Đánh giá và mô phỏng các hệ số đàn hồi đa tintinh thể hỗn độn tt tiếng anh

Therefore, author has selected the topic "Estimates and simulation for the elastic moduli of random polycrystals " as the research thesis.. Numerical simulation: Large-scale FEM results

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

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

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

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

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

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:

d Self- consistent value(SC)

SC value is the solution C0 of the equation:

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 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 C mn , S S mn , , 1, 6

m n or 4 index C C ijkl , S S ijkl , i j k l, , , 1,d

2.1.2 Elastic coeficicents of polycrystals

Elastic moduli are determined by the folowing fomulae:

We consider RVE has volume V=1, vis corresponding

volume ratio of V V Three-point correlation parameters:

  are harmonic and biharmonic functions Geometric

parameters f 1 , f 3 , g 1 , g 3 are restricted by:

2.2.2 Upper bound of bulk elastic modulus

HS polarization trial field has form:

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:

After putting trial strain field in to minimum energy expression,

transforming it, we have:

d d

b b

eff Ud 

1 1, , ,

ensure the smallest bulk modulus

 Choose maximum over f 1 , g 1: these are two parameters representing the geometry of polycrystals, so select the biggest values to ensure the upper bound

2.2.3 Lower bound of bulk elastic modulus

Similarly, we select general trial stress field:

Iis the geometric indicator function of α-phase

Putting this trial feld in to minimum complemantary energy expression , optimizing over variables aij, b, f 1 , g 1 restricted,

we obtain the lower bound:

1 1

1 1 ,

2.3.2 Lower bound of shear elastic modulus

We choose general trial stress field as:

1 1

1 1 ,

moduli of d-dimensional polycrystalline materials:

 This estimates are complexly dependent on the geometric

parameters f 1 , g 1 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 bulkS and shear S k moduli:

, , ,

k 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

b Lower bound of area elastic modulus

Similarly, from (2.73) we receive:

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, L

K K are thesis’ estimates; U

b , f ,1U g and 1U b , L f ,1L g are values of b and f1L 1 , g 1, at which the respective extrema in the thesis’ bounds; U, L

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 valuesS k LA are almost equal S k cirand much smaller the S k VR, proving that the thesis evaluation is close to the circle cell and much better than V-R

S

(%)

ir

c k

S

(%)

VR k

S

(%)

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

5 4.10 4.105 0.08

c Result and comparison

Calculating for datas in Table 3.3, comparing with V-R, HS bounds (K HS U ,K HS L ,U HS,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

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 Uir, Lir

0 0

b Estimate for shear elastic modulus

Our estimates for circular cell crystals  Uir, Lir

11 22 0

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

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

3.2.3 Result and comparison

Calculating for data in Table 3.8, comparing with V-R, HS, PDC bounds (k k S u, S l, S u, l S), SC value, we obtain the specific results in Tables 3.9 and 3.10

Table 3.8: Elastic constants of some 3D tetragonal crystals

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 f 1 =g 1 =0: the new

estimates of the thesis equal to PDC bounds, which proves that this result is completely convincing

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

Y is unit cell size;   ij

e is unit test train; C y is local elasticity that varies arccording to the location in the unit cell;  ij is characteristic displacement corresponding to   ij

e

In the basic coordinate system, Hooke's law:

eff:

nodes, each node has 2 degrees of freedom Thus, RVE 4 4

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 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 overall node displacement,  q e is the element nodal displacement,  L is the element's positioning matrix, K e   is the overall stiffness matrix,  P is the load vector The total potential has form:

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:

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  

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

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