Development of new finite elements based on consecutive interpolation for 2d and 3d thermal mechanical problems

VIETNAM NATIONAL UNIVERSITY HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY NGUYỄN NGỌC MINH DEVELOPMENT OF NEW FINITE ELEMENTS BASED ON CONSECUTIVE-INTERPOLATION FOR 2D AND 3D THERMAL-MECHANICAL PROBLEMS PHD THESIS IN ENGINEERING HO CHI MINH CITY - 2020 VIETNAM NATIONAL UNIVERSITY HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY NGUYỄN NGỌC MINH DEVELOPMENT OF NEW FINITE ELEMENTS BASED ON CONSECUTIVE-INTERPOLATION FOR 2D AND 3D THERMAL-MECHANICAL PROBLEMS Major: Engineering Mechanics (Cơ kỹ thuật) Codes: 62520101 Independent Examiner 1: Associate Prof Dr NGUYỄN VĂN HIẾU Independent Examiner 2: Associate Prof Dr NGUYỄN QUỐC HƯNG Examiner 1: Dr TRỊNH ANH NGỌC Examiner 2: Associate Prof Dr CHÂU ĐÌNH THÀNH Examiner 3: Associate Prof Dr LƯƠNG VĂN HẢI SCIENTIFIC SUPERVISORS: Assoc Prof Dr TRƯƠNG TÍCH THIỆN Assoc Prof Dr BÙI QUỐC TÍNH LỜI CAM ĐOAN / DECLARATION Tác giả xin cam đoan cơng trình nghiên cứu thân tác giả Các kết nghiên cứu kết luận luận án trung thực, không chép từ nguồn hình thức Việc tham khảo nguồn tài liệu (nếu có) thực trích dẫn ghi nguồn tài liệu tham khảo quy định This doctoral dissertation is the outcome of my original research, conducted at the Ho Chi Minh city University of Technology, VNU – HCM, Viet Nam I declare that this document is my own work and has not submitted for any other degree or qualification except as specified All the references are cited in the document Tác giả luận án / Author Chữ ký / Signature i TĨM TẮT LUẬN ÁN Luận án trình bày nhóm phần tử hữu hạn dựa tích hợp kỹ thuật nội suy liên tiếp vào phương pháp phần tử hữu hạn truyền thống Với kỹ thuật này, không giá trị nút mà giá trị đạo hàm trung bình nút sử dụng q trình xấp xỉ Nhờ trường đạo hàm thu từ nhóm phần tử hữu hạn trường liên tục, thay bất liên tục nút (không hợp lý mặt vật lý) phương pháp truyền thống Sự cải thiện tính liên tục mang đến độ xác cao lời giải xấp xỉ Tuy nhiên khác với phương pháp có độ liên tục bậc cao hành phương pháp Đẳng hình học hay phương pháp khơng lưới, phương pháp phần tử hữu hạn nội suy liên tiếp trì thuộc tính Kronecker quan trọng tính tốn số Thêm vào đó, phương pháp đề xuất sử dụng lưới phần tử với phương pháp phần tử hữu hạn truyền thống không làm tăng số lượng bậc tự Kỹ thuật nội suy liên tiếp ban đầu giới thiệu riêng lẻ cho phần tử tam giác nút phần tử tứ giác nút áp dụng với toán đàn hồi tuyến tính hai chiều Trong luận án này, phương pháp hệ thống hóa phát triển nâng cao để tạo nhóm phần tử hữu hạn phù hợp với nhiều miền toán từ chiều đến ba chiều, áp dụng để phân tích toán tương tác cơ-nhiệt Phương pháp đề xuất tiếp tục mở rộng để khảo sát ứng xử miền chứa dạng bất liên tục vết nứt, với vật liệu đẳng hướng vật liệu trực hướng ii ABSTRACT This dissertation presents a new group of finite elements based on the integration of consecutive-interpolation procedure into the traditional Finite Element Method With this technique, not only the nodal values but also the averaged nodal gradients are included in approximation process As a result, the gradient fields obtained by the new group of finite elements are smooth, unlike the fields which are (non-physically) discontinuous at nodes delivered by traditional Finite Element Method The improvement on continuity results in higher accuracy of approximated solution as well On the other hand, unlike the other higher-order methods such as the Isogeometric Analysis and the Meshfree methods, the novel Consecutive-interpolation Finite Element Method possesses the important Kronecker-delta property Furthermore, the proposed method employs the same discretization mesh with the traditional Finite Element Method and does not increase the number of degrees of freedom The consecutive-interpolation procedure was initially introduced separately for the 3node triangular element and the 4-node quadrilateral element to be used in analysis of two-dimensional linear elastic problems In this dissertation, the method is further developed to form a new class of finite elements which is suitable for domains from 1D to 3D The new group of finite elements (being integrated with consecutive-interpolation procedure) is applied to analyze the thermo-mechanical problems The proposed method is also extended to study behaviors of bodies containing discontinuities such as cracks, for both isotropic and orthotropic materials iii CONTENTS LỜI CAM ĐOAN / DECLARATION .i TÓM TẮT LUẬN ÁN ii ABSTRACT iii CONTENTS iv LIST OF FIGURES viii LIST OF TABLES xii NOMENCLATURE xiii CHAPTER INTRODUCTION Heat transfer and thermo-mechanical problems Finite element method (FEM) and its issues Trends in development of numerical methods Original contributions of the dissertation Scientific and practical meaning of the contributions by the dissertation Methodology Scope of the dissertation Outline of the dissertation CHAPTER LINEAR THERMO-ELASTIC PROBLEMS 11 Formulation 11 Discrete form 13 Time integration scheme 15 2.3.1 Backward Euler scheme 15 2.3.2 Newmark scheme 16 CHAPTER CONSECUTIVE-INTERPOLATION PROCEDURE FOR 1D AND 2D PROBLEMS 17 Issue of non-physically discontinuous nodal gradient in Finite Element Method (FEM): An example of two-node bar element (L2 element) 17 The consecutive-interpolation procedure (CIP) for two-node bar element: CL2 element 19 3.2.1 Calculation of CIP-based shape functions 21 3.2.2 First order derivative of CIP-based shape functions 23 3.2.3 Modification to retain the C0-continuity 24 iv 3.2.4 Numerical example 25 The consecutive-interpolation procedure (CIP) for three-node triangular element (CT3) and four-node quadrilateral element (CQ4) 27 3.3.1 CIP-enhanced formulation for 2D domain 27 3.3.2 The CT3 element 28 3.3.3 The CQ4 element 28 3.3.4 Numerical examples 31 Conclusion .37 CHAPTER TWO-DIMENSIONAL DYNAMIC AND QUASI-STATIC THERMOELASTIC FRACTURE PROBLEMS IN ISOTROPIC MATERIALS .38 Introduction 38 Numerical modeling of cracks 39 The extended consecutive-interpolation four-node quadrilateral element (XCQ4) 40 4.3.1 Enriched formulation for displacement field 41 4.3.2 Enriched formulation for temperature field 43 Computation of (dynamic) stress intensity factors (DSIFs) for thermo-elastic fracture problems .45 Crack growth modeling 48 Numerical results and discussion .49 4.6.1 Edge crack under constant flux (mode-I) 49 4.6.2 Static SIFs analysis: Square plate with a center crack 55 4.6.3 Static SIFs analysis: Rectangular plate with a slant center crack (mixed mode) 59 4.6.4 Quasi-static crack propagation simulation of a slant edge crack in a cruciform panel 63 4.6.5 Dynamic SIFs analysis: Edge crack under quasi-static thermal shock .66 4.6.6 Dynamic SIFs analysis: Center crack under quasi-static thermal shock 68 4.6.7 Dynamic SIFs analysis: Curved crack under dynamic thermal shock 70 Conclusion .76 CHAPTER TWO-DIMENSIONAL QUASI-STATIC THERMO-ELASTIC FRACTURE PROBLEMS IN ORTHOTROPIC MATERIALS 78 Introduction 78 v Formulation of XCQ4 element for linear thermo-elastic fracture problems in orthotropic media .79 5.2.1 The characteristic equation of an arbitrary orthotropic material 80 5.2.2 Enriched formulation for displacement 81 5.2.3 Enriched formulation for temperature 83 Evaluation of SIFs by Interaction integral 83 Crack growth modeling 84 Numerical examples 86 5.5.1 Static SIFs analysis: Single edge notched specimen under mechanical tensile load .86 5.5.2 Static SIFs analysis: Rectangular epoxy/glass plate with a horizontal edge crack under constant flux 89 5.5.3 Static SIFs analysis: Rectangular epoxy/glass plate with a slant center crack 92 5.5.4 Static SIFs analysis: An anisotropic square plate with two parallel isothermal cracks 94 5.5.5 Quasi static crack propagation in an anisotropic cracked disc 97 5.5.6 Quasi static crack propagation of an edge crack in a rectangular plate under constant flux 100 5.5.7 Quasi static crack propagation in a perforated panel with a circular hole under constant heat flux .102 Conclusions 106 CHAPTER DEVELOPMENT OF 3D CIP-BASED FINITE ELEMENTS .107 Generalized formulation to determine auxiliary functions 107 CIP-enhanced FEM for 3D linear heat transfer problems .110 6.2.1 Steady-state heat convection in a 3D complicated domain 110 6.2.2 Transient heat transfer in a plate with cylindrical hole 113 CIP-enhanced FEM for 3D linear elastic problems .116 6.3.1 Static analysis of a cantilever beam with T-shaped cross-section 116 6.3.2 Free vibration analysis of a hollow cylinder 120 6.3.3 Free vibration analysis of a composite sandwich beam 122 Conclusions 124 CHAPTER CONCLUSIONS AND OUTLOOKS 125 vi Conclusions 125 7.1.1 XCQ4 element for analysis of linear thermo-elastic fracture problems 125 7.1.2 General formulation of CIP-enhanced elements .126 Outlooks 127 PUBLICATIONS 128 BIBLIOGRAPHY .129 APPENDIX A 140 APPENDIX B 144 vii LIST OF FIGURES Figure 1.1 Thermal crack in an asphalt pavement [7] Figure 1.2 Illustration of a finite element mesh for a two-dimensional heat transfer problem Figure 1.3 Convergence of thermal energy corresponding to mesh size Figure 3.1 Illustration of a two-node bar element (L2) in global system of coordinates (left) and in natural coordinates (right) 17 Figure 3.2 A one-dimensional domain being discretized by two L2 elements 18 Figure 3.3 Shape function associated with node (global) of the Example provided in Figure 3.2, computed by traditional FEM and by CIP-enhanced FEM 22 Figure 3.4 CIP-based shape functions R1, R2 and R3 associated with the nodes given in Figure 3.2 23 Figure 3.5 First-order derivative of the shape functions associated with node of the Example given in Figure 3.2, computed by traditional FEM and CIP-enhanced FEM .24 Figure 3.6 Example 3.2.4: Sketch of 1D bar subjected to body load The bar is uniformly discretized by eleements 25 Figure 3.7 Example 3.2.4: Comparison of stress calculated by standard FEM and the proposed CIP-enhanced FEM 26 Figure 3.8 Example 3.2.4: Convergence rate with respect to number of degrees of freedom (DOFs) of the standard linear FEM, quadratic FEM and CIP approach 27 Figure 3.9 Visualization of the shape function for a CQ4 element 29 Figure 3.10 Visualization of the first order derivative of shape function for a CQ4 element 29 Figure 3.11 Sketch of the CIP-enhanced four-node quadrilateral element (CQ4) in a 2D finite element mesh [42] 30 Figure 3.12 Example 3.3.2.1: Cantilever beam being subject to parabolic shear load31 Figure 3.13 Example 3.3.2.1: (a) The triangular mesh and (b) quadrilateral mesh at coarsest level 32 Figure 3.14 Example 3.3.2.1: Convergence rate of elastic energy with respect to mesh size, presented in log-log scale 33 Figure 3.15 Example 3.3.2.1: The field of normal stress component σxx computed by four types of element: CQ4, Q4, CT3 and T3 33 Figure 3.16 Example 3.3.2.2: Quarter model of the thick cylinder pipe and boundary conditions 35 Figure 3.17 Example 3.3.2.2: Temperature distribution in the cylinder pipe .36 Figure 3.18 Example 3.3.2.2: Hoop stress distribution in the cylinder pipe .37 Figure 4.1 Signed distance function 42 Figure 4.2 The set of nodes enriched by Heaviside function (denoted by x symbol and the set of nodes enriched by branch functions (denoted by square symbol) 42 viii [26] W K Liu, S Jun, J Adee and T Belytschko, "Reproducing kernel particle methods for structural dynamics," International Journal for Numerical Methods in Engineering, vol 38, pp 1655-1679, 1995 [27] M Hillman and J.-S Chen, "Nodally integrated implicit gradient reproducing kernel particle method for convection dominated problems," Computer Methods in Applied Mechanics and Engineering, vol 299, pp 381-400, 2016 [28] J G Wang and G R Liu, "A point interpolation method based on radial basis functions," International Journal for Numerical Methods in Engineering, vol 54, pp 1623-1648, 2002 [29] T N Nguyen, Q T Bui, C Zhang and T T Truong, "Crack growth modeling in elastic solids by the extended meshfree Galerkin radial interpolation method," Engineering Analysis with Boundary Elements, vol 44, pp 87-97, 2014 [30] A A Becker, The Boundary Element Method in Engineering: A complete course, McGraw - Hill, 1992 [31] L C Wrobel and C A Brebbia, Boundary Element Methods in Heat Transfer, Springer, 1992 [32] G R Liu and T Nguyen-Thoi, Smoothed Finite Element Methods, CRC Press, 2010 [33] N Nguyen-Minh, N Tran-Van, T Bui-Xuan and T Nguyen-Thoi, "Static analysis of corrugated panels using homogenization models and a cell-based smoothed mindlin plate element (CS-MIN3)," Frontiers of Structural and Civil Engineering, vol 13, no 2, pp 251-272, 2019 [34] T He, "The cell-based smoothed finite element method for viscoelastic fluid flows using fractional-step schemes," Computers & Structures, vol 222, pp 133-147, 2019 [35] T Nguyen-Thoi, H C Vu-Do, T Rabczuk and H Nguyen-Xuan, "A nodebased smoothed finite element method (NS-FEM) for upper bound solution to visco-elastoplastic analyses of solids using triangular and tetrahedral meshes," Computer Methods in Applied Mechanics and Engineering, vol 199, no 45-48, pp 3005-3027, 2010 [36] H Yang, X Y Cui, S Li and Y H Bie, "A stable node-based smoothed finite element method for metal forming analysis," Computational Mechanics, vol 63, no 6, pp 1147-1164, 2019 [37] Z C He, G R Liu, Z H Zhong, S C Wu, G Y Zhang and A G Cheng, "An edge-based smoothed finite element method (ES-FEM) for analyzing threedimensional acoustic problems," Computer Methods in Applied Mechanics and Engineering, vol 199, no 1-4, pp 20-33, 2009 131 [38] Y Jiang, T E Tay, L Chen and X S Sun, "An edge-based smoothed XFEM for fracture in composite materials," International Journal of Fracture, vol 179, pp 179-199, 2013 [39] T Nguyen-Thoi, G R Liu, K Y Lam and G Y Zhang, "A face-based smoothed finite element method (FS-FEM) for 3D linear and geometrically non-linear solid mechanics problems using 4-node tetrahedral elements," International Journal for Numerical Methods in Engineering, vol 78, pp 324353, 2009 [40] S Z Feng, X Y Cui, A M Li and G Z Xie, "A face-based smoothed point interpolation method (FS-PIM) for analysis of nonlinear conduction in multimaterial bodies," International of Thermal Sciences, vol 100, pp 430-437, 2016 [41] C Zheng, S C Wu, X H Tang and J H Zhang, "A novel twice-interpolation finite element method for solid mechanics problems," Acta Mechanica Sinica, vol 26, pp 265-278, 2010 [42] Q T Bui, Q D Vo, C Zhang and D D Nguyen, "A consecutive-interpolation quadrilateral element (CQ4): Formulation and Applications," Finite Element in Analysis and Design, vol 84, pp 14-31, 2014 [43] Z Kang, Q T Bui, D D Nguyen, T Saitoh and S Hirose, "An extended consecutive-interpolation quadrilateral element (XCQ4) applied to linear elastic fracture mechanics," Acta Mechanica, vol 226, no 12, pp 3991-4015, 2015 [44] Z Kang, Q T Bui, T S and S Hirose, "Quasi-static crack propagation simulation by an enhanced nodal gradient finite element with different enrichments," Theoretical and Applied Fracture Mechanics, vol 87, pp 61-77, 2017 [45] H L Ton-That, H Nguyen-Van and T Chau-Dinh, "An improved four-node element for analysis of composite plate/shell structures based on twice interpolation strategy," International Journal of Computational Methods, vol 16, p 1950020, 2019 [46] N Moes, J Dolbow and T Belytschko, "A finite element method for crack growth without re-meshing," International Journal for Numerical Methods in Engineering, vol 46, pp 131-150, 1999 [47] T Hughes, The Finite Element Method - Linear Static and Dynamic Finite Element Analysis, New Jersey: Prentice Hall, 1987 [48] K.-J Bathe, Finite Element Procedures, New Jersey: Prentice Hall, 1996 [49] O C Zienkiewicz and R L Taylor, The Finite Element Method - Volume 1: The Basis, fifth edition ed., Butterworth - Heinemann, 2000 132 [50] G R Liu and S S Quek, The Finite Element Method: A practical course, Butterworth-Heinemann, 2003 [51] N M Nguyen, Q T Bui, T T Truong, A N Trinh, I V Singh, T Yu and H D Doan, "Enhanced nodal gradient 3D consecutive-interpolation tetrahedral element (CTH4) for heat transfer analysis," International Journal of Heat and Mass Transfer, vol 103, pp 14-27, 2016 [52] N M Nguyen, Q T Bui, T T Truong, S Tanaka and S Hirose, "Numerical analysis of 3-D solids and composite structures by an enhanced 8-node hexahedral element," Finite Elements in Analysis and Design, vol 131, pp 116, 2017 [53] T T Truong, T N Nguyen, N M Nguyen and K B Tran, Các phương pháp số phân tích kết cấu (in Vietnamese), Nhà xuất Đại học Quốc gia Thành phố Hồ Chí Minh, 2018 [54] N M Nguyen, Q T Bui, T N Nguyen and T T Truong, "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 [55] N M Nguyen, T N Nguyen, T T Truong and Q T Bui, "Thermalmechanical crack propagation in orthotropic composite materials by the extended four-node consecutive-interpolation element (XCQ4)," Engineering Fracture Mechanics, vol 206, pp 89-113, 2019 [56] N M Nguyen, T N Nguyen and T T Truong, "A novel numerical approach for fracture analysis in orthotropic media," Tạp chí phát triển Khoa học & Cơng nghệ, vol 20, no K2, pp 5-13, 2017 [57] S P Timoshenko and J N Goodier, Theory of Elasticity, 3rd ed., New York: McGraw-Hill, 1970 [58] M Kassir, Applied Elasticity and Plasticity, CRC Press, 2018 [59] E E Gdoutos, Fracture Mechanicss: An Introduction, Springer, 2005 [60] G C Sih, "On the singular character of thermal stress near a crack tip," Journal of Applied Mechanics, vol 29, no 3, pp 587-589, 1962 [61] S Mohammadi, Extended Finite Element Method for Fracture Analysis of Structures, Blackwell Publishing, 2008 [62] T.-P Fries and T Belytschko, "Review: The extended/generalized element method: An overview of the method and its applications," International Journal for Numerical Methods in Engineering, vol 84, pp 253-304, 2010 [63] C Miehe, M Hofacker and F Welschinger, "A phase-field model for rateindependent crack propagation: Robust algorithmic implementation based on 133 operator splits," Computer Methods in Applied Mechanics and Engineering, vol 199, pp 2765-2778, 2010 [64] C Kuhn and R Müller, "A continuum phase field model for fracture," Engineering Fracture Mechanics, vol 77, pp 3625-3634, 2010 [65] R J H Peerlings, R de Borst, W A M Brekelmans and J H P de Vree, "Gradient-enhanced damage for quasi-brittle materials," International Journal for Numerical Methods in Engineering, vol 39, pp 3391-3403, 1996 [66] C Giry, F Dufour and J Mazars, "Stress-based nonlocal damage model," International Journal of Solids and Structures, vol 48, pp 3431-3443, 2011 [67] G Ventura, E Budyn and T Belytschko, "Vector level sets for description of propagating cracks in finite element," International Journal for Numerical Methods in Engineering, vol 58, pp 1571-1592, 2003 [68] M Duflot, "A study of the representation of cracks with level sets," International Journal for Numerical Methods in Engineering, vol 70, pp 12611302, 2007 [69] M Duflot, "The extended finite element method in thermoelastic fracture mechanics," International Journal for Numerical Methods in Engineering, vol 74, pp 827-847, 2008 [70] A Hansbo and P Hansbo, "A finite element method for the simulation of strong and weak discontinuities in solid mechanics," Computer Methods in Applied Mechanics and Engineering, vol 193, pp 3523-3540, 2004 [71] N Moes, E Bechet and M Tourbier, "Imposing Dirichlet boundary conditions in the extended finite element method.," International Journal for Numerical Methods in Engineering, vol 67, pp 1641-1669, 2006 [72] L Banks-Sills and O Dolev, "The conservative M-integral for thermal-elastic problems," International Journal of Fracture, vol 125, pp 149-170, 2004 [73] S H Song and G H Paulino, "Dynamic stress intensity factor for homogeneous and smooth heterogeneous materials using the interaction integral method," International Journal of Solids and Structures, vol 43, pp 48304866, 2006 [74] A Zamani and M R Eslami, "Implementation of the extended finite element method for dynamic thermoelastic fracture initiation," International Journal of Solids and Stuctures, vol 47, pp 1392-1404, 2010 [75] N N V Prasad, M H Aliabadi and D P Rooke, "The dual boundary element method for thermoelastic crack problems," International Journal of Fracture, vol 66, pp 255-272, 1994 134 [76] L Bouhala, A Makradi and S Belouettar, "Thermal and thermo-mechanical influence on crack propagation using an extended mesh free method," Engineering Fracture Mechanics, vol 88, pp 35-48, 2012 [77] V L Le and V T Do, "Plastic stress singularity near interface edge of elastoplastic/elastic bi-material," Computational Materials Science, vol 78, pp 140146, 2013 [78] T Shimada, V L Le, K Ouchi, Y Chihara and T Kitamura, "Critical dimensional limit of continuum fracture mechanics for dislocation emission," Engineering Fracture Mechanics, vol 163, pp 108-116, 2016 [79] J Qian and A Fatemi, "A mixed mode fatigue crack growth: a literature survey," Engineering Fracture Mechanics, vol 55, no 6, pp 969-990, 1996 [80] S Adden, M Merzbacher and P Horst, "Material forces as simple criterion for the description of crack-turning problems," Aerospace Science and Technology, vol 10, no 6, pp 519-526, 2006 [81] M Baydoun and T P Fries, "Crack propagation criteria in three dimensions using the XFEM and an explicit-implicit crack description," International Journal of Fracture, vol 178, pp 51-70, 2012 [82] F Erdogan and G C Sih, "On the crack extension in plates under plane loading and transverse shear," Journal of Basic Engineering, vol 85, pp 519-527, 1963 [83] M Stolarska, D L Chopp, N Moes and T Belytschko, "Modeling crack growth by level sets in the extended finite element method," International Journal of Numerical Methods in Engineering, Vols 943-60, p 51, 2001 [84] M Pant, I V Singh and B K Mishra, "Numerical simulatoin of thermo-elastic fracture problems using element free Galerkin method," International Journal of Mechanical Sciences, vol 52, pp 1745-1755, 2010 [85] H Chen, Q Wang, G R Liu, Y Wang and J Sun, "Simulation of thermoelastic crack problems using singular edge-based smoothed finite element method," International Journal of Mechanical Sciences, Vols 115-116, pp 123-134, 2016 [86] W K Wilson and I W Yu, "The use of the J-integral in thermal stress crack problems," International Journal of Fracture, vol 15, no 4, pp 377-387, 1979 [87] H Bayesteh, A Afshar and S Mohammadi, "Thermo-mechanical fracture study of inhomogeneous cracked solids by the extended isogeometric analysis method," European Journal of Mechanics A/Solids, vol 51, pp 123-139, 2015 [88] D Tsang, S Oyadiji and A Leung, "Two-dimensional fractal-like finite element method for thermoelastic crack analysis," International Journal of Solids and Structures, vol 44, no 24, pp 7862-76, 2007 135 [89] H S Wang, "A meshfree variational multiscale methods for thermo-mechanical material failure," Theoretical and Applied Fracture Mechanics, vol 75, pp 1-7, 2015 [90] N N V Prasad and M H Aliabadi, "Incremental crack growth in thermoelastic problems," International Journal of Fracture, vol 66, pp 45-50, 1994 [91] K Y Lee and K.-B Sim, "Thermal shock stress intensity factor by Bueckner's weight function method," Engineering Fracture Mechanics, vol 37, no 4, pp 799-804, 1990 [92] D E Katsareas and N K Anifantis, "On the computation of mode I and II thermal shock stress intensity factors using a boundary-only element method," International Journal for Numerical Methods in Engineering, vol 38, pp 41574169, 1995 [93] M Wünsche, A Sáez, F García-Sánchez and C Zhang, "Transient dynamic crack analysis in linear magnetoelectroelastic solids by a hypersingular timedomain BEM," European Journal of Mechanics - A/Solids, vol 32, pp 118130, 2012 [94] Q T Bui and C Zhang, "Analysis of generalized dynamic intensity factors of cracked magnetoelectroelastic solids by XFEM," Finite Element Analysis and Design, Vols 19-36, p 69, 2013 [95] N N V Prasad, M H Aliabadi and D P Rooke, "The dual boundary element method for transient thermoelastic crack problems," International Journal of Solids and Structures, vol 33, no 19, pp 2695-2718, 1996 [96] A Zamani, R B Hetnarski and M R Eslami, "Second sound in a cracked layer based on Lord-Shulman theory," Journal of Thermal Stresses, vol 34, pp 181200, 2011 [97] G C Sih, P C Paris and G R Irwin, "On cracks in rectilinearly anisotropic bodies," International Journal of Fracture Mechanics, vol 1, pp 189-203, 1965 [98] O L Bowie and C E Freese., "Central crack in plane orthotropic rectangular sheet," International Journal of Fracture Mechanics, vol 1, pp 189-203, 1972 [99] A Viola, A Piva and E Radi, "Crack propagation in an orthotropic medium under general loading," Engineering Fracture Mechanics, vol 34, no 5, pp 1155-1174, 1989 [100] L Nobile and C Carloni, "Fracture analysis for orthotropic cracked plates," Composite Structures, vol 68, no 3, pp 285-293, 2005 136 [101] L O Jernkvist, "Fracture of wood under mixed mode loading II Experimental investigation of Picea abies," Engineering Fracture Mechanics, vol 68, pp 565-576, 2001 [102] L M A Cahill, S Natarajan, S P A Bordas, R M O'Higgins and C T McCarthy, "An experimental/numerical investigation into the main driving force of crack propagation in uni-directional fiber-reinforced composite laminae," Composite Structures, vol 107, pp 119-130, 2014 [103] C.-C Ke, C.-S Chen and C.-H Tu, "Determination of fracture toughness of anisotropic rocks by boundary element method," Rock Mechanics and Rock Engineering, vol 41, no 4, pp 509-538, 2008 [104] M H Aliabadi and P Sollero, "Crack growth analysis in homogeneous orthotropic laminates," Composite Science and Technology, vol 58, no 10, pp 1697-1703, 1998 [105] J.-H Kim and G H Paulino, "Finite element evaluation of mixed mode stress intensity factors in functionally graded materials," International Journal for Numerical Methods in Engineering, vol 53, pp 1903-1935, 2002 [106] A Asadpoure and S Mohammadi, "Developing new enrichment functions for crack simulation in orthotropic media by the extended finite element method," International Journal for Numerical Methods in Engineering, vol 69, p 2150:2172, 2007 [107] D Motamedi and S Mohammadi, "Dynamic crack propagation analysis of orthotropic media by the extended finite element method," International Journal of Fracture, vol 161, pp 21-29, 2010 [108] S G Lekhnitskii, Theory of an Anisotropic Elastic Body, San Francisco: Holden-Day, 1963 [109] S H Sajjadi, J O A Ghorabi and D Salimi-Majid, "A novel mixed mode brittle fracture criterion for crack prediction under static and fatigue loading," Fatigue & Fracture of Engineering Materials & Structures, vol 38, no 11, pp 1372-1382, 2015 [110] E D Leonel and W S Venturini, "Multiple random crack propagation using a boundary element formulation," Engineering Fracture Mechanics, vol 78, pp 1077-1090, 2011 [111] V E Saouma, M L Ayari and D A Leavell, "Mixed mode crack propagation in homogeneous anisotropic solids," Engineering Fracture Mechanics, vol 27, no 2, pp 171-184, 1987 [112] L Bouhala, A Makradi and S Belouetta, "Thermo-anisotropic crack propagation by XFEM," International Journal of Mechanical Sciences, vol 103, pp 235-246, 2015 137 [113] C Carloni and L Nobile, "Maximum circumferential stress criterion applied to orthotropic materials," Fatigue and Fracture of Engineering Materials & Structures, vol 28, pp 825-833, 2005 [114] I Pasternak, "Boundary integral equations and the boundary element method for fracture mechanics analysis in 2D anisotropic thermoelasticity," Engineering Analysis with Boundary Elements, vol 36, pp 1931-1941, 2012 [115] C.-C Ke, C.-S Chen, C.-Y Ku and C.-H Chen, "Modeling crack propagation path of anisotropic rocks using boundary element method," International Journal for Numerical and Analytical Methods in Geomechanics, vol 33, pp 1227-1253, 2009 [116] S M Hasheminejad and Y Mirzaei, "Three dimensional vibration analysis of a class of traction-free solid elastic bodies with an eccentric cavity," Shock and Vibration, vol 19, pp 1341-1357, 2012 [117] B O Baba, S Thoppul and R F Gibson, "Experimental and numerical investigation of free vibrations of composite sandwich beams with curvature and debonds," Experimental Mechanics, vol 51, pp 857-868, 2011 [118] K Lee, Y Son and S Im, "Three-dimensional variable-node elements based upon CS-FEM for elastic-plastic analysis," Computers and Structures, vol 158, pp 308-332, 2015 [119] W Lai, Y T., Q T Bui, Z Wang, J L Curiel-Sosa, R Das and S Hirose, "3D elasto-plastic large deformations: IGA simulation by Bézier extraction of NURBS," Advances in Engineering Software, vol 108, pp 68-82, 2017 [120] H D Huynh, P Tran, X Zhuang and H Nguyen-Xuan, "An extended polygonal finite element method for large deformation fracture analysis," Engineering Fracture Mechanics, vol 209, pp 344-368, 2019 [121] A Afshar, S H Ardakhani and S Mohammadi, "Transient analysis of stationary interface cracks in orthotropic bi-materials using oscillatory crack tip enrichments," Composite Structures, vol 142, pp 200-214, 2016 [122] Z.-H Teng, D M Liao, S C Wu, F Sun, T Chen and Z B Zhang, "An adaptively refined XFEM for the dynamic fracture problems with microdefects," Theoretical and Applied Fracture Mechanics, vol 103, p 102255, 2019 [123] J Bobinski and J Tejchman, "A coupled constitutive model for fracture in plain concrete based on continuum theory with non-local softening and eXtended Finite Element Method," Finite Elements in Analysis and Design, vol 114, pp 1-21, 2016 [124] Y Wang and H Waisman, "From diffuse damage to sharp cohesive cracks: A coupled XFEM framework for failure analysis of quasi-brittle materials," 138 Computer Methods in Applied Mechanics and Engineering, vol 299, pp 57-89, 2016 [125] V B Pandey, I V Singh, B K Mishra, S Ahmad, A V Rao and V Kumar, "A new framework based on continuum damage mechanics and XFEM for high cyclic fatigue crack growth simulations," Engineering Fracture Mechanics, vol 206, pp 172-200, 2019 [126] Y Sun and K M Liew, "Modeling of thermo-mechanical fracture behaviors based on cohesive segments formulation," Engineering Analysis with Boundary Element, vol 77, pp 81-88, 2017 [127] X F Hu, X Lu and T E Tay, "Modelling delamination migration using virtual embedded cohesive elements formed through floating nodes," Composite Structures, vol 204, pp 500-512, 2018 139 APPENDIX A Equation Chapter Section Here, the auxiliary functions for CTH4 elements, see Equations (6.11 – 6.14), are shown to satisfy the conditions in Equation (6.3) Recall the Lagrange shape functions associated to the four local nodes i, j, k, m of a fournode tetrahedral element written in natural coordinates (ξ, η, ζ) N i       , N j   , N k   , N m   (A.1) The derivatives of Lagrange shape functions with respect to the physical coordinates are obtained by chain rule,      x     y    z         N i      Nj Nk   N  i      N N m   J 1  i      N   i     N j  N j  N j  N k  N k  N k   N m    N m     N m       (A.2) where the derivatives of Lagrange shape functions with respect to natural coordinates are as follows   N  i    N  i    N  i     N j  N j  N j  N k  N k  N k     N m       1 0   N m     1 0 ,       1 0 1 N m            (A.3) and J is the Jacobian matrix, which is the derivative of physical coordinates with respect to natural coordinates 140    x    y J    z     x  y  z  x  y  z    x    y     z     (A.4) Using the isoparametric concept, the same approximation scheme is used for geometry and field variables, i.e     x      x =  y   N i    z          Nj  xi  xj  Nm  xk   x  m   Nk yi yj yk ym zi   zi   zk   zm     (A.5) Hence, the Jacobian matrix is calculated by   N  i    N J i    N  i     N j  N j  N j  N k  N k  N k   x N m   i    x j  N m    x    k N m   x m       yi yj yk ym zi     zi   x j  xi   zk   xk  xi   z m  x m  x i         z j zi   z k  z i  (A.6)  zm zi     y j  yi yk  yi ym  yi Now, the Kronecker-delta property of fucntion ϕi is proved, i.e ϕi (xp) = δip, in which xp is the nodal coordinates of any node p within the element, i.e p = i, j, k, m Substitute xp into Equation (6.5)     i ( x p )  N i ( x p )  N i2( x p ) 1( x p )  N i ( x p )  2( x p )  N i2( x p ) 141 (A.7) Due to the Kronecker-delta property of the Lagrange basis functions, i.e Ni (xp) = δip, one obtains i ( x p )  N i ( x p )  , if i  p and i ( x p )  N i ( x p )  , if i  p (A.8) The proof of Kronecker-delta property for 𝜙𝑗 , 𝜙𝑘 and 𝜙𝑚 is similar Next, the derivatives of ϕi with respect to the physical coordinates are proved to vanish at nodes, i.e ϕi,x (xp) = 0, ϕi,y (xp) = and ϕi,z (xp) = for p being the indices of any local node i, j, k and m Derivative of ϕi with respect to w-direction (w = x, y, z) is taken from Equation (6.5) as  i ,w  N i ,w  2N i N i ,w  1  N i   N i2 N j ,w  N k ,w  N m ,w      N i ,w 2  N i2  2N i N j N j ,w  N k N k ,w  N m N m ,w (A.9)  For p = i, Equation (A.9) becomes i ,w ( x i )  N i ,w ( x i )  N j ,w ( x i )  N k ,w ( x i )  N m ,w ( x i ) , (A.10) in which Hence: N i ,w  N i  N i  N i           w  w  w w w w (A.11) N j ,w  N j  N j  N j       w  w  w w (A.12) N k ,w  N k  N k  N k       w  w  w w (A.13) N m ,w  N m  N m  N m       w  w  w w (A.14) i ,w ( x i )   For p = j or p = k or p = m 142 (A.15) i ,w ( x p )  N i ,w ( x p )  N i ,w ( x p )  (A.16) Similarly, the rest of conditions in Equation (6.3) will be obtained straightforwardly 143 APPENDIX B Asymptotic displacements and stresses for pure mode I and pure mode II in isotropic materials [59] B.1 Pure mode I ux  KI r   cos     2sin2  2 2 2 2 uy  K II 2 r   sin     2cos2  2 2 2 (B.1) (B.2)   3  cos   sin sin  2 2 2 r (B.3)   3 sin cos cos 2 2 r (B.4)   3  cos   sin sin  2 2 2 r (B.5)  xx  KI  xy  KI  yy  KI B.2 Pure mode II r   cos     2cos2  2 2 2 ux  K II 2 uy  K II r   sin     2sin2  2 2 2 2  xx    xy    3  sin   cos cos  2 2 2 r K II   3 sin cos cos 2 2 r K II (B.6) (B.7) (B.8) (B.9) 144  yy    3  cos   sin sin  2 2 2 r K II (B.10) Here κ is the Kolosov parameter     , plane stress       3  4 , plane strain   (B.11) 145 ...VIETNAM NATIONAL UNIVERSITY HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY NGUYỄN NGỌC MINH DEVELOPMENT OF NEW FINITE ELEMENTS BASED ON CONSECUTIVE- INTERPOLATION FOR 2D AND 3D THERMAL- MECHANICAL PROBLEMS. .. dissertation presents a new group of finite elements based on the integration of consecutive- interpolation procedure into the traditional Finite Element Method With this technique, not only the... assumed For cracked solids, currently only two-dimensional problems are considered For intact solids, the formulation is generalized and is applicable for 1D, 2D, and 3D problems Outline of the