Anisotropic Elasticity with Matlab

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

Chapter 2: Complex Variable Formalism

2.1 Two-Dimensional Analysis

2.1.1 Lekhnitskii Formalism

2.1.3 Extended Stroh Formalism for Thermoelastic Problems

2.1.4 Expanded Stroh Formalism for Piezoelectric Materials

2.2 Plate Bending Analysis

2.2.2 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

2.4.2 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

2.5.4 Common Symbols

2.5.5 Extended Symbols

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

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

Chapter 5: Wedges and Interface Corners

5.1 Uniform Tractions on the Wedge Sides

5.1.1 Non-Critical Wedge Angles

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 - 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

5.4 Stress Intensity Factors of Interface Corners

5.4.1 Near Tip Solutions

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 A Single Wedge

5.6.2 Multi-Material Wedges

5.6.3 Interface Corners

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

6.3.1 Function - mapEP

6.3.2 Function - logBranch

6.4 Examples

6.4.1 Elliptical Holes

6.4.2 Polygon-like Holes

7.1 Near Tip Solutions

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 Inplane Bending at Infinity - s722crackIB

          7.2.3 Arbitrary Load on the Crack Surfaces - s723crackAL

          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 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