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 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)
eff eff
,k
: macroscopic bulk and shear elastic moduli;
, , ,
V V R R
kk

: Voigt, Reuss estimates;
are the stiffness and compliant elastic tensors of α- orientation
crystal, respectively:
22
1 1 1
;2
V iijj V ijij iijj
k C C C
d d d d



(1.1)
1
1
2
41
;2
R iijj R ijij iijj
k S S S
d
dd



(1.2)
eff eff
;
V R V R
k 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
HS
k
and lower bound
L
HS
k
:
11 11 33
122
9
UL
HS HS V R
k k k k C C C
(1.5)
HS estimates for shear uper
U
HS
and lower bounds
L
HS
:
0
( ,k , )
U eff U
HS HS
P

C
,
0
( ,k , )
L eff L
HS HS
P

C
,
11 12 * 44 *
0 0 *
11 12 44 *
2 ( )
( ,k , ) 5 3( ) 4 10
C C C
PC C C



C
,