Mathematical Methods and Models in Composites

Description

Mathematical Methods and Models in Composites (Second Edition) provides an in-depth treatment of modern and rigorous mathematical methods and models applied to composites modeling on the micro-, meso-, and macro scale. There has been a steady growth in the diversity of such methods and models that are used in the analysis and characterization of composites, their behavior, and their associated phenomena and processes. This second edition expands upon the success of the first edition, and has been substantially revised and updated.Written by well-known experts in different areas of applied mathematics, physics, and composite engineering, this book is mainly focused on continuous fiber reinforced composites and their ever increasing range of applications (for example, in the aerospace industry), though it also covers other kind of composites. The chapters cover a range of topics including, but not limited to: scaling and homogenization procedures in composites, thin plate and wave solutions in anisotropic materials, laminated structures, fiber-reinforced nonlinearly elastic solids, buckling and postbuckling, fracture and damage analysis of composites, and highly efficient methods for simulation of composites manufacturing such as resin transfer molding. The results presented are useful for the design, fabrication, testing and industrial applications of composite components and structures.This book is an essential reference for graduate and doctoral students, as well as researchers in mathematics, physics and composite engineering. Explanations and references in the book are sufficiently detailed so as to provide the necessary background to further investigate the fascinating subject of composites modeling and explore relevant research literature. It is also suitable for non-experts who wish to have an overview of the mathematical methods and models used for composites, and of the open problems in this area that require further research. Contents Preface About the Editor 1. Micromechanical Modeling of Advanced Composites and Smart Composite Structures Using the Asymptotic Homogenization Method 1.1. Introduction 1.2. Asymptotic Homogenization Method 1.3. Unit-Cell Problems 1.4. Three-Dimensional Grid-Reinforced Composites 1.4.1. Examples of 3D grid-reinforced composite structures 1.5. Asymptotic Homogenization of Thin-Walled Composite Reinforced Structures 1.6. Generally Orthotropic Grid-Reinforced Composite Shell 1.6.1. Calculation of the effective elastic coefficients 1.7. Examples of Grid-Reinforced Composite Shells with Orthotropic Reinforcements 1.8. Sandwich Composite Shells with Cellular Cores 1.8.1. Examples of sandwich shells 1.9. Smart Composite Materials and Structures 1.9.1. Asymptotic homogenization of 3D smart composite materials 1.9.2. Asymptotic homogenization of smart composite shells and plates 1.10. Conclusion References 2. Scaling Functions in Spatially Random Composites 2.1. Introduction 2.2. Conductivity of Random Polycrystals 2.2.1. The Hill–Mandel condition 2.2.2. Bounds on the conductivity 2.2.3. Scaling function in heat conduction 2.2.4. Some properties of and bounds on the scaling function 2.2.5. Numerical simulations 2.2.6. Constructing the scaling function 2.3. Conductivity of Planar Random Checkerboards 2.3.1. Governing equations 2.4. Elastic Properties of Random Polycrystals 2.4.1. The Hill–Mandel condition 2.4.2. Bounds on the elastic response 2.4.3. Elastic scaling function 2.5. Elastic Properties of Planar Random Checkerboards 2.6. Scaling in Inelastic and Nonlinear Materials 2.6.1. Thermoelasticity 2.6.2. Elasto-plasticity 2.6.3. Finite elasticity 2.6.4. Permeability of porous media 2.6.5. Comparative numerical results 2.7. Conclusions Acknowledgments References 3. Stroh-Like Formalism for General Thin-Laminated Plates and Its Applications 3.1. Introduction 3.2. Stroh-Like Formalism 3.3. Extended Stroh-Like Formalism — Hygrothermal Stresses 3.4. Expanded Stroh-Like Formalism — Electro-Elastic Laminates 3.5. Holes and Cracks 3.5.1. Holes in laminates under uniform stretching and bending moments 3.5.2. Holes in laminates under uniform heat flow and moisture transfer 3.5.3. Holes in electro-elastic laminates under uniform loads and charges 3.5.4. Cracks in laminates 3.6. Numerical Examples 3.6.1. Holes 3.6.2. Thermal environment 3.6.3. Electro-elastic coupling 3.6.4. Cracks 3.7. Conclusions Acknowledgments References 4. Classical, Refined, Zig-Zag, Layer-Wise Models and Best Theory Diagrams for Laminated Structures 4.1. Introduction 4.2. Who First Proposed a Zig-Zag Theory? 4.3. The Lekhnitskii–Ren Theory 4.4. The Ambartsumian–Whitney–Rath–Das Theory 4.5. The Reissner–Murakami–Carrera Theory 4.6. Remarks on the Theories 4.7. A Brief Discussion on Layer-Wise Theories 4.8. Best Theory Diagrams via the Axiomatic/Asymptotic Method 4.8.1. The axiomatic/asymptotic method 4.8.2. The Best Theory Diagram 4.9. CUF Shell Finite Elements 4.9.1. Geometry of cylindrical shells 4.9.2. MITC method 4.9.3. Governing equations 4.10. Numerical Examples 4.11. Best Theory Diagrams 4.11.1. Plates 4.11.2. Shells 4.12. Conclusions References 5. A Modeling Framework for the Analysis of Instabilities and Delamination in Composite Shells 5.1. Introduction 5.1.1. Review of shell formulations 5.1.2. Finite element formulations for shells 5.1.3. Instabilities in thin-walled composite engineering systems 5.1.4. Overview 5.2. Shell Formulation: 7-Parameter Model 5.2.1. Differential geometry and fundamental equations 5.2.2. Three-dimensional shell parametrization 5.2.3. Solid shell parametrization 5.3. Constitutive Formulations for the Shell 5.3.1. Layered composite shells 5.3.2. Functionally graded isotropic shells 5.4. Cohesive Interface for Large Deformation Analysis 5.5. Computational Framework and Finite Element Formulation 5.5.1. Variational basis 5.5.2. Shell finite element discretization 5.5.2.1. Displacement formulation supplemented by EAS 5.5.2.2. Interpolation of the incompatible strains 5.5.2.3. The ANS method 5.5.3. Interface finite element discretization 5.6. Representative Applications 5.6.1. Postbuckling analysis of composite stiffened panel 5.6.2. Wrinkling–delamination analysis of composite systems 5.7. Concluding Remarks Acknowledgments References 6. Bifurcation of Elastic Multilayers 6.1. Introduction 6.2. Notations and Governing Equations 6.3. Uniaxial Tension/Compression of an Elastic Multilayer 6.3.1. Equations for a layer 6.3.1.1. Traction free at the external surface of the multilayer 6.3.1.2. Bonding to an elastic half-space at the external surface of the multilayer 6.3.1.3. Bonding to an undeformable substrate at the external surface of the multilayer 6.3.1.4. Bonding to an undeformable substrate with a compliant interface at the external surface of the multilayer 6.3.2. Bifurcation criterion 6.3.3. Results and discussion 6.3.3.1. Layer bonded to a half-space 6.3.3.2. Periodic multilayered structures 6.4. Bending of Elastic Multilayers with Imperfect Interfaces 6.4.1. Kinematics 6.4.2. Stress 6.4.3. Incremental bifurcations superimposed on finite bending of an elastic multilayered structure 6.4.4. An example: Bifurcation of a bilayer 6.4.5. Experiments on coated and uncoated rubber blocks under bending 6.5. Conclusions Acknowledgments References 7. Instabilities Associated with Loss of Ellipticity in Fiber-Reinforced Nonlinearly Elastic Solids 7.1. Introduction 7.2. Compressible Materials in Three Dimensions 7.2.1. Kinematics 7.2.2. Elasticity 7.2.3. Equilibrium and ellipticity considerations 7.2.4. Reinforcing models 7.3. Specialization to Plane Strain Deformations 7.3.1. Ellipticity of the reinforcing model F(I4) 7.3.1.1. Failure of ellipticity 7.3.2. Ellipticity of a hybrid reinforcing model 7.4. Incompressible Materials 7.4.1. Plane strain 7.5. Strong Discontinuities 7.5.1. Piecewise homogeneous deformation gradients 7.5.2. Energy considerations 7.5.3. Illustrations 7.5.3.1. Example 1: An orthogonal kink 7.5.3.2. Example 2: A weak kink 7.6. Concluding Remarks References 8. Propagation of Rayleigh Waves in Anisotropic Media and an Inverse Problem in the Characterization of Initial Stress 8.1. Introduction 8.2. Basic Elasticity in Anisotropic Materials with Initial Stress 8.3. The Stroh Formalism for Dynamic Elasticity 8.4. Rayleigh Waves in Anisotropic Materials 8.5. Perturbation of the Phase Velocity of Rayleigh Waves in Prestressed Anisotropic Media When the Base Material is Orthotropic 8.6. An Inverse Problem on Recovery of Initial Stress 8.7. Perturbation of the Polarization Ratio of Rayleigh Waves in Prestressed Anisotropic Media When the Base Material is Orthotropic Acknowledgments References 9. Advanced Mathematical Models and Efficient Numerical Simulation in Composite Processes 9.1. Introduction 9.2. Reinforced Polymers 9.2.1. Fiber suspensions in Newtonian fluids 9.2.2. Fiber suspensions in non-Newtonian fluids 9.3. Multi-Physics in Laminates 9.3.1. PGD at a glance 9.3.2. Heat transfer in laminates 9.3.2.1. Computing R(x) from S(z) 9.3.2.2. Computing S(z) from R(x) 9.3.3. 3D RTM 9.3.4. The elastic problem defined in plate domains 9.3.5. 3D elastic problem in a shell domain 9.3.5.1. Shell representation 9.3.5.2. Weak form 9.3.5.3. In-plane–out-of-plane separated representation 9.3.6. Squeeze flow in composite laminates 9.3.6.1. Stokes model 9.3.6.2. Power-law fluid 9.3.6.3. Brinkman’s model 9.3.6.4. Squeeze flow of multiaxial laminates 9.3.7. Electromagnetic models in laminates 9.4. Coupled Physics at Interfaces 9.4.1. Surface representation 9.4.2. High-resolution numerical solution 9.4.3. Surface evolution during the in-situ consolidation 9.4.4. Consolidation simulation strategy 9.5. Conclusions References 10. Modeling Fracture and Complex Crack Networks in Laminated Composites 10.1. Introduction: Damage Idealization and Scale 10.2. Crack Initiation and Propagation 10.2.1. Linear elastic fracture of composites 10.2.2. Cohesive laws 10.2.2.1. Length of the fracture process zone 10.2.2.2. Size effects 10.2.2.3. Softening law and the R-curve effect 10.2.2.4. Mixed-mode cohesive laws 10.3. Continuum Representation of Material Response 10.3.1. Distributed damage vs. localization of fracture 10.3.2. Idealization of damage modes in composite materials 10.3.3. Failure criteria and strength 10.3.4. Crack tunneling and in situ strength 10.3.5. Continuum damage models for composite materials 10.4. CDM: Limitations 10.5. Bridging the Gap between DDM and CDM 10.6. Regularized x-FEM (Rx-FEM) Framework 10.6.1. Matrix crack modeling using Rx-FEM 10.7. Rx-FEM Simulations 10.7.1. Transverse crack tension test 10.7.2. Effect of ply thickness 10.7.3. Internal delamination vs. edge delamination 10.8. Conclusions Acknowledgments References 11. Delamination and Adhesive Contacts, Their Mathematical Modeling and Numerical Treatment 11.1. Introduction 11.2. Concepts in Quasistatic Rate-Independent Evolution 11.3. Mathematical Concepts to Solve the System (11.4) 11.3.1. General weak solutions — Local solutions 11.3.2. Weak solutions conserving energy: Energetic solutions 11.3.3. Weak solutions of stress-driven types 11.4. Quasistatic Brittle Delamination, The Griffith Concept 11.5. Elastic-Brittle Delamination 11.5.1. The model and its asymptotics to brittle delamination 11.5.2. Numerical implementation 11.5.3. Illustrative examples 11.6. Various Refinements and Enhancements 11.6.1. Cohesive contacts 11.6.2. Delamination in Modes I, II and mixed modes 11.6.3. Multi-threshold delamination 11.6.4. Combinations with other inelastic processes in bulk 11.6.5. Another inelastic process on the surface: Friction 11.6.6. Dynamical adhesive contact in visco-elastic materials 11.6.7. Thermodynamics of adhesive contacts 11.7. Applications to Fiber-Reinforced Composites 11.8. Conclusion Acknowledgments References 12. Interaction of Cracks with Interfaces 12.1. Introduction 12.2. The Coupled Criterion 12.3. Matched Asymptotic Expansions 12.4. Application to the Crack Onset at a V-notch in a Homogeneous Material 12.5. Application to the Deflection of Transverse Cracks 12.6. The Cook and Gordon Mechanism 12.7. The Interface Crack Growing Along the Interface — Delamination 12.8. The Crack Kinking out of the Interface 12.9. The Interface Corner 12.10. Conclusion References 13. Computational Procedure for Singularity Analysis of Anisotropic Elastic Multimaterial Corners — Applications to Composites and Their Joints 13.1. Introduction 13.2. Lekhnitskii–Stroh Formalism for Linear Elastic Anisotropic Materials 13.2.1. Basic equations 13.2.2. Sextic eigen-relation: Stroh orthogonality and closure relations 13.2.3. Representation of displacement and stress function vectors 13.3. Elastic Multimaterial Corner 13.3.1. Corner configuration 13.3.2. Boundary and interface conditions: Matrix formalism 13.3.2.1. Coordinate systems 13.3.2.2. Boundary condition matrices 13.3.2.3. Interface condition matrices 13.4. Singular Elastic Solution in a Single-Material Wedge: Transfer Matrix 13.4.1. Non-degenerate materials 13.4.2. Degenerate materials 13.4.3. Extraordinary degenerate materials 13.5. Characteristic System for the Singularity Analysis of an Elastic Multimaterial Corner 13.5.1. Transfer matrix for a multimaterial wedge 13.5.2. Characteristic system assembly 13.5.2.1. Open multimaterial corner 13.5.2.2. Closed multimaterial corner (periodic corner) 13.5.3. Solution of the characteristic system — Singular elastic solution 13.6. Evaluation of GSIFs 13.6.1. Least-squares fitting technique 13.6.2. Implementation, accuracy and robustness 13.7. Examples of Singularity Analysis 13.7.1. Transverse crack terminating at the interface in a [0/90]S laminate 13.7.2. Bimaterial corner in an adhesively bonded double-lap joint 13.7.2.1. Singularity analysis of a closed corner 13.7.2.2. Singularity analysis of a corner including an interface crack with sliding friction contact 13.8. Failure Criterion for a Multimaterial Closed Corner Based on Generalized Fracture Toughness 13.9. Removal of Stress Singularities in Bimaterial Joints 13.9.1. Tensile and shear strength in bimaterial samples 13.9.2. Removal or reduction of stress singularities associated to tabs bonding in standard composite testing 13.9.2.1. Compression test of thick composite laminates 13.9.2.2. Off-axis test of unidirectional composite materials 13.10. Conclusions Acknowledgments References Index