Estimates and simulations for the elastic moduli of random polycrystals

he best ones for the macroscopic properties of polycrystals as well as composites. 1.2.2. Typical estimates a. Voigt- Ruess- Hill estimate (first order) eff eff,k  : macroscopic bulk and shear elastic moduli; , , ,V V R Rk k  : Voigt, Reuss estimates; ij ij, ( , , , 1, )kl klC S i j k l d    are the stiffness and compliant elastic tensors of α- orientation crystal, respectively: 2 2 1 1 1 ; 2 V iijj V ijij iijjk C C C d d d d              (1.1)   1 1 2 4 1 ; 2 R iijj R ijij iijjk S S S dd d                (1.2) eff eff;V R V Rk k k       (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 estimates for bulk uper U HSk and lower bound L HSk :  11 11 33 1 2 2 9 U L HS HS V Rk k k k C C C      (1.5) HS estimates for shear uper UHS and lower bounds L HS : 0( ,k , ) U eff U HS HSP  C , 0( ,k , ) L eff L HS HSP  C ,  11 12 * 44 * 0 0 * 11 12 44 * 2 ( ) ( ,k , ) 5 3( ) 4 10 C C C P C C C               C , 4 0 0 * 0 0 0 9 8 6 12 k k        , 11 120 44max , 2 U HS C C C          , 11 12 0 44min , 2 L HS C C C          . (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: * 0 0 ij ij * * * 0 * 0 0 0 9 84 (k , );k ; 3 6 12 kl kl k C T k            (1.10)   1 0 0 0 * 1 * 0: : : ) :eff     ε C ε ε C C ε , 0 C C (1.24)   1 1 0 1 0 0 * 1 * 0: ( ) : : ) :eff       σ C σ σ C C σ , 1 0 1  C C (1.26) d. Self- consistent value(SC) SC value is the solution 0 C of the equation:   1 1 0 * *    C 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 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  mnCC ,  mnSS , , 1,6m n  or 4 index  ijklCC ,  ijklSS , , , , 1,i j k l 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: :effσ C ε (2.22) b. Minimum energy princilple (ε is compatible) 0 0 0: : inf : :eff V W d    ε ε ε ε C ε ε C ε x (2.29) c. Minimum complementary energy princilple(σ is balanced)   0 1 0 0 1: : inf : :eff V W d      σ σ σ σ C σ σ C σ x (2.34) 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: ij ij V A d         x , , , 1 ij ij ij V d v           x , ijkl ijkl V B d         x , , , 1 ijkl ijkl ijkl V d v           x . (2.50) ,   are harmonic and biharmonic functions. Geometric parameters f1, f3, g1, g3 are restricted by: 1 3 1   d f f d , 1 3 1 3 2      ( )( ) ( ) d d g g d d (2.52) 1 1 0    d f d , 2 1 1 1 6 1 6 4 2 4 4        ( )( ) d f g f d d d d (2.54) 2.2.2. Upper bound of bulk elastic modulus HS polarization trial field has form: 0 0 , , 0 0 0 0 3 1 ( ) (3 4 ) ij kl ijkl m j m k p p i k                     (2.55) 7 This field has only 2 free coefficients 0 0,k  . Refering to HS field, PDC ’s thesis, PhD selects diffirent general polar fields for upper and lower bounds, specifically with the upper bound:  0 , , , 1 1 1 2 n ij ij ik kj jk ki kl ijkla a ba d                       ε (2.56) 0ε is volumetric strain field;  ija a are free scalar constants restricted by 1 0 n v    a ; 0 2  b is free parameter. After putting trial strain field in to minimum energy expression, transforming it, we have:   2 0 0 1 1 2 : : : n n V KW k v v               ε ε C a a A a (2.60)  ij ij ij2 1 2 1 2 2 K V K kk bkb C C d dd                 C ,  ApqCA 'A A pq pq pqC C D  ,    'ij ijkl 1 ikjl jkil 2 iipp klpp ij 3 1 1 2 2 A kl klC C B C C B C C B         ipjp kplp ij 4 ipkp jpkp il jplp ik iplp 5 1 1 2 4 kl jl jkC C B C C C C B            ikpp jlpp ik jkpp il ilpp 6 1 4 jl jkC C C C B       ,  ij ij 1 ik il 2 1 2 kl kl jl jkD D D        ,    1 3 1 3 1 3 2 3 2 2 V V VD k f F g G f F g G d                 2 3 3 3 3 3 4 3 4 2 V V V d d dk f F g G k f F g G d             , 8        2 7 7 2 1 31 2 2 V d dd b F G k d d d d        ,    2 3 1 3 1 3 2 3 2 2 2 V V V d D f F g G k f F g G d                  2 3 5 3 5 3 6 3 6 8 2 1 V V V d d d k f F g G dk f F g G F d d               ,      8 1 3 2 d d G d d     , 2 1 1 1 1 12 1 2 1 2 b B f F g G dd          , 2 1 2 1 2B f F g G  ,     2 3 1 3 1 32 22 42 2 2 bb B f F g G d d d d       4 1 4 1 4B f F g G  , 5 1 5 1 5B f F g G  , 6 1 6 1 6B f F g G  (2.61) Optimizing (2.60) over the free variables ija  restricted by (2.59), using Lagrange multiplier method, we recive:  eff Ud 1 1, , ,k k f g b C , Ud -1: :V Kk k      C A 1 -1 -1 1: : : :K K K          A A C C A C (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:   1 1 eff Ud 1 1 , ax min , , , bf g k m K f g b C (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. 9 2.2.3. Lower bound of bulk elastic modulus Similarly, we select general trial stress field:  0 , , , 1 1 n ij ij ik kj jk ki ij kl kla a b a                     ,ij kl ijkla ba         (2.65) where a are free scalar constants restricted by 1 0 n v    a ; I is the geometric indicator function of α-phase. Putting this trial feld in to minimum complemantary energy expression , optimizing over variables ija  , b, f1, g1 restricted, we obtain the lower bound:   1 1 eff Ld 1 1 , min ax , , f g b k m K f g C , Ld 1 -1: :R KK k       C A 1 1 -1 -1 1: : : :K K K              A A C C A C (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:  0ij , , , 1 1 2 n ij ik kj jk ki kl ijkla a ba                        (2.75) Similarly, we have upper bound:   1 1 eff Ud 1 1 , ax min , , , bf g m f g b  C , Ud 2 1 1 32 V ijij iijjM M d d                , -1: :TM   M C A 1 -1 -1 1: : : :TM M M          A A C C A C (2.79) 2.3.2. Lower bound of shear elastic modulus We choose general trial stress field as: 10  0 , , , 1 1 n ij ij ik kj jk ki ij kl kla a b a                    ,ij kl ijkla ba         (2.80) Transforming it silmilarly, we obtain:   1 1 eff Ld 1 1 , min ax , , f g b m f g  C , 1 Ld 1 2 1 5 3 R ijij iijjM M             ,   -1: :Tijkl MM     M C A 1 -1 -1 1: : : :TM M M          A A C C A C (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 ijC  .  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 kS and shear S moduli: U L k U L k k S k k    , U L U L S        (3.1) 11 , , ,U L U Lk k   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 11 22 K U AC AC A CAC V K K R pq KK K C C S C    (3.11) b. Lower bound of area elastic modulus Similarly, from (2.73) we receive:     1 2 1 1 11 22 1 K 4 Lfgb AC AC A CAC R K K V pq KK K C C C C           (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, ,U LK K are thesis’ estimates; Ub , 1 Uf , 1 Ug and Lb , 1 Lf , 1 Lg are values of b and f1, g1, at which the respective extrema in the thesis’ bounds; ,U Lcir cirK K are estimates for circle cell crystals; LA kS , irc kS , VR kS 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 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 LA kS are almost equal irc kS and much smaller the VR kS , proving that the thesis evaluation is close to the circle cell and much better than V-R. RK LK LcirK U cirK UK VK Lb 1 Lf 1 Lg Ub 1 Uf 1 Ug LA kS (%) irc kS (%) VR kS (%) S(1) 1.9928 2.1365 2.1365 2.1612 2.1612 2.5150 -1.40 0.06 0.51 -0.67 0 0.20 0.57 0.57 11.5 S(2) 1.7604 1.7678 1.7678 1.7680 1.7774 1.7775 -0.52 0 0.41 -0.88 0.01 0.04 0.27 0.01 0.48 U(1) 16.554 16.739 16.7399 16.7489 16.7489 17.022 -1.02 0.16 0.51 -0.97 0.31 0.41 0.03 0.03 1.39 U(2) 12.637 12.643 12.6434 12.64341 12.64341 12.657 -0.05 0 0.31 -1.25 0.16 0.14 54.10 54.10 0.08 13 3.1.2. Square a. Estimate for area elastic modulus  11 12 1 2 effK C C  (3.17) b. Estimate for shear elastic modulus   1 1 eff 11 12 33 , 1 ax min 2 4 CAC CAC CAC V M M M bf g m C C C          1 2 11 12 33 11 12 33 1 1 2 4 2 A A A AC AC AC M M MS S S C C C              (3.22)   1 1 eff 1 11 12 33 , min max 2CAC CAC CACR M M M f g b C C C          1 1 2 11 12 33 11 12 332 2 A A A AC AC AC M M MC C C C C C         (3.25) c. Result and comparison Calculating for datas in Table 3.3, comparing with V-R, HS bounds ( , , , )U L U LHS HS HS HSK K   , SC value ( , )SC SCK  , 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 eff V R HSK K K K   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 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 ir ir ,U Lc cK K : ir ir  L eff Uc cK K K ,  ir   *, L c K R RK P ,  ir   *, U c K V VK P , 11 22 12 0 11 2 0 2 33 2 4              * * * * * ( , )K C C C P C C C . (3.27) where: 0 0 0 0 2      * K K , 2      * V V V V V K K , 2      * R R R R R K K , 11 11 0     * *C C , 12 12 0     * *C C , 22 22 0     * *C C , 33 33    * *C C . Square R L HS L SC U U HS V LAS HSS VRS 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 15 b. Estimate for shear elastic modulus Our estimates for circular cell crystals ir ir , U L c c :    L eff UC C ,     *, L C R RP ,     *, U C V VP , 1 11 22 0 11 22 12 3 12 3 2 4 1 2                   * * * * * ** ),( C C C P C C C C . (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 Table 3.6: Estimates for area elastic modulus of tetragonal 2D 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 Crystal VK UK LK RK VR kS LA kS 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 44.10 53.10 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 3.2. 3D crystals Similarly calculate for 3D tetragonal, we get the below results. 3.2.1. Bulk elastic modulus     2 11 332 2 U AC AC A CAC V K K R pq KK k C C S C     (3.34)     1 2 1 1 11 33 1 2 2 9 L AC AC A CAC R K K V pq KK k C C C C            (3.39) 3.2.2. Shear elastic modulus       1 1 eff 2 , ax min 4M M MA AC CACV R pq V Mpq V Mpq bf g m S C C       (3.44)       1 1 eff 1 1 2 , min max 4M MA CAR V pq V Mpq f g b C C        1 4M CACV MpqC    (3.48) 3.2.3. Result and comparison Calculating for data in Table 3.8, comparing with V-R, HS, PDC bounds ( , , , )u l u lS S S Sk k   , 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 Crystal V  U C  L C R VRS LAS 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 17 Table 3.9: Estimates for bulk elastic modulus of tetragonal 3D Table 3.10: Estimates for shear elastic modulus of tetragonal 3D 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. Tinh thể R k L HSk Lk L l S Sk k SCk U u S Sk k Uk UHSk Vk LA kS HS kS VR kS 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 Tinh thể R L HS L L l S S  SC U u S S  U UHS V LAS HSS VRS 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 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: Macro elastic moduli effijklC are determined by general formula:            ij ijeffij 1 kl klkl y y Y C e y e dy Y     C (4.1) Y is unit cell size;  ij e is unit test train;  yC is local elasticity that varies arccording to the location in the unit cell;  ij is characteristic displacement corresponding to  ije . In the basic coordinate system, Hooke's law: eff :σ C 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, 8m  ); Grid element is quadrangle, each element has 4 19 nodes, each node has 2 degrees of freedom. Thus, RVE 4 4 has    8 8 4 4 1.024    elements, RVE 64 64 has    8 8 64 64 262.144    elements, 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 eN 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 overall node displacement,   e q is the element nodal displacement,   e L is the element's positioning matrix, K   is the overall stiffness matrix,  P is the load vector. The total potential has form:                 1 1 2 eN T T T e e e ee e e q L K L q q L P    (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     11 22 eff 11 222 V V dx k dx          , 12 12eff 12 12 2 2 V V dx dx          (4.14) Attaching each crystal to a rotation angle φ, ( 0 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:    0   U x d ε 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 3. 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 elastic modulus for square Cu, 0.77%S  , convergence RVE size 64x64 Figure 4.4: : FE result of shear elastic modulus for squar Pb, 1.82%S  , convergence RVE 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 4 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