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

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 materia

VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY

GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY

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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

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:

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 methods to study several representative material models

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

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 Shtrikman 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 Applications of the eff

Hashin-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 μ from eff

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

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 α⊂V of volume

1 ( , ,

vα α = N ; the volume of V is assumed to be the unity)

The stress field satisfies equilibrium equation in V:

The effective elastic moduli C(x)=T( ,k α μ α), where T is the

isotropic fourth-rank tenser with components:

The relationship between the average value of the stress and strain

on V is given by effective elastic moduli Ceff:

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):

or via the minimum complementary energy principle (where the

static field σ is equilibrated):

 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

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 vIregion (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

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

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

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 eff

k from (1.7), we choice the trial field as form:

0 1

Substituting the trial field (2.1) into energy functional (1.7), one obtains:

α = ϕ ϕ∫ is three-point correlation parameters

We minimize (2.2) over the free variables aα have restriction with the help of Lagrange multiplier λ and get the equations:

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 eff

k from (1.8) we take the following equilibrated stress trial field:

0 1

1,

N

α α=

with Iαis an indicator function

Substituting the trial field (2.5) into (1.8) and following procedure similar to that form, one obtains:

a v d

v k

k

α α=

=∑ is called Reuss harmonic average ,

k d

The best possible lower bound on eff

k has been identified:

(c) Figure 2.1 Bounds on the elastic bulk modulus of two-phase coated spheres and symmetric spherical cell mixture at

1=1,μ1=0.3, 2=20,μ2=10, 2=0.1→0.9

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 ζ2=1 và ζ1=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

2 =0.1→0.99,k1=1, 1=0.3, 2 =20, 2=10,

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

(a)

1 2 3 4 5 6 7 8 9

(b)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

(a)

0 2 4 6 8 10 12 14 16 18 20

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

(b)Hình 2.4 Bounds on the elastic bulk modulus of doubly coated spheres at the range 1 0 1 0 9 2 3 1 1 1

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])

v 1

HS TDX 3D

(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)

1 1, 1 0 3 , 2 12, 2 8, 3 30, 3 15

k = μ = k = μ = k = μ = (a) A symmetric cell mixture; (b) The bounds

2.4 Conclusions

On the variational way the author has constructed the upper and lower bounds k effof 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 doubly-coated 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