Extended galerkin meshfree methods for fracrure modeling in advanced functional composite materials

VIETNAM NATIONAL UNIVERSITY HO CHI MINH CITY HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY NGUYỄN THANH NHÃ EXTENDED GALERKIN MESHFREE METHODS FOR FRACTURE MODELING IN ADVANCED FUNCTIONAL COMPOSITE MATERIALS PHD THESIS IN ENGINEERING HO CHI MINH CITY - 2018 VIETNAM NATIONAL UNIVERSITY HO CHI MINH CITY HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY NGUYỄN THANH NHÃ EXTENDED GALERKIN MESHFREE METHODS FOR FRACTURE MODELING IN ADVANCED FUNCTIONAL COMPOSITE MATERIALS Major Subject: Engineering Mechanics Codes: 62 52 02 01 Independent Examiner 1: Assoc Prof Dr Nguyễn Xuân Hùng Independent Examiner 2: Assoc Prof Dr Nguyễn Mạnh Cường Examiner 1: Assoc Prof Dr Nguyễn Đình Kiên Examiner 2: Assoc Prof Dr Lê Văn Cảnh Examiner 3: Assoc Prof Dr Lương Văn Hải SCIENTIFIC SUPERVISORS: Assoc Prof Dr Trương Tích Thiện Assoc Prof Dr Bùi Quốc Tính DECLARATION The thesis content is based on my original research work in the Department of Engineering Mechanics, Faculty of Applied Science, Ho Chi Minh city University of Technology, VNU – HCM, Vietnam I declare surely that this document has been created by myself and that has not been submitted for any other degree or qualification except as specified Author Nguyễn Thanh Nhã i ABSTRACT This thesis deals with the numerical computation of 2-D linear fracture problems using the two extended Galerkin meshless methods including radial point interpolation method (RPIM) and improved moving Kriging (MK) interpolation method Enrichment techniques including the use of step function for crack faces, standard branch functions and new linear ramp function for crack tip are first applied in RPIM and MK meshless frameworks The meshless moving Kriging method is improved by using three types of correlation function (i.e quartic polynomial, truncated quartic polynomial and Gaussian functions) to eliminate the effect of the user numerical experience parameter and applied to crack problems The developed methods are applied for crack analyzing in several types of material including isotropic, orthotropic and functionally graded composite materials Various crack problems such as static, dynamic behavior of crack models and quasi-static crack propagation are numerically investigated and compared with solutions given by analytical, experiment or other numerical methods The agreements between the obtained results using extended meshless methods and those of other methods show the correction of the developed approaches ii ACKNOWLEDGEMENTS This doctoral dissertation is the outcome of many years working at the Department of Engineering Mechanics (DEM), Faculty of Applied Science, Ho Chi Minh City University of Technology I would like to sincerely thank my principal supervisor Assoc Prof Truong Tich Thien for his helpful advices and guidance during my study work I express my special gratitude to my scientific supervisor Assoc Prof Bui Quoc Tinh from the Department of Civil and Environmental Engineering, Tokyo Institute of Technology, Japan, for his invaluable support, guidance and mentoring I deeply thank all my supervisors for offering me the opportunity to conduct this research work I also would like to thank Assoc Prof Nguyen Luong Dung, Prof Ngo Kieu Nhi, Assoc Prof Vu Cong Hoa and other members for their useful advice and supports, and for creating such a friendly and comfortable working atmosphere In addition, I am grateful for the support from my department colleagues, Nguyen Thai Hien, Tran Thai Duong, Nguyen Duy Khuong, Tran Kim Bang, Le Duong Hung Anh who has been always with me in difficult times Special thanks go to my close colleague and friend, Nguyen Ngoc Minh, for his useful discussions, ideas and programming experience Last but not least, I would like to express my profound gratefulness to my family, especially my parents, my wife Nguyen Thi My Hien and my sons Nguyen Quang Khai, Nguyen Minh Quoc Without their continuous encouragement, support and love, I would not have been able to pursue my work and ambition Ho Chi Minh City, July 2018 Nguyen Thanh Nha iii CONTENTS DECLARATION i ABSTRACT ii ACKNOWLEDGEMENTS iii LIST OF FIGURES ix LIST OF TABLES xvi NOMENCLATURE xvii CHAPTER INTRODUCTION AND OBJECTIVE 1.1 Statement of crack problems 1.2 Advanced functional composite materials 1.3 Literature review 1.3.1 Extended Finite Element method (XFEM) 1.3.2 Extended Meshfree approach 1.4 Fundamental of Fracture Mechanics 1.4.1 Crack behavior in isotropic 1.4.2 Crack behavior in orthotropic materials .12 1.4.3 Crack behavior in functionally graded materials 14 1.5 Objective of the dissertation 15 1.6 Outline of the thesis 16 CHAPTER EXTENDED MESHFREE GALERKIN METHODS FOR FRACTURE MECHANICS 18 2.1 The Radial Point Interpolation method (RPIM) 18 iv 2.2 Enrichment methods 21 2.2.1 Enrichment for discontinue crack faces .21 2.2.2 Standard enrichment for crack tip using branch functions 22 2.2.3 New enrichment for crack tip using ramp function .24 2.2.4 Apply to crack propagation problems 26 2.3 Meshfree Galerkin method for fracture problems and solution procedure28 2.3.1 Fundamental equations of elastic problems 28 2.3.2 Discrete equations for fracture problem .30 2.4 General J-integral (static and dynamic) 32 2.5 Numerical integration 36 2.6 Numerical implementation procedure 36 2.6.1 Implementation procedure for quasi-static crack growth problem 36 2.6.2 Implementation procedure for dynamic crack problem (stationary state) 38 2.7 Summary 39 CHAPTER X-RPIM FOR QUASI-STATIC CRACK GROWTH SIMULATION OF 2-D SOLIDS 40 3.1 Introduction 40 3.2 Crack growth and the SIFs implementation in isotropic material 43 3.3 Accuracy study 46 3.3.1 Mode I: Single edge-crack plate under tensile loading 46 3.3.2 Mixed-mode: Single edge-crack plate under uniform shear loading 49 3.4 Numerical examples for crack growth problems 51 v 3.4.1 Crack growth from a fillet 53 3.4.2 Crack growth in a perforated panel with a circular hole 57 3.5 Conclusions 59 CHAPTER TRANSIENT DYNAMIC CRACK ANALYSIS OF ISOTROPIC AND COMPOSITE MATERIALS 61 4.1 Introduction 61 4.2 Evaluation of dynamic stress intensity factors for isotropic solids 64 Remark 1: .66 4.3 Transient dynamic crack analysis of isotropic solids 66 Remark 2: .67 4.3.1 Accuracy study of the SIFs in cracked isotropic plates .67 4.3.2 A semi-infinite edge crack under dynamic loading .72 4.3.3 Mixed-mode analysis of a slanted edge-cracked rectangular plate .80 4.3.4 Mixed-mode analysis of a cracked pipe .84 4.3.5 A complex structure with an edge crack 89 4.4 Transient dynamic crack analysis of orthotropic composites 91 4.4.1 Orthotropic enrichment functions for crack 92 4.4.2 Evaluation of dynamic stress intensity factors for orthotropic composites 92 4.5 Numerical results and discussion 94 4.5.1 Accuracy study of the SIFs of orthotropic composite 94 4.5.2 An edge crack in an orthotropic composite plate under dynamic loading 95 4.5.3 A center crack in an orthotropic composite plate under dynamic loading 98 vi 4.6 Crack growth in orthotropic model 101 4.6.1 Criterion for crack growth direction in orthotropic model 101 4.6.2 Predicting for propagation angle in an edge crack orthotropic plate 101 4.7 Conclusions 103 CHAPTER EXTENDED MESHLESS RADIAL POINT INTERPOLATION METHOD FOR FRACTURE ANALYSIS OF FGMs 105 5.1 Introduction 105 5.2 The interaction integral formulation for non-homogenous materials 108 5.2.1 Non-equilibrium formulation for FGM model 109 5.2.2 Extract SIFs for FGM model .111 5.3 Accuracy study of SIFs in FGM crack models 113 5.3.1 Single edge crack plate under mode I 113 5.3.2 Mixed-mode edge crack problem 116 5.3.3 Slant edge crack problem 119 5.4 Dynamic SIFs calculation for FGM crack models 121 5.4.1 FGM plate with center crack under dynamic tensile loading (case 1: x1-x2 FGM) 121 5.4.2 FGM plate with center crack under dynamic tensile loading (case 2: x2 FGM) 124 5.5 Conclusions 129 CHAPTER IMPROVED EXTENDED MESHLESS MOVING KRIGING FOR FRACTURE MODELING OF SOLIDS AND FGMs 130 6.1 Introduction to the moving Kriging method 130 vii 6.1.1 The moving Kriging shape function 131 Remark 3: .133 6.1.2 The improved moving Kriging shape functions 133 6.2 Improved X-MK for crack analysis of isotropic material 136 6.2.1 Accuracy study on static SIFs in solid .137 6.2.2 Dynamic crack analysis of isotropic material 141 6.3 Improved X-MK for dynamic crack analysis of FGM material 146 6.3.1 Rectangular x1-x2-FGM plate with center crack under dynamic tensile loading 146 6.3.2 Inclined center crack FGM plate under dynamic tensile loading 149 6.3.3 Dynamic crack in complex FGM model 152 6.4 Discussion 155 CHAPTER CONCLUSIONS AND OUTLOOKS 157 7.1 Conclusions 157 7.2 Outlooks 158 LIST OF PUBLICATIONS 160 REFERENCES 162 viii functions totally removes the effect of the user numerical experience parameter for MK shape function The proposed X-RPIM method has shown a good performance for static SIFs calculation and quasi-static crack growth analysis in isotropic model, as mentioned in Chapter In Chapter 4, the X-RPIM approach once again shows its potential application for dynamic SIFs computation for both isotropic and orthotropic materials The quasi-static crack growth simulation in orthotropic media using the X-RPIM approach is also presented in Chapter Finally, the obtained results of static and dynamic SIFs in crack problems of nonhomogenous material in Chapter and Chapter show the completion of X-RPIM and X-MK for such problems To perform the analysis for FGM models, the proposed XRPIM and X-MK methods are more convenient than the traditional FEM, for instance, there is no need to the interpolation to obtain the graded material properties at a point of interest The graded material properties can be directly computed from the known material function at that point 7.2 Outlooks According to the results obtained in this thesis, the present methods have shown to be very dominant for crack analysis in various types of material such as isotropic, orthotropic and functionally graded materials There are some further development related to this study can be addressed as follows:  The present method is very potential in dealing with the crack propagation problems in other advanced materials and complicated structures in real life problem  Basing on the numerical algorithm developed in this study, the proposed methods can be developed for 3-D crack problems  The present work can be further improved either by integrating the adaptive meshfree integration technique [128] to enhance the accuracy  The integration task in the proposed X-RPIM and X-MK can be improved by using the Cartesian transformation method (CTM) [33, 129] into, which is to avoid the need for a background cell and to make them become truly meshless method 158  More challenging tasks based on this work to be tackled may fall into the analysis of transient cracks response of smart functional materials for thermal shock crack or interface cracks, for instance, see [60-62, 130-132]  In addition to the future works, dynamic crack propagation in solids and composites, especially dynamic crack modeling in quasi-brittle materials like concrete or rock, e.g., see [133-135] would also be the other interesting research direction 159 LIST OF PUBLICATIONS ISI peer-reviewed Journals N T Nguyen, T Q Bui, C Zhang, and T T Truong, "Crack growth modeling in elastic solids by the extended meshfree Galerkin radial point interpolation method," Engineering Analysis with Boundary Elements, vol 44, pp 87-97, 2014 N T Nguyen, T Q Bui, and T T Truong, "Transient dynamic fracture analysis by an extended meshfree method with different crack-tip enrichments," Meccanica, vol 52, pp 2363-2390, 2017 T Q Bui, N T Nguyen, L V Le, M N Nguyen and T T Truong, “Analysis of transient dynamic fracture parameters of cracked functionally graded composites by improved meshfree methods,” Theoretical and Applied Fracture Mechanics, vol 96, pp 642-657, 2017 ISI peer-reviewed Journal (not directly related to the thesis) M N Nguyen, T.Q Bui, N T Nguyen, T T Truong, V.L Le, “Simulation of dynamic and static thermoelastic fracture problems by extended nodal gradient finite elements,” International Journal of Mechanical Sciences, vol 134, pp 370-386, 2017 National Journals N T Nguyen, B K Tran, T Q Bui, and T T Truong, “Extended Radial Point Interpolation Method for dynamic crack analysis in functionally graded materials,” Science & Technology Development Journal, vol 18, pp 59-64, 2015 N T Nguyen, B K Tran, T Q Bui, and T T Truong, “Extended Radial Point Interpolation Method for crack analysis in orthotropic media,” Science & Technology Development Journal, vol 18, pp 5-13, 2015 N T Nguyen, T Q Bui, and T T Truong, “Enriched radial point interpolation method for stress intensity factors calculation in 2D elasto-dynamic cracked problem,” Journal of Science and Technology, vol 52(5C), pp 48-59, 2014 160 N T Nguyen, B K Tran, T Q Bui, and T T Truong, “Elastostatic analysis of isotropic and orthotropic functionally graded structures by meshfree radial point interpolation method,” Journal of Science and Technology, vol 52(5C), pp 1-15, 2014 International Conference paper N T Nguyen, M N Nguyen, T T Truong and T Q Bui, “Dynamic crack analysis of fgm using an extended meshfree moving kriging method,” in The 11th SEATUC Symposium, HCM City - Vietnam, 2017 National Conference papers 10 N T Nguyen, M N Nguyen, T Q Bui and T T Truong, “Extended radial point interpolation method for static and dynamic crack problems in orthotropic material,” in Hội nghị Khoa học toàn quốc Vật liệu và Kết cấu Composite - Cơ học, Công nghệ và ứng dụng, Nha Trang - Việt Nam, 2016 11 N T Nguyen, T Q Bui and T T Truong “Extended moving Kriging interpolation for crack growth in functionally graded materials,” in Hội nghị Khoa học toàn quốc Cơ học Vật rắn biến dạng lần thứ XII, Đà Nẵng - Việt Nam, 2015 12 N T Nguyen, B K Tran, T Q Bui, and T T Truong “Extend radial point interpolation method for analysis of stress intensity 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CHI MINH CITY HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY NGUYỄN THANH NHÃ EXTENDED GALERKIN MESHFREE METHODS FOR FRACTURE MODELING IN ADVANCED FUNCTIONAL COMPOSITE MATERIALS Major Subject: Engineering... extended Galerkin meshless methods including radial point interpolation method (RPIM) and improved moving Kriging (MK) interpolation method Enrichment techniques including the use of step function for. .. functions for modeling strong discontinuities like crack in solids using XFEM was developed in [44] The ramp functions were later applied to the crack modeling of for instance general inelastic materials