Phát triển phương pháp tính toán phi tuyến đa tỉ lệ ngẫu nhiên cho vật liệu tổng hợp sợi ngắn ứng dụng cho nghiên cứu ảnh hưởng của cấu trúc vi mô biến đổi đến quá trình phát triển phá hỏng vật liệu

Luận Án Tiến Sỹ Kỹ Thuật

Phát triển phương pháp tính toán phi tuyến đa tỉ lệ ngẫu

nhiên cho vật liệu tổng hợp sợi ngắn ứng dụng cho

nghiên cứu ảnh hưởng của cấu trúc vi mô biến đổi đến

quá trình phát triển phá hỏng vật liệu

8/2020

Khoa Sau Đại Học Về Khoa Học Và Kỹ Thuật

Đại Học Keio

Hoàng Tiến Đạt

I

LUẬN ÁN

Đã nộp tới Đại học Keio và thỏa mãn hết các tiêu chí của bằng tiến sỹ

Nay, luận án này được nộp tới Đại học Thái Nguyên

II

Abstract

The mechanical properties of fiber reinforced composite materials are scattered

especially in the development of a new and cost-effective manufacturing process. The main

reason lies in the microstructural variability expressed by physical parameters of constituent

materials and geometrical parameters to express the morphology at microscale. The short fiber

reinforced composite materials can be fabricated easily by injection molding method, but they

have random microstructures. To solve the problem considering variability, there have been

many studies on the stochastic finite element method. The first-order perturbation based

stochastic homogenization (FPSH) method has been developed based on the multiscale theory

and verified for porous materials and multi-phase composite materials considering the

variability in physical parameters. However, its applications were limited to linear elastic

problems. Therefore, this study aims at the development of a stochastic nonlinear multiscale

computational method. In its application to short fiber reinforced composites, the final goal of

this study is to clarify the important random factors in the microstructure that give significant

influence on the damage propagation.

Firstly, the above FPSH method was extended for the stochastic calculation of

microscopic strain. This theory enabled us to analyze the damage initiation and propagation in

a stochastic way. Since huge scenarios exist in the nonlinear behaviors, however, a sub￾sampling scheme was proposed in the analysis by FPSH method together with the sampling

scheme for geometrical random parameters. Furthermore, to reduce the computational time for

practical and large-scale analyses of stochastic damage propagation problems, a numerical

algorithm to accelerate the convergence of element-by-element scaled conjugate gradient

(EBE-SCG) iterative solver for FPSH method was developed. The efficiency was demonstrated

for spherical particulate-embedded composite material considering the damage in the coating

layer and the variability in physical parameter.

Finally, the developed computational method was applied to short fiber reinforced

composite materials. The fiber length distribution, fiber orientation denoted by two angles and

III

fiber arrangement were considered as the geometrical parameters in addition to a physical

random parameter. 11 models were analyzed having different fiber orientation and fiber

arrangement with variability. In the sub-sampling, 2 scenarios with 50% and 0.3% probabilities

were analyzed, which resulted in totally 22 cases. The differences among 22 possible damage

patterns were discussed deeply. It was figured out that very largely scattered degradation of

homogenized macroscopic properties was mostly affected by the fiber arrangement rather than

the fiber orientation. This finding was different from the result in linear elastic region. The

physical random parameters were more influential on the macroscopic properties. Also in these

analyses, the accelerated EBE-SCG method was again shown to be efficient.

IV

Acknowledgements

First and foremost, I would like to give my deep gratitude to my advisor Prof. Naoki Takano,

who is one of my great and respected teachers in my life. He always gives me very clear guidance

and suggestions from the first contact before entering to Keio University until now. When I was

stuck in some difficult tasks, he encouraged, taught and provided very good ideas to motivate me.

He is very close and smiling with students in the discussion. Besides asking about my research, he

sometimes asks about my family and my student life and understands the difficulties of international

students in Japan. Without him, I could not complete my research and learn much new knowledge.

I am deeply grateful to him for being my advisor.

I am sincerely grateful to Prof. Fukagata, Prof. Omiya, and Assist. Prof. Muramatsu for

being the reviewers of my dissertation. From their valuable questions and comments, I could

improve the dissertation well and understand more about the limitations as well as the potential

applications of my proposed method in the future works.

I would like to extend my thankfulness to Assoc. Prof. Akio Otani (Kyoto Institute of

Technology) for supporting the measured fiber length data, and Prof. Heoung-Jae Chun (Yonsei

University) for giving me necessary knowledge about composite materials. I also would like to

thank all my labmates for three years at Keio University. Especially, Mr. Kohei Hagiwara came to

take me in Haneda airport and helped me prepared for my student life on the first days in Japan;

And, Mr. Daichi Kurita, Mr. Yutaro Abe, Mr. Lucas Degeneve, and Mr. Ryo Seino joined to discuss

my research topic and assisted me to use some Japanese software in our Lab. Besides, Mr. Yuki

Nakamura, Mr. Tatsuto Nose, and Ms. Mizuki Maruno also helped me to easily get involved in the

student life, and Japanese culture. I also would to thank Dr. Akio Miyoshi and Mr. Shinya Nakamura

from Insight, Inc., company for helping us to develop the Meshman Particle Packing software.

Next, I greatly thank the Ministry of Education, Culture, Sports, Science and Technology

of Japan for supporting the full scholarship (Monbukagakusho – MEXT) to me in 3 years at Keio

University. In addition, I would like to thank Keio University for providing the Keio Leading-Edge

Laboratory of Science and Technology (KLL) funding. I also want to express my gratitude to my

home university, Thai Nguyen University of Technology for giving me an opportunity to study in

Japan, keeping my lecturer position when I come back, and also paying a part of my salary.

Finally, I would like to give millions of love to my parents, my wife, my daughter, and my

son. They always encourage and look forward every single day of my life. Especially, I could not

express my words to describe the sacrifice of my wife to take care of the children when I was not

at home (2 internships in USA, 2 years in Taiwan and 3 years in Japan). To show my gratitude

towards everybody, I tried to hard study every day to complete the tasks as well as possible. This

dissertation is the achievement that I would like to give to everyone.

V

Chapter of Contents

Abstract ..................................................................................................................................... II

Acknowledgements..................................................................................................................IV

Chapter of Contents................................................................................................................... V

List of Figures ....................................................................................................................... VIII

List of Tables............................................................................................................................XI

Abbreviation............................................................................................................................XII

Nomenclatures....................................................................................................................... XIII

Chapter 1 Introduction ............................................................................................................... 1

1.1 Motivation ................................................................................................................... 1

1.2 Short fiber reinforced composite materials ................................................................. 6

1.2.1. Injection molding and conventional micromechanical model ............................. 6

1.2.2. Fiber length, fiber orientation and fiber arrangement .......................................... 8

1.3 Aims and scopes of research ..................................................................................... 11

1.4 Structure of dissertation............................................................................................. 12

Chapter 2 Literature review and methodologies ...................................................................... 14

2.1 Micromechanics, multiscale approach and homogenization with composite

materials............................................................................................................................... 14

2.2 Damage model and damage criteria .......................................................................... 20

2.3 Variability, uncertainty or randomness ..................................................................... 26

2.3.1 Variability of physical properties....................................................................... 26

2.3.2 Variability of geometrical parameters................................................................ 27

2.3.3 Variability of loading and boundary conditions................................................. 28

2.3.4 Variability of manufacturing process parameters .............................................. 28

VI

2.4 Stochastic finite element methods............................................................................. 29

2.5 First-order perturbation based stochastic homogenization method for composite

materials............................................................................................................................... 31

2.6 Stochastic modeling of fiber reinforced composites ................................................. 34

Chapter 3 Stochastic calculation of microscopic strain ........................................................... 36

3.1 Microscopic displacement ......................................................................................... 36

3.2 Derivation of microscopic strain in stochastic way................................................... 38

3.3 Numerical example of microscopic strain considering only variability of physical

parameters ............................................................................................................................ 41

Chapter 4 Stochastic nonlinear multiscale computational scheme with accelerated EBE-SCG

iterative solver.......................................................................................................................... 50

4.1 Sampling and sub-sampling for stochastic nonlinear multiscale computational

scheme.................................................................................................................................. 50

4.2 Acceleration of EBE-SCG iterative solver................................................................ 56

4.3 Numerical example.................................................................................................... 60

4.3.1 Verification of the accelerated EBE-SCG solver by characteristic displacement

visualization ..................................................................................................................... 60

4.3.2 Stochastic damage propagation.......................................................................... 65

4.3.3 Degradation of homogenized properties............................................................ 68

4.3.4 Acceleration of solution for nonlinear simulation.............................................. 68

Chapter 5 Application to short fiber reinforced composites to study the influence of

microstructural variability on damage propagation ................................................................. 71

5.1 Microstructural modeling and sampling.................................................................... 71

5.2 Numerical results....................................................................................................... 77

5.2.1 Probable damage patterns................................................................................... 77

VII

5.2.2 Microscopic strain during damage propagation ................................................. 79

5.2.3 Homogenized properties in linear and nonlinear analysis ................................. 80

5.3 Acceleration of EBE-SCG during damage propagation............................................ 81

5.4 Discussion of influence level of variability in physical and geometrical parameters 83

Chapter 6 Concluding remarks................................................................................................. 87

List of publications................................................................................................................... 91

References................................................................................................................................ 92

VIII

List of Figures

Fig. 1. 1 Distribution of mechanical properties of constituent materials................................... 4

Fig. 1. 2 Influence of order on the variance of outcome result .................................................. 4

Fig. 1. 4 Injection molding process............................................................................................ 7

Fig. 1. 5 Model of an injection molded structure ..................................................................... 10

Fig. 1. 6 Main points and their relations in structure of dissertation........................................ 13

Fig. 2. 1 Multiscale framework ................................................................................................ 15

Fig. 2. 2 Unit cell array ............................................................................................................ 17

Fig. 2. 3 Classification of composite materials........................................................................ 19

Fig. 2. 4 Rate of use for different failure criteria for composite materials in published papers

by others................................................................................................................................... 23

Fig. 2. 5 General framework of stochastic finite element approach ........................................ 30

Fig. 2. 6 Two-scale problem of heterogeneous media ............................................................. 33

Fig. 3. 1 Flowchart of damage analysis formulation................................................................ 36

Fig. 3. 2 Definition of SVE for short fiber-reinforced composite material.............................. 42

Fig. 3. 3 Microscopic strain

33 

in interphase and short fibers when random physical

parameters for all constituent materials are considered ........................................................... 45

Fig. 3. 4 RVE models of glass fiber reinforced plastics composite based on the lognormal

distribution ............................................................................................................................... 47

Fig. 3. 5 Definition of two cross-sections ................................................................................ 48

Fig. 3. 6 Microscopic effective strain distributions on the cross-section 1-2 under macroscopic

strain

2

31 E 0 02 10

   . ............................................................................................................ 49

Fig. 3. 7 Microscopic effective strain distributions on the cross-section 2-3 under macroscopic

strain

2

31 E = 0 02 10

  . ............................................................................................................ 49

IX

Fig. 4. 1 General flowchart of research algorithm ................................................................... 53

Fig. 4. 2 Detail algorithm of stochastic damage propagation analysis..................................... 54

Fig. 4. 3 Accelerated procedure for the EBE-SCG solver in Fig. 4.1 ...................................... 57

Fig. 4. 4 Visualization of characteristic displacements of two successive sub-cycles............. 58

Fig. 4. 6 SVE model of coated particle-embedded composite material................................... 60

Fig. 4. 7 Finite element model of coated particle-embedded composite material ................... 61

Fig. 4. 8 Zeroth-order terms of characteristic displacements in x direction

 

11 0

x



of two

successive sub-cycles at (a) cycle 1 and (b) cycle 17 .............................................................. 62

Fig. 4. 9 Zeroth-order terms of characteristic displacements in y direction

 

11 0

y



of two

successive sub-cycles at (a) cycle 1 and (b) cycle 17 .............................................................. 63

Fig. 4. 10 Zeroth-order terms of characteristic displacements in z direction

 

11 0

z



of two

successive sub-cycles at (a) cycle 1 and (b) cycle 17 .............................................................. 63

Fig. 4. 11 First-order terms of characteristic displacements in x direction

 

11 1

3

x



of two

successive sub-cycles at (a) cycle 1 and (b) cycle 17 .............................................................. 64

Fig. 4. 12 First-order terms of characteristic displacements in y direction

 

11 1

3

y



of two

successive sub-cycles at (a) cycle 1 and (b) cycle 17 .............................................................. 64

Fig. 4. 13 First-order terms of characteristic displacements in z direction

 

11 1

3

z



of two

successive sub-cycles at (a) cycle 1 and (b) cycle 17 .............................................................. 65

Fig. 4. 14 Stochastic damage propagation of coated particle-embedded composite material . 66

Fig. 4. 15 Damage element visualization of a half of coating.................................................. 67

Fig. 4. 16 Damage element visualization of a half of coating.................................................. 67

Fig. 4. 17 Degradation of homogenized macroscopic properties DH when ............................ 68

Fig. 4. 18 Acceleration ratios and convergence histories in EBE-SCG iterations to calculate 69

Fig. 4. 19 Acceleration ratios and convergence histories in EBE-SCG iterations to calculate 70

X

Fig. 5. 1 Microstructure modeling for skin-core-skin layered specimen of injection molded

short fiber reinforced composite material ................................................................................ 72

Fig. 5. 2 Example model of Meshman Particle Packing .......................................................... 73

Fig. 5. 3 Microstructure models with different fiber arrangements for given fiber angel

variabilities by modified Meshman ParticlePacking................................................................ 74

Fig. 5. 4 Definition of fiber orientation and SVE model of short fiber reinforced composites75

Fig. 5. 5 Initial configuration of samples for cycle 1,

1 S X ...................................................... 76

Fig. 5. 6 Some damage patterns predicted at cycle 3, E = 0.005,

3 S X

n

(n = 0 or 3)................. 78

Fig. 5. 7 Influence of physical parameters on damage pattern when the properties of matrix 78

Fig. 5. 11 Strain distributions in matrix highlighting on high strain value and on the............. 80

Fig. 5. 12 Degradation of apparent stiffness ............................................................................ 81

Fig. 5. 13 Number of EBE-SCG iterations and damaged volume evolution of a sample

3 1 2  | 

S A X X .................................................................................................................................. 82

Fig. 5. 14 Number of EBE-SCG iterations and damaged volume evolution of a sample

3 1 4  | 

S A X X .................................................................................................................................. 83

Fig. 5. 15 SVE specification in degraded apparent stiffness with enlarged views on the........ 84

Fig. 5. 16 Influence of fiber arrangement on damaged volume evolution ............................... 85

Fig. 5. 17 Homogenized properties without damage ............................................................... 86

Fig. 5. 18 Variability influence level of physical and geometrical parameters........................ 86