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This book provides the theory of anisotropic elasticity with the computer program for analytical solutions as well as boundary element methods. It covers the elastic analysis of two-dimensional, plate bending, coupled stretching-bending, and three-dimensional deformations, and is extended to the piezoelectric, piezomagnetic, magnetic-electro-elastic, viscoelastic materials, and the ones under thermal environment. The analytical solutions include the solutions for infinite space, half-space, bi-materials, wedges, interface corners, holes, cracks, inclusions, and contact problems. The boundary element solutions include BEMs for two-dimensional anisotropic elastic, piezoelectric, magnetic-electro-elastic, viscoelastic analyses, and their associated dynamic analyses, as well as coupled stretching-bending analysis, contact analysis, and three-dimensional analysis. This book also provides source codes and examples for all the presenting analytical solutions and boundary element methods. The program is named as AEPH (Anisotropic Elastic Plates – Hwu), which contains 204 MATLAB functions.
Chapter 1: Anisotropic Elasticity
1.1 Theory of Elasticity
1.2 Linear Anisotropic Elastic Materials
1.2.1 Three-Dimensional Constitutive Relations
1.2.2 Two-Dimensional Constitutive Relations
1.2.3 Laminate Constitutive Relations
1.3 Thermoelastic Problems
1.4 Piezoelectric Materials
Chapter 2: Complex Variable Formalism
2.1 Two-Dimensional Analysis
2.1.1 Lekhnitskii Formalism
2.1.2 Stroh Formalism2.1.3 Extended Stroh Formalism for Thermoelastic Problems
2.1.4 Expanded Stroh Formalism for Piezoelectric Materials
2.2 Plate Bending Analysis
2.2.1 Lekhnitskii Bending Formalism2.2.2 Stroh-Like Bending Formalism
2.3 Coupled Stretching-Bending Analysis
2.3.1 Stroh-Like Formalism
2.3.2 Extended Stroh-Like Formalism for Thermal Stresses in Laminates
2.3.3 Expanded Stroh-Like Formalism for Electro-Elastic Laminates
2.4 Explicit Expressions
2.4.1 Fundamental Matrix N2.4.2 Material Eigenvector Matrices A and B
2.4.3 Barnett-Lothe Tensors S, H and L
2.5 General Remarks
2.5.1 Degeneracy of Material Eigenvectors
2.5.2 Units, Scaling Factors, and Dimensions
2.5.3 Sign Convention2.5.4 Common Symbols
2.5.5 Extended Symbols
Chapter 3: Computer Program with Matlab
3.1 Program Structures
3.1.1 Computational Procedure
3.1.2 Control Parameters
3.1.3 Global Variables
3.1.4 Input
3.1.5 Output
3.2 Main Program and Functions
3.2.1 Main program
3.2.2 Function Description
3.3 Input and Calculation of Material Properties
3.3.1 Function - elastic
3.3.2 Function - thermal
3.3.3 Function - piezoM
3.4 Calculation of Material Eigenvalues and Eigenvectors
3.4.1 Function - material_eigen
3.4.2 Function - thermal_eigen
3.5 Calculation of Analytical Solutions
3.5.1 Function - internal, positionTime
3.5.2 Function - uphi_bank
3.6 Functions for Double Check
3.6.1 Function – piezo2, piezoM2
3.6.2 Function - fundamental_N
3.6.3 Function - eigen_muAB
3.6.4 Function – identities
3.7 Functions for Output
3.7.1 Function – output_caption
3.7.2 Function - printTF
3.7.3 Function – TableFig, TableFig3D
3.8 Examples
3.8.1 Elastic Properties
3.8.2 Thermal Properties
3.8.3 Piezoelastic Properties
Chapter 4: Infinite Space, Half Space and Bi-materials4.1 Infinite Space
4.1.1 Uniform Load - s411infUL
4.1.2 Inplane Bending - s412infIB
4.1.3 Point Force - s413infPF
4.1.4 Point Moment - s414infPM
4.1.5 Dislocation - s415infDL
4.2 Half Space
4.2.1 Point Force - s421halfPF
4.2.2 Point Force on Surface - s422halfPFs
4.2.3 Distributed Load - s423halfDT
4.2.4 Point Moment - s424halfPM
4.2.5 Dislocation - s425halfDL
4.3 Bi-materials
4.3.1 Point Force and Dislocation - s431bimatPFD
4.3.2 Point Force and Dislocation on the Interface - s432bimatPFDi
4.4 Functions for Common Use
4.4.1 Function - Stroh_matrices
4.4.2 Function - Gauss
4.5 Examples
4.5.1 Infinite Space
4.5.2 Half Space
4.5.3 Bi-materials
Chapter 5: Wedges and Interface Corners
5.1 Uniform Tractions on the Wedge Sides
5.1.1 Non-Critical Wedge Angles
5.1.2 Critical Wedge Angles - s512wedgeUT5.2 Forces at the Wedge Apex
5.2.1 A Single Wedge Under a Point Force - s521wedgePF
5.2.2 A Single Wedge Under a Point Moment - s522wedgePM
5.2.3 Multi-material Wedge Spaces - s523MwedgesPFD
5.2.4 Multi-material Wedges - s524MwedgePF
5.3 Stress Singularities
5.3.1 Multi-Material Wedge Spaces
5.3.2 Multi-Material Wedges
5.3.3 Eigenfunctions - s533MwedgesSOE5.4 Stress Intensity Factors of Interface Corners
5.4.1 Near Tip Solutions
5.4.2 A Unified Definition - s542MwedgeNTP5.4.3 H-Integral for 2D Interface Corners - s543MwedgeSIF2d
5.4.4 H-Integral for 3D Interface Corners - s544MwedgeSIF3d
5.5 Functions for Common Use
5.5.1 Function - multiwedge
5.5.2 Function - muller
5.5.3 Function - s5_ut
5.5.4 Function – MLS
5.6 Examples
5.6.1 A Single Wedge
5.6.2 Multi-Material Wedges
5.6.3 Interface Corners
Chapter 6: Holes
6.1 Elliptical Holes
6.1.1 Uniform Loading - s611EholeUL
6.1.2 Inplane Bending - s612EholeIB
6.1.3 Arbitrary Loading - s613EholeAL
6.1.4 Point Force - s614EholePF
6.1.5 Dislocation - s615EholeDL
6.2 Polygon-like Holes
6.2.1 Transformation
6.2.2 Uniform Loading - s622PholeUL
6.2.3 In-plane Bending - s623PholeIB
6.3 Functions for Common Use6.3.1 Function - mapEP
6.3.2 Function - logBranch
6.4 Examples
6.4.1 Elliptical Holes
6.4.2 Polygon-like Holes
Chapter 7: Cracks
7.1 Near Tip Solutions
7.1.1 Cracks in Homogeneous Materials - s711crackNTS7.1.2 Interfacial Cracks - s712IFcrackNTS
7.1.3 Cracks Terminating at the Interfaces – s713crackTI
7.2 A Finite Straight Crack
7.2.1 Uniform Load at Infinity - s721crackUL
7.2.2 Inplane Bending at Infinity - s722crackIB
7.2.3 Arbitrary Load on the Crack Surfaces - s723crackAL
7.2.4 Point Force at Arbitrary Location - s724crackPF7.2.5 Dislocation at Arbitrary Location - s725crackDL
7.3 Collinear Cracks
7.3.1 General Solutions
7.3.2 Two Collinear Cracks - s732CO2crackUL
7.3.3 Collinear Periodic Cracks - s733COPcrackUL
7.4 Collinear Interface Cracks
7.4.1 General Solutions - s741IFcrack
7.4.2 A Semi-infinite Interface Crack - s742SIFcrackPFs
7.4.3 A Finite Interface Crack - s7431IFcrackPFs, s7432IFcrackUL
7.4.4 Two Collinear Interface Cracks - s744CO2IFcrackUL
7.5 Examples
7.5.1 Near Tip Solutions
7.5.2 A Finite Straight Crack
7.5.3 Collinear Cracks
7.5.4 Collinear Interface Cracks
Chapter 8: Inclusions
8.1 Elliptical Elastic Inclusions
8.1.1 Uniform Loading - s811EEincluUL
8.1.2 Point Force at the Matrix - s812EEincluPFm
8.2 Rigid Inclusions
8.2.1 Elliptical Inclusions - s821_1ERincluUL, s821_2ERincluPF
8.2.2 Line Inclusions - s822_1RLincluUL
8.2.3 Polygon-like Inclusions - s823PRincluUL
8.3 Interactions Between Inclusions and Dislocations
8.3.1 Dislocations Outside the Inclusions - s831EEincluDLo
8.3.2 Dislocations Inside the Inclusions - s832EEincluDLi
8.3.3 Dislocations on the Interfaces - s833EEincluDLf
8.4 Interactions Between Inclusions and Cracks
8.4.1 Cracks Outside the Inclusions - s841EEincluCo
8.4.2 Cracks Inside the Inclusions - s842EEincluCi8.4.3 Cracks Penetrating the Inclusions - s843EEincluCp
8.4.4 Curvilinear Cracks Lying Along the Interfaces - s844EEincluCc8.5 Functions for Common Use
8.5.1 Function – TGCEF
8.5.2 Function – Gauss_elimination8.5.3 Function – s84_CoeffUniform
8.5.4 Function – s84_abcEFG
8.5.5 Function – s84_Kt
8.5.6 Function – s84_F12
8.5.7 Function – s84_Kbeta
8.5.8 Function – s84_uphi
8.6 Examples
8.6.1 Elliptical Elastic Inclusions
8.6.2 Rigid Inclusions
8.6.3 Inclusions and Dislocations
8.6.4 Inclusions and CracksChapter 9: Contact Problems
9.1 Rigid Punches on a Half-Plane
9.1.1 General Solutions
9.1.2 A Flat-Ended Punch Indented by a Load - s912FpunchL9.1.3 A Flat-Ended Punch Tilted by a Moment - s913FpunchM
9.1.4 A Parabolic Punch Indented by a Load - s914PpunchL
9.2 Rigid Stamp Indentation on a Curvilinear Hole Boundary
9.2.1 General Solutions
9.2.2 Elliptical Hole Boundaries - s922Estamp 9.2.3 Polygonal Hole Boundaries - s923Pstamp9.3 Rigid Punches on a Perturbed Surface
9.3.1 Straight Boundary Perturbation
9.3.2 Elliptical Boundary Perturbation
9.3.3 Illustrative Examples – s9331_Cpunch, s933_2Tstamp
9.4 Sliding Punches with or without Friction
9.4.1 General Solutions - s941SpunchGS
9.4.2 A Sliding Wedge-Shaped Punch - s942SWpunch9.4.3 A Sliding Parabolic Punch - s943SPpunch
9.4.4 Two Sliding Flat-Ended Punches - s944S2punch
9.5 Contact Between Two Elastic Bodies
9.5.1 Contact in the Presence of Friction - s951P2Fcontact
9.5.2 Contact in the Absence of Friction - s952P2contact
9.5.3 Contact in Complete Adhesion
9.6 Functions for Common Use
9.6.1 Function – s9_delLam
9.6.2 Function – s9_fzp
9.6.3 Function – s9_uphi
9.6.4 Function – s9_Plemelj
9.7 Examples
9.7.1 Rigid Punches on a Half-Plane
9.7.2 Rigid Stamp Indentation on a Curvilinear Hole Boundary
9.7.3 Rigid Punches on a Perturbed Surface
9.7.4 Sliding Punches with or without Friction
9.7.5 Contact Between Two Elastic Bodies
Chapter 10: Thermoelastic Problems
10.1 Extended Stroh Formalism
10.2 Holes and Cracks
10.2.1 Elliptical Holes under Uniform Heat Flow - s1021EholeUH
10.2.2 Cracks under Uniform Heat Flow - s1022crackUH
10.3 Rigid Inclusions
10.3.1 Elliptical Rigid Inclusions under Uniform Heat Flow - s1031ERincluUH
10.3.2 Rigid Line Inclusions under Uniform Heat Flow - s1032RLincluUH
10.4 Collinear Interface Cracks
10.4.1 General Solutions
10.4.2 Uniform Heat Flow - s1042IFcrackUH
10.5 Multi-Material Wedges
10.5.1 Stress and Heat Flux Singularity10.5.2 Near Tip Solutions - s1052MwedgeTH
10.6 Function for Common Use
10.6.1 Function – s10_gamma
10.7 Examples
10.7.1 Holes, Cracks and Inclusions
10.7.2 Multi-Material Wedges
Chapter 11: Piezoelectric and Magneto-Electro-Elastic Materials
11.1 Constitutive Laws
11.1.1 Piezoelectric Materials
11.1.2 Magneto-Electro-Elastic Materials – MEE, MEExy, MEE3Dto2D
11.2 Expanded Stroh Formalism
11.2.1 Piezoelectric Materials
11.2.2 Magneto-Electro-Elastic Materials
11.3 Holes
11.3.1 Elliptical Holes – s1131piezoEhole
11.3.2 Polygon-Like Holes - s1132piezoPhole
11.4 Multi-Material Wedges
11.4.1 Orders of Stress/Electric Singularity
11.4.2 Near Tip Solutions
11.4.3 H-integral
11.5 Singular Characteristics of Cracks
11.5.1 Cracks
11.5.2 Interface Cracks
11.6 Some Crack Problems
11.6.1 Cracks - s1161piezoCOcrack
11.6.2 Interface Cracks - s1162piezoIFcrack
11.7 Examples
11.7.1 Holes and Cracks
11.7.2 Multi-Material Wedges
11.7.3 Inclusions
11.7.4 Contact Problems
11.7.5 Thermoelastic Problems
Chapter 12: Viscoelastic Materials
12.1 Linear Anisotropic Viscoelasticity12.1.1 Stroh Formalism in Laplace Domain
12.1.2 Material Eigenvalues and Eigenvectors - visco
12.1.3 Numerical inversion of the Laplace transform - Laplace_inv
12.2 Linear Anisotropic Thermo-Viscoelasticity
12.3 Problems with Viscoelastic Materials - s1221visco, visco_load
12.4 Examples
12.4.1 Holes, Cracks and Inclusions
12.4.2 Wedges and Interface Corners
12.4.3 Contact Problems
Chapter 13: Plate Bending Analysis
13.1 Bending Theory of Anisotropic Plates
13.2 Holes/Inclusions/Cracks
13.2.1 Elliptical Holes - s1321EholeUB
13.2.2 Elliptical Rigid Inclusions - s1322ERincluUB, s1420LAMincluUSB
13.2.3 Cracks - s1323crackUB
13.2.4 Elliptical Elastic Inclusions – s1324EEincluUB, s1423LAMEEincluUSB
13.3 Examples
Chapter 14: Coupled Stretching-Bending Analysis
14.1 Coupled Stretching-Bending Theory of Laminates
14.2 Holes in Laminates
14.2.1 Uniform Stretching and Bending Moments - s1421LAMholeUSB
14.2.2 Uniform Heat Flow - s1422LAMholeUH
14.3 Holes in Electro-Elastic Laminates
14.4 Green’s Functions for Laminates - s1441LAMinfPFM
14.5 Green’s Functions for Laminates with Holes/Cracks
14.5.1 Holes - s1451LAMholePFM
14.5.2 Cracks - s1452LAMcrackPFM
14.6 Green’s Functions for Laminates with Elastic Inclusions
14.6.1 Outside the Inclusions - s1461LAMincluPFMo
14.6.2 Inside the Inclusions - s1462LAMincluPFMi
14.7 Functions for Common Use
14.7.1 Function – s14_mdinf
14.7.2 Function – s14_eck
14.8 Examples
14.8.1 Holes in Laminates14.8.2 Green’s Functions
Chapter 15: Boundary Element Analysis
15.1 An Overview
15.1.1 Boundary Integral Equations15.1.2 Fundamental Solutions – Greenbank
15.1.3 Interpolation Functions
15.1.4 Boundary Element Formulation
15.1.5 Boundary-based Finite Element
15.1.6 Computational Procedure15.1.7 Program Structure – BEMbankB, BEMbankIN
15.2 Fundamental Solutions for Two-Dimensional Anisotropic Elastic Analysis
15.2.1 An Infinite Plane – G1inf2D
15.2.2 A Half Plane – G2half2D
15.2.3 Interfaces – G3interface2D
15.2.4 Holes – G4hole2D
15.2.5 Cracks
15.2.6 Rigid Inclusions – G6Rinclusion2D
15.2.7 Elastic Inclusions – G7Einclusion2D
15.3 Fundamental Solutions for Coupled Stretching-Bending Analysis
15.3.1 An Infinite Laminate – G1infCouple
15.3.2 Holes – G4holeCouple
15.3.3 Cracks
15.3.4 Inclusions – G7inclusionCouple
15.4 Two-Dimensional Anisotropic Elastic Analysis – Basic Version
15.4.1 Mesh Generation of Boundary Element – BEMmesh
15.4.2 Influence Matrices – BEMinfluence, BEMinfluence_YG15.4.3 Computation of Singular Integrals – BEMinfluence_G2
15.4.4 Solutions at the Boundary Nodes – BEM2DelasticB
15.4.5 Solutions at the Internal Points – BEM2DelasticIN, BEMinfluenceIN
15.4.6 Multiple Holes/Cracks/Inclusions - BFEM
15.5 Two-Dimensional Anisotropic Elastic Analysis – Extended Version
15.5.1 Piezoeletric/MEE Analysis
15.5.2 Viscoelastic Analysis – BEM2DviscoB, BEM2DviscoIN, BFEMv
15.5.3 Thermoelastic Analysis – BEMload_thermo, thermal_BEM
15.6 Two-Dimensional Anisotropic Dynamic Analysis
15.6.1 Particular Solutions – BEMload_dynamic
15.6.2 Boundary Element Formulation – BEM_YGMVsplit
15.6.3 Free vibration
15.6.4 Steady-state forced vibration
15.6.5 Transient analysis - BEM2DdynamicB, BEMdynamicIN15.7 Coupled Stretching-Bending Analysis
15.7.1 Boundary Element Formulation –BEMcoupleB, BEMload_couple, BEMinfluence_Cc, BEMinfluence_Yt
15.7.2 Computation of Singular Integrals
15.7.3 Auxiliary Relations for the Multiple Nodes of Corners – BEM_aux
15.7.4 Solutions at the Boundary Nodes – BEMstrainstressB
15.7.5 Solutions at the Internal Points – BEMcoupleIN
15.8 Contact Analysis
15.8.1 Contact of Two Elastic Solids – BEM2Dcontact2ElaB
15.8.2 Indentation by Multiple Rigid Punches – BEM2DcontactMReB
15.8.3 Contact of Viscoelastic Solids - BEM2Dcontact2VisB, BEM2DcontactMRvBc, BEM2DcontactMRvBt
15.8.4 Functions for common use - BEM2Dcontact_BCv, BEM2Dcontact_CCR, BEM2Dcontact_Cstatus, BEM2Dcontact_Dfq, BEM2Dcontact_DT, BEM2Dcontact_localC, BEM2Dcontact_MRB, BEM2Dcontact_ut12, BEM2Dcontact_vtv, BEM2Dcontact_YGtoKf, BEM2DviscoINt
15.9 Three-Dimensional Analysis
15.9.1 Radon-Stroh Formalism - CijkstoCik
15.9.2 Fundamental Solutions - G1inf3D
15.9.3 Boundary Element Formulation - BEM3DelasticB, BEM3DelasticIN,
BEMinfluence3D_YG, BEMstrainstressB3D
15.9.4 Extension to Piezoelectric and MEE materials
15.10 Functions for Common Use
15.10.1 Function - GreenCouple15.10.2 Function - BEM_YGtoVg
15.10.3 Function - CSABD_star
15.10.4 Function - s15_pgzV
15.11 Examples
15.11.1 Two-Dimensional Anisotropic Elastic Analysis
15.11.2 Two-Dimensional Piezoelectric/Viscoelastic/Thermoelastic Analysis
15.11.3 Two-Dimensional Anisotropic Dynamic Analysis
15.11.4 Coupled Stretching-Bending Analysis15.11.5 Contact Analysis
15.11.6 Three-Dimensional Analysis
Appendix A: Numerical Algorithms
A.1 Numerical Integration
A.1.1 Gaussian Quadrature Rule
A.1.2 Weakly Singular Integration - GaussLog
A.1.3 Strongly Singular Integration - GaussInv
A.2 Solving Systems of Linear Equations
A.2.1 Gaussian Elimination
A.3 Finding Zeros of FunctionsA.3.1 Newton’s Method
A.3.2 Muller’s Method
Appendix B: Loops and Vectorization
B.1 Array vs. Matrix Operations
B.2 “for loop” Vectorization
B.3 “if statement” Vectorization
Appendix C: List of Functions
Appendix D: List of Global Variables
Appendix E: List of Input Files
E.1 Input Files for All Cases
E.2 Input Files for Material Properties
E.3 Input files for the Arrangement of Internal Points
E.4 Input Files for Load and Structural Information
E.5 Additional Input Files for BEM
Appendix F: AEPH Source Code
References
Author Index
Subject Index
Chyanbin Hwu obtained his B.S. degree of Civil Engineering from National Taiwan University in 1981, M.S. degree of Power Mechanical Engineering from National Tsing-Hua University in 1985, and Ph.D. degree of engineering mechanics from University of Illinois at Chicago in 1988. He joined Department of Aeronautics and Astronautics, National Cheng Kung University, Taiwan, as an associate professor in 1988, became a full professor in 1992 and a distinguished professor in 2003, and a chair professor since 2008. He was elected to be the president of the Society of Theoretical and Applied Mechanics (R.O.C.) in 2008. He is a fellow of the Aeronautical and Astronautical Society (R.O.C.), and the Society of Theoretical and Applied Mechanics (R.O.C.). He was the recipient of academic award (Ministry of Education, R.O.C.) in 2014, Sun Fang-Duo medal in 2011, and outstanding research awards (National Science Council, R.O.C.) in 1995-1996, 1998-1999, 2002-2004. He served as member of editorial board for International Journal of Solids and Structures in 2000-2005, editorial advisor for Journal of Mechanics of Materials and Structures in 2005-2015, and associated editor for Transactions of JSASS (The Japan Society for Aeronautical and Space Sciences) in 2014-present. He has published 114 referred journal papers, authored 1 book and 3 book chapters, edited 4 books, and presented 151 conference papers. His current research interests include mechanics of composite materials, fracture mechanics, anisotropic elasticity, and nanomaterials.
This book provides the theory of anisotropic elasticity with the computer program for analytical solutions as well as boundary element methods. It covers the elastic analysis of two-dimensional, plate bending, coupled stretching-bending, and three-dimensional deformations, and is extended to the piezoelectric, piezomagnetic, magnetic-electro-elastic, viscoelastic materials, and the ones under thermal environment. The analytical solutions include the solutions for infinite space, half-space, bi-materials, wedges, interface corners, holes, cracks, inclusions, and contact problems. The boundary element solutions include BEMs for two-dimensional anisotropic elastic, piezoelectric, magnetic-electro-elastic, viscoelastic analyses, and their associated dynamic analyses, as well as coupled stretching-bending analysis, contact analysis, and three-dimensional analysis. This book also provides source codes and examples for all the presenting analytical solutions and boundary element methods. The program is named as AEPH (Anisotropic Elastic Plates – Hwu), which contains 204 MATLAB functions.
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