(Luận án tiến sĩ) 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 sĩ) 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 sĩ) 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 sĩ) 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 sĩ) 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 sĩ) 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 sĩ) 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 sĩ) 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 sĩ) 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 sĩ) 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 sĩ) 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 sĩ) 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 sĩ) 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 sĩ) 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 sĩ) 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 vvvvvvvvvv

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

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

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 sub-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

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

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

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

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

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

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  in interphase and short fibers when random physical 33 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 E31 0 02 10.  2 49

Fig 3 7 Microscopic effective strain distributions on the cross-section 2-3 under macroscopic strain 2 31 0 02 10 E = .   49

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  0 11 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  0 11 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  0 11 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  1 11 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  1 11 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  1 11 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

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 composites 75 Fig 5 5 Initial configuration of samples for cycle 1, 1X 76 S Fig 5 6 Some damage patterns predicted at cycle 3, E = 0.005, 3X (n = 0 or 3) 78 n S 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

List of Tables

Table 1 Expected values of components in stress–strain matrices 42

Table 2 Engineering constants 47

Table 3 Random parameters 47

Table 4 Engineering constants 61

Table 5 Setting of samplings 76

Table 6 Properties of constituent materials 76

Table 7 Legend of characteristic displacements 81

BBA: Building block approach

BC: Boundary condition

CDF: Cumulative density function

CDM: Continuum damage mechanics

Cov: Coefficient of variance

DEM: Discrete element method

EBE-SCG: Element-by-element scaled conjugate gradient

FEM: Finite element method

FPSH: First-order perturbation based stochastic homogenization

FRC: Fiber reinforced composite

IM: Injection molding

KL: Karhunen-Loeve

MCS: Monte Carlo simulation

PC: Polynomial Chaos

PDF: Probability density function

RVE: Representative volume element

SFEM: Stochastic finite element method

SFRC: Short fiber reinforced composite

SFRP: Short fiber reinforced plastic

SVE: Statistical volume element

UD: Unidirectional

X-FEM: Extended finite element method

CTE: Coefficient of thermal expansion

Chapters 1, 2, 3: Multiscale method and first-order perturbation based stochastic homogenization method

L N Average fiber length, N is number of fibers

f ( ) Probability density function

kl Deformation mode of characteristic displacement

B r Strain-displacement matrix of constituent material r

P A matrix to assign a scalar or random variable to

component of stress-strain matrix D

Q A vector to extract each column of stress-strain matrix D

material r

matrix D of constituent material r

  The first order term of homogenized properties

corresponding to random variable α r,mn

f (X b ) Probability of model X b

Chapters 4, 5: Stochastic nonlinear multiscale computational scheme

n Level of standard deviation of microscopic strain

n = 0 - 3

i max Total number of fiber arrangement sample

j max Total number of fiber orientation sample

X X Sub-sample corresponding to n level of standard

deviation and fiber arrangement sample A |

i j X

X Microscopic strain of model X i j |

 ε

Zero-order term of characteristic displacement vector

for kl mode in x, y, z direction

 1

, ,

χ r kl

x y z

First-order term of characteristic displacement vector

for kl mode in x, y, z direction of constituent material r

p Initial search direction

Chapter 1 Introduction

1.1 Motivation

Heterogeneous and composite materials, like hardened steel, bronze or wood were

valued since ancient times because they provide better performance compared to the individual

phases or components which they consist of Nowadays the idea of combining eligible materials

to form a composite material with new and superior properties compared to its individual

components is still subject to ongoing research For example, polymers, which by nature have

a low density, can be reinforced by highly stiff and strong glass or carbon fibers, both

continuous and discontinuous Such fiber reinforced composites excel in high specific

mechanical properties For lightweight structures, high specific stiffness and strength are crucial

requirements The higher the specific mechanical properties are the lighter a part or construction

can be designed Hence, composite materials are increasingly attractive for fulfilling industrial

needs because their mechanical and physical-chemical properties can be adapted to meet

specific design requirements To improve the performance of composites, multiscale study is

now a popular topic in computational mechanics The real composite materials are

heterogeneous and characterized by various degrees of inherent variability or randomness

Variability exists on all scales from the arrangement of a material microstructure to the structure

at the macroscale Particularly, there may exist variability in the constituent material properties,

and geometries of the composite materials at various scales The variability in these parameters

induces variability of the mechanical behaviour and damage evolution in the materials which

may cause severe random reflections of composite structures Besides, the variability of the

materials may result in huge unpredicted scenarios of damage evolution Damages in a

composite structure may remain hidden below the surface, undetectable by visual inspection

until the entire structure has failed

For the improvements in the design processing technologies on cost-efficient design,

the building block approach (BBA) is applied to test scale-up from coupons, elements, and

subcomponents to establish final composite structures [1, 2] This approach implies decreasing

the number of tests when moving from coupon tests level to testing of the entire products such

as aircraft or automobiles This approach still results in a large number of mechanical tests

required for certification of new material because of variability or uncertainty (~104) [3] The total number of tests becomes enormous when several candidate materials are considered At

the lower level of the BBA, the composite microstructures of coupons are often investigated by

virtual testing The engineering constants, strengths, and strain-to-failure of a coupon are

estimated by tension, compression, and shear tests [2] During manufacturing processes,

composite materials involve many randomness or variability in microstructures affecting the

quality of the component This BBA needs to involve many experiments to achieve a safe

design It results in excessive cost and time-consuming processes Modelling of mechanical

behaviour can facilitate the development of new materials at all stages of the design chain and

reduce the number of real experiments during the certification process And, the replacement

of a huge number of experimental tests of composite materials by the stochastic numerical

simulation considering the variability or uncertainty is a big matter of concern recently

However, only a few of these variabilities have been studied in detail

Multiscale modeling is a useful tool for predicting the effective macroscopic behaviour

of materials having a periodic structure at the microscale Most of the modern approaches to

composites modelling are based on the well-established multiscale approach which suggests to

build a hierarchy of scale levels starting from the micro-scale of individual fibers up to the

macro-scale of components Several analytical and computational models have been proposed

to calculate the mechanical properties of composite materials, such as multiphase materials [4–

6], short fiber reinforced composite (SFRC) structure [7, 8], metal/ceramic composites [9],

textile composite materials [10, 11], or particulate composite structures [12, 13] The

asymptotic homogenization-based finite element method, a popular multiscale method, has

been widely used in the analysis of the mechanical properties of composite material [14–17]

However, conventional multiscale modeling does not explicitly account for variability

and uncertainty in the physical and geometrical parameters of microstructures A few recent

studies have used a variety of numerical methods to address this Xu [18] developed a multiscale

stochastic finite element method (SFEM) to resolve scale-coupling stochastic elliptic problems

Savvas et al [19] showed the effect that uncertainty in the constituent materials and the

geometry of the microstructure has on the macroscopic properties of composite materials, using

Monte Carlo simulation (MCS) Zhou et al [20] proposed a perturbation-based stochastic

multiscale finite element method for calculating the mean values and coefficients of variation

of the effective elasticity properties of composite materials given the uncertainties of the

mechanical properties of the constituent materials Wen et al [21] presented a practical

first-order perturbation-based stochastic homogenization (FPSH) method that considers many

random physical parameters by using a finite element method (FEM) But these works have

only focused on the prediction of macroscopic properties of composite materials

Stochastic multiscale computational approach should be developed to investigate not

only properties but also other behaviors, especially damage propagation in composite materials

considering the variability or uncertainty in their microstructures based on stochastically

analyzing microscopic strain or stress Only a few papers seem to have examined microscopic

strain/stress or analyzed damage stochastically Sakata et al [22] reported a method of linear

analysis that uses first-order perturbation technique, analyzing a simple unidirectional

fiber-reinforced plastic to predict the initial probability of damage Ma et al [23] applied MCS to a

2D particle-matrix composite material Ju et al [24] took a micromechanical approach and

solved the problem in two dimensions for a fiber-reinforced titanium-alloy matrix composite

material to study the evolution of progressive damage from fiber breaking Only a few works

were presenting the stochastic multiscale framework for initial damage prediction with simple

2D models to avoid high computational cost

Applying a suitable method in nonlinear simulation to avoid high computational cost is

a big matter of concern, especially in stochastic nonlinear simulation Mechanical properties of

constituent materials of composites considering uncertainty are often assumed distributed as

normal distributions with small covariance, i.e less than 5% as shown in Fig 1.1(a) [25, 26]

The influence of higher orders in perturbation method or polynomial expansion method on the

variation of the outcome result Φ is illustrated in Fig 1.2(a) The first-order approach works

well in a small range of covariance less than 3% [27] Figure 1.2(b) shows the first-order

perturbation function with respect to random variable ζ Since to shorten the entire

computational process in nonlinear simulation, and since the results of the higher-order

approach are not always give higher accuracy than that of the order approach, the

first-order perturbation was used in this dissertation The general formulation of the method will be

provided in Section 2.5

Fig 1 1 Distribution of mechanical properties of constituent materials

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

Stochastic multiscale approach can be used to represent the macroscopic nonlinear at a

point in a composite structure When investigating damage behavior at the macroscale, the

average properties of the materials are employed Thus, the variability in microstructure is not

taken into account or the influence of microstructural variability cannot be observed by

macroscopic damage analysis However, the failure appears at the micro scale and initiates from

the interface between fiber and matrix or interface between laminae The debonding between

fiber and matrix as well as the small transverse cracks in matrix leads to delamination between

layers Finally, the composite material and structure are totally broken by fiber breakage Thus,

 

2

1 2 1 2

the microscopic damage is followed by macroscopic structural fracture It is well known that

the multiscale methods can consider the microscopic damage behaviors, but there is a problem

when the variability or uncertainty at the micro scale is taken into consideration The influence

of microstructural variability on the behaviors is limited only to local behavior, but will not be

clearly observed in macroscopic behavior Some slight differences in microscopic behaviors

will result in almost the same macroscopic behavior because the macroscopic quantities are the

averaged ones of microscopically distributed quantities In other words, slight differences will

be vanished by the averaging process On the other hand, when variability or scattering was

seen in macroscopic behavior, to predict such behavior, since the parameter or index in

commonly used damage models (see Section 2.2) can be applied to macroscopic behavior of

fiber bundles and composite structures, the variability of those parameters must be determined

It is easy to understand that the reliable distribution of damage parameters is hardly measured

Moreover, it will be impossible to correlate with physical meaning in real phenomena, for

instance, by analyzing the posterior probability On the other hand, the stochastic multiscale

analysis enables us to understand the variability in real phenomena and to analyze the source

of variability in measured data When the interaction between macro- and microscopic

behaviors with variability is considered as mentioned above, it implies the possibility of

numerous scenarios for microscopic behaviors due to variability, although the macroscopic

behavior seems to be unique Figure 1.3 illustrates this complexity more clearly For example,

there are n models (samplings) when considering random geometrical parameters Each sample

has m sub-models (sub-samplings) considering random physical parameters The macroscopic

displacement-loading plot is achieved during the microscopic damage propagation by

stochastic nonlinear simulation The number of obtained curves are the same number of the

multiple n × m Due to the effect of random variables, the knee points are varied and the

variation of those curves will become larger after the knee points or in nonlinear zone because

of uncertainty propagation It is very challenging to capture this phenomenon Each response

curve is the combination of multi-phenomena from the microscale to macroscale during damage

propagation from time t 1 to t k Moreover, although the damage behaviors of different

sub-samplings are not the same, their macroscopic behaviors can be the same This phenomenon

can be similar in case of different samplings For this complicated multiscale problem, the

stochastic nonlinear simulation is needed Specifically, the first-order perturbation based

stochastic homogenization method must be improved and extend to nonlinear simulation

Recently, fiber reinforced composite (FRC) materials have many applications in a

broad range of engineering fields and industries such as aerospace, automobile, aviation, and

light industrial products, because of their low density, high strength and design flexibility [28]

Especially, SFRCs are being used extensively since they can be used to make large and complex

body parts [29] In this dissertation, the development of the stochastic homogenization method

for damage prediction can be applied for all types of composite materials However, particle

composites and short fiber reinforced plastics (SFRPs) are used as illustrations for the efficiency

of the new proposed scheme

1.2 Short fiber reinforced composite materials

1.2.1 Injection molding and conventional micromechanical model

When fibers in a composite are discontinuous and are shorter than a millimeter, the

composite is called SFRC Short fiber reinforced plastic (SFRP) composites have found

extensive applications in automobiles, business machines, durable consumer items, sporting

goods, and electrical industries, etc., because of their low cost, easy processing and their

superior mechanical properties over the parent polymers Extrusion compounding and IM

processes are frequently employed to make SFRP composites During the injection molding

(IM) process, fibers breakage also usually occurs, and a random orientation distribution, a fiber

length distribution, random fiber arrangement would result in the final product (see Fig 1 4)

The mechanical properties, such as strength, elastic modulus of SFRPs are critically dependent

on these morphological structures [30, 31] The fiber orientation in SFRPs during the IM has

been the subject of several investigations [28] To maximize the effectiveness of short glass

fiber structures in mechanical properties, damage behaviors, the variabilities or uncertainties of

their microstructures will be deeply considered in this work The mechanical and physical

properties of SFRP composites have been the subject of much attention and are influenced to a

great degree by the type, amount and morphology of the reinforcing fibers, and the interfacial

bonding efficiency between the fibers and polymer matrix Variables such as properties of

constituents, fiber volume fraction, orientation, arrangement, length or aspect ratio, and

interfacial strength are of prime importance to the final mechanical properties and performances

exhibited by injection molded polymer composites [32]

Fig 1 3 Injection molding process

During the IM process, fibers breakage also usually occurs, and a random orientation

distribution, a fiber length distribution, random fiber arrangement would result in the final

product Mortazavin and Faremi [28] investigated the anisotropic effects on tensile strength of

short fiber reinforced plastics (SFRP) and found the variation of these materials caused by fiber

orientation Ioannis et al [30] showed the larger effect of microstructure parameters such as

aspect ratio, fiber orientation on the macroscopic behavior of short fiber reinforced

thermoplastic composites The mechanical properties of SFRPs are critically dependent on

these morphological structures, especially fiber orientation distribution which can result in the

unreliable determination of the deterministic computation [33] The fiber orientation in SFRCs

made by IM has been optimized by Chen et al [34] Additionally, fiber arrangement in

microstructures of SFRPs should also be considered To maximize the effectiveness of short

glass fiber structures in mechanical properties, damage behaviors, fiber orientation, and fiber

arrangement were deeply considered in this research

For structural design of short fiber reinforced parts, micromechanical or numerical

models are often used to predict their properties and behavior Many micromechanical models

Gate Region

Lubrication Region

Fountain Region

Inlet

Mold wall Frozen layer

2h Flow

have been developed but they are often based on idealized composite morphologies, a matrix

comprising aligned fibers with equal size [35], or a single ellipsoid in an infinite matrix [36]

In order to predict the properties and behaviour of realistic composite morphologies, it is

necessary to use models that take into account the composite morphology as realistic as possible

There exist numerous micromechanics-based models that were developed to predict a

complete set of elastic constants for aligned short-fiber composites One of the most popular

ones is the Halpin-Tsai model which was initially developed for continuous fiber composites

and which was derived from the self-consistent models of Hermans [37] and Hill [38] A well

established and theoretically well founded micromechanical model is the one of Tandon and

Weng [35] which is based on the Eshelby’s solution of an ellipsoidal inclusion in an infinite

matrix [36] and Mori-Tanaka’s average stress [39] This model is applicable to spherical,

fiber-as well fiber-as to disk-shaped particles

Because the overall effective properties of a SFRP can vary between isotropic (3D

randomly oriented fibers) and highly anisotropic (aligned fibers) it is of great importance to

design the mold and to control the process parameters, in a way that everywhere locally across

the finished part the short fibers act along the axes of principal stresses There exist commercial

software packages (Moldflow, Sigmasoft, ) that are used to simulate the mold filling process

and at the same time to determine the local fiber orientation states in a finished injection molded

part after cooling The elastic tensor for the unidirectional reinforced unit is subsequently used

in orientation averaging or orientation tensor to determine the elastic tensor for all the actual

fiber orientation states which are present in the simulated injection molded part It is, however,

this approach is not accurate enough in order to be used in practice for designing injection

molded short fiber reinforced structures Therefore, one of the goals of this dissertation is to

propose the different way to consider random orientation in a micromechanical model of

SFRPs

1.2.2 Fiber length, fiber orientation and fiber arrangement

The mechanical properties of short fiber reinforced polymers depend on the dispersion

of fiber length in the finished part One aspect of short fiber composites which can be difficult

to address analytically is the distribution of fiber lengths that are normally present in a real

material The most popular approach is to replace the fiber length distribution with a single

length, normally the number average length L N

The most common method used for fiber length measurement is a direct measurement

of fiber lengths after resin burnout The fibers are cast onto glass slides and dispersed in an aqueous solution like saline/lubricant solution or natural water The fiber dispersion is then dried, leaving an even fibers distribution on the glass slides The fibers samples are then used for measurement of fibers length by optical photos The length distribution of fibers in a short fibers reinforced polymer composite can be described with a probability density function On the other hand, fiber length distribution is considered deterministically by a simple way that the fibers are chosen randomly from the measured fiber length distribution

The fiber orientation pattern is the dominant structural feature of injection molded short

fiber reinforced polymer composites The composite is stiffer and stronger in the direction of

the major orientation while much weaker in the transverse direction Fiber orientation can be

strongly influenced by the processing condition and molded geometry For large samples like

injection molded plates, the composite molding has a typical skin–core–skin structure with two

skin layers The fibers are highly oriented in the flow direction and a core layer whereas fibers

are mainly aligned transversely to the flow direction as shown in Fig 1 5 [40, 41]

Fig 1 4 Model of an injection molded structure

Fiber orientation can be measured using an image analyzer system [42] The system

works by imaging directly from a polished and etched section taken from the SFRP composite,

in which each fiber image appears as an ellipse The reflection microscopy of a polished

composite section easily lends itself to automation allowing a large number of fiber images to

be processed in a short time at the order of 10,000 images in 20 minutes The analysis method

allows three-dimensional (3D) fiber orientation distribution functions to be determined To

quantify the fiber orientation in modelling, the fiber orientation tensor or averaging scheme is

proposed [33, 43] The orientation averaging scheme is one of the methods to predict the overall

properties of a known orientation state of fibersby averaging the UD property tensor T(p) or

the orientation averaged elastic tensor <C ijkl> over all directions weighted by the orientation

distribution function ψ(p) [43] It is, however, too cumbersome for numerical calculations and

therefore efforts have been made to find alternative ways of describing orientation states A

new proposed way to generate and consider the fiber orientation in micromechanical model of

SFRP will be presented in Section 5.1

Both regular and irregular fiber arrangements on the cross-section of UD composites

was studied in [44] In regular fiber arrangement models, the failure paths are perpendicular to

the loading direction In irregular fiber arrangement models, the failure paths are more

complicated and tortuous than regular fiber arrangement Charles and Tucker [45] identified

the best model for predicting the stiffness of aligned short-fiber composites with different fiber

packing arrangements Bhaskara et al [46] studied the effect of fiber geometry including fiber

arrangement on elastic and thermal properties of unidirectional fiber reinforced composites

Zixing et al [47] show that the fiber arrangement has an important influence on the yarn’s

Flow direction Skin layer

Core layer

z

x

y

damage process and the final strength of woven fabric composites The study of the

investigation of the influence of fiber arrangement on the dynamic response of a particular

composite structure has been performed in [33] The importance of the fiber arrangement was

shown by Swolfs et al [48] who conducted numerical studies of stress concentrations near a

broken fiber The finite element simulations were used to show that stress concentration factors

in the neighboring fibers in the presence of a broken fiber are higher in a fiber array with random

packing than in a fiber array with regular packing Trias et al [49] utilized a uniform distribution

of fiber positions to generate an RVE and shown that the von Mises stresses in a random UD

composite are typically 2.5 times higher than in the periodic UD composite with hexagonal

packing and the same fiber volume fraction Mishnaevsky [50] studied the tensile strength of a

UD composite combining a Weibull distribution and random arrangement of fibers using 3D

finite element simulations A random number of damageable zones per fiber was used in an

improvement to the model to capture gradual damage propagation in fibers [51] However, no

framework was developed to consider random fiber arrangement in computational micro-scale

models for multiscale damage modelling In this work, new multiscale computational scheme

is proposed to point out the influence of randomness in fiber arrangement on damage

propagation

1.3 Aims and scopes of research

In this research, there are two main aims Firstly, an FPSH method is developed to

build a stochastic nonlinear multiscale computational scheme considering both physical and

geometrical variability and/or uncertainty on damage propagation in composite materials,

especially short fiber-reinforced composites The formulation of stochastic calculation of

microscopic strain is derived and applied as a damage criterion Since the characteristic

displacements, which represents the heterogeneity, play an important role in FPSH, it was

visualized to show the physical meaning and used to verify the code in this study In the

formulation of FPSH, most of the computational time is spent obtaining the characteristic

displacement, which makes it impractical for large-scale problems For problems, an iterative

algebraic equation solver should be used Thus, the element-by-element scaled conjugate

gradient (EBE-SCG) method [52, 53] was employed in this research However, for nonlinear

analysis, the equations must be solved many times after each increment In this work, the

characteristic displacements and their extensive use as the initial vectors in the incremental

steps, and carry out that investigation to increase the convergence rate of EBE-SCG solver The

application of the acceleration of EBE-SCG solver is proved through the numerical example of

a coated particle-embedded composite model

Secondly, the influence of microstructural variability on the damage propagation of

SFRC, one example of composite materials, is investigated by the stochastic nonlinear

multiscale computational scheme Microstructure modeling of a SFRP made by IM is

introduced Following that, the sampling using for generating random geometrical parameters,

and the sub-sampling using for a huge scenario in the stochastic nonlinear analysis are

proposed The sub-sampling is an effective approach to capture small damage probability at the

response distribution tail The application of the proposed scheme for the SFRP is carried out

to predict probable damage patterns under a complex strain condition For an application, the

damage criterion is assumed to be deterministic one Since the effective microscopic strain of

each element corresponding to its certain sub-sampling based on the upper limit of the strain

distribution for instance, exceeds the strain threshold value, that element is damaged Finally,

the discussion of the influence of physical and geometrical parameters on this material is shown

in this investigation The efficiency of the accelerated element-by-element scaled conjugate

gradient (EBE-SCG) solver is also performed This present work represents the first step

towards developing a robust simulation framework for the prediction of practical damage

evolution of composite materials

1.4 Structure of dissertation

The dissertation is divided into 6 Chapters Literature review of the research is shown

in Chapter 2 including micromechanics, randomness of composite microstructures, stochastic

finite element methods, and the first-order perturbation based stochastic homogenization

method The damage model and criteria are provided And then, the sources of variability and

the stochastic computational approaches are presented Chapter 3 addresses the derivation of

microscopic strain in a stochastic way which will be used in the stochastic nonlinear scheme of

damage propagation analysis and a numerical example Chapter 4 shows the stochastic

nonlinear multiscale computational scheme on damage propagation The scheme are figured

out from general to detail algorithm by flowcharts Following that, to apply the scheme more

practically, an acceleration of EBE-SCG solver in FPSH method during damage simulation and

its verification and application on simple multiphase particle composite material are performed

Chapter 5 conducts the application to random SFRPs to study the influence of microstructural

variability on damage propagation Based on the observed results, a discussion of the influential

level of variability in physical and geometrical parameters is appended Finally, Chapter 6

remarks some important conclusions and proposes the potential future works The main points

and their relations in the structure of dissertation are shown in Fig 1.6

Fig 1 5 Main points and their relations in structure of dissertation

Motivation

Multiscale(2.1), Variability(2.2), Stochastic FEM(2.4), First order perturbation(2.5)

Short FRP(1.2), Stochastic modeling(2.6)

Chapter 4

Sampling for nonlinear simulation Fast computing scheme

(4.1) (4.2)

Example (4.3)

Review (1.1)

Chapter 3

- Prediction of probable damage patterns

- Degradation of macroscopic properties

- Variability influence level of microstructural parameters

Acceleration (5.3)

Chapter 5

Chapter 2 Literature review and

methodologies

2.1 Micromechanics, multiscale approach and

homogenization with composite materials

Nowadays, the multiscale approach in various forms has become a standard approach

for composites modelling among researchers Prediction of mechanical behaviour FRCs is a

complicated task due to their complex structure As mentioned, the homogenization procedure

assumes substitution of a heterogeneous structure with a homogeneous medium exhibiting

equivalent properties Central to the homogenization of composite materials is the concept of a

representative volume element (RVE) An RVE is a part of the heterogeneous structure that can

be considered instead of the whole composite structure by the means of mechanical or other

properties The RVE should be large enough to contain a sufficient number of geometrical

features in order to represent typical properties at the chosen level On the other hand, the RVE

should be small enough to be considered as a typical region of heterogeneous medium and to

reduce computational cost [54] The multiscale framework is shown in Fig 2.1 The

homogenization technique provides the properties or response of a structure at higher scale given the properties or response of the structure’s constituent materials at lower scale Conversely, localization techniques provides the response of the constituents given the response

of the structure During multiscale analysis, a particular stage in the analysis procedure can

function on both levels simultaneously

Fig 2 1 Multiscale framework

The homogenized properties in Eqs (2.1) and (2.2) of a heterogeneous medium relate

the average applied strain to the average stress in the medium through tensor of average stiffness

where σ and ε are stress and strain tensors in the RVE respectively  and E the volume

averaging stress and strain V is the volume of RVE domain Ω

The micro-scale of an FRC is the scale of individual fibers bound together with matrix

material A composite with all the fibers aligned in one direction is called unidirectional (UD)

composite which is presented at the micro-scale as a fiber array Two idealizations are usually

made for simplification of the modelling of UD composites The first common assumption for

models at this scale is to assume an infinite length of fibers [55] Obviously, this is not the case

in real structures in which fibers have a finite length However, the ratio of fiber diameter to

length of fiber is small and the influence of a fiber end due to stress concentrations is negligibly

small in most regions of the FRC In addition, UD composite is assumed that fibers are perfectly

straight and parallel to each other

One of the first models for the prediction of elastic properties of UD composites were

suggested by Voigt and Reuss [55] It was suggested that the components of a composite

structure (fibers and matrix) can be represented as springs connected in parallel or series with

weights proportional to their volume fraction The elastic properties of these two models can

be estimated applying uniform strain or stress respectively These approaches are widely known

as the rule of mixtures In the case of parallel connection, the stiffness in a specified direction

is equal to the weighted average of the individual stiffness components in this direction In the

case of serial connection, the compliance is equal to weighted average of compliances The UD composite’s elastic modulus can be expressed by Eq (2.3)

where V i is volume fraction of the i-th component, n is number of components and K i is the

elastic modulus of i-th component, respectively But it should be used only for longitudinal

Young’s modulus

It was shown by Christensen [55] that these formulae provide lower and upper bounds

for the elastic properties for a real composite In general, these bounds are usually far from

experimental values However, the rule of mixtures for the Young’s modulus in the fiber

direction predicts the experimental value with a good accuracy Another attempt to derive closer

bounds for the properties of UD composites was made by Hashin and Rosen [56] An RVE was

constructed as a single fiber surrounded by a cylindrical bulk of matrix material for their study

Two sets of boundary condition (BCs) were applied to the RVE: traction and displacement (von

Neumann and Dirichlet BCs) That allowed prediction of theoretical lower and upper bounds

for the effective elastic properties of UD composites These predictions are more precise than

the rule of mixtures but still cannot predict properties sufficiently close to experimental data

Chamis [57] modified the rule of mixtures formulae for engineering constants to fit

experimental data This semi-empirical approach does not require any modelling and is widely

used due to its simplicity [58] The Chamis formulae are reproduced below

where the index 1 corresponds to the longitudinal direction and indices 2 and 3 correspond to

the transversal directions Equation (2.4) which is the original rule of mixtures gives good

precision and is often used to estimate the longitudinal Young’s modulus

Another idealization is often made when an RVE of UD composites is constructed

Fibers are assumed to be arranged in one of two regular patterns: square or hexagonal, as shown

in Fig 2.2 These two patterns are periodic and the smallest period of the patterns is called a

unit cell The unit cell of a periodic pattern allows recreation of a whole pattern using

translations only A periodic representation of UD composite requires correctly formulated BCs

Fig 2 2 Unit cell array

z

y x

z

y x

(a) Square array (b) Hexagonal array

A periodic representation of UD composite requires correctly formulated BCs Von

Neumann and Dirichlet BCs both satisfy the Hill-Mandel principle of homogeneity [59]

However, von Neumann and Dirichlet boundary conditions do not provide a purely periodic

solution in the case of a periodic unit cell For periodic unit cells, periodic BCs are required to

satisfy both periodicity of the stress-strain field and the Hill-Mandel condition General periodic

BCs for various periodic fiber arrangements were presented by Li [60]

Despite there is a large number of models dealing with a regular arrangement of fibers

the concept of regular arrangement contradicts all the experimental observations which show

that a realistic fiber arrangement is inherently stochastic The approaches for modelling random

fiber arrangement have been big concerned and will be reviewed in Section 5.1

Recently, composite materials have many applications in a broad range of engineering

fields and industries such as aerospace, automobile, civil engineering and so on Composite

materials are often clarified into 3 types: particle reinforced composites, fiber reinforced

composites, and structural composites as shown in Fig 2.3 However, modelling and predicting

the properties of these materials remains a challenge due to their heterogeneities in

microstructures Homogenization techniques are well-known tools to efficiently derive those

homogenized composite properties analytically or numerically from the constituent properties

and from the microstructures of heterogeneous materials Various micromechanical models

have been developed to predict the macroscopic properties and simulate the behaviour of

composite materials In the analytical homogenization branch, Eshelby proposed a method to

predict the effective moduli of composite considering the ellipsoidal inclusion in 1957 [61]

From this solution for the ellipsoidal inclusion, many special cases such as sphere, elliptic

cylinder, spheroid, crack, etc., can be derived Later, Mori and Tanaka [39] developed a technique based on Eshelby’s inclusion method to estimate the average internal stress in a matrix containing inclusion with eigenstrains For commercialization, the Eshelby and Mori-

Takana methods were integrated into CAE software such as Digimat and Geodict which are

used widely in various engineering fields However, these analytical homogenization methods

can only consider uniform strain or stress inside microstructures

Fig 2 3 Classification of composite materials

Among numerical homogenization methods with the help of FEM as a numerical tool,

asymptotic homogenization method is very popular and is used to model, predict the material

properties, and compute the behavior of complex composite materials accurately In 1990,

Guedes and Kikuchi [62] derived a rigorous formulation of asymptotic homogenization method

in a weak form and applied the formulation to analyse some practical applications of composites

such as fiber reinforced plastics, sandwich honeycomb plate, or woven textile composites

Hence, it has been broadly applied to show macro-micro coupling behavior of composites [9,

63, 64] Terada and Kikuchi have been employed the homogenization method to characterize

the response of heterogeneous solids undergoing inelastic deformations in three-dimensional

elasto-plastic problems [65] and presented a rigorous computational tool of using digital

image-based modeling with asymptotic homogenization method to study complex micromechanical

characteristics of composite materials [66] Takano et al [67] have also presented a formulation

of the homogenization method to analyse the mechanical behaviours of knitted fiber reinforced

thermoplastics under larger deformation In a later work, the results in Ref [67] was

experimentally studied furthermore [68] Chun et al [69, 70] also found the microstructure

effects of geometrical parameters of multiaxial warp knit fabric composites and has validated

the proposed method through the comparison of the elastic properties of the materials to the

experimental values Additionally, asymptotic homogenization can be used to simulate and

Particle Reinforced Fiber Reinforced Structural

Continuous

Discontinuous

Laminates

Sandwich Panels

…

Nylon, PP, ABS, PC,

…

Epoxy, Polyester, Vinylester,

…

OF COMPOSITES

predict the permeability of reinforced composites considering micro-macro coupling effect for

the flow through porous media [71] Fish et al [72] have developed a nonlocal damage theory

for brittle composite materials using homogenization method and localization techniques based

on the double scale asymptotic expansion of damage Especially, asymptotic homogenization

was integrated into a CAE software named VOXELCON which is popularly applied in

industrial and medical fields together with the image-based modeling capability

In homogenization methods with the help of FEM, representative volume element

(RVE) is used to analyze complex composite materials However, modelling RVE when

considering uncertainty or variability is very difficult and still challenging for complex

microstructures Therefore, digital image-based RVE model was used and validated in

multiscale analysis in VOXELCON [53, 73, 74] Nevertheless, in these works, the RVE models

of the materials were only investigated after the materials were fabricated In designing and

manufacturing new materials, there is a need to predict the characteristics and behaviors of the

materials before fabrications

2.2 Damage model and damage criteria

Damage in composite materials occurs through different mechanisms that are complex

and usually involve interaction between constituents at the microscale During the past two

decades, a number of damage models have been developed to simulate damage and failure

process in composite materials, among which the damage mechanics approach is particularly

attractive in the sense that it provides a viable framework for the description of distributed

damage including material stiffness degradation, initiation, and propagation From the

mathematical formulation standpoint, in the macromechanical approach, homogenization is

performed first followed by application of damage mechanics principles to homogenized

anisotropic medium, while in the micromechanical approach, damage mechanics is applied to

each phase followed by homogenization

The strength of composite material at the microscale level depends on the strength of

the fibers, the matrix and the bonding between the fibers and the matrix Usually, the

longitudinal strength of fibers is higher than the strength of the matrix Disregarding imperfect

when a composite is under tensile loading in the fiber direction The elastic behavior of a UD

composite under longitudinal load can be found by assuming uniform strain in the fibers and

matrix applying the rule of mixtures Assuming a uniform strength of all the fibers to be S f, the

longitudinal strength S L can be found by the rule of mixtures for strengths:

The strength of a UD composite under transverse tension or shear cannot be correctly

estimated by the rule of mixtures An attempt to provide simple empirical formulae for strength

of UD composites was made by Chamis [57]:

The transverse or shear loading of UD composites usually results in damage initiation

in the matrix material or debonding of matrix from fibers Brittle failure can be described by a

maximum stress criterion or maximum principal stress criterion However, failure under a

complex loading is better described by an interactive criterion such as von Mises:

1 2 1 3 2 3 2S m

where S m is the strength of the matrix, determined by a unidirectional tensile test

One of the many methods for numerical modelling the damage or failure propagation in

the mechanics of composites have been developed: continuum damage mechanics (CDM) and

extended finite element method (X-FEM) CDM was initially suggested by Kachanov [75] as

a method for modelling damage in isotropic materials The main idea of the concept was to

represent damaged media with microcracks as a homogeneous media with the reduced

properties CDM was used by Ernst et al [76] for multiscale analysis of textile composite

including a comparison of non-linear behaviour of UD composites with square and hexagonal

packing Both models predicted reasonable results but the hexagonal model predicted strength

values closer to experimental results and the model with square packing yielded elastic

properties close to experimental results However, the question of which packing allows better

predictions remains open Alternative approaches that consider random packing of fibers should

be widely considered By contrast with CDM which represents discontinuities by degrading the

properties of continuous material, X-FEM makes it possible to model discontinuities using a

special FE formulation, using developed fracture mechanics for simulation of damage

propagation The transverse strength of a UD composite was studied by Bouhala et al [77] via

the use of X-FEM Two-scale damage modeling of brittle composites [78] The mathematical

homogenization method based on double-scale asymptotic expansion is generalized to account

for damage effects in heterogeneous media It can be seen that the numerical simulation results

are in good agreement with the experimental data in terms of predicting the overall behavior

Both numerical simulation and experimental data predict that the dominant failure mode is

tension/compression Mishnaevsky et al [50] proposed a numerical algorithm to investigate

damage evolution and analyzed the interplay of damage mechanisms in unidirectional fiber

reinforced composites Sasayama et al [79] introduced the tensile failure of injection molded

short glass fiber reinforced polyamide 6,6 by using a multiscale mechanistic model

Notta-Cuvier et al [80] presented a damage model for SFRCs with random fiber orientation subjected

to uniaxial tension An analysis of fracture progress in unidirectional composites under tension

using the extended finite element method (X-FEM) was performed by Wang et al [81] Jha et

al [82] introduced a computational modelling framework for investigating the damage effects

into fiber reinforced matrix composite materials The micromechanical model based on

mathematical homogenization for damage simulation of composite materials is commonly used

[83], [84]