimate for area elastic modulus K eff   C11  C12  (3.17) b Estimate for shear elastic modulus  CAC CAC  eff  max  V  CMCAC 11  CM 12  2CM 33 b  f1 , g1  1    S11A  S12A  S33A  4  1 C AC M 11 2  CMAC12  2CMAC33     CAC CAC  eff  max  R1  CMCAC 11  CM 12  2CM 33 f1 , g1 b    C11A  C12A  2C33A  C 1 AC M 11  (3.22)    CMAC12  2CMAC33   1 (3.25) c Result and comparison Calculating for datas in Table 3.3, comparing with V-R, HS U L U L bounds ( K HS , K HS , HS , HS ) , SC value ( K SC , SC ) , we obtain the specific results in Tables 3.3 and 3.4 Table 3.3: Estimates for area elastic modulus of square Square C11 C12 C33 K eff  KV  K R  K HS Ag 123 92 45.3 107.5 Ca 16 12 Cu 169 122 75.3 145.5 Ni 247 153 122 200 Pb 123 92 45.3 45.1 Li 13.6 11.4 9.8 12.5 12 14 Table 3.4: Estimates for shear elastic modulus of square Square R L  HS L  SC U UHS V S LA S HS S VR Ag Ca Cu Ni Pb Li 23.1 6.0 35.82 67.86 5.92 1.98 25.17 6.462 39.41 72.43 6.772 2.49 25.63 6.545 40.26 73.24 7.04 2.73 25.76 6.563 40.51 73.41 7.152 2.90 25.94 6.60 40.89 73.71 7.302 3.19 26.36 6.667 41.64 74.42 7.556 3.41 30.40 8.0 49.40 84.50 9.250 5.45 0.61 0.41 0.77 0.32 1.82 7.77 2.31 1.56 2.75 1.35 5.47 15.59 13.64 14.29 15.94 10.92 21.95 46.7 Comment of Table 3.3, 3.4: Our area elastic modulus of square equals to V-R, HS bounds, our shear elastic modulus is better than previos ones, proving that the thesis results are completely reasonable 3.1.3 Tetragonal 2D a Estimate for area elastic modulus Our third order estimates for tetragonal 2D made from circular cell crystals K cUir , K cLir : K cLir  K eff  K cUir , K cLir  PK  R , * R  , K cUir  PK  V , *V  , PK ( 0 , * )  where: *  * C11*C 22  C12*  0 C11  C 22  2C33  4* K 0 KV V K R R * , *V  , * R  , C11  C11  0  * , K  0 KV  2V K R  2 R * *  C33  *  C22  0  * , C33 C12*  C12  0  * , C22 (3.27) 15 b Estimate for shear elastic modulus Our estimates for circular cell crystals cUir , cLir : CL   eff  CU , CL  P  R , * R  , CU  P  V , *V  , 1  C  C  2C  4  P ( 0 , * )   11 *22 * 12 * *  *   * C11 C 22  C12 C33   (3.28) c Result and comparison Calculating for tetragonal 2D in Table 3.5, comparing with V-R bounds, we obtain the similar results in Tables 3.6 and 3.7 Table 3.5: Elastic constants of some 2D tetragonal crystals Tetragonal 2D BaTiO3 ZrSiO4 Sn TiO2 In Hg2Cl2 SnO2 Urea C11 275 73.5 75.3 273 44.5 18.8 262 21.7 C12 151 -5.4 44.1 149 40.5 15.6 156 24 C22 165 46 95.5 484 44.4 80.1 450 53.2 C33 54.3 13.8 21.9 125 6.5 85.3 103 6.26 Table 3.6: Estimates for area elastic modulus of tetragonal 2D Crystal KV KU KL KR SkVR SkLA BaTiO3 185.5 173.78 173.083 163.58 6.279 0.201 ZrSiO4 27.175 26.0262 26.0009 25.724 2.743 0.049 Sn 64.75 63.9885 63.9843 63.515 0.963 0.003 TiO2 263.75 248.078 247.672 239.501 4.818 0.082 In 42.475 42.4749 42.4749 42.4747 Hg2Cl2 32.525 24.4991 22.3135 18.6487 4.104 27.12 3.105 4.669 Urea 30.725 25.2314 24.7086 21.5033 17.66 1.047 16 Table 3.7: Estimates for shear elastic modulus of tetragonal 2D Crystal V CU CL R S VR S LA BaTiO3 ZrSiO4 Sn TiO2 In Hg2Cl2 44.4 23.187 21.275 119.87 4.2375 51.112 40.924 20.176 21.1092 115.821 3.5787 29.292 40.7742 20.081 21.1087 115.744 3.49441 24.4153 38.997 19.066 21.046 113.65 3.0294 17.42 6.479 9.752 0.541 2.663 16.62 49.15 0.183 0.236 0.001 0.033 1.192 9.08 3.2 3D crystals Similarly calculate for 3D tetragonal, we get the below results 3.2.1 Bulk elastic modulus  AC K U  kV  2CKAC 11  2CK 33    AC K L   kR1  2CKAC 11  2CK 33    A  R S pq  CKCAC     A V1 C pq  CKCAC   3.2.2 Shear elastic modulus       4M C  M C  (3.34) 1 (3.39)  A AC CAC  (3.44)  eff  max  V  M R S pq MV2 CMpq  MV CMpq f1 , g1 b   eff  max  R1 f1 , g1 b  4 MV 1 V A pq C  CAC Mpq V CA Mpq 1 (3.48) 3.2.3 Result and comparison Calculating for data in Table 3.8, comparing with V-R, HS, PDC bounds (kSu , kSl , Su , Sl ) , SC value, we obtain the specific results in Tables 3.9 and 3.10 Table 3.8: Elastic constants of some 3D tetragonal crystals Tinh thể BaTiO3 ZrSiO4 Sn TiO2 In Hg2Cl2 C11 275 73.5 75.3 273 44.5 18.8 C33 165 46 95.5 484 44.4 80.1 C12 179 61.6 176 39.5 173 C13 151 -5.4 44.1 149 40.5 15.6 C44 54.3 13.8 21.9 125 6.5 85.3 C66 113 16 23.7 194 12.2 12.6 17 Table 3.9: Estimates for bulk elastic modulus of tetragonal 3D Tinh thể BaTiO3 ZrSiO4 Sn TiO2 In Hg2Cl2 kR L k HS kL kSL  kSl k SC kSU  kSu kU U k HS kV SkLA SkHS SkVR 162.82 19.056 606.200 210.61 41.600 17.8 174.3 19.6 606.315 213.4 41.601 18.3 177.6 19.74 606.325 214.7 41.605 18.82 178.2 19.75 60.633 214.7 41.608 18.82 178.8 19.78 60.635 215.0 41.612 19.61 179.3 19.82 60.637 215.1 41.615 19.99 179.3 19.82 606.338 215.2 41.617 20.24 181.9 20.1 606.341 216.0 41.619 21.3 186.33 21.04 606.342 219.78 41.620 22.3 0.476 0.202 0.001 0.116 0.014 3.635 2.134 1.259 0.002 0.605 0.022 7.575 6.733 4.948 0.012 2.131 0.024 11.22 Table 3.10: Estimates for shear elastic modulus of tetragonal 3D Tinh thể R L  HS BaTiO3 ZrSiO4 Sn TiO2 In Hg2Cl2 47.77 18.37 15.67 101.2 3.716 2.930 51.4 19.5 17.6 111.4 4.4 4.9 L SL  Sl  SC US  Su U UHS V S LA S HS S VR 53.28 19.71 18.35 114.7 4.770 6.184 53.48 19.71 18.43 115.0 4.770 6.407 53.80 19.77 18.56 115.7 4.90 7.655 54.08 19.84 18.61 116.1 4.980 8.057 54.12 19.85 18.61 116.1 4.990 8.057 55.5 20.3 18.8 118.1 5.3 9.0 59.92 21.71 19.92 125.9 5.900 10.54 0.782 0.354 0.703 0.607 2.254 13.15 3.835 2.01 3.297 2.919 9.278 29.5 11.282 8.3333 11.942 10.876 22.712 56.496 Comment of Table 3.9 and 3.10: Similar to the comments of 2D case, in addition, when f1=g1=0: the new estimates of the thesis equal to PDC bounds, which proves that this result is completely convincing 18 3.3 Conclusion of chapter Applying the estimates built in chapter 2, PhD has achieved:  Construct specific evaluation formulae for some 2D and 3D crystals; Calculate for some actual polycrystalline materials and compare with V-R, HS, PCDC, SC  These results are reasonable and better than previous ones CHAPTER 4: APPLICATION OF FINITE ELEMENT METHOD AND COMPARISON WITH ESTIMATES FOR SOME SPECIFIC POLYCRYSTALLINE MODELS This chapter uses FEM to simulate the effective elastic coefficients of 2D polycrystalline, calculates for some specific crystals and compares with VR, HS, SC, new estimates of the thesis 4.1 Begining fomulas: eff Macro elastic moduli Cijkl are determined by general formula: Cijeffkl  Y  e  ij    ij kl kl   y    C  y  e    y    dy (4.1) Y ij Y is unit cell size; e  is unit test train; C  y  is local elasticity that varies arccording to the location in the unit ij ij cell;    is characteristic displacement corresponding to e  In the basic coordinate system, Hooke's law: σ  Ceff : ε (4.3) 4.2 FEM calculate process 4.2.1 Mesh RVE Denote: nxn is RVE size, mxm is mesh size (n: number of hexagonals per RVE size, m: number of elements per hexagonal size, m  ); Grid element is quadrangle, each element has 19 nodes, each node has degrees of freedom Thus, RVE  8  8    4  1.024 elements, 8  8   64  64  262.144 elements, has RVE 64  64 has this is not a small number, so we need much time and mainframe resources RVE 4x4 RVE 8x8 RVE 16x16 RVE 32x32 RVE 64x64 Figure 4.1: Mesh RVE 4.2.2 Determine matrices, vectors RVE is divided into N e quadrilateral elements with R nodes, each element has r nodes, each node has s degrees of freedom To calculate the elastic coefficients, we select the displacement as variable, the stress and the deformation will be determined after knowing the nodal displacements q  is the q e overall node displacement, is the element nodal displacement,  L e is the element's positioning matrix,  K  is the overall stiffness matrix, P is the load vector The total potential has form:  Ne  q L  K  L q  q L  P e 1 T e e T e e T e e (4.10) Applying Lagrange's principle about equilibrium conditions of the whole system at the nodes, we have:  K  q  P (4.13) 4.2.3 Determine the elastic moduli values With average stress and strain, from (4.3) we calculate the bulk and shear elastic moduli, respectively: 20 k eff    V 11   22  dx  11   22  dx  ,  eff V 12 dx  12   12 2 12 dx V (4.14) V Attaching each crystal to a rotation angle φ, (    2 ) In the calculation program, select the "random" command for φ to ensure randomization in the direction of the crystal Periodic boundary conditions of the problem: U  x  d   ε0  d  U  x  (4.16) d is the boundary distance between two adjacent elements, U is the displacement of the element 4.3 Applying to specific symetric crystals Calculating for orthorhombic 2D, square, tetragonal 2D with hexagonal shape as discussed in chapter 4.4 Numerical simulation and comparison Choosing randomly 20 rotation angles, calculation time for each case (corresponding to each figure)) is about 18 hours 4.4.1 Results for square Figure 4.3: FE result of area Figure 4.4: : FE result of shear elastic modulus for square Cu, elastic modulus for squar Pb, S  0.77% , convergence RVE S  1.82% , convergence RVE size 64x64 size 32x32 21 4.4.2 Results for orthorhombic 2D Figure 4.6: FE result of area elastic modulus S(1) Figure 4.10: FE result of shear elastic modulus S(3) 4.4.3 Results for tetragonal 2D Figure 4.12: FE result of shear elastic modulus Hg2Cl2 Figure 4.15: FE result of shear elastic modulus In General comments of FE results:  FE results scatter around V-R, HS, SC, thesis, proving that the results of FEM are completely reasonable  When the number of test samples is larger, FE values tend to focus around the analytic values, that is, when the number of crystal directions is increased, the macroscopic properties of polycrystals are shown more clearly 22  RVE size is increased, FE results fall in the better bounds However, time and computer configuration are major obstacles  Crystals with large scatter parameters have convergence speed faster than crystals with the small ones  When considering the relationship between the convergence RVE size and the scatter parameter, we should compare the crystals in the same elastic property 4.5 Conclusion of chapter Using FEM to simulate effective elastic moduli of 2D random polycrystals and compare with the analytical results, the FEM results converge to the thesis's evaluation with RVE 64x64 crystals; This FEM used is not new, but the calculation approach for the specific elastic moduli of the thesis is new, can be used to simulate other crystals, and determine the better estimates for macro elastic moduli CONCLUSION AND NEXT RESEARCH Approaching the problem by using the variational principle and applying both analytic and numerical methods, thesis has achieved: Theory result  The thesis has built general estimate formulae for effective elastic coefficients of d-dimensional random polycrystals New points of the thesis are the informations on the geometry of the material and possible test fields more general than HS ones  Constructing specific estimates for some 2D, 3D crystals; Calculating and comparing with V-R, HS, SC, PĐC  The results of thesis are completely reasonable and better than previous estimates 23 Simulation result  Applying FEM to calculate some 2D typical hexagonal polycrystals and compare them with analytical estimates (including estimate of thesis)  FE results are reasonable with existing results and almost converge to our new bounds  Concluding about the relationship between the convergence RVE size and the scatter parameters of crystals  The results of the thesis are new and can be used in subsequent studies Further studies of the thesis From the achieved results, PhD and research team will still combine analytic and numerical methods to search for better results Analytical method  Researching with other symmetric crystals such as 3D orthorhombic, trigonal, hecxagonal  Constructing better estimates (narrower upper-lower bounds) for macro elastic coefficients is not a simple problem and still needs further research  It is very useful to determine the correlation equation between the scatter parameters and the converged RVE size, it helps to show mathematical quantification more clearly as well as the close relationship between analytical method and FEM Numerical method  Applying FEM to simulate elastic coefficients of other 2D symmetric crystals with hecxagonal shape and approach the 24 Voronoi polycrystals (completely random polycrystals) This is a very complicated problem  Calculating for larger RVE size may yield better results, but due to slow computer configuration, and there are no better approximation results currently so comparing is verry difficult  Applying FEM to calculate and simulate for 3D polycrystals is very complex task, not only for domestic scientists but also for those anywhere in the world PUBLISHED SCIENTIFIC WORKS (1) P.D Chinh, L.H Chau, V.T.M Hanh, Estimates for the elastic moduli of d-dimensional random cell polycrystals, Acta Mechanica, 2016, 227, 2881-2897 (2) Vuong Thi My Hanh, Pham Duc Chinh, Vu Lam Dong, Improved estimates for the effective elastic bulk modulus of random tetragonal crystal aggregates, Vietnam Journal of Mechanics, VAST, 2016, Vol 38, No 3, 181 -192 (3) Vũ Lâm Đơng, Phạm Đức Chính, Vương Thị Mỹ Hạnh, Xây dựng đánh giá mô đun trượt hiệu vật liệu đa tinh thể hỗn độn, Tuyển tập Hội nghị Khoa học toàn quốc Cơ học Vật rắn biến dạng lần thứ XII, 2016, tập 1, 458-464, ISBN: 978-604-913-458-6, Đà Nẵng (4) Vũ Lâm Đông, Phạm Đức Chính, Vương Thị Mỹ Hạnh Lê Hồi Châu, Mơ đánh giá mô đun đàn hồi vật liệu đa tinh thể ngẫu nhiên 2D, Tuyển tập Hội nghị Cơ học toàn quốc lần thứ X, Cơ học Vật rắn biến dạng, 8-9/12/2017, tập 3, tr.276-283, ISBN: 978-604-913-722-8, Hà Nội (5) Vuong Thi My Hanh, Le Hoai Chau, Pham Duc Chinh, Vu Lam Dong, Estimates for the elastic moduli of 2D hecxagonal- shape orthohombic crystals with in-plane random crystalline orientations, Vietnam Journal of Mechanics, VAST, 2019, Vol 41, No 2, 171177 (6) Vương Thị Mỹ Hạnh, Lê Hoài Châu, Phạm Đức Chính, Vũ Lâm Đơng, Đánh giá mô số phần tử hữu hạn số đa tinh thể tetragonal có hướng tinh thể phân bố hỗn độn, Hội nghị Cơ học Kỹ thuật toàn quốc, 2019, tập 1, tr 119-126, Hà Nội ... 171177 (6) Vương Thị Mỹ Hạnh, Lê Hồi Châu, Phạm Đức Chính, Vũ Lâm Đông, Đánh giá mô số phần tử hữu hạn số đa tinh thể tetragonal có hướng tinh thể phân bố hỗn độn, Hội nghị Cơ học Kỹ thuật toàn... 978-604-913-458-6, Đà Nẵng (4) Vũ Lâm Đơng, Phạm Đức Chính, Vương Thị Mỹ Hạnh Lê Hồi Châu, Mơ đánh giá mơ đun đàn hồi vật liệu đa tinh thể ngẫu nhiên 2D, Tuyển tập Hội nghị Cơ học toàn quốc lần thứ X, Cơ... 181 -192 (3) Vũ Lâm Đơng, Phạm Đức Chính, Vương Thị Mỹ Hạnh, Xây dựng đánh giá mô đun trượt hiệu vật liệu đa tinh thể hỗn độn, Tuyển tập Hội nghị Khoa học toàn quốc Cơ học Vật rắn biến dạng lần