Luận án tiến sĩ: Extended Galerkin Meshfree cho mô hình nứt vật liệu composite

VIETNAM NATIONAL UNIVERSITY HO CHI MINH CITY HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY NGUYӈN 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 NGUYӈN 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. NguyӉn Xuân Hùng Independent Examiner 2: Assoc. NguyӉn MҥQK&ѭӡng Examiner 1: Assoc.LrQ Examiner 2: Assoc. /r9ăQ&ҧnh Examiner 3: Assoc.

/ѭѫQJ9ăQ+ҧi SCIENTIFIC SUPERVISORS: 1. Trѭѫng Tích ThiӋn 2. Bùi Quӕc 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 NguyӉn 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. 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. I also 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. iii LIST OF FIGURES. ix LIST OF TABLES.

INTRODUCTION AND OBJECTIVE. Statement of crack problems. Advanced functional composite materials. Extended Finite Element method (XFEM).

Extended Meshfree approach. Fundamental of Fracture Mechanics. Crack behavior in isotropic. Crack behavior in orthotropic materials.

Crack behavior in functionally graded materials. Objective of the dissertation. Outline of the thesis. EXTENDED MESHFREE GALERKIN METHODS FOR FRACTURE MECHANICS.

The Radial Point Interpolation method (RPIM). Enrichment for discontinue crack faces. Standard enrichment for crack tip using branch functions. New enrichment for crack tip using ramp function.

Apply to crack propagation problems. Meshfree Galerkin method for fracture problems and solution procedure28 2. Fundamental equations of elastic problems. Discrete equations for fracture problem.

Numerical implementation procedure. 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.

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 growth problems.

Crack growth from a fillet. Crack growth in a perforated panel with a circular hole. TRANSIENT DYNAMIC CRACK ANALYSIS OF ISOTROPIC AND COMPOSITE MATERIALS. Evaluation of dynamic stress intensity factors for isotropic solids.

Transient dynamic crack analysis of isotropic solids. Accuracy study of the SIFs in cracked isotropic plates. A semi-infinite edge crack under dynamic loading. Mixed-mode analysis of a slanted edge-cracked rectangular plate.

Mixed-mode analysis of a cracked pipe. A complex structure with an edge crack. 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. 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. 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 OF FGMs. The interaction integral formulation for non-homogenous materials. Non-equilibrium formulation for FGM model. Extract SIFs for FGM model.

Accuracy study of SIFs in FGM crack models. Single edge crack plate under mode I. Mixed-mode edge crack problem. Slant edge crack problem.

Dynamic SIFs calculation for FGM crack models. FGM plate with center crack under dynamic tensile loading (case 1: x1-x2 FGM). FGM plate with center crack under dynamic tensile loading (case 2: x2 FGM). IMPROVED EXTENDED MESHLESS MOVING KRIGING FOR FRACTURE MODELING OF SOLIDS AND FGMs.

Introduction to the moving Kriging method. The moving Kriging shape function. The improved moving Kriging shape functions. Improved X-MK for crack analysis of isotropic material.

Accuracy study on static SIFs in solid. Dynamic crack analysis of isotropic material. Improved X-MK for dynamic crack analysis of FGM material. Rectangular x1-x2-FGM plate with center crack under dynamic tensile loading.

Inclined center crack FGM plate under dynamic tensile loading. Dynamic crack in complex FGM model. CONCLUSIONS AND OUTLOOKS. 158 LIST OF PUBLICATIONS.

162 viii 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 windshield.

The three basic modes of fracture. Cracks observed on an aircraft body that made from composite material. Crack growth in a FGM specimen [4]. Definition of the coordinate axis ahead of a crack tip.

2D orthotropic body with crack. 2-D FGM body with crack. Schematic representation of the distance r and angle T at a crack-tip. Definition of the sets of nodes Wb and WS , respectively, in our meshfree method.

Schematic of a crack tip coordinates in terms of linear ramp function. Crack is represented by a black curve. (X1, X2) is the global coordinate system whereas (x1, x2) is the local coordinate system at the crack-tip. In practice, for simplicity the value of lc 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. 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 S.

The projection of a point x I 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 coordinate systems. Sub-triangle cell description for crack edge (a) and crack tip (b) .36 where T is the crack angle with respect to the X 1 axis (see Figure 2.

(a) Schematic configuration of an edge-crack plate subjected to the uniformed tensile loading. The deformed shape and normal stress V yy distribution of the cracked plate with 20 u 40 nodes, a=3.5, enlarged by a factor of 50. Schematic configuration of mixed-mode single edge-crack in a rectangular plate under a uniform shear loading. Distributed nodes and deformed shape of the cracked plate subjected to a uniform shear loading.

(a) 10 u 20 and (b) 20 u 40 enlarged by 10 times .5 A center crack plate subjected to the uniformed tensile loading. Distributed nodes and deformed shape of the square plate with center inside crack subjected to a uniform tensile loading. Schematic configuration of a fillet with crack. Type-1 (a) and type-2 (b) boundary conditions.

Distributed nodes and evolution of the crack path from a fillet: the type-1 boundary condition. Distributed nodes and evolution of the crack path from a fillet: the type-2 boundary condition. 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 loading.

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 loading. 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 loading.

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 approaches. 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 approaches. Edge crack problem of tensile loading. Condition numbers of the system matrix produced by approaches based on either the standard or ramp functions versus the number of nodes per side.

The crack length is a=3. Dash lines are averaged curves of the two methods. Geometry and boundary conditions of an infinite plate with a semi-infinite mode I crack. Regular meshless distributions of an infinite plate with a semi-infinite edge crack.

Convergence of the normalized mode-I DSIF as a function of time with respect to nodal densities using the standard branch function. Convergence in percentage errors for the normalized mode-I DSIF as a function of time with respect to nodal densities using the standard branch function. Large error can be found for the coarsest model as usual. Schematic of elastic stress waves (von Mises) propagating in an infinite plate with a semi-infinite edge crack at different steps using the present formulation with the linear ramp function.

It graphically shows that the elastic stress waves start, touch and open the crack, then reflect, repeatedly.