Luận văn thạc sĩ Cơ kỹ thuật: The extended meshfree method for cracked hyperelastic materials

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 we

VIET NAM NATIONAL UNIVERSITY HO CHI MINH CITY

HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY

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)

KHOA HỌC ỨNG DỤNG

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

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 biếndạng lớn của vật liệu siêu đàn hồi, bài toán nứt trong vật liệu siêu đàn hồi Tính toántrường 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

i

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

ii

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

iii

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

iv

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

v

Contents

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

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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

vii

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

viii

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

ix

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

x

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

xi

List of Abbreviations and Nomenclatures

Abbreviation

iRBF integrated Radial Basis Functions

RPIM Radial Point Interpolation Method

VDQ Variational Differential Quadrature

XRPIM Extended Radial Point Interpolation Method

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

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

1

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

2

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

3

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,

4

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

5

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

6

Chapter 2

METHODOLOGY

2.1 Hyperelastic material

2.1.1 Constitutive equations of hyperelastic material

Consider a general solid of the hyperelastic material that is subjected to external forces and displacements so that its geometry is changed from the initial (undeformed) to current (deformed) state as show in Fig 2.1

Figure 2.1: Undeformed and deformed geometries of a body

The strain energy density function  exists naturally and it can be constructed

by right Cauchy-Green deformation tensor C Stress can be obtained from the order derivative of the strain energy density function with respect to the Lagrangian strain The deformation gradient tensor at the current configuration of solid is defined as

2 S



In the hyperelasticity, the constitutive tensor D is a function of deformation and

it is achieved by differentiating the second Piola-Kirchhoff stress S

S D E

11 12

21 22

33

00

0

t C T

CS

2 2

12 22 11 12 11 22 12 12

2 2

22 12 1 2

12 22 11 12 11 22 12 2

22 11 12 12 22

11 1 3

det

2 det det

where κ is the bulk modulus Based on Eq (2.4) the second Piola-Kirchhoff stress

of the compressible neo-Hookean model can be written as

 2  1 ( 1 )

S   J  J C   I C   (2.24)2.1.2 Fracture analysis of hyperelastic material

In 1953, Rivlin and Thomas proposed an extension of the energy release rate initially proposed by Griffith [16] to rubber-like materials [17] In their work, they

1 WT

13

Figure 2.2: Contour used for J-intergal

2.2 Meshfree shape functions construction

2.2.1 Radial Point Interpolation Method (RPIM)

Consider a field function u (x) defined by a set of arbitrary distributed nodes xi

(i = 1, 2, , n) in a domain Ω (Fig 2.3) Using n nodes in the local support domain

of interested point xQ (red color), the well-known RPIM interpolation uh (x) is defined as

14

Figure 2.3: Local support domains and field node for RPIM

For stability, the minimum number of terms of polynomial basis and more terms of radial basis (m < n) [20] is used in this study There are a number of sort of radial basis functions as listed in Table 2.1 [20] with ri the distance between point

 1, 2

x x x andxix x For 2D problem, r1 i, 2 i i is defined as

  2 2

1 1i 2 2ii

In this study, the thin plate spline function is employed to simulate the nonlinear behavior of cracked hyperelastic materials and the shape parameter η = 2.01 is used for construction of RPIM shape functions

15

Table 2.1: Some radial basis functions

d is usually the average nodal spacing for all nodes in the local support domain

By enforcing uh (x) into Eq (2.29) to pass through all the nodal values at n nodes surrounding the point xQ, a system of n linear algebraic equations is then obtained, one for each node, which can be written in the matrix form as

 1 2 

where the moment matrix of the RBF RQ and the polynomial moment matrix Pm

have been given as follows

2.2.2 integrated Radial Basis Functions Method (iRBF)

The approximation based on integrated radial basis functions is performed in contrast to other methods like polynomial basis functions and radial basis functions due to the fact that it is started with the highest-order derivative of the original function Consider a problem domain, which is presented by a set of arbitrarily distributed nodes as Fig 2.4 The approximation of functions u (x) in a local support domain with an interested point xI (red color) of the problem domain is conducted

by all nodes in it To reduce the error of derivative, this study uses the second-order derivative of the original function as a started point The first-order derivative and the original function are obtained by integration as follows

 

2 2

where n is the number of nodes in the local support domain, xj the j - component of

x (j = [1, 2] for 2D problem), CI1 and CI2 are the constants of integration which are univariate function of the variable other than xj [i.e., xk (k ≠ j)]

18

Figure 2.4: Local support domains and field node for iRBF

In this study, G is the multiquadric (MQ) function which is used to simulate ithe nonlinear behavior of hyperelastic materials like rubber material in compressible and nearly-incompressible states G is defined as i

 

2 2

1

Nk

i i i

where P x is the product of Y and the ii k th column of V-1

Substituting Eq (2.58) into Eqs (2.46) and (2.47) yields

n Nk

I

i i i

and substituting Eq (2.74) into Eqs (2.61), (2.62), and (2.63), one can have

2.3 The XRPIM for crack problem in hyperelastic bodies

2.3.1 Enriched approximation of the displacement field by XRPIM

Based on extended finite element method (XFEM), the crack path in the extended radial point interpolation method (XRPIM) is modeled by using two sorts

of enrichment functions, including the Heaviside function and an asymptotic function, which present the discontinuity (jump) of the displacement filed along the crack and the singular stress filed at the crack tip The displacement approximation

is expressed as

where i and ui are the RPIM shape functions and the nodal displacement at the ith

node, aj and bk are the nodal enriched degree of freedom associated with the Heaviside function Hj (x) and asymptotic crack-tip enrichment, respectively WI is the set of all nodes in local support domain, the set WJ named split nodes is formed

by the nodes whose support contains the point x and is bisected by the crack line, and WK named tip nodes is the set surrounding the crack tip, see Fig 2.5 For each node in set WJ or set WK, two degrees of freedom (dof) of this node are added

Figure 2.5: Field node surrounding the crack line

1 0

x

if fH