FINITE ELEMENT ANALYSIS OF COMPOSITE MATERIALS

Transcription

1 FINITE ELEMENT ANALYSIS OF COMPOSITE MATERIALS Ever J. Barbero Department of Mechanical and Aerospace Engineering West Virginia University USA CRC Press Taylor &.Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business

2 Contents Preface Acknowledgments List of Symbols List of Examples xiii xvii xix xxv 1 Mechanics of Orthotropic Materials Material Coordinate System Displacements Strain Stress Contracted Notation Alternate Contracted Notation Equilibrium and Virtual Work Boundary Conditions Traction Boundary Conditions Free Surface Boundary Conditions Continuity Conditions Traction Continuity Displacement Continuity Compatibility Coordinate Transformations Stress Transformation Strain Transformation Transformation of Constitutive Equations D Constitutive Equations Anisotropic Material Monoclinic Material Orthotropic Material Transversely Isotropic Material Isotropic Material Engineering Constants 25 vn

3 viii Finite Element Analysis of Composite Materials Restrictions on Engineering Constants From 3D to Plane Stress Equations Apparent Laminate Properties 31 Suggested Problems 32 References 34 2 Introduction to the Finite Element Method Basic FEM procedure Discretization Element Equations Approximation over an Element Interpolation Functions Element Equations for a Specific Problem Assembly of Element Equations Boundary Conditions Solution of the Equations Solution Inside the Elements Derived Results General FEM Procedure FE Analysis with CAE systems Pre-process: Model Generation Model Geometry Load States Boundary Conditions Loads Solution Procedure Post-process: Analysis and Results Visualization 55 Suggested Problems 57 References 58 3 Elasticity and Strength of Laminates Kinematic of Shells First-Order Shear Deformation Theory Kirchhoff Theory FE Analysis of Laminates Shell Element Types in FE codes A-B-D-H Input Data for Laminate FEA Equivalent Orthotropic Input for Laminate FEA LSS for Multidirectional Laminate FEA FEA of Ply Drop-Off Laminates FEA of Sandwich Shells Element Coordinate System Failure Criteria D Failure Criteria for Unidirectional Laminae D Failure Criteria 97

4 Table of Contents ix Suggested Problems 103 References Buckling Bifurcation Methods Imperfection Sensitivity Asymmetric Bifurcation Post-Critical Path Continuation Methods 115 Suggested Problems 118 References Free Edge Stresses Poisson's Mismatch Interlaminar Force Interlaminar Moment Coefficient of Mutual Influence Interlaminar Stress due to Mutual Influence 131 Suggested Problems 135 References Computational Micromechanics Analytical Homogenization Reuss Model Voigt Model Periodic Microstructure Model Transversely Isotropic Averaging Numerical Homogenization Local-Global Analysis Laminated RVE 162 Suggested Problems 165 References Viscoelasticity Viscoelastic Models Maxwell Model Kelvin Model Maxwell-Kelvin Model Power Law Prony Series Generalized Kelvin Model Nonlinear Power Law Boltzmann Superposition Linear Viscoelastic Material Unaging Viscoelastic Material 175

5 x Finite Element Analysis of Composite Materials 7.3 Correspondence Principle Frequency Domain Spectrum Representation Micromechanics of Viscoelastic Composites One-Dimensional Case Three-Dimensional Case Macromechanics of Viscoelastic Composites Balanced Symmetric Laminates General Laminates FEA of Viscoelastic Composites 185 Suggested Problems 188 References Damage Mechanics One-Dimensional Damage Mechanics Damage Variable Damage Threshold and Activation Function Kinetic Equation Statistical Interpretation of the Kinetic Equation One-Dimensional Random-Strength Model Fiber-Direction, Tension Damage Fiber-Direction, Compression Damage Multidimensional Damage and Effective Spaces Thermodynamics Formulation First Law Second Law Kinetic Law in Three-Dimensional Space Return-Mapping Algorithm Damage and Plasticity 225 Suggested Problems 227 References A Damage Model for Fiber Reinforced Composites Theoretical Formulation Damage and Effective Spaces Thermodynamic Formulation Damage and Plastic Strain Evolution Functions Numerical Implementation Model Identification Damage Surface Shear Coefficients Damage Surface Normal Coefficients Plastic-Strain Surface Hardening Functions Laminate Damage 251

6 Table of Contents xi References Delaminations Two-Dimensional Delamination Energy Release Rate (ERR) Modes of Fracture Crack Propagation Delamination in Composite Plates Sublaminate Modeling Delamination Modeling Unilateral Contact and Damaging Interface ERR-Interface Model Mixed Mode Analysis 282 Suggested Problems 291 References 293 A Tensor Algebra 299 A.l Principal Directions of Stress and Strain 299 A.2 Tensor Symmetry 299 A.3 Matrix Representation of a Tensor 300 A.4 Double Contraction 301 A.5 Tensor Inversion 301 A.6 Tensor Differentiation 302 A.6.1 Derivative of a Tensor With Respect to Itself 302 A.6.2 Derivative of the Inverse of a Tensor With Respect to the Tensor 303 В Strain Concentration Tensors 305 С Second-Order Diagonal Damage Models 309 C.l Effective and Damaged Spaces 309 C.2 Thermodynamic Force Y 310 C.3 Damage Surface 312 C.4 Unrecoverable-Strain Surface 313 D Numerical Inverse Laplace Transform 315 E Introduction to the Software Interface 319 E.l ANSYS 319 E.1.1 ANSYS USERMAT, Compilation and Execution 321 E.2 BMI3 322 E.2.1 Stand Alone BMI3 322 E.2.2 BMI3 within ANSYS 322 References 324 Index 325