Yosibash Singularities in Elliptic Boundary Value Problems and Elasticity and Their Connection with Failure Initiation

1. Auflage 2011
ISBN: 978-1-4614-1508-4
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 37, 462 Seiten

Reihe: Interdisciplinary Applied Mathematics

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

Autoren/Hrsg.

Yosibash, Zohar

Inhaltsverzeichnis

1;Singularities in Elliptic Boundary Value Problems and Elasticity and Their Connection with Failure Initiation;3
1.1;Preface;7
1.2;Contents;11
1.3;List of Main Symbols;19
1.4;Chapter 1 Introduction
;23
1.4.1;1.1 What Is It All About?;23
1.4.2;1.2 Principles and Assumptions;27
1.4.3;1.3 Layout;29
1.4.4;1.4 A Model Problem;31
1.4.4.1;1.4.1 A Path-Independent Integral;35
1.4.4.2;1.4.2 Orthogonality of the ``Primal'' and ``Dual''Eigenfunctions;36
1.4.4.3;1.4.3 Particular Solutions;37
1.4.4.4;1.4.4 Curved Boundaries Intersecting at the Singular Point;39
1.4.5;1.5 The Heat Conduction Problem: Notation;39
1.4.6;1.6 The Linear Elasticity Problem: Notation;42
1.5;Chapter 2 An Introduction to the p- and hp-Versions of the Finite Element Method
;48
1.5.1;2.1 The Weak Formulation;48
1.5.2;2.2 Discretization;50
1.5.2.1;2.2.1 Blending Functions, the Element Stiffness Matrix and Element Load Vector;52
1.5.2.2;2.2.2 The Finite Element Space;53
1.5.2.3;2.2.3 Mesh Design for an Optimal Convergence Rate;57
1.5.3;2.3 Convergence Rates of FEMs and Their Connection to the Regularity of the Exact Solution;57
1.5.3.1;2.3.1 Algebraic and Exponential Rates of Convergence;59
1.5.3.1.1;2.3.1.1 Numerical Examples;61
1.6;Chapter 3 Eigenpair Computation for Two-Dimensional Heat Conduction Singularities
;67
1.6.1;3.1 Overview of Methods for Computing Eigenpairs;67
1.6.2;3.2 Formulation of the Modified Steklov Eigenproblem;69
1.6.2.1;3.2.1 Homogeneous Dirichlet Boundary Conditions;73
1.6.2.2;3.2.2 The Modified Steklov Eigen-problemfor the Laplace Equation with Homogeneous Neumann BCs;74
1.6.3;3.3 Numerical Solution of the Modified Steklov Weak Eigenproblem by p-FEMs;74
1.6.4;3.4 Examples on the Performance of the ModifiedSteklov Method;78
1.6.4.1;3.4.1 A Detailed Simple Example;78
1.6.4.2;3.4.2 A Crack with Homogeneous Newton BCs(Laplace Equation);83
1.6.4.3;3.4.3 A V-Notch in an Anisotropic Material with Homogeneous Neumann BCs.;85
1.6.4.4;3.4.4 An Internal Singular Point at the Interface of Two Materials;86
1.6.4.5;3.4.5 An Anisotropic Flux-Free Bimaterial Interface;90
1.7;Chapter 4 GFIFs Computation for Two-Dimensional Heat Conduction Problems
;93
1.7.1;4.1 Computing GFIFs Using the Dual Singular Function Method;93
1.7.2;4.2 Computing GFIFs Using the Complementary Weak Form;96
1.7.2.1;4.2.1 Derivation of the Complementary Weak Form;96
1.7.2.2;4.2.2 Using the Complementary Weak Formulation to Extract GFIFs ;99
1.7.2.3;4.2.3 Extracting GFIFs Using the Complementary Weak Formulation and Approximated Eigenpairs;104
1.7.3;4.3 Numerical Examples: Extracting GFIFs Using the Complementary Weak Form;106
1.7.3.1;4.3.1 Laplace equation with Newton BCs;107
1.7.3.2;4.3.2 Laplace Equation with Homogeneous Neumann BCs: Approximate eigenpairs;109
1.7.3.3;4.3.3 Anisotropic Heat Conduction Equation with Newton BCs;112
1.7.3.4;4.3.4 An Internal point at the Interface of Two Materials;113
1.8;Chapter 5 Eigenpairs for Two-Dimensional Elasticity
;116
1.8.1;5.1 Asymptotic Solution in the Vicinity of a Reentrant Corner in an Isotropic Material;117
1.8.2;5.2 The Particular Case of TF/TF BCs;125
1.8.2.1;5.2.1 A TF/TF Reentrant Corner (V-Notch);126
1.8.2.2;5.2.2 A TF/TF Crack;130
1.8.2.3;5.2.3 A TF/TF Crack at a Bimaterial Interface;134
1.8.3;5.3 Power-Logarithmic or Logarithmic Singularities with Homogeneous BCs;140
1.8.4;5.4 Modified Steklov Eigenproblem for Elasticity;141
1.8.4.1;5.4.1 Numerical Solution by p-FEMs;145
1.8.4.2;5.4.2 Numerical Investigation: Two Bonded Orthotropic Materials;148
1.8.4.3;5.4.3 Numerical Investigation: Power-LogarithmicSingularity;150
1.9;Chapter 6 Computing Generalized Stress Intensity Factors (GSIFs)
;152
1.9.1;6.1 The Contour Integral Method, Also Known as the Dual-Singular Function Method or the Reciprocal Work Contour Method;152
1.9.1.1;6.1.1 A Path-Independent Contour Integral;152
1.9.1.2;6.1.2 Orthogonality of the Primal and Dual Eigenfunctions;154
1.9.1.3;6.1.3 Extracting GSIFs (Ai's) Using the CIM;156
1.9.1.3.1;6.1.3.1 Extracting GSIFs for a TF/TF V-Notch Using the CIM;157
1.9.1.3.2;6.1.3.2 Extracting SIFs for a TF/TF Crack Using the CIM;160
1.9.2;6.2 Extracting GSIFs by the Complementary EnergyMethod (CEM);161
1.9.2.1;6.2.0.3 FE Implementation of the CEM for Extracting GSIFs;164
1.9.3;6.3 Numerical Examples: Extracting GSIFs by CIM and CEM;166
1.9.3.1;6.3.1 A Crack in an Isotropic Material: Extracting SIFs by the CIM and CEM;166
1.9.3.2;6.3.2 Crack at a Bimaterial Interface: Extracting SIFs by the CEM;168
1.9.3.3;6.3.3 Nearly Incompressible L-Shaped Domain: Extracting SIFs by the CEM;171
1.10;Chapter 7 Thermal Generalized Stress Intensity Factors in 2-D Domains
;176
1.10.1;7.1 Classical (Strong) and Weak Formulationsof the Linear Thermoelastic Problem;177
1.10.1.1;7.1.1 The Linear Thermoelastic Problem;177
1.10.1.2;7.1.2 The Complementary Energy Formulation of the Thermoelastic Problem;180
1.10.1.3;7.1.3 The Extraction Post-solution Scheme;181
1.10.1.4;7.1.4 The Compliance Matrix, Load Vector and Extraction of TGSIFs;182
1.10.1.5;7.1.5 Discretization and the Numerical Algorithm;184
1.10.2;7.2 Numerical Examples;185
1.10.2.1;7.2.1 Central Crack in a Rectangular Plate;185
1.10.2.2;7.2.2 A Slanted Crack in a Rectangular Plate;190
1.10.2.3;7.2.3 A Rectangular Plate with Cracks at an Internal Hole;191
1.10.2.4;7.2.4 Singular Points Associated with MultimaterialInterfaces;197
1.10.2.4.1;7.2.4.1 An Inclusion Problem;197
1.10.2.4.2;7.2.4.2 Two 90 Dissimilar Bonded Wedges;200
1.11;Chapter 8 Failure Criteria for Brittle Elastic Materials
;203
1.11.1;8.1 On Failure Criteria Under Mode I Loading;206
1.11.1.1;8.1.1 Novozhilov-Seweryn Criterion;206
1.11.1.2;8.1.2 Leguillon's Criterion;208
1.11.1.3;8.1.3 Dunn et al. Criterion;209
1.11.1.4;8.1.4 The Strain Energy Density (SED) Criterion;209
1.11.1.4.1;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
1.11.2;8.2 Materials and Experimental Procedures;214
1.11.2.1;8.2.1 Experiments with Alumina-7%Zirconia;214
1.11.2.2;8.2.2 Experiments with PMMA;218
1.11.3;8.3 Verification and Validation of the Failure Criteria;221
1.11.3.1;8.3.1 Analysis of the Alumina-7%Zirconia Test Results;223
1.11.3.2;8.3.2 Analysis of the PMMA Tests;225
1.11.4;8.4 Determining Fracture Toughness of Brittle Materials Using Rounded V-Notched Specimens;228
1.11.4.1;8.4.1 The Failure Criterion for a Rounded V-Notch Tip;229
1.11.4.2;8.4.2 Estimating the Fracture Toughness From Rounded V-Notched Specimens;230
1.11.4.3;8.4.3 Experiments on Rounded V-Notched Specimens in the Literature;232
1.11.4.3.1;8.4.3.1 Experiments on Alumina-7% Zirconia from YoBuGi04;233
1.11.4.3.2;8.4.3.2 Experiments on PMMA DuSu97a;233
1.11.4.3.3;8.4.3.3 Experiments on PMMA Reported in GoEl05;234
1.11.4.4;8.4.4 Estimating the Fracture Toughness;234
1.11.4.4.1;8.4.4.1 Estimated Fracture Toughness for Alumina-7%Zirconia YoBuGi04;234
1.11.4.4.2;8.4.4.2 Estimated Fracture Toughness for PMMA DuSu97a;235
1.11.4.4.3;8.4.4.3 Estimated Values for PMMA Reported in GoEl05;236
1.12;Chapter 9 A Thermoelastic Failure Criterion at the Micron Scalein Electronic Devices
;239
1.12.1;9.1 The SED Criterion for a Thermoelastic Problem;242
1.12.2;9.2 Material Properties;245
1.12.2.1;9.2.1 Material Properties of Passivation Layers;246
1.12.2.2;9.2.2 Aluminum Lines and Dielectric Layers;248
1.12.3;9.3 Experimental Validation of the Failure Criterion;248
1.12.3.1;9.3.1 Computing SEDs by p-Version FEMs;249
1.13;Chapter 10 Singular Solutions of the Heat Conduction (Scalar) Equation in Polyhedral Domains
;255
1.13.1;10.1 Asymptotic Solution to the Laplace Equationin a Neighborhood of an Edge;258
1.13.2;10.2 A Systematic Mathematical Algorithm for the Edge Asymptotic Solution for a General Scalar Elliptic Equation;264
1.13.2.1;10.2.1 The Eigenpairs and Computation of ShadowFunctions;265
1.13.2.2;10.2.2 Eigenfunctions, their Shadow Functions and Duals for Cases 1-4 (Dirichlet BCs);267
1.13.2.2.1;10.2.2.1 Summary of Cases (1-4): Eigenfunctions, Shadows, and Duals (Dirichlet BCs);272
1.13.2.3;10.2.3 The Primal and Dual Eigenfunctions and Shadows for Case 5 (Dirichlet BCs);272
1.13.3;10.3 Eigenfunctions, Shadows and Duals for Cases 1-5 with Homogeneous Neumann Boundary Conditions;275
1.13.3.1;10.3.0.1 Primal and Dual Eigenfunctions and Shadows for Case 1;276
1.13.3.2;10.3.0.2 Primal and Dual Eigenfunctions and Shadows for Case 2;277
1.13.3.3;10.3.0.3 Primal and Dual Eigenfunctions and Shadows for Case 3;278
1.13.3.4;10.3.0.4 Primal and Dual Eigenfunctions and Shadows for Case 4;279
1.13.3.5;10.3.0.5 Primal and Dual Eigenfunctions and Shadows for Case 5;281
1.14;Chapter 11 Extracting Edge-Flux-Intensity Functions (EFIFs) Associated with Polyhedral Domains
;283
1.14.1;11.1 Extracting Pointwise Values of the EFIFs by the L2 Projection Method;283
1.14.1.1;11.1.1 Numerical Implementation;286
1.14.1.2;11.1.2 An Example Problem and Numerical Experimentation;288
1.14.2;11.2 The Energy Projection Method;291
1.14.3;11.3 A Quasidual Function Method (QDFM) for Extracting EFIFs;293
1.14.3.1;11.3.0.1 The Quasidual Extraction Functions;294
1.14.3.2;11.3.1 Jacobi Polynomial Representationof the Extraction Function;295
1.14.3.3;11.3.2 Jacobi Extraction Polynomials of Order 2;297
1.14.3.4;11.3.3 Analytical Solutions for Verifying the QDFM;297
1.14.3.5;11.3.4 Numerical Results for (BC4) Using K2(1);298
1.14.3.6;11.3.5 A Nonpolynomial EFIF;300
1.14.3.7;11.3.6 A Domain with Edge and Vertex Singularities;303
1.15;Chapter 12 Vertex Singularities for the 3-D Laplace Equation
;309
1.15.1;12.1 Analytical Solutions for Conical Vertices;310
1.15.1.1;12.1.1 Homogeneous Dirichlet BCs;312
1.15.1.2;12.1.2 Homogeneous Neumann BCs;313
1.15.2;12.2 The Modified Steklov Weak Form and Finite Element Discretization;315
1.15.2.1;12.2.0.1 An Asymmetric Weak Eigenform;317
1.15.2.2;12.2.1 Application of p/Spectral Finite Element Methods;319
1.15.2.2.1;12.2.1.1 The Basis Functions;320
1.15.3;12.3 Numerical Examples;321
1.15.3.1;12.3.1 Conical Vertex, /2=3/4, Homogeneous Neumann BCs;321
1.15.3.2;12.3.2 Conical Vertex, /2=3/4, Homogeneous Dirichlet BCs;322
1.15.3.3;12.3.3 Vertex at the Intersection of a Crack Front with a Flat Face, Homogeneous Neumann BCs;324
1.15.3.4;12.3.4 Vertex at the Intersection of a V-Notch Front with a Conical Reentrant Corner, Homogeneous Neumann BCs;325
1.15.4;12.4 Other Methods for the Computation of the Vertex eigenpairs, and Extensions to the Elasticity System;325
1.15.4.1;12.4.1 Extension of the Method to the Elasticity System;329
1.16;Chapter 13 Edge EigenPairs and ESIFs of 3-D Elastic Problems
;333
1.16.1;13.1 The Elastic Solution for an Isotropic Material in the Vicinity of an Edge;335
1.16.1.1;13.1.1 Differential Equations for 3-D Eigenpairs;335
1.16.1.2;13.1.2 Boundary Conditions for the Primal, Dualand Shadow Functions;339
1.16.1.2.1;13.1.2.1 Traction-Free Boundary Conditions;339
1.16.1.2.2;13.1.2.2 Clamped Boundary Conditions;340
1.16.1.3;13.1.3 Primal and Dual Eigenfunctions and Shadow Functions for a Traction-Free Crack;340
1.16.1.4;13.1.4 Primal and Dual Eigenfunctions and Shadow Functions for a Clamped 3/2 V-notch;347
1.16.2;13.2 Extracting ESIFs by the J[R]-Integral;351
1.16.2.1;13.2.1 Jacobi Extraction Polynomials of Order 4;353
1.16.2.1.1;13.2.1.1 Numerical Computation of the J[R] Integral;354
1.16.2.2;13.2.2 Numerical Example: A Cracked Domain (=2) with Traction-Tree Boundary Conditions;355
1.16.2.3;13.2.3 Numerical Example: A Clamped V-notched Domain (=32);357
1.16.2.4;13.2.4 Numerical Example of Engineering Importance: Compact Tension Specimen;358
1.16.2.4.1;13.2.4.1 The Relation Between the SIFs KI, KII and the ESIF;359
1.16.2.4.2;13.2.4.2 Compact Tension Specimen (CTS) Under a Constant Tension Along x3;361
1.16.3;13.3 Eigenpairs and ESIFs for Anisotropicand Multimaterial Interfaces;364
1.16.3.1;13.3.1 Computing Eigenpairs;370
1.16.3.1.1;13.3.1.1 p-FEMs for the Solution of the Weak Eigenformulation;372
1.16.3.2;13.3.2 Computing Complex Primal and Dual ShadowFunctions;375
1.16.3.2.1;13.3.2.1 The Weak Form for the Computation of Primal and Dual Shadow Functions;375
1.16.3.2.2;13.3.2.2 p-FEMs for the Solution of (13.105);377
1.16.3.3;13.3.3 Difficulties in Computing Shadows and Remedies for Several Pathological Cases;378
1.16.3.4;13.3.4 Extracting Complex ESIFs by the QDFM;382
1.16.3.5;13.3.5 Numerical Example: A Crack at the Interface of Two Isotropic Materials;384
1.16.3.6;13.3.6 Numerical Example: CTS, Crack at the Interface of Two Anisotropic Materials;389
1.17;Chapter 14 Remarks on Circular Edges and Open Questions
;394
1.17.1;14.1 Circular Singular Edges in 3-D Domains:The Laplace Equation;394
1.17.1.1;14.1.1 Axisymmetric Case, 0;396
1.17.1.1.1;14.1.1.1 A Specific Example Problem: Penny-Shaped Crackwith Axisymmetric Loading and Homogeneous Neumann BCs;398
1.17.1.1.2;14.1.1.2 A Specific Example Problem: Penny-Shaped Crackwith Axisymmetric Loading and Homogeneous Dirichlet BCs;401
1.17.1.1.3;14.1.1.3 A Specific Example Problem: Circumferential Crackwith Axisymmetric Loading and Homogeneous Neumann BCs;401
1.17.1.2;14.1.2 General Case;402
1.17.1.2.1;14.1.2.1 A Specific Example Problem: Penny-Shaped Crackfor a Nonaxisymmetric Loading and Homogeneous Neumann BCs;404
1.17.1.2.2;14.1.2.2 A Specific Example Problem: Penny-Shaped Crackfor a Nonaxisymmetric Loading and Homogeneous Dirichlet BCs;405
1.17.1.2.3;14.1.2.3 A Specific Example Problem: Hollow Cylinderwith Nonaxisymmetric Loading and Homogeneous Neumann BCs;406
1.17.1.2.4;14.1.2.4 A Specific Example Problem: Exterior Circular Crack wich Nonaxisymmetric Loading and Homogeneous Neumann BCs;406
1.17.2;14.2 Circular Singular Edges in 3-D Domains:The Elasticity System;407
1.17.3;14.3 Further Theoretical and Practical Applications;409
1.18;Appendix A Definition of Sobolev, Energy, and Statically Admissible Spaces and Associated Norms
;411
1.19;Appendix B Analytic Solution to 2-D Scalar Elliptic Problemsin Anisotropic Domains
;416
1.19.1;B.1 Analytic Solution to a 2-D Scalar Elliptic Problemin an Anisotropic Bimaterial Domain;419
1.19.1.1;B.1.1 Treatment of the Boundary Conditions;421
1.19.1.2;B.1.2 An Example;422
1.20;Appendix C Asymptotic Solution at the Intersection of Circular Edges in a 2-D Domain
;426
1.21;Appendix D Proof that Eigenvalues of the Scalar Anisotropic Elliptic BVP with Constant Coefficients Are Real
;432
1.22;Appendix E A Path-Independent Integral and Orthogonalityof Eigenfunctions for General Scalar Elliptic Equationsin 2-D Domains
;435
1.23;Appendix F Energy Release Rate (ERR) Method, its Connectionto the J-integral and Extraction of SIFs
;440
1.23.1;F.1 Derivation of the ERR;440
1.23.1.1;F.1.1 The Energy Argument KeSi95;440
1.23.1.2;F.1.2 The Potential Energy Argument KeSi95;441
1.23.2;F.2 Griffith's Energy Criterion Grif20, Grif24;443
1.23.3;F.3 Relations Between the ERR and the SIFs;449
1.23.3.1;F.3.1 Symmetric (Mode I) Loading;449
1.23.3.2;F.3.2 Antisymmetric (Mode II) Loading;450
1.23.3.3;F.3.3 Combined (Mode I and Mode II) Loading;451
1.23.3.4;F.3.4 Computation of G by the Stiffness Derivative Method;451
1.23.3.5;F.3.5 The Stiffness Derivative Method for 3-D Domains;455
1.23.4;F.4 The J-Integral and its Relation to ERR;455
1.24;References;460
1.25;Index;470