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Luận Án Tiến Sỹ Kỹ Thuật Phát triển phương pháp tính tốn phi tuyến đa tỉ lệ ngẫu nhiên cho vật liệu tổng hợp sợi ngắn ứng dụng cho nghiên cứu ảnh hưởng cấu trúc vi mô biến đổi đến 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 thỏa mãn hết tiêu chí tiến sỹ Nay, luận án 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, 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 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, years in Taiwan and 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 Introduction 1.1 Motivation 1.2 Short fiber reinforced composite materials 1.2.1 Injection molding and conventional micromechanical model 1.2.2 Fiber length, fiber orientation and fiber arrangement 1.3 Aims and scopes of research 11 1.4 Structure of dissertation 12 Chapter 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 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 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 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 Concluding remarks 87 List of publications 91 References 92 VII List of Figures Fig 1 Distribution of mechanical properties of constituent materials Fig Influence of order on the variance of outcome result Fig Injection molding process Fig Model of an injection molded structure 10 Fig Main points and their relations in structure of dissertation 13 Fig Multiscale framework 15 Fig 2 Unit cell array 17 Fig Classification of composite materials 19 Fig Rate of use for different failure criteria for composite materials in published papers by others 23 Fig General framework of stochastic finite element approach 30 Fig Two-scale problem of heterogeneous media 33 Fig Flowchart of damage analysis formulation 36 Fig Definition of SVE for short fiber-reinforced composite material 42 Fig 3 Microscopic strain  33 in interphase and short fibers when random physical parameters for all constituent materials are considered 45 Fig RVE models of glass fiber reinforced plastics composite based on the lognormal distribution 47 Fig Definition of two cross-sections 48 Fig Microscopic effective strain distributions on the cross-section 1-2 under macroscopic strain E31  0.02 102 49 Fig Microscopic effective strain distributions on the cross-section 2-3 under macroscopic strain E31=  0.02 102 49 VIII should be further investigated The stochastic stress should be investigated for strength prediction considering variability in composite microstructures When the developed method is used to simulate the material test under uniform tension/compression and bending test with Neumann boundary condition, the stress is free from material type In this case, strain is the main quantity of interest However, if the Dirichlet boundary condition is considered macroscopically, the stress becomes the main quantity of interest rather than strain from macroscopic viewpoint In the framework of multiscale problem setup, further discussion on stochastic formulation is required In addition, the stress/strain concentration at the macroscale and the microscopic stress/strain under high gradient of macroscopic stress/strain should be considered in the future calculations Also, the experiment should be carried out to deeply investigate the research aims in this dissertation 90 List of publications Articles on periodicals 1) Tien-Dat Hoang and Naoki Takano, First-order perturbation-based stochastic homogenization method applied to microscopic damage prediction for composite materials, Acta Mechanica, ISSN 0001-5970, 230(3):1061-1076, 2019 2) Tien-Dat Hoang, Yutaro Abe, Shinya Nakamura, Akio Miyoshi, and Naoki Takano, Stochastic nonlinear multiscale computational scheme for SFRCs to study the influence of microstructural variability on damage propagation, SN Applied Sciences, ISSN 2523-3963, 2(2), 16 pages, 2020 https://doi.org/10.1007/s42452-020-1961-7 Presentations at international conferences 1) Tien-Dat Hoang and Naoki Takano, Stochastic homogenization analysis of multi-phase composite materials based on the first-order perturbation method, The 4th International Conference on Computational Design in 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damage probability at the response distribution tail The application of the proposed scheme for the SFRP is carried... in FPSH method during damage simulation and its verification and application on simple multiphase particle composite material are performed Chapter conducts the application to random SFRPs to... 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