Anisotropic Elasticity with Matlab

Preface
	Reference
Contents
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
	References
2 Complex Variable Formalism
	2.1 Two-Dimensional Analysis
		2.1.1 Lekhnitskii Formalism
		2.1.2 Stroh Formalism
		2.1.3 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 Formalism
		2.2.2 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 Elasticity Matrix N
		2.4.2 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 Convention
		2.5.4 Common Symbols
		2.5.5 Extended Symbols
	References
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 Piezoelectric Properties
	References
4 Infinite Space, Half Space and Bi-materials
	4.1 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
	References
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—s512wedgeUT
	5.2 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—s523MwedgePFD
		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—s533MwedgeSOE
	5.4 Stress Intensity Factors
		5.4.1 Near Tip Field Solutions
		5.4.2 Unified Definition—s542MwedgeNTS
		5.4.3 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 Single Wedge
		5.6.2 Multi-material Wedges
		5.6.3 Interface Corners
	References
6 Holes
	6.1 Elliptical Holes
		6.1.1 Uniform Load at Infinity—s611EholeUL
		6.1.2 In-plane Bending at Infinity—s612EholeIB
		6.1.3 Arbitrary Load Along the Hole Boundary—s613EholeAL
		6.1.4 Point Force at an Arbitrary Location—s614EholePF
		6.1.5 Dislocation at an Arbitrary Location—s615EholeDL
	6.2 Polygon-Like Holes
		6.2.1 Transformation Function
		6.2.2 Uniform Load at Infinity—s622PholeUL
		6.2.3 In-plane Bending at Infinity—s623PholeIB
	6.3 Functions for Common Use
		6.3.1 Function—mapEP
		6.3.2 Function—logBranch
	6.4 Examples
		6.4.1 Elliptical Holes
		6.4.2 Polygon-Like Holes
	References
7 Cracks
	7.1 Singular Characteristics of Cracks
		7.1.1 Cracks in Homogeneous Materials—s711crackNTS
		7.1.2 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 In-plane Bending at Infinity—s722crackIB
		7.2.3 Arbitrary Load on the Crack Surfaces—s723crackAL
		7.2.4 Point Force at Arbitrary Location—s724crackPF
		7.2.5 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 Semi-infinite Interface Crack—s742SIFcrackPFs
		7.4.3 Finite Interface Crack—s743_1IFcrackPFs, S743_2IFcrackUL
		7.4.4 Two Collinear Interface Cracks—s744CO2IFcrackUL
	7.5 Examples
		7.5.1 Near Tip Solutions
		7.5.2 Finite Straight Crack
		7.5.3 Collinear Cracks
		7.5.4 Collinear Interface Cracks
	References
8 Inclusions
	8.1 Elliptical Elastic Inclusions
		8.1.1 Uniform Load at Infinity—s811EEincluUL
		8.1.2 A Point Force at the Matrix—s812EEincluPFm
	8.2 Rigid Inclusions
		8.2.1 Elliptical Rigid Inclusions—s821_1ERincluUL, s821_2ERincluPF
		8.2.2 Rigid Line Inclusions—s822_1RLincluUL
		8.2.3 Polygon-Like Rigid 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—s842EEincluCi
		8.4.3 Cracks Penetrating the Inclusions—s843EEincluCp
		8.4.4 Curvilinear Cracks Lying Along the Interfaces—s844EEincluCc
	8.5 Functions for Common Use
		8.5.1 Function—TGCEF
		8.5.2 Function—Gauss_elimination
		8.5.3 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 Cracks
	References
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—s912FpunchL
		9.1.3 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—s923Pstamp
	9.3 Rigid Punches on a Perturbed Surface
		9.3.1 Straight Boundary Perturbation
		9.3.2 Elliptical Boundary Perturbation
		9.3.3 Illustrative Examples—s933_1Cpunch, s933_2Tstamp
	9.4 Sliding Punches with or without Friction
		9.4.1 General Solutions
		9.4.2 A Sliding Wedge-Shaped Punch—s942SWpunch
		9.4.3 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
	References
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 Singularity
		10.5.2 Eigenfunctions—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
	References
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 Field 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
	References
12 Viscoelastic Materials
	12.1 Linear Anisotropic Viscoelasticity
		12.1.1 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
	References
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
	References
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 Inclusion—s1461LAMincluPFMo
		14.6.2 Inside the Inclusion—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 Laminates
		14.8.2 Green’s Functions
	References
15 Boundary Element Analysis
	15.1 An Overview
		15.1.1 Boundary Integral Equations
		15.1.2 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 Procedure
		15.1.7 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_YG
		15.4.3 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, BEM2DdynamicIN
	15.7 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—GreenCouple
		15.10.2 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 Analysis
		15.11.5 Contact Analysis
		15.11.6 Three-Dimensional Analysis
	References
Appendix_1
	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 Functions
		A.3.1 Newton’s Method
		A.3.2 Muller’s Method
A.3.2 Muller’s Method
	B.1 Array Versus Matrix Operations
	B.2 “for loop” Vectorization
	B.3 “if statement” Vectorization
B.3 “if statement” Vectorization
B.3 “if statement” Vectorization
B.3 “if statement” Vectorization
	E.1 Input Files for All Cases (Sect. 3.1.4)
	E.2 Input Files for Material Properties (Sect. 3.1.4)
	E.3 Input Files for the Arrangement of Internal Points (Sect. 3.1.4)
	E.4 Input Files for Load and Structural Information (Sects. from 4.1.1 to 15.1.7)
	E.5 Additional Input Files for BEM (Sect. 15.1.7)
E.5 Additional Input Files for BEM (Sect. 15.1.7)
Author Index
Subject Index