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研究生:陳家豪
研究生(外文):Chia-Hau Chen
論文名稱:邊界元素法於異向性岩石破壞力學之研究
論文名稱(外文):Application of Boundary Element Method on Fracture Mechanics Analysis of Anisotropic Rocks
指導教授:陳昭旭陳昭旭引用關係Ernian Pan
指導教授(外文):Chao-Shi ChenErnian Pan
學位類別:博士
校院名稱:國立成功大學
系所名稱:資源工程學系碩博士班
學門:工程學門
學類:材料工程學類
論文種類:學術論文
論文出版年:2008
畢業學年度:96
語文別:英文
論文頁數:164
中文關鍵詞:表面裂紋應力強度因子破裂韌度裂縫開口位移對偶邊界元素法橫向等向性單域邊界元素法
外文關鍵詞:Stress intensity factor (SIF)single-domain BEMfracture toughnesstransversely isotropysurface crackdual-BEMcrack opening displacement (COD)
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在求解各種有關岩石破壞力學的科學及工程問題上,數值技術已成為不可缺少的工具。許多的研究中,著重在發展新的數值方法來求得應力強度因子,其在工程設計的彈性構造物安全因子考量下具有直接實質上的應用性。由於現實中具有多變的工程條件,使得發展一套新的數值方法或替代的技術能夠提高現存或新的數值方法以有效與準確的解決較具複雜的問題。

本論文提出一套對偶邊界元素法(或稱單域邊界元素法),用來分析線彈性異向性岩石的混合模態應力強度因子。探討二維環型裂縫試體之混合模態I及II型應力強度因子(KI及KII),以及三維矩形體含有嵌式裂縫或邊緣裂縫之混合模態I、II及III型應力強度因子(KI、KII及KIII)有關之破壞力學問題。並考慮材料的橫向等向性平面及裂縫面皆可在固定的座標系統下呈現任意的方位。

採用五種類型的三節點二次元素和九種類型的九節點四邊形元素來近似裂縫尖端(前緣)及外部邊界,且混合模態應力強度因子運用應力強度因子與藉由Barnett-Lothe張量得到的相對裂縫開口位移之漸近關係來求值。在三維案例中,首次將特殊的九個節點四邊形形狀函數應用在口狀邊界上,且此數值方法的發展更可應用到具有傾斜表面裂縫的有限橫向等向性矩形體之應力強度因子計算上。這種近似的應力強度因子計算方法在異向性彈性構造體上具有良好的模型建立與設計價值。

在二維的應用中,結合對偶邊界元素法及環型裂縫試驗的新研究方法來測定異向性岩石之混合模態I及II型的破裂韌度(KIC及KIIC),而且所提出的對偶邊界元素法程式可以準確的計算異向性裂縫平板之應力強度因子。採用具有清楚異向性層理之台灣花蓮大理岩來進行環型裂縫試驗,藉由量測試驗的破壞載重,則可求得花蓮大理岩的的混合模態破裂韌度。試驗結果顯示,岩石之破裂韌度會隨著半徑比、材料傾角及裂縫角度而產生變化。

在三維的應用上,由嵌式裂縫的案例結果顯示,當材料層面傾向及傾角為45度及裂縫傾向及傾角為0度時,模態I的應力強度因子可到達最大值。且當裂縫呈現垂直或近似垂直方位時,模態I的應力強度因子將變成負值,表示裂縫由於受到全面壓縮載重正向於裂縫表面而造成閉合。沿著裂縫前緣的II及III型應力強度因子變化,在受到材料及裂縫方位變換的組合下也顯示著有趣的特色。同樣的,在邊緣裂縫的案例中,所提出的數值結果也清楚顯示混合模態應力強度因子會受到材料及裂縫方位變化的影響。在嵌式及邊緣裂縫的模態I應力強度因子比較上,當水平矩形裂縫完全嵌入矩形體內部時,其沿著裂縫前緣的應力強度因子值會大於當裂縫愈來愈趨於矩形體邊界時的值。
Numerical techniques in rock fracture mechanics have become indispensable tools for solving all kinds of science and engineering problems. Extensive research has been carried out for the development of new numerical methods to determine the stress intensity factors (SIFs). The evaluation of the SIFs may have direct practical applications in obtaining the safety factor of the elastic structures in engineering design. Due to the varied engineering conditions in this research area, it is imperative to develop new numerical methods or to explore alternative techniques for the purpose of solving the complicated problems and to improve the efficiency and accuracy of the existing or new numerical methods.

This thesis presents the dual boundary element method (dual-BEM) or single-domain BEM to analyze the mixed-mode SIFs in a linear anisotropic elastic rock. The mixed-mode (I-II) SIFs (KI and KII) for two-dimensional (2-D) cracked ring disk, and the mixed-mode (I-II-III) SIFs (KI, KII and KIII) for a three-dimensional (3-D) cuboid with either an embedded crack or edge surface crack are selected to explore the fracture mechanics problems. The transversely isotropic plane of the material and the crack surface can both orient arbitrarily with respect to a fixed global coordinate system.

Five types of three-node quadratic elements and nine types of nine-node quadrilateral elements are utilized to approximate the crack tip (front) as well as the outer boundary, and the mixed-mode SIFs are evaluated using the asymptotical relation between the SIFs and the relative crack opening displacements (COD) via the Barnett-Lothe tensor. In the 3-D case, it is for the first time that the special nine-node quadrilateral shape function is applied to the boundary containing the crack mouth. The numerical method developed can be applied to the SIF calculation in a finite transversely isotropic cuboid within an inclined surface crack. The computational approach and the results of SIFs are of great value for the modeling and design of anisotropic elastic structures.

A new methodology by combining the dual-BEM and the cracked ring test are presented to determine the mixed-mode (I-II) fracture toughness (KIC and KIIC) in 2-D anisotropic rocks. It has been proved that the proposed dual-BEM program can be used to calculate the SIFs of cracked anisotropic plate with good accuracy. An anisotropic Hualien marble of Taiwan with clear foliation was selected to carry out the cracked ring tests. Based on the measurement of failure load during the test, the mixed-mode fracture toughness of Hualien marble can be determined. Experimental results show that fracture toughness of rocks varies with the radius ratio, material inclined angle and crack angle.

For the embedded crack case in 3-D, our numerical results show that the mode-I SIF arrives at the largest possible value when the material inclined angle and dig angle are equal to 45 degree, and the crack inclined angle and dip angle are equal to 0 degree. It is further observed that when the crack is oriented vertically or nearly vertically, the mode-I SIF becomes negative, indicating that the crack closes due to an overall compressive loading normal to the crack surface. Variation of the SIFs for modes II and III along the crack fronts also shows some interesting features for different combinations of the material and crack orientations. For the edge crack case, the numerical results show clearly the influence of the material and crack orientations on the mixed-mode SIFs. For comparison, we have also calculated the mode-I SIF when a horizontal rectangular crack is embedded entirely within the cuboid. It is observed that the SIF values along the crack front are larger when the crack is closer to the surface of the cuboid than those when the crack is further away from the surface.
Abstract (English).................................................................................I
Abstract (Chinese)................................................................................IV
Acknowledgement...............................................................................VII
Table of Contents.................................................................................IX
List of Tables.......................................................................................XII
List of Figures.....................................................................................XIII
List of Notations..................................................................................XIX
Chapter 1
Introduction
1.1 Background and Motivation..........................................................1
1.2 Scope and Objectives....................................................................4
1.3 Organization..................................................................................6
Chapter 2
Literature Review
2.1 Introduction...................................................................................10
2.2 Anisotropic Behaviors of Fractured Rocks...................................11
2.3 Boundary Element Method...........................................................15
2.4 Stress Intensity Factor and Fracture Toughness............................18
2.5 Summary......................................................................................24
Chapter 3
Theoretical Background
3.1 Introduction...................................................................................25
3.2 Material Coordinate System and Basic Equations.........................25
3.3 Fundamental Solutions of Anisotropic Elasticity...........................33
3.3.1 Green's Functions for 3-D Anisotropic Elasticity.........................33
3.3.2 Green's Functions for 2-D Anisotropic Elasticity.........................34
3.4 Boundary Integral Equations.........................................................35
3.5 Numerical Discretization...............................................................37
3.6 Stress Intensity Factor Expression................................................45
3.7 Numerical Examples....................................................................47
3.7.1 Brazilian Disk with a Central Crack.............................................47
3.7.2 Ring Disk with a Single Crack.....................................................49
3.7.3 Ring Disk with a Paired Crack.....................................................52
3.8 Summary......................................................................................54
Chapter 4
Two-dimensional Fracture Toughness Analysis on Cracked Ring Test of Anisotropic Rocks
4.1 Introduction...................................................................................55
4.2 Rock Description and Sample Preparation....................................56
4.3 Elastic Constants Determined by Brazilian Test............................58
4.4 Fracture Toughness by Cracked Ring Test....................................67
4.5 Summary.......................................................................................79
Chapter 5
Three-dimensional Analysis of Stresses and Displacements in Linear Anisotropic Rocks
5.1 Introduction....................................................................................80
5.2 Anisotropic Rocks under a Uniaxial Compression.........................81
5.3 Anisotropic Rocks under a Pure Shear Stress.................................86
5.4 Displacements of Anisotropic Rocks Subject to the Gravity or a
Uniform Vertical Compression.......................................................88
5.5 Rock Stress Variation with Material Angles....................................91
5.6 Summary.........................................................................................98
Chapter 6
Three-dimensional Stress Intensity Factors in an Anisotropic Cuboid with a Central Square Crack
6.1 Introduction......................................................................................99
6.2 Description of the Geometry and Material Property.........................99
6.3 Effect of Material Anisotropy and Crack Orientations on SIFs........102
6.4 Behavior of the Relative Crack Opening Displacements..................120
6.5 Summary..........................................................................................124
Chapter 7
Three-dimensional Stress Intensity Factors in an Anisotropic Cuboid with an Inclined Surface Crack
7.1 Introduction.......................................................................................125
7.2 Description of the Geometry and Material Property.........................125
7.3 Stress Intensity Factors for a Horizontal Crack with Different
Inclined Material Angles...................................................................130
7.4 Stress Intensity Factors in Different Materials with Different
Crack Orientations.............................................................................136
7.5 Edge Crack vs. Embedded Crack......................................................140
7.6 Summary..........................................................................................143
Chapter 8
Conclusions and Future Work
8.1 Conclusions.......................................................................................144
8.2 Future Work......................................................................................147
References.................................................................................................149
Appendix A Six Shape Function for the Nine Special Element Types....158
Vita............................................................................................................161
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