VIETNAM NATIONAL UNIVERSITY HO CHI MINH CITY HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY NGUYEN THANH NHA 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 NGUYEN THANH NHA 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. Nguyễn Xuân Hùng Independent Examiner 2: Assoc. Nguyễn Mạnh Cường Examiner 1: Assoc. Nguyén Dinh Kién Examiner 2: Assoc.
Lé Van Cảnh Examiner 3: Assoc. Luong Van Hai SCIENTIFIC SUPERVISORS: 1. Truong Tich Thién 2. Bui Quéc Tinh 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 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.
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. Truong Tich Thien for his helpful advices and guidance during my study work. I express my special gratitude to my scientific supervisor Assoc.
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. Ialso would like to thank Assoc. Nguyen Luong Dung, Prof.
Ngo Kieu Nhi, Assoc. 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 1. INTRODUCTION AND OBJECTIVE.
Statement of crack problems 1 1. Advanced functional composite materials 3 1. Extended Einite Element method (XFEM). Extended Meshfree approach.--‹- ‹ + + + k2 HH nh it 8 1.
Fundamental of Fracture Mechanics 9 1. Crack behavior in 1SOfTOpIC .-- - ¿+ 1S 31 1 1E 11T HH ng it 9 1. Crack behavior in orthotropIC maf€TIaÌS. Crack behavior In functionally graded mafteriaÌs.
Objective of the dissertation 15 1. Outline of the thesis 16 CHAPTER 2. EXTENDED MESHFREE GALERKIN METHODS FOR FRACTURE MECHANICS 18 2. The Radial Point Interpolation method (PM).
Enrichment for disconfInue cracK ÏaC€S. Standard enrichment for crack tip using branch functions. New enrichment for crack tip using ramp function. Apply to crack propagation probÌÏe1ms.
Meshfree Galerkin method for fracture problems and solution procedure28 2. Fundamental equations of elastic proble1ms. --- ¿+ + sc+x+s£sc+xz+ 28 2. Discrete equations for fracture problemm.
Numerical implementation procedure 36 2. Implementation procedure for quasi-static crack growth problem. Implementation procedure for dynamic crack problem (stationary state). X-RPIM FOR QUASI-STATIC CRACK GROWTH SIMULATION OF 2-D SOLIDS 40 3.
Crack growth and the SIFs implementation in isotropic material. Mode I: Single edge-crack plate under tensile loading. Mixed-mode: Single edge-crack plate under uniform shear loading. Numerical examples for crack ørowfh proÌÏ©INS.
Crack growth from a ẨilÏef. Crack growth in a perforated panel with a circular hole. TRANSIENT DYNAMIC CRACK ANALYSIS OF ISOTROPIC AND COMPOSITE MATERIALS 61 4. Evaluation of dynamic stress intensity factors for isotropic solids.
Transient dynamic crack analysis of isotropi€ solids. Accuracy study of the SIEs In cracked 1sotropic plafes. A semi-infinite edge crack under dynamic loading. Mixed-mode analysis of a slanted edge-cracked rectangular plate.
Mixed-mode analysis of a cracked pIpG. A complex sfructure with an edge CTaCKK. Transient dynamic crack analysis of orthotropic composites. Orthotropic enrichment functions for craCK.
Evaluation of dynamic stress intensity factors for orthotropic composites. Numerical results and discussion 94 4. Accuracy study of the SIFs of orthotropic composite. An edge crack in an orthotropic composite plate under dynamic loading.
A center crack in an orthotropic composite plate under dynamic loading. Crack growth in orthotropic model 101 4. Criterion for crack growth direction in orthotropic model. Predicting for propagation angle in an edge crack orthotropic plate.
EXTENDED MESHLESS RADIAL POINT INTERPOLATION METHOD FOR FRACTURE ANALYSIS OE EGÌMs. The interaction integral formulation for non-homogenous materials. Non-equilibrium formulation for FGM model. Extract SIFs for FGM model.
Accuracy study of SIEs in FGM crack mordeÌS. Single edge crack plate under mode Ì. Mixed-mode edge crack problÏeim. -¿-¿- + + kx St riey 116 5.
Slant edge crack probÏeim.- - ¿+ + kh HH TT HH it 119 5. Dynamic SIEs calculation for FGM crack modeÌs. FGM plate with center crack under dynamic tensile loading (case 1: x1-x2 5. IMPROVED EXTENDED MESHLESS MOVING KRIGING FOR FRACTURE MODELING OF SOLIDS AND FGMs.
Introduction to the moving Kriging me€(hO( .--s- <5 55s 5< se sesse 130 Vii 6. The moving Kriging shape functiOII.-- -- 5-5 + + sxsserereee Cu GEN. The improved moving Kriging shape functions. Improved X-MK for crack analysis of isotropic material.
Accuracy study on static SIFs 1n $OÏid. Dynamic crack analysis OŸ 1sOfTOpIC Tmaf€T1Ì. Improved X-MK for dynamic crack analysis of FGM material. Rectangular x1-x2-FGM plate with center crack under dynamic tensile ÏOACÏNB.
SG TH TT TH TT HH nọ TH HT TH HT TH HT TH ch nè 6. Inclined center crack FGM plate under dynamic tensile loading. Dynamic crack in complex FGM mO€Ì. CONCLUSIONS AND OUTLOOKS.
Outlooks LIST OF PUBLICATIONS REFERENCES Vili 155 157 157 158 160 162 LIST OF FIGURES Figure 1. Cracks observed in Song Tranh hydropower dam. Cracks observed on road surface of Thang Long Bridge. Cracks in an airplane windshieÏi.
The three basic modes O ÍTaCfUI€. Cracks observed on an aircraft body that made from composite material. Crack growth in a FGM specimen [4] oo. eee eee cseesetseseseeseseeees 4 Figure 1.
Definition of the coordinate axis ahead of a cracK tp. 2D orthotropic body with CTaCK. - «6 << 1xx TH, 13 Figure 1. 2-D FGM body with CraCĂK.-¿- «6 + x1 T HH TH Hư, 15 Figure 2.
Schematic representation of the distance r and angle 0 at a crack-tip. Definition of the sets of nodes W, and W,, respectively, in our meshfree Figure 2. Schematic of a crack tip coordinates in terms of linear ramp function. Crack is represented by a black curve.
(X;, X2) is the global coordinate system whereas (x), x2) is the local coordinate system at the crack-tip. In practice, for simplicity the value of /. is taken as the radius of the support domain. Green nodes are the split nodes enriched by the Heaviside function only, the red nodes enriched by the linear ramp function associated with Heaviside function.
- ¿+ 6+1 x1 HH TH HH Hit 25 Figure 2. Visualization of enrichments for crack tip location using Heaviside function associated with linear ramp function. The Heaviside function (left), the linear ramp function (middle) and the Heaviside function along with linear ramp function (right). Geometric description for set 9°.
The projection of a point X, belonging to S“ onto the advance vector t". Notation representation of a cracked model. Definition of integral paths around the crack tip, normal unit vectors and lU0001151/53/3/00) TS ",š§53ä3ïnAẢẲỲỮỮ. Sub-triangle cell description for crack edge (a) and crack tip (b) where 0 is the crack angle with respect to the XÃ, axis (see Eigure 2.
(a) Schematic configuration of an edge-crack plate subjected to the uniformed tensile loading. The deformed shape and normal stress o,,, distribution of the cracked plate with 20x40 nodes, a=3.5, enlarged by a factor Of 50. cece eee e cesses eeteeeeseeeees 48 Figure 3. Schematic configuration of mixed-mode single edge-crack in a rectangular plate under a uniform shear ÏOadIng .- - -¿- ¿+ + St 1E 11T HT ng rưn 50 Figure 3.
Distributed nodes and deformed shape of the cracked plate subjected to a uniform shear loading. (a) 10 x 20 and (b) 20 x 40 enlarged by 10 times.5 A center crack plate subjected to the uniformed tensile loading Figure 3. Distributed nodes and deformed shape of the square plate with center inside crack subjected to a uniform tensile ÏOad1ng. Schematic configuration of a fillet with crack.
Distributed nodes and evolution of the crack path from a fillet: the type-1 boundary CORIfTOH. ¿<< 1191 E91 1 1 11 1 1 11111 HT TT TT TH TH 55 Figure 3. Distributed nodes and evolution of the crack path from a fillet: the type-2 boundary CORIfTOH. ¿<< 1191 E91 1 1 11 1 1 11111 HT TT TT TH TH 56 Figure 3.
Zoom of the crack paths for both types of the boundary conditions at the vicinity of the fillet. Schematic configuration of a perforated plate with a circle hole subjected to a uniform tensile ÏOad1nE.---¿- «6 << 1S 121 19121 1 11111 HH TH TH ngư 58 Figure 3. Distributed nodes and propagation of the crack path of a perforated panel with a circular hole subjected to a uniform tensile loading. Zoom of the crack paths of a perforated panel with a circular hole subjected to a uniform tensile ÏOad1nE.---¿- «6 << 1S 121 19121 1 11111 HH TH TH ngư 59 Figure 4.
Schematic of different dynamic loadings used in the analysis. Geometry of two static crack problems: (a) single mode with an edge- cracked plate subjected to tensile loading and (b) mixed-mode with an edge-cracked plate subjected to shear ÏOading.- ‹ ¿+ + tàn HH TH HT HH HH ngàn 68 Figure 4. Convergence rate of the relative error vs number of nodes in each direction of an edge-cracked plate under tensile loading between the standard and ramp APPTOACHES.¿- (2c 2E E2 21121 1 1121101 1101101 H1 TH HT TH TH TH TH TT HH ng 70 Figure 4. Comparison of convergence of computational time (in second) versus several nodal distributions in each direction of an edge-cracked plate under tensile loading between standard and ramp apprOaces.