Singularities in Elliptic Boundary Value Problems and Elasticity and Their Connection with Failure Initiation (eBook)
XXII, 462 Seiten
Springer New York (Verlag)
978-1-4614-1508-4 (ISBN)
This introductory and self-contained book gathers as much explicit mathematical results on the linear-elastic and heat-conduction solutions in the neighborhood of singular points in two-dimensional domains, and singular edges and vertices in three-dimensional domains. These are presented in an engineering terminology for practical usage. The author treats the mathematical formulations from an engineering viewpoint and presents high-order finite-element methods for the computation of singular solutions in isotropic and anisotropic materials, and multi-material interfaces. The proper interpretation of the results in engineering practice is advocated, so that the computed data can be correlated to experimental observations.
The book is divided into fourteen chapters, each containing several sections.
Most of it (the first nine Chapters) addresses two-dimensional domains, where
only singular points exist. The solution in a vicinity of these points admits an asymptotic expansion composed of eigenpairs and associated generalized flux/stress intensity factors (GFIFs/GSIFs), which are being computed analytically when possible or by finite element methods otherwise. Singular points associated with weakly coupled thermoelasticity in the vicinity of singularities are also addressed and thermal GSIFs are computed. The computed data is important in engineering practice for predicting failure initiation in brittle material on a daily basis. Several failure laws for two-dimensional domains with V-notches are presented and their validity is examined by comparison to experimental observations. A sufficient simple and reliable condition for predicting failure initiation (crack formation) in micron level electronic devices, involving singular points, is still a topic of active research and interest, and is addressed herein.
Explicit singular solutions in the vicinity of vertices and edges in three-dimensional domains are provided in the remaining five chapters. New methods for the computation of generalized edge flux/stress intensity functions along singular edges are presented and demonstrated by several example problems from the field of fracture mechanics; including anisotropic domains and bimaterial interfaces. Circular edges are also presented and the author concludes with some remarks on open questions.
This well illustrated book will appeal to both applied mathematicians and engineers working in the field of fracture mechanics and singularities.
Zohar Yosibash is a Professor of Mechanical Engineering at Ben-Gurion University of the Negev in Beer-Sheva, Israel
This introductory and self-contained book gathers as much explicit mathematical results on the linear-elastic and heat-conduction solutions in the neighborhood of singular points in two-dimensional domains, and singular edges and vertices in three-dimensional domains. These are presented in an engineering terminology for practical usage. The author treats the mathematical formulations from an engineering viewpoint and presents high-order finite-element methods for the computation of singular solutions in isotropic and anisotropic materials, and multi-material interfaces. The proper interpretation of the results in engineering practice is advocated, so that the computed data can be correlated to experimental observations. The book is divided into fourteen chapters, each containing several sections.Most of it (the first nine Chapters) addresses two-dimensional domains, whereonly singular points exist. The solution in a vicinity of these points admits an asymptotic expansion composed of eigenpairs and associated generalized flux/stress intensity factors (GFIFs/GSIFs), which are being computed analytically when possible or by finite element methods otherwise. Singular points associated with weakly coupled thermoelasticity in the vicinity of singularities are also addressed and thermal GSIFs are computed. The computed data is important in engineering practice for predicting failure initiation in brittle material on a daily basis. Several failure laws for two-dimensional domains with V-notches are presented and their validity is examined by comparison to experimental observations. A sufficient simple and reliable condition for predicting failure initiation (crack formation) in micron level electronic devices, involving singular points, is still a topic of active research and interest, and is addressed herein. Explicit singular solutions in the vicinity of vertices and edges in three-dimensional domains are provided in the remaining five chapters. New methods for the computation of generalized edge flux/stress intensity functions along singular edges are presented and demonstrated by several example problems from the field of fracture mechanics; including anisotropic domains and bimaterial interfaces. Circular edges are also presented and the author concludes with some remarks on open questions.This well illustrated book will appeal to both applied mathematicians and engineers working in the field of fracture mechanics and singularities.
Zohar Yosibash is a Professor of Mechanical Engineering at Ben-Gurion University of the Negev in Beer-Sheva, Israel
Singularities in Elliptic Boundary Value Problems and Elasticity and Their Connection with Failure Initiation 3
Preface 7
Contents 11
List of Main Symbols 19
Chapter 1 Introduction
23
1.1 What Is It All About? 23
1.2 Principles and Assumptions 27
1.3 Layout 29
1.4 A Model Problem 31
1.4.1 A Path-Independent Integral 35
1.4.2 Orthogonality of the ``Primal'' and ``Dual''Eigenfunctions 36
1.4.3 Particular Solutions 37
1.4.4 Curved Boundaries Intersecting at the Singular Point 39
1.5 The Heat Conduction Problem: Notation 39
1.6 The Linear Elasticity Problem: Notation 42
Chapter 2 An Introduction to the p- and hp-Versions of the Finite Element Method
48
2.1 The Weak Formulation 48
2.2 Discretization 50
2.2.1 Blending Functions, the Element Stiffness Matrix and Element Load Vector 52
2.2.2 The Finite Element Space 53
2.2.3 Mesh Design for an Optimal Convergence Rate 57
2.3 Convergence Rates of FEMs and Their Connection to the Regularity of the Exact Solution 57
2.3.1 Algebraic and Exponential Rates of Convergence 59
2.3.1.1 Numerical Examples 61
Chapter 3 Eigenpair Computation for Two-Dimensional Heat Conduction Singularities
67
3.1 Overview of Methods for Computing Eigenpairs 67
3.2 Formulation of the Modified Steklov Eigenproblem 69
3.2.1 Homogeneous Dirichlet Boundary Conditions 73
3.2.2 The Modified Steklov Eigen-problemfor the Laplace Equation with Homogeneous Neumann BCs 74
3.3 Numerical Solution of the Modified Steklov Weak Eigenproblem by p-FEMs 74
3.4 Examples on the Performance of the ModifiedSteklov Method 78
3.4.1 A Detailed Simple Example 78
3.4.2 A Crack with Homogeneous Newton BCs(Laplace Equation) 83
3.4.3 A V-Notch in an Anisotropic Material with Homogeneous Neumann BCs. 85
3.4.4 An Internal Singular Point at the Interface of Two Materials 86
3.4.5 An Anisotropic Flux-Free Bimaterial Interface 90
Chapter 4 GFIFs Computation for Two-Dimensional Heat Conduction Problems
93
4.1 Computing GFIFs Using the Dual Singular Function Method 93
4.2 Computing GFIFs Using the Complementary Weak Form 96
4.2.1 Derivation of the Complementary Weak Form 96
4.2.2 Using the Complementary Weak Formulation to Extract GFIFs 99
4.2.3 Extracting GFIFs Using the Complementary Weak Formulation and Approximated Eigenpairs 104
4.3 Numerical Examples: Extracting GFIFs Using the Complementary Weak Form 106
4.3.1 Laplace equation with Newton BCs 107
4.3.2 Laplace Equation with Homogeneous Neumann BCs: Approximate eigenpairs 109
4.3.3 Anisotropic Heat Conduction Equation with Newton BCs 112
4.3.4 An Internal point at the Interface of Two Materials 113
Chapter 5 Eigenpairs for Two-Dimensional Elasticity
116
5.1 Asymptotic Solution in the Vicinity of a Reentrant Corner in an Isotropic Material 117
5.2 The Particular Case of TF/TF BCs 125
5.2.1 A TF/TF Reentrant Corner (V-Notch) 126
5.2.2 A TF/TF Crack 130
5.2.3 A TF/TF Crack at a Bimaterial Interface 134
5.3 Power-Logarithmic or Logarithmic Singularities with Homogeneous BCs 140
5.4 Modified Steklov Eigenproblem for Elasticity 141
5.4.1 Numerical Solution by p-FEMs 145
5.4.2 Numerical Investigation: Two Bonded Orthotropic Materials 148
5.4.3 Numerical Investigation: Power-LogarithmicSingularity 150
Chapter 6 Computing Generalized Stress Intensity Factors (GSIFs)
152
6.1 The Contour Integral Method, Also Known as the Dual-Singular Function Method or the Reciprocal Work Contour Method 152
6.1.1 A Path-Independent Contour Integral 152
6.1.2 Orthogonality of the Primal and Dual Eigenfunctions 154
6.1.3 Extracting GSIFs (Ai's) Using the CIM 156
6.1.3.1 Extracting GSIFs for a TF/TF V-Notch Using the CIM 157
6.1.3.2 Extracting SIFs for a TF/TF Crack Using the CIM 160
6.2 Extracting GSIFs by the Complementary EnergyMethod (CEM) 161
6.2.0.3 FE Implementation of the CEM for Extracting GSIFs 164
6.3 Numerical Examples: Extracting GSIFs by CIM and CEM 166
6.3.1 A Crack in an Isotropic Material: Extracting SIFs by the CIM and CEM 166
6.3.2 Crack at a Bimaterial Interface: Extracting SIFs by the CEM 168
6.3.3 Nearly Incompressible L-Shaped Domain: Extracting SIFs by the CEM 171
Chapter 7 Thermal Generalized Stress Intensity Factors in 2-D Domains
176
7.1 Classical (Strong) and Weak Formulationsof the Linear Thermoelastic Problem 177
7.1.1 The Linear Thermoelastic Problem 177
7.1.2 The Complementary Energy Formulation of the Thermoelastic Problem 180
7.1.3 The Extraction Post-solution Scheme 181
7.1.4 The Compliance Matrix, Load Vector and Extraction of TGSIFs 182
7.1.5 Discretization and the Numerical Algorithm 184
7.2 Numerical Examples 185
7.2.1 Central Crack in a Rectangular Plate 185
7.2.2 A Slanted Crack in a Rectangular Plate 190
7.2.3 A Rectangular Plate with Cracks at an Internal Hole 191
7.2.4 Singular Points Associated with MultimaterialInterfaces 197
7.2.4.1 An Inclusion Problem 197
7.2.4.2 Two 90 Dissimilar Bonded Wedges 200
Chapter 8 Failure Criteria for Brittle Elastic Materials
203
8.1 On Failure Criteria Under Mode I Loading 206
8.1.1 Novozhilov-Seweryn Criterion 206
8.1.2 Leguillon's Criterion 208
8.1.3 Dunn et al. Criterion 209
8.1.4 The Strain Energy Density (SED) Criterion 209
8.1.4.1 Computation of the Critical SED[R]crack for a Crack and SED[R]straight for a Straight Edge, and the Material Characteristic Integration Radius Rmat 212
8.2 Materials and Experimental Procedures 214
8.2.1 Experiments with Alumina-7%Zirconia 214
8.2.2 Experiments with PMMA 218
8.3 Verification and Validation of the Failure Criteria 221
8.3.1 Analysis of the Alumina-7%Zirconia Test Results 223
8.3.2 Analysis of the PMMA Tests 225
8.4 Determining Fracture Toughness of Brittle Materials Using Rounded V-Notched Specimens 228
8.4.1 The Failure Criterion for a Rounded V-Notch Tip 229
8.4.2 Estimating the Fracture Toughness From Rounded V-Notched Specimens 230
8.4.3 Experiments on Rounded V-Notched Specimens in the Literature 232
8.4.3.1 Experiments on Alumina-7% Zirconia from YoBuGi04 233
8.4.3.2 Experiments on PMMA DuSu97a 233
8.4.3.3 Experiments on PMMA Reported in GoEl05 234
8.4.4 Estimating the Fracture Toughness 234
8.4.4.1 Estimated Fracture Toughness for Alumina-7%Zirconia YoBuGi04 234
8.4.4.2 Estimated Fracture Toughness for PMMA DuSu97a 235
8.4.4.3 Estimated Values for PMMA Reported in GoEl05 236
Chapter 9 A Thermoelastic Failure Criterion at the Micron Scalein Electronic Devices
239
9.1 The SED Criterion for a Thermoelastic Problem 242
9.2 Material Properties 245
9.2.1 Material Properties of Passivation Layers 246
9.2.2 Aluminum Lines and Dielectric Layers 248
9.3 Experimental Validation of the Failure Criterion 248
9.3.1 Computing SEDs by p-Version FEMs 249
Chapter 10 Singular Solutions of the Heat Conduction (Scalar) Equation in Polyhedral Domains
255
10.1 Asymptotic Solution to the Laplace Equationin a Neighborhood of an Edge 258
10.2 A Systematic Mathematical Algorithm for the Edge Asymptotic Solution for a General Scalar Elliptic Equation 264
10.2.1 The Eigenpairs and Computation of ShadowFunctions 265
10.2.2 Eigenfunctions, their Shadow Functions and Duals for Cases 1-4 (Dirichlet BCs) 267
10.2.2.1 Summary of Cases (1-4): Eigenfunctions, Shadows, and Duals (Dirichlet BCs) 272
10.2.3 The Primal and Dual Eigenfunctions and Shadows for Case 5 (Dirichlet BCs) 272
10.3 Eigenfunctions, Shadows and Duals for Cases 1-5 with Homogeneous Neumann Boundary Conditions 275
10.3.0.1 Primal and Dual Eigenfunctions and Shadows for Case 1 276
10.3.0.2 Primal and Dual Eigenfunctions and Shadows for Case 2 277
10.3.0.3 Primal and Dual Eigenfunctions and Shadows for Case 3 278
10.3.0.4 Primal and Dual Eigenfunctions and Shadows for Case 4 279
10.3.0.5 Primal and Dual Eigenfunctions and Shadows for Case 5 281
Chapter 11 Extracting Edge-Flux-Intensity Functions (EFIFs) Associated with Polyhedral Domains
283
11.1 Extracting Pointwise Values of the EFIFs by the L2 Projection Method 283
11.1.1 Numerical Implementation 286
11.1.2 An Example Problem and Numerical Experimentation 288
11.2 The Energy Projection Method 291
11.3 A Quasidual Function Method (QDFM) for Extracting EFIFs 293
11.3.0.1 The Quasidual Extraction Functions 294
11.3.1 Jacobi Polynomial Representationof the Extraction Function 295
11.3.2 Jacobi Extraction Polynomials of Order 2 297
11.3.3 Analytical Solutions for Verifying the QDFM 297
11.3.4 Numerical Results for (BC4) Using K2(1) 298
11.3.5 A Nonpolynomial EFIF 300
11.3.6 A Domain with Edge and Vertex Singularities 303
Chapter 12 Vertex Singularities for the 3-D Laplace Equation
309
12.1 Analytical Solutions for Conical Vertices 310
12.1.1 Homogeneous Dirichlet BCs 312
12.1.2 Homogeneous Neumann BCs 313
12.2 The Modified Steklov Weak Form and Finite Element Discretization 315
12.2.0.1 An Asymmetric Weak Eigenform 317
12.2.1 Application of p/Spectral Finite Element Methods 319
12.2.1.1 The Basis Functions 320
12.3 Numerical Examples 321
12.3.1 Conical Vertex, /2=3/4, Homogeneous Neumann BCs 321
12.3.2 Conical Vertex, /2=3/4, Homogeneous Dirichlet BCs 322
12.3.3 Vertex at the Intersection of a Crack Front with a Flat Face, Homogeneous Neumann BCs 324
12.3.4 Vertex at the Intersection of a V-Notch Front with a Conical Reentrant Corner, Homogeneous Neumann BCs 325
12.4 Other Methods for the Computation of the Vertex eigenpairs, and Extensions to the Elasticity System 325
12.4.1 Extension of the Method to the Elasticity System 329
Chapter 13 Edge EigenPairs and ESIFs of 3-D Elastic Problems
333
13.1 The Elastic Solution for an Isotropic Material in the Vicinity of an Edge 335
13.1.1 Differential Equations for 3-D Eigenpairs 335
13.1.2 Boundary Conditions for the Primal, Dualand Shadow Functions 339
13.1.2.1 Traction-Free Boundary Conditions 339
13.1.2.2 Clamped Boundary Conditions 340
13.1.3 Primal and Dual Eigenfunctions and Shadow Functions for a Traction-Free Crack 340
13.1.4 Primal and Dual Eigenfunctions and Shadow Functions for a Clamped 3/2 V-notch 347
13.2 Extracting ESIFs by the J[R]-Integral 351
13.2.1 Jacobi Extraction Polynomials of Order 4 353
13.2.1.1 Numerical Computation of the J[R] Integral 354
13.2.2 Numerical Example: A Cracked Domain (=2) with Traction-Tree Boundary Conditions 355
13.2.3 Numerical Example: A Clamped V-notched Domain (=32) 357
13.2.4 Numerical Example of Engineering Importance: Compact Tension Specimen 358
13.2.4.1 The Relation Between the SIFs KI, KII and the ESIF 359
13.2.4.2 Compact Tension Specimen (CTS) Under a Constant Tension Along x3 361
13.3 Eigenpairs and ESIFs for Anisotropicand Multimaterial Interfaces 364
13.3.1 Computing Eigenpairs 370
13.3.1.1 p-FEMs for the Solution of the Weak Eigenformulation 372
13.3.2 Computing Complex Primal and Dual ShadowFunctions 375
13.3.2.1 The Weak Form for the Computation of Primal and Dual Shadow Functions 375
13.3.2.2 p-FEMs for the Solution of (13.105) 377
13.3.3 Difficulties in Computing Shadows and Remedies for Several Pathological Cases 378
13.3.4 Extracting Complex ESIFs by the QDFM 382
13.3.5 Numerical Example: A Crack at the Interface of Two Isotropic Materials 384
13.3.6 Numerical Example: CTS, Crack at the Interface of Two Anisotropic Materials 389
Chapter 14 Remarks on Circular Edges and Open Questions
394
14.1 Circular Singular Edges in 3-D Domains:The Laplace Equation 394
14.1.1 Axisymmetric Case, 0 396
14.1.1.1 A Specific Example Problem: Penny-Shaped Crackwith Axisymmetric Loading and Homogeneous Neumann BCs 398
14.1.1.2 A Specific Example Problem: Penny-Shaped Crackwith Axisymmetric Loading and Homogeneous Dirichlet BCs 401
14.1.1.3 A Specific Example Problem: Circumferential Crackwith Axisymmetric Loading and Homogeneous Neumann BCs 401
14.1.2 General Case 402
14.1.2.1 A Specific Example Problem: Penny-Shaped Crackfor a Nonaxisymmetric Loading and Homogeneous Neumann BCs 404
14.1.2.2 A Specific Example Problem: Penny-Shaped Crackfor a Nonaxisymmetric Loading and Homogeneous Dirichlet BCs 405
14.1.2.3 A Specific Example Problem: Hollow Cylinderwith Nonaxisymmetric Loading and Homogeneous Neumann BCs 406
14.1.2.4 A Specific Example Problem: Exterior Circular Crack wich Nonaxisymmetric Loading and Homogeneous Neumann BCs 406
14.2 Circular Singular Edges in 3-D Domains:The Elasticity System 407
14.3 Further Theoretical and Practical Applications 409
Appendix A Definition of Sobolev, Energy, and Statically Admissible Spaces and Associated Norms
411
Appendix B Analytic Solution to 2-D Scalar Elliptic Problemsin Anisotropic Domains
416
B.1 Analytic Solution to a 2-D Scalar Elliptic Problemin an Anisotropic Bimaterial Domain 419
B.1.1 Treatment of the Boundary Conditions 421
B.1.2 An Example 422
Appendix C Asymptotic Solution at the Intersection of Circular Edges in a 2-D Domain
426
Appendix D Proof that Eigenvalues of the Scalar Anisotropic Elliptic BVP with Constant Coefficients Are Real
432
Appendix E A Path-Independent Integral and Orthogonalityof Eigenfunctions for General Scalar Elliptic Equationsin 2-D Domains
435
Appendix F Energy Release Rate (ERR) Method, its Connectionto the J-integral and Extraction of SIFs
440
F.1 Derivation of the ERR 440
F.1.1 The Energy Argument KeSi95 440
F.1.2 The Potential Energy Argument KeSi95 441
F.2 Griffith's Energy Criterion Grif20, Grif24 443
F.3 Relations Between the ERR and the SIFs 449
F.3.1 Symmetric (Mode I) Loading 449
F.3.2 Antisymmetric (Mode II) Loading 450
F.3.3 Combined (Mode I and Mode II) Loading 451
F.3.4 Computation of G by the Stiffness Derivative Method 451
F.3.5 The Stiffness Derivative Method for 3-D Domains 455
F.4 The J-Integral and its Relation to ERR 455
References 460
Index 470
| Erscheint lt. Verlag | 2.12.2011 |
|---|---|
| Reihe/Serie | Interdisciplinary Applied Mathematics | Interdisciplinary Applied Mathematics |
| Zusatzinfo | XXII, 462 p. |
| Verlagsort | New York |
| Sprache | englisch |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| Mathematik / Informatik ► Mathematik ► Angewandte Mathematik | |
| Mathematik / Informatik ► Mathematik ► Statistik | |
| Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik | |
| Technik ► Bauwesen | |
| Technik ► Maschinenbau | |
| Schlagworte | Continuum Mechanics • Elasticity • Elliptic boundary problems • Finite Element Methods |
| ISBN-10 | 1-4614-1508-X / 146141508X |
| ISBN-13 | 978-1-4614-1508-4 / 9781461415084 |
| Haben Sie eine Frage zum Produkt? |
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