Tài liệu Phttri~1

.. 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 I 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 subsampling 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 II 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. III 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. IV 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 V 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 VI 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 VII 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 E31  0.02 102 ............................................................................................................ 49 Fig. 3. 7 Microscopic effective strain distributions on the cross-section 2-3 under macroscopic strain E31=  0.02 102 ............................................................................................................ 49 VIII 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  of two x 11 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  y of two 11 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  of two z 11 successive sub-cycles at (a) cycle 1 and (b) cycle 17 .............................................................. 63 Fig. 4. 11 First-order terms of characteristic displacements in x direction   31 of two x 11 successive sub-cycles at (a) cycle 1 and (b) cycle 17 .............................................................. 64 Fig. 4. 12 First-order terms of characteristic displacements in y direction   31 of two y 11 successive sub-cycles at (a) cycle 1 and (b) cycle 17 .............................................................. 64 Fig. 4. 13 First-order terms of characteristic displacements in z direction   31 of two z 11 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 IX 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, 1 X S ...................................................... 76 Fig. 5. 6 Some damage patterns predicted at cycle 3, E = 0.005, 3 X nS (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   X 3S X 1A|2 .................................................................................................................................. 82 Fig. 5. 14 Number of EBE-SCG iterations and damaged volume evolution of a sample   X 3S X 1A|4 .................................................................................................................................. 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 X 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 XI Abbreviation 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 XII Nomenclatures Chapters 1, 2, 3: Multiscale method and first-order perturbation based stochastic homogenization method LN f() Σ Average fiber length, N is number of fibers Probability density function Volume average stress DXH Volume average stress Volume of the RVE Domain Elastic modulus Index, i = 1, 2, 3 Index, j = 1, 2, 3 Index, k = 1, 2, 3 Index, l = 1, 2, 3 Young’s modulus Shear’s modulus Poisson ratio Strength Stress Microscopic Strain Stress-strain matrix Mean value Standard deviation Variance Average Random variable Four-order elastic tensor Scale ratio Homogenized properties of model X u0 u1 δu Macroscopic displacement Perturbed displacement Virtual displacement uimicro Microscopic displacement Γ Boundary E V Ω K i j k l Ei Gij νij Sij σ ε D Exp( ) S.D.( ) Var( ) Ave( ) α Dijkl λ XIII t x y MTOT Φ( ) Traction Macroscopic scale Microscopic scale Total number of constituent material Arbitrary function Volume of an RVE Characteristic displacement matrix Deformation mode of characteristic displacement |Y | [χ] kl χ  0 χ  1 kl = 11, 22, 33, 23, 31, 12 Zero-order term of characteristic displacement vector for kl mode First-order term of characteristic displacement vector for kl mode Total number of constituent materials kl kl NMAT r Constituent material number, 1  r  NMAT m Index, 1  m  6 n Index, 1  n  6 N Br Shape function Strain-displacement matrix of constituent material r A matrix to assign a scalar or random variable to component of stress-strain matrix D A vector to extract each column of stress-strain matrix D Component mn of stress-train matrix D of constituent material r Random variable for component mn of stress-train matrix D of constituent material r P Q Dr,mn αr,mn Identity matrix Expected value of homogenized properties I  D H  0  DrH,mn  f (Xb) 1 The first order term of homogenized properties corresponding to random variable αr,mn Probability of model Xb XIV Chapters 4, 5: Stochastic nonlinear multiscale computational scheme n X Oj Level of standard deviation of microscopic strain n=0-3 Cycle Sub-cycle Constituent material number Fiber arrangement sample index Total number of fiber arrangement sample Fiber orientation sample index Total number of fiber orientation sample Packing fiber orientation sample j X iA| j Fiber arrangement sample i in group orientation j c Sample at cycle c c s r i imax j jmax XS c X nS  X nS X iA| j  Sub-sample corresponding to n level of standard deviation at cycle c Sub-sample corresponding to n level of standard deviation and fiber arrangement sample X iA| j DXHi | j Homogenized properties of model X i | j εXi | j Microscopic strain of model X i | j DH Mixture homogenized properties Ave  ε  Average microscopic strain A b  Stiffness matrix Right-hand side vector Effective microscopic strain  cr Effective microscopic strain damage threshold χ  0 kl ini χ  kl 1 r ini χ  0 χ  kl x , y ,z kl 1 r x , y ,z x Initial vector of zero-order term of characteristic displacement vector in EBE-SCG solver Initial vector of first-order term of characteristic displacement vector in EBE-SCG solver Zero-order term of characteristic displacement vector for kl mode in x, y, z direction First-order term of characteristic displacement vector for kl mode in x, y, z direction of constituent material r Displacement XV p  δcr γ e β P E ΔE np Lf d g Initial search direction Residual vector Convergence threshold Search factor Iteration Updated search factor Diagonal matrix of A Macroscopic strain Macroscopic strain increment Number of particle Fiber length Diameter of fiber Distance between two successive particle XVI 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 1 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. 2