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 |  موضوع: كتاب Flow Analysis of Injection Molds السبت 02 نوفمبر 2019, 4:33 pm | |
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أخوانى فى الله
أحضرت لكم كتاب
Flow Analysis of Injection Molds
2nd Edition
Peter Kennedy, Rong Zheng
و المحتوى كما يلي :
Contents
Preface . IX
Notation XXI
I The Current Status of Simulation 1
1 Introduction 3
1.1 The Injection Molding Process 3
1.2 Molding Terminology . 4
1.3 What is Simulation? . 5
1.4 The Challenges for Simulation . 6
1.4.1 Basic Physics of the Process 7
1.5 Why Simulate Injection Molding? . 7
1.6 How Good is Simulation? . 8
2 Stress and Strain in Fluid Mechanics . 11
2.1 Stress in Fluids . 11
2.1.1 The Stress Tensor 11
2.1.2 The Extra Stress Tensor . 14
2.1.3 Rate of Strain Tensor 14
2.2 Newtonian and Non-Newtonian Fluids 15
2.3 The Generalized Newtonian Fluid 16
3 Material Properties of Polymers . 19
3.1 Types of Polymers . 19
3.2 Amorphous Polymers . 20
3.3 Semi-Crystalline Polymers . 20
3.4 Overview of Material Properties for Simulation . 21
3.5 Viscosity 22
3.6 Modeling Viscosity 23
3.6.1 The Viscosity Function 23XII Contents
3.6.2 The Power Law Model . 23
3.6.3 The Carreau Model 23
3.6.4 The Cross Model . 24
3.6.5 Incorporation of Temperature Effects 24
3.6.6 The Solidification Problem . 25
3.7 Thermal Properties 26
3.7.1 Specific Heat Capacity 26
3.7.2 Thermal Conductivity . 27
3.8 Thermodynamic Relationships 29
3.8.1 Expansivity and Compressibility 29
3.9 Pressure-Volume-Temperature (PVT) Data 31
3.10 Fiber Orientation 31
3.11 Shrinkage and Warpage . 32
4 Governing Equations 35
4.1 Introduction 35
4.2 Mathematical Preliminaries . 35
4.2.1 The Material Derivative . 35
4.2.2 The Gauss Divergence Theorem . 36
4.2.3 Reynolds Transport Theorem 37
4.2.4 Integration by Parts 37
4.3 Conservation of Mass . 38
4.4 Conservation of Momentum . 38
4.5 Conservation of Energy . 40
4.5.1 Relating Specific Energy to Temperature . 43
4.5.2 The Energy Equation in Terms of Temperature . 45
4.6 Boundary Conditions . 46
4.6.1 Pressure and Flow Rate Boundary Conditions . 47
4.6.2 Temperature Boundary Conditions . 48
4.6.3 Mold Deformation Boundary Conditions 48
4.6.3.1 Thin Cavities 48
4.6.3.2 Long Cores and Mold Inserts . 49
4.7 Fiber-Filled Materials . 49
4.7.1 Fiber Concentration . 50
4.7.2 Jeffery’s Equation 50
4.7.3 A Statistical Approach . 51
4.7.4 Mechanical Properties 52
4.8 Shrinkage and Warpage . 52
4.9 Runners . 53Contents XIII
5 Approximations for Injection Molding 55
5.1 Introduction 55
5.2 Material Property Approximations 56
5.3 Filling, Packing, and Cooling Analysis 56
5.3.1 The Thermal Source Term in the Energy Equation . 57
5.3.2 Viscosity Modeling 57
5.3.3 Specific Heat Capacity 58
5.3.4 Thermal Conductivity . 58
5.3.4.1 Unfilled Amorphous . 58
5.3.4.2 Unfilled Semi-Crystalline . 59
5.3.4.3 Filled Materials . 59
5.3.5 No-Flow or Transition Temperature 59
5.3.6 Pressure-Volume-Temperature (PVT) Data 61
5.3.7 Fiber Orientation, Shrinkage, and Warpage 62
5.3.7.1 Fiber Orientation Analysis 62
5.3.7.2 Shrinkage and Warpage Analysis 63
5.4 Summary of Material Assumptions . 63
5.5 Governing Equations 64
5.6 The 2.5D Approximation . 65
5.6.1 Governing Equations in Cartesian Coordinates . 66
5.6.1.1 Conservation of Mass 66
5.6.1.2 Conservation of Momentum . 68
5.6.1.3 Conservation of Energy . 68
5.6.2 Estimation of Relevant Terms 69
5.6.3 Velocity in the z Direction 71
5.6.4 Integration of the Momentum Equations 72
5.6.5 Integration of the Continuity Equation . 75
5.6.5.1 Summary of the 2.5D Approximation . 77
5.7 Mold Cooling Analysis 78
5.8 Fiber Orientation 80
5.8.1 Orientation Tensors . 80
5.8.2 Folgar-Tucker Equation . 81
5.8.3 Closure Approximations 81
5.8.3.1 Linear Closure 82
5.8.3.2 Quadratic Closure 82
5.8.3.3 Hybrid Closure . 82
5.8.3.4 Orthotropic Closure . 83
5.8.3.5 The Interaction Coefficient . 83XIV Contents
5.9 Shrinkage and Warpage . 84
5.9.1 Shrinkage Prediction 85
5.9.1.1 Residual Strain Methods 85
5.9.1.2 Residual Stress Models 87
5.10 The 2.5D Approximation for Runners 91
5.10.1 Conservation of Mass for Runners 92
5.10.2 Conservation of Momentum for Runners 93
5.10.3 Conservation of Energy for Runners 93
5.10.4 Integration of the Momentum Equation for Runners 94
5.10.5 Integration of the Continuity Equation for Runners 96
6 Numerical Methods for Solution . 99
6.1 Midplane Methods 99
6.1.1 Extraction of a Midplane from a 3D Model 100
6.1.2 Dual Domain Analysis for Flow 101
6.1.3 Dual Domain Structural Analysis 103
6.1.4 Warpage Analysis Using the Dual Domain FEM . 107
6.2 3D Analysis . 107
6.2.1 Finite Volume Methods . 107
6.2.2 A Pseudo-3D Approach . 108
6.3 Warpage and Shrinkage Analysis in 3D . 108
6.4 3D Analysis of Runner Systems 109
II Improving Molding Simulation 111
7 Improved Fiber Orientation Modeling .113
7.1 Introduction 113
7.2 ARD Model . 114
7.2.1 Evolution Equation 114
7.2.2 Direct Simulation 116
7.2.3 Calculation of CI . 117
7.3 RSC Model 117
7.4 Suspension Rheology . 118
7.5 Brownian Dynamics Simulation . 120Contents XV
8 Improved Mechanical Property Modeling .123
8.1 Introduction 123
8.2 Unidirectional Composites . 124
8.2.1 Effective Stiffness 124
8.2.2 Effective Thermal Expansion Coefficients . 126
8.2.3 Effects of Fiber Concentration and Aspect Ratio 126
8.2.3.1 Effect of Fiber Concentration . 126
8.2.3.2 Effect of Fiber Aspect Ratio . 127
8.3 Fiber Orientation Averaging 130
9 Long Fiber-Filled Materials .131
9.1 Fiber Orientation Evolution Model . 131
9.2 Flow-Induced Fiber Migration Model 132
9.3 Fiber Length Attrition Model . 134
9.4 Uniaxial Tensile Strength Model . 135
9.5 Flexible Fiber Modeling . 136
9.5.1 Direct Simulation Methods . 136
9.5.2 Continuum Modeling . 139
10 Crystallization 141
10.1 Quiescent Crystallization . 141
10.1.1 The Kolmogoroff-Avrami-Evans Model . 142
10.1.2 The Rate Equations of Schneider 143
10.1.3 Quiescent Nuclei Number Density 144
10.1.4 Growth Rate of Spherulites . 145
10.1.5 Material Characterization 146
10.1.5.1 Half-Crystallization Time . 146
10.1.5.2 Equilibrium Melting Temperature . 146
10.1.5.3 Crystal Growth Rate 148
10.2 Flow-Induced Crystallization 149
10.2.1 Enhanced Nucleation . 150
10.2.2 Critical Parameters 151
10.2.3 Shish-Kebab Structure 152
10.2.4 Material Characterization 152
11 Effects of Crystallization on Rheology and Thermal Properties 155
11.1 Effects of Crystallization on Rheology 155
11.1.1 Viscosity-Enhancement-Factor Model . 155XVI Contents
11.1.2 Two-Phase Model 157
11.2 Effect of Crystallization on PVT 159
11.3 Effect of Crystallization on Specific Heat Capacity . 160
11.4 Effect of Crystallization on Thermal Conductivity 161
11.4.1 Non-Fourier Thermal Conduction 161
11.4.2 Van den Brule’s Law for Amorphous Polymers 162
11.4.3 Extending the Van den Brule Approach to Semi-Crystalline Polymers . 162
11.5 Effect of Crystallization on Heat Transfer 164
11.5.1 Stefan’s Solution . 164
11.5.2 Numerical Solution with Crystallization Kinetics . 165
11.6 Modification to the Hele-Shaw Equation 166
12 Colorant Effects 167
12.1 Introduction 167
12.2 Material Characterization 168
12.2.1 Morphology 168
12.2.2 Specific Heat 169
12.2.3 Half-Crystallization Time . 169
12.2.3.1 Quiescent Crystallization . 169
12.2.3.2 Flow-Induced Crystallization . 169
12.3 Effect on Shrinkage 171
13 Prediction of Post-Molding Shrinkage and Warpage .175
13.1 Introduction 175
13.2 Governing Equations 176
13.3 Constitutive Equations . 177
13.3.1 Viscoelastic Effect 177
13.3.2 Thermal Expansion Effect 178
14 Additional Issues of Injection-Molding Simulation .181
14.1 Weldlines . 181
14.2 Core Shift . 182
14.3 Non-Conventional Injection Molds . 182
14.3.1 Overmolding . 182
14.3.2 Gas-Assisted Injection Molding 183
14.3.3 Microcellular Injection Foaming Molding . 186
14.3.4 Micro-Injection Molding . 188
14.4 Viscoelastic Effects 191Contents XVII
14.4.1 Flow-Induced Residual Stress and Birefringence . 191
14.4.2 Viscoelastic Instability 193
14.4.3 Viscoelastic Suspensions . 194
14.5 Other Numerical Methods . 196
14.5.1 Molecular Dynamics Simulation 196
14.5.2 Meshless Methods . 197
15 Epilogue .201
Appendices 203
A History of Injection-Molding Simulation 205
A.1 Early Academic Work on Simulation 205
A.2 Early Commercial Simulation . 206
A.3 Simulation in the Eighties 208
A.3.1 Academic Work in the Eighties . 209
A.3.1.1 Mold Filling . 209
A.3.1.2 Mold Cooling . 211
A.3.1.3 Warpage Analysis . 212
A.3.2 Commercial Simulation in the Eighties . 212
A.3.2.1 Codes Developed by Large Industrials and Not for Sale 214
A.3.2.2 Codes Developed by Large Industrials for Sale in the Marketplace 214
A.3.2.3 Companies Devoted to Developing and Selling Simulation Codes 215
A.4 Simulation in the Nineties 216
A.4.1 Academic Work in the Nineties 217
A.4.2 Commercial Developments in the Nineties 218
A.4.2.1 SDRC . 218
A.4.2.2 Moldflow 219
A.4.2.3 AC Technology/C-MOLD . 220
A.4.2.4 Simcon . 220
A.4.2.5 Sigma Engineering . 220
A.4.2.6 Timon 221
A.4.2.7 Transvalor . 221
A.4.2.8 CoreTech Systems 221
A.5 Simulation Science Since 2000 221
A.5.1 Commercial Developments Since 2000 . 223
A.5.1.1 Moldflow 224
A.5.1.2 Timon 225XVIII Contents
A.5.1.3 CoreTech Systems 225
A.5.1.4 Autodesk . 225
A.5.2 Note for Students 225
B Tensor Notation 227
B.1 Index Notation . 227
B.2 Einstein Summation Convention . 228
B.3 Kronecker Delta 229
B.4 Alternating Tensor . 229
B.5 Product Operations of Two Tensors . 230
B.6 Transpose Operation 230
B.7 Transformation of Principal Axes . 231
B.8 Gradient of a Field . 233
B.9 Unit Vector p and Operator ∂/∂p 233
B.10 Identities 234
C Derivation of Fiber Evolution Equations 235
C.1 The Langevin Equation . 235
C.2 Probability Density Function and Orientation Tensors 237
C.3 Equations of Change for the Orientation Tensors . 238
C.3.1 Isotropic Rotary Diffusion Model (Folgar-Tucker Model) 239
C.3.2 Anisotropic Rotary Diffusion Model 241
D Dimensional Analysis of Governing Equations .243
D.1 Conservation of Mass . 244
D.2 Conservation of Momentum . 245
D.3 The Energy Equation 248
D.4 Summary . 250
D.4.1 Conservation of Mass . 250
D.4.2 Conservation of Momentum . 250
D.4.3 Energy Equation . 251
E The Finite Difference Method .253
E.1 Introduction to the Finite Difference Method . 253
E.1.1 A Simple Example . 255
E.2 Application to Temperature Calculation . 257
E.2.1 Explicit Methods . 257
E.2.1.1 Stability Criteria for Explicit Methods . 258
E.2.2 Implicit Methods . 258Contents XIX
F The Finite Element Method 261
F.1 Basic Terminology . 261
F.2 The Finite Element Approach 262
F.2.1 Geometric Modeling of the Solution Domain . 262
F.2.2 Meshing 263
F.2.3 Derivation of Element Equations 263
F.2.4 Assembly of Element Equations . 263
F.2.5 Application of Boundary Conditions . 264
F.2.6 Solution of the System Equations . 264
F.2.7 Display of Results 264
F.3 The Nature of a Finite Element Solution . 265
F.4 Shape Functions . 267
F.5 Approximating Nodal Values . 267
F.5.1 Weighted Residual Methods 268
F.6 Constraint Equations . 268
F.6.1 Special Case 1: Two Unknowns Equal 271
F.6.2 Special Case 2: One Known Constraint . 272
F.7 A One-Dimensional Problem Solved Using the FEM . 273
F.7.1 Meshing . 273
F.7.2 Derivation of Element Equations 274
F.7.3 Assembly 278
F.7.4 Application of Boundary Conditions . 280
F.7.5 Solution of System Equations 281
G Numerical Methods for the 2.5D Approximation 283
G.1 Overview of Solution Process 283
G.1.1 Numerical Methods . 284
G.2 Finite Element Formulation for the Pressure Field . 285
G.2.1 Interpolation Functions 285
G.2.2 Area Coordinates 286
G.3 Finite Element Derivation 287
G.3.1 Assembly of Element Equations and Solution . 295
G.4 Solution of the Energy Equation . 296
G.4.1 Finite Difference Discretization . 296
G.4.2 Solution of the Conduction Problem . 297
G.4.3 Explicit Method 297
G.5 Flow Front Advancement . 298
G.6 Runners . 298XX Contents
H Three-Dimensional FEM for Mold Filling Analysis .303
H.1 Governing Equations 303
H.2 Weak Formulations 304
H.3 Finite Element Matrix Formulations 305
H.4 Solution Procedures . 309
H.5 Flow-Front Advancement 310
H.6 Numerical Solution For Temperature Field 311
I Level Set Method 313
J Full Form of Mori-Tanaka Model .317
J.1 Eshelby Tensor Components . 317
J.1.1 Material with Isotropic Matrix and Inclusions . 317
J.1.2 General Anisotropic Materials . 318
J.2 Expanded Mori-Tanaka Equation . 319
J.2.1 Contracted Notation for Stiffness Tensor and Compliance Tensor . 319
J.2.2 Inverse of a Matrix 319
J.2.3 Expanded Expression of the Mori-Tanaka Equation 320
Bibliography .321
Index .345Notation
Symbols which have more than one meaning are listed with a semicolon dividing the meanings. To avoid being too lengthy and jumbled, not all symbols and their definitions used in the
book are included in the notation list, but they are defined throughout the text.
Roman symbols
a thermal diffusivity k/(ρcp )
ai j second-order fiber orientation tensor
ai jkl fourth-order fiber orientation tensor
ar fiber aspect ratio
aT time-temperature shift factor
A area
b FENE-P model parameter
A (Ai j kl ) strain-concentration tensor
c pseudo-concentration parameter
c
p specific heat capacity under constant pressure
cv specific heat capacity under constant volume
C heat capacity
C g
1 , C2g WLF universal constants (C1g = 17.44, C2g = 51.6 K)
C 0
1 , C20 WLF constants
CB stress-optical coefficient
CI interaction coefficient in Folgar-Tucker model
Ct stress-thermal coefficient
C (Ci j ) interaction coefficient tensor in anisotropic rotary diffusion model
C (Ci jkl ) stiffness tensor (also called elasticity tensor)
d diameter; distance function
D(r ) diffusion coefficient (isotropic)
D (Di j ) rate-of-deformation tensor 1 2 ‡ ∂∂xvij + ∂∂vxij ·
D(r ) (D(r )
i j ) diffusion coefficient tensor
Ea activation energy
E (Ei jkl ) Eshelby tensor
f function
△F f flow-induced free energy changeXXII Notation
△F
q difference of free energies between melt and crystalline phases under quiescent condition
F (Fi ) force vector
g (gi ) acceleration vector due to gravity
G radial growth rate of spherulite; shear modulus
GN melt plateau modulus
H cavity half-thickness
H(t) Heaviside unit step function
Hˆ specific enthalpy
△Hc latent heat of crystallization for perfect crystals
I (δi j ) unit tensor (also called the Kronecker tensor)
I (Ii jkl ) fourth-order unit tensor
J Jocobian of coordinate transformations
k thermal conductivity
kB Boltzmann’s constant (1.380658 × 10−23 J/K)
k (ki j ) thermal conductivity tensor
l length
L (Li j ) velocity gradient tensor ∂vi /∂x j
Mn number-average molecular weight
Mw weight-average molecular weight
n power-law exponent
n0 number of molecules per unit volume
n (ni ) outward-pointing unit normal vector
N nuclei number density
Nf flow-induced nuclei number density
Np
particle number
Nq
quiescent nuclei number density
N0 constant nuclei number density
N (Ni ) particle flux
O(A) mathematical symbol reading as the order of magnitude of A
p pressure
pt thermodynamic pressure
p (pi ) orientation unit vector
q (qi ) heat flux vector; orientation vector q = |q|p
Q heat
R radius
R
g gas constant (8.3143 J / mol·K)
S1 one-dimensional fluidity for 2.5D runner approximationNotation XXIII
S2 two-dimensional fluidity for 2.5D cavity approximation
S3 three-dimensional fluidity for pseudo 3D approximation
S∥ shrinkage parallel to flow direction
S⊥ shrinkage perpendicular to flow direction
Sˆ specific entropy
S (Si j kl ) elastic compliance tensor
t, t′ time
t1/2 half-crystallization time
t (ti ) traction vector (also called stress vector)
T temperature; as superscript denotes transpose of a tensor
Tg
glass transition temperature
T o
m equilibrium melting temperature
u (ui ) velocity vector; unit orientation vector; displacement vector
Uˆ internal specific energy
U ∗ activation energy
v (vi ) velocity vector
V volume
Vˆ specific volume
W i Weissenberg number
W (Wi j ) vorticity tensor 1 2 ‡ ∂∂xvij − ∂∂vxij ·
Greek symbols
α relative crystallinity
α (αi j ) linear thermal expansion coefficient tensor
β coefficient of volume expansion; empirical parameter of some equations
γ shear strain
γ˙ generalized strain rate
γ˙ (γ˙i j ) shear strain rate tensor
δ(t) Dirac delta function (also called impulse function)
δi j Kronecker tensor (also called unit tensor)
ε (εi j ) strain tensor
ζ dimensionless drag coefficient
η shear viscosity
η0 zero shear rate viscosity
λ time constant
µ viscosity
µd dilatational viscosityXXIV Notation
ξ slip parameter 2/(ar2 +1)
ξ(t) pseudo time
ρ mass density
σ surface tension coefficient, tensile strength
σb tensile strength at perfectly bounded interface
σw tensile strength at weldline interface
σ (σi j ) stress tensor
τ (τi j ) extra stress tensor
φ volume fraction
χ absolute crystallinity
χ∞ ultimate absolute crystallinity
ψ probability density
ω angular velocity; frequency
L (Li j ) effective velocity gradient Li j −ξDi j
Operator symbols
D/Dt material derivative ∂/∂t + vk ∂∂xk
△/△t upper upper convected derivative defined as
△( )i j
△t
=
∂( )i j
∂t
+ vk
∂( )i j
∂xk
−Lik ( )k j −L jk ( )kiList of Figures
1.1 A simple two-cavity mold . 5
1.2 A simple two-cavity mold showing runners and gates 6
2.1 Definition of the stress vector 12
2.2 Resolution of stress vector 13
2.3 Stress components . 13
3.1 Steady simple shear flow 22
3.2 In unbalanced flow, some regions of the molding may be in the packing phase
with very low shear rates and high pressures, while material near the flow front is
at low pressure but high shear rate 24
3.3 Definition of thermal conductivity 28
3.4 PVT surface . 30
4.1 Boundary conditions for simulation of filling, packing, and cooling . 47
5.1 Need for a transition or no-flow temperature 60
5.2 Mold for which material is in packing and filling phases . 61
5.3 Thin-walled cavity with coordinates systems defined at two points 66
5.4 Definition of frozen layer thickness . 72
5.5 Schematic representations of fiber orientation distributions (a) fully aligned in
the 1-direction; (b) random in the 1-2 plane; (c) random in 3D space . 81
5.6 A comparison of the simulated CI for ar = 10,16.9,20,30, and 31.9 [289] with
experimental data of Folgar and Tucker [122] (reproduced from Phan-Thien et
al. [289] with permission from Elsevier) 84
5.7 Actual sample for shrinkage measurement 86
5.8 Simulated and measured packing pressure versus time results for different transition temperatures 90
5.9 Calculated shrinkage in the parallel direction for different no-flow or transition
temperatures. The measured value from Luye [233] is 0.8% . 90
5.10 Geometry and coordinate system for runners . 91
6.1 Generation of a midplane mesh . 100
6.2 3D representation of a complex injection molding . 101XXVI List of Figures
6.3 Dual Domain flow analysis; (a) depicts injection into the center of a rectangular
plate; (b) shows the flow in the cross-section of the plate; (c) shows the flow front
advancement on the surface mesh, and (d) shows the use of a connector element
to ensure physical agreement with the true flow shown in (b) 102
6.4 Dual Domain flow analysis for a part with two ribs . 103
6.5 A simple plate may be decomposed into two parts, each of half the original thickness, and perfectly bonded together 104
6.6 Eccentric shell element for structural analysis . 104
6.7 Structural elements matched for Dual Domain analysis . 105
6.8 Elements on top and bottom surfaces are generally not coincident; that is, the
normal from node n of the bottom element, intersects the top element at some
point p within the element. In this case, interpolation is required . 106
6.9 In 3D analysis, temperature is correctly convected around changes in direction
in runners. In these cases, the temperature differences due to shear heating may
result in an imbalanced filling of cavities despite the naturally balanced feed system 110
7.1 Direct simulation results of the fiber configuration at different strains (reproduced from Fan et al. [103] with permission of Elsevier) . 116
8.1 Reduced effective moduli scaled by Em vs. fiber volume fraction for aR = 20.
Predicted using the Mori-Tanaka model . 127
8.2 Effective Poisson’s ratios vs. fiber volume fraction for aR = 20. Predicted using
the Mori-Tanaka model . 127
8.3 Effective coefficients of thermal expansion vs. fiber volume fraction for aR = 20.
Predicted using the Rosen-Hanshin model 128
8.4 Reduced effective moduli scaled by Em vs. fiber aspect ratio for φ = 0.20. Predicted using the Mori-Tanaka model 128
8.5 Effective Poisson’s ratios vs. fiber aspect ratio for φ = 0.20. Predicted using the
Mori-Tanaka model 129
8.6 Effective coefficients of thermal expansion vs.fiber aspect ratio for φ = 0.20. Predicted using the Rosen-Hanshin model 129
9.1 Schematic representations of (a) short-fiber pellet and (b) long-fiber pellet used
for injection molding 132
9.2 Schematic representations of flexible fiber models . 138
9.3 Bead-rod model of Strautins and Latz [346] 139
10.1 Schematic representations of (a) spherulite structure, and (b) shish-kebab structure (from Zhao et al. [417] with permission from Cambridge University Press) . 142
10.2 Nuclei number density as a function of supercooling temperature for a sample
of industrial iPP (reproduced from Koscher and Fulchiron [211] with permission
from Elsevier) . 144List of Figures XXVII
10.3 Isothermal crystallization curve of the Borealis iPP sample at 132◦C. Inset: Variation of half-crystallization time with crystallization temperature 147
10.4 Heat flow curves for Borealis iPP . 147
10.5 Determination of equilibrium melting temperature using Hoffman-Weeks
method, for Borealis iPP. Melting point data measured on samples isothermally
crystallized at different Tc s were used to determine the equilibrium melting temperature . 148
10.6 Isothermal crystal growth for Borealis iPP sample at 132◦C under quiescent condition 149
10.7 Two-dimensional SAXS image patterns at different distances from the skin surface to the mid-surface for an iPP (from Zhu and Edward [428], with permission
from American Chemical Society) . 153
11.1 PVT diagram for different cooling rates (from Luyé et al. [234], with permission
from John Wiley and Sons) . 160
11.2 Undisturbed equilibrium thermal conductivity against temperature for
polypropylene (from Speight et al. [339]) . 163
11.3 Temperature evolution at the core region of an injection molded part 165
12.1 The molecular structures of two types of blue pigments: (a) the UB-colorant; (b)
the CuPc-colorant (reproduced from Lee Wo and Tanner [404], with permission
from Springer) 168
12.2 Morphologies of (a) virgin iPP at T = 132◦C, t = 180 s; (b) iPP mixed with UB
colorant at T = 132◦C, t = 180 s, and (c) iPP mixed with CuPc colorant at T =
140◦C, t = 150 s, during quiescent crystallization . 169
12.3 Specific heat capacities of three samples: virgin iPP, UB-colored iPP (0.8% colorant by weight), and CuPc-colored iPP (0.8% colorant by weight), denoted by
PP, PP+08U, and PP+08P, respectively (Zheng et al. [425]) 170
12.4 Half-crystallization time vs. crystallization temperature of three samples: virgin iPP, UB-colored iPP (0.8% colorant by weight), and CuPc-colored iPP (0.8%
colorant by weight), denoted by PP, PP+08U and PP+08P, respectively (Zheng et
al. [425]) . 170
12.5 Half-crystallization time vs. short-term shear rate for virgin iPP. Shearing time 1
sec; temperatures: 132◦C and 136◦C. Symbols are experimental data, and solid
and dotted lines are from modeled results (Zheng et al. [425]) 171
12.6 Half-crystallization time vs. short-term shear rate for 0.8% UB-colored iPP.
Shearing time 1 sec; temperatures: 132◦C and 136◦C. Symbols are experimental
data, and the solid and dotted lines are modeled results (Zheng et al. [425]) . 172
12.7 Half-crystallization time vs. short-term shear rate for 0.8% CuPc-colored iPP.
Shearing time 1 sec; temperatures: 144◦C and 148◦C. Symbols are experimental
data, and solid and dotted lines are modeled results (Zheng et al. [425]) 172
12.8 Experimental and predicted parallel and perpendicular shrinkage for the virgin
iPP (Zheng et al. [425]) 173XXVIII List of Figures
12.9 Experimental and predicted parallel and perpendicular shrinkage for the iPP
with UB colorant (Zheng et al. [425]) 173
12.10 Experimental and predicted parallel and perpendicular shrinkage for the iPP
with CuPc colorant (Zheng et al. [425]) . 174
14.1 Pressure development during filling in (a) conventional injection molding and
(b) gas-injection molding (adapted from Turng [373]) . 184
14.2 Dynamic contact angle 190
14.3 Unstable fountain flow (reproduced from Grillet et al. [132]) . 194
A.1 Flow progresses faster in the thick rim of the box and creates an air trap on the
front (shown) and rear sides 207
A.2 The “layflat” is created by unfolding the box to lie in a plane. Note though that
the correct thickness for each surface of the box is retained. Dark lines represent
possible flow paths for analysis 207
A.3 An automotive component and its associated layflat model 208
E.1 A finite difference mesh in the x-y plane . 253
E.2 A simple mesh for a one-dimensional finite difference solution 255
F.1 Approximation of a simple curve 265
F.2 Finite element solution of a two-dimensional problem 266
F.3 Exact solution to 1D FEM sample problem 273
F.4 Mesh for sample problem . 274
F.5 Finite element solution for sample problem using linear interpolation . 275
F.6 Comparison of exact and approximate FEM solution for the sample problem . 282
G.1 Area (barycentric) coordinates for triangular elements 286
G.2 Geometry of triangular element for the pressure field solution . 291
H.1 MINI finite element with linear interpolation and bubble enrichment for velocity,
and linear interpolation for pressure . 308
I.1 Evolution of a free surface simulated by the level set method (provided by Dr.
Huagang Yu) 315List of Tables
3.1 Specific Heat of Some Polymers and Metals . 27
3.2 Thermal Conductivity of Polymers and Metals 28
5.1 Molding Conditions . 89
7.1 Asymptotic Values of Ai , i = 1 to 4 . 119
8.1 Property Data for Components of a Short Glass Fiber-Reinforced Composite 126
J.1 Relation Between Indices in Contracted and Tensor Notations . 319
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