ESTIMATION AND COMPUTATIONAL SIMULATION FOR THE EFFECTIVE ELASTIC MODULI OF MULTICOMPONENT MATERIALS

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 three-point correlation
parameters.

We

use

numerical

representative material models.

methods

to

study

several


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 three-point 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, multi-coated sphere, random and

periodic materials.

•

Construction of three-point 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 third-order bounds on the effective
bulk modulus of isotropic multicomponent materials
The author uses new a trial field that is more general than HashinShtrikman polarization ones to derive three-point bounds on the

effective elastic bulk modulus tighter than the previous ones and to
construct upper and lower bounds of k eff . Applications of the
bounds are given to composite structures.
Chapter 3: Construction of third-order bounds on the effective
shear modulus of isotropic multicomponent materials
The author constructs upper and lower bounds of μeff 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

Vα ⊂ V

of

volume

vα (α = 1, . . . , N ; the volume of V is assumed to be the unity).
The stress field satisfies equilibrium equation in V:
∇⋅ σ = 0 ,

x ∈V

(1.1)

The local elastic tensor C(x) relates the local stress and strain
tensor fields:
σ(x) = C(x) : ε(x)

(1.2)


5
 

The effective elastic moduli C(x) = T(kα , μα ) , where T is the
isotropic fourth-rank tenser with components:
2
Tijkl (k, μ ) = kδ ij δ kl + μ (δ ikδ jl + δ ilδ jk − δ ij δ kl )
d

(1.3)

,

δ ij is Kronecker symbol.

The strain field ε(x) is expressible via the displacement field

u( x )
1
ε(x) = ⎣⎡∇u + (∇u )T ⎦⎤ ; x ∈ V
2

(1.4)

The average value of the stress and strain has form:
1

1
σ =
σ dx , ε =
ε dx
VV
VV

∫

∫

(1.5)

The relationship between the average value of the stress and strain
on V is given by effective elastic moduli Ceff :
σ = Ceff : ε ,
Ceff = T(k eff , μ eff ).  

 

(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):
ε0 : Ceff : ε0 = inf0 ∫ ε : C : εdx
〈 ε 〉= ε

(1.7)

V

or via the minimum complementary energy principle (where the
static field σ is equilibrated):
σ 0 : (Ceff )−1 : σ 0 = inf 0 ∫ σ : (C)−1 : σd x
〈 σ〉 =σ

(1.8)

V

 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 multi-phase materials received
great attention from the leading scientists in the world.
In the case of the model is two-phase 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 Hill-Paul.
Of domestic studies, Pham Duc Chinh’s works considered the
problem for the multi-phase materials when considering the

difference of phase volume ratio, micro-geometries of the
components that are characterized by three-point 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
high-speed computers. Approximation techniques smart elements
(element-wise) 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 sub-domains called elements.
This process is designed to keep the results of algebraic computation
and memory management efficiency as possible.


9
 
CHAPTER 2. CONSTRUCTION OF THIRD-ORDER BOUNDS
ON THE EFFECTIVE BULK MODULUS OF
ISOTROPIC MULTICOMPONENT MATERIALS
Three-point correlation parameters have been constructed and
used by many authors in the evaluation and approximation of
effective elastic composite materials. By choosing more general
multi-free parameter trial fields than the ones of Hashin – Shtrikman,
we constructed tighter three-point 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 k eff from (1.7), we
choice the trial field as form:
⎛ δij N
⎞
(2.1) 
εij = ⎜ + aα ϕ,αij ⎟ ε0 ; i , j = 1,…, d
⎝ d α=1
⎠
δ

Where ε ij0 = ij ε 0 is a constant volumetric strain, ϕ α   is hamornic
d
potential; aα are free scalar that satisfy the restrictions [for the trial

∑

field to satisfy the restriction ε = ε 0 ], Latin indices after comma
designate differentiation with respective Cartesian coordinates.
Substituting the trial field (2.1) into energy functional (1.7), one
obtains:
N
N
⎡
⎤
Wε = ε : C( x ) : εd x = ⎢ kV + vα kα ( 2aα + aα2 ) +
Aαβγ 2μα aβ aγ ⎥ (ε0 )2
α=1
α ,β, γ=1
V
⎦⎥
⎣⎢
(2.2)
Where:

∑

∫

N

-

∑

kV = ∑ vα kα  is called Voigt arithmetic average.
α =1


10
 
Aαβγ =

-

∫ϕ

βα γα
ij ϕij d x

is three-point correlation parameters.

Vα

We minimize (2.2) over the free variables aα have restriction
with the help of Lagrange multiplier λ and get the equations:
N
⎡
⎤
Wε = ε : C : εd x = ⎢ kV + vα kα aα ⎥ (ε0 )2 = ⎡⎣kV − v 'k ·Ak−1 ·vk ⎤⎦ (ε0 )2
α=1

⎣
⎦
V
(2.3)
Where:

∑

∫

v 'k = {v1k1 ," , vN k N } ;

vk = {v1 (k1 − kR )," , vN (k N − kR )}

T

-

{ }

k
Ak = Aαβ

α, β = 1," , N

,

k
Aαβ
= vα kα δαβ +

T

N

⎛

∑⎜ A
γ=1 ⎝

αβ
γ

− vα kR

N

∑k

−1 δβ
δ Aγ

δ=1

⎞
⎟2μ γ
⎠

Taking in account (1.7) with (2.3), one obtains the upper bound
on k eff :

k eff ≤ K UA ({kα , μα , vα },{ Aαβγ }) = kV − v′k ·Ak −1 ·vk

(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 k eff from (1.8) we take
the following equilibrated stress trial field:
N
⎡
⎤
σij = ⎢δij + aα (ϕ,αij − δij I α )⎥ σ0 ; i , j = 1," , d ; (2.5)
α=1
⎣
⎦
α
with I is an indicator function.

∑

Substituting the trial field (2.5) into (1.8) and following procedure
similar to that form, one obtains:
⎡
−1
d − 1 N aα vα ⎤ 0 2 ⎡ −1
⎤ 0 2
Wσ = ⎢ kR−1 −
⎥ (σ ) = ⎢ kR − v 'k ·A k ·vk ⎥ (σ ) (2.6)
⎣
⎦
d
k
α ⎦
α=1
⎣

∑


11
 
Where:

-

kR−1 =

N

vα

∑k

is called Reuss harmonic average ,

α

α=1

T

-

1− d
⎧1 − d
⎫
vk = ⎨
v1 (k1−1 − kV−1 )," ,
vN (k N−1 − kV−1 )⎬ ,
d
⎩ d
⎭

{ }
k

A k = Aαβ

-

k

A αβ =

N
⎛ αβ vα
(1 − d )2
−1
v
k
δ
+
⎜ Aγ −
α α αβ
2
kV
d
γ=1 ⎝

∑

N

∑k A
δ

δβ

γ

δ=1

⎞
−1
⎟( 2μ γ ) ,
⎠

T

-

1− d
⎧1 − d
⎫
v'k = ⎨
v1k1−1 ," ,
vN k N−1 ⎬ .
d
⎩ d
⎭

The best possible lower bound on k eff has been identified:
′

−1

k eff ≥ K AL ({kα , μα , vα },{ Aαβγ }) = (kR−1 − v k ·A k ·vk )−1 (2.7)

2.3. Applications
2.3.1. Two-phase coated spheres model

(a)

(b)


12
 
18
16
HS

14

DXC 3D
12

k eff

10
8
6
4
2
0
0.1

0.2

0.3

0.4

0.5
v2

0.6

0.7

0.8

0.9

(c)
Figure 2.1 Bounds on the elastic bulk modulus of two-phase coated
spheres

and

symmetric

spherical

cell

mixture

at

k1 = 1, μ1 = 0.3, k2 = 20, μ2 = 10, v2 = 0.1 → 0.9 . (a) Coated spheres; (b)
Symmetric spherical cell mixture; (c) HS - Hashin-Strikman upper
and lower bounds and also the respective exact effective bulk moduli
of the coated spheres at ζ 2 = 1 và ζ 1 = 0 , DXC 3D - upper and lower
bounds for the symmetrical spherical cell mixtures.
2.3.2. Two-phase random suspensions of equisized spheres

Now consider the two-phase random suspensions of equisized
hard spheres (Fig. 2.5a) and overlapping spheres (Fig. 2.6a) in a base
phase-1. The bounds (2.4) and (2.7) for the models at

v2 = 0.1 → 0.99, k1 = 1, μ1 = 0.3, k2 = 20, μ2 = 10,

together

Hashin-Shtrikman bounds are projected in Fig.2.2b, Fig.2.3b.

with


13
 
9
8
HS
7

KCL 3D

k eff

6
5
4
3
2
1
0.1

0.15

0.2

0.25

0.3

0.35
v2

0.4

0.45

0.5

0.55

0.6

(b)

(a)

Figure 2.2 Hashin-Strikman bounds (HS) and the bounds (KCL
3D) on the elastic bulk modulus of the random suspension of
equisized hard spheres
20
18
HS

16

CL 3D
14

k eff

12
10
8
6
4
2
0
0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

v2

(a)

(b)

Figure 2.3 Hashin-Strikman 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 v2 = 0.99 .
2.3.3. Three-phase doubly-coated sphere model

We come to the three-phase doubly-coated 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 Hashin-Shtrikman two-phase model, at the range
k1 = 12, μ1 = 8, k2 = 1, μ 2 = 0.3, k3 = 30, μ3 = 15.
12
11
10
9

k eff

8
7
6
5
HS

4

PDC 1996
3

2
0.1

(a)

NCX 3D
0.2

0.3

0.4

0.5
v1

0.6

0.7

0.8

0.9

(b)

Hình 2.4 Bounds on the elastic bulk modulus of doubly coated
1
spheres at the range v1 = 0.1 → 0.9, v2 = v3 = (1− v1) .
2
(a) Doubly coated spheres; (b) Hashin-Strikman (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 three-phase doubly
coated spheres model. In the case of two-phase coated spheres
model, the PDC 1996 bounds are convergence, but it is not
convergence in the case of three-phase, also three-point 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]).
k eff fall inside Hashin-Shtrikman bounds for the large class of

isotropic composites.
16
14

HS
TDX 3D

12

k eff

10
8
6
4
2
0
0.1

(a)

0.2

0.3

0.4

0.5
v1

0.6

0.7

0.8

0.9

(b)

Figure 2.5 Hashin-Strikman bounds (HS) and bounds (SYM) on
the elastic bulk modulus of three-phase symmetric cell mixtures
(TDX3D), v1 = 0.1 → 0.9, v2 = v3 = 0.5(1 − v1 )
k1 = 1, μ1 = 0.3, k2 = 12, μ 2 = 8, k3 = 30, μ3 = 15 (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 keff 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 three-phase doublycoated sphere model.
Some models built in case of d-dimensional space so the results
are used in the general case and the bounds contain the properties,
volume fractions of the component materials and the three-point
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
two-phase coated spheres model, two-phase random suspensions of
equisized hard spheres, three-phase doubly-coated 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 THIRD-ORDER 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 μeff , we choice the trial
field as form:
εij =εij0 +

N

⎡

∑ ⎢⎣a

α

α=1

Where

ε0ij

1 α 0
⎤
(ϕ,ikεkj + ϕ,αjkε0ki ) + bα ψ ,αijklε0kl ⎥, i , j = 1,..., d ; (3.1)
2
⎦

=εij0 (εii0 = 0) is a constant deviatoric strain, ψ α is

biharmonic potential, aα , bα are free variables that have restricted.
Substituting (3.1) into (1.7), one obtains:
Wε = ε : C : εd x = μV − v′μ ·Aμ−1 ·vμ 2ε0ijε 0ij .

∫

(

)

(3.2)

V

From (1.7) and (3.2), finally we obtain the upper bound on the
effective shear modulus μ eff :
μ eff ≤ M UAB ({kα , μ α , vα },{ Aαβγ , Bαβγ }) = μV − v′μ ·Aμ−1 ·vμ

(3.3)

′

We have introduced vectors v μ , vμ and matrix Aμ in 2N-space:
T

v
2 v (μ − μ R )
2 v (μ − μ R ) ⎫
⎧v
vμ = ⎨ 1 (μ1 − μ R )," , N (μ N − μ R ), 1 1
," , N N
⎬ ,
d
d
d
(
d
+
2
)
d ( d + 2) ⎭
⎩

{ }

Aμ = Aαμβ ,

α, β = 1," , 2 N ,
T

N
v μ
2v1μ1
2v μ ⎫
⎧v μ
v′μ = ⎨ 1 1 ," , N N ,
," , N N ⎬ , μV = vα μ α .
d
d ( d + 2)
d ( d + 2) ⎭
⎩ d
α =1

∑


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 μ eff , we
take the admissible equilibrated stress trial field:
σij = σij0 +

N

∑ ⎡⎣a ( ϕ
α

α=1

α
0
,ik σkj



)

0
0
0
⎤ , i , j = 1,..., d ;
+ ϕ,αjkσki

− I ασij0 − (aα + bα )δij ϕ,αklσkl
+ bα ψ ,αijklσkl
⎦

(3.4)

Where σ0ij = σij0 (σii0 = 0) 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 μ eff :

(

)

′

−1

L
μ eff ≥ M AB
{kα , μ α , vα },{ Aαβγ , Bαβγ } = (μ −R1 − v μ ·A μ ·vμ )−1 (3.5)

Where:
T

2v (μ −1 − μV−1 )
2v (μ −1 − μV−1 ) ⎪⎫
( 2 − d )vN −1
⎪⎧ ( 2 − d )v1 −1
vμ = ⎨

(μ1 − μV−1 )," ,
(μ N − μV−1 ), 1 1
," , N N
⎬
d
d
d ( d + 2)
d(d + 2) ⎭⎪
⎩⎪
⎧⎪ ( 2 − d )v1μ1−1
′
( 2 − d )vN μ −N1 2v1μ1−1
2v μ −1 ⎫⎪
vμ =⎨
," ,
,
," , N N ⎬ ,
d
d
d ( d + 2)
d(d + 2) ⎭⎪
⎩⎪

{ }
μ

A μ = Aαβ ,

α, β = 1," , 2 N

;

μ −R1 =

N

vα

∑μ
α =1

.

α

3.3. Applications
3.3.1. Symmetric cell material model

This material model without distinct inclusion and matrix phase
Pham [50] in 3D-space (Fig.3.1a), the three-point correlation
parameters Aαβγ , Bαβγ have particular forms [50-51].


19
 
10
9

HS

8

TDX 3D
DXC 3D

7

μeff

6
5
4
3
2
1
0
0.1

0.2

0.3

(a)

0.4

0.5
v1

0.6

0.7

0.8

0.9

(b)

Figure 3.1 Bounds on the effective shear modulus of threecomponent symmetric cell materials (TDX 3D), compared to bounds
for the specific symmetric spherical cell materials (DXC 3D) and
Hashin-Shtrikman (HS) bounds; v1 = 0.1 → 0.9, v2 = v3 = 0.5(1 − v1 )
with k1 = 1, μ1 = 0.3, k2 = 12, μ 2 = 8, k3 = 30, μ3 = 15 . (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 v1 = 0.1 → 0.4 , the upper

bound μUDXC and upper bound μUTDX are the same. According to the
opposite direction in range v1 = 0.5 → 0.9 the lower bound μ LDXC and
L
lower bound μTDX
are also the same.

3.3.2. Periodic two-phase model with hexagonal in shape

Lastly we come to periodic two-phase model with hexagonal in
shape

(LGD)

(Fig.

3.2a)

in

range

v2 = 0.1 → 0.7 with k1 = 1, μ1 = 0.5, k2 = 10, μ 2 = 5 . Two parameters

ζ1 (or ζ 2 ) and η1 (or η2 ) for this material are given in [77].


20
 
3

HS

2.5

LGD

μ

eff

2

1.5

1

0.5

0
0.1

(a)

0.2

0.3

0.4
v2

0.5

0.6

0.7

(b)

Figure 3.2 Bounds on the effective shear modulus of periodic
hexagonal material (LGD) and Hashin-Shtrikman (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 threepoint correlation bounds on the effective conductivity and elastic
moduli of statistically isotropic N-component materials from
minimum energy principles, using multi-free-parameter trial fields,
which are generalizations of Hashin–Shtrikman two-free-parameter
polarizationfields [1], [49].
The trial fields include 2 N − 2 free parameters compared with 2
parameters ( k0 , μ0 ) of [1], [49], to construct new tighter bounds
at N ≥ 3 . Bounds are specified to the practical class of symmetric

cell materials and two-phase 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 field E . 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 transverse-isotropic 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 (e1 , e2 ) K eff , effective shear modulus (e1 , e2 ) μeff ,  Poisson's
ratio in (e1 , e3 )   or (e2 , e3 ) ν eff ,  axial elastic modulus E eff and axial
shear modulus in (e1 , e3 )  or  (e2 , e3 ) μeff . 
The transverse-isotropic 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 vi that will be presented from
figure 4.3 to figure 4.7.
The

Fig.

(a)

corresponds

to

the

data:

Em = 1, ν m = 0.25, Ei = 10, ν i = 0.35 and Fig.  (b) corresponds to the
data: Em = 10, ν m = 0.35, Ei = 1, ν i = 0.25 . FEM results are indicated
by bold dashed line and two solid lines performance the upper and
lower bounds.
Results for K eff , μeff are taken by chapter 2 and 3 for 2D-space
problem. Estimations of E eff and ν eff rely on Hill’s relations for twophase transverse-isotropic unidirectional composites. Estimation of
μeff is the same with effective conductivity of two-phase isotropic
composite that uses results in [45].