Using MATLAB program to set the vector, matrix in calculating the new estimations to optimize geometric
parameters for specific materials. CAST3M program is
used to calculate the moduli of twophase transverseisotropic unidirectional composites (cross section is
hexagonal symmetry) for comparisons with the evaluations. The thesis constructed the numerical
program which can help to designed and predict new material properties, as desired.
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VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY
GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY
o0o
VU LAM DONG
ESTIMATION AND COMPUTATIONAL SIMULATION
FOR THE EFFECTIVE ELASTIC MODULI OF
MULTICOMPONENT MATERIALS
Major: Engineering Mechanics
Code: 62 52 01 01
SUMMARY OF PhD THESIS
Hanoi – 2016
The thesis has been completed at:
Vietnam Academy of Science and Technology
Graduate University of Science and Technology
Supervisors: Assoc. Prof. Dr.Sc. Pham Duc Chinh
Reviewer 1:
Reviewer 2:
Reviewer 3:
Thesis is defended at:
on , Date Month Year 2016
Hardcopy of the thesis can be found at:
1
INTRODUCTION
Homogenization for composite material properties has made great
progress in scientific research. The construction of the material
models was made very early and from the basic ones. The
macroscopic properties of materials depend on many factors such as
the nature of the material components, volume ratio of components,
contact between elements, geometrical characteristics ... Therefore,
the thesis is done with the purpose of building evaluations for
macroscopic elastic moduli of isotropic multicomponent materials
which yield results better than the previous ones.
Topicality and significance of the thesis
Multicomponent materials (also known as composite materials) are
used widely life today. We may see composite materials that will be
the key ones in the future because of flexibility and multipurpose
material, however the identification of macroscopic materials is not
easy because we often have only limited information about the
structure of the composites.
The objective of the thesis
Construction of bounds on the elastic moduli of isotropic
multicomponent materials which involve threepoint correlation
parameters. We use numerical methods to study several
representative material models.
2
Study method
• Using the variational approach based on minimum energy
principles to construct upper and lower bounds for the
effective elastic moduli of isotropic multicomponent
materials.
• The numerical method using MATLAB program to set
the formula, matrix ... optimal geometric parameters for
particular composite materials. CAST3M program
(established by finite element method) has been applied
to several periodic material models for comparisons with
the bounds.
New findings of the dissertation
• Construction of threepoint correlation bounds on the
effective elastic bulk modulus of composite materials and
applications of the bounds are given to some composites
such as symmetric cell, multicoated sphere, random and
periodic materials.
• Construction of threepoint correlation bounds on the
effective elastic shear modulus of composite materials
and applications of the bounds are given to some
composites.
• FEM is applied to some compact composite periodic
multicomponent materials for comparisons with the
bounds.
3
Structure of the thesis
The content of the thesis includes an introduction, four chapters
and general conclusions, namely:
Chapter 1: An overview of homogeneous material
Chapter 1 presents a literature review of related work on
homogeneous material previously obtained by the domestic and
abroad researchers. Two methods to find effective elastic moduli,
which are direct equation solving and variational approach based on
energy principles, are briefly reviewed.
Chapter 2: Construction of thirdorder bounds on the effective
bulk modulus of isotropic multicomponent materials
The author uses new a trial field that is more general than Hashin
Shtrikman polarization ones to derive threepoint bounds on the
effective elastic bulk modulus tighter than the previous ones and to
construct upper and lower bounds of effk . Applications of the
bounds are given to composite structures.
Chapter 3: Construction of thirdorder bounds on the effective
shear modulus of isotropic multicomponent materials
The author constructs upper and lower bounds of effμ from
minimum energy and minimum complementary energy principles.
Applications of the bounds are given to some composites.
Chapter 4: FEM applies for homogeneous material
Calculations by the constructed FEM program for a number of
problems with periodic boundary conditions are compared with the
results from the two previous chapters.
General conclusions: present the main results of the thesis and
discuss further study.
4
CHAPTER 1. AN OVERVIEW OF HOMOGENEOUS
MATERIAL
1.1. Properties of isotropic multicomponent materials
Representative Volume Element (RVE) of multicomponent
materials is given by Buryachenko [11], Hill [30]; RVE is “entirely
typical of the whole mixture on average”, and “contains a sufficient
number of inclusions for the apparent properties to be independent of
the surface values of traction and displacement, so long as these
values are macroscopically uniform”.
Figure 1.1 Representative Volume Element (RVE)
Consider a RVE of a statistically isotropic multicomponent
material that occupies spherical region V of Euclidean space,
generally, in d dimensions (d = 2, 3). The centre of RVE is also the
origin of the Cartesian system of coordinates {x}. The RVE consists
of N components occupying regions V Vα ⊂ of volume
1 , . .( . ,v Nα α = ; the volume of V is assumed to be the unity).
The stress field satisfies equilibrium equation in V:
V∇⋅ = ∈xσ 0 , (1.1)
The local elastic tensor C(x) relates the local stress and strain
tensor fields:
( ) : ( )=x C(x) xσ ε (1.2)
5
The effective elastic moduli ( , )kα αμ=C(x) T , where T is the
isotropic fourthrank tenser with components:
2( , ) ( )ijkl ij kl ik jl il jk ij klT k k d
= + + −μ δ δ μ δ δ δ δ δ δ , (1.3)
ijδ is Kronecker symbol.
The strain field ( )ε x is expressible via the displacement field
( )u x
1
2
( ) ( ) ;T V⎡ ⎤= ∇ + ∇ ∈⎣ ⎦x u u xε (1.4)
The average value of the stress and strain has form:
1 1,= =∫ ∫σ ε
V V
d
V
d
V
σ εx x (1.5)
The relationship between the average value of the stress and strain
on V is given by effective elastic moduli effC :
: , ( , ).= = efeff eff ffe fk μC C Tσ ε (1.6)
This is called the direct solving of the equation.
In addition, a different approach to determine the macroscopic
elastic moduli may be defined via the minimum energy principle
(where the kinematic field ε is compatible):
0
0 0: : inf : :eff
V
d
〈 〉=
= ∫ε εε ε ε εC C x (1.7)
or via the minimum complementary energy principle (where the
static field σ is equilibrated):
0
0 1 0 1: ( ) : inf : ( ) :eff
V
d− −
〈 〉=
= ∫ xσ σσ σ σ σC C (1.8)
The variational approach can give the exact results but it will be
upper and lower bounds, this is a possible result when we apply to
the specific material that we do not have all information of material
geometry.
6
1.2. An overview of homogeneous material
From the late 19th century to early 20th century, the study of the
nature of the ongoing environment of multiphase materials received
great attention from the leading scientists in the world.
In the case of the model is twophase materials with inclusion
particles as spherical shape beautiful, oval (ellipsoid) distribution
platform apart in consecutive phases (phase aggregate ratio is small),
Eshelby [20] took out an inclusion of infinite domain of the
background phase, and precisely calculated the stress and strain. On
that basis, he found effective elastic moduli in a volume ratio vI
region (inclusions apart each other).
For models with the component materials distributed chaotically
(indeterminate phase), it is difficult for pathway directly solving
equations. Therefore, several methods are proposed. A typical model
is differential diagram method (differentials scheme) in which
stresses and strains calculated in step with the background of the
previous step phase contain a small percentage of spherical
inclusions or oval (using results of Eshelby). Finally, effective moluli
of the mixture for the steps can be calculated.
Besides getting answers through solving equations, there is
another method to find out macroscopic properties of composite
materials based on finding extreme points of energy function.
Although not getting the stress field and strain field accurately
corresponding to the extreme point, we still receive the
corresponding bounds for extreme values of energy functions and the
macroscopic properties of the material which is relatively close to the
true value.
7
Hashin and Shtrikman (HS) [28] have built variational principle
by using the possible polarization (polarization fields) with average
values various across different phases. Their results for isotropic
composite materials were much better than those of HillPaul.
Of domestic studies, Pham Duc Chinh’s works considered the
problem for the multiphase materials when considering the
difference of phase volume ratio, microgeometries of the
components that are characterized by threepoint correlation
parameters. In some cases, he found the optimal results (achieved by
a number of specific geometric models).
For the evaluation narrower than the rated HS, the following
authors have researched and built the variational inequalities
containing random function describing additional information about
the geometry of the particular material phase. The random function
of degree n (n  point correlation functions) depending on the
probability of any n points is taken incidentally (with certain
distance) and points fall into the same phase between them. Not from
the principle of HS, but from the minimum energy principle and the
HS polarization trial fields, Pham found a narrower HS’s estimations
though part that contains information about geometry of materials.
Another study on the homogeneous materials using numerical
method with classic digital technique has built approximately from
kinetic field possible. But there are also obstacles where it is difficult
to find the simplest possible field over the entire survey area. In case
the field is found, the system of equations may be large and complex
to solve. These problems have been overcome by the fact that the
local approximation, on a small portion of the survey area, has
explanation and simultaneously and leads to neat equations and
8
calculations extent consistent with the possibility the system features
highspeed computers. Approximation techniques smart elements
(elementwise) have been recognized for at least 60 years ago by
Courant [17]. There have been many approximation methods for
solving elastic equations. The most popular is probably finite
element method (FEM). The significance of this approach is the
partition object into a set of discrete subdomains called elements.
This process is designed to keep the results of algebraic computation
and memory management efficiency as possible.
9
CHAPTER 2. CONSTRUCTION OF THIRDORDER BOUNDS
ON THE EFFECTIVE BULK MODULUS OF
ISOTROPIC MULTICOMPONENT MATERIALS
Threepoint correlation parameters have been constructed and
used by many authors in the evaluation and approximation of
effective elastic composite materials. By choosing more general
multifree parameter trial fields than the ones of Hashin – Shtrikman,
we constructed tighter threepoint correlation bounds.
2.1. Construction of upper bound on the effective bulk modulus
of isotropic multicomponent materials via minimum energy
principle
To construct the effective bulk modulus effk from (1.7), we
choice the trial field as form:
0
1
1, ; , , ,
N
ij ij
ij a i j d
d
α
α
α=
⎛ ⎞ε = + ϕ ε = ⎜ ⎟δ⎝ ⎠∑ (2.1)
Where 0 0= ijij d
δε ε is a constant volumetric strain, αϕ is hamornic
potential; aα are free scalar that satisfy the restrictions [for the trial
field to satisfy the restriction 0ε = ε ], Latin indices after comma
designate differentiation with respective Cartesian coordinates.
Substituting the trial field (2.1) into energy functional (1.7), one
obtains:
2 0 2
1 1
C x x 2 2
, ,
: ( ) : ( ) ( )
N N
V
V
W d k v k a a A a aβγα α α α α α β γ
α= α β γ=
= = + + + μ⎡ ⎤⎢⎢ ⎥⎣
ε⎥
⎦∑ ∑∫ε ε ε
(2.2)
Where:

1=
=∑NVk kvα α
α
is called Voigt arithmetic average.
10
 xij ij
V
A d
α
βγ βα γα
α = ϕ ϕ∫ is threepoint correlation parameters.
We minimize (2.2) over the free variables aα have restriction
with the help of Lagrange multiplier λ and get the equations:
0 2 1 0 2
1
C x v v: : ( ) ' · · ( )
N
V V k k k
V
W d k v k a k −ε α α α
α=
⎡ ⎤ ⎡ ⎤= = + ε = − ε⎢ ⎥ ⎣ ⎦⎣ ⎦∑∫ε ε A
(2.3)
Where:
 { } { }1 1 1 1v v' , , ( ), , ( );T Tk N N k R N N Rv k v k v k k v k k= = − −" "
{ }
1
1 1
1
2
, , , ,kk
N N
k
R
A N
A v k A v k k A
αβ
αβ − δβ
αβ α α αβ γ α δ γ γ
γ= δ=
= α β =
⎛ ⎞= δ + − μ⎜ ⎟⎝ ⎠∑ ∑
"A
Taking in account (1.7) with (2.3), one obtains the upper bound
on effk :
1v v({ , , },{ }) · ·eff UA V k k kk K k v A k
βγ ′ −
α α α α≤ μ = − A (2.4)
2.2. Construction of lower bound on the effective bulk modulus
of isotropic multicomponent materials via minimum
complementary energy principle
To find the best possible lower bound on effk from (1.8) we take
the following equilibrated stress trial field:
0
1
1,( ) ; , , ,
N
ij ij ij ija I i j d
α α
α
α=
⎡ ⎤σ = δ + ϕ −δ σ =⎢ ⎥⎣ ⎦∑ " ; (2.5)
with I α is an indicator function.
Substituting the trial field (2.5) into (1.8) and following procedure
similar to that form, one obtains:
11 0 2 1 0 2
1
1 v v( ) ' · · ( )
N
kkR R k
a vd
W k k
d k
−− −α α
σ
αα=
⎡ ⎤− ⎡ ⎤= − σ = − σ⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦∑ A (2.6)
11
Where:
 1
1
N
R
v
k
k
− α
αα=
=∑ is called Reuss harmonic average ,
 1 1 1 11 1
1 1v ( ), , ( )
T
k V N N V
d d
v k k v k k
d d
− − − −− −⎧ ⎫= − −⎨ ⎬⎩ ⎭" ,

{ }
2
1 1
2
1 1
1 2( ) ( ,)
k
k
N N
k
V
A
vd
A v k A k A
kd
αβ
− αβ δβ −ααβ α α αβ γ δ γ γ
γ= δ=
=
⎛ ⎞−= δ + − μ⎜ ⎟⎝ ⎠∑ ∑
A
 1 11 1
1 1v ' , ,
T
k N N
d d
v k v k
d d
− −− −⎧ ⎫= ⎨ ⎬⎩ ⎭" .
The best possible lower bound on effk has been identified:
11 1v v({ , , },{ }) ( · · )eff L k kkA Rk K k v A k
′ −βγ − −
α α α α≥ μ = − A (2.7)
2.3. Applications
2.3.1. Twophase coated spheres model
(a)
(b)
12
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
2
4
6
8
10
12
14
16
18
v2
ke
ff
HS
DXC 3D
(c)
Figure 2.1 Bounds on the elastic bulk modulus of twophase coated
spheres and symmetric spherical cell mixture at
1 1 2 2 21, 0.3, 20, 10, 0.1 0.9= = = = = →μ μk k v . (a) Coated spheres; (b)
Symmetric spherical cell mixture; (c) HS  HashinStrikman upper
and lower bounds and also the respective exact effective bulk moduli
of the coated spheres at 2 1=ζ và 1 0=ζ , DXC 3D  upper and lower
bounds for the symmetrical spherical cell mixtures.
2.3.2. Twophase random suspensions of equisized spheres
Now consider the twophase random suspensions of equisized
hard spheres (Fig. 2.5a) and overlapping spheres (Fig. 2.6a) in a base
phase1. The bounds (2.4) and (2.7) for the models at
2 1 1 2 20.1 0.99, 1, 0.3, 20, 10,= → = = = =kv kμ μ together with
HashinShtrikman bounds are projected in Fig.2.2b, Fig.2.3b.
13
(a)
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
1
2
3
4
5
6
7
8
9
v2
ke
ff
HS
KCL 3D
(b)
Figure 2.2 HashinStrikman bounds (HS) and the bounds (KCL
3D) on the elastic bulk modulus of the random suspension of
equisized hard spheres
(a)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
2
4
6
8
10
12
14
16
18
20
v2
ke
ff
HS
CL 3D
(b)
Figure 2.3 HashinStrikman bounds (HS) and the bounds (CL
3D) on the elastic bulk modulus of the random suspension of
equisized overlaping spheres
14
Comment: In Figure 2.2b shows the lower bound that approaches
bound HS and the upper bound also quite far apart because kα
differences between inclusion (spheres) and the matrix. In figure
2.3b, lower bound still tend to approach the lower bound of HS but
upper bound also closes to the upper HS bound at 2 0 99.v = .
2.3.3. Threephase doublycoated sphere model
We come to the threephase doublycoated sphere model (Fig.
2.4a), where the composite spheres of all possible sizes but with the
same volume proportions of phases fill all the material space  an
extension of HashinShtrikman twophase model, at the range
1 1 2 2 3 312 8 1 0 3 30 15, , , . , , .k k k= μ = = μ = = μ =
(a)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
2
3
4
5
6
7
8
9
10
11
12
v1
ke
ff
HS
PDC 1996
NCX 3D
(b)
Hình 2.4 Bounds on the elastic bulk modulus of doubly coated
spheres at the range 1 2 3 1
10 1 0 9 1
2
. . , ( )v v v v= → = = − .
(a) Doubly coated spheres; (b) HashinStrikman (HS), old
bounds (PDC 1996) and the new bounds (NCX 3D) which
converges to the exact value of the effective bulk modulus
15
Comment: Figure 2.4b shows the results of threephase doubly
coated spheres model. In the case of twophase coated spheres
model, the PDC 1996 bounds are convergence, but it is not
convergence in the case of threephase, also threepoint correlation
parameters of the material has considered. The results are the same
between the upper and lower bounds, a new contribution of the
thesis.
2.3.4. Symmetric cell material model
Lastly we come to the symmetric cell material (Fig. 2.5a) without
distinct inclusion and matrix phase (Pham [50], Torquato [77]).
effk fall inside HashinShtrikman bounds for the large class of
isotropic composites.
(a)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
2
4
6
8
10
12
14
16
v1
ke
ff
HS
TDX 3D
(b)
Figure 2.5 HashinStrikman bounds (HS) and bounds (SYM) on
the elastic bulk modulus of threephase symmetric cell mixtures
(TDX3D), 1 2 3 10 1 0 9 0 5 1. . , . ( )v v v v= → = = −
1 1 2 2 3 31 0 3 12 8 30 15, . , , , ,k k k= μ = = μ = = μ = (a) A symmetric cell
mixture; (b) The bounds
16
2.4. Conclusions
On the variational way the author has constructed the upper and
lower bounds effk of isotropic effective elastic material through the
minimum energy and minimum complementary energy principles.
Lagrange multiplier method is used to optimize the energy
function with free variables aα have restriction.
It was found that the trial fields which are chosen (contain N  1
free parameters) more general than those in [1] contain only one free
parameter, as compared in detail in the case of threephase doubly
coated sphere model.
Some models built in case of ddimensional space so the results
are used in the general case and the bounds contain the properties,
volume fractions of the component materials and the threepoint
correlation parameters that contain information about the geometry
of the material phase to give the better results.
The results were applied to some specific material models such as
twophase coated spheres model, twophase random suspensions of
equisized hard spheres, threephase doublycoated sphere, symmetric
cell material in space 2D and 3D. To be clear, in the calculation of
comparison, the author chose the material properties varying widely.
The small difference of estimations comes closer to each other for an
approximate value of macroscopic material properties.
Results in this chapter are published by the author in the scientific
works 1, 2, 4 and 5.
17
CHAPTER 3. CONSTRUCTION OF THIRDORDER BOUNDS
ON THE EFFECTIVE SHEAR MODULUS OF
ISOTROPIC MULTICOMPONENT MATERIALS
Similar to chapter 2, the method is also based on energy principle
to help identify the upper and lower bounds of effective shear
modulus of isotropic multicomponent materials.
3.1. Construction of upper bound on the effective shear modulus
of isotropic multicomponent materials via minimum energy
principle
To construct the effective shear modulus effμ , we choice the trial
field as form:
0 0 0 0
1
1 1
2 , , ,
( ) , , ,... ;,
N
ij ij ik kj jk ki ijkl kla b i j d
α α α
α α
α=
⎡ ⎤ε = ε + ϕ ε +ϕ ε + ψ ε =⎢ ⎥⎣ ⎦∑ (3.1)
Where 0 0 0 0( )ij ij iiε = ε ε = is a constant deviatoric strain, αψ is
biharmonic potential, ,a bα α are free variables that have restricted.
Substituting (3.1) into (1.7), one obtains: ( )1 0 0C x v v 2 : : · · .V ij ij
V
W d −ε μ μ μ′= = μ − ε ε∫ ε ε A (3.2)
From (1.7) and (3.2), finally we obtain the upper bound on the
effective shear modulus effμ :
1v v({ , , },{ , }) · ·eff UAB VM k v A B
βγ βγ ′ −
α α α α α μ μ μμ ≤ μ = μ − A (3.3)
We have introduced vectors v v,′μ μ and matrix μA in 2Nspace:
{ }
1 1 1
1
1 1 1 1
1
2 2v
2 2
1 2
2 2v
2 2
( ) ( )
( ), , ( ), , , ,
( ) ( )
, , , , ,
, , , , ,
( )
,
( )
.
T
N R N N R
R N R
T
N N N N
N
V
v v v v
d d d d d d
A N
v v v v
d d d d d
v
d
μ
μ
α α
α
μ α
=
β
μ
μ −μ μ −μ ⎫⎧= μ −μ μ −μ⎨ ⎬+ +⎩ ⎭
= α β =
μ μ μ μ μ⎧ ⎫′ == ⎨ ⎬+ +⎩ ⎭ μ∑
" "
"
" "
A
18
3.2. Construction of lower bound on the effective shear modulus
of isotropic multicomponent materials via minimum
complementary energy principle
To find a lower bound on the effective shear modulus effμ , we
take the admissible equilibrated stress trial field:
( )0 0 0 0 0 0
1
1, , , ,( ) , , ,.. , ;.
N
ij ij ik kj jk ki ij ij kl kl ijkl kla I a b b i j d
α α α α α
α α α α
α=
⎡ ⎤σ = σ + ϕ σ +ϕ σ − σ − + δ ϕ σ + ψ σ =⎦⎣∑ (3.4)
Where 0 0 0 0( )ij ij iiσ = σ σ = is a constant deviatoric stress.
Substituting the trial field (3.4) into (1.8) and following procedure
similar, one obtains the best possible lower bound on effμ : ( ) 11 1v v{ , , },{ , } ( · · )eff LAB RM k v A B ′ −βγ βγ − −μ μμα α α α αμ ≥ μ = μ − A (3.5)
Where:
{ }
1 1 1 1
1 1 1 1 1 11
1
1 1 1 1
1 1 1 1
1
2 22 2
v
2 2
2 2 2 2
v
2 2
1 2
( ) ( )( ) ( )
( ), , ( ), , ,
( ) ( )
( ) ( )
, , , , ,
( ) ( )
, , , ,
,
;
T
V N N VN
V N V
N N N N
R
v vd v d v
d d d d d d
d v d v v v
d d d d d d
v
A N
− − − −
− − − −
μ
− − − −′
μ
μ − ααβμ
αα
⎧ ⎫μ −μ μ −μ− −⎪ ⎪= μ −μ μ −μ⎨ ⎬+ +⎪ ⎪⎩ ⎭
⎧ ⎫− μ − μ μ μ⎪ ⎪= ⎨ ⎬
μ
+ +⎪ ⎪⎩ ⎭
= α β = = μ
" "
" "
"A
1
.
N
=
∑
3.3. Applications
3.3.1. Symmetric cell material model
This material model without distinct inclusion and matrix phase
Pham [50] in 3Dspace (Fig.3.1a), the threepoint correlation
parameters ,A Bβγ βγα α have particular forms [5051].
19
(a)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
1
2
3
4
5
6
7
8
9
10
v1
μef
f
HS
TDX 3D
DXC 3D
(b)
Figure 3.1 Bounds on the effective shear modulus of three
component symmetric cell materials (TDX 3D), compared to bounds
for the specific symmetric spherical cell materials (DXC 3D) and
HashinShtrikman (HS) bounds; 1 2 3 10 1 0 9 0 5 1. . , . ( )v v v v= → = = −
with 1 1 2 2 3 31 0 3 12 8 30 15, . , , , ,k k k= μ = = μ = = μ = . (a) A symmetric
cell mixture; (b) Bounds
Comment: The thesis’s estimations are tighter than those before
(HS bounds), and in 3D space in range 1 0 1 0 4. .v = → , the upper
bound UDXCμ and upper bound UTDXμ are the same. According to the
opposite direction in range 1 0 5 0 9. .v = → the lower bound LDXCμ and
lower bound LTDXμ are also the same.
3.3.2. Periodic twophase model with hexagonal in shape
Lastly we come to periodic twophase model with hexagonal in
shape (LGD) (Fig. 3.2a) in range
2 0 1 0 7. .v = → with 1 1 2 21 0 5 10 5, . , ,k k= μ = = μ = . Two parameters
1ζ (or 2ζ ) and 1η (or 2η ) for this material are given in [77].
20
(a)
0.1 0.2 0.3 0.4 0.5 0.6 0.7
0
0.5
1
1.5
2
2.5
3
v2
μef
f
HS
LGD
(b)
Figure 3.2 Bounds on the effective shear modulus of periodic
hexagonal material (LGD) and HashinShtrikman (HS) bounds
The results show that LGD bounds are tighter than HS bounds.
3.4. Conclusions
The author has developed an unified approach to construct three
point correlation bounds on the effective conductivity and elastic
moduli of statistically isotropic Ncomponent materials from
minimum energy principles, using multifreeparameter trial fields,
which are generalizations of Hashin–Shtrikman twofreeparameter
polarizationfields [1], [49].
The trial fields include 2 2N − free parameters compared with 2
parameters ( 0 0,k μ ) of [1], [49], to construct new tighter bounds
at 3N ≥ . Bounds are specified to the practical class of symmetric
cell materials and twophase periodic material.
Results in this chapter are published by the author in the scientific
works 1, 3 and 5.
21
CHAPTER 4. APPLICATION OF FINITE ELEMENT
METHOD TO PERIODIC MULTICOMPONENT
MATERIALS
This chapter describes homogeneous theory for periodic materials
with and the assumptions for the application of the FEM which runs
on the open source code of the program CAST3M (France). Results
are calculated for a specific periodic model and they are compared
with the estimations in chapter 2 and chapter 3.
4.1. Homogenization of periodic materials
The idea of this theory is the basic information that is related to
the physical properties of the microstructure and stored in a base
structure (periodic cell). Then, a periodic pattern for the actual
material can be achieved by filling the entire space with the base
structure in a cyclic.
Figure 4.1 Basic structure of the periodic material
Based on the theory of homogenization, the calculations make on
specific element. To solve the problem we considered characteristic
elements of the research material symbols Ω affected by a strain
homogeneous fieldE . This strain field is produced by an average
stress field across the region Σ .
22
4.2. Introduction to CAST3M program
CAST3M program is supported by technology research
organization under the French government with a history of 20 years.
This program contains the necessary elements to simulate the object
calculated by FEM. Scope of application includes mechanical
behavior of elastic materials, elastic  visco  plastic...
As well as the structure of a program computed by the finite
element method, CAST3M is a open source code for the analysis of
structures that also includes the steps of:
• Create the respective finite element meshes.
• Provide the respective material and mechanical
properties.
• Establish the periodic boundary conditions
• Impose a macroscopic uniform stress
• Calculate the macroscopic elastic properties
4.3. Application for specific material
In the case of transverseisotropic unidirectional composites is
studied, RVE is shown in Fig 4.2
Figure 4.2 Unit cell of periodic material
23
The fiber reinforced cylindrical shaped shaft runs vertical axis
and the cross section is arranged in hexagonal shape. Five effective
elastic moduli of this material are specified as follows: effective area
modulus 1 2( , )
effe e K , effective shear modulus 1 2( , )
effe e μ , Poisson's
ratio in 1 3( , )e e or 2 3( , )
effe e ν , axial elastic modulus effE and axial
shear modulus in 1 3( , )e e or 2 3( , )
effe e μ .
The transverseisotropic unidirectional composites have matrix
phase and inclusion phase denoted m and i, respectively. Two phases
are linear elastic and isotropic, and characterized by E andν. The
value of 5 effective elastic moduli depends on the percentage of the
volume fraction of inclusion phase iv that will be presented from
figure 4.3 to figure 4.7.
The Fig. (a) corresponds to the data:
1 0 25 10 0 35, . , , .m m i iE E= ν = = ν = and Fig. (b) corresponds to the
data: 10 0 35 1 0 25, . , , .m m i iE E= ν = = ν = . FEM results are indicated
by bold dashed line and two solid lines performance the upper and
lower bounds.
Results for effK , effμ are taken by chapter 2 and 3 for 2Dspace
problem. Estimations of effE and effν rely on Hill’s relations for two
phase transverseisotropic unidirectional composites. Estimation of
effμ is the same with effective conductivity of twophase isotropic
composite that uses results in [45].
24
Figure 4.3 Relationship eff iK v−
Figure 4.4 Relationship eff ivμ −
Figure 4.5 Relationship eff ivν −
25
Figure 4.6 Relationship eff iE v−
Figure 4.7 Relationship eff ivμ −
Comment: From figure 4.3 to 4.7, numerical results (bold dashed
line) line between the evaluations (solid line), especially effE and
effν are coincided (Figure 4.5, 4.6).
The results of effE and effν are determined after we obtain effK .
It is noted that upper bound of effK may not give upper bound
of effE , effν but may give lower bounds of effK (surveying function
is monotonous!).
26
4.4. Conclusions
The author has presented theoretical homogenization for periodic
composite material within the framework of the thesis. It can be the
basis for further studies. The author has calculated several effective
elastic moduli by FEM for transverseisotropic unidirectional
composites with hexagonal cross section. The problems of periodic
composite material on the boundary force and displacement
conditions have differences compared with the common FEM
calculations. Here, the material is periodic so problem should be
taken on periodic cell with boundary condition that is periodic
displacement while condition of forces is antiperiodic.
The results were displayed and compared with the ones calculated
by estimations which built in the previous chapters. The results of
FEM line between the evaluations (which are almost coincident)
confirming the reliability of the method.
The drawback of the method is calculated primarily for periodic
materials. For materials with randomly arranged forms such as
random spheres mentioned in the previous chapter, it will be difficult
for calculation because we must consider for RVE with size rather
than unit cell.
The results presented in this chapter are published in references 5
and 6.
27
GENERAL CONCLUSIONS
The thesis has constructed estimations for effective elastic moduli
of isotropic multicomponent materials, which involve threepoint
correlation parameters. FEM is applied to a periodic composite
material for comparisons with the bounds.
New findings of the dissertation
• Construction of new estimations for effective elastic
moduli of isotropic Ncomponent materials in d
dimensional space. The thesis has used the general
polarization fields contain more free variables than the
fields in [1], to get the evaluation simpler and better in the
3N ≥ . Lagrange multiplier method is applied to
optimize the energy functional containing free variables
that are subjected to the constraints. The new estimations
contain information on the properties, volume proportions
of the component materials and threepoint correlation
parameters about the geometry of the material.
• The new estimations are applied to some models with
known threepoint correlation parameters: symmetric cell
material model, twophase random suspensions of
equisized spheres, multiphase coated spheres model (a
rather interesting finding is when applied to threephase
doublycoated sphere model, the results for the upper and
lower bound are coincident, to be the exact moduli),
periodic models in 2Dspace and 3Dspace.
28
• Using MATLAB program to set the vector, matrix in
calculating the new estimations to optimize geometric
parameters for specific materials. CAST3M program is
used to calculate the moduli of twophase transverse
isotropic unidirectional composites (cross section is
hexagonal symmetry) for comparisons with the
evaluations. The thesis constructed the numerical
program which can help to designed and predict new
material properties, as desired.
Issues for further research
• Construct the trial fields for better evaluation (narrower
upper and lower bound).
• Build further evaluations for more sophisticated materials
such as random cell polycrystals.
• Combine estimations with the FEM simulation and
approximation methods to study the properties of
complex geometry materials.
List of the Author’s publications
1. Pham, D.C., Vu, L.D., Nguyen, V.L. (2013), Bounds on the
ranges of the conductive and elastic properties of randomly
inhomogeneous materials. Philosophical Magazine 93,
22292249.
2. Pham Duc Chinh and Vu Lam Dong (2012), Threepoint
correlation bounds on the effective bulk modulus of
isotropic multicomponent materials. Vietnam Journal of
Mechanics 34, pp. 6777.
3. Vu Lam Dong and Pham Duc Chinh (2013), Construction of
bounds on the effective shear modulus of isotropic
multicomponent materials. Vietnam Journal of Mechanics
35, 275283.
4. Vũ Lâm Đông, Phạm Đức Chính (2012). Đánh giá bậc 3 mô
đun đàn hồi diện tích của vật liệu đẳng hướng hai chiều
nhiều thành phần. Hội nghị Cơ học toàn quốc lần thứ IX Hà
Nội, 89/12/2012, 303312.
5. Vũ Lâm Đông, Phạm Đức Chính và Trần Bảo Việt (2013).
Đánh giá biến phân và tính toán số PTHH cho các hệ số đàn
hồi vật liệu tổ hợp đẳng hướng ngang. Hội nghị Khoa học
toàn quốc Cơ học Vật rắn biến dạng lần thứ XI Thành phố
Hồ Chí Minh, 79/11/2013.
6. Trần Bảo Việt, Vũ Lâm Đông và Phạm Đức Chính (2014).
Mô phỏng số PTHH và đánh giá các hệ số đàn hồi vật liệu
cốt sợi dọc trục đẳng hướng ngang. Tuyển tập công trình
Hội nghị Cơ học Kỹ thuật toàn quốc Kỷ niệm 35 năm thành
lập Viện Cơ học, 10/4/197910/4/2014,Tập 2, 443448.
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