Table of Contents:

Preface

Acknowledgment

List of Symbols

List of Examples

1 Mechanics of Orthotropic Materials

1.1 Material Coordinate System           

1.2 Displacements               

1.3 Strain                 

1.4 Stress                 

1.5 Contracted Notation             

1.5.1 Alternate Contracted Notation        

1.6 Equilibrium and Virtual Work          

1.7 Boundary Conditions             

1.7.1 Traction Boundary Conditions        

1.7.2 Free Surface Boundary Conditions       

1.8 Continuity Conditions             

1.8.1 Traction Continuity           

1.8.2 Displacement Continuity          

1.9 Compatibility               

1.10 Coordinate Transformations           

1.10.1 Stress Transformation           

1.10.2 Strain Transformation           

1.11 Transformation of Constitutive Equations       

1.12 3D Constitutive Equations            

1.12.1 Anisotropic Material           

1.12.2 Monoclinic Material           

1.12.3 Orthotropic Material           

1.12.4 Transversely Isotropic Material         

1.12.5 Isotropic Material            

1.13 Engineering Constants             

1.13.1 Restrictions on Engineering Constants      

1.14 From 3D to Plane Stress Equations         

1.15 Apparent Laminate Properties          

Suggested Problems                

References                  

2 Introduction to the Finite Element Method

2.1 Basic FEM procedure             

2.1.1 Discretization             

2.1.2 Element Equations            

2.1.3 Approximation over an Element        

2.1.4 Interpolation Functions          

2.1.5 Element Equations for an Specific Problem     

2.1.6 Assembly of Element Equations        

2.1.7 Boundary Conditions           

2.1.8 Solution of the Equations          

2.1.9 Solution Inside the Elements         

2.1.10 Derived Results             

2.2 General FEM Procedure            

2.3 FE Analysis with CAE systems          

2.3.1 Pre-process: Model Generation        

2.3.2 Model Geometry            

2.3.3 Load States              

2.3.4 Boundary Conditions            

2.3.5 Loads               

2.3.6 Solution Procedure            

2.3.7 Post-process: Analysis and Results Visualization   

Suggested Problems               

References                  

3 Elasticity and Strength of Laminates

3.1 Kinematic of Shells              

3.1.1 First Order Shear Deformation Theory      

3.1.2 Kirchhoff Theory            

3.2 FE Analysis of Laminates            

3.2.1 Shell Element Types in FE codes        

3.2.2 A-B-D-H Input Data for Laminate FEA      

3.2.3 Equivalent Orthotropic Input for Laminate FEA   

3.2.4 LSS for Multidirectional Laminates FEA     

3.2.5 FEA of Ply Drop-off Laminates        

3.2.6 FEA of Sandwich Shells          

3.2.7 Element Coordinate System         

3.3 Failure Criteria                

3.3.1 2D Failure Criteria for Unidirectional Laminae    

3.3.2 3D Failure Criteria            

Suggested Problems               

References                  

4 Buckling

4.1 Bifurcation Methods             

4.1.1 Imperfection Sensitivity           

4.1.2 Asymmetric Bifurcation          

4.1.3 Post Critical Path            

4.2 Continuation Methods             

Suggested Problems               

References                  

5 Free Edge Stresses

5.1 Poisson's Mismatch              

5.1.1 Interlaminar Force            

5.1.2 Interlaminar Moment           

5.2 Coeffcient of Mutual Influence          

5.2.1 Interlaminar Stress due to Mutual Influence     

Suggested Problems               

References                  

6 Computational Micromechanics

6.1 Analytical Homogenization           

6.1.1 Reuss Model             

6.1.2 Voigt Model              

6.1.3 Periodic Microstructure Model        

6.1.4 Transversely Isotropic Averaging        

6.2 Numerical Homogenization           

6.3 Local-global Analysis             

6.4 Laminated RVE               

Suggested Problems               

References                  

7 Viscoelasticity 167

7.1 Viscoelastic Models              

7.1.1 Maxwell Model             

7.1.2 Kelvin Model              

7.1.3 Maxwell-Kelvin Model           

7.1.4 Power Law              

7.1.5 Prony Series             

7.1.6 Generalized Kelvin Model          

7.1.7 Non-linear Power Law           

7.2 Boltzmann Superposition            

7.2.1 Linear Viscoelastic Material         

7.2.2 Unaging Viscoelastic Material         

7.3 Correspondence Principle            

7.4 Frequency Domain              

7.5 Spectrum Representation            

7.6 Micromechanics of Viscoelastic Composites       

7.6.1 One-dimensional case           

7.6.2 Three-dimensional case          

7.7 Macro-mechanics of Viscoelastic Composites       

7.7.1 Balanced Symmetric Laminates        

7.7.2 General Laminates            

7.8 FEA of Viscoelastic Composites          

Suggested Problems               

References                  

8 Damage Mechanics

8.1 One-dimensional Damage Mechanics         

8.1.1 Damage Variable            

8.1.2 Damage Threshold and Activation Function     

8.1.3 Kinetic Equation            

8.1.4 Statistical Interpretation of the Kinetic Equation   

8.1.5 One-dimensional Random-strength Model     

8.1.6 Fiber-direction, Tension Damage        

8.1.7 Fiber-direction, Compression Damage      

8.2 Multi-dimensional Damage and Effective Spaces      

8.3 Thermodynamics Formulation           

8.3.1 First Law              

8.3.2 Second Law              

8.4 Kinetic Law in Three-dimensional Space        

8.4.1 Return-mapping Algorithm         

8.5 Damage and Plasticity             

Suggested Problems                

References                  

9 A Damage Model for Fiber Reinforced Composites

9.1 Theoretical Formulation            

9.1.1 Damage and Effective Spaces         

9.1.2 Thermodynamic formulation         

9.1.3 Damage and Plastic Strain          

9.1.4 Evolution Laws             

9.2 Numerical Implementation            

9.3 Model Identification             

9.3.1 Damage Surface Shear Coefficients       

9.3.2 Damage Surface Normal Coefficients       

9.3.3 Plastic-strain surface            

9.3.4 Hardening laws             

9.4 Laminate Damage              

Suggested Problems

References                                  

10 Delaminations

10.1 Two-dimensional Delamination          

10.1.1 Energy Release Rate (ERR)         

10.1.2 Modes of Fracture            

10.1.3 Crack Propagation            

10.2 Delamination in Composite Plates         

10.2.1 Sublaminate Modeling           

10.2.2 Delamination Modeling          

10.2.3 Unilateral Contact and Damaging Interface     

10.2.4 ERR-Interface Model           

10.2.5 Mixed Mode Analysis           

Suggested Problems               

References                  

A Tensor Algebra

A.1 Principal Directions of Stress and Strain        

A.2 Tensor Symmetry              

A.3 Matrix Representation of a Tensor         

A.4 Inner-product Tensor Multiplication         

A.5 Tensor Inversion              

A.6 Tensor Differentiation             

A.6.1 Derivative of a tensor with respect to itself     

A.6.2 Derivative of the Inverse of a Tensor with respect to the Tensor

B Strain Concentration Tensors

C Second Order Diagonal Damage Models

C.1 Effective and Damaged Spaces          

C.2 Thermodynamic force Y            

C.3 Damage Surface               

C.4 Unrecoverable-strain Surface           

D Numerical Inverse Laplace Transform

E Introduction to the Software Interface

E.1 ANSYS                 

E.1.1 ANSYS USERMAT, Compilation and Execution   

E.2 BMI3                 

E.2.1 Stand Alone BMI3            

E.2.2 BMI3 within ANSYS           

References