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Neil McCartney. Properties for Design of Composite Structures
Properties for Design of Composite Structures. Theory and Implementation Using Software
Contents
List of Figures
List of Tables
Guide
Pages
Preface
About the Companion Website
1 Introduction
Reference
2 Fundamental Relations for Continuum Models
2.1 Introduction
2.2 Vectors
2.3 Tensors
2.3.1 Fourth-order Elasticity Tensors
2.4 Displacement and Velocity Vectors
2.5 Material Time Derivative
2.6 Continuity Equation
2.7 Equations of Motion and Equilibrium
2.8 Energy Balance Equation
2.8.1 Conservative Body Forces
2.9 Equations of State for Hydrostatic Stress States
2.9.1 Global Thermodynamic Relations
2.9.2 Local Thermodynamic Relations
2.10 Strain Tensor
2.11 Field Equations for Infinitesimal Deformations
2.12 Equilibrium Equations
2.13 Strain–Displacement Relations
2.14 Constitutive Equations for Anisotropic Linear Thermoelastic Solids
2.14.1 Isotropic Materials
2.15 Introducing Contracted Notation
2.16 Tensor Transformations
2.17 Transformations of Elastic Constants
2.17.1 Transverse Isotropic and Isotropic Solids
2.17.2 Introducing Familiar Thermoelastic Constants
2.18 Analysis of Bend Deformation
2.18.1 Geometry and Basic Equations
2.18.2 Stress and Displacement Fields
2.18.3 Some Special Cases
2.18.3.1 Four-point Bending Tests
2.18.3.2 Plane Strain Bending
References
3 Maxwell’s Far-field Methodology Applied to the Prediction of Effective Properties of Multiphase Isotropic Particulate Composites
3.1 Introduction
3.2 General Description of Maxwell’s Methodology Applied to Thermal Conductivity
3.2.1 Description of Geometry
3.2.2 Temperature Distribution for an Isolated Sphere Embedded in an Infinite Matrix
3.2.3 Maxwell’s Methodology for Estimating Conductivity
3.3 Bulk Modulus and Thermal Expansion Coefficient. 3.3.1 Spherical Particle Embedded in Infinite Matrix Subject to Pressure and Thermal Loading
3.3.2 Applying Maxwell’s Methodology to Isotropic Multiphase Particulate Composites
3.4 Shear Modulus. 3.4.1 Spherical Particle Embedded in Infinite Matrix Material Subject to Pure Shear Loading
3.4.2 Application of Maxwell’s Methodology
3.5 Summary of Results. 3.5.1 Multiphase Composites
3.5.2 Two-phase Composites
3.6 Bounds for Two-phase Isotropic Composites
3.7 Comparison of Predictions with Known Results
References
4 Maxwell’s Methodology for the Prediction of Effective Properties of Unidirectional Multiphase Fibre-reinforced Composites. Overview:
4.1 Introduction
4.2 General Description of Maxwell’s Methodology Applied to Thermal Conductivity
4.2.1 Temperature Distribution for an Isolated Fibre
4.2.2 Maxwell’s Methodology for Estimating Transverse Conductivity
4.3 The Basic Equations for Thermoelastic Analysis
4.3.1 Properties Defined from Axisymmetric Distributions
4.3.2 Solution for an Isolated Fibre Perfectly Bonded to the Matrix
4.3.3 Solution in the Absence of Fibre
4.3.4 Applying Maxwell’s Approach to Multiphase Fibre Composites
4.4 Axial Shear of Anisotropic Fibres
4.4.1 Solution for an Isolated Fibre Perfectly Bonded to the Matrix
4.4.2 Solution in the Absence of Fibre
4.4.3 Applying Maxwell’s Approach to Multiphase Fibre Composites
4.5 Transverse Shear of Multiphase Fibre Composites
4.5.1 Representation for Displacement Strain and Stress Distributions
4.5.2 Stress Field in the Absence of Fibre
4.5.3 Displacement and Stress Fields in Fibre
4.5.4 Displacement and Stress Fields in Matrix
4.5.5 Applying Maxwell’s Approach to Multiphase Fibre Composites
4.6 Other Effective Elastic Properties for Multiphase Fibre-reinforced Composites
4.7 Relationship between Two-phase and Multiphase Formulae
4.8 Summary of Results for Multiphase Composites
4.9 Results for Two-phase Fibre-reinforced Composites
4.10 Bounds for Two-phase Fibre-reinforced Composites
4.10.1 Thermal Conductivity
4.10.2 Axial Young’s Modulus
4.10.3 Axial Poisson’s Ratio
4.10.4 Transverse Bulk Modulus
4.10.5 Transverse Shear Modulus
4.10.6 Axial Shear Modulus
4.10.7 Axial Thermal Expansion
4.10.8 Transverse Thermal Expansion
4.11 Comparison of Predictions with Known Results
References
5 Reinforcement with Ellipsoidal Inclusions
5.1 Stress-Strain Relations
5.2 Theorems for Mean Strain and Mean Stress
5.2.1 Mean Strain
5.2.2 Mean Stress
5.3 Eshelby Theory for an Isolated Particle
5.4 Isolated Ellipsoidal Inclusion
5.5 Multiple Ellipsoidal Inclusions
5.6 Dilute Approximation
5.7 General Case
5.8 Walpole’s Notation
References
6 Properties of an Undamaged Single Lamina
6.1 Notation for the Properties of a Single Lamina
6.2 Lamina Stress-Strain Relations
6.3 Inverted Form of Lamina Stress-Strain Relations
6.4 Generalised Plane Stress Conditions
6.5 Generalised Plane Strain Conditions
6.6 Extending the Contracted Notation for Tensors
6.7 Thermoelastic Constants for Angled Laminae
6.8 Inverse Approach
6.9 Shear Coupling Parameters and Reduced Stress-Strain Relations
6.10 Mixed Form of Stress-Strain Relations
6.11 Special Case of 0o and 90o Plies
7 Effective Thermoelastic Properties of Undamaged Laminates. Overview:
7.1 Laminate Geometry (Symmetric Laminates)
7.2 Equilibrium Equations
7.3 Interfacial and Boundary Conditions
7.4 Displacement and Strain Distributions
7.5 Effective In-plane Properties for Laminate
7.6 Out-of-Plane Shear Properties
7.7 Combined Stress-Strain Relations
7.8 Stress-Strain Equations for Transverse Isotropic Materials
7.9 Accounting for Bend Deformation (Nonsymmetric Laminates)
7.10 A More Limited Explicit Formulation
7.10.1 The Stress Field
7.10.2 Calculation of Effective Through-thickness Strain
References
8 Energy Balance Approach to Fracture in Anisotropic Elastic Material. Overview:
8.1 Introduction
8.2 Thermodynamics for Isothermal Deformations
8.2.1 Local Energy Balance Equation Based on Helmholtz Energy
8.2.2 Local Energy Balance Equation Based on Gibbs Energy
8.3 Linear Thermoelasticity
8.4 Global Energy Balance Equations
8.5 Energy-based Global Fracture Criteria
8.6 Energy-based Local Fracture Criteria
8.7 Fracture Involving Cohesive Zones
8.8 Isolated Single Crack
8.8.1 Anisotropic Stress-Strain Relations
8.8.2 A Representation for Stress and Displacement Fields
8.8.3 Chebyshev Polynomial Expansion
8.8.4 Traction Distribution on the Crack
8.8.5 Stress and Displacement Fields around the Crack
8.8.6 Displacement Discontinuity across the Crack
8.8.7 Stress Intensity Factors
8.8.8 Integral Representations for the Solution of Matrix Crack Problems
8.8.9 Energy Balance Calculation
8.8.10 Special Case of Long Ply Cracks
8.8.11 Concluding Remarks
References
9 Ply Crack Formation in Symmetric Cross-ply Laminates
9.1 Fundamental Equations and Conditions
9.1.1 Basic Field Equations
9.1.2 Boundary and Interface Conditions
9.1.3 Generalised Plane Strain Conditions
9.2 Solution for Undamaged Laminates
9.3 Shear-lag Theory for Cross-ply Laminates
9.4 Generalised Plane Strain Theory for Cross-ply Laminates
9.4.1 Solution for Ply Cracks
9.5 Calculation of In-plane Thermoelastic Constants for Damaged Laminate
9.5.1 Approach 1
9.5.2 Approach 2
9.6 Through-thickness Properties of Damaged Laminates
9.7 Consideration of Ply Crack Closure
9.7.1 Uniaxial Loading in Axial Direction
9.7.2 Uniaxial Loading in In-plane Transverse Direction
9.7.3 Uniaxial Loading in Through-thickness Direction
9.7.4 Derivation of Important Interrelationships
9.7.5 An Alternative Derivation of Interrelations
9.7.5.1 Uniaxial Loading in Axial Direction
9.7.5.2 Uniaxial Loading in In-plane Transverse Direction
9.7.5.3 Uniaxial Loading in Through-thickness Direction
9.7.6 An Observation
9.8 Example Predictions
References
10 Theoretical Basis for a Model of Ply Cracking in General Symmetric Laminates
10.1 Introduction
10.2 Geometry and Basic Field Equations
10.3 Boundary Conditions for Uniformly Cracked Laminates
10.4 Generalised Plane Strain Conditions
10.5 Reduced Stress-Strain Relations for a Cracked Laminate
10.6 Interrelationships for Thermoelastic Constants
10.6.1 Ply Crack Closure for Constrained Uniaxial Loading in Axial Direction
10.6.2 Ply Crack Closure for Constrained Uniaxial Loading in In-plane Transverse Direction
10.6.3 Ply Crack Closure for Constrained Uniaxial Loading in Through-thickness Direction
10.6.4 Useful Independent Interrelationships
10.7 Predicting Crack Formation under Fixed Applied Stresses
10.8 Accounting for Nonuniform Cracking during Ply Crack Simulation
10.9 Progressive Ply Cracking
References
11 Ply Cracking in Cross-ply Laminates Subject to Biaxial Bending
11.1 Introduction
11.2 Geometry and Basic Equations
11.3 In-plane Transverse Loading and Bending
11.4 Interfacial and Boundary Conditions
11.5 Effective Stress-Strain Relations
11.6 Reduced Stress-Strain Relations for Constrained Triaxial Loading
11.7 Ply Crack Closure for Uniaxial Loading
11.7.1 Useful Independent Relationships
11.8 Energy for a Cracked Laminate Subject to Biaxial Bending
11.9 Predicting First Ply Cracking
11.10 Progressive Cracking
11.10.1 Ply Crack Formation during Simple Bending
11.10.2 Simple Bending with Thermal Residual Stresses
11.11 An Alternative Approximate Approach to Ply Cracking
References
12 Energy-based Delamination Theory for Biaxial Loading in the Presence of Thermal Stresses
12.1 Introduction
12.2 Geometry and Mode of Loading
12.3 Undamaged Laminates
12.4 Consideration of Crack Closure
12.5 Analysis for Unconstrained Conditions
12.5.1 Bonded Region of Laminate
12.5.2 Debonded Region of Laminate
12.5.3 Self-similar Region
12.5.4 Laminate Stress-Strain Relations
12.6 Analysis for Generalised Plane Strain Conditions
12.6.1 Bonded Region of Laminate
12.6.2 Debonded Region of Laminate
12.6.3 Self-similar Region
12.6.4 Laminate Stress-Strain Relations
12.7 Calculation of the Gibbs Energy
12.8 Calculation of Energy Release Rates for Delamination Growth
12.8.1 Unconstrained Delamination (σA, σT Fixed)
12.8.2 Generalised Plane Strain (σA, σT Fixed)
12.8.3 Constrained Delamination (εA, εT Fixed)
12.9 Results
12.10 Discussion
References
13 Energy Methods for Fatigue Damage Modelling of Laminates
13.1 Introduction
13.2 Defining Preexisting Damage
13.3 Fatigue Crack Growth Laws for Cracks in Homogeneous Anisotropic Materials
13.4 Ply Crack Instability Criteria for Monotonic Loading
13.5 Stress Intensity Factors and Energy Release Rates for Long Ply Cracks
13.6 Determination of Crack Bridging Parameters
13.7 Alternative Method of Calculating Energy Release Rates
13.8 Parameters Defining Cyclic Crack Tip Deformation
13.9 Fatigue Crack Growth Law for Ply Cracks in Cross-ply Laminates
13.10 Predicting Fatigue Damage and Property Degradation (First-order Model)
13.11 Material Properties for Example Simulations
13.12 Relationship of the Model to Experimental Data
13.13 Discussion
References
Note: Modelling reversed plasticity using the Dugdale model
14 Model of Composite Degradation Due to Environmental Damage
14.1 Introduction
14.2 Model Geometry
14.3 Basic Mechanics for the Parallel Bar Model of a Composite
14.4 Accounting for Defect Growth
14.5 Prediction of Maximum load
14.6 Prediction of Progressive Damage
14.7 Predicting the Failure Stress and Time to Failure
14.8 Predicting Residual Strength
14.9 Example Results
14.10 Conclusion
References
15 Maxwell’s Far-field Methodology Predicting Elastic Properties of Multiphase Composites Reinforced with Aligned Transversely Isotropic Spheroids
15.1 Introduction
15.2 General Description of Maxwell’s Methodology Applied to Spheroidal Inclusions
15.2.1 Description of Geometry
15.2.2 Maxwell’s Methodology for Estimating Elastic Constants
15.3 Isolated Spheroidal Inclusion
15.4 Far-field Displacement Distribution
15.5 Estimating Shear Properties
15.6 Far-field Solution for Nonshear Case
15.7 Solving for Parameters Defining Properties of the Effective Medium
15.7.1 Uniaxial Axial Loading
15.7.2 Plane-strain Equibiaxial Transverse Loading
15.7.3 Defining a Soluble Set of Nonlinear Algebraic Equations
15.8 Determination of Effective Composite Properties
15.9 Composites Reinforced with Isotropic Spherical Inclusions
15.10 Composites Reinforced with Aligned Transversely Isotropic Cylindrical Fibres
15.11 Discussion of Results
References
NOTE: This chapter is based on the publication
16 Debonding Models and Application to Fibre Fractures and Matrix Cracks
16.1 Introduction
16.2 Field Equations
16.3 Interfacial and Radial Boundary Conditions
16.4 Shear-lag Theory
16.4.1 Fibre Fractures
16.4.2 Matrix Cracks
16.5 More Accurate Stress-transfer Model
16.6 Determination of the Integration Functions
16.7 Derivation of Differential Equation for a Perfectly Bonded Interface
16.8 The Average Axial Displacement Functions
16.9 Axial Boundary Conditions
16.10 Perfectly Bonded Fibre/Matrix Interfaces
16.10.1 Fibre Fractures
16.10.2 Matrix Cracks
16.11 Frictionally Slipping Interfaces with Uniform Interfacial Shear Stress
16.11.1 A Simplified Model
16.11.2 An Improved Stress-transfer Model for Interface Debonding
16.12 Solution for a Debonded Interface with Coulomb Friction
16.13 Example Predictions for Carbon-fibre Composites
16.14 Prediction of Matrix Cracking
16.14.1 Solution of the Bridged Crack Problem
16.14.2 Special Case of Long Matrix Cracks
16.14.3 Consideration of Matrix Cracking for Perfectly Bonded Fibre/Matrix Interfaces
16.15 Conclusion
References
17 Interacting Bridged Ply Cracks in a Cross-ply Laminate
17.1 Introduction
17.2 Crack-bridging for Long Isolated Cracks in Cross-ply Laminates
17.3 Method of Solution for Isolated Bridged Cracks of Any Length
17.3.1 Special Case of Long Ply Cracks
17.4 Multiple Crack Problems
17.4.1 Stress, Displacement Fields and Stress Intensity Factors
17.4.2 A Uniform Preexisting Stress Distribution with Stress-free Cracks
17.4.2.1 Method Using Orthogonality
17.4.2.2 Collocation Method
17.4.3 An Arbitrary Preexisting Stress Distribution with Bridged Cracks
17.4.3.1 Method Using Orthogonality
17.4.3.2 Collocation Method
17.5 Numerical Results
17.5.1 Stress-free Cracks in an Isotropic Plate
17.5.2 Bridged Cracks
References
18 Theoretical Basis for a Model of Ply Cracking in General Symmetric Laminates
18.1 Introduction
18.2 Geometry and Basic Field Equations
18.3 Edge Boundary Conditions for Uniformly Cracked Laminates
18.4 Generalised Plane Strain Conditions
18.5 Solution for Undamaged Laminates
18.6 Stress and Displacement Fields for Cracked Laminates
18.7 Averaged Boundary Conditions and Stress-Strain Relations
18.8 Calculation of Thermoelastic Constants for Cracked Laminate
18.9 General Description of the Homogenisation Approach
18.9.1 Geometry and Basic Field Equations
18.9.2 Development of Homogenisation Procedure
Defining the thermoelastic constants of a damaged laminate
References
19 Stress-transfer Mechanics for Biaxial Bending
19.1 Introduction
19.2 Representation for Stress and Displacement Fields
19.2.1 The Stress Field
19.2.2 The Displacement Field
19.2.3 The Recurrence Relations
19.3 Derivation of Differential Equations
19.3.1 Averaging
19.3.2 The Integrated Moments
19.3.3 Additional Expressions Involving the Interfacial Displacements
19.3.4 Solving the Recurrence Relations
19.4 Application of the Boundary Conditions
19.5 Determination of Effective Constants for Undamaged and Damaged Laminates
References
Appendix A: Solution for Shear of Isolated Spherical Particle in an Infinite Matrix. 1. Spherical Shell Subject to Pure Shear Loading
2. Deformation and Stress Fields in a Spherical Particle
3. Deformation and Stress Fields in an Infinite Matrix
4. Isolated Spherical Particle Embedded in an Infinite Matrix
References
Appendix B: Elasticity Analysis of Two Concentric Cylinders
Separation Condition for a Sliding Interface
Appendix C: Gibbs Energy per Unit Volume for a Cracked Laminate
1. In-plane Biaxial Plus Shear Deformation
2. Bend Deformation
Appendix D: Crack Closure Conditions for Laminates
1. Crack Closure Analysis for Cross-ply Laminates (in the Presence of Bending)
2. Crack Closure Analysis for General Symmetric Laminates (in the Absence of Bending)
3. Crack Closure Analysis for Cross-ply Laminates with Delaminations but Without Bending
3.1 Uniaxial Axial Loading
3.2 Uniaxial in-plane Transverse Loading
Appendix E: Derivation of the Solution of Nonlinear Equations
Appendix F: Analysis for Transversely Isotropic Cylindrical Inclusions
Appendix G: Recurrence Relations, Differential Equations and Boundary Conditions. 1. Recurrence Relations
2. Derivation of Differential Equations
3. Calculation of the Coefficients
4. Boundary Conditions for the Differential Equations
Appendix H: Solution of Differential Equations
1. The Auxiliary/Characteristic Equation
2. Series Solution of the Differential Equations
Appendix I: Energy Balance Equation for Delamination Growth
Appendix J: Derivation of Energy-based Fracture Criterion for Bridged Cracks
Appendix K: Numerical Solution of Integral Equations for Bridged Cracks
1. Nonlinear Case
2. Linear Case
Index
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