Sumary of doctoral thesis in Mechanics: Estimates and simulation for the elastic moduli of random polycrystals

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Homogenization of materials is a long-term research field of supervisor Pham Duc Chinh and Material Mechanics team with many published results. Therefore, author has selected the topic "Estimates and simulation for the elastic moduli of random polycrystals " as the research thesis.

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Nội dung Text: Sumary of doctoral thesis in Mechanics: Estimates and simulation for the elastic moduli of random polycrystals

MINISTRY OF EDUCATION VIETNAM ACADEMY OF ANDTRAINING 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 2 : 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 Technology- Vietnam 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
1 PREFACE 1. 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. 2. 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. 3. 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.
2 4. 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. 5. Thesis layout Chapter 1 presents the development history and research methods of the previous authors. Chapter 2 constructs general estimates for macroscopic elastic moduli. Chapter 3 applies Chapter 2 results to 2D and 3D polycrystals; calculates and compares thesis estimates with V-R, HS, PĐC, SC estimates. Chapter 4 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
3 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: 1  1   1   kV  Ciijj ; V  2  Cijij  Ciijj  (1.1) d2 d d 2 d  1   1     1 4  kR  Siijj ; R   Sijij  Siijj  (1.2) d d 2 2 d  kV  k eff  kR ; V   eff  R (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 U L estimates for bulk uper k HS and lower bound k HS : U kHS  kHS L  kV  kR  1 9  2C11  2C11  C33  (1.5) HS estimates for shear uper  UHS and lower bounds  HS L : UHS  P (C,k eff , 0UHS ) , HS L  P (C,k eff , 0LHS ) , P (C, k 0 , 0 )  5 C 11   C12  2* (C44  * )  * , 3(C11  C12 )  4C44  10*
4 9k0  80   C  C12   *  0 , 0UHS  max  11 , C44  , 6k0  120   2     C11  C12   0LHS  min  , C44  . (1.8)   2   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: 4 9 k0  8  0 Cij*kl  Tijkl (k* , * );k*  0 ; *  0 (1.10) 3 6k0  120  ε0 : Ceff : ε 0  ε 0 : C * )1 1    C* : ε 0 , C0  C (1.24)   1 1 1 1 σ 0 : (Ceff )1 : σ 0  σ 0 : C * )1  C* : σ0 , C  C 0 (1.26)  d. Self- consistent value(SC) SC value is the solution C0 of the equation: 1 C0  C   * 1  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
5 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 1 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 2 index-Voigt notation C  Cmn  , S  Smn  ,   m, n  1,6 or 4 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:
6 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 0  ε : C : εdx (2.29) ε ε V c. Minimum complementary energy princilple( σ is balanced)   1 Wσ  σ 0 : Ceff : σ 0  inf 0  σ : C1 : σdx (2.34) σ σ 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: 1 A   ij ij dx , ij  ,ij    ,ij dx , V v V 1 B    ijkl     ijkl dx ,  ijkl   ,ijkl    ,ijkl dx . (2.50) V v V   ,  are harmonic and biharmonic functions. Geometric parameters f1, f3, g1, g3 are restricted by: d 1 (d  1)(d  3) f1  f3  , g1  g3  (2.52) d d(d  2) d 1 6 d2 1 6  f1  0 , f1   g1  f1 (2.54) d d4 (d  2)(d  4) d4 2.2.2. Upper bound of bulk elastic modulus HS polarization trial field has form: 3k0  0  p   p (i ) 1  ij   (2.55) 0 (3k0  40 ) 0 kl ,ijkl m ,j m
7 This field has only 2 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: 1     n 1  ij   ij ε0   aik ,kj  ajk,ki  bakl ,ijkl  (2.56) d  1  2  ε 0 is volumetric strain field; a  aij  are free scalar constants n restricted by   v a  0 ; 1 0  b  2 is free parameter. After putting trial strain field in to minimum energy expression, transforming it, we have:   n n   2 W  kV ε0  2ε0 v CK : a  v a : A : a (2.60) 1  1  1  2b  bkV    CK  CijK  Cijkk 2 1   d  d  2    ij  d  2  , A  C pq  A   A C pq  C 'pqA  Dpq , 1 Cij' Akl  Cijkl B1  2  1   Cikjl  C jkil B2  Ciipp kl  Cklppij B3 2  1  1   Cipjp kl  Ckplpij B4  Cipkp jl  C jpkpil  C jplpik  Ciplp jk B5 2 4  1   Cikpp jl  C jlpp ik  C jkpp il  Cilpp jk B6 , 4  1  Dijkl   ij kl D1   ik jl   il jk D2 , 2   2  D1   kV  V   f3 F1  g3G1   V  f3 F2  g3G2   d   d2  d  2   dkV  f3 F3  g3G3    kV  V   f3 F4  g3G4  ,  d 
8 d 1  d  1 d  3 b2  F7  G7  kV , d d  d  2  d  2 2  d 2  D2  2V  f3 F1  g3G1    kV  V   f3 F2  g3G2    d   d2  d  2  d 1   kV  V   f3 F5  g3G5   dkV  f3 F6  g3G6   F8 ,  d  d  d  1 d  3 , 1  2b  2  G8 B1  2 1    f1 F1  g1 G1 , d  d  2 d  d 2 2b 4b2 B2  f1 F2  g1 G2 , B3    f1 F3  g1 G3 d 2  d  2  d 2  d  2 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 :  1 A-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 min K Ud  C, f1 , g1 , b  (2.64) f1 , g1 b  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.
9 2.2.3. Lower bound of bulk elastic modulus Similarly, we select general trial stress field: n  ij   ij 0    aik ,kj  a jk,ki   b  1  ij akl ,kl  1 aij   bakl ,ijkl  (2.65) n where a are free scalar constants restricted by   v a  0 ; 1  I is the geometric indicator function of α-phase. Putting this trial feld in to minimum complemantary energy expression , optimizing over variables aij , b, f1, g1 restricted, we obtain the lower bound: k eff  min max K Ld  C, f1 , g1  , K Ld   kR1  CK : A-1 : f1 , g1 b   1 A-1 1 : A-1 : CK  CK : A1 : CK  (2.73)     2.3. Shear elastic modulus of d-dimensional polycrystals 2.3.1. Upper bound of shear elastic modulus General trial strain field has form: 1      n  ij   ij0    aik,kj  ajk,ki  bakl ,ijkl  (2.75)  1  2  Similarly, we have upper bound:  eff  max min  Ud  C, f1 , g1 , b  , f1 , g1 b  1  1   Ud   V   M ijij  M iijj   , M  CM  : A T -1 :  d d 2 2 3   1 A-1 : A-1 : CM  CTM : A1 : CM (2.79)    2.3.2. Lower bound of shear elastic modulus We choose general trial stress field as:
10 n  ij   ij0    aik ,kj  a jk,ki   b  1 ij akl ,kl  1 aij   bakl ,ijkl  (2.80) Transforming it silmilarly, we obtain:  eff  min max  Ld  C, f1 , g1  , f1 , g1 b 1  2   Ld  5 1 3      R1   M ijij  M iijj   , M  M ijkl  CTM  : A-1  : 1 A-1 : A-1 : CM  CTM : A1 : CM (2.84)    2.4. Conclusion of chapter 2 Starting from energy principles, with trial fields being more general than HS, thesis have built new estimates for elastic moduli of d-dimensional polycrystalline materials:  This estimates are complexly dependent on the geometric parameters f1, g1 and component elastic coefficients Cij .  Without these geometric informations, the estimates are V-R bounds. The second term in our evaluation expressions makes the results of the thesis better. CHAPTER 3: ESTIMATES FOR EFFECTIVE ELASTIC MODULI OF SPECIFIC POLYCRYSTAL CLASSES This chapter will apply the general evaluation formulae in chapter 2 for some 2D, 3D polycrystals. We use Matlab to calculate the bounds for some actual polycrystals and compare with the previous results. For comparison, thesis uses scatter measure parameters of bulk S k and shear S  moduli: kU  k L U   L Sk  , S   (3.1) kU  k L U   L
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     2 K U  KV  CKAC 11  CK 22 AC A K R S pq  CKCAC (3.11) b. Lower bound of area elastic modulus Similarly, from (2.73) we receive: 1   1     2 K Lfgb   K R1  CKAC11  CKAC22 KV1 C pq A  CKCAC  (3.15)  4  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 the respective extrema in the thesis’ bounds; Kcir U L , Kcir are 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 bL bU SkLA Skcir SkVR KR KL L K cir U K cir KU KV f1L f1U (%) (%) (%) g1L g1U -1.40 -0.67 S(1) 1.9928 2.1365 2.1365 2.1612 2.1612 2.5150 0.06 0 0.57 0.57 11.5 0.51 0.20 -0.52 -0.88 S(2) 1.7604 1.7678 1.7678 1.7680 1.7774 1.7775 0 0.01 0.27 0.01 0.48 0.41 0.04 -1.02 -0.97 U(1) 16.554 16.739 16.7399 16.7489 16.7489 17.022 0.16 0.31 0.03 0.03 1.39 0.51 0.41 -0.05 -1.25 U(2) 12.637 12.643 12.6434 12.64341 12.64341 12.657 0 0.16 4.105 4.105 0.08 0.31 0.14 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 1 K eff   C11  C12  (3.17) 2 b. Estimate for shear elastic modulus  f1 , g1 b  1  eff  max min  V  CMCAC 4 CAC  11  CM 12  2CM 33 CAC  2 1 1  4 1   S11A  S12A  S33A   C AC M 11  CMAC12  2CMAC33  (3.22) 2   eff  min max  R1  CMCAC f1 , g1 b   11  CM 12  2CM 33 CAC CAC  1   C  CMAC12  2CMAC33   1 2  C11A  C12A  2C33A AC (3.25) M 11  c. Result and comparison Calculating for datas in Table 3.3, comparing with V-R, HS U bounds ( K HS L , K HS , HS U , HS L ) , 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 8 12 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
14 Table 3.4: Estimates for shear elastic modulus of square Square R  HS L L  SC U UHS V S LA S HS S VR Ag 23.1 25.17 25.63 25.76 25.94 26.36 30.40 0.61 2.31 13.64 Ca 6.0 6.462 6.545 6.563 6.60 6.667 8.0 0.41 1.56 14.29 Cu 35.82 39.41 40.26 40.51 40.89 41.64 49.40 0.77 2.75 15.94 Ni 67.86 72.43 73.24 73.41 73.71 74.42 84.50 0.32 1.35 10.92 Pb 5.92 6.772 7.04 7.152 7.302 7.556 9.250 1.82 5.47 21.95 Li 1.98 2.49 2.73 2.90 3.19 3.41 5.45 7.77 15.59 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 : C11*C 22 *  C12* K cLir  K eff  K cUir , K cLir  PK  R , * R  , K cUir  PK  V , *V  , PK ( 0 , * )   0 . (3.27) C11  C 22  2C33  4* K 0 0 KV V K R R where: *  , *V  , * R  , C11  C11  0  * , * K 0  2 0 KV  2V K R  2 R C12*  C12  0  * , C22 *  C22  0  * , C33 *  C33  * .
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 1  P ( 0 , * )  2  11 *22 * 12 * *  *   * . (3.28)  C11 C 22  C12 C33  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 C11 C12 C22 C33 BaTiO3 275 151 165 54.3 ZrSiO4 73.5 -5.4 46 13.8 Sn 75.3 44.1 95.5 21.9 TiO2 273 149 484 125 In 44.5 40.5 44.4 6.5 Hg2Cl2 18.8 15.6 80.1 85.3 SnO2 262 156 450 103 Urea 21.7 24 53.2 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 4.104 3.105 Hg2Cl2 32.525 24.4991 22.3135 18.6487 27.12 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 44.4 40.924 40.7742 38.997 6.479 0.183 ZrSiO4 23.187 20.176 20.081 19.066 9.752 0.236 Sn 21.275 21.1092 21.1087 21.046 0.541 0.001 TiO2 119.87 115.821 115.744 113.65 2.663 0.033 In 4.2375 3.5787 3.49441 3.0294 16.62 1.192 Hg2Cl2 51.112 29.292 24.4153 17.42 49.15 9.08 3.2. 3D crystals Similarly calculate for 3D tetragonal, we get the below results. 3.2.1. Bulk elastic modulus     2 K U  kV  2CKAC 11  2CK 33 AC  R S pq A  CKCAC (3.34) 1  1      2 K L   kR1  2CKAC 11  2CK 33 AC V1 C pq A  CKCAC  (3.39)  9  3.2.2. Shear elastic modulus  eff  max min  V  4 M R S pq f1 , g1 b  A      MV2 CMpq AC CAC   MV CMpq (3.44)   eff  min max  R1  4M C  M C  1 A 2 CA f1 , g1 b  V pq V Mpq C  1 4 MV CAC Mpq (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 C11 C33 C12 C13 C44 C66 BaTiO3 275 165 179 151 54.3 113 ZrSiO4 73.5 46 9 -5.4 13.8 16 Sn 75.3 95.5 61.6 44.1 21.9 23.7 TiO2 273 484 176 149 125 194 In 44.5 44.4 39.5 40.5 6.5 12.2 Hg2Cl2 18.8 80.1 173 15.6 85.3 12.6
17 Table 3.9: Estimates for bulk elastic modulus of tetragonal 3D Tinh kR L k HS kL kSL  kSl k SC kSU  kSu kU U k HS kV SkLA SkHS SkVR thể BaTiO3 162.82 174.3 177.6 178.2 178.8 179.3 179.3 181.9 186.33 0.476 2.134 6.733 ZrSiO4 19.056 19.6 19.74 19.75 19.78 19.82 19.82 20.1 21.04 0.202 1.259 4.948 Sn 606.200 606.315 606.325 60.633 60.635 60.637 606.338 606.341 606.342 0.001 0.002 0.012 TiO2 210.61 213.4 214.7 214.7 215.0 215.1 215.2 216.0 219.78 0.116 0.605 2.131 In 41.600 41.601 41.605 41.608 41.612 41.615 41.617 41.619 41.620 0.014 0.022 0.024 Hg2Cl2 17.8 18.3 18.82 18.82 19.61 19.99 20.24 21.3 22.3 3.635 7.575 11.22 Table 3.10: Estimates for shear elastic modulus of tetragonal 3D Tinh thể R  HS L L SL  Sl  SC US  Su U UHS V S LA S HS S VR BaTiO3 47.77 51.4 53.28 53.48 53.80 54.08 54.12 55.5 59.92 0.782 3.835 11.282 ZrSiO4 18.37 19.5 19.71 19.71 19.77 19.84 19.85 20.3 21.71 0.354 2.01 8.3333 Sn 15.67 17.6 18.35 18.43 18.56 18.61 18.61 18.8 19.92 0.703 3.297 11.942 TiO2 101.2 111.4 114.7 115.0 115.7 116.1 116.1 118.1 125.9 0.607 2.919 10.876 In 3.716 4.4 4.770 4.770 4.90 4.980 4.990 5.3 5.900 2.254 9.278 22.712 Hg2Cl2 2.930 4.9 6.184 6.407 7.655 8.057 8.057 9.0 10.54 13.15 29.5 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 3 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:  e    1  ij Cijeffkl    y    C  y  e    y    dy ij kl kl Y (4.1) Y Y is unit cell size; e  is unit test train; C  y  is local ij elasticity that varies arccording to the location in the unit cell;    is characteristic displacement corresponding to e  . ij ij 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  8 ); Grid element is quadrangle, each element has 4