臺灣博碩士論文加值系統

一般而言，應力在裂縫及缺角尖端會產生無限大之現象，對於二維的裂縫及缺角問題，目前已有理論方法可完整求解，但對三維的裂縫及缺角問題，例如一平版含有缺角，則目前還沒有合理的理論解。本論文將利用無元素法求解材料裂縫及缺角的力學問題，最主要原因是無元素法可容易的將應力或變位函數加到其要推導之勁度矩陣中，目前無元素法已大量的應用在裂縫開裂問題上，但目前卻沒有相關文獻用在缺角的力學問題上。
本論文首先將從無元素法的基礎理論作進一步延伸，以解決目前無元素法在數值分析上的一些缺點；接著使用理論公式推導二維非均質材料之缺角應力與變位場，再將此理論結果應用在無元素法上，並建立一奇異性缺角元素，如此可用較疏的節點求得較精確的分析結果。
另外，本論文也將提出三維彈性體材料含無限裂縫之理論公式，此理論公式包含主要解( primary solution )及次要解( shadow solution )，其中主要解即為傳統平面應變理論下的解析解，而次要解則是根據三維彈性力學理論平衡方程式來求得，最小二乘法結合有限元素法的數值解則被使用來確認未定係數；本論文研究的結果顯示，只要包含足夠的主要解及次要解項次，則可以得到非常準確的三維彈性體裂縫問題的變位場，本方法最主要的優點在於可以求得裂縫尖端附近三維變位場的奇異影響，而不需要有材料特性、邊界條件及施加載重的任何限制。
本論文最後會將無元素法初步應用到三維破壞力學解析理論中，其分析結果將與正常但極密網格之有限元素法之分析結果相比較，此法最大之優點為可得到裂縫的應力奇異性質，此是有限元素法所忽略且無法求得之項目。

Generally speaking, the stress in crack and notch tip will produce infinitely great phenomenon. For the two-dimensional crack and notch problems, stress near the crack and notch tip can solved exactly by the theoretical analysis, but for the three-dimensional crack and notch ones, for example a plate with a notch, there is no rational solutions at present. In this thesis, the element-free method will be used to solve the mechanics problem of crack and notch, and the main reason is that the element-free method can add stress and strain functions in the strength matrix easily. The element-free method has a wild application on crack problem, but there are no relevant documents using this method on the notch mechanics problem at present.
First, the basic theory of the element-free method will be further extended to solve the numerical problems associated with this method in this thesis. Second, the theoretical formula will be used to derive the stress and displacement fields on in-plan notch problems with anisotropic materials, and then this theory results will be applied to the element-free method. Finally, a singularity notch element is set up in this thesis, and this method can obtain more accurate analysis result with the node that is relatively dredged.
Furthermore, this thesis proposes the theoretical solution for the three-dimensional problem containing a semi-infinite crack with primary and shadow solutions, where the primary solution is the traditional plane-strain solution and the shadow ones can be obtained from the 3D equilibrium equation of the theory of elasticity. A least-squares method incorporating the finite element results was used to determine these factors in this theoretical solution. The results of this research show that if enough primary and shadow solutions are included, numerical simulations indicate that the proposed method can obtain an accurate displacement field for 3D crack problems. The major advantage of this method is that a 3D whole displacement field with the analytic singular effect near the crack tip can be obtained, without any limitation with regard to the material properties, boundary conditions, and applied loads.
Finally, the element-free method is applied to the three-dimensional fracture mechanics preliminarily, and the result was compared with the normal finite element method result using a fine mesh. The major advantage is that the crack stress singularity, which is ignored in the finite element method, can be obtained in the proposed method.

CONTENTS
摘 要(CHINESE ABSTRACT)……………………………………………………I
ABSTRACT…………………………………………………………………………III
誌 謝(CHINESE ACKNOWLEDGEMENTS)……………………………………V
CONTENTS………………………………………………………………………VI
LIST OF TABLES…………………………………………………………………X
LIST OF FIGURES………………………………………………………………XI
Chapter 1 INTRODUCTION………………………………………………………1
1.1 Background…………………………………………………………………1
1.2 Objective and Scope of Research…………………………………………2
1.3 Organization of Dissertation………………………………………………3
Chapter 2 LITERATURE REVIEW………………………………………………7
2.1 Development of element-free method……………………………………7
2.2 Error analysis of element-free method……………………………………8
2.3 Element-free method applied to crack and notch problem……………10
2.4 Analysis solution investigation for 3D semi-infinite crack problem……12
Chapter 3 THEORY AND APPLICATION OF ELEMENT-FREE METHOD……15
3.1 Solutions to decrease numerical errors in EFM………………………15
3.2 OpenMP parallel scheme for EFM………………………………………17
3.2.1 Parallel calculation of stiffness matrices and selected nodes for Gaussian points………………………………………………………18
3.2.2 Parallel generation and solution of global stiffness matrix………20
3.3 Numerical examples………………………………………………………23
3.3.1 Investigation of the truncation error in EFM……………………23
3.3.2 Study of efficiency in EFM………………………27
3.4 Chapter summary…………….30
Chapter 4 ANALYTIC AND NUMERICAL SOLUTION FOR CRACK PROBLEM……33
4.1 Relation between stress and strain in an anisotropic elastic material…33
4.2 Introduction of Lekhnitskii formulation and Stroh formulation employ to crack problem…………………………………………………… 34
4.3 Introduction of William’s solution for crack problem…………………40
4.4 Introduction of current element-free method for crack problem………47
Chapter 5 ELEMENT-FREE METHOD FOR IN-PLANE NOTCH PROBLEM WITH ANISOTROPIC MATERIAL……50
5.1 In-plane displacement and stress fields near notches…………………… 50
5.2 Element-free Galerkin method with MLS formulation for Sharp-V notches……53
5.2.1 Shape functions and their derivatives in the notch coordinates……53
5.2.2 Transformation of matrices from local directions to global direction………………………………………………………………55
5.2.3 Stiffness matrix based on the element-free Galerkin method……57
5.3 Illustration of numerical validation and tested cases……………………58
5.3.1 H-integral and least-squares schemes to evaluate notch SIFs……58
5.3.2 Details of V-notches examples………………………59
5.4 Numerical results and comparisons……………63
5.4.1 Effect of the number of Gaussian quadrature points and displacement terms…………………………………………………… 63
5.4.2 Comparison between FEM and EFGM with the mesh size effect… 65
5.5 Chapter summary……………68
Chapter 6 DISPLACEMENT FIELD ANALYSIS SOLUTION OF 3D BODY WITH A SEMI-INFINITE CRACK………………………………69
6.1 Recursive solutions of 3D elasticity bodies…………………… 69
6.2 Displacement fields of a 3D body with a semi-infinite crack……………70
6.2.1 Primary solutions……………………………………………………71
6.2.2 Shadow solutions……………………………………………………72
6.2.2.1 Solving the Mode-II shadow solutions {U1} under {t}={1,0,0}T
for {U0}……………………………………………………............ 72
6.2.2.2 Solving the Mode-II shadow solutions {U2} under {t}={1,0,0}T for {U0}……………………………………………………72
6.3 Least-squares method to fit the 3D finite element results………………74
6.4 Numerical simulations……………76
6.4.1 A plate with a central horizontal crack………………………… 77
6.4.2 A plate with a central slant crack…………………………………… 81
6.5 Chapter summary……………84
Chapter 7 PRELIMINARY STUDY OF ELEMENT-FREE METHOD FOR 3D CRACK PROBLEMS……………………………………………86
7.1 Element-free method applied for 3D crack problem with analytic solutions…….86
7.2 Numerical simulations……96
7.3 Chapter summary………… 98
Chapter 8 CONCLUSIONS AND RECOMMENDATIONS……………………100
8.1 Conclusions………………………………………………………………100
8.2 Recommendations for further research…………………………………102
REFERENCES……………………………………………………………………103
APPENDIX………………………………………………………………………116
自 述(CHINESE VITA)…………………………………………………………122

1. Lucy LB. A numerical approach to the testing of the fission hypothesis. The Astron. J. 1977; vol. 8(12):1013-1024.
2. Libersky LD and Petschek AG. Calculation of oblique impact and fracture of tungsten cubes using smoothed particle hydrodynamics. International Journal of Impact Engineering 1995; vol. 17:661.
3. Lancaster P and Salkauskas K. Surfaces Generated by Moving Least Squares Method. Mathematics of Computation 1981; vol. 37:141-158.
4. Nayroles B, Touzot G, Villon P. Generalizing the Finite Element Method: Diffuse Approximation and Diffuse Element. Computational mechanics 1992; vol. 10:307-318.
5. Belytschko T, Lu YY and Gu L. Element-free Galerkin Method. International Journal for Numerical Methods in Engineering 1994; vol. 37:229-256.
6. Chen JS, Pan C, Wu CT and Liu WK. Reproducing Kernel Particle Method for Large Deformation Analysis of non-linear structures. Computer methods in applied mechanics and engineering 1996; vol. 139:195-227.
7. 盛若磐. 元素釋放法積分法則與權函數之改良. 近代工程計算論壇(2000)論文集，國立中央大學土木系,2000.
8. Mendonca P de TR, Barcellos CS de, Duarte A. Investigation on the hp-Method by solving Timoshenko beam problems. Computer Mechanics 2000; vol. 25:286-395.
9. Belytschko T, Krongauz Y, Flemming M, Organ D and Liu WK. Smoothing and Accelerated Computations in the Element free Galerkin Method. Journal of Computational and Applied Mechanics 1996; vol. 74:111-126.
10. Modaressi H and Aubert P. A Diffuse Element-Finite Element Technique for Transient Coupled Anlysis. International Journal for Numerical Methods in Engineering 1996; vol. 39:3809-3838.
11. Bobaru F and Mukherjee S. Shape Sensitivity Analysis and Shape Optimization in Planar Elasticity Using the Element-free Galerkin Method. Computer methods in applied mechanics and engineering 2001; vol. 190:4319-4337.
12. Belytschko T, Organ D and Krongauz Y. A coupled finite element-element-free Galerkin method. Computer Mechanics 1995; vol. 17:186-195.
13. Flemming M, Chu YA, Moran AB and Belytschko T. Enriched element-free Galerkin methods for crack-tip field. Int. J. Number. Methods Engrg. 1997; vol. 40:1483-1504.
14. Rosolen A, Mill´an D, Arroyo M. On the optimum support size in meshfree methods: A variational adaptivity approach with maximum-entropy approximants. Int. J. Numer. Method Engrg. 2009; vol. 82:868-895.
15. Gu YT, Liu GR. Meshless techniques for convection dominated problems. Comput. Mech. 2006; vol. 38:171-182.
16. Netuzhylov H, Zilian A. Space-time meshfree collocation method: methodology and application to initial value problems. Int. J. Numer. Method Engrg. 2009; vol. 80:355-380.
17. Zhang XH, Ouyang J, Wang JY. Stabilization meshless method for convection-
dominated problems. Appl. Math. Mech. 2008; vol. 29:1067–1075.
18. Tu W, Gu YT, Wen PH. Effective shear modulus approach for two dimensional solids and plate bending problems by meshless point collocation method. Engrg. Anal. Boundary Elem. 2012; vol. 36:675–684.
19. Ooi EH, Popov V. An efficient implementation of the radial basis integral equation method. Engrg. Anal. Boundary Elem. 2012; vol. 36:716–726.
20. Wu XH, Chang ZJ, Lu YL, Tao WQ, Shen SP. An analysis of the convection-
diffusion problems using meshless and meshbased methods. Engrg. Anal. Boundary Elem. 2012; vol. 36:1040–1048.
21. Han W, Meng X. Error analysis of the reproducing kernel particle method. Comput. Methods Appl. Mech. Engrg. 2001; vol. 190:6157-6181.
22. Cheng RJ, Cheng YM. Error estimates for the finite point method. Appl. Numer. Math. 2008; vol. 58:884-898.
23. Hu HY, Chen JS, Hu W. Error analysis of collocation method based on reproducing kernel approximation. Numer. Methods for Partial Differ. Equ. 2009; DOI 10.1002/num 554-580.
24. Cheng RJ, Cheng YM. Error estimate of element-free Galerkin method for elasticity. Acta Phys. Sin. 2011; vol. 60:070206-1~070206-6.
25. Zhuang X, Heaney C, Augarde C. On error control in the element-free Galerkin method. Engrg. Anal. Boundary Elem. 2012; vol. 36:351–360.
26. Jin X, Li G, Aluru NR. Positivity conditions in meshless collocation methods. Comput. Methods Appl. Mech. Engrg. 2004; vol. 193:1171–1202.
27. Liu WK, Han W, Lu H, Li S, Cao J. Reproducing kernel element method. Part I: Theoretical formulation. Comput. Methods Appl. Mech. Engrg. 2004; vol. 193:933–951.
28. Liu GR, Gu YT. An introduction to meshfree methods and their programming. New York: Springer 2005.
29. Liu WK, Jun S, Zhang YF. Reproducing kernel particle methods. Int. J. Numer. Methods Fluids 1995; vol. 20:1081-1106.
30. Jin X, Li G, Aluru NR. On the equivalence between least-squares and kernel approximations in meshless methods. CMES 2001; vol. 2:447-462.
31. Singha IV. Parallel implementation of the EFG method for heat transfer and fluid flow problems. Comput. Mech. 2004; vol. 34:453-463.
32. Singha IV, Jainb PK. Parallel EFG algorithm for heat transfer problems. Adv. Engrg. Softw 2005; vol. 36:554-560.
33. Singha IV, Jainb PK. Parallel meshless EFG solution for fluid flow problems. Numer. Heat Transfer B 2005; vol. 48:45-66.
34. Danielson KT, Hao S, Liu WK, Aziz R, Li S. Parallel computation of meshless methods for explicit dynamic analysis. Int. J. Numer. Method Engrg. 2000; vol. 47:1323-1341.
35. Danielson KT, Adley MD. A meshless treatment of three-dimensional penetrator targets for parallel computation. Comput. Mech. 2000; vol. 25:267-273.
36. Danielson KT, Uras RA, Adley MD, Li S. Large-scale application of some modern CSM methodologies by parallel computation. Adv. Engrg. Softw. 2000; vol. 31:501-509.
37. Cartwright C, Oliveira S, Stewart DE. Parallel support set searches for meshfree methods. SIAM J. Sci. Comput. 2006; vol. 28:1318-1334.
38. Nakata S. Parallel meshfree computation for parabolic equations on graphics hardware. Int. J. Comput. Math. 2011; vol. 88:1909-1919.
39. Nakata S, Takeda Y, Fujita N, Ikuno S. Parallel algorithm for meshfree radial point interpolation method on graphics hardware. IEEE Trans. Magn. 2011; vol. 47:1206-1209.
40. Zhang LT, Wagner GJ, Liu WK. A Parallelized Meshfree Method with Boundary Enrichment for Large-Scale CFD. J. Comput. Phy. 2002; vol. 176:483-506.
41. Zhang LT, Wagner GJ, Liu WK. Modelling and simulation of fluid structure interaction by meshfree and FEM. Commun. Numer. Methods Engrg. 2003; vol. 19:615-621.
42. Wang H, Li G, Han X, Zhong ZH. Parallel point interpolation method for three-dimensional metal forming simulations. Engrg. Anal. Boundary Elem. 2007; vol. 31:326-342.
43. Wang H, Li G, Han X, Zhong ZH. Development of parallel 3D RKPM meshless bulk forming simulation system. Adv. Engrg. Softw 2007; vol. 38:87-101.
44. Trobec R, Sterk M, Robic B, Belytschko T. Computational complexity and parallelization of the meshless local Petrov-Galerkin method. Comput. and Struct. 2009; vol. 87:81-90.
45. Ventura1 G, Xu JX, Belytschko T. A vector level set method and new discontinuity approximations for crack growth by EFG. International Journal for Numerical Methods in Engineering 2002; vol. 54:923–944.
46. Lee SH, Yoon YC. An improved crack analysis technique by element-free Galerkin method with auxiliary supports. International Journal for Numerical Methods in Engineering 2003; vol. 56:1291-1314.
47. Marc D, Hung ND. A meshless method with enriched weight functions for fatigue crack growth. International Journal for Numerical Methods in Engineering 2004; vol. 59:1945–1961.
48. Thomas M. A natural neighbour-based moving least-squares approach for the element-free Galerkin method. International Journal for Numerical Methods in Engineering 2007; vol. 71:224–252.
49. Hildebrand G. Fracture analysis using an enriched meshless method. Meccanica 2009; vol. 44:535–545.
50. Gu YT. An enriched radial point interpolation Method Based on Weak-Form and Strong-Form. Mechanics of Advanced Materials and Structures 2011; vol. 18:578–584.
51. Zhuang X, Augarde1 Charles, Bordas Stéphane. Accurate fracture modelling using meshless methods, the visibility criterion and level sets: Formulation and 2D modeling. International Journal for Numerical Methods in Engineering 2011; vol. 86:249–268.
52. Zhu H, Zhuang X, Cai Y. High rock slope stability analysis using the enriched meshless shepard and least squares method. International Journal of Computational Methods 2011; vol. 8(2):209–228.
53. Gu YT, Wanga W, Zhang LC, Feng XQ. An enriched radial point interpolation method (e-RPIM) for analysis of crack tip fields. Engineering Fracture Mechanics 2011; vol. 78:175–190.
54. Singh IV, Mishra BK, Mohit P. An enrichment based new criterion for the simulation of multiple interacting cracks using element free Galerkin method. Int J Fract 2011; vol. 167:157–171.
55. Rao BN, Rahman S. An enriched meshless method for non-linear fracture mechanics. International Journal for Numerical Methods in Engineering 2004; vol. 59:197–223.
56. Wen PH, Aliabadi MH. A variational approach for evaluation of stress intensity factors using the element free Galerkin method. International Journal of Solids and Structures 2011; vol. 48:1171–1179.
57. Zhang H. Simulation of crack growth using cohesive crack method. Applied Mathematical Modelling 2010; vol. 34:2508–2519.
58. Ettore B, Michele M. A meshless cohesive Segments Method for Crack Initiation and propagation in composites. Appl Compos Mater 2011; vol. 18:45–63.
59. Barbieri1 E, Petrinic1 N, Meo M, Tagarielli VL. A new weight-function enrichment in meshless methods for multiple cracks in linear elasticity. International Journal for Numerical Methods in Engineering 2012; vol. 90:177–195.
60. Wang H, Li L, Liu S. Phenomenological method for fracture. Meccanica 2012; vol. 47:163–173.
61. Liew KM, Cheng Y, Kitipornchai S. Boundary element-free method (BEFM) and its application to two-dimensional elasticity problems. International Journal for Numerical Methods in Engineering 2006; vol. 65:1310–1332.
62. Abdollahifar A, Nami MR, Shafiei AR. A new MLPG method for elastostatic problems. Engineering Analysis with Boundary Elements 2012; vol. 36:451–457.
63. Rajesh KN, Rao BN. Two-dimensional analysis of anisotropic crack problems using coupled meshless and fractal finite element method. Int J Fract 2010; vol. 164:285–318.
64. Mohit P, Singh IV, Mishra BK. Evaluation of mixed mode stress intensity factors for interface cracks using EFGM. Applied Mathematical Modelling 2011; vol. 35:3443–3459.
65. Hegen D. Element-free Galerkin methods in combination with finite element approaches. Comput. Methods Appl. Mech. Engrg. 1996; vol. 135:143-146.
66. Rao BN, Rahman S. A coupled meshless-finite element method for fracture analysis of cracks. International Journal of Pressure Vessel and Piping 2001; vol. 78:647-657.
67. Xiao QZ, Dhanasekara M. Coupling of FE and EFG using collocation approach. Advances in Engineering Software 2002; vol. 33:507–515.
68. Gu YT, Zhang LC. Coupling of the meshfree and finite element methods for determination of the crack tip fields. Engineering Fracture Mechanics 2008; vol. 75:986–1004.
69. Rajesh KN, Rao BN. Coupled meshfree and fractal finite element method for mixed mode two-dimensional crack problems. International Journal for Numerical Methods in Engineering 2010; vol. 84:572–609.
70. Luo H, Zhu HP, Miao Y, Chen CY. Simulation of top-down crack propagation in asphalt pavements. J Zhejiang Univ-Sci A (Appl Phys ＆ Eng) 2010 ; vol. 11(3):223-230.
71. Krysl P, Belytschko T. The element free galerkin method for dynamic propagation of arbitrary 3-D cracks. International Journal for Numerical Methods in Engineering 1999; vol. 44:767-800.
72. Marc D. A meshless method with enriched weight functions for three-dimensional crack propagation. International Journal for Numerical Methods in Engineering 2006; vol. 65:1970–2006.
73. Yagawa Genki. Computational performance of Free Mesh Method applied to continuum mechanics problems. Proc. Jpn. Acad. 2011; Ser. B 87.
74. Portela A, Aliabadi MH, Rooke DP. Efficient boundary element analysis of sharp notched plates. International Journal for Numerical Methods in Engineering 1991; vol. 32: 445-470.
75. Ping XC, Chen MC, Xie J-L. Singular stress analyses of V-notched anisotropic plates based on a novel finite element method. Engineering Fracture Mechanics 2008; vol. 75:3819–3838.
76. Treifi M, Oyadiji SO, Tsang DKL. Computations of the stress intensity factors of double-edge and centre V-notched plates under tension and anti-plane shear by the fractal-like finite element method. Engineering Fracture Mechanics 2009; vol. 76: 2091-2108.
77. Niu Z, Cheng C, Ye J, Recho N. A new boundary element approach of modeling singular stress fields of plane V-notch problems. International Journal of Solids and Structures 2009; vol. 46:2999–3008.
78. Chen MC, Ping XC. A novel hybrid finite element analysis of inplane singular elastic field around inclusion corners in elastic media. International Journal of Solids and Structures 2009; vol. 46:2527–2538.
79. Passieux JC, Gravouil A, Rethore J, Baietto MC. Direct estimation of generalized stress intensity factors using a three-scale concurrent multigrid X-FEM. International Journal for Numerical Methods in Engineering 2011; vol. 85:1648-1666.
80. Kachanov M and Laures JP. Three-dimensional problems of strongly interacting arbitrarily located penny-shaped cracks. International Journal of Fracture 1989; vol. 41: 289-313.
81. Mear ME, Sevostianov I, Kachanov M. Elastic compliances of non-flat cracks. International Journal of Solids and Structures 2007; vol. 44:6412-6427.
82. Sih GC. A review of the three-dimensional stress problem for a cracked plate. International Journal of Fracture Mechanics 1971; vol. 7(1):39–61.
83. Bercial-Velez JP, Antipov YA, Movchan AB. High-order asymptotics and perturbation problems for 3D interfacial cracks. Journal of the Mechanics and Physics of Solids 2005; vol. 53:1128-1162.
84. Pindra N, Lazarus V, Leblond JB. The deformation of the front of a 3D interface crack propagating quasistatically in a medium with random fracture properties. Journal of the Mechanics and Physics of Solids 2008; vol. 56:1269-1295.
85. She CM, Zhao JH, Guo WL. Three-dimensional stress fields near notches and cracks. International Journal of Fracture 2008; vol. 151:151-160.
86. Chaudhuri RA, Chiu SHJ. Three-dimensional asymptotic stress field in the vicinity of an adhesively bonded scarf joint interface. Composite Structures 2009; vol. 89:475–483.
87. Kotousov A, Berto F, Lazzarin P, Pegorin F. Three dimensional finite element mixed fracture mode under anti-plane loading of a crack. Theoretical and Applied Fracture Mechanics 2012; vol. 62:26–33.
88. Pook LP. A 50-year retrospective review of three-dimensional effects at cracks and sharp notches. Fatigue and Fracture of Engineering Materials and Structures 2013; vol. 36(8):699-723.
89. Kanaun SK. Fast solution of the elasticity problem for a planar crack of arbitrary shape in 3D-anisotropic medium. International Journal of Engineering Science 2009; vol. 47:284-293.
90. Nikishkov GP and Atluri SN. Calculation of fracture mechanics parameters for an arbitrary three-dimensional crack by the equivalent domain integral method. International Journal for Numerical Methods in Engineering 1987; vol. 24:1801-1821.
91. Huber O, Nickel J and Kuhn G. On the decomposition of the J-integral for 3D crack problems. Int. J. of Fracture 1993; vol. 64:339-348.
92. Chessa J, Smolinski P, Belytschko T. The extended finite element method (XFEM) for solidification problems. International Journal for Numerical Methods in Engineering 2002; vol. 53(8):1959-1977.
93. Liang J, Huang R, Prevost JH, Suo Z. Evolving crack patterns in thin films with the extended finite element method. International Journal of Solids and Structures 2003; vol. 40(10):2343-2354.
94. Shen Y and Lew AJ. A locking-free and optimally convergent discontinuous- Galerkin-based extended finite element method for cracked nearly incompressible solids. Computer Methods in Applied Mechanics and Engineering 2014; vol. 273:119-142.
95. Treifi M and Oyadiji SO. Strain energy approach to compute stress intensity factors for isotropic homogeneous and bi-material V-notches. International Journal of Solids and Structures 2013; vol. 50:2196-2212.
96. Tsang DKL, Oyadiji SO, Leung AYT. Applications of numerical eigenfunctions in the fractal-like finite element method. International Journal for Numerical Methods in Engineering 2004; vol. 61(4):475-495.
97. Murotani K, Yagawa G, Choi JB. Adaptive finite elements using hierarchical mesh and its application to crack propagation analysis. Computer Methods in Applied Mechanics and Engineering 2013; vol. 253:1-14.
98. Belytschko T and Tabbara M. Dynamic fracture using element-free galerkin method. International Journal for Numerical Methods in Engineering 1996; vol. 39:923-938.
99. Sun Y, Zhang Z, Kitipornchai S, Liew KM. Analyzing the interaction between collinear interfacial cracks by an efficient boundary element-free method. Int. J. Engrg. Sci. 2006; vol. 44:37-48.
100.Ju SH. OpenMp solvers for parallel finite element and meshless analyses. accepted by Engineering Computations.
101.Ju SH. A simple OpenMP scheme for parallel iteration solvers in finite element analysis. CMES-Computer Modeling in Engineering ＆ Sciences 2010; vol. 64:91-108.
102.Lekhnitskii SG. Theory of elasticity of an anisotropic body. Holden-Day, San Francisco; 1963.
103.Stroh AN. Steady state problems in anisotropic elasticity. Journal of Math. Phys. 1962; vol.41:77-103.
104.Ting TCT. Anisotropic elasticity: theory and application. Oxford University Press, New York; 1996.
105.Williams ML. Stress singularities resulting from various boundary conditions in angular corners of plates in extension. Journal of Applied Mechanics 1952; vol.19:526–528.
106.Ting TCT. Barnett-Lothe tensors and their associated tensors for monockunuc materials with the symmetry plane at x3=0. Journal of Elasticity 1992; vol.27:143-165.
107.Wu KC, Chang FT. Near-tip field in a notched body with dislocations and body forces, ASME Journal of Applied Mechanics 1993; vol.60:936-941.
108.Press WH, Flannery BP, Teukolsky SA, Vettenling WT. Numerical recipes, the art of scientific computing. Cambridge University Press, New York; 1986.
109.Krongauz Y, Belytschko T. Enforcement of essential boundary conditions in meshless approximations using finite elements. Comput. Methods Appl. Mech. Engrg. 1996; vol.131:133-145.
110.Ju SH, Chiu CY, Jhao BJ. Experimental calculation of mixed-mode notch stress intensity factors for anisotropic materials. Engineering Fracture Mechanics 2009; vol.76(14):2260-2271.
111.Ayatollahi MR, Nejati M. Determination of NSIFs and coefficients of higher order terms for sharp notches using finite element method. International Journal of Mechanical Sciences 2011; vol.53:164-177.
112.Ju SH. Finite element calculation of stress intensity factors for interface notches. Computer Methods in Applied Mechanics and Engineering 2010; vol.199(33-36): 2273-2280.
113.Chen DA. Stress intensity factors for V-notched strip under tension or in-plane bending. International Journal of Fracture 1995; vol.70:81-97.
114.Omer N and Yosibash Y. Edge singularities in 3-D elastic anisotropic and multi-material domains. Computer Methods in Applied Mechanics and Engineering 2008; vol.197:959-978.
115.Henshell RD and Shaw KG. Crack tip elements are unnecessary. International Journal for Numerical Methods in Engineering 1975; vol.9:495-509.
116.Ju SH. Simulating stress intensity factors for anisotropic materials by the least-squares method. Int. J. of Fracture 1996; vol.81:283-297.