Tài liệu The extended meshfree method for cracked hyperelastic materials

VIET NAM NATIONAL UNIVERSITY HO CHI MINH CITY HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY -------------------- VU VAN THAI THE EXTENDED MESHFREE METHOD FOR CRACKED HYPERELASTIC MATERIALS PHƯƠNG PHÁP KHÔNG LƯỚI MỞ RỘNG CHO BÀI TOÁN NỨT TRONG VẬT LIỆU SIÊU ĐÀN HỒI MAJOR: ENGINEERING MECHANICS MAJOR CODE: 8520101 MASTER THESIS HO CHI MINH CITY, January 2022 CÔNG TRÌNH ĐƯỢC HOÀN THÀNH TẠI TRƯỜNG ĐẠI HỌC BÁCH KHOA – ĐHQG – HCM Cán bộ hướng dẫn khoa học: TS. Nguyễn Thanh Nhã Cán bộ chấm nhận xét 1: PGS.TS. Nguyễn Hoài Sơn Cán bộ chấm nhận xét 2: TS. Nguyễn Ngọc Minh Luận văn thạc sĩ được bảo vệ tại Trường Đại học Bách Khoa, ĐHQG Tp. HCM ngày 15 tháng 01 năm 2022 Thành phần Hội đồng đánh giá luận văn thạc sĩ gồm: 1. Chủ Tịch Hội Đồng: PGS. TS. Trương Tích Thiện 2. Thư Ký Hội Đồng: TS. Phạm Bảo Toàn 3. Phản Biện 1: PGS. TS. Nguyễn Hoài Sơn 4. Phản Biện 2: TS. Nguyễn Ngọc Minh 5. Ủy Viên: TS. Nguyễn Thanh Nhã Xác nhận của Chủ tịch Hội đồng đánh giá LV và Trưởng Khoa quản lý chuyên ngành sau khi luận văn đã được sửa chữa (nếu có). CHỦ TỊCH HỘI ĐỒNG TRƯỞNG KHOA KHOA HỌC ỨNG DỤNG PGS. TS. Trương Tích Thiện PGS. TS. Trương Tích Thiện ĐẠI HỌC QUỐC GIA TP.HCM TRƯỜNG ĐẠI HỌC BÁCH KHOA CỘNG HÒA XÃ HỘI CHỦ NGHĨA VIỆT NAM Độc lập - Tự do - Hạnh phúc NHIỆM VỤ LUẬN VĂN THẠC SĨ Họ tên học viên: VŨ VĂN THÁI MSHV: 1970187 Ngày, tháng, năm sinh: 28/11/1991 Nơi sinh: Kiên Giang Chuyên ngành: CƠ KỸ THUẬT Mã số: 8520101 I. TÊN ĐỀ TÀI : The extended meshfree method for cracked hyperelastic materials (Phương pháp không lưới mở rộng cho bài toán nứt trong vật liệu siêu đàn hồi) II. NHIỆM VỤ VÀ NỘI DUNG: Xây dựng phương pháp không lưới cho bài toán biến dạng lớn của vật liệu siêu đàn hồi, bài toán nứt trong vật liệu siêu đàn hồi. Tính toán trường chuyển vị, ứng suất, tích phân J, hệ số k và so sánh với các lời giải tham khảo. Đánh giá các kết quả thu được từ phương pháp được đề xuất. III. NGÀY GIAO NHIỆM VỤ : 06/09/2021 IV. NGÀY HOÀN THÀNH NHIỆM VỤ: 22/05/2022 V. CÁN BỘ HƯỚNG DẪN: TS. Nguyễn Thanh Nhã Tp. HCM, ngày 09 tháng 03 năm 2022 . CÁN BỘ HƯỚNG DẪN CHỦ NHIỆM BỘ MÔN ĐÀO TẠO TS. Nguyễn Thanh Nhã PGS. TS Vũ Công Hòa TRƯỞNG KHOA KHOA HỌC ỨNG DỤNG Acknowledgement The completion of this thesis could not has been possible without guidance of my thesis supervisor Dr. Nha Thanh Nguyen. I would like to express my sincere gratitude to him for his continuous support, patience, enthusiasm during the process of my Master study. Besides my thesis supervisor, I am very grateful to the lecturers of Department of Engineering Mechanics for their lectures, advice while I am studying Master program. I am also thankful to my friends Master Vay Siu Lo, Master student Dung Minh Do, Master student Binh Hai Hoang for their listening and comments, which help me have more ideas to write my thesis. Finally, I sincerely and genuinely thank my dear parents, my siblings, my beautiful wife, and my lovely daughter for their love, care, and giving me motivation throughout my life. This thesis is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02-2019.237 i Abstract The simulation of finite strain fracture is still an open problem and appeal to many researchers in computational engineering field due to its complication of modeling and finding solution. In this thesis, the non-linear fracture analysis of rubber-like materials is studied. The extended radial point interpolation method (XRPIM), which combines both the Heaviside function and the branch function is employed to capture the discontinuous deformation field, as well as stress singularity around the crack tip in a hyperelastic material with incompressible state. The support domains are generated to approximate displacement field and its derivatives using shape function of radial point interpolation method (RPIM). For the analysis implementation, total Largange formulation is taken into XRPIM and the numerical integration is performed by Gaussian Quadrature. The tearing energy that controls the fracture of rubber-like materials is investigated by computing J-integral which is commonly used in linear fracture mechanics. k parameter that is constant for a given state of strain and the displacement field surrounding two crack edges are also studied. Moreover, the behavior of a hyperelastic solid with both compressible and nearly-incompressible state are analyzed by using integrated radial basis functions (iRBF) meshfree method. The efficiency and accuracy of the presented method are demonstrated by several numerical examples, in which results are compared with the reference solutions. ii Tóm tắt luận văn Mô phỏng phá hủy biến dạng lớn vẫn là một vấn đề mở và thu hút nhiều nhà nghiên cứu ở lĩnh vực cơ học tính toán do sự phức tạp trong việc mô hình hóa và tìm lời giải. Luận văn thực hiện nghiên cứu sự phá hủy phi tuyến của các vật liệu như cao su bằng việc sử dụng phương pháp nội suy điểm hướng kính mở rộng (XRPIM), trong đó có sự kết hợp hàm “Heaviside” và hàm “Branch” để biểu diễn sự bất liên tục của trường chuyển vị và sự suy biến của trường ứng suất xung quanh đỉnh vết nứt trong vật liệu siêu đàn hồi ở trạng thái không nén được. Các miền phụ trợ được tạo ra để xấp xỉ trường chuyển vị và các đạo hàm của chúng thông qua việc sử dụng hàm dạng của phương pháp nội suy điểm hướng kính (RPIM). Để thực thi sự phân tích, XRPIM được áp dụng vào công thức “Largange” tổng và tích phân số được thực hiện bằng “Gaussian Quadrature”. Năng lượng xé kiểm soát sự phá hủy của các vật liệu như cao su được khảo sát thông qua việc tính tích phân J, đại lượng được sử dụng rộng rãi trong cơ học phá hủy tuyến tính. Luận văn cũng thực hiện khảo sát về trường chuyển vị lân cận 2 mép vết nứt và thông số k, đại lượng là hằng số đối với một trạng thái biến dạng được cho. Ngoài ra, luận văn cũng trình bày về ứng xử của một vật rắn siêu đàn hồi ở cả hai trạng thái nén được và gần như không nén được bằng phương pháp không lưới sử dụng các hàm cơ sở hướng kính tích phân (iRBF). Sự hiệu quả và chính xác của phương pháp được giải thích thông qua các ví dụ số, trong đó kết quả được so sánh với các lời giải tham chiếu. iii Declaration I declare that this thesis is the result of my own research except as cited in the references which has been done after registration for the degree of Master in Engineering Mechanics at Ho Chi Minh city University of Technology, VNU – HCM, Viet Nam. The thesis has not been accepted for any degree and is not concurrently submitted in candidature of any other degree. Author Vũ Văn Thái iv Contents List of Figures vii List of Tables x List of Abbreviations and Nomenclatures xi 1. INTRODUCTION 1 1.1 State of the art ............................................................................................... 1 1.2 Scope of the study ......................................................................................... 3 1.3 Research objectives ....................................................................................... 3 1.4 Author’s contributions ................................................................................... 4 1.5 Thesis outline ................................................................................................ 4 2. METHODOLOGY 6 2.1 Hyperelastic material ..................................................................................... 6 2.1.1 Constitutive equations of hyperelastic material ....................................... 6 2.1.2 Fracture analysis of hyperelastic material ............................................. 11 2.2 Meshfree shape functions construction ........................................................ 13 2.2.1 Radial Point Interpolation Method (RPIM) ........................................... 13 2.2.2 integrated Radial Basis Functions Method (iRBF) ................................ 17 2.3 The XRPIM for crack problem in hyperelastic bodies ................................. 22 2.3.1 Enriched approximation of the displacement field by XRPIM .............. 22 2.3.2 Weak form for nonlinear elastic problem and discrete equations .......... 24 3. IMPLEMENTATION 29 v 3.1 Numerical implementation procedure .......................................................... 29 3.2 Computation procedure of K maxtrix and fint matrix .................................... 30 3.3 Computation procedure of B matrix and O matrix ....................................... 31 4. NUMERICAL EXAMPLES 34 4.1 Non-cracked hyperelastic solid .................................................................... 34 4.1.1 Inhomogeneous compression problem .................................................. 34 4.1.2 Curved beam problem .......................................................................... 39 4.2 Cracked hyperelastic solid ........................................................................... 42 4.2.1. Rectangular plate with an edge crack under prescribed extension ........ 42 4.2.2. Square plate with an edge crack under prescribed extension ................ 44 4.2.3. Nonlinear Griffith problem .................................................................. 47 4.2.4. Square plate with an inclined central crack .......................................... 52 5. CONCLUSION AND OUTLOOK 55 5.1 Conclusions ................................................................................................. 55 5.2 Future works ............................................................................................... 56 List of Publications 57 REFERENCES 58 vi List of Figures 2.1: Undeformed and deformed geometries of a body ............................................ 6 2.2: Contour used for J-intergal ............................................................................ 13 2.3: Local support domains and field node for RPIM ........................................... 14 2.4: Local support domains and field node for iRBF ............................................ 18 2.5: Field node surrounding the crack line ............................................................ 23 2.6: Distance r and angle  of xk in local coordinate system ................................ 24 2.7: 2D hyperelastic solid with a crack and boundary conditions .......................... 24 3.1: The algorithm of Numerical implementation procedure ................................ 32 3.2: The algorithm for computing B matrix and O matrix..................................... 33 4.1: Inhomogeneous compression problem .......................................................... 35 4.2: Percent of compression at point M for various values of distributed force in the compressible inhomogeneous compression problem................................ 36 4.3: Percent of compression at point M for various values of distributed force in the nearly-incompressible inhomogeneous compression problem.................. 36 4.4: Deformed configuration of the plate in the compressible state with f = 200 N/mm2 (magenta grid indicates the undeformed configuration of the plate) ............................................................................................................. 37 4.5: Deformed configuration of the plate in the nearly-incompressible state with f = 250 N/mm2 (magenta grid indicates the undeformed configuration of the plate) ............................................................................................................. 37 vii 4.6: The first Piola-Kirchhoff stress P in the compressible state with f = 200 N/mm2 .......................................................................................................... 38 4.7: The convergence rate in the compressible state with f = 200 N/mm2 ............. 39 4.8: The convergence rate in the nearly-incompressible state with f = 250 N/mm2 ...................................................................................................................... 39 4.9: Curved beam problem ................................................................................... 40 4.10: Vertical displacement at point O for various values of shearing force in the compressible curved beam problem .............................................................. 41 4.11: Vertical displacement distribution in the compressible curved beam problem (magenta grid indicates the undeformed configuration of the beam) .............. 41 4.12: Rectangular plate with an edge crack (a), Nodal distribution (b) ................... 42 4.13: Comparision of two crack vertical displacements ......................................... 43 4.14: Deformed configuration of the rectangular plate with an edge crack (a) 10 × 30 nodes, (b) 14 × 42 nodes, (c) 20 × 60 nodes (magenta grid and colors indicate the un-deformed configuration and values of von Mises stress at each node, respectively) ........................................................................................ 44 4.15: Square plate with an edge crack (a), Nodal distribution (b) .......................... 45 4.16: Variations of J-integral with respect to the elongation of four sets of scatter nodes in the case of square plate. Comparison of XFEM solution [22] with XRPIM results. ............................................................................................. 46 4.17: J-integral domains ....................................................................................... 46 4.18: Nonlinear Griffith problem: (a) uniaxial extension; (b) equibiaxial extension ...................................................................................................................... 48 4.19: Nodal distribution of nonlinear Griffith problem.......................................... 48 4.20: Variations of J-integral with respect to the elongation in the case of uniaxial extension. Comparison of XFEM solution [3] with XRPIM method results. . 49 viii 4.21: Variations of k with respect to the elongation in the case of uniaxial extension. Comparison of XRPIM method results with Lake [23] and Yeoh [24] .......... 50 4.22: Yeoh’s assumsion of the crack’s deformation ............................................... 50 4.23: Deformed configuration surrounding two crack edges in the case of equabiaxial extension. Comparison of XRPIM method results with Yeoh [24] ...................................................................................................................... 51 4.24: Variations of k with respect to the elongation in the case of equibiaxial extension. Comparison of XRPIM method results with Legrain [3] and Yeoh [24] ............................................................................................................... 52 4.25: Square plate with an inclined central crack (a), Nodal distribution (b) .......... 53 4.26: Variations of J-integral of the right crack tip with respect to applied force.... 53 4.27: Variations of normalized J-integral value of the right crack tip with respect to skew angles. .................................................................................................. 54 ix List of Tables 2.1: Some radial basis functions ............................................................................ 15 4.1: The effect of domain size chosen to compute J-integral on its results ............. 47 x List of Abbreviations and Nomenclatures Abbreviation 2D two dimensional dof degree of freedom FEM Finite Element Method iRBF integrated Radial Basis Functions RBF Radial Basis Functions RPIM Radial Point Interpolation Method VDQ Variational Differential Quadrature XFEM Extended Finite Element Method XRPIM Extended Radial Point Interpolation Method Nomenclatures  angle between the tangent of the crack line and the segment x k  xtip κ bulk modulus  principal extension ratios μ shear modulus σ Cauchy stress (real stress)  strain energy density function B matrix of derivatives of shape functions C right Cauchy-Green deformation tensor D Constitutive tensor xi E Lagrangian strain F deformation gradient tensor H (x) Heaviside function I identity matrix I1, I2, I3 three invariants of right Cauchy-Green deformation tensor J determinant of deformation gradient tensor K tangent stiffness matrix n number of nodes in local support domain P the first Piola-Kirchhoff stress Pm polynomial moment matrix RQ moment matrix of radial basis functions S the second Piola-Kirchhoff stress t final thickness of the sample T0 initial thickness of the sample WI The set of all nodes in local support domain WJ The set of nodes whose support contains the point x and is bisected by the crack line WK the set surrounding the crack tip X initial Cartesian coordinate x current Cartesian coordinate xii Chapter 1 INTRODUCTION 1.1 State of the art Hyperelastic materials are special elastic materials for which the stress is derived by the strain energy density function that determined by the current state of deformation. One of the attractive properties of these rubber-like materials is their ability to have large strains under small loads and retains initial configuration after unloading. Moreover, hyperelastic materials have lightweight and good form-ability so they are widely used in various engineering applications such as shock-absorbing matters in transport vehicles, sport devices and buildings protection from earthquakes. There are various forms of strain energy potentials to model the nonlinear stress-strain relationship of such materials including Neo-Hookean, Mooney-Rivlin, Yeoh, Ogden and so on. Because these materials mainly work in large strain condition, so fracture analyses are usually considered as nonlinear fractures. In practice, experiments are usually adopted to verify the behavior of hyperelastic structures but their costs are high and it takes too much time to do a lot of tests for obtaining an optimal design. For several decades, together with the rapidly developing of computer and numerical methods, the extended finite element methods (XFEM) are very strong and popular method in computational engineering. It is introduced by Moës at al. and Dolbow at al. for the first time [1, 2], XFEM has been successful in presenting the geometry of the crack through some level set functions. And then, some linear elastic problems of fracture mechanics were solved by this approach. The extension of XFEM to non-linear fracture mechanics has attracted many researchers. In 2005, Legrain et al. used the XFEM to analyze the stress around the crack tips in an incompressible rubber-like material at large strain with classical Neo-Hookean model [3]. Later, an extension of XFEM has been 1 presented for large deformation of cracked hyperelastic bodies [4]. Recently, Huynh at al. has proposed an extended polygonal finite element method for large deformation fracture analysis [5]. Although re-meshing is avoided in crack propagation, XFEM also has disadvantages because of the existence of the mesh of elements. Especially in geometrical non-linear problems, when large deformation cannot be passed over, the elements can be distorted and they cannot give good approximated results. In order to overcome the drawbacks of mesh-based methods, several meshless or meshfree approaches have been developed, the main purpose is to remove the depending on mesh of finite element models. In meshfree methods, there is no finite element required for the domain but a system of scattered nodes is used for the approximation. The most advantage of meshfree approach is that field nodes can be removed, added or changed position easily in each computation step, it is useful in problems that the domain changing occurs continuously. The enrichment techniques are integrated into the approximation spaces of meshless methods to accurately describe the discontinuities and the singular field at the crack-tips. On the other hand, the vector level set method is also used as a useful tool in representing crack geometry. There were some studies of crack problems based on linear fracture mechanics using meshless methods [6-10]. One of them is the extended radial point interpolation method (XRPIM) [10]. Similar to the formulation of XFEM, Nguyen at al. introduced and successfully applied XRPIM for crack growth modeling in elastic solids by combining radial point interpolation method (RPIM) and enrichment functions. However, the number of studies on non-linear fracture mechanics using meshless methods is still limited [11, 12]. So using meshfree methods for the crack problem in large deformation is hopeful. In this study, XRPIM is employed to investigate the behavior of crack problems with incompressible hyperelastic solid. The incompressible Neo-Hookean model is used for simulation and problems are considered in plane stress condition. Some results of simulation for non-cracked hyperelastic solid using integrated radial basis 2 functions (iRBF) are also presented in this thesis. According to Mai at al. [13], using iRBF can improve the accuracy for the approximation of the derivative of a function. Phuc at al. [14] has successfully applied iRBF to develop a meshfree method for quasi-lower bound shakedown analysis of structures. It is interesting that among meshfree approaches, the radial point interpolation method (RPIM) and integrated radial basis functions (iRBF), automatically satisfies the Kronecker property, and thus the direct enforcement of boundary conditions can be taken. 1.2 Scope of the study In this study, the author concentrates on the following contents  The integrated Radial Basis Function Meshfree Method: this method is employed to analyze the behavior of 2D non-cracked hyperelastic solid.  The eXtended Radial Point Interpolation Method: the Radial Point Interpolation Method is used as the cardinal method and XRPIM based on RPIM is used for analysis of cracked hyperelastic solid under plane stress condition.  The behavior of non-cracked hyperepastic solid: the displacement field and stress are taken into account.  The behavior of cracked hyperepastic solid: the displacement field surrounding two crack edges, the evolution of J-integral and k parameter are considered. Other issues not mentioned above are beyond the scope of this study and will not be discussed in this thesis. 1.3 Research objectives The goal of this study is investigate the behavior of cracked hyperelastic solid with incompressible state under plane stress condition using XRPIM. In addition, 3 the displacement field and stress of 2D non-cracked hyperelastic solid are taken into account using iRBF method. To obtained these targets, the following tasks must be completed:  Build the stress-strain relation of the hyperelastic material.  Construct the iRBF formulation for analyzing 2D non-cracked hyperelastic solid.  Construct the XRPIM formulation for analyzing the crack problem of hyperelastic solid with incompressible state under plane stress condition.  Develop the program to analyze the behavior of non-cracked and cracked hyperelastic solid. 1.4 Author’s contributions Author’s contributions for scientific aspects are  Formulating 2D non-cracked hyperelastic solid with compressible and nearly-incompressible state using iRBF  Formulating the crack problem of hyperelastic solid with incompressible state under plane stress condition using XRPIM  Build the program for analysing the behavior of non-cracked and cracked hyperelastic solid. 1.5 Thesis outline This thesis is constructed as follows. After introduction, Chapter 2 presents the methodology of this thesis. First is the constitutive laws and fracture analysis of hyperelastic materials. Next, a brief review on radial point interpolation method and integrated radial basis functions are given. Finally, the extended radial point interpolation method is provided for crack problem in hyperelastic bodies. Chapter 4 3 shows the implementation of XRPIM for analysis cracked hyperelastic bodies. Some numerical examples are investigated in Chapter 4 to demonstrate the performance of the proposed method. Finally, Chapter 5 presents main conclusions and remarks about the presented method. 5