Smoothed methods for fracture problems and application to composite materials

Smoothed methods for fracture problems and application to composite materials Jiang Yong B.Eng., Nanjing University of Aeronautics and Astronautics M.Eng., Nanjing University of Aeronautics and Astronautics A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2013 i Acknowledgements Acknowledgements I would like to express my deepest gratitude to my supervisors, Prof. Tay TongEarn and Prof. Liu Gui-Rong for their dedicated support and guidance during my four years study in National University of Singapore. Their serious research attitude, enlightening instructions and patience are invaluable to me and will benefit me in my whole life. I would also like to say thanks to Associate Prof. Zhang Yong-Wei, who is my first supervisor, for his kindness and help. Many thanks to my friends and seniors in “Center for Advanced Computations in Engineering and Science (ACES)” for their help. Special thanks to Associate Prof. Tan Vincent, Dr. Sun Xiu-Shan, Dr. Ridha Muhammad, Dr. Su Zhou-Cheng, Dr. Chen Yu, Miss Li Si-Xuan Christabelle, Mr. Chen Bo-Yang, Mr. Mao Jia-Zhen, Mr. Song ShaoNing, Mr. Umeyr Kureemun and Mr. Mohammad Ravandi for their help and friendship during my last two and a half years in Impact and Strength lab. To my parents, my sister and brother, I appreciate their support and encouragement in the four years. Without their understanding and support, I cannot finish this thesis. Last but not least, I would express my deepest thanks to National University of Singapore for the scholarship during these four years. ii Table of Contents Table of Contents Declaration …………………………………………………………i Acknowledgements…………………………………………………… ii Table of Contents .……………………………………………………iii Summary .…………………………………………………………… .ix Nomenclature .………………………… …………………………… xii List of acronyms .………………………… …………………………xvi List of figures .………………………………………………………xviii List of tables .……………………………………………….……… xxv Chapter Introduction………………………………………………….1 1.1 Background………………………………………………… …… .1 1.2 Linear elastic fracture mechanics…………………………… …… 1.3 Plastic fracture mechanics…………………………………… …… .6 1.4 Numerical methods for fracture mechanics………………………….10 1.4.1 Finite element method………………………………………….10 1.4.2 Extended finite element method……………………………… 15 1.5 Smoothing technique… ……………… .…… .………………….20 1.6 Comparison of different numerical methods for fracture mechanics .24 1.7 Objectives and significance of the thesis…………………… 26 1.8 Organization………………………… ………………………… 28 iii Table of Contents Chapter A singular cell-based smoothed radial point interpolation method for linear elastic fracture problems………………………… 30 2.1 Introduction………………………………………………… …… .30 2.2 Cell-based radial point interpolation method………………… .… 31 2.2.1 Basic equations for 2D solids………………………………… 31 2.2.2 Edge based T-schemes of node selection………………………34 2.2.3 Displacement approximation using RPIM…………………… 35 2.2.4 Singular shape function……………………………………… .37 2.2.5 Cell-based smoothed strain…………………………………….38 2.2.6 Different schemes of strain smoothing in the singular element .40 2.3 Domain interaction integral methods for 2D fracture problems….….44 2.4 Numerical implementation………………………………………… 47 2.5 Numerical examples……………………………………………… 48 2.5.1 Rectangular plate with an edge crack under tension………… .48 2.5.1.1 Influence of the number of the Gauss points .…………… 49 2.5.1.2 Domain independence study……………………………….50 2.5.1.3 The sensitivity of RPIM to the nodes distribution………….51 2.5.1.4 Result… ………….……………………………………….52 2.5.1.5 Comparison between SCS-RPIM and SES-FEM………56 2.5.2 Rectangular plate with an edge crack under shear…………… 58 2.5.3 An inclined crack in rectangular plate under tension………… 62 iv Table of Contents 2.5.4 Convergence rate and efficiency of SCS-RPIM……………… 70 2.6 Conclusion………………………………………………………… .74 Chapter A singular ES-FEM for plastic fracture mechanics….… .76 3.1 Introduction………………………………………………… … 76 3.2 Edge-based smoothed finite element method……………………… 78 3.2.1 A brief on edge-based smoothed finite element method… … .78 3.2.2 Stress calculation for edge-based smoothed finite element method 80 3.3 Singular element…………………………………………………… 81 3.3.1 Singular shape functions……………………………………….82 3.3.2 Smoothing scheme for singular element……………………….85 3.4 Material model……………………………………………… 89 3.4.1 Consistent elastoplastic tangent moduli……………………… 89 3.4.2 Incremental plasticity scheme……………………………… .91 3.4.3 Stress computation by incremental plasticity………………… 92 3.5 Numerical model……………………………………………….… 94 3.5.1 First example………………………………………………… .94 3.5.1.1 Model…………………………………………………… 94 3.5.1.2 Convergence study……………………………………… 97 3.5.1.3 Result .………………………………………………… .97 3.5.1.4 Condition number……………………………………… .105 v Table of Contents 3.5.2 Second example………………………………………………106 3.6 Conclusion……………………………………………………… .108 Chapter An edge-based smoothed XFEM for fracture in composite materials…………………………………………………………….…110 4.1 Introduction……………………………………………………… 110 4.2 Fracture mechanics for anisotropic material………………….… .111 4.2.1 Background knowledge about anisotropic material………… 111 4.2.2 Domain interaction integral method for anisotropic materials 115 4.3 Formulation of edge-based smoothed extended finite element method…………………… …… .117 4.3.1 Edge-based smoothed extended finite element method (ES- XFEM)……………………………………………………….117 4.3.1.1 The formulation of the ES-XFEM……………………… 117 4.3.1.2 Crack tip enrichment functions for anisotropic fracture mechanics………………………………………………………120 4.3.2 Numerical integration for ES-XFEM…………………………123 4.4 Numerical examples……………………………….……………….129 4.4.1 Rectangular plate with a central crack under tension…………129 4.4.1.1 Result…………………………………………………… 131 4.4.1.2 Convergence rate and efficiency of ES-XFEM………… .133 vi Table of Contents 4.4.1.3 The influence of radius for the selection of crack tip functions enriched nodes………………………………………………….137 4.4.2 Delamination of the composite plate under compression…… 141 4.5 Conclusion……… .…………………………………….………….143 Chapter A face-based smoothed XFEM for three-dimensional fracture problems…………………………………………………… 145 5.1 Introduction…………………………………………….………… 145 5.2 Face-based smoothed FEM (FS-FEM)………………….………….146 5.2.1 Smoothing domain formation……………………………… .146 5.2.2 The formulation of FS-FEM………………………………….147 5.3 Face-based smoothed XFEM (FS-XFEM)…………….………… .150 5.3.1 Level set presentation…………………………………………150 5.3.2 The formulation of FS-XFEM……………………………… 151 5.3.3 Smoothing scheme……………………………………………155 5.4 Three-dimensional stress intensity factor calculation…….……… .161 5.5 Numerical examples……………………………………….……….167 5.5.1 A plate with a thorough edge crack under tension……………167 5.5.1.1 Result…………………………………………………… 168 5.5.1.2 Convergence rate and efficiency of FS-XFEM………… .168 5.5.2 A cylinder with a penny-shaped crack under remote tension…171 vii Table of Contents 5.5.3 A cylinder with an inclined penny-shaped crack under remote tension……………………………………………………… 174 5.6 Conclusion……………………………………………….…………177 Chapter Conclusions and recommendations… ………………… 178 6.1 Conclusions……………………………………………….……… 178 6.2 Recommendations…………….……………………………………181 References……………………………………… ………………… .183 Publications……………………………………… ………………… 200 viii Summary Summary Fracture is one of the most common failures in structures and components. Analytical solutions provided by fracture mechanics only exist for a few cases. Therefore, numerical methods are usually used for fracture and fatigue analysis of structures. In this thesis, smoothing technique is applied to develop numerical methods for fracture mechanics and the application of smoothed method is extended to composite materials. A singular cell-based smoothed radial point interpolation method (SCS-RPIM) is developed for linear elastic fracture problems. The strain smoothing is performed over the background triangular cells. A five-node singular element, which can produce singular strain field around the crack tip, is devised in this study. With a layer of these five-node singular elements laid around the crack tip, singular strain and stress fields can be captured. Different schemes are devised in the five-node elements to perform the strain smoothing. Several examples are presented to validate the newly developed method. The results are found in excellent agreement with the exact (or reference) solutions. The plastic stress and strain fields around the crack tip for power hardening material, which are singular as r approaches zero, are crucial to fracture and fatigue of structures. Traditional finite element method cannot produce the singularity around the crack tip. To simulate effectively the ix References 21. D.B. Bogy. The plane solution for anisotropic elastic wedges under normal and shear traction. Journal of Applied Mechanics 1972; 39:1103-1109. 22. O.L. Bowie, C.E. Freese. Central crack in plane orthotropic rectangular sheet. International Journal of Fracture Mechanics 1972; 8:49-58. 23. D.M. Barnett, R.J. Asaro. The fracture mechanics of slit-like cracks in anisotropic elastic media. Journal of the Mechanics and Physics of Solids 1972; 20: 353-366. 24. M.C. Kuo, D.B. Bogy. Plane solutions for the displacement and tractiondisplacement problem for anisotropic elastic wedges. 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A face-based smoothed XFEM for threedimensional fracture problems. (to be submitted to Computer methods in applied mechanics and engineering). 200 [...]... Introduction applied to diverse fields for their superior mechanical and lightweight properties compared to metallic materials Several analytical investigations [16-27] have been reported on the fracture behavior of composite materials Since 1980, threedimensional fracture mechanics has been a hot topic Guo [28-29] has made great contribution to this field Guo analyzed the structure of stress and strain fields... cracks and gave the asymptotic solution for the fields Based on whether plastic deformation is included, fracture mechanics is usually classified into two major categories: linear elastic fracture mechanics (LEFM) which was developed on the basis of linear elastic theory and plastic fracture mechanics which was established by taking the crack-tip plastic deformation into account 1.2 Linear elastic fracture. .. distribution along the  direction for n=9 of different numerical methods and the HRR solution Table 3.16   /  0 distribution along the  direction for n=9 of different numerical methods and the HRR solution Table 3.17  r /  0 distribution along the  direction for n=9 of different numerical methods and the HRR solution Table 3.18 Condition number of ES-FEM and SES-FEM for three different types of... b Body force vector Baj (xk ) Smoothed strain gradient matrix corresponding to Heaviside function enrichment Smoothed strain gradient matrix corresponding to branch Bbj (xk ) function enrichment D Matrix of material constants E Young’s modulus f Force vector fa Load vector associated with Heaviside function enriched node fb Load vector associated with branch function enriched node fu Load vector associated... of different numerical methods and the HRR solution Table 3.12  r /  0 distribution along the  direction for n=5 of different numerical methods and the HRR solution Table 3.13   /  0 distribution along the  direction for n=5 of different numerical methods and the HRR solution Table 3.14  r /  0 distribution along the  direction for n=5 of different numerical methods and the xxvii List of... become unstable and fracture occurs Griffith’s theory was the first model proposed to analyze the relationship between strength and flaw size In Griffith’s theory, the strain energy released from crack propagation is only to produce new crack surface Before fracture failure, metals undergo intensive plastic deformation, during which strain energy can also be stored in the metals Therefore, Griffith’s... applicable to brittle fracture, not applicable to metals Considering plastic deformation around 1 Chapter 1 Introduction the crack tip, Irwin [3] and Orowan [4] independently extended the application of Griffith theory to metals Westergaard [5] developed a semi-inverse method to analyze the stress and displacement fields around crack tip Irwin [6] introduced a single constant, which was related to energy... Hutchison [14] and Rice and Rosengren [15] related the J integral to crack tip stress fields in nonlinear materials These analyses showed that J integral can be viewed as a nonlinear, stress-intensity parameter as well as an energy release rate As technology develops, more requirements for high-performance materials have been proposed by different areas of industries Composite materials are developed and 2... interaction integral for states 1 and 2 J J integral J act The actual state J integrals J aux The auxiliary state J integrals KI Stress intensity factor of the first fracture mode K II Stress intensity factor of the second fracture mode K III Stress intensity factor of the third fracture mode K Stiffness matrix Ln Matrix of components of outward unit vector Ld Matrix of differential operator M The interaction... nodes for n=9 of different numerical methods and the reference solution of different locations on the   0 edge o Table 3.9  r /  0 distribution along the  direction for n=3 of different numerical methods and the HRR solution Table 3.10   /  0 distribution along the  direction for n=3 of different numerical methods and the HRR solution Table 3.11  r /  0 distribution along the  direction for . Smoothed methods for fracture problems and application to composite materials Jiang Yong B.Eng., Nanjing University of Aeronautics and Astronautics M.Eng.,. usually used for fracture and fatigue analysis of structures. In this thesis, smoothing technique is applied to develop numerical methods for fracture mechanics and the application of smoothed. Kureemun and Mr. Mohammad Ravandi for their help and friendship during my last two and a half years in Impact and Strength lab. To my parents, my sister and brother, I appreciate their support and