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Loading...Injection Molding – Integration of Theory and Modeling Methods
Rong Zheng , Roger I. Tanner , Xi-Jun Fan
Contents
1 Introduction 1
1.1 Injection Molding . 1
1.1.1 Injection Molding Machine . 1
1.1.2 Injection Molding Cycle . 1
1.1.3 Feed System 3
1.1.4 The Need for Rheological Information . 3
1.2 Classification of Polymers . 4
1.3 Rheological Characterization of Polymer Fluids 5
1.3.1 Subject and Goals 5
1.3.2 Some Rheological Phenomena . 6
1.3.3 Solidification Rheology 8
1.4 Development of Numerical Simulations
for Injection Molding . 9
2 Fundamentals of Rheology 11
2.1 Introduction to Basic Concepts 11
2.1.1 Kinematics . 11
2.1.2 Viscometric and Extensional Flows 13
2.1.3 Conservation Equations 14
2.2 Constitutive Equations 17
2.2.1 Newtonian Fluids 17
2.2.2 Generalized Newtonian Fluids . 18
2.2.3 Linear Viscoelastic Models . 19
2.2.4 Viscoelastic Fluid Models 20
2.3 Time–Temperature Superposition 28
2.4 The Pressure–Volume–Temperature (PVT) Relation . 32
2.5 Lubrication Approximation . 33
3 Mold Filling and Post Filling 35
3.1 Hele-Shaw Equation . 35
3.1.1 Flow in Thin Cavity of Arbitrary
In-plane Dimensions 35
ix3.1.2 Axisymmetric Flow in a Tube . 41
3.2 Frozen Layer . 41
3.3 Mold Deformation . 42
3.4 Wall Slip . 44
4 Crystallization . 47
4.1 Introduction . 47
4.2 Crystallization Kinetics . 48
4.2.1 The Kolmogoroff-Avrami Model . 48
4.2.2 Growth Rate 49
4.2.3 Nuclei Number Density 51
4.2.4 Molecular Orientation . 55
4.3 Effect of Crystallization on Physical Properties . 56
4.3.1 Effect of Crystallization on Rheology 56
4.3.2 Effect of Crystallization on
Pressure–Volume–Temperature Relations 58
4.3.3 Effect of Crystallization on Thermal Conductivity . 59
4.4 Influence of Colorants 60
4.5 Molecular Dynamics Simulation . 63
5 Flow-Induced Alignment in Short-Fiber Reinforced Polymers 65
5.1 Concentration Regimes of Fiber Suspensions 65
5.2 Evolution Equations 66
5.2.1 Jeffery’s Orbit 66
5.2.2 Orientation Characterization . 67
5.2.3 Fiber–Fiber Interactions 68
5.3 Closure Approximations . 70
5.3.1 Linear Closure 70
5.3.2 Quadratic Closure 71
5.3.3 Hybrid Closure 71
5.3.4 Composite Closure . 71
5.3.5 Orthotropic Fitted Closure 71
5.3.6 Natural Closure 73
5.3.7 Invariant-Based Optimal Fitting (IBOF) Closure . 73
5.4 Interaction Coefficient 74
5.5 Modifications to Folgar–Tucker Model 75
5.5.1 Anisotropic Rotary Diffusion Model . 75
5.5.2 Reduced-Strain Closure Model . 76
5.6 Rheological Equations for Fiber Suspensions 77
5.6.1 Transversely Isotropic Fluid (TIF) Model . 77
5.6.2 Dinh–Armstrong Model 78
5.6.3 Phan-Thien–Graham Model . 78
5.7 Tucker’s Flow Classification for Fiber Suspension
in Thin Cavities 79
x Contents5.8 Fiber Migration in Inhomogeneous Flow Fields 80
5.9 Brownian Dynamics Simulation . 81
5.10 Non-Newtonian Matrix Suspensions . 83
6 Shrinkage and Warpage 87
6.1 Introduction . 87
6.2 Mechanical and Thermal Properties of Short-Fiber
Composites 89
6.2.1 Effective Stiffness Tensor of Unidirectional
Composites 89
6.2.2 Effective Thermal Expansion Coefficients
of Unidirectional Composites 93
6.2.3 Orientation Averaging . 94
6.3 Thermally and Pressure-Induced Stresses . 95
6.3.1 Stress Development . 95
6.3.2 Viscous-Elastic Model and Viscoelastic Model 97
6.3.3 Assumptions and Boundary Conditions . 100
6.4 Displacement Calculation 101
6.5 Empirical Approach 102
6.6 Corner Deformation 103
7 Mold Cooling 105
7.1 Mold Cooling System 105
7.2 Transient Heat Transfer in Mold . 105
7.3 Cycle-Average Simplification . 108
8 Computational Techniques 111
8.1 Introduction . 111
8.2 Flow Analyses . 112
8.2.1 Midplane Approach . 112
8.2.2 Advancement of the Flow Front 117
8.2.3 Fountain Flow Effect 122
8.2.4 Dual Domain Approach 124
8.2.5 Three-Dimensional Finite Element Method 126
8.2.6 Smoothed Particle Hydrodynamics (SPH) Method . 131
8.3 Structural Analysis for Shrinkage and Warpage Prediction . 134
8.3.1 Shell Finite Elements . 134
8.3.2 Dual-Domain Structural Analysis . 135
8.3.3 3D Structural Analysis . 137
8.4 Boundary Element Method for Mold Cooling Analysis . 138
8.4.1 Transient Mold Cooling 138
8.4.2 Steady-State Mold Cooling . 143
8.4.3 Modified Boundary Integral Equations
for Closely Spaced Surfaces . 143
Contents xi8.4.4 Boundary Discretization 145
8.5 Overall Conclusion 147
Appendix A 149
Appendix B 153
Appendix C 159
References 163
Author Index 179
Subject Index
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