Flow Analysis of Injection Molds

Preface

Notation

I The Current Status of Simulation

1 Introduction

1.1 The Injection Molding Process

1.2 Molding Terminology

1.3 What is Simulation?

1.4 The Challenges for Simulation

1.4.1 Basic Physics of the Process

1.5 Why Simulate Injection Molding?

1.6 How Good is Simulation?

2 Stress and Strain in Fluid Mechanics

2.1 Stress in Fluids

2.1.1 The Stress Tensor

2.1.2 The Extra Stress Tensor

2.1.3 Rate of Strain Tensor

2.2 Newtonian and Non-Newtonian Fluids

2.3 The Generalized Newtonian Fluid

3 Material Properties of Polymers

3.1 Types of Polymers

3.2 Amorphous Polymers

3.3 Semi-Crystalline Polymers

3.4 Overview of Material Properties for Simulation

3.5 Viscosity

3.6 Modeling Viscosity

3.6.1 The Viscosity Function

3.6.2 The Power Law Model

3.6.3 The Carreau Model

3.6.4 The Cross Model

3.6.5 Incorporation of Temperature Effects

3.6.6 The Solidification Problem

3.7 Thermal Properties

3.7.1 Specific Heat Capacity

3.7.2 Thermal Conductivity

3.8 Thermodynamic Relationships

3.8.1 Expansivity and Compressibility

3.9 Pressure-Volume-Temperature (PVT) Data

3.10 Fiber Orientation

3.11 Shrinkage and Warpage

4 Governing Equations

4.1 Introduction

4.2 Mathematical Preliminaries

4.2.1 The Material Derivative

4.2.2 The Gauss Divergence Theorem

4.2.3 Reynolds Transport Theorem

4.2.4 Integration by Parts

4.3 Conservation of Mass

4.4 Conservation of Momentum

4.5 Conservation of Energy

4.5.1 Relating Specific Energy to Temperature

4.5.2 The Energy Equation in Terms of Temperature

4.6 Boundary Conditions

4.6.1 Pressure and Flow Rate Boundary Conditions

4.6.2 Temperature Boundary Conditions

4.6.3 Mold Deformation Boundary Conditions

4.6.3.1 Thin Cavities

4.6.3.2 Long Cores and Mold Inserts

4.7 Fiber-Filled Materials

4.7.1 Fiber Concentration

4.7.2 Jeffery's Equation

4.7.3 A Statistical Approach

4.7.4 Mechanical Properties

4.8 Shrinkage and Warpage

4.9 Runners

5 Approximations for Injection Molding

5.1 Introduction

5.2 Material Property Approximations

5.3 Filling, Packing, and Cooling Analysis

5.3.1 The Thermal Source Term in the Energy Equation

5.3.2 Viscosity Modeling

5.3.3 Specific Heat Capacity

5.3.4 Thermal Conductivity

5.3.4.1 Unfilled Amorphous

5.3.4.2 Unfilled Semi-Crystalline

5.3.4.3 Filled Materials

5.3.5 No-Flow or Transition Temperature

5.3.6 Pressure-Volume-Temperature (PVT) Data

5.3.7 Fiber Orientation, Shrinkage, and Warpage

5.3.7.1 Fiber Orientation Analysis

5.3.7.2 Shrinkage and Warpage Analysis

5.4 Summary of Material Assumptions

5.5 Governing Equations

5.6 The 2.5D Approximation

5.6.1 Governing Equations in Cartesian Coordinates

5.6.1.1 Conservation of Mass

5.6.1.2 Conservation of Momentum

5.6.1.3 Conservation of Energy

5.6.2 Estimation of Relevant Terms

5.6.3 Velocity in the z Direction

5.6.4 Integration of the Momentum Equations

5.6.5 Integration of the Continuity Equation

5.6.5.1 Summary of the 2.5D Approximation

5.7 Mold Cooling Analysis

5.8 Fiber Orientation

5.8.1 Orientation Tensors

5.8.2 Folgar-Tucker Equation

5.8.3 Closure Approximations

5.8.3.1 Linear Closure

5.8.3.2 Quadratic Closure

5.8.3.3 Hybrid Closure

5.8.3.4 Orthotropic Closure

5.8.3.5 The Interaction Coefficient

5.9 Shrinkage and Warpage

5.9.1 Shrinkage Prediction

5.9.1.1 Residual Strain Methods

5.9.1.2 Residual Stress Models

5.10 The 2.5D Approximation for Runners

5.10.1 Conservation of Mass for Runners

5.10.2 Conservation of Momentum for Runners

5.10.3 Conservation of Energy for Runners

5.10.4 Integration of the Momentum Equation for Runners

5.10.5 Integration of the Continuity Equation for Runners

6 Numerical Methods for Solution

6.1 Midplane Methods

6.1.1 Extraction of a Midplane from a 3D Model

6.1.2 Dual Domain Analysis for Flow

6.1.3 Dual Domain Structural Analysis

6.1.4 Warpage Analysis Using the Dual Domain FEM

6.2 3D Analysis

6.2.1 Finite Volume Methods

6.2.2 A Pseudo-3D Approach

6.3 Warpage and Shrinkage Analysis in 3D

6.4 3D Analysis of Runner Systems

II Improving Molding Simulation

7 Improved Fiber Orientation Modeling

7.1 Introduction

7.2 ARD Model

7.2.1 Evolution Equation

7.2.2 Direct Simulation

7.2.3 Calculation of CI

7.3 RSC Model

7.4 Suspension Rheology

7.5 Brownian Dynamics Simulation

8 Improved Mechanical Property Modeling

8.1 Introduction

8.2 Unidirectional Composites

8.2.1 Effective Stiffness

8.2.2 Effective Thermal Expansion Coefficients

8.2.3 Effects of Fiber Concentration and Aspect Ratio

8.2.3.1 Effect of Fiber Concentration

8.2.3.2 Effect of Fiber Aspect Ratio

8.3 Fiber Orientation Averaging

9 Long Fiber-Filled Materials

9.1 Fiber Orientation Evolution Model

9.2 Flow-Induced Fiber Migration Model

9.3 Fiber Length Attrition Model

9.4 Uniaxial Tensile Strength Model

9.5 Flexible Fiber Modeling

9.5.1 Direct Simulation Methods

9.5.2 Continuum Modeling

10 Crystallization

10.1 Quiescent Crystallization

10.1.1 The Kolmogoroff-Avrami-Evans Model

10.1.2 The Rate Equations of Schneider

10.1.3 Quiescent Nuclei Number Density

10.1.4 Growth Rate of Spherulites

10.1.5 Material Characterization

10.1.5.1 Half-Crystallization Time

10.1.5.2 Equilibrium Melting Temperature

10.1.5.3 Crystal Growth Rate

10.2 Flow-Induced Crystallization

10.2.1 Enhanced Nucleation

10.2.2 Critical Parameters

10.2.3 Shish-Kebab Structure

10.2.4 Material Characterization

11 Effects of Crystallization on Rheology and Thermal Properties

11.1 Effects of Crystallization on Rheology

11.1.1 Viscosity-Enhancement-Factor Model

11.1.2 Two-Phase Model

11.2 Effect of Crystallization on PVT

11.3 Effect of Crystallization on Specific Heat Capacity

11.4 Effect of Crystallization on Thermal Conductivity

11.4.1 Non-Fourier Thermal Conduction

11.4.2 Van den Brule's Law for Amorphous Polymers

11.4.3 Extending the Van den Brule Approach to Semi-Crystalline Polymers

11.5 Effect of Crystallization on Heat Transfer

11.5.1 Stefan's Solution

11.5.2 Numerical Solution with Crystallization Kinetics

11.6 Modification to the Hele-Shaw Equation

12 Colorant Effects

12.1 Introduction

12.2 Material Characterization

12.2.1 Morphology

12.2.2 Specific Heat

12.2.3 Half-Crystallization Time

12.2.3.1 Quiescent Crystallization

12.2.3.2 Flow-Induced Crystallization

12.3 Effect on Shrinkage

13 Prediction of Post-Molding Shrinkage and Warpage

13.1 Introduction

13.2 Governing Equations

13.3 Constitutive Equations

13.3.1 Viscoelastic Effect

13.3.2 Thermal Expansion Effect

14 Additional Issues of Injection-Molding Simulation

14.1 Weldlines

14.2 Core Shift

14.3 Non-Conventional Injection Molds

14.3.1 Overmolding

14.3.2 Gas-Assisted Injection Molding

14.3.3 Microcellular Injection Foaming Molding

14.3.4 Micro-Injection Molding

14.4 Viscoelastic Effects

14.4.1 Flow-Induced Residual Stress and Birefringence

14.4.2 Viscoelastic Instability

14.4.3 Viscoelastic Suspensions

14.5 Other Numerical Methods

14.5.1 Molecular Dynamics Simulation

14.5.2 Meshless Methods

15 Epilogue

Appendices

A History of Injection-Molding Simulation

A.1 Early Academic Work on Simulation

A.2 Early Commercial Simulation

A.3 Simulation in the Eighties

A.3.1 Academic Work in the Eighties

A.3.1.1 Mold Filling

A.3.1.2 Mold Cooling

A.3.1.3 Warpage Analysis

A.3.2 Commercial Simulation in the Eighties

A.3.2.1 Codes Developed by Large Industrials and Not for Sale

A.3.2.2 Codes Developed by Large Industrials for Sale in the Marketplace

A.3.2.3 Companies Devoted to Developing and Selling Simulation Codes

A.4 Simulation in the Nineties

A.4.1 Academic Work in the Nineties

A.4.2 Commercial Developments in the Nineties

A.4.2.1 SDRC

A.4.2.2 Moldflow

A.4.2.3 AC Technology/C-MOLD

A.4.2.4 Simcon

A.4.2.5 Sigma Engineering

A.4.2.6 Timon

A.4.2.7 Transvalor

A.4.2.8 CoreTech Systems

A.5 Simulation Science Since 2000

A.5.1 Commercial Developments Since 2000

A.5.1.1 Moldflow

A.5.1.2 Timon

A.5.1.3 CoreTech Systems

A.5.1.4 Autodesk

A.5.2 Note for Students

B Tensor Notation

B.1 Index Notation

B.2 Einstein Summation Convention

B.3 Kronecker Delta

B.4 Alternating Tensor

B.5 Product Operations of Two Tensors

B.6 Transpose Operation

B.7 Transformation of Principal Axes

B.8 Gradient of a Field

B.9 Unit Vector p and Operator /p

B.10 Identities

C Derivation of Fiber Evolution Equations

C.1 The Langevin Equation

C.2 Probability Density Function and Orientation Tensors

C.3 Equations of Change for the Orientation Tensors

C.3.1 Isotropic Rotary Diffusion Model (Folgar-Tucker Model)

C.3.2 Anisotropic Rotary Diffusion Model

D Dimensional Analysis of Governing Equations

D.1 Conservation of Mass

D.2 Conservation of Momentum

D.3 The Energy Equation

D.4 Summary

D.4.1 Conservation of Mass

D.4.2 Conservation of Momentum

D.4.3 Energy Equation

E The Finite Difference Method

E.1 Introduction to the Finite Difference Method

E.1.1 A Simple Example

E.2 Application to Temperature Calculation

E.2.1 Explicit Methods

E.2.1.1 Stability Criteria for Explicit Methods

E.2.2 Implicit Methods

F The Finite Element Method

F.1 Basic Terminology

F.2 The Finite Element Approach

F.2.1 Geometric Modeling of the Solution Domain

F.2.2 Meshing

F.2.3 Derivation of Element Equations

F.2.4 Assembly of Element Equations

F.2.5 Application of Boundary Conditions

F.2.6 Solution of the System Equations

F.2.7 Display of Results

F.3 The Nature of a Finite Element Solution

F.4 Shape Functions

F.5 Approximating Nodal Values

F.5.1 Weighted Residual Methods

F.6 Constraint Equations

F.6.1 Special Case 1: Two Unknowns Equal

F.6.2 Special Case 2: One Known Constraint

F.7 A One-Dimensional Problem Solved Using the FEM

F.7.1 Meshing

F.7.2 Derivation of Element Equations

F.7.3 Assembly

F.7.4 Application of Boundary Conditions

F.7.5 Solution of System Equations

G Numerical Methods for the 2.5D Approximation

G.1 Overview of Solution Process

G.1.1 Numerical Methods

G.2 Finite Element Formulation for the Pressure Field

G.2.1 Interpolation Functions

G.2.2 Area Coordinates

G.3 Finite Element Derivation

G.3.1 Assembly of Element Equations and Solution

G.4 Solution of the Energy Equation

G.4.1 Finite Difference Discretization

G.4.2 Solution of the Conduction Problem

G.4.3 Explicit Method

G.5 Flow Front Advancement

G.6 Runners

H Three-Dimensional FEM for Mold Filling Analysis

H.1 Governing Equations

H.2 Weak Formulations

H.3 Finite Element Matrix Formulations

H.4 Solution Procedures

H.5 Flow-Front Advancement

H.6 Numerical Solution For Temperature Field

I Level Set Method

J Full Form of Mori-Tanaka Model

J.1 Eshelby Tensor Components

J.1.1 Material with Isotropic Matrix and Inclusions

J.1.2 General Anisotropic Materials

J.2 Expanded Mori-Tanaka Equation

J.2.1 Contracted Notation for Stiffness Tensor and Compliance Tensor

J.2.2 Inverse of a Matrix

J.2.3 Expanded Expression of the Mori-Tanaka Equation

Bibliography

Index