a studyof casting distortion and residual stresses in die casting ...

A STUDYOF CASTING DISTORTION AND RESIDUAL

STRESSES IN DIE CASTING

DISSERTATION

Presented in Partial Fulfillment of the Requirements for

the Degree Doctor of Philosophy in the Graduate School

of The Ohio State University

By

Abelardo Garza-Delgado, M.S.

*****

The Ohio State University 2007

Dissertation Committee: Approved by: Professor R. Allen Miller, Adviser Professor Jerald Brevick _______________________ Dr. Khalil Kabiri-Bamoradian Adviser

Industrial and Systems Engineering Graduate Program

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ABSTRACT

The use of numerical methods to analyze the design and performance of

mechanical components has been widely used in industry for many years. The

results obtained have been used to improve the design of the products by

providing useful insights into the critical areas of the component during operation.

However, the numerical analysis of the manufacturing process that integrates

machine, tooling and products has not been widely done due to the greater

complexity of the physical phenomena involved. This dissertation work presents

a computer modeling methodology developed to predict the final dimensions and

residual stresses in a die casting. The determination of the temperature and

strain-rate dependent mechanical properties of the casting material needed for

the computer model is also presented here. Furthermore, to validate the

adequacy of the modeling methodology computer model predictions are

compared against experimental measurements taken on productions castings.

The methodology uses the finite element method to analyze the

solidification and cooling conditions of a casting during the die casting process.

The die and machine components were incorporated into the analysis and were

modeled as deformable bodies. An innovative method that uses a shell mesh is

presented that allows tracking the elastic deflections in the die cavity resulting

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from the die casting process loads. A fully coupled thermal-mechanical analysis

was done to model the die casting process. The finite element model was solved

using the commercially available finite element package Abaqus.

The determination of the casting constitutive model needed for the

computer model was done as part of this dissertation work. Tensile bars made of

die casting aluminum alloy A380.0 in compliance with industrial testing standards

were produced using an existing tooling. A series of tensile tests at different

combinations of temperatures and strain rates were conducted to determine the

casting constitutive behavior. The tests were performed using a Gleeble 1500

thermo-mechanical simulator.

An experimental Design of Experiments was done to validate the

adequacy of the computer model predictions and to study the effect of process

variables on final casting dimensions. Castings were produced and different

features were selected to characterize the in-cavity and across parting plane

distortion. The production castings were then precisely measured using a

Coordinate Measurement Machine. An Analysis of Variance was performed

using the experimental data and the statistical significance of the main variables

was determined. The experimental results were compared against computer

models simulating the same runs. The comparisons between the experimental

and computer model distortion results are discussed. Computer model residual

stress predictions are discussed as well.

iv

Dedicated to my family, my father Alfonso, my mother Ana Maria, my

brother Ismael and my sister Ana Patricia

v

ACKNOWLEDGMENTS

I wish to thank my adviser, Dr. R. Allen Miller, for the opportunity he gave

me to pursue my graduate studies here at The Ohio State University under his

guidance. I am extremely grateful for his dedication towards developing good

people with sound engineering judgment and for providing the means and the

freedom that let me explore and experiment my ideas throughout the course of

my studies. I am particularly thankful for the countless hours of intellectual

discussion we had during all these years.

I would like to thank my coadviser Dr. Khalil Kabriri-Bamoradian for the

unconditional support he provided throughout these years. I appreciate

intellectual exchanges we had in discussing the finite element models, the

casting experiments as well as the tensile tests. I owe much of my success in

my studies to him.

I am grateful to my colleagues and friends at the Center for Die Casting. I

would like to thank Adham Ragab, Karthik Murugesan, Jeeth Kinatingal and

Hung Yu Xu. The countless hours we spend discussing our difficulties and

enjoying our studies will be always in my mind.

I would like to thank Dr. Jerald Brevick for the joy he always brought while

advising me throughout all these years. I am appreciative of the effort he made

revising this dissertation and also in guiding me while conducting my elevated

vi

temperature tensile tests. I would like to thank him for motivating me and my

colleagues to study and contribute towards the improvement of the die casting

industry.

I am extremely grateful to Cedric Sze and to Shih-Kwang Chen for the

outstanding job they do in keeping the CAD/CAM Laboratory running. I

appreciate all the help provided that allowed me to run my finite element models.

I would like to thank the following people for helping during the various

stages of the tensile tests project. I am grateful to Professor John Lippold for

allowing me to use the Gleeble 3800 at the Welding Engineering facilities. To

Morgan Gallagher for instructing me on how to operate the Gleeble. To

Professor Hamish Fraser at the Materials Science and Engineering Department

for letting me use the Gleeble 1500 to conduct my experiments. To Rick

Tomazin for modifying the tooling to die cast the tensile bars. To Mark Kubicki

from Empire Die Casting for casting the tensile bars and for providing the cycle

design for the production.

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VITA June 23, 1979 Born, Monclova, Coah, Mexico Dec. ‘00 B.S. Mechanical Engineering ITESM Campus Monterrey Dec. ‘03 M.S. Industrial Engineering The Ohio State University Sept. ‘01-Sept. ’07 Graduate Research Associate The Ohio State University

PUBLICATIONS Research Publication 1. Garza, A., Kabiri-Bamoradian, K., Miller, R.A., “Using Die Distortion Modeling to Predict Component Failure in a Miniature Zinc Die”, NADCA Transactions, 2003 2. Garza, A., Kabiri-Bamoradian, K., Miller, R.A., “Finite Element Modeling of Die Casting Die Distortion by Coupled Thermal-Fluid-Structural Analysis”, NADCA Transactions, 2004 3. Garza, A., Kabiri-Bamoradian, K., Miller, R.A., “Finite Element Modeling of Casting Distortion in Die Casting”, NADCA Transactions, 2007

FIELDS OF STUDY Major Field: Industrial and Systems Engineering Minor Field: Manufacturing Engineering

viii

TABLE OF CONTENTS Page Abstract ii

Dedication iv

Acknowledgments v

Vita vii

List of Tables xi

List of Figures xii

1 CHAPTER 1 ................................................................................................................. 1

1.1 Motivation ............................................................................................................. 1 1.2 Die casting process .............................................................................................. 3 1.3 Problem statement ............................................................................................... 7 1.4 Research objectives ............................................................................................. 9 1.5 Research contributions ...................................................................................... 10 1.6 Dissertation outline ............................................................................................. 11

2 CHAPTER 2 ............................................................................................................... 13 2.1 Basic concepts of stress development in castings ............................................. 13

2.1.1 Cooling ........................................................................................................... 13 2.1.2 Development of stresses ............................................................................... 15 2.1.3 Shrinkage ....................................................................................................... 19

2.2 Casting distortion ............................................................................................... 22 2.2.1 Mold restraint ................................................................................................. 22 2.2.2 Casting restraint ............................................................................................. 26

2.3 Casting and die distortion in die casting ............................................................ 28 2.3.1 Die distortion .................................................................................................. 28 2.3.2 Casting distortion ........................................................................................... 30

2.4 Computer modeling of solidification ................................................................... 32 2.4.1 Fluid flow modeling ........................................................................................ 35 2.4.2 Stress modeling ............................................................................................. 39

2.5 Finite element method in solidification modeling ............................................... 43 2.6 Computer modeling of casting and die distortion in die casting ......................... 47

2.6.1 Die distortion modeling .................................................................................. 48 2.6.2 Casting distortion modeling ........................................................................... 51

2.7 Casting distortion studies ................................................................................... 55

ix

2.8 Summary ............................................................................................................ 59 3 CHAPTER 3 ............................................................................................................... 61

3.1 Introduction ......................................................................................................... 61 3.2 Die distortion modeling ....................................................................................... 62

3.2.1 Clamping force modeling ............................................................................... 64 3.2.2 Thermal load modeling .................................................................................. 67 3.2.3 Intensification pressure modeling .................................................................. 72

3.3 Modeling of part distortion .................................................................................. 73 3.3.1 Modeling the tracking of cavity distortion ....................................................... 74 3.3.2 Modeling the cooling of casting inside the die ............................................... 77 3.3.3 Modeling cooling of casting post-ejection ...................................................... 80

3.4 Constitutive model for the casting material ........................................................ 80 3.4.1 Finite element selection for casting ............................................................... 83

3.5 Modeling die distortion using Fluid-Structure-Interaction (FSI) in ADINA .......... 83 3.5.1 Fluid Structure Interaction (FSI) Model .......................................................... 85

4 CHAPTER 4 ............................................................................................................... 89 4.1 Introduction ......................................................................................................... 89 4.2 Background ........................................................................................................ 91 4.3 Literature review ................................................................................................. 95

4.3.1 Die casting alloys ........................................................................................... 96 4.3.2 Aluminum casting alloys ................................................................................ 98 4.3.3 Gleeble testing ............................................................................................. 105

4.4 Determination of aluminum A380.0 mechanical properties ............................. 108 4.4.1 Machine and specimen selection ................................................................ 108 4.4.2 Design of experiments matrix ...................................................................... 114 4.4.3 Specimen production and preparation ......................................................... 117 4.4.4 Testing methodology ................................................................................... 118

4.4.4.1 Velocity controlled ................................................................................. 119 4.4.4.2 Force controlled .................................................................................... 128 4.4.4.3 Results .................................................................................................. 137

4.5 Summary .......................................................................................................... 164 5 CHAPTER 5 ............................................................................................................. 166

5.1 Introduction ....................................................................................................... 166 5.2 Part distortion experiments .............................................................................. 167

5.2.1 Dimensional measurements on production castings ................................... 171

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5.2.2 Statistical analysis ....................................................................................... 174 5.2.3 In-cavity dimensions .................................................................................... 175 5.2.4 Across parting plane dimensions ................................................................. 182

5.3 Computer model predictions ............................................................................ 190 5.3.1 Model preparation ........................................................................................ 190 5.3.2 Distance calculations ................................................................................... 191

5.3.2.1 Coordinate transformation ..................................................................... 191 5.3.2.2 Distance calculation .............................................................................. 197

5.3.3 In cavity dimensions .................................................................................... 198 5.3.4 Across parting plane dimensions ................................................................. 201

5.4 Comparison of experimental and computer model results .............................. 204 5.5 Testing the adequacy of part distortion modeling assumptions ....................... 210 5.6 Stress results .................................................................................................... 216 5.7 Summary .......................................................................................................... 227

6 CHAPTER 6 ............................................................................................................. 230 6.1 Introduction ....................................................................................................... 230 6.2 Research contributions .................................................................................... 231 6.3 Conclusions ...................................................................................................... 232

6.3.1 Finite element modeling ............................................................................... 232 6.3.2 Determination of mechanical properties for casting material ...................... 235 6.3.3 Design of Experiments ................................................................................. 238

6.3.3.1 Experimental results .............................................................................. 238 6.3.3.2 Computer model results ........................................................................ 239

6.3.4 Adequacy of different sets of modeling assumptions .................................. 241 6.3.5 Analysis of residual stress profiles .............................................................. 242

REFERENCES 260

APPENDIX A Test bars insert 267

APPENDIX B Tensile bars material chemical composition and die casting process

control parameters 269

APPENDIX C Collected data during tensile tests 272

APPENDIX D Sample measurements obtained from coordinate measurement

machine 276

APPENDIX E Research casting insert dimensions 284

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LIST OF TABLES Table Page Table 3.1 Boundary conditions applied on the simulated thermal cycles ......................... 69

Table 3.2 Physical properties for H-13 tool steel .............................................................. 70

Table 3.3 Physical properties for a typical 4140 steel alloy .............................................. 71

Table 3.4 Physical properties for aluminum A380.0 die casting alloy .............................. 71

Table 4.1 Experimental array for tensile tests ................................................................ 116

Table 5.1 Matrix design for experimental DOE ............................................................... 170

Table 5.2 ANOVA results for distance D1 ...................................................................... 176



Table 5.5 ANOVA results for distance H2 ...................................................................... 184



Table 5.8 Proposed sets of modeling assumptions ........................................................ 212

Table C.1 Data collection for tensile tests 273

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LIST OF FIGURES Figure Page Fig. 1.1 Schematic of a hot chamber die casting process wild [1 ] ..................................... 4

Fig. 1.2 Schematic of a cold chamber die casting process [1 ] .......................................... 5

Fig. 1.3 Process stages in a typical die casting cycle [1 ] ................................................... 6

Fig. 2.1 Temperature profile across casting/mold interface [4] ......................................... 14

Fig. 2.2 Mechanical behavior of solidifying aluminum alloy under tension [5] .................. 17

Fig. 2.3 Stress-elongation curves at different temperatures [5] ........................................ 18

Fig. 2.4 Residual stresses as a function of ejection time [4] ............................................. 19

Fig. 2.5 Types contractions experienced by a solidifying material [4] .............................. 21

Fig. 2.6 Contraction of three different casting shapes [4] ................................................. 23

Fig. 2.7 Pattern's maker contraction as a function of casting envelope density [4] .......... 24

Fig. 2.8 Contraction of steel castings for different degrees of mold constrain [4] ............. 25

Fig. 2.9 Effect of mold constrain in casting distortion [4] .................................................. 26

Fig. 2.10 Effect of casting geometry in casting distortion pattern [4] ................................ 28

Fig. 2.11 Schematic of different types of analyses for solidification modeling [10] .......... 34

Fig. 2.12 Typical stages in a comprehensive solidification model [10] ............................. 35

Fig. 2.13 Relationship between specific heat and enthalpy [11] ....................................... 39

Fig. 3.1 Machine model finite element mesh .................................................................... 63

Fig. 3.2 Schematic of clamping method ............................................................................ 65

Fig. 3.3 Location of clamping pressure on platens ........................................................... 66

Fig. 3.4 Casting finite element mesh ................................................................................. 74

Fig. 3.5 Shell element mesh ............................................................................................. 77

Fig. 3.6 Temperature dependence of Young’s Modulus [45 ]. .......................................... 82

Fig. 3.7 FSI cavity displacement predictions .................................................................... 86

xiii

Fig. 4.1 Creep properties for die casting aluminum alloy A380.0 [52] .............................. 98

Fig. 4.2 Testing devices used for determining flow stress a) below 300 ºC and b) above

300 ºC [53] ...................................................................................................................... 100

Fig. 4.3 Flow stress for Al-Si alloys at temperatures below the solidus [53] ................... 101

Fig. 4.4 Flow stress for Al-Si alloys at temperatures above the solidus [53] .................. 102

Fig. 4.5 Material properties for an Al-7%Si-0.3%Mg alloy [54] ....................................... 103

Fig. 4.6 Schematic of a typical Gleeble test [57] ............................................................. 110

Fig. 4.7 Schematic of the test bar used for tensile tests ................................................. 112

Fig. 4.8 Strain and strain rate plots for a 0.001mm/s jaw velocity .................................. 122

Fig. 4.9 Strain rate distribution for a 0.001mm/s jaw velocity ......................................... 123

Fig. 4.10 Strain and strain rate plots for a 1mm/s jaw velocity ....................................... 124

Fig. 4.11 Strain rate distribution for a 1mm/s jaw velocity .............................................. 125

Fig. 4.12 Strain and strain rate plots for a 100mm/s jaw velocity ................................... 126

Fig. 4.13 Strain rate distribution for a 100mm/s jaw velocity .......................................... 127

Fig. 4.14 Strain rate vs. jaw velocity correlation plot ....................................................... 128

Fig. 4.15 Strain vs. time curve for a 1.0KN force at 381 ºC ............................................ 130



Fig. 4.18 Strain rate force vs. force correlation plot at 300 ºC ........................................ 133





Fig. 4.23 Stress vs. strain curves at 500 ºC and 1x10-4 s-1 strain rate ............................ 138


xiv






Fig. 4.30 Stress vs. strain curves at 81 ºC and 1x10-2 s-1 strain rate .............................. 145













Fig. 4.43 Stress vs strain curves for temperatures below 300 ºC ................................... 159

Fig. 4.44 Stress vs strain curves for temperatures above 300 ºC .................................. 160

Fig. 4.45 Simulation material properties for strain rate of 1x10-6 s-1 ............................... 161





xv

Fig. 5.1 Selected casting ................................................................................................. 168

Fig. 5.2 Casting dimensions in millimeters (2º draft on interior walls formed by the insert

and 1º draft on walls formed by the insert and die shoe, see Appendix E) .................... 169

Fig. 5.3 Location of fixture gauging points ...................................................................... 172

Fig. 5.4 Selected casting dimensions ............................................................................. 173

Fig. 5.5 Experimental main effect plots for in-cavity dimensions .................................... 179

Fig. 5.6 Experimental interaction plots for in-cavity dimensions ..................................... 181

Fig. 5.7 Box plots for in-cavity dimensions ..................................................................... 182

Fig. 5.8 Experimental main effect plots for across parting plane dimensions ................. 187

Fig. 5.9 Experimental interaction plots for across parting plane dimensions .................. 189

Fig. 5.10 Box plots for across parting plane dimensions ................................................ 190

Fig. 5.11 Sampling order of casting nodes for coordinate transformation ...................... 192

Fig. 5.12 Simulation main effect plots for in-cavity dimensions ...................................... 199

Fig. 5.13 Simulation interaction plots for in-cavity dimensions ....................................... 200

Fig. 5.14 Simulation main effect plots for across parting plane dimensions ................... 202

Fig. 5.15 Simulation interaction plots for across parting plane dimensions .................... 203

Fig. 5.16 Comparison of results for in-cavity dimensions ............................................... 205

Fig. 5.17 Comparisons of results for across parting plane dimensions .......................... 206

Fig. 5.18 Distribution of across parting plane experimental dimensions ........................ 208

Fig. 5.19 Distribution of across parting plane simulation dimensions ............................. 210

Fig. 5.20 Comparisons of results for in-cavity dimensions predicted by the different sets

of modeling assumptions ................................................................................................ 214

Fig. 5.21 Comparison of results for across parting plane dimensions predicted by the

different sets of modeling assumptions........................................................................... 215

Fig. 5.22 Analyzed stress locations at the symmetry plane ............................................ 217

xvi

Fig. 5.23 Stress and temperature profiles at location 1 .................................................. 219




Fig. 5.27 Strain profiles at location 1 ............................................................................... 224




1

1 CHAPTER 1

INTRODUCTION

1.1 Motivation

The ever-demanding quest for lighter components that help reduce weight

in these energy-driven times places a great deal of pressure on mechanical

component designers. Designers are often challenged by selecting the

appropriate manufacturing process that would yield products in the least number

of steps and with the dimensional and functional requirements that the given

application demands. Castings are often among the most sophisticated

mechanical components that due to their processing routes lend themselves to a

large degree of component consolidation, allowing designers to reduce parts

weight.

Casting processes are widely known for their ability to manufacture

products with a very attractive interplay of product complexity and dimensional

accuracy. Among the many casting processes, die casting is often selected

because of its ability to produce thin-walled parts with a great degree of

complexity, accuracy, and at the same time meeting the economies of scale

2

needed for competing with other mass manufacturing processes. However, the

freedom that component designers exercise by merging many parts into one

comes often with the price of very long lead times before a casting can be

successfully produced. Dimensional non-conformance, or casting distortion, is

among the most critical factors driving these long lead times.

Casting distortion is among the many defects that cause a product to be

scraped. It has been estimated that more than $50 million are spent in distortion

related casting defects. Casting distortion arises from two main sources, namely,

uneven cooling of the different sections of the casting due to their different

geometries, and the constraint imposed by the mold walls that limits the ability of

the casting to shrink freely. Distortions may also be induced by post-processing

operations such as fixturing, machining, and quenching.

In die casting the majority of the long lead times afore mentioned is often

spent reworking the tooling. The main source of this reworking is the so called

shrinkage or allowance factor, which tool designers usually apply uniformly

throughout the cavity, implicitly assuming that the casting shrinks evenly.

However, because castings are often made of complex irregular shapes, cooling

of these sections is often uneven. Additionally, due to the high degree of

restraint imposed by the die, die castings may be ejected with residual stresses,

a key factor that plays a major role in post-ejection casting distortion.

Thus, in order to position die casting in a more competitive position among

the other net shape manufacturing processes, it becomes of paramount

importance that lead times be reduced. A significant reduction of lead times can

3

only be achieved when the prediction of casting dimensions is done before the

tooling is produced and all the tool reworking can be eliminated. The prediction

of casting dimensions requires a good understanding of the factors that drive

casting distortion, so shrinkage factors can be applied in a more intelligent

fashion, rather than by trial and error.

1.2 Die casting process

Die casting is high production rate manufacturing process which consists

of the injection of non-ferrous alloys under high pressures and high velocities into

metallic reusable dies. The main characteristics that differentiate this process

from other casting processes are: the high injection velocities, high injection

pressures, short cycle times, the use of metallic reusable dies, thin-walled

casting designs, excellent surface finish attainable, good casting mechanical

properties, dimensional repeatability, etc. Die castings are also prone to higher

porosity contents when compared to other castings mainly due to the entrapment

of air bubbles during the turbulent filling of the die cavity. The majority of the

alloys processed by die casting include aluminum, magnesium, and zinc.

The die casting process presents two main variants, namely, hot chamber

and cold chamber. The schematics of these two processes are presented in Fig.

1.1 and Fig. 1.2 [1 ]. As illustrated in Fig. 1.1, in the hot chamber process the

injection mechanism is in direct contact with the molten metal. On the other

hand, in the cold chamber process the molten metal is in direct contact with the

injection mechanism only during a very small fraction of time, usually in the order

4

of one to two seconds. The differentiation of these two processes comes due the

inability of the injection mechanism to handle highly chemically attacking alloys

such as aluminum, which tends to dissolve iron fairly quickly, and thus, due to the

short exposure times, cold chamber machines are designed to process these

types of alloys, whereas magnesium and zinc alloys are processed in hot

chamber machines.

Fig. 1.1 Schematic of a hot chamber die casting process wild [1 ]

Plunger

HydraulicCylinder

Liquid Metal

Furnace

Gooseneck

Nozzle

Holding Pot

Ejector Platen

Cover DieEjector Die

Ejector Box

Stationary Platen

Die Cavity

Plunger

HydraulicCylinder

Liquid Metal

Furnace

Gooseneck

Nozzle

Holding Pot

Ejector Platen


Ejector Box

Stationary Platen

Die Cavity

5

Fig. 1.2 Schematic of a cold chamber die casting process [1 ]

Both die casting processes follow similar processing stages. Fig. 1.3

depicts the most common steps during a cold chamber injection cycle. The

process starts with the metal ladling in which the material is transported from the

holding furnace to the injection chamber by the use of a ladle. Injection of the

metal is carried out in two phases, namely slow shot and fast shot. During slow

shot the metal is carried from the injection chamber to the runner under small

velocities. The fast shot stage is triggered to get the metal at the gate under

velocities that range from 30 to 40 m/s (1200 to 1600 in/s). During this stage the

Ladle

Hydraulic Cylinder

Plunger

Shot Sleeve


Ejector Box

Stationary PlatenEjector Platen

Die CavityLadle

Hydraulic Cylinder

Plunger

Shot Sleeve


Ejector Box


Die CavityLadle

Hydraulic Cylinder

Plunger

Shot Sleeve


Ejector Box


Die Cavity

6

metal actually fills the die cavity in the order of a few milliseconds. Immediately

after the end of filling the metal is pressurized. Pressurization is achieved in a

small time window before the gate freezes and is carried out by the release of

stored energy in an accumulator located by the injection end of the machine.

After the casting has solidified and cooled, the die is open and the injection

plunger is retracted as illustrated. The casting is finally ejected by the use of

ejector pins driven by a mechanism located behind the movable die half.

Fig. 1.3 Process stages in a typical die casting cycle [1 ]

(d)

(a)

(c)

(e) (f)

(b)

(d)

(a)

(c)

(e) (f)

(b)

7

As already mentioned, one of the characteristics of die casting is the use

of metallic dies. It is precisely the metallic nature of the die that enables the

process to be carried out with short cycle times due to its relatively high thermal

conductivity. The die consists of two halves, namely the cover or fixed half, and

the ejector or movable half. Each half of the die is mounted onto a machine

platen. The machine provides a sturdy frame with three main functions:

repeatedly clamp the die with proper alignment and withstand the high injection

pressures, provide the means for injecting the metal into the cavity, and lastly,

provide the means of extracting the part.

1.3 Problem statement

Die casting as a manufacturing and forming process is subjected to two

sources of errors that can cause parts to be produced out of tolerances and be

rejected. Karve [2] identified these two sources as: random variability in process

conditions and incorrect die dimensions. Typical sources of random variability in

die casting can be attributed to: variations in cycle time, injection pressure,

injection velocity, metal temperature, spraying patterns, cooling line conditions,

etc. All these factors have a definite effect on the repeatability of the process

and nowadays more than ever efforts are being made by machine and peripheral

equipment builders to control them more precisely so they can be kept within the

desired magnitudes. On the other hand, incorrect die dimensions represent a

systematic source of error leading to the production of castings with dimensions

that are consistently off target magnitudes regardless of any variability in the

8

process conditions. Because of the trial and error approach used in dealing with

this issue, long lead times are usually experienced before a die can be released

to the production floor.

The systematic nature of this source of error is usually attributed to the

incorrect oversizing of the die cavity. Because metals experience shrinkage

when they cool down, die cavities are usually oversized to compensate for this

physical phenomenon. Thus, when built die cavities are usually enlarged by an

amount commonly known as the shrinkage or allowance factor. For simplicity

tool builders apply this factor uniformly throughout the cavity shape, inherently

assuming that the casting would cool down evenly. However, because of the

irregular shapes usually encounter in die castings, cooling does not progress

uniformly and some sections of the casting cool faster than others. This uneven

cooling behavior is one of the major causes of casting distortion that represents a

major challenge for tool builders when trying to size die cavities properly.

Another factor that poses a major challenge in properly sizing the die

cavity is the distortion the die cavity experiences due to the operating conditions.

The causes and effects of die distortion in die casting have been identified by

Ahuett [3]. This research work and many others that followed from the same

research group have supported the claim that, as a result of the combined effect

of some mechanical and thermal loads, dies in fact elastically deflect and their

operating dimensions are somewhat different, and usually unknown, from the

intended ones by the tool builder. This cavity distortion complicates matters

further for tool builders when trying to size cavities because of their inability to

9

predict what the operating conditions of the die are, and for that matter, what the

operating cavity dimensions are.

In addition to the cavity distortion issue, the restrained cooling the casting

experiences while cooling inside the die leads to generation of thermal strains.

At high temperatures, when the casting material yields easily, these thermal

strains result in plastic deformation. However, when the material has cooled

down enough and is no longer able to yield, these thermal strains lead to the

appearance of elastic residual stresses, which at the point of ejection may have a

marked effect on the way the casting distorts while cooling to room temperature.

Elastic residual stresses may also be caused as a result of thermal strains

generated while the casting cools outside the die due to uneven cooling of

different sections.

As can be appreciated, the complex interplay of the described physical

phenomena that are involved when trying to accurately size die cavities poses a

major challenge for tool builders. The large complexity of this problem has been

traditionally resolved by tool builders iteratively, usually modifying the cavity

dimensions many times before a die can be released to the shop floor. Clearly,

this approach is lengthy and cumbersome limiting to a large degree the

competitiveness of the die casting industry due to the long lead times involved.

1.4 Research objectives

This research work was focused on the development of a computer model

to simulate the die casting process in order to predict the final deformation and

10

residual stresses in a die casting. Experimental validation of the computer model

predictions formed part of the main objective of this research work as well. The

results of this investigation have the main goal of providing practitioners and tool

designers a better understanding of how process and tool design related factors

affect the final deformation and residual stresses in a die casting.

Among many other things, computer model predictions rely heavily upon

the correct representation of the system being analyzed as well the use of

appropriate constitutive models and physical properties that describe the

physical/mechanical behavior of the components involved. An important side

objective of this research work was the determination of some of the high

temperature mechanical properties of the die casting alloy employed in the

model. The availability and use of these properties was of paramount

importance to have a high degree confidence in the predictions provided by the

computer model.

1.5 Research contributions

Modeling of part distortion in die casting has been a subject of study for

many years. Nonetheless, all of the existing approaches made use of simplifying

assumptions that limit to varying degrees the respective predictions. Accounting

for the contact conditions between the casting and the die, and thus the

development of important deformations and residual stresses in the casting, has

been among the most common simplifications taken. Proper consideration of the

described situation requires incorporating the deforming die into the model. The

11

main contribution of this work is the development of a modeling methodology that

allows the proper establishment of the contact conditions between the casting

and the die prior to solidification and cooling. This modeling approach takes into

account the cavity deformations caused by the elastic deflections experienced by

the die during operation, and through a proper spatial mapping applies this

deformation to the casting shape at the end of filling; thus, accurately

establishing the contact conditions between the casting and the die walls at the

onset of cooling and solidification.

The degree of confidence in computer model predictions was increased by

the determination of the high-temperature mechanical properties of the die

casting aluminum alloy A380.0 used in this study. In particular, temperature and

strain rate dependent stress-strain curves were obtained experimentally using a

thermo-mechanical simulator available at the Materials Science and Engineering

facilities at The Ohio State University. The generated data was used as input

data in the computer model developed as part of this research work. The

benefits of this experimental work will help the modeling community by providing

a reliable set of experimental data that can be readily used in part distortion

models.

1.6 Dissertation outline

The dissertation is structured as follows. Chapter 2 presents a literature

review on casting distortion and the computer modeling techniques developed to

analyze it. Chapter 3 describes the methodology developed in this research

12

work to model residual stresses and distortion in die castings. Chapter 4

presents the experimental work done to determine the elevated temperature

mechanical properties for the die casting alloy used in this study. Chapter 5

describes and compares the results of an experimental design of experiments

with the computer model results. Lastly, the conclusions of this dissertation are

presented in Chapter 6.

13

2 CHAPTER 2

LITERATURE REVIEW

2.1 Basic concepts of stress development in castings

2.1.1 Cooling

In its most crude description, casting involves the pouring of a hot liquid

metal into a mold, which provides a negative print of the desired final product so

that when the liquid metal solidifies a shaped product is obtained. Very intricate

shapes can be obtained via this route, provided properly designed passages are

used to get the liquid metal to fill all the regions of the mold. From a broad

perspective, it can be seen that understanding of a casting process requires the

understanding of the thermal exchange phenomena that take place and govern

the described operation.

Therefore, in order for a casting to be produced the heat content stored in

the hot liquid metal must be taken out by a shaped mold whose main functions

are to extract the heat from the casting and to provide a structurally sound and

stable shape to be filled by the liquid metal. From a thermal point of view, the

mold must be kept at a lower temperature than that of the liquid metal, so a

te

m

ta

a

emperature

mold. Witho

ake place.

rrangemen

e gradient is

out a tempe

A typical de

nt is illustrat

Fig. 2

s imposed a

erature grad

epiction of

ted in Fig. 2

2.1 Temperat

14

and heat ca

dient, no he

the temper

2.1 [4]

ture profile ac

4

an flow from

eat flux is p

rature profil

cross casting/

m the liquid

resent and

e in a casti

mold interfac

metal to th

no cooling

ng-mold

e [4]

he

can

15

2.1.2 Development of stresses

Upon cooling, the hot liquid metal starts to freeze and becomes solid at

the melting point if it is a pure substance or a eutectic, or over a temperature

range if it is an alloy. As the liquid is turning into a solid it starts to develop

stresses as a result of the denser and more crystalline regularity present in the

atomic arrangement, which is held by stronger interatomic binding forces than in

the liquid state. An illustration of the progression of the stress development in an

aluminum alloy subject to tension is shown in Fig. 2.2 [5]. In order to understand

this process, Fig. 2.3 depicts the stress-elongation curves for this material as a

function of the temperatures experienced at the same locations as those

highlighted in Fig. 2.2.

In stage 1 when the metal is still completely liquid the applied stress would

just cause the liquid to follow the end plates, since a liquid cannot develop any

stresses. Looking at Fig. 2.3, it can be predicted that any stress applied at this

stage would cause it to elongate quite substantially. Following the next stage,

when a small amount of solid is present in the liquid, the application of a tensile

stress will cause this mixture to deform quite substantially since there is no

coherency among the solid particles that can provide any obstruction to the

tensile load. In stage 3, when the mixture is mostly solid with some liquid, a

coherent network between the solid particles or grains has already been

established, thus upon application of the tensile load the weak solid would

deform and break easily as shown in Fig. 2.3. The rupture tears apart the grains

and since the small amount of liquid present is isolated, it cannot fill the cracks

16

thus the very well known defect of hot tears develops. Stretching of the metal

below the solidus temperature as shown in stage 4 causes the metal to deform

plastically under very low stress magnitudes. In this stage the ductility of the

material is high and cracking is not observed unless there exist a brittle phase

that can cause it. At this stage recrystallization of the grains may be obtained if

the grains are deformed such that they begin to recrystallize. A twin effect that is

difficult to distinguish but which is also present in this stage is the phenomenon of

creep, which is characterized by a continuous elongation of the material with time

under the application of a constant load. Creep causes the material to further

deform plastically and contributes to the total plastic elongation of the material.

Finally, once the material has cooled enough such that the flow stress is

substantially large, any applied stress smaller in magnitude than the achieved

flow stress will produce only elastic strains, which are not relieved by any plastic

deformation but rather when the load is removed causing the material to spring

back.

17

Fig. 2.2 Mechanical behavior of solidifying aluminum alloy under tension [5]

18

Fig. 2.3 Stress-elongation curves at different temperatures [5]

The time the casting remains under the application of a given stress

magnitude, either due to a restraint offered by the mold or due to stronger

sections of the casting that have cooled earlier as will be shown later, determines

the magnitude of the elastic residual stresses with which the casting is ejected

from the mold. This behavior is depicted in Fig. 2.4, which shows the effect of

the time the casting remains in the mold on the magnitude of the residual

stresses.

19

Fig. 2.4 Residual stresses as a function of ejection time [4]

2.1.3 Shrinkage

As is well known in foundries, the mold is made slightly larger than the

size of the final product because the liquid metal occupies more volume than the

final casting. This behavior is illustrated in Fig. 2.5 [4]. Three different types of

shrinkage are experienced when a liquid molten metal becomes solid. The first

20

one is liquid-to-liquid shrinkage, which represents no challenge for the

foundryman since the contraction experienced in this stage can be easily made

up for by pouring slightly more metal. The second stage is liquid-to-solid

shrinkage. This shrinkage is experienced due to the denser arrangement of the

atoms in the solid state as compared with the liquid state. Problems may arise

due to this type of shrinkage mainly if the feeding of liquid or solid metal is not

adequate, leading to the formation of shrinkage cavities. The last type of

shrinkage is solid-to-solid shrinkage. This type of solid contraction represents no

problem to the foundryman if the casting is free to contract by itself. Typically

this is not the case, and due to the complex shapes usually cast, the cooling of

some regions of the casting is often constrained by other regions that have

cooled earlier and thus are stronger, causing the weaker regions to plastically

deform. Another type of constrain typically present while the solid casting is

cooling is that offered by the mold walls. The mold, which is usually made of a

stronger material than the casting, exerts a force in the opposite direction to that

exerted by the casting upon it while contracting, thus offering a rather strong

obstacle to the free contracting action of the casting causing it to enlarge and

build up residual stresses at the latter stages of cooling. After removal of the

casting from the mold has taken place, it can be expected that the casting being

perhaps with some shrinkage porosity, somewhat distorted, and out of tolerances

in some regions making the predictability of the dimensions troublesome for the

foundryman. This leads to difficulties in estimating the size of the mold since the

21

contraction and deformation experienced by each region of the casting might be

really challenging to predict.

Fig. 2.5 Types contractions experienced by a solidifying material [4]

22

2.2 Casting distortion

2.2.1 Mold restraint

As was mentioned before, as a result of the solid-to-solid contraction, the

casting shrinks in a magnitude proportional to the temperature difference and

also proportional to the coefficient of linear thermal expansion. However, the

amount of shrinkage a given casting can experience in actuality depends on the

degree of restraint imposed by other regions of the casting cooling faster at

earlier stages and also the degree of restraint imposed by the mold walls. Thus,

the amount of allowance applied to the die in order to account for the shrinkage

of the casting while cooling can be rather cumbersome since different regions of

the casting may require different allowance, depending on the contribution of

each of the restraints to which the regions of the casting are subjected to while

cooling.

Campbell [4] proposed a rather simplistic method to assess the degree of

restraint offered by sand molds and in this way predict the pattern/mold

allowance based purely on the geometry of the casting. It is proposed that a

straight bar such as that shown in Fig. 2.6 would experience zero restraint and

would be allowed to cool and shrink freely. On the other hand, a box-shaped

casting with thin walls cast around a rather big and stiff sand core would

experience theoretically infinity constrain as the walls of the casting get thinner

and thinner. In this case the casting contraction would be theoretically zero. The

results of these two cases can be thought to provide the points for a calibration

curve in which the first one represents the case of a fully dense casting, whereas

th

b

d

d

c

T

F

he second o

e obtained

ensities of

ivided by th

ontraction v

The results o

Fig. 2.7.

one represe

by using d

the geome

he overall v

values are

of such ana

Fig

ents the ca

ifferent geo

tries are co

volume occu

measured o

alysis can b

g. 2.6 Contrac

23

se of a null

ometries su

omputed by

upied by th

once the pa

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ction of three

3

l dense cas

uch as those

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art as reach

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different cas

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rent points

Fig. 2.6. T

the casting

ting, and th

emperature

hose shown

4]

can

The

he

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n in

d

a

re

te

F

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The r

egrees of r

re shown in

esults of the

emperature

Fig. 2.8.

ig. 2.7 Pattern

results of an

restraint offe

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The results

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predicted c

24

ntraction as a

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contraction

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gation in w

molds in stee

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70 years a

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ections of t

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Fig. 2.8 Contr

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be restrain

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o is the cast

25

el castings for

gs are comp

in Fig. 2.6,

nted by diffe

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5

r different deg

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thus one m

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in Fig. 2.9.

grees of mold

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Upon coo

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mple

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26

wants to sh

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of mold const

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on the volum

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at is extract

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ifficult to pr

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rink and pu

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27

As an illustration of the effect of casting design on the casting distortion,

Fig. 2.10 shows two scenarios that can yield quite different results as far as

casting distortion is concerned. The distortion or eventual crack that results in

the top case can be explained as follows. Upon cooling, the thinner sections of

the casting solidify the first, and their supply of material upon shrinkage is fed by

the thicker section that remains hotter for longer time. Once these sections cool

enough, they become strong, and the shrinkage action of the thicker section can

cause bending of the thin section or even produce a crack if the thicker section is

weak enough not being able to withstand the tensile load. The bottom case

shows a slightly different scenario. In this case, the outer sides bow because the

inner section cools the slowest due to the fact that it is surrounded by hotter

cores and thus the dissipation of the heat from the cores takes longer. Upon

cooling, the inner section starts to pull the outer ones, and cause them to bow, or

if it is weak enough it may crack as shown.

28

Fig. 2.10 Effect of casting geometry in casting distortion pattern [4]

2.3 Casting and die distortion in die casting

2.3.1 Die distortion

During a typical die casting cycle a certain set of loads are present that

undoubtedly affect the performance of the die, the machine, and ultimately the

productivity of the process. In this section, a description of the loads that affect

the performance of the die are highlighted with the goal of setting the background

for the even more complex phenomena of part distortion.

As a broad classification, the loads experienced by the dies are classified

as mechanical or thermal. Due to their nature, the mechanical loads can be

29

subdivided into static and dynamic. Among the static loads, the clamping force

exerted by the toggle mechanism is the simplest. This force is applied,

depending on the toggle location relative to the ejector platen back surface,

either on the top and bottom or nearly equally distributed on the center of the

ejector platen back surface. This load strains the tie bars in tension so that a

constant, but non-uniform, pressure keeps the die halves in contact during the

injection and dwell stages. As a simplification to the characterization of the static

loads, the intensification pressure is considered in this category. After the filling

of the cavity is completed, high pressure stored in an accumulator is released so

that the injection piston can further push the molten metal into the cavity,

guaranteeing filling of intricate regions in the cavity and also compressing the

pores of entrapped air caused by the traveling of the molten metal.

In the dynamic loads category, the well-known pressure spike at the end

of filling is definitely the most important one. This phenomenon has been

observed and documented in the die casting literature for years. It is all a result

of the inertia of moving masses coming to rest in a very short period of time. At

the start of filling of the die cavity the plunger is accelerated to achieve the fast

shot speed. However, the filling time does not last long (on the order of

milliseconds) and suddenly all moving masses including hydraulic oil, piston, and

the rod come to rest once the cavity is filled. This sudden stop imparts a

tremendous pressure spike onto the die and machine structure that sometimes

has been identified to be the cause of flashing problems.

30

The thermal load affecting the dies is undoubtedly very critical. When built

at room temperature the die halves come perfectly in contact. However, due to

the continuous injection and cooling of the metal inside the die cavity, the dies

absorb heat, causing some elastic deflection. Moreover, since during the

injection and dwell stages the die is not free to expand, the thermal distortion

usually causes the die to grow inside the cavity, affecting its dimensions.

The combined effect of all these loads imparts some distortion in the die,

which under most circumstances is not easy to predict. As a result of this, the

process of producing a die is usually iterative requiring many trials before a die

can be released to production, leading to a decrease in the productivity of the

industry due to the long lead times incurred.

2.3.2 Casting distortion

The process of producing a die casting starts with the injection of the

molten metal into the die. As was already discussed, due to the thermal and

mechanical loads experienced by the die during operation, the cavity shape the

liquid encounters once it fills it is somewhat distorted from its nominal shape at

room temperature. These important small cavity deformations have an important

effect in that they determine the initial casting shape just prior to the onset of

solidification and cooling.

During solidification and cooling, the casting releases its heat content to

the die. The effect of this heat exchange produces reversed effects on the

casting and die, leading to shrinkage of the casting while causing expansion of

the die, thus affecting once more the cavity dimensions. In die casting as in any

31

other casting process, self induced as well as mold induced restraints occur while

the casting is cooling inside the mold. It should be mentioned that because die

casting relies in using metallic dies, the degree of restraint imposed by the die

walls is much higher than other processes such as sand casting, having a

marked effect on casting shrinkage. This phenomenon leads to plastic

deformation in casting during the early stages of cooling, while during the later

stages it leads to the formation of important residual stresses as was already

discussed.

One interesting feature that differentiates die casting from the rest of the

casting processes is the use of an intensification pressure that helps in reducing

porosity levels as well as in feeding material to intricate regions. As a result of

the high pressures applied, the hot liquid metal is left pressurized before it starts

to solidify. It is conjectured that this pressurization may affect the early plastic

and visco-plastic behavior of the newly formed solid, which experiences higher

plastic deformation due to the large hydrostatic state of stress to which it is

initially subjected. Thus, the fact that the liquid experiences a phase change

under pressure may lead to additional plastic deformation at the early stages of

solidification.

After a predetermined time the casting is ejected from the die. At the point

of ejection the casting may possess some residual stresses as well as a non-

uniform temperature profile due to its cooling process. Under these conditions,

die castings are usually left to cool at room temperature. Sometime, parts are

quenched or trimmed for subsequent post-processing operations such as

32

machining. The final casting obtained is the result of all deformations induced

during the various process stages it goes through.

2.4 Computer modeling of solidification

From a metallurgical standpoint the phenomenon of solidification is a

process of nucleation and growth as outlined by Wallace et al [6]. A wide variety

of physical phenomena take place when a liquid metal solidifies. The range of

these physical phenomena can vary from a few atomic distances when the

clustering of the atoms is taking place during the growth of a solid grain, to a few

millimeters or centimeters when the solid transport of a few solid particles or

grains takes place, leading to macrosegregation problems in castings. From a

mechanical engineering point of view, solidification might be regarded as a fluid

flow, heat transfer, and stress development problem, with their respective

physical equations developed based on continuum mechanics principles. The

solution of these “macroscopic” phenomena provide the basis on which the

analysis and study of a mold filling, heat flow, and residual stress and distortion

of casting can be done to gain understanding in the design of molds and parts for

instance. Due to the scope of these analyses, little or no attention is paid to the

nucleation and growth of solid grains and other microscopic phenomena that take

place and in most instances, the solution of the macroscopic phenomena is used

to make gross predictions of the “microscopic” ones and vice versa. Because of

the nature of the wide variety physical phenomena as well as the wide range of

length and time scales in which these take place, when one speaks of computer

33

modeling solidification a clear boundary ought to be drawn to isolate the system

one wishes to analyze and the simplifying assumptions that will be considered.

This section is intended to provide a review of the literature of computer modeling

of the macroscopic phenomena of solidification.

The mathematical modeling of some of the macroscopic as well as the

microscopic phenomena present in solidification is discussed by Dantzig [7],

Overfelt [ 8] and Upadhya [9]. Fig. 2.11 depicts the different types of analyses

that can be performed as well as the outputs and results they may provide. Fig.

2.12 shows the typical areas that a comprehensive solidification modeling system

may consist of.

F

ig. 2.11 Scheematic of diffe

34

erent types of

4

analyses for solidification

modeling [100]

2

c

s

c

e

lin

o

.4.1 Fluid

Filling

asting proc

queeze cas

oupled with

nough to h

nes. Also,

f filling repr

Fig. 2.12

flow mode

g of the mo

cesses. Fo

sting and sa

h the heat t

inder prope

the temper

resent the i

2 Typical stag

ling

ld by the m

r slow filling

and casting

ransfer, sin

er filling of t

rature profil

nitial condit

35

es in a comp

molten meta

g processes

g, modeling

nce heat los

the mold, le

le of the liqu

tion for a su

5

rehensive sol

l usually re

s such as lo

g of the fluid

sses during

eading to co

uid casting

ubsequent

lidification mo

epresents th

ow pressur

d flow must

filling migh

old shuts an

and the mo

solidificatio

odel [10]

he first step

re die castin

be done

ht be large

nd/or weld

old at the e

on analysis,

p in

ng,

end

,

36

thus the accuracy of the solidification analysis is dictated to some extend by the

accuracy of the thermal conditions obtained at the end of the filling analysis.

Since most metals can be represented as incompressible Newtonian fluids, their

flow is governed by the Continuity and Navier-Stokes equations [7]

0 Eq. 2.1

Eq. 2.2

where:

is the density

is the velocity vector

is time

is pressure

is viscosity

is the body force

Besides proper predictions of velocities and pressure profiles in the liquid,

the filling of the mold is an important feature that can be included in the fluid flow

analysis. To properly model the advancement of the interface between the liquid

molten metal and the air in the mold cavity, Eq. 2.2 must be augmented to

describe the movement of the free surface. To model this situation, the majority

37

of the fluid flow software uses the volume-of-fluid method (VOF), which uses a

filling function which is advected with the fluid satisfying the equation [7]

· 0

Eq. 2.3

The value of within a given computational cell is associated with the

state of the cell: when =1 the cell is full, when =0, it is empty, and cells where

0 < < 1 contain the interface [7].

The other important part of the filling analysis is the heat transfer. Since

conduction is metals can be properly represented by Fourier’s law, the energy

transport equation is described by the heat conduction equation

· ·

Eq. 2.4

where:

is the specific heat

is the temperature

is the thermal conductivity

is the internal heat generation

The advective term · in Eq. 2.4 couples the energy transport equation

to the velocity field. To model the “phase change” of the liquid metal, the term

38

in Eq. 2.4 is used to represent the evolution of the latent heat in the liquid. One

of the expressions that can be used is

Eq. 2.5

where:

is the latent heat

is the fraction of solid

The evolution of the fraction of solids for a particular alloy must be

determined so that its mathematical representation can be used in Eq. 2.5. An

easier way to model the release of the latent heat is the so called “specific heat

method”. The release of the latent heat produces an increase in the internal

energy or enthalpy of the liquid, and since the specific heat and the enthalpy for

an alloy are related by

… … … … … … … … … …

1 ∆ … … … .

∆ … … … … … … …

Thus, knowing the evolution of the enthalpy from measurements, one can

backwards compute the specific heat in the region between the liquidus and

solidus, preserving the area under the curve as illustrated by Fig. 2.13. This

m

b

2

th

ty

la

re

fi

method prod

e easily im

.4.2 Stres

The m

he fluid flow

ype of analy

arge numbe

esults obtai

nal dimens

duces a pie

plemented

Fig. 2

ss modeling

modeling of

w problem.

ysis is that

er of compu

ined from a

sions in cas

ece-wise co

in the ener

2.13 Relations

g

f stresses in

One of the

the govern

utational iss

a stress mod

tings as we

39

ontinuous fu

rgy transpo

ship between

n castings r

e main reas

ing phenom

sues. Howe

del can be

ell as to imp

9

unction for t

rt equation

specific heat

represents

ons for the

mena are ve

ever, when

used to pre

prove mold

the specific

.

and enthalpy

a greater c

greater dif

ery comple

properly so

edict residu

design to m

c heat that c

y [11]

challenge th

fficulty in th

x and pose

olved the

ual stresses

match casti

can

han

is

e a

s and

ng

40

shrinkage to the so called pattern maker allowance, mostly regarded as an

empirical factor. This topic has been an active research area in many casting

processes for more than a decade and the underlying theory behind this

modeling work has been outlined in a series of papers [11-16].

Stress modeling is a thermo-mechanical analysis that involves solving the

equilibrium, constitutive and compatibility equations, which relate force to stress,

stress to strain, and strain to displacement, respectively. The equilibrium

equations can be represented by

∆ ∆ Eq. 2.6

where

∆ ∆ , ∆ , ∆ , ∆ , ∆ , ∆

∆ ∆ , ∆ , ∆

The compatibility equations that relate strain to displacements can be

represented by

∆ ∆ Eq. 2.7

where

∆ ∆ , ∆ , ∆ , ∆ , ∆ , ∆

∆ ∆ , ∆ , ∆

41

The matrix representation of the differential operators is written in the

matrix as

0 0

0 0

0 0

0

0

0

The constitutive equations that relate stress to strain can be represented

by

∆ ∆ Eq. 2.8

where the matrix contains the elastic constants, which are a function of only

two material properties ( , Young’s modulus, and , Poisson’s ratio) for isotropic

materials

42

As stated in Eq. 2.8, only elastic strains are responsible for stress

development in castings. This just confirms the experimental results of different

stress-strain curves at different temperatures such as those shown in Fig. 2.3,

since plastic strains are easily accommodated by plastic deformation of the

casting, whereas elastic strains produce elastic stresses which represent the

residual stresses with which the casting is left after is taken out of the mold, or in

some instances at room temperature.

As already mentioned, the stress modeling problem is a thermo-

mechanical one. The temperature differences between the casting and the mold

that are present when the casting is cooling inside the mold are the source of

volumetric contractions that happen in the casting. Since most castings are

comprised of intricate shapes of different volume-to-surface-area ratios which

usually cool at different rates and which interact with the mold walls in many

different ways, the volumetric shrinkage induced by the cooling of the casting is

usually restrained by either, other sections of the casting or by the mold walls.

This restraining effect causes the development of strains that lead to casting

distortion or the development residual stresses during the later stages of cooling.

It should be mentioned that thermal strains by themselves do not cause any

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43

stresses, they just promote a “thermal load” or a “mismatch” that are eventually

translated into elastic or plastic strains.

As far as modeling of casting stresses is concerned, thermal strains are

the main source of elastic and plastic strains. The stress model is usually

composed of a thermal and a mechanical model. Since the conduction in most

metals can be accurately represented by Fourier’s law, the thermal model

involves solving the energy equation (Eq. 2.4) without the advective term.

The stress model can be solved in a sequential or a coupled manner. The

thermal model can be solved first and its results can be “loaded” into the

mechanical one. On the other hand, both models can be solved in a coupled

manner, solving for the temperature field first and the displacement field second,

and then back checking whether the produced displacement field affected in any

way the thermal field. Coupled models are usually employed when the

development of a gap due to shrinkage of the casting affects the heat transfer

rate between the casting and the mold. This phenomenon can affect the thermal

field in the casting because the formation of the gap reduces the heat transfer

coefficient at the casting/mold interface, which retards cooling of the casting and

produces higher ejection temperatures that may lead to larger distortions.

2.5 Finite element method in solidification modeling

Because of the many challenges that the modeling of macroscopic

phenomena of solidification involves, the topic has attracted the attention of

many researchers throughout the world. Finite elements have been widely

44

accepted for solving the governing equations previously presented. The

formulation and solution of the governing equations by finite elements has been

presented in a wide variety of publications [10-13, 17-24].

Dantzig [12] presents the formulation and solution technique for modeling

the development of the thermal stresses in castings. A detailed description of the

treatment of the different strains present during the cooling of a casting is

presented with enough emphasis on the constitutive relations that govern each

type of deformation. The formulation presented treats the problem as a thermal-

mechanical one, thus no account is given to the effects of the thermally coupled

fluid flow effects on the stress development in the casting. Lewis et al [11]

discuss the finite element algorithms to model the solidification of a metal from

the mold filling stage to the final cooling of the casting. The proposed formulation

considers the thermally coupled fluid flow and mechanical conditions that govern

the whole solidification problem. Considerable attention is given to the way the

latent heat release at the solid-liquid interface takes place. A free surface

tracking method is implemented in the formulation so the position of the fluid front

can be predicted in the mold filling stage. Trovant et al [17] present a numerical

algorithm to model the shrinkage of a metal while it undergoes solidification.

Latent heat release, fluid front tracking as well as other fluid flow conditions are

taken into account in this work. Seetharamu et al [10] discuss the thermal stress

generation problem in castings and the formulation of a thermally coupled elasto-

viscoplastic model is presented. The proposed model results are validated

mainly with experimental results for large steel ingots. A numerical scheme that

45

uses shell elements to represent the casting is presented by Hetu et al [18] to

simulate the filling of thin-walled die castings. Owing to their very thin design, the

filling analysis of die castings is simulated using the also known 2.5D approach,

in which a thickness is given to the shell elements, reducing the numerical

complexity and running times usually faced with full three dimensional meshes.

A numerical scheme to treat three dimensional meshes for casting processes

such as sand casting is also presented. A free surface tracking method is also

implemented to accurately predict the advancement of the fluid front during mold

filling stage. The formulation of an elastic-viscoplastic stress model is also

discussed. The formulation of a very sophisticated coupled thermoplastic model

is presented by Agelet et al [19]. The model considers temperature dependent

material properties. Latent heat effects are also incorporated into the formulation

through a free energy function. Plastic response of the material has been

modeled using a temperature dependent model that takes into account nonlinear

hardening due to plastic deformation and thermal linear softening.

Implementation of gap dependent thermal conditions is also considered in the

formulation. A very original formulation of a thermal-mechanical model with

considerations for microstructure evolution in aluminum alloys is presented by

Celentano [20]. This original work presents several important relevant

contributions such as: the unified formulation to describe the liquid, mushy and

solid phases present during solidification, gap dependent thermal conditions, the

effect of volumetric expansions due to phase transformations, the possibility to

capture large geometric distortions especially at high temperatures and the

46

influence of the coupled dendritic/eutectic microstructure formation on the

temperature dependent constitutive laws.

Of special consideration is the work presented by Bellet et al [21-22]. The

proposed work discusses a very comprehensive formulation of a fully coupled

thermo-mechanical model that uses an elastic-viscoplastic material formulation to

describe the casting. The proposed formulation is justified on the basis of the

importance of the gap-dependent heat transfer conditions present at the

casting/mold interface. A unified temperature dependent elastic-viscoplastic

formulation to describe the material behavior in the liquid, mushy and solid states

is presented, so that the solidification of the liquid metal can be accurately

described throughout the whole cooling process. An important contribution of

this work is the incorporation of contact conditions between the casting and the

mold even though the mold is considered a rigid body. This important feature

allows the modeling of stress development due to contact restraints imposed by

the mold during solid-to-solid shrinkage. The proposed formulation has been

implemented in the finite element code THERCAST targeted for solidification

analyses. A recent enhancement to the code was made by incorporating an

Arbitrary Lagrangian Eulerian (ALE) method as described in [23]. The added

capabilities allow the tracking of the free surface of the fluid front in a filling

analysis with better resolution than the to the commonly used VOF method for

Reynold’s number on the order of 103 to 104. The ALE scheme is also used for

handling convection effects due to density gradients induced in the remaining

47

liquid pool at the end of filling. The implications of the proposed formulation are

explained in large detail therein.

The formulation of a model that accounts for fluid-solid thermal-

mechanical interactions is presented by Cruchaga et al [24]. A coupled multi-

physics model for the analysis of solidification processes is presented. The

proposed formulation has been developed to accurately describe the material

description during the liquid, mushy and solid phases that occur during the filling,

solidification and cooling of the casting. The important contribution of this

formulation is that it is aimed at being able to describe the influence of fluid

motion on the temperature field and casting/mold evolutions when advective

effects caused by the liquid pool are taken into account. An additional

contribution sought is to be able to study the effect of non-uniform initial

temperature fields in the solidification and cooling stages. This modeling feature

enables to suppress the commonly used assumption of uniform temperature

distribution at the end of filling, which might be invalid for slow filling processes

such as sand and permanent mold casting. As noted by the authors, the

consideration of buoyancy effects and phase change during the whole process

couples the fluid phase motion with the thermal and mechanical responses

through the fluid velocity, temperature field and gap formation [24].

2.6 Computer modeling of casting and die distortion in die casting

Because of the high cost involved in building a die, computer modeling

has emerged as a useful tool in helping designers to predict potential operational

48

problems in dies. The field of computer modeling in die casting has being an

active research area for more than two decades. Early computer models have

focused on predicting thermal profiles as well as fluid patterns. These models

have been widely popular and the area has become relatively mature. However,

because of its complexity the subject of part and die distortion in die casting has

been explored for more than a decade only. This section provides a review of

the literature that has been published in this field.

2.6.1 Die distortion modeling

Die casting die distortion modeling has been analyzed in a series of

publications [25-34]. Barone et al [25] proposed a modeling technique that

accounts for the clamping, temperature and pressure loads using the boundary

element method. The proposed scheme is particularly unique, in the sense that

the thermal load resulting from the continuous injection and cooling of molten

metal that causes the die to elastically distort, has been characterized to affect

only the region of the die near the cavity. It is claimed that the effects of this

thermal load are concentrated near this region, and based on that the boundary

element method is used. The proposed modeling approach considers structural

elements such as inserts, dies, and platens. Machine toggle and tie bars are

incorporated by additional representation of their stiffness. The results obtained

for an automotive transmission case claimed to correlate well with experimental

evidence.

Another modeling approach based on the boundary element method is

proposed by Milroy et al [26]. The model attempts to predict elastic deformations

49

at the cavity surface that result from thermal expansion and cavity pressure

effects. Clamping force effects are not incorporated on the model, but instead

the die is anchored at the corners. It is claimed that based on the model

predictions, dimensional changes that compensate for the thermal expansion on

the cavity can be proposed, so the cavity dimensions come to be correct during

operational conditions and flashing problems can be eliminated.

A normalized approach designed to predict thermal stress in a die casting

die is proposed by Dour [27]. The model represents the die casting die as a slab,

which is subjected to a heat flux load on one face, while the other face is kept

isolated. The model is rather simple and its results clearly do not reflect the

effects of important mechanical loads.

The bulk of the literature as far as die distortion is concerned is presented

in references [28-34]. These publications emerged as a result of the

investigations performed at the Center for Die Casting at The Ohio State

University. The main purpose of the research was to: a) assess and characterize

the important loads that contribute to the deformation of the die, b) provide

design guidelines to machine and die builders in order to minimize and/or

attenuate die deflection, parting plane separation, excessive machine distortion

that may result from unbalanced tie bar loading, etc, c) provide guidelines to

practitioners to assess the influence of machine and die interaction in the final

distortion of die casting parts. Several research cases have been performed and

their results have provided a clearer understanding of the important factors that

determine the deflection in die casting dies.

50

The results of the research work at The Ohio State University have

established the main causes of die deflection. It has been determined that die

deflections are caused by the mechanical and thermal loads experienced by the

dies during operation. Research findings have suggested that the majority of the

die distortion is caused by the uneven thermal profiles experienced by the die

during operation. These uneven thermal profiles are caused by the asymmetric

and irregular shapes normally used in casting design. Process control

parameters such as total cycle time, dwell time and cooling line configuration

have been proven to substantially affect the way thermal profiles develop in the

die. Clamping force effects have been also proven to be significant. Due to the

uneven thermal expansion of the die, once it is brought in contact during

clamping, an uneven sealing at the parting plane might be caused. The

described situation can give rise to spitting of molten metal outside the cavity.

Flashing problems might reduce the productivity of the casting process, since

post-processing operations may need to be incorporated to trim the extra

material. Another side effect of the described situation is the uneven loading of

the tie bars, thus tie bar breakage or severe wearing of bushings might be also

experienced. The intensification pressure used at the end of filling has also

being characterized as important in die deflection modeling. Research findings

suggest that this pressure load may affect cavity dimensions and also can

worsen flashing problems if uneven sealing of the parting plane is present.

51

2.6.2 Casting distortion modeling

The prediction of residual stresses and part distortion in die casting has

been analyzed by Ragab [35], Sequeria et al [36], and Caulk et al [37]. The first

two studies rely in the use of commercially available Computer Aided

Engineering (CAE) packages, the first one based on a finite element formulation

using Abaqus, whereas the second one uses the finite difference based code

tailored for casting analyzes MAGMASoft. Caulk et al [37] present a formulation

that employs a combination of boundary element and finite element methods. A

description of the procedures followed in each case study is presented.

The understanding of phenomena governing die deflection modeling in the

Center for Die Casting brought up the issue of incorporating the possibility of

modeling casting distortion. The case study presented by Ragab [35] was aimed

at predicting the distortion and residual stresses in a ribbed plate. The modeling

stages were as follows. At the start of the simulation, the casting was placed in

contact with the undeformed die. Subsequently, the clamping, thermal and

pressure loads were applied all instantaneously. After these loads were applied,

the casting was allawed to cool inside the die. The cooling of the casting to room

temperature was also simulated, taking the displacements, temperatures and

stresses at the point of ejection as the initial conditions for this analysis. A

sensitivity analysis in which several parameters such as: material constitutive

model, gap dependency of heat transfer coefficient was also performed to test

their impact on the final distortion results. Dimensional results of selected

features of the casting from the simulation model were compared against

52

experimental results. A fair amount of disagreement was found between the

experimental results and the simulation predictions, having the greatest

difference for casting features across the parting plane. Three main factors were

identified as the main source of disagreement. The first one was the inability of

the casting solid elements to follow the distorted shape of the die as the

clamping, thermal and pressure loads were applied, leading to inaccuracies in

the initial shape of the casting as well as to some contact stresses induced in the

casting from contact with the die walls. The second factor was the modeling

inaccuracy in representing the intensification pressure load. This load, which is

in actuality transferred by the liquid casting to the cavity walls, was modeled as a

boundary condition applied at the cavity surface. In doing this, the casting and

the cavity wall were not placed in tight contact at the beginning of the cooling

period, thus initial contact conditions of the casting features with those of the die

were not properly represented. A side effect of the intensification pressure used

in die casting is the fact that the phase change of the liquid metal takes place

under the action of high pressures. This condition means that a hydrostatic

compressive stress of state should be applied to the solidifying liquid elements,

translating into initial plastic deformation of the weak metal, producing high

distortions. This complex phenomenon was also not modeled. The lack of

material properties that can describe the mechanical behavior of the material at

high temperatures, especially near the solidus, was the third factor that was

identified as contributor for the differences between the modeling and the

experimental results.

53

The distortion of an aluminum outboard housing was modeled by Sequeria

et al [36]. The modeling work was done using the module MAGMAstress

included in the commercial casting software MAGMASoft. The procedure to

predict residual stresses and distortions in MAGMA is carried out in two stages.

During the first stage the die quasi steady-state temperature profile is obtained

by simulating a repeated number of thermal cycles. Once this quasi steady-state

profile is reached, the casting is added to the analysis and the cooling period of

the casting cycle is simulated. It is worth mentioning that this analysis is purely

thermal; thus, no account for any deformations and stresses due to contact

interactions between the casting and the die is being taken. At the end of this

step, the casting final temperature profile is extracted and is taken as the initial

condition for the second stage. This part of the analysis consists in performing a

thermal-mechanical analysis on the casting under free convection thermal

conditions. The results of this part produce the distorted pattern that results from

the cooling of the casting to room temperature. Typical outputs from this kind of

simulation are displacements and residual stresses. As far as the case study is

concerned, it was stated that the numerical displacement predictions obtained

from the analysis correlate well with experimental measurements. However, the

modeling procedure just described does not take into account any deformation of

the die that results from thermal and mechanical loads present during normal

operation, thus any dimensional changes in the cavity resulting from such loads

are not accounted for. Furthermore, residual stresses and distortion of the

casting that results from hindered shrinkage by the die walls is also not being

54

considered, since the modeling procedure does not model contact between the

solid bodies. Phase change under pressurized conditions is likely to be also not

incorporated, thus one more source of inaccuracy is present as well. Based on

the above mentioned factors, it is doubtful that the reported simulation results

could correlate well with the experimental measurements.

The modeling procedure described in [25] for modeling of die distortion is

part of the numerical scheme used by Caulk et al [37] to model distortion and

residual stresses in die casting parts. The procedure followed can be described

in two stages. In the first stage a die distortion analysis is performed in order to

obtain the deformed operating conditions of the die. Modeling of die distortion is

obtained by considering the thermal load that results from the continuous

injection and solidification of the casting inside the die, the clamping load, and

the intensification pressure. The casting is not incorporated in this stage of the

model. It is assumed that the casting ejects with the same shape as the

deformed die cavity. It is claimed that because the yield stress of the casting is

so low while inside the die, most of the deviatoric stresses are insignificant. The

second stage consists of a thermo-mechanical model that represents the cooling

of the casting after ejection. It is assumed that the casting responds elastically

as it cools outside the die. It is worth mentioning that the casting is modeled

using linear shell elements, and the analysis is solved using a finite element

formulation.

55

2.7 Casting distortion studies

A series of research projects aimed at determining the variability of casting

dimensions in regard to process variables and casting geometric features were

done in Professor’s Voigt group at the Pennsylvania State University [2, 38-39].

The studies done covered three casting processes, die casting [2], sand casting

[38] and investment casting [39]. The major goal in all these studies was to

predict, by using statistical methods, the pattern allowance that should be applied

to the mold based on different casting geometric features. The projects

consisted in performing a statistical dimensional analysis of a series of casting

geometric features, the results of which provided a measure of the dimensional

variability of the feature in regard to different process variables as well as its own

geometric shape.

As far as the die casting study is concerned [2], several important

conclusions were drawn that are worth mentioning. A major conclusion of this

study was the fact that current dimensional tolerance standards in die casting do

not represent the actual capabilities of the process. Underestimation of

tolerances for small features and overestimation of tolerances for large features

figured among the most common practices in industry. The study also concluded

that casting geometry variables and process variables significantly affect the

dimensional stability of die casting features. The following variables were

identified as major sources of dimensional variability: casting feature length, shot

weight, feature restraint, hold time, and die locking force. The degree of restraint

imposed by a core or a die wall was also identified as a contributor in this regard.

56

Numerical predictions of dimensional changes in steel castings using

MAGMASoft were conducted by Professor Beckermann’s group [40]. The goal

of the study was to assess the capabilities of MAGMA to predict pattern

allowance factors for steel sand castings by comparing the results of the

computer models with measurements done by Professor’s Voigt group at the

Pennsylvania State University. The geometries analyzed corresponded to one of

those analyzed by Peters [38] and a shovel adapter produced by an industrial

partner. A thermal-mechanical analysis of the solidification of each sand casting

was done and the results were compared with experimental measurements. This

case study concluded that for features that undergo unrestrained shrinkage, the

results provided by the computer model showed fairly good agreement with the

experimental ones. However, large disagreement was reported for features that

experienced partial or complete restraint during shrinkage. Among the possible

reasons identified were the inability of MAGMA to incorporate the mold material

in the stress analysis, leading to inaccurate modeling of phenomenon such as

gap formation and/or development of stresses and deformations in castings due

to contact interactions between the casting and the mold. Another reason was

the inaccurate modeling of irreversible sand expansion experienced by some

types of sand when a certain temperature range is reached.

The results of an industrial study performed by Honda engineers were

presented in [41]. The goal of this project was to determine the most significant

process variables that dictate the dimensional stability of die castings. A

statistical analysis of casting geometric features was done in order to assess the

57

influence of process variables such as: melting temperature, die filling velocity,

water quenching temperature, spraying time and holding time. The study

concluded that holding time was the most significant factor that affects the

dimensional variability of die castings. It was stated that all other factors had a

minor effect in this regard.

A study of the effects of process control parameters on casting

dimensional variability for die casting was done by Osborne [42]. The research

study focused on determining the effect of process parameters on across parting

plane dimensions and in dimensions within the ejector die. The selected factors

were: injection velocity, dwell time, cycle time, cooling line, spray time and metal

temperature. To assess the effects of these process variables two die designs

were selected. For across parting plane dimensions a flat plate, the parting

plane of which was placed in the middle of the plate thickness, was selected.

The plate thickness and its flatness were chosen as response variables. To

study dimensions within the ejector die, a flat plate with four ribs protruding into

the ejector die was selected. The three distances formed between the four ribs

were chosen as response variables for this case. A fractional factorial design

was used as a guide to conduct the experiments. Analysis of variance results

showed that the gate velocity, cooling line and the cycle time have a significant

effect on the plate thickness. Flatness measurements showed no significant

dependence on any of the selected factors. Statistical results showed that

dimensions within the ejector die are significantly affected by the dwell time only,

no other factor showed any significant effect on the analyzed features.

58

Computer models have also been recently used for the prediction of

tooling allowance factors for investment casting [43]. Investment casting

represents a more challenging case due to the complexity of the many physical

phenomena involved in producing a final casting. Production of the wax pattern,

shell, dewaxing, and final casting all add more complexity to the already difficult

task of predicting tooling shrinkage factors. In essence, two different molds are

used before a casting is actually produced, and thus tooling factors for each mold

have to be accurately predicted before a dimensionally sound casting is

obtained. The study only considered the die-wax and shell-alloy systems, and

ignored the shell-wax system. It was stated that because the fused silica shell

experiences null deformation before the casting is poured, this part of the

process could actually be ignored in the analysis.

The die-wax model was developed using Abaqus. A visco-elastic

constitutive model was used to characterize the wax, the properties of which had

been previously determined by the authors. An eight-node thermally coupled

brick hybrid element able to carry hydrostatic pressure was used to represent the

wax. The reported results for the die-wax system showed fairly large

disagreement between the computer model and the experimental data.

Shrinkage factors for the computer model were 2.5 times larger than the

measured ones. Regarding the shell-alloy system, limited modeling details were

given in the report. The model consisted in predicting the deformation of an

aluminum A356 casting. It was claimed that good agreement was found between

59

the computer model predictions and the experimental measurements; however,

numerical results were not reported.

2.8 Summary

Castings distort and develop residual stresses mainly due to the effect of

two restraints. One source of restraint is the mold walls that prevent the casting

from freely shrinking while cooling inside the mold. The other set of restraints is

developed due to the differential thermal cooling of different regions of the

casting. This differential thermal cooling is the result of the irregularity of the

different shapes that make up a casting.

Analyzing casting distortion for any casting process represents a

challenging endeavor due to the complex interplay of the many physical

phenomena involved. From a macroscopic point of view, it involves solving the

fluid flow, heat transfer and stress equations in a coupled manner. It is the stress

aspects of the analysis that represent the greater challenge due to complex

nonlinearities in material behavior and contact conditions during solidification and

cooling.

Finite elements have emerged as a reliable tool for solving the complex

set of nonlinear equations afore mentioned. The wide variety of the different

techniques developed is based on different sets of simplifying assumptions to

make the problem solvable. The insights provided by this numerical analysis

toolset have helped product designers and tool builders produce better castings.

60

In die casting finite elements and finite difference techniques have been

employed to solve the complex problem of casting distortion. The problem has

been mainly formulated as thermal-mechanical, without accounting for the fluid

flow aspects due to the fast filling times characteristic of the process. The

developed methodologies differ from each other based on the different

assumptions taken and also based on the limitations of the formulation used.

61

3 CHAPTER 3

MODELING METHODOLOGY

3.1 Introduction

This chapter describes the modeling approach taken to predict the

distortion of the casting. Because die casting is a forming process that relies on

the use of a rather stiff metallic die to shape the final product, deformations on

tooling shape cavity become significantly important when trying to predict final

product dimensions. Therefore, the analysis and prediction of final casting shape

must include the prediction of the die cavity shape at operating conditions.

This chapter is divided into two main parts: methodology for modeling die

distortion and the methodology for modeling part distortion. The first part

describes in detail all the modeling steps followed to predict the operating die

conditions, thermal and mechanical. The second part uses the predictions given

by the die distortion model and incorporates the casting into the analysis in order

to predict the shape of the casting while it cools inside the die and after ejection.

A detailed description of each of these two sections is provided in the following

pages.

62

3.2 Die distortion modeling

Modeling of die distortion has been an active research area at the Center

for Die Casting at The Ohio State University. The results of the studies done

have been published in a series of papers [28-34]. As a result of the early

research findings it was determined that the main causes of the elastic

deflections experienced by the die are: the clamping force developed by the tie

bar stretching, the uneven thermal profile of the die that results from the

continuous injection and cooling of material cycle after cycle and the

intensification pressure applied to increase material feeding and reduce pore size

across the casting volume. It has been found that the result of the interaction of

all these static loads leads to uneven elastic deflections of the die that have

proved to affect its performance as well as the cavity dimensions. The

description of the model and the approach followed to model the mentioned

loads is provided in the following paragraphs.

Modeling of die distortion requires the inclusion of the die and machine

elements. In this research project the model was built based on a Buhler SC-250

ton cold chamber die casting machine available at Ohio State. The model

included the die, ejector support block, ejector and cover platens. Tie bars were

not included due to the procedure used to model the clamping force which is

described later. Fig. 3.1 shows the described model.

h

th

Beca

alf of the m

he nodes ly

use of the l

model was u

ying at the s

Fig. 3.1 M

left-to-right

used. A sym

symmetry p

63

Machine mode

symmetry

mmetric bo

plane, const

3

el finite eleme

of the mach

undary con

training the

ent mesh

hine and th

ndition was

eir motion in

he casting, o

specified o

n the directi

only

on all

ion

64

normal to that plane. In order to simulate the resting of the cover platen on the

machine foundation and of the ejector platen on the rails, the motion in the

vertical direction of some nodes on these surfaces was restrainted. The contact

interaction among all the structural elements was specified as small sliding,

having a coulomb friction coefficient of 0.1.

3.2.1 Clamping force modeling

Modeling of clamping force required a particular procedure. To model the

effects of clamping force a pressure load was applied on both platens as shown

in Fig. 3.2. Fig. 3.3 shows the areas where the pressure was applied. The

region at the back of the ejector platen was selected as the area where the

toggle pads act, whereas the area in the cover platen represents the regions

where the reaction from the tie bar locking nuts is developed. The application of

both pressure loads was done during the same step. The clamping force

magnitude was equally divided between the top and bottom regions on each side

of the machine, and thus, any unbalance due to die location was not accounted

for. The unbalanced between the top and bottom tie bars has been estimated to

be about 8%, and it is thought to be negligible [44].

The procedure described was followed in order to arrest an observed

semi-rigid body motion that resulted from tie bar stretching when tie bars were

included in preliminary models. The observed motion caused divergence when

contact was being established between the casting and the die surfaces. A more

detailed description of this situation will be provided when the procedure to model

the cooling of the casting inside the die is described in the following sections.

65

Fig. 3.2 Schematic of clamping method

66

Fig. 3.3 Location of clamping pressure on platens

Modeling clamping force as a constant pressure load may cause

suboptimal machine deflection patterns. The assumption of equal load between

the top and the bottom might be reflected in a platen distortion patterns that may

not represent the machine characteristics and die sealing at the parting plane

adequately. In reality, when the die is moved down the bottom tie bars take up

more load and greater contact pressure magnitudes are obtained at the bottom

67

of the parting plane. The assumption of equal load therefore, may be expected

to alter this profile, possibly affecting die opening magnitudes when the pressure

load is applied. Additionally, tie bar load redistribution due to the effects of

intensification pressure cannot be account for, since the pressure magnitude is

kept constant throughout the analysis. The inability to account for this load

redistribution may lead to slightly different platen distortion patterns that could

possibly change across parting plane dimensions.

3.2.2 Thermal load modeling

The thermal load represents the uneven thermal profile that develops in

the die as a result of the continuous injection and cooling of fresh metal cycle

after cycle. Thus, in order to model this load a series of cycles simulating the

different thermal conditions experienced by the die had to be modeled in a

transient thermal analysis. The thermal model included the casting, die and

cover platen.

A series of twenty thermal cycles were run in order to develop the die

quasi steady-state thermal profile. A description of the boundary conditions

specified on each of the different stages in each of the cycles is presented in

Table 3.1. All the structural elements are assumed to start at a room

temperature of 30 ºC in the first cycle. The liquid metal is assumed to be injected

in each cycle at a uniform temperature of 600 ºC. The heat transfer coefficient

between the casting and the die, as well as all of the structural elements was

kept constant and with a magnitude of 5000 W/m2-K. Although this coefficient is

time and temperature dependent for the case of casting/die interaction,

68

simulation results have proved to be rather insensitive to these effects. This

might be attributed to the finite capacity of the die to extract heat because of the

drop of its thermal conductivity as its temperature rises. Additionally, as the

liquid metal releases its latent heat during solidification, the surface temperature

of the die rises and in some regions may be equal to the casting temperature,

which limits the amount of heat flux through the interface.

69

Stage Boundary conditions

Die is closed

• Contact between the casting/die is disabled

• Contact between the die halves is enabled

• Free convection on exposed surfaces of die

• Free convection on the die cavity surface

• Free convection on exposed surface of cover platen

Dwell time • Contact between casting/die is enabled

• Free convection on die cavity surface is disabled

Die is open

• Casting/die contact on the cover side is disabled

• Contact between the die halves is disabled

• Free convection on exposed surfaces on the die

• Free convection on exposed surface on the casting

Casting is ejected • Casting/die contact on the ejector side is disabled

• Free convection on exposed surfaces on the die

Spray • Forced convection on the die cavity

• Free convection on remaining surfaces of the die

Idle • Free convection on exposed surfaces on the die

Table 3.1 Boundary conditions applied on the simulated thermal cycles

70

Table 3.2 to Table 3.4 show the physical and thermal material properties

used in the analysis for each of the different materials. The die was assigned H-

13 material properties, whereas the cover platen was characterized using the

steel 4140 material properties. The casting material was aluminum A380.0,

having the typical material properties found in the die casting literature.

Property Magnitude

Density 7820 kg/m3

Coefficient of thermal expansion 1.7x10-5

Thermal conductivity 29 W/m2-K

Specific heat

23 °C 458.8 J/kg-K

200 °C 518.5 J/kg-K

400 °C 587.7 J/kg-K

600 °C 726.2 J/kg-K

700 °C 905.4 J/kg-K

760 °C 1151.1 J/kg-K

800 °C 885 J/kg-K

850 °C 792.7 J/kg-K

900 °C 747.9 J/kg-K

1000 °C 733 J/kg-K

Table 3.2 Physical properties for H-13 tool steel

71

Property Magnitude

Density 7820 kg/m3

Coefficient of thermal expansion 1.7x10-5

Thermal conductivity 40 W/m2-K

Specific heat

150 °C 473 J/kg-K

200 °C 473 J/kg-K

350 °C 520 J/kg-K

400 °C 520 J/kg-K

550 °C 561 J/kg-K

600 °C 561 J/kg-K

Table 3.3 Physical properties for a typical 4140 steel alloy

Property Magnitude

Specific heat 419 J/kg-K

Latent heat 120000 J/kg

Liquidus temperature 580 C

Solidus temperature 538 C

Density 2700 kg/m3

Table 3.4 Physical properties for aluminum A380.0 die casting alloy

72

3.2.3 Intensification pressure modeling

The intensification pressure was modeled as a constant pressure load. It

was applied on all the die cavity surfaces where the pressurized liquid casting

would act upon the die. Even though the intensification pressure is time and

space dependent, modeling it in a dynamic analysis would make the running time

extensively long, making the analysis rather impractical for the scope of this

research work. Additionally, the uncertainty in the spatial and temporal variation

of the pressure may make the analysis rather inconclusive.

The die distortion model is assembled in a single Abaqus input file. The

nodes and element sets of all the structural components are all listed in the input

file. Material properties are defined and assigned to each of the different

components. In addition to the listed thermal and physical properties shown in

Table 3.2 to Table 3.4 typical magnitudes for Young’s Modulus and Poisson’s

ratio for steel are included in the material properties as well.

The loads as well as the boundary conditions are defined in the different

steps of the analysis. For the die distortion model, a static analysis is performed

and the described loads are sequentially applied in three different steps. The

first step applies the clamping force. The application of the thermal load is done

during the second step by specifying the nodal temperatures at the point of

injection as predicted by the transient thermal analysis. Lastly, in the third step

the cavity pressure is applied.

73

3.3 Modeling of part distortion

The modeling of part distortion was divided into three different models.

The first model determines the initial casting shape after the metal has filled the

die cavity. The second model simulates the cooling of the casting inside the die.

The third model simulates the cooling of the casting after ejection. The models

are run in the described sequence and the results provided by each model are

subsequently used by the following one. A detailed description of the modeling

procedure used in each case is provided in the following sections.

Fig. 3.4 shows the casting used for this research project. The geometry of

the casting was designed for a process control study done by Osborne [ 42] as

part of his research work. All of the casting ribs are formed in the ejector half,

with the cover contributing only to form the back of the plate. The design of this

casting was done in order to use the distance between the ribs as a measure for

in-cavity distortion, whereas the depth of the ribs was used to provide a measure

for across parting plane distortion.

3

st

th

d

.3.1 Mode

The p

tep towards

he die expe

ifferent pro

eling the tra

predictions

s the mode

eriences ela

ocess loads

Fig. 3

acking of ca

obtained by

eling of cast

astic deflect

. These de

74

3.4 Casting fin

avity distorti

y the die di

ting distortio

tions result

eflections ca

4

nite element m

ion

stortion mo

on. As has

ing from the

ause small

mesh

odel represe

s been alrea

e combined

but importa

ent the first

ady describ

d effects of

ant

t

bed,

the

75

dimensional changes in the die cavity, changes that must be captured since they

represent the initial shape the liquid casting acquires at the end of the filling.

In order to know what the initial casting shape is prior to the onset of

cooling, an accurate description of the deformed die cavity shape must be

obtained first. Ragab [ 35] experimented tracking the distorted die shape by tying

the casting surfaces to the die cavity surfaces. The model included the casting,

die, platens, ejector support block. All parts in the model were discretized using

solid brick finite elements. It was reported that because the casting was

represented by solid elements, limitations in the element deformations prevented

it from accurately tracking the die shape as the different loads were applied.

The methodology used in this research work relies in the use of a shell

mesh to track the distortions in the cavity. The same structural components for

the die distortion model are used in this model. The application of the three

different process loads is sequentially done as described before. In order to

avoid having a solid casting tracking the cavity distortions, a shell mesh is used

instead. The shell mesh is built using the surface elements of the casting mesh,

sharing the same nodes as the casting surface elements. By sharing the same

nodes, the displacements obtained from the shell mesh can be readily mapped

onto the casting mesh surface.

The dimensional changes in the cavity are tracked by tying the shell mesh

to the die cavity mesh. Tying the shell to the die cavity provides a description of

the distorted die cavity shape after the application of the already mentioned static

loads. After the die has been distorted, the predicted displacements of the shell

76

mesh can be applied to the casting mesh, providing a casting shape that

matches that of the deformed die cavity. The underlying assumption in this

procedure is that the distortions in the cavity are small enough that they can be

mapped only to the casting surface without affecting its interior structure.

Fig. 3.5 shows the shell mesh used for this model. With a three-

dimensional mesh of the casting, the shell mesh can be readily obtained using

the meshing capabilities of any pre-processor. For this model, a shell thickness

of 0.0254mm was specified. To avoid any modeling limitations due to rigidity on

the shell, the Young’s Modulus of the shell material was specified to be three

orders of magnitude smaller than that of regular steel.

The model is run as a static analysis. As has been already described,

clamping, thermal and pressure load are sequentially applied in three different

steps. At the end of the analysis, the displacement predictions from the shell are

extracted and used in the following model. The description for modeling the

cooling of casting inside the die is provided in the next section.

3

th

ej

d

b

.3.2 Mode

Mode

he casting t

jector supp

imensional

y the shell

eling the co

eling the co

to the die d

port block a

casting me

mesh.

F

oling of cas

oling of the

istortion mo

nd the cast

esh are mo

77

ig. 3.5 Shell e

sting inside

e casting ins

odel. This

ting. The in

odified using

7

element mesh

the die

side the die

model inco

nitial nodal

g the displa

h

e requires th

orporates th

coordinates

acement pre

he addition

e die, plate

s of the thre

edictions gi

of

ens,

ee-

iven

78

The analysis is run as fully-coupled thermal-mechanical in two different

steps. During the first step the contact between the casting and the die is

disabled and the whole structure is elastically deformed by applying the same

clamping, thermal and pressure loads as in the preceding shell model. The

objective is to reproduce the die cavity deflections that were obtained in the shell

model, but now in a fully-coupled thermal-mechanical model. In the second step,

once the structure has already been deformed and the casting shape comes into

perfect contact with the distorted cavity shape, the contact between the casting

and the die is enabled and the casting is left to cool for a specified dwell time.

Based on the procedure just described, it can be noticed that the modeling

of the intensification pressure required a decoupling between the casting and the

die at the contact surfaces. The intensification pressure is modeled as a

pressure load on the faces of the die cavity surface elements, where in actuality

comes from the loading action of the casting onto the die. This decoupling

means that the pressurized conditions under which the casting cools inside the

die are not present, and as such, the contact conditions are not being properly

represented. This pressurization represents also an initial hydrostatic state of

stress which may lead to initial plastic strains at the early stages of cooling and

possibly to elastic strains at the later stages when the casting has acquired

enough strength. This modeling limitation is currently faced by the lack of finite

elements that are able to carry hydrostatic pressure and develop multi-phase

behavior depending on their temperature. Although Abaqus does list a “hybrid”

element in its element library possessing the characteristics just described,

79

preliminary tests performed confirmed the inability to transfer the hydrostatic

pressure uniformly throughout the casting volume. It was also observed that the

deformations induced due to thermal loads behaved unrealistically.

Another modeling item that deserves a more detailed explanation is the

clamping force modeling procedure. As it is well known, the clamping force is

developed due to tie bar stretching. As the tie bars stretch there usually is some

quasi-rigid body motion created along the clamping direction. During the course

of this research work a model that included the tie bars was initially used. As

already described, the displacements obtained by the shell mesh are used to

provide a description of the distorted cavity, a distortion that is then mapped to

the three-dimensional casting mesh using the shell nodal displacements.

However, in the fully-coupled thermal-mechanical model that simulates the

cooling of casting in the die, convergence problems were faced because of the

inability of the contact algorithm in Abaqus to properly establish contact between

the casting and the die cavity surfaces. This modeling difficulty was experienced

in spite of the fact that there was a perfect match between the casting and the die

cavity surfaces at the point at which the contact between the surfaces was

enabled.

The source of the divergence was found to be the rigid body motion of the

structure prior to the establishment of contact between the casting and the die.

Abaqus/Standard, which uses an implicit algorithm to solve for the equilibrium

equations, has limited capabilities if such contact conditions are present during

80

the analysis. This solver requires that all contacting surfaces meet at the

beginning of the analysis in order to establish contact properly.

3.3.3 Modeling cooling of casting post-ejection

Modeling the cooling of the casting after ejection required a model with the

casting being the only component. As the previous case, the model is defined as

a fully-coupled thermal-mechanical analysis. The predicted nodal displacements,

nodal temperatures and stresses provided by the previous analysis are used as

initial conditions in this model. The initial coordinates of the casting are modified

by adding the displacements at ejection to the casting starting coordinates. In

order to guarantee mechanical equilibrium, a series of weak spring elements are

used to restraint the casting motion in all three directions. The stiffness of the

springs is set to be 10N/m to avoid any artificial loading onto the casting. A

convective boundary condition of 20W/m2-K is applied to the casting surface to

simulate the heat transfer conditions between the casting and the surrounding

air. The analysis is run for sufficient time to guarantee the casting temperature

reaches that of the environment.

3.4 Constitutive model for the casting material

The casting was represented using an elastic-plastic constitutive model.

The constitutive model describes the mechanical behavior of the casting material

and should include appropriate representations over the range of temperatures

relevant for the analysis. For this analysis, temperature dependent elastic

81

properties and temperature and strain rate dependent plastic properties were

used.

The elastic properties included the definition of the Young’s Modulus and

Poisson’s ratio over a range of temperatures. Fig. 3.6 shows the temperature

dependency of pure aluminum, AA1201 and AA3104. As can be noticed from

the curves, Young’s modulus for aluminum alloys is independent of the alloy

chemistry. The results shown confirm the understanding that Young’s Modulus is

an intrinsic material property that is a function of the strength of the atomic

bonding. The results of Fig. 3.6 were used to specify the elastic properties for

the casting alloy used in this study.

82

Fig. 3.6 Temperature dependence of Young’s Modulus [45 ].

The plastic behavior of the die casting alloy was experimentally

determined as part of this research work. A series of tensile tests at different

combinations of temperatures and strain rates were conducted and the results

were used as input data in the finite element model. The details of this

experimental research project are presented in Chapter 4.

83

3.4.1 Finite element selection for casting

The casting was represented by using reduced integration finite elements

with temperature and displacement degrees of freedom. Reduced integration

was employed because volumetric locking was observed in some regions of the

casting when fully integrated finite elements were used. Volumetric locking is a

numerical phenomenon characteristic of plastic analysis that rely on

approximating the incompressibility of the material response by making the

Poisson’s ratio close to ½. This incompressibility response brings the constraint

of no change in volume into the finite elements that must be satisfied in addition

to meeting the displacement field that results from the kinematics of the problem

being analyzed. These constraints must be satisfied at all integrations points in

the finite elements. Since the fully integrated coupled thermal-mechanical finite

elements in Abaqus (C3D8T) use 2 integration points along each direction (a

total of 8 per element), the constraints cannot be met and the element is said to

“lock”. Reduced integrated elements use one lesser integration point along each

direction facilitating meeting the mentioned constraints.

3.5 Modeling die distortion using Fluid-Structure-Interaction (FSI) in ADINA

The distorted die cavity at the end of filling represents the initial shape

conditions for the casting at the onset of solidification. This distorted die cavity

shape results from the effects of mainly three process loads, clamping,

temperature and intensification pressure. Accurate predictions of casting final

dimensions require modeling the distorted die cavity shape adequately, since the

84

elastic deflections experienced produced dimensional changes in the die cavity

that affect the casting dimensions.

In die distortion models the intensification pressure effects have been

traditionally represented by using a constant pressure boundary condition applied

to the die cavity surface. Ideally, the intensification pressure should be the result

of the loading action of the pressurized casting acting onto the die cavity

surfaces. However, this hydrostatic loading cannot be modeled because

continuum solid elements lack a hydrostatic pressure degree of freedom and are

rendered inadequate for these purposes.

The latest developments in finite element modeling algorithms allow

modeling the interaction of fluid and structural elements. These modeling

capabilities represent the state-of-the-art in finite element codes and were initially

adopted for modeling casting and die distortion. The finite element package

ADINA was selected because it allows multi-physics modeling in one single

integrated code.

The adoption of this modeling technique was thought to augment casting

and die distortion modeling efforts due to the following. First, since the casting

could be represented using liquid elements that possess a pressure degree of

freedom, modeling intensification pressure effects could be readily done by

applying a pressure load to the biscuit and letting the casting fluid elements load

the deformable die. Second, the distorted die cavity shape at the end of filling

could be readily obtained from the distorted casting mesh because the fluid-

structure-interaction algorithm requires the liquid elements to always follow the

85

distorted shape. Therefore, the initial casting shape at the onset of solidification

could be readily obtained from the distorted fluid mesh.

3.5.1 Fluid Structure Interaction (FSI) Model

The model was divided into two domains, namely the solid and the liquid

domains. The solid domain was comprised of the structural elements which

included the inserts, die, ejector support block, ejector and cover platens and tie

bars. The liquid domain was on the other hand comprised of only the casting. In

the structural elements the usual boundary conditions applied to die distortion

models were considered. Contact between all the deformable bodies was

incorporated as well. In the fluid domain, the casting was represented with liquid

elements, which have pressure and velocity degrees of freedom. A fluid-

structure-interaction boundary condition was specified for all the surfaces in both

the casting and the die where they were expected to interact. Clamping and

thermal force was simulated by applying a pressure load on the ejector platen.

Thermal load was modeled by prescribing a temperature load on the inserts and

die. Intensification pressure was modeled by normal surface traction load on the

biscuit region of the casting. The loads were applied sequentially in a total of

three steps.

The displacements on the casting mesh were extracted at the end of the

analysis. Fig. 3.7 shows the predictions given by the FSI model. These results

show that the cavity distortion is not negligible and the predicted displacement

magnitudes emphasize the importance of incorporating the contributions of the

die deflections in casting distortion analyses.

86

Fig. 3.7 FSI cavity displacement predictions

This displacement field represented the distorted cavity shape at the end

of filling and was considered as the initial casting shape for a subsequent

thermal-mechanical solidification/cooling model. The same mesh for all the

different components used in the FSI model was used in a fully coupled thermal-

mechanical model. The model was comprised of the same structural

components as before with the addition of the casting. The final coordinates of

87

the casting in the FSI model were taken as the initial coordinates for the thermal-

mechanical model.

Since modeling casting solidification/cooling must consider the interaction

between the casting and the die, the deformed state predicted in the FSI model

had to be reproduced using the same loading conditions. Clamping and thermal

loads were modeled as described, whereas the intensification pressure was now

modeled as a pressure load applied to the die cavity surfaces. Contact between

the casting and the die was not enabled until the deformation was reproduced. It

is worth mentioning that during this model all the finite elements used to

represent all the components had temperature and displacements degrees of

freedom. The model was run in Abaqus, using the C3D8T coupled temperature-

displacement three dimensional elements.

While running this model convergence difficulties were experienced.

Examination of the results showed a mismatch between the casting shape and

die cavity surfaces after the initial loading was applied. The semi-rigid body

motion due to tie bar stretching after clamping was identified as the source of

divergence in the thermal-mechanical model. The displacement predictions after

clamping between the FSI and the thermal-mechanical model differed by as

much as 0.07mm and this difference was large enough to prevent proper

establishment of contact.

It was conjectured that the different contact algorithms used by each code

produced different displacements. Since coupled thermal-mechanical models

can be analyzed in ADINA it was decided to set up the solidification model and

88

run it all in ADINA. The same loading and boundary conditions as described

before were reproduced in an ADINA thermal-mechanical model. The results

showed that the displacement predictions of the thermal-mechanical model were

different when compared with the FSI predictions. The magnitude of these

differences was close to 0.07mm as well. These differences in displacement

predictions within ADINA prevented further modeling efforts using the code and it

was abandoned.

89

4 CHAPTER 4

DETERMINATION OF CASTING ALLOY CONSTITUTIVE

MODEL

4.1 Introduction

Physical and mechanical material properties are a fundamental piece of

information needed in computer modeling. The accuracy in the properties

magnitude and their dependencies on different material characteristics determine

to a large degree the predictions given by the computer models. Different kinds

of properties are needed depending on the type of analysis being performed. For

instance, thermal analyses require physical and thermo-physical properties such

as thermal conductivity, density, specific heat and latent heat. Depending on the

type of analysis, mechanical finite element models usually require properties

such as Young’s Modulus and Poison’s ratio. If the material deformation is

expected to exceed the elastic range, definition of the plastic behavior must be

part of the material properties as well.

As outlined by Ludwig [46], casting distortion computer models require a

particular set of thermal and mechanical material properties. On one hand,

90

the thermal properties are needed because they are used to compute the

evolution of temperatures in the casting as it solidifies and cools. On the other

hand, the mechanical properties are needed because they are used to predict

the strains and stresses in the casting as it cools and develops strength. The

accuracy in the magnitudes of the required properties and the adequate

expression of their dependencies on physical quantities such as temperature, or

on material characteristics such as strain rate, determine the accuracy of the

predictions given by the computer models.

Currently, the lack of high temperature mechanical properties for die

casting alloys is limiting the advancement of casting distortion computer models

in the die casting industry. The thermal properties are thermo-physical and

physical properties for which well-known procedures are available and have been

widely determined already. At the present time, computer codes have been

developed that can compute theses properties using thermo-dynamical principles

as presented by Saunders et al [47], Miettinen [48] and Miettinen et al [49].

These codes have been readily incorporated in casting CAE packages such as

ProCAST [50]. However, the mechanical properties at high temperatures are not

available and their determination remains open. The need is justified because it

can help increase the productivity of the industry by providing tools to predict

machine tooling allowances that can produce castings within tolerances at the

first shot, eliminating the high cost trial and error procedure. High-temperature

mechanical properties can also be used to predict casting residual stresses and

91

give product designers a valuable input to their designs before a single casting is

produced.

This chapter describes the activities done to determine the elevated

temperature mechanical properties for the die casting aluminum alloy A380.0.

First, a background regarding the finite element procedures used to predict

distortion and stresses in castings is provided. This section highlights the

importance and relevance of the use of accurate mechanical material properties

from a computational point of view. Next, a literature review of the available

high-temperature mechanical properties for some casting alloys and the testing

methods used for their determination is presented. The procedures followed to

obtain the elevated temperature mechanical properties for the die casting

aluminum alloy A380.0 are presented next. The section describes the selected

specimen, machine, testing temperatures and methodology. The results of the

tensile tests performed to the material are presented in the following section.

4.2 Background

Finite element analyses are mathematical models aimed at representing

the performance of a given isolated natural system. The mathematical

expressions represent the differential equations that mathematically describe the

nature of the physical phenomenon under study. As discussed in Chapter 2

section 2.4.2, the set of mathematical differential equations get assembled in an

algebraic form that for a mechanical system lead to the following three sets of

equations: mechanical equilibrium equations that relate forces and stresses,

92

compatibility equations that relate strains to displacements and finally the

constitutive equations that relate stresses to strains. It is in the last set of

equations that material properties are used to map the strains to stresses using

the elasticity and/or plasticity matrices. This chapter presents the finite element

procedures followed to compute the stresses in a typical casting analysis.

As was described in Chapter 2 section 2.4.2 elastic strains are responsible

for residual stresses in castings. However, it has been stated by Thomas [13]

that the expansions and contractions due to temperature change and phase

transformations, plasticity and creep contribute to generation of small but

important elastic strains. Therefore, the total increment strain vector is

composed of the elastic, thermal and inelastic strain components

∆ ∆ ∆ ∆ Eq. 4.1

where:

∆ is the incremental elastic strain

∆ is the incremental thermal strain

∆ is the incremental inelastic strain

As presented by Ludwig [46], viscoplasticity occurs when the comparison

stress in the casting material reaches the yield stress, which is temperature

dependent . Usually, the von Mises criterion is employed as the

comparison stress for the analysis, given the following flow condition

93

0,0,

with

12

12

12 3

The visco-plastic flow model can be expressed in different mathematical

models. Abaqus offers among many others the Johnson-Cook hardening model,

Drucker-Prager model and Hyperbolic-sine law model. Assuming a Power-law

model as shown in [46], the inelastic incremental strain rate can be expressed as

∆∆

Eq. 4.2

with

where

is the hardening exponent

The incremental thermal strain is expressed as:

∆ ∆ 1,1,1,0,0,0

94

where

is the thermal expansion coefficient

With the expressions for the elastic, thermal and inelastic incremental

strains available, the generalized constitutive equation can be expressed as

follows [46]

∆ · ∆ ∆ ∆ Eq. 4.3

∆

0,

∆,

With ∆ being a fictive stress that accounts for the temperature

dependence of the yield stress .

The plasticity matrix is defined as

( )

( )⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡+

+

⎥⎦

⎤⎢⎣

⎡+

==

2'6

2'6

2'5

2'5

2'6

2'4

2'5

2'4

2'4

2'6

2'3

2'5

2'3

2'4

2'3

2'3

2'6

2'2

2'5

2'2

2'4

2'2

2'3

2'2

2'2

2'6

2'1

2'5

2'1

2'4

2'1

2'3

2'1

2'2

2'1

2'1

2_

2

123

129

σσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσ

σγ

γ

KKKKK

M

M

M

M

En

E

Dplastic

with

95

( ) 3/3211'1 σσσσσ ++−=

( ) 3/3212'2 σσσσσ ++−=

( ) 3/3213'3 σσσσσ ++−=

As can be seen from Eq. 4.2 the incremental inelastic strain component

depends on the material properties. The strain hardening exponent and the

yield stress are both a function of temperature. Therefore, in order to compute

accurately the contribution of the inelastic strain, the temperature dependence

the plastic material properties must be defined in the casting constitutive model.

The accuracy on the material properties will determine the accuracy of the total

strain, which in turn will determine the accuracy of the stress predictions as has

been shown above.

4.3 Literature review

This section presents a review of the available relevant literature regarding

elevated temperature material properties of some casting alloys. The section is

divided into three parts. Part one introduces some of the available mechanical

properties of die casting alloys. The second part describes the different tests

performed by several researchers in determining some of the mechanical

properties of a variety of aluminum casting alloys. Lastly, the third one presents

the high-temperature tensile test results obtained using a Gleeble machine for a

variety of direct chill casting alloys done by several research groups.

96

4.3.1 Die casting alloys

The microstructures and mechanical properties of a wide variety of

common die casting alloys were determined in a comprehensive research project

done at Worcester’s Polythecnic Institute Materials Processing Institute [51]. The

results of the study were intended to be used by product designers to provide

them with more accurate tools while selecting die casting materials. A total of 26

different alloys were tested. Tensile test and mechanical test specimens were

produced following required ASTM standards. The tensile tests were performed

at room temperature, 100 ºC and 200 ºC. Axial extensometers were used for

both, the room temperature and the higher temperature cases. For the higher

temperature cases, the specimens were heated using a heating chamber,

monitoring the specimen’s temperature using thermocouples attached to the

specimen’s length. A total of 35 specimens were tested at room temperature,

while a total of 5 were tested for the higher temperature conditions. The machine

test velocity for both all the temperature conditions was 0.05in/min. A series of

tensile tests were generated using the recorded data for all the tested alloys.

The results can be seen in the cited report.

The creep behavior of three aluminum die casting alloys was determined

by Jaglinski et al [52]. The selected aluminum alloys were B-390 (Al-17%Si-

4%Cu-0.5%Mg), eutectic Al-Si alloy (Al-13%Si-3%Cu-0.2%Mg) and Al-17%Si-

0.2%Cu-0.5%Mg alloy. The objective of the study was to provide product

designers with creep material data for engine design, as many of these

components work under high temperatures and creep and relaxation might be

97

observed during engine operation. As-cast tensile test specimens were used to

perform the creep tests. Constant creep load uniaxial tests were done using a

dead weight lever frame, where strain was measured using axial extensometers

securely attached to the specimen’s gage section. The testing temperatures

were 220 ºC, 260 ºC and 280 ºC. The results obtained are shown in Fig. 4.1.

From the results it was observed that the eutectic alloy experienced larger strains

to failure and longer rupture times than the other two tests alloys.

98

Fig. 4.1 Creep properties for die casting aluminum alloy A380.0 [52]

4.3.2 Aluminum casting alloys

Singer et al [53] conducted an experimental work to determine the

mechanical properties of a variety of hypoeutectic Al-Si alloys below and in the

solidus temperatures. The purpose of the study was to provide experimental

data that would assist in explaining the phenomenon of hot-shortness. A total of

99

ten different alloys were tested, including super-purity aluminum. The selected

specimen design varied depending on whether the temperature was below or

above the solidus. Fig. 4.2 shows the two designs used. The specimens were

heated using a resistance tube furnace that slid on the testing apparatus,

providing uniform heating throughout the specimen’s length. Fig. 4.3 and Fig. 4.4

show the results obtained. The experimental results showed that the tensile

strength of the tested Al-Si alloys decreases gradually as the temperature

increases above 400 ºC, reaching finite magnitudes at the solidus temperature.

As can be seen from the figures below, the slope of the strength-temperature

curves increases as the silicon content increases. The results showed that the

reduction in area and elongation remained high for all the tested alloys up to the

solidus temperature. At the solidus temperature however, the elongation and

reduction in area decreased to zero and the maximum strength reached a very

low magnitude.

100

a)

b)

Fig. 4.2 Testing devices used for determining flow stress a) below 300 ºC and b) above 300 ºC [53]

101

Fig. 4.3 Flow stress for Al-Si alloys at temperatures below the solidus [53]

102

Fig. 4.4 Flow stress for Al-Si alloys at temperatures above the solidus [53]

Kim et al [54] performed a numerical analysis on the deformation and

shrinkage of small rectangular castings. The dimensions of the castings were

50x24, 100x24 and 150x24mm. A really simple two dimensional analysis,

assuming four-folded symmetry of the casting was done. Four-node quadratic

elements were used to represent the castings. A useful piece of information

103

encountered in the article is the temperature-dependent behavior for the Yield

Strength and Young’s Modulus for the selected casting material shown in Fig.

4.5. The material under study corresponded to an Al-7%Si-0.3%Mg alloy. It is

not clear from the results obtained whether the computer model results

correlated well with experimental observations.

Fig. 4.5 Material properties for an Al-7%Si-0.3%Mg alloy [54]

The mechanical behavior of various Al-Cu alloys at solidifying and just-

solidified temperature conditions was determined by Wisniewski et al [55]. The

goal of the study was to develop an adequate fracture criterion to predict hot

104

tearing for Al-Cu solidifying alloys. The percentage Cu content was varied from

2.5 to 7.5%. Rectangular bars 13x51mm were tested at strain rates that ranged

from 1x10-5 to 1x10-1 s-1. The specimens were mounted onto a fixture in an

Instron machine. A quartz lamp furnace was used to heat the specimens at the

required temperatures. The tests were carried at constant extension rates. Load

and extension magnitudes during testing were recorded and the obtained data

were converted to stress and strain respectively. The obtained results show that,

at all strain rates, the maximum stress decreased as the Cu content was

increased. Also, it was found that the maximum strength decreased sharply as

the liquid fraction increased.

The mechanical behavior of an Al-4.5%Cu-Mg-Ti alloy in the mushy zone

was determined in an experimental study conducted by Vicente-Hernandez et al

[56]. The main objective was to determine an adequate testing procedure to

determine the visco-plastic mechanical behavior of aluminum alloys in the mushy

zone in order to provide solidification computer models with reliable material data

for gap formation predictions. The developed test consisted in the pushing of a

needle at a constant rate into a solidifying ingot. The cooling conditions in the

mold were controlled carefully to obtain vertical isotherms. The described test

provided records of applied force versus needle displacement at different

temperatures. The Norton-Hoff power law model describing the strain-rate

dependency of the mechanical behavior was then fitted using the experimental

data. The fitted model was used in a thermal-mechanical analysis of a solidifying

ingot aimed at predicting gap formation. The results showed a remarkable match

105

between the measured displacements and the displacements predicted by the

simulation. It was claimed that the lack of accurate material data, especially

visco-plastic mechanical behavior, in the mushy zone can limit to a great extend

the validity of computer models in predicting air gap formations, heat transfer

conditions and thermal stresses in solidification analyses.

4.3.3 Gleeble testing

Computer modeling of residual stresses and deformations in castings has

gained popularity across casting industries because the general understanding of

the governing phenomena and the computer algorithms have progressed

considerably. However, a lacking piece of information in all computer models is

the mechanical behavior of alloys at the cooling conditions dictated by the

casting process being modeled. The Direct Chilling (DC) casting process has

received increased attention from the modeling community and as a result, the

determination of material properties for typical DC aluminum alloys has been

done over the past years. A common denominator across all studies conducted

is the use of Gleeble testing for performing elevated-temperature tensile tests at

various strain rates characteristic of the DC process. A review of some of some

of the studies published is provided in the following paragraphs.

Van Haaften et al [57] conducted an experimental study to determine the

mechanical properties of AA1050, 113104 and AA5182 DC aluminum casting

alloys. The objective was to obtain a Power law model using the experimental

data that can be used in computer models to predict thermal stresses and

deformation in DC ingots. A series of tensile tests at different temperatures and

106

strain rates were performed using a Gleeble 1500 and a Gleeble 3500. The

specimens were machined from as-cast rolling slabs. Non-cylindrical specimens

10mm in diameter and 95mm long, with a reduced 8mm diameter at the center

were used for the testing. The selected temperatures ranged from room

temperature to the solidus temperature, whereas the strain rates ranged from

1x10-5 to 1x10-1 s-1. It is stated that most of the experiments were conducted in a

stroke controlled mode, using the load controlled mode for the highest

temperature cases. From the described procedures, it is not clear however how

the constant strain rates were achieved, just the obtained range as mentioned

above was stated. Three replications were done for all the testing cases. The

obtained data was fitted to the extended Ludwik power law and to the Garofalo’s

hyperbolic-sine models.

Using a Gleeble machine the flow stress at steady-state creep conditions

for AA3103 aluminum alloy was determined by Farup et al [58]. The goal of the

experimental study was to obtain the parameters of the Garofalo’s model for use

in DC thermal-mechanical finite element models. Non-cylindrical specimens

being 10mm in diameter, 90mm long and a reduced 8mm diameter at the

specimen’s center were used. The temperatures ranged from 325 ºC to 550 ºC,

whereas the strain rates varied from 1x10-6 to 1x10-2 s-1. In order to achieve

steady state creep conditions at the intended strain rates in the range of 1x10-2 to

1x10-4 s-1a constant jaw velocity was used, whereas for the smallest strain rates

1x10-5 to 1x10-6 s-1a constant force was used instead. The obtained results

showed a marked scattered in the data. It was stated that differences in

107

specimens’ circularity measured at the specimen’s center were as high as 50 to

80%, causing an uncertainty of ±30% in the steady state strain rate.

Furthermore, noise in the load cell readings at the low stress levels was identified

as another source of error in the measurements. The obtained results were

successfully used to determine the parameters of the Garofalo’s equation.

The mechanical properties under compression, tension and cyclical

compression for AA582 DC aluminum alloy were determined by Alhassan-Abu et

al [59]. The objective was to determine the parameters of the Garofalo’s

equation for use in computer modeling of thermal-mechanical stress in DC

ingots. The tests were carried out using a Gleeble 1500 machine. The

temperatures ranged from 250 ºC to 500 ºC, whereas strain rates varied from

1x10-5 to 1x10-1 s-1. For the tensile tests the specimens had a gauge length of

27mm and a gauge diameter of 5mm. It is stated that the intended strain rates

were achieved using constant displacement rates, yielding mean strain rates in

the above mentioned range. The results obtained were used to determine the

parameter of the Garofalo’s hyperbolic-sine model. It was concluded that the

fitted Garofalo’s model predicted accurately the mechanical behavior of the

material for temperatures above 350 ºC, whereas it failed to predict the strain

hardening behavior at the lower temperatures.

Tensile tests were conducted to determine the effects of strain rate and

thermal history on the constitutive behavior of an Al-Mg AA5182 DC alloy [60]. A

Gleeble 3500 thermo-mechanical simulator was used to conduct the

experiments. The thermal histories included as-cast, heated and homogenized

108

conditions. The strain rates ranged from1x10-4 to 1x10-3 s-1. The selected

specimen shape and testing procedures were the same as in [57]. The results

obtained suggested that the constitutive behavior is independent of the thermal

history. The fitted Garofalo’s equation was able to accurately predict the

mechanical behavior of the alloy in the tested conditions.

4.4 Determination of aluminum A380.0 mechanical properties

This section describes the procedures followed for the determination of

the elevated temperature mechanical properties for A380.0 aluminum alloy. A

detailed description of all the steps followed for carrying out the project is

provided. Issues such as machine and specimen selection, the selected design

of experiments matrix, production and preparation of specimens and testing

methodology are presented in the following paragraphs.

4.4.1 Machine and specimen selection

The Gleeble machine was selected to carry out the elevated temperature

tensile tests. The decision was taken based on the information found in the

literature and presented in the previous section. As was reported in the cited

articles, the Gleeble thermo-mechanical simulator can readily perform tensile

tests at the required temperatures and strain rates. The reliability of the control

system in controlling temperature, jaw velocity and loading force made the

Gleeble an excellent choice for the project. Currently, a Gleeble 3800 and a

109

Gleeble 1500 are available at The Ohio State University at the Welding

Engineering and Material’s Science Engineering departments respectively.

The Gleeble system is a high-strain-rate, high temperature testing

machine where a solid specimen is held horizontally by water cooled grips,

through which electric current is introduced to resistance heat the test specimen

[61]. Fig. 4.6 shows the schematic of a typical Gleeble test. Specimen

temperature is monitored by a thermocouple that is welded to the specimen’s

surface. The temperature of the specimen is controlled by a function generator

using the readings provided by the thermocouple. The direct-resistance heating

system of the Gleeble machine can heat specimens at rates of up to 10,000 ºC/s.

Due to this capability, this machine can perform hot tensile tests several times

faster than conventional methods. Radial strain measurements are readily done

by attaching a dilatometer that monitors the reduction in area as the specimen is

being tensioned.

110

Fig. 4.6 Schematic of a typical Gleeble test [57]

Gleeble machines are commonly used for thermo-mechanical physical

modeling of a wide variety of processes. Applications range from phase

transformation analyses where heat treatment conditions ought to be controlled

really accurately, plastic deformation of materials at high strain rates such as

high velocity forging, friction stir welding where material is subjected to a

torsional state of stress at high temperatures, etc. The different machine models

provide different capabilities in the maximum developed force, torque, maximum

and minimum jaw velocities, etc. The 3800 model available at the Welding

Engineering facilities was designed for high velocity and high force applications,

whereas the 1500 model was designed for small forces and slow velocity

applications. A key differentiator between these two models is the valve that

controls the flow rate of oil to displace the jaws, determining the minimum and

111

maximum attainable speeds. The 1500 model with a 60GPM control valve was

selected because of its capabilities in displacing the jaws at velocities that would

generate the smallest strain rates in the order of 1x10-5 and 1x10-6 s-1

characteristic of the cooling conditions in die castings.

The specimen design shown in Fig. 4.7 was selected based on the shapes

described in the articles cited in section 4.3.3. The common denominator in

those designs was the use of a cylindrical specimen with a reduced diameter at

the center. The reduction in area was purposely done in order to guarantee that

the maximum strained region is at the specimen’s center. Additionally, as

reported by Walsh et al [62], quadratic temperature profiles are typical of a

Gleeble test, where the maximum temperature is located at the specimen’s

center and the water cooled grips define the lowest temperature points. Thus,

the quadratic temperature profile together with the reduction in area guarantees

that the region of failure is located at the specimen’s center, eliminating the need

of a gage length.

112

Fig. 4.7 Schematic of the test bar used for tensile tests

The radial strain at this region can be readily obtained by placing a

dilatometer that monitors the reduction in area as shown in Fig. 4.6.

Eq. 4.4 shows how the radial strains are computed

2

Eq. 4.4

where

is the current diameter

is the starting diameter

113

The radial strains can be converted to longitudinal strains as expressed in

Eq. 4.5

Eq. 4.5

where

is the Poisson’s ration

The stresses can be readily computed by dividing the applied force by the

reduced cross-sectional area as shown

Eq. 4.6

where

is the applied force

is the reduced cross-sectional area

A couple of assumptions are used behind the computation of the

longitudinal strains and stresses. One is the assumption of a homogeneous

temperature distribution. The other assumption is a homogenous cross-sectional

shape throughout the specimen’s length. Both assumptions generate non-

uniform flow stresses along the specimen’s length, leading to inaccuracies in the

computed stresses and strain rates. This was recognized by Farup et al [63]

where they investigated the effects of those in-homogeneities in the computed

114

strain fields. An axis-symmetric model representing the specimen in their tests

was modeled in Abaqus. The obtained visco-plastic properties from the

experiments were assigned to the material. Three conditions were studied: one

represents the same temperature and shape in-homogeneities as in the

experiments, another one assumes a non-cylindrical specimen but with a uniform

longitudinal temperature profile and the last one assumes a cylindrical specimen

with the parabolic temperature profile characteristic of their tests. The obtained

results showed that the variation in the visco-plastic strain rate along the radius

varies as much as 12.5%, with a maximum at the center. It was recognized that

the major source of variation was the parabolic temperature profile, contributing

more than the non-cylindrical shape to the observed radial differences. It can

therefore be expected that the obtained strain-rates in the experiments carry an

error of at least 12.5% due to the above mentioned in-homogeneities.

4.4.2 Design of experiments matrix

A design of experiments was proposed where the specimen temperature

and strain rate were the design variables. The selected temperatures ranged

from 25 ºC to 500 ºC having ten different levels. The strain rates were

determined by analyzing the cooling conditions on the casting following the

expression

Eq. 4.7

where

115

is the strain rate

α is the coefficient of thermal expansion

is the cooling rate

Cooling rates from a computational thermal analysis of different regions of

a solidifying casting were extracted and Eq. 4.7 was used to calculate the strain

rates. The obtained results produced strain rates that ranged from 1x10-2 to

1x10-6 s-1. Given the results obtained, five different levels for the strain rate were

selected. Because of the different levels obtained for the two design variables, a

Hexagonal design was selected [64]. Table 4.1 shows the design matrix.

116

Run Temperature (C) Strain rate (s-1)

1 500 3.16E-04

2 486 2.26E-03

3 444 1.28E-02

4 381 4.62E-02

5 304 9.16E-02

6 221 9.16E-02

7 144 4.63E-02

8 81 1.28E-02

9 39 2.27E-03

10 25 3.17E-04

11 25 3.17E-04

12 39 4.42E-05

13 81 7.83E-06

14 144 2.16E-06

15 221 1.09E-06

16 304 1.09E-06

17 381 2.16E-06

18 444 7.80E-06

19 486 4.40E-05

20 263 3.16E-04

Table 4.1 Experimental array for tensile tests

117

4.4.3 Specimen production and preparation

Specimen were produced using an existing insert available at The Ohio

State University. The insert design allowed the production of two specimen per

shot. A photograph of the insert used for the production of the specimen is

shown in Appendix A. The production was carried out continuously during one

shift at Empire Die Casting Co., Inc. located in Macedonia, Ohio. A total of 800

specimen were die cast with an average cycle time of 40s and a shot weight of

0.5kg. The information regarding alloy chemistry and cycle time parameters

used for production is presented in Appendix B.

After production, specimen were trimmed. Runners, biscuit and any

excessive flash were manually removed. The runner and biscuit were removed

using a rubber hammer, while the flash was removed using a vertical band saw.

Radiographs were taken on all specimen to guarantee that only the best

would be tested. The specimen were x-rayed at Ultra Labs Inc., located in

Cleveland, Ohio. The radiographs showed that the majority of the specimens

had shrinkage porosity at the grip ends, while very little porosity was detected at

the reduced cross-sectional area. Out of the 800 specimen produced 370 were

selected for testing.

The selected specimen were machine-threaded using a threading die.

The specific thread used was ½-20. The finer pitch was selected to maximize

the number of threads engaged during testing to avoid stripping the threads. The

machine-threading was also done to guarantee proper alignment between the

axes produced by the threads on both grip ends.

118

After threading the specimen were polished. Polishing was done using a

rotating brush, with the main intention of removing residual flash at the parting

plane. A smooth surface finish was obtained after polishing was completed.

4.4.4 Testing methodology

As can be seen in Table 4.1, the 20 cases required different combinations

of temperatures and strain rates. Two different approaches, namely constant jaw

velocity and constant force, were used to perform the tests and they were

developed based on the way the required strain rates were achieved. A

description of the procedures followed on each of the approaches is provided in

the following paragraphs.

The strain rate is a mechanical behavior material characteristic that

provides a measure of the rate of strengthening or softening of the material under

a given set of loading conditions. This rate of strengthening or softening may or

may not be constant depending on the way the material responds as the plastic

deformation is taking place. Constant strain rates at temperatures below

creeping are usually achieved by loading the material under a constant rate of

displacement and can be readily computed by using the following expression

Eq. 4.8

where

is the jaw velocity

is the deformed length

119

Because most aluminum alloys start creeping at temperatures around 300

ºC, all the test cases at temperatures below 300 ºC were conducted under

constant jaw velocities. As can be seen from Eq. 4.8 the magnitude of the strain

rate is proportional to the magnitude of the jaw velocity. A detailed description of

the correlations obtained between applied jaw velocity and achieved strain rate

will be provided in the next section.

For temperatures above 300 ºC, the tests were conducted by applying a

constant force. Under steady state creep conditions, the constant force is

translated into a constant stress because the rate of elongation remains

constant. That constant rate of elongation is a guarantee that the strain rate

remains constant throughout the test. A detailed description of the obtained

correlations between the applied force and achieved strain rate for the different

temperatures above 300 ºC is provided in section 4.4.4.2

4.4.4.1 Velocity controlled

A series of room temperature tests were performed to determine the

correlation between the jaw velocity and the obtained radial strain rate measured

at the specimen’s center. The jaw velocity was varied progressively by orders of

magnitude and the strain was measured by a dilatometer and recorded by the

Gleeble’s data acquisition system. With the time history of the recorded strain,

MATLAB’s curve fitting toolbox was used to filter the data and to compute the

first derivative.

Fig. 4.8 shows the strain-time and strain rate-time plots for a case where

the jaw velocity was 0.001mm/s. The top figure shows that the radial strain came

120

to be in the order of 1x10-3 mm/mm, while the strain rate was about 1x10-6 s-1.

The fact that the strain rate is 3 orders of magnitude smaller can be explained by

the order of magnitude of the time scale, which is about 1x103 s. The strain rate

results were plotted in a histogram and the results are shown in Fig. 4.9. The

histogram clearly depicts that the radial strain rates at the specimen’s center

were all in the order of 1x10-6 s-1.

Fig. 4.10 shows the results obtained for a jaw velocity of 1mm/s. As can

be seen from the top plot the radial strain remained in the same order of

magnitude as in the previous case, whereas the time scale was reduced by 3

orders of magnitude. This resulted in an average radial strain rate in the order of

1x10-3 s-1 as shown by the histogram in Fig. 4.11. From the observed trends it

can be expected that a higher jaw velocity would cause even shorter failure

times, resulting in higher radial strain rates.

The results for a jaw velocity of 100mm/s are shown in Fig. 4.12. This jaw

velocity magnitude caused failure times to be in the order of 1x10-3 s, generating

radial strain rates in the order of 1x101 s-1 as shown in the histogram in Fig. 4.13.

As can be noticed from the results shown, there exists a clear correlation

between the jaw velocity and the achieved radial strain rates. A series of more

tests with various jaw velocities were done and the results were processed as

described before. From the log-log scatter plot shown in Fig. 4.14, a linear

correlation between the jaw velocity and the achieved radial strain rate can be

observed. Although the correlation plot shown in Fig. 4.14 was generated using

the results of room temperature tests, this correlation was found to be valid and

121

vary only slightly for temperatures up to 260 ºC. Therefore, the obtained plot was

used as a basis to carry out all the test cases where the specimen’s temperature

was under 300 ºC.

122

Fig. 4.8 Strain and strain rate plots for a 0.001mm/s jaw velocity

123

Fig. 4.9 Strain rate distribution for a 0.001mm/s jaw velocity

124

Fig. 4.10 Strain and strain rate plots for a 1mm/s jaw velocity

125

Fig. 4.11 Strain rate distribution for a 1mm/s jaw velocity

126

Fig. 4.12 Strain and strain rate plots for a 100mm/s jaw velocity

127

Fig. 4.13 Strain rate distribution for a 100mm/s jaw velocity

4

w

a

p

to

T

.4.4.2 Forc

As wa

where the sp

luminum al

rocedure.

o steady sta

Thus, it was

ce controlle

as mention

pecimen’s t

lloy at this t

Under a co

ate creep, g

s decided to

Fig. 4.14 Stra

ed

ed already,

temperature

temperature

onstant forc

generating

o perform a

12

ain rate vs. ja

, a force co

e was abov

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ce, the rate

the constan

series of te

28

w velocity co

ontrol schem

ve 300 ºC.

as the reaso

of elongatio

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ensile tests

rrelation plot

me was use

The creep

on for this te

on remains

tes needed

s at differen

ed for the ca

behavior o

esting

s constant d

for the test

t temperatu

ases

of the

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ts.

ures

129

and constant forces and obtain a correlation between the applied constant force

and the achieved radial strain rate. The procedure followed to obtain such

correlation plots is presented using some sample results for a particular set of

temperature conditions.

Fig. 4.15 shows the strain-time plot for a tensile test carried out at 381 ºC

under a constant force of 1KN. The figure clearly shows a linear behavior of the

radial strain, depicting the steady state creep. A linear polynomial was fit to the

linear portion of the curve and the obtained slope was taken as the radial strain

rate. For this test, a strain rate of 1x10-6 s-1 was obtained.

The results for a test performed at the same temperature but under a

1.5KN constant force are presented in Fig. 4.16. The curve shows a slightly

different behavior as compared with the previous one, but a large linear portion

can still be seen. After fitting a linear polynomial to the linear portion of the curve

a radial strain rate of 1x10-4 s-1 was obtained.

Fig. 4.17 shows the results of a test done using a constant force of 2.0KN

and at the same 381 ºC temperature. The obtained results show a similar trend

of the strain-time curve as in the previous case, but with a time to failure

reduction of about an order of magnitude. Taking the linear portion of the curve

from 65 to 85s, the slope of a linear polynomial fitted came to be about 1x10-3 s-1.

From the results presented thus far a correlation between the applied

force and the achieved radial strain rate can be observed. It can be noticed that

at a given specimen’s temperature, the higher the applied force the smaller the

obtained radial strain rate. Based on this, a series of correlation plots were

130

generated for the temperatures at and above 300 ºC. At those temperature

levels, the constant force was varied and the radial strain rates were computed

as described before. The obtained correlation plots are shown in Fig. 4.18 to Fig.

4.22.

Fig. 4.15 Strain vs. time curve for a 1.0KN force at 381 ºC

131


132


133

Fig. 4.18 Strain rate force vs. force correlation plot at 300 ºC

1.00E‐06

1.00E‐05

1.00E‐04

1.00E‐03

1.00E‐02

1.00E‐01

1.00E+00

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

Strain ra

te (1

/s)

Force (KN)

Correlation between Force and Strain Rate at 300 C

134


1.00E‐06

1.00E‐05

1.00E‐04

1.00E‐03

1.00E‐02

1.00E‐01

1.00E+00

0.5 1 1.5 2 2.5 3 3.5 4

Strain ra

te (1

/s)

Force (KN)


135


1.00E‐06

1.00E‐05

1.00E‐04

1.00E‐03

1.00E‐02

1.00E‐01

1.00E+00

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6

Strain ra

te (1

/s)

Force (KN)


136


1.00E‐06

1.00E‐05

1.00E‐04

1.00E‐03

1.00E‐02

1.00E‐01

0.5 0.6 0.7 0.8 0.9 1 1.1

Strain ra

te (1

/s)

Force (KN)


137


4.4.4.3 Results

The tensile tests were carried out using the correlation plots shown in the

previous section. Five replications were performed for each of the twenty

different cases. All one hundred tests were randomized to minimize the effect of

the many sources of error. The details of the randomization of the runs can be

found in Appendix C. Selection of the specimens for each test was done

randomly as well to minimize the effect of specimen quality on the results. The

tests were completed in the period from May 10 to June 8, 2007, five months

1.00E‐07

1.00E‐06

1.00E‐05

1.00E‐04

1.00E‐03

1.00E‐02

1.00E‐01

1.00E+00

0.5 0.6 0.7 0.8 0.9 1 1.1

Strain ra

te (1

/s)

Force (KN)


138

after they were produced. Fig. 4.23 to Fig. 4.42 show the true Stress vs. true

strain curves for each of the twenty runs.

Fig. 4.23 Stress vs. strain curves at 500 ºC and 1x10-4 s-1 strain rate

139


140


141


142


143


144


145


146


147


148


149


150


151


152


153


154


155


156


157


For Run 18 as shown in Fig. 4.40 only three replications were done. This

was due to the fact the first two replications were carried out using a greater

force, while for the last three a smaller force was used. At the completion of the

one hundred runs it was decided to re-run the first two replications. The

dilatometer could not be borrowed again and the tests were not completed.

Because five replications were done for each of the twenty runs, the mean

was selected to represent the five different sets of data. The mean was

158

computed by averaging the five different stress magnitudes strain-wise. Fig. 4.43

and Fig. 4.44 show the computed average curves for the cases with

temperatures below 300 ºC and above 300 ºC respectively.

159

Fig. 4.43 Stress vs strain curves for temperatures below 300 ºC

Fig.

4.4

3 S

tress

vs

stra

in c

urve

s fo

r tem

pera

ture

s be

low

300

ºC

160

Fig. 4.44 Stress vs strain curves for temperatures above 300 ºC

Fig.

4.4

4 S

tress

vs

stra

in c

urve

s fo

r tem

pera

ture

s ab

ove

300

ºC

161

The mechanical properties shown in the figures above had to be

processed for use in computer models. In Abaqus, the elastic and plastic

properties need to be defined in separate tables. The elastic properties must be

defined as a table of Young’s Modulus and Poisson’s ration at different

temperatures. These properties were extracted from Fig. 3.6 as mentioned. The

plastic properties must be defined as tables of Yield Stress, plastic strain as a

function of temperature for each of the different strain rates. Since the obtained

material properties include the elastic and plastic behavior, the elastic part was

removed for defining the required tables. Fig. 4.45 to Fig. 4.49 show final curves

used for the computer models.

Fig. 4.45 Simulation material properties for strain rate of 1x10-6 s-1

0.00E+00

5.00E+07

1.00E+08

1.50E+08

2.00E+08

2.50E+08

3.00E+08

0 0.02 0.04 0.06 0.08 0.1 0.12

Yield Stress (P

a)

Plastic strain

Simulation material properties for strain rate of 1x10‐6 s‐1

81 C

144 C

220 C

304 C

381 C

444 C

162



0.00E+00

5.00E+07

1.00E+08

1.50E+08

2.00E+08

2.50E+08

3.00E+08

0 0.01 0.02 0.03 0.04 0.05 0.06

Yield Stress (P

a)

Plastic strain


39 C

486 C

0.00E+00

5.00E+07

1.00E+08

1.50E+08

2.00E+08

2.50E+08

3.00E+08

3.50E+08

0 0.02 0.04 0.06 0.08 0.1 0.12

Yield Stress (P

a)

Plastic strain


25 C

263 C

500 C

163



0.00E+00

5.00E+07

1.00E+08

1.50E+08

2.00E+08

2.50E+08

3.00E+08

0 0.01 0.02 0.03 0.04 0.05 0.06

Yield Stress (P

a)

Plastic strain


39 C

486 C

0.00E+00

5.00E+07

1.00E+08

1.50E+08

2.00E+08

2.50E+08

3.00E+08

0 0.02 0.04 0.06 0.08 0.1 0.12

Yield Stress (P

a)

Plastic strain


81 C

144 C

221 C

304 C

381 C

444 C

164

4.5 Summary

Material properties are one of the most important pieces of information

when building a finite element model. The constitutive equations that relate

stress to strain are a function of the material constitutive model used. If the

constitutive model used does not represent adequately the response of the

analyzed part, the stress and force predictions will be highly questionable.

The lack of mechanical properties at temperatures relevant for

solidification analysis was the motivation to carry out a project to determine the

constitutive model for the die casting alloy being used in this research project.

Test bars were die cast and specimens were qualitatively selected based

radiography results. A series of stress-strain curves at different combinations of

temperature and strain rate were obtained. Five replications were done at each

combination of temperature and strain rates in order to obtain statistical

significance. The tests were carried out using displacement and force control

modes in a Gleeble thermal-mechanical simulator. The displacement control

mode was used for temperatures below creep (about 300 ºC), whereas the force

control model was used for temperature above creep. Calibration curves for both

modes of control were developed to carry out the tests.

The results obtained showed the large dependency of the flow stress on

both, temperature and strain rate. Temperature and strain rate effects on flow

stress showed a more marked effect at temperatures greater than 150 ºC.

Perfect plasticity was obtained at temperatures about 220 ºC. The obtained

trends suggested that strain rate has a greater impact on flow stress at

165

temperature above 300 ºC. For instance, a reduction of 4 orders of magnitude

on strain rate at 381 ºC caused the flow stress to be reduced by two-thirds. The

obtained experimental curves were summarized in a series of tables that were

used as input data to the simulation models.

166

5 CHAPTER 5

RESULTS

5.1 Introduction

This chapter presents the results of a dimensional analysis performed on

a research casting. Experimental measurements taken on actual production

castings are compared against computer model predictions. The comparisons

are made using the results of a Design of Experiments (DOE) used to assess the

effects of process parameters on final casting distortion. Computer models

simulating the same process conditions as the DOE were prepared and the

results were used to test the adequacy of the computer model predictions for in-

cavity and across parting plane dimensions.

The chapter is divided into four sections. The first section presents the

results of the statistical analysis performed using the measurements taken on

experimental castings. The results of computer models simulating the same

combination of process conditions as in the DOE are presented in the second

section. In this section comparisons between the two sets of results are

presented as well. The third section is devoted to the analysis of the adequacy

167

of different sets of modeling assumptions used for part distortion modeling in die

casting. Lastly, an analysis of the residual stress predictions given by the

computer models is presented in the fourth section

5.2 Part distortion experiments

A design of experiments was done with the objective of determining the

effects of typical process parameters on final casting distortion. The parameters

selected were intensification pressure and dwell time. These two factors have

been commonly used to explain part distortion behavior in actual production

castings. Dwell time is a control parameter typically used for controlling casting

dimensions. It is a common belief among die casters that leaving the casting in

the die longer produces castings with less distortion. Intensification pressure on

the other hand has been commonly linked to across parting plane dimensional

issues in castings, having a direct correlation with the pressure magnitude. The

practical relevance and the simplicity of adjusting them during production was a

key factor in selecting these two factors for this experimental set up.

Three levels for each factor were selected, making a total of nine different

runs. Table 5.1 shows the matrix design. The casting selected for this study is

shown in Fig. 5.1 and corresponds to the ribbed plate shown in Chapter 3. This

casting is formed using a simple open-close die, with the ribs all formed within

the ejector side, whereas the ribs height is formed between the ejector and cover

sides. Fig. 5.2 shows the casting dimensions. The dimensions shown represent

the room temperature cavity dimensions without considering any shrinkage

fa

th

w

d

actor. Beca

he ribs were

whereas the

istortion.

ause the rib

e selected a

e ribs heigh

bs are forme

as features

t was selec

16

ed within th

s that chara

cted to char

Fig. 5.1 Sele

68

he ejector s

acterize the

racterize th

cted casting

side, the dis

in-cavity d

e across pa

stance betw

istortion,

arting plane

ween

e

c

w

e

b

ru

p

Fig. 5an

The c

onditions u

was decided

arlier and 2

ars represe

un tests wit

ressure at

5.2 Casting dind 1º draft on

combination

sed for pro

d to test the

2 seconds l

ents the up

th half mach

the lowest

imensions in walls formed

n of process

oducing the

e effect of d

ater. The n

per limit the

hine injectio

pressure le

16

millimeters (2d by the insert

s paramete

casting in t

dwell time b

nominal inte

e machine c

on capacity

evel.

69

2º draft on intet and die shoe

ers of run 5

the previou

y ejecting t

ensification

can provide

y and nearly

erior walls fore, see Appen

represents

us research

the casting

n pressure l

e; thus, it w

y no intensi

rmed by the inndix E)

s the nomin

studies. It

2 seconds

evel of 680

was decided

ification

nsert

nal

t

0

d to

170

Run Pressure (bars) Time (s)

1 170 11

2 170 7

3 170 9

4 340 11

5 680 9

6 680 7

7 340 9

8 340 7

9 680 11

Table 5.1 Matrix design for experimental DOE

The different runs were randomized and were run in the order listed on

Table 5.1. A series of castings were initially produced using the parameters of

run 1 to get the die to a quasi-steady thermal state. Batches of 30 castings were

produced for each of the different 9 runs. Although 30 castings were produced

for each run, only the last 20 were used for the dimensional analysis to allow the

die to reach quasi-steady thermal state after the change in process conditions.

During production a series of process parameters were recorded. Actual

cycle times were taken because a rather high variability was introduced due to

the difficulty in controlling spraying time, machine dwell time, closing time and

ladling time. Actual casting weights with and without runner and biscuit were

171

recorded as well because of the inconsistency in pouring volume during the

experiments. The furnace temperature was also observed to vary during the

experiments and it was therefore decided to record it.

5.2.1 Dimensional measurements on production castings

The scope of this research required precise, accurate and repeatable

measurements on the production castings. A Coordinate Measurement Machine

(CMM) was selected to perform the measurements. Additionally, a fixture was

designed and built to secure the castings properly during the measurements.

The fixture was designed considering the 3-2-1 fixturing principle, establishing

this way the coordinate system relative to which all measurements were

expressed consistently. Fig. 5.3 shows the six gauging locations on the casting

that were contacted by the fixture.

Fig. 5.3

17

Location of fi

72

xture gaugingg points

s

c

b

h

a

d

5

th

The C

eries of poi

asting featu

etween the

eight was u

For th

long each o

imensions

.4. The five

he points at

CMM meas

ints on the

ures used t

e ribs were

used to stud

he in-cavity

of the fins w

were recor

e points we

t the middle

surements c

different ca

o character

used to stu

dy the acro

Fig. 5

y dimension

width. The

rded at a dis

ere equally

e. Five indi

17

consisted in

asting featu

rize the dist

udy the in-c

oss-parting

.4 Selected c

ns the coord

coordinate

stance 25m

spaced alo

ividual dista

73

n recording

res. Fig. 5

tortion. As

avity dimen

plane dime

casting dimen

dinates of 5

s of all the

mm from the

ong the widt

ances were

the coordin

.4 shows th

mentioned

nsions, whe

ensions.

sions

5 points wer

points for t

e bottom as

th, always l

e computed

nates of a

he selected

d, the distan

ereas the rib

re recorded

the in-cavity

s shown in

locating on

using the

d

nces

bs

d

y

Fig.

e of

174

coordinates of each of the five pairs of points. The final distance between the

ribs was taken as the average of those computed distances.

Coordinates at the top surface of each of the ribs were recorded to

determine the across-parting plane dimensions. As for the in-cavity dimensions,

five points were sampled along the fins width. Rib # 1 was discarded from the

measurements because of excessive flashing. The recorded coordinate along

the ribs height was used to determine the across-parting plane distance for each

fin. The final distance was taken as the average of five different coordinates.

5.2.2 Statistical analysis

An Analysis of Variance (ANOVA) was performed using the obtained

CMM measurements. The intensification pressure and dwell time were specified

as the main factors during the analysis. The additional recorded data for total

cycle time, shot weight, casting weight and furnace temperature was treated

considering these parameters as blocking factors.

A General Linear Model was selected to perform the statistical analysis in

MINITAB. The two main factors and the additional recorded data were all

included in the model. The furnace temperature, cycle time, shot weight and

casting weight were treated as random factors in the model. It was assumed that

the interactions between the random factors were not statistically significant and

only their main effects were considered. Eq. 5.1 shows the model form used for

this analysis as specified in MINITAB.

Eq. 5.1

175

where

corresponds to the analyzed distance

represents the effects of the intensification pressure

represents the effects of the dwell time

represents the effects of the interaction between pressure and dwell

time

represents the effects of the furnace temperature

represents the effects of the cycle time

represents the effects of the shot weight

represents the effects of the casting weight

represents the error term

The ANOVA table, main effect plots, interaction plots and box plots were

requested as outputs for the statistical analysis. The results of this analysis are

presented next for the in-cavity and across parting plane dimensions.

5.2.3 In-cavity dimensions

Table 5.2 to Table 5.4 show the ANOVA results for three selected

distances. As can be seen, for all three distances the intensification pressure,

dwell time and their interaction are all statistically significant. Regarding the

blocking factors, shot weight and casting weight are significant for D1 and D2,

whereas for D3 only the shot weight turned out to be significant. Total cycle time

and the furnace temperature did not have a statistically significant effect on the

final casting dimensions as shown in the ANOVA tables.

176

Source DF Seq SS Adj SS Adj MS F P

Pressure 2 2.46975 0.34399 0.17199 35.25 0

Dwell time 2 0.73519 0.38066 0.19033 39.00 0

Pressure*Dwell time 4 0.49068 0.30921 0.07730 15.84 0

Furnace temp 1 0.01079 0.01662 0.01662 3.41 0.067

Cycle time 2 0.00763 0.01446 0.00723 1.48 0.230

Shot weight 4 0.21162 0.15678 0.03920 8.03 0

Casting weight 2 0.06845 0.06845 0.03422 7.01 0.001

Error 162 0.79051 0.79051 0.00488

Total 179 4.78642

Table 5.2 ANOVA results for distance D1

177


Pressure 2 0.02224 0.001695 0.000847 3.27 0.040

Dwell time 2 0.159563 0.097493 0.048747 188.38 0


Furnace temp 1 0.000375 0.000028 0.000028 0.11 0.742

Cycle time 2 0.006029 0.000835 0.000417 1.61 0.202

Shot weight 4 0.04748 0.044191 0.011048 42.69 0

Casting weight 2 0.004126 0.004126 0.002063 7.97 0

Error 162 0.041923 0.041923 0.000259

Total 179 0.362680


178


Pressure 2 0.0647878 0.020 0.010 14.42 0

Dwell time 2 0.0245811 0.0125068 0.006253 9 0


Furnace temp 1 0.0001343 0.0000318 0.0000318 0.05 0.831

Cycle time 2 0.006718 0.0019448 0.0009724 1.40 0.25

Shot weight 4 0.0248597 0.0224419 0.0056105 8.08 0

Casting weight 2 0.001697 0.0016978 0.0008489 1.22 0.297

Error 162 0.1125 0.1125 0.0006945

Total 179 0.275561


Fig. 5.5 shows the main effect plots obtained from the statistical analysis.

To facilitate interpretation of distortion results the experimental measurements

have been normalized against the nominal distance at room temperature. The Y

axis scale shows this normalized magnitude multiplied by 100. The closest the

magnitude is to 100, the lesser the distortion.

The observed pressure trends seem to suggest that for all distances, the

greater the intensification pressure the bigger the distortion. It is conjectured that

as the pressure is increased the ribs thickness increases, coupled with more die

warpage leading to a greater reduction in the selected in-cavity distances. On

the other hand, the dwell time curves show rather inconsistent trends, suggesting

179

only for D2 that the longer the casting remains in the die the smaller the

distortion. For D1 and D3 the middle dwell time level produces the smallest

distortion. A possible explanation for this behavior is the fact that the distance

D2 is formed in the middle section of the insert where there is more mass

distribution and a hot spot is developed, leaving the warpage of the casting on

that region more sensitive to the dwell time. As the casting warps it is

conjectured that the distance between the fins that form D2 gets reduced more

than D1 or D3.

Fig. 5.5 Experimental main effect plots for in-cavity dimensions

180

Fig. 5.6 shows the obtained interaction plots. The lowest intensification

pressure level curves show consistent behavior for distances D1 and D3,

suggesting only for D2 that the longer the dwell time the smaller the distortion. At

this lowest pressure level, the middle dwell time level produces the smallest

distortion for distances D1 and D3. The middle level pressure curves on the

other hand show that for D1 and D2 the longer the dwell time the smaller the

distortion, whereas for D3 the pressure curve has a maximum at 9s dwell time

rather than a minimum as in the previous set of pressure results. The highest

pressure level curves show a consistent trend for D1, D2 and D3, suggesting a

minimum at the 9s dwell time.

The rather high inconsistency in the interaction plot results seems to

indicate that distortion is a more complex phenomena than what would be

expected, being a complex interplay of physical phenomena such as residual

stresses and temperature at ejection. On one hand it might be possible to have

more distortion on a casting that gets ejected after a longer dwell time just

because of the higher residual stress built-up even though the ejection

temperatures are lower. On the other hand it might be possible to produce more

distortion on a casting ejected sooner as a result of the combination of higher

ejection temperatures and very low residual stresses. Therefore, although the

trends depicted in Fig. 5.6 may not be generalized, they do bring the attention to

the fact that distortion is far more dependent on the interaction of process control

parameters than on the individual action of them. Moreover, it can be expected

that different features or dimensions on castings respond differently to the

181

process parameters. Thus, adjusting process parameters might bring some

dimensions within tolerances while it may cause others to fall out of tolerances.

Fig. 5.6 Experimental interaction plots for in-cavity dimensions

The box plots for these measurements are shown in Fig. 5.7. As can be

seen, the spread of the data is rather large in some cases especially for distance

D3, but overall the means can be distinguished somewhat clearly. The spread of

182

the data might be attributed to the lack of accurate control in total cycle time and

pour volume during the casting session.

Fig. 5.7 Box plots for in-cavity dimensions

5.2.4 Across parting plane dimensions

Because across parting plane dimensions are typically reported to be

harder to control, it was decided to analyze some of these features on this

183

casting. These dimensional features are typically more sensitive to variations in

intensification pressure magnitudes and slides positioning; thus, it was thought

that the DOE used for this study could provide some insights into the

dependency of these features to the proposed design variables.

The ANOVA results for these dimensions are presented in Table 5.5 to

Table 5.7 for the three different distances. The statistical results show that for H2

and H4, the pressure, dwell time and their interaction are all statistically

significant, whereas for H3 only the pressure and the interaction term are

significant. Regarding the blocking factors, only the shot weight and casting

weight resulted to be statistically significant as seen in the tables.

The ANOVA results for the across parting plane dimensions came in good

agreement with the in-cavity dimensions results. Both sets of results show that

the design variables, intensification pressure and dwell time, and their interaction

term resulted to be statistically significant for all but H3 dimension. It is worth

mentioning that although the ANOVA results show that the design variables have

a statistically significant effect on the dimensions studied, the practical effect may

or may not be significant enough and the dimensional changes may be

considered all within the expected variability of the process.

184


Pressure 2 7.67 0.96983 0.48491 45.25 0

Dwell time 2 0.09110 0.21502 0.10751 10.03 0

Pressure*Dwell time

4 3.70939 1.27241 0.3181 29.68 0

Furnace temp 1 0.00283 0.00090 0.00090 0.08 0.773

Cycle time 2 0.01951 0.06119 0.03059 2.85 0.060

Shot weight 4 0.25488 0.16029 0.04007 3.74 0.006

Casting weight 2 0.31945 0.31945 0.15972 14.90 0

Error 162 1.73607 1.73607 0.01072

Total 179 13.808

Table 5.5 ANOVA results for distance H2

185


Pressure 2 3.42595 0.47073 0.23536 29.80 0

Dwell time 2 0.02434 0.02398 0.01199 1.52 0.222

Pressure*Dwell time

4 1.51288 0.43658 0.10914 13.82 0

Furnace temp 1 0.00815 0.00628 0.00628 0.80 0.374

Cycle time 2 0.00649 0.01968 0.00984 1.25 0.290

Shot weight 4 0.28381 0.15766 0.03942 4.99 0.001

Casting weight 2 0.33147 0.33147 0.16574 20.98 0

Error 162 1.27954 1.27954 0.00790

Total 179 6.87262


186


Pressure 2 3.78156 0.78919 0.3936 34.05 0

Dwell time 2 0.00883 0.10098 0.05049 4.36 0.014

Pressure*Dwell time

4 1.35853 0.48737 0.12184 10.51 0

Furnace temp 1 0.00030 0.00117 0.00117 0.1 0.751

Cycle time 2 0.03882 0.01100 0.00550 0.47 0.623

Shot weight 4 0.69171 0.43277 0.10819 9.34 0

Casting weight 2 0.41198 0.41198 0.20599 17.77 0

Error 162 1.87746 1.87746 0.01159

Total 179 8.16198


Fig. 5.8 shows the main effect plots for selected across parting plane

distances. Only for H2 do the plots show that the higher the intensification

pressure the larger the distance. The data seems to suggest that only for this

case, does the common belief that the higher pressure the larger the across

parting plane distance may hold. For distances H3 and H4, the highest pressure

level produced a smaller distance than the middle pressure level as shown in the

plots. The dwell time main effect plots depict a rather consistent trend for all

three distances, suggesting that the 9s dwell time produces the longest ribs.

187

Fig. 5.8 Experimental main effect plots for across parting plane dimensions

The interaction plots for the across parting plane dimensions are

presented in Fig. 5.9. The obtained trends clearly show the effects of the

intensification pressure on the across parting plane dimensional behavior. The

pressure plots also show the rather consistent behavior across all the analyzed

distances.

At the lowest pressure level, where almost no intensification pressure was

used, minimal distortion was obtained at a dwell time of 9 seconds. This

behavior was reversed at the highest intensification pressure level, suggesting

188

that distortion is the maximum at the 9 seconds dwell time. When intensification

pressure is used at the middle level, the results show that the longer the dwell

time the greater the obtained shrinkage.

The interaction results show the intricate interrelation of the effects of the

intensification pressure and dwell time on the across parting plane dimensions.

These trends reinforced the claim that in-cavity and across parting plane casting

distortion is far greater the result of a complex interplay of residual stresses and

temperature profiles and it becomes rather difficult to isolate the effects of these

two process variables when their interaction effects drive casting distortion to

such a large extent. Only at one pressure level did the results suggest that the

longer the dwell time the greater the shrinkage.

The interaction plots may also indicate that pressure gradients within the

solidifying casting played some role in across parting plane dimensions. This

claim can be supported by looking at the Y scale of the interaction plot for

distance H4, which shows magnitudes closer to 100, indicating that the distance

at this location is larger. Smaller distances at locations far from the gate might

have been obtained due to premature solidification before the intensification

pressure took effect.

189

Fig. 5.9 Experimental interaction plots for across parting plane dimensions

The box plots for the across parting plane measurements are shown in

Fig. 5.10. Even though a large amount of outliers can be seen, some general

trends can be observed from the data. For a fixed dwell time level, the

intensification pressure follows an upward trend across all three distances. By

the same token, at the same pressure level, the dwell time curves follow similar

trends for the three distances. Three main trends can be noticed resembling the

observed behaviors shown in Fig. 5.9.

190

Fig. 5.10 Box plots for across parting plane dimensions

5.3 Computer model predictions

5.3.1 Model preparation

Computer models simulating the process conditions listed in Table 5.1

were prepared following the modeling methodology described in Chapter 3. The

three modeling stages were followed to predict the final casting distortion for

each of the 9 different experimental runs.

191

Individual thermal models for each of the three different dwell times were

prepared in order to account for the differences in die thermal profiles resulting

from the different cycle times. The obtained thermal results were used to

simulate the thermal load during the first two modeling stages. The three

different intensification pressure levels used in the DOE were modeled by

applying three different pressure loads correspondingly.

The fully-coupled thermal-mechanical models were run to completion

using 64-bit Windows-based work stations. The running time for the computer

models varied based on the simulated dwell time, taking 15, 19 and 21 days for

the 7, 9 and 11 seconds simulated dwell time respectively. The running times

were rather long because the maximum step size was controlled not to exceed

0.010 seconds. This restriction in step size was imposed by design in order to

avoid numerical errors in the stress field due to large temperature differences

within steps.

5.3.2 Distance calculations

5.3.2.1 Coordinate transformation

The distortion predictions given by the computer models were analyzed

taken the CMM measurements as the basis for the analysis. A coordinate

transformation was required in order to express the final distorted nodal

coordinates of the casting relative to the same coordinate system defined by the

fixture used in the experimental measurements. The coordinate system

established by the fixture during the measurements was mathematically

constructed using the final distorted coordinates of six nodes in the casting

lo

c

c

d

ocated the c

ontacted by

oordinates.

Three

istorted coo

closest to th

y the fixture

.

Fig. 5.11 S

e datum pla

ordinates o

he same ga

e. Fig. 5.11

Sampling orde

anes were m

of those six

19

auging poin

1 shows the

er of casting n

mathematic

sampled no

92

nts were the

e sampling

nodes for coo

cally constru

odes. The

e actual cas

order of the

ordinate trans

ucted using

normal of t

stings were

e nodal

sformation

g the final

the first dat

e

tum

193

plane was obtained using the coordinates of the nodes 1, 2 and 3 and was

computed as follows

where

corresponds to the normal of the first datum plane

, 1 … 3 corresponds to the , , nodal coordinates of the sampled

nodes

With the first’s plane normal known, the complete plane equation for the

first datum plane was obtained by using the coordinates of node 1 and solving

the following dot product

,

The normal of the second datum plane was obtained by using the

coordinates of the nodes 4 and 5, and the normal of the first datum plane by

solving the following cross product

where

corresponds to the normal of the first datum plane

194

, 4,5 corresponds to the , , nodal coordinates of the sampled

nodes

With the second’s plane normal known, the complete plane equation for

the second datum plane was obtained by using the nodal coordinates of node 4

and solving the following dot product

,

The third’s datum plane normal was determined to be the cross product of

the normals of the first and second datum planes

The complete plane equation for the third datum plane was obtained by

using the coordinates of node 6 and solving the following dot product

,

Knowing the parameters of the plane equation for the three datum planes,

the origin was readily determined by solving the following linear system of

equations

195

where

, , correspond to the , , coordinates of the origin

, , , 1 … 3 correspond to the parameters of the plane’s normal

Expressing the distorted nodal coordinates in the coordinate system

determined by the datum planes consisted in performing an affine transformation.

An affine transformation is a map between two vector spaces and consists of a

linear transformation and a translation. The fixture basis determined by its datum

planes were expressed as vectors in the computer model coordinate system as

defined by the three computed normal vectors

The order of these basis vectors is defined by the original orientation of

the coordinate system used in the computer models. The last computer normal

corresponds to the axis, the second computed normal corresponds to the

axis and the first computed normal corresponds to the axis.

The orientation of the basis in the fixture coordinate system was not

coincident with the orientation of the basis in the computer model. The axis in

the fixture basis corresponded to the negative axis in the computer model

basis, the axis corresponded to the axis, respectively, while both axes were

oriented in the same direction. The rotation matrix R shown below was used to

orient the fixture basis properly

196

0 1 01 0 00 0 1

The affine transformation of coordinates may be done either way, from the

fixture coordinate system to the computer model coordinate system or viceversa.

Considering expressing the CMM experimental readings in terms the computer

model coordinate system, the following affine transformation may be used

1 0 1 1

where

corresponds to a vector of ( , , ) coordinates expressed in terms of

the computer model coordinate system

corresponds to a vector of ( , , ) coordinates expressed in terms of

the fixture coordinate system

, , is the vector containing the coordinates of the origin

Since the objective is the transformation of computer model results in

terms of the fixture coordinate system, the following transformation was used

1 0 1 1

197

5.3.2.2 Distance calculation

The in-cavity and across parting plane distances were computed using the

same procedures as the CMM measurements. The nodal coordinates of points

located the closest to the same locations of the sampled points during the CMM

measurements were selected to compute the distances. Since a symmetric

model was used in the simulations only three nodes were sampled along the

casting width. The nodes were chosen to be equally spaced as in the CMM

measurements with the middle point always located at the symmetry plane.

As in the CMM measurements, for the in-cavity dimensions the nodes

were selected on opposing sides of the ribs forming the corresponding distance.

For each of the in-cavity dimensions three pairs of nodal coordinates were

selected and the individual distance between each pair of nodes was computed.

The final in-cavity dimension was taken to be the average of the three individual

distances.

As in the CMM measurements, the coordinate along the ribs height was

used to estimate the across-parting plane dimension for each rib. For each

across parting plane dimension, three individual coordinates corresponding to the

three selected nodes along the ribs width were used and the final distance was

taken to be the average of those three coordinates. The dimensions H2, H3 and

H4 used in the experimental measurements were considered in this analysis as

well.

198

5.3.3 In cavity dimensions

Fig. 5.12 shows the main effect plots obtained using the computer model

predictions. As for the experimental result plots, all computer model results are

expressed as a ratio of the final predicted dimension to the nominal dimension,

expressed in percentage. On the one hand, the pressure and dwell times curves

for distances D2 and D3 show the same trends all suggesting that maximum

distortion is obtained at the middle level for each factor. On the other hand, the

results for distance D1 are different. The pressure curve in this case shows a

downward trend indicating that the higher the intensification pressure the greater

the distortion, whereas the dwell time curve suggests that the longer the dwell

time the lesser the distortion.

As far as trends are concerned, computer model results show a rather

considerable disagreement when compared to the experimental results. The

main effect results come in good agreement only for distance D1 for the

intensification pressure factor and in somewhat close agreement for distance D3

for the dwell time factor. All other plots show opposite trends.

199

Fig. 5.12 Simulation main effect plots for in-cavity dimensions

Fig. 5.13 shows the interaction plots using the simulation results. In most

cases the interaction plots show different behaviors across the three different

distances, depicting more repeatability for distances D2 and D3. The plots

showed that at the lowest pressure curve, lesser distortion was obtained at a

dwell time of 9 seconds for distance D1, whereas it was the opposite for

distances D2 and D3. The same trends were obtained at the middle pressure

level. At the highest pressure level, the pressure curves for distances D1 and D2

200

suggested that the higher the intensification pressure the lesser the distortion,

whereas for distance D3 a minimum was observed at the middle dwell time level.

As with the main effect plots, the interaction plots using the computer

model results came in noticeable disagreement with the experimental results.

The two sets of results showed good agreement only for distances D1 and D3 at

the highest pressure level. The results for the remaining plots differed notably.

Fig. 5.13 Simulation interaction plots for in-cavity dimensions

201

5.3.4 Across parting plane dimensions

Fig. 5.14 shows the main effect plots for the across parting plane

dimensions. The main effect pressure plots show similar trends for distances H3

and H4, depicting a minimum at the middle level. Only for distance H2 do the

pressure curve follow an upward trend, suggesting the higher the intensification

pressure the smaller the across parting plane distortion. The main effect

pressure plot for distance H2 shows a good correlation with the experimental

case, following the same trend. On the other hand, the plots for distances H3

and H4 follow reversed trends when compared with the experimental results.

The computer model dwell time main effect plots show rather expected

trends. For two of the three analyzed dimensions, the plots suggest that the

longer the dwell time the greater the distortion. These trends might be somewhat

expected since leaving the casting in longer contact with the die would potentially

lead to greater across parting plane shrinkage. The general trends of these

results do not correlate well with the experimental trends; however, the physical

interpretation of the computer model results agrees better with field observations.

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Fig. 5.14 Simulation main effect plots for across parting plane dimensions

Fig. 5.15 shows the interaction plots obtained from the computer models

results. The obtained plots show somewhat consistent trends for distances H3

and H4 only. For these two distances, the lowest and mid pressure level plots

follow a downward trend suggesting the longer the dwell time the greater the

shrinkage. At the highest pressure, maximum shrinkage was obtained at the

middle level dwell time. On the contrary, reversed behaviors were obtained for

distance H2. This subset of results show reversed trends for all pressure levels

when compared to the results for distances H3 and H4.

203

When compared to the experimental results, relatively good agreement

was observed. For distances H3 and H4, both the experimental and computer

model results follow similar trends for the middle and highest pressure level

curves. For these two distances at the lowest pressure, the results do not

correlated well. The largest differences were observed for distance H2, where

for all pressure levels the computer models followed reversed trends when

compared to the experimental plots. A more detailed comparison of the results is

presented in the next section.

Fig. 5.15 Simulation interaction plots for across parting plane dimensions

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5.4 Comparison of experimental and computer model results

Simulation results for in-cavity and across parting plane dimensions were

compared against the experimental measurements. Because computer models

generate only one numerical result for each distance, a single numerical

experimental value had to be selected for comparisons purposes. It was decided

to use the median of each run and compare it with the simulation results. The

results for each run are expressed as a ratio of the difference between the

experimental and computer model predictions to the experimental distance,

expressed in percentage. The following equation was used for computing the

results

100

Eq. 5.2

Fig. 5.16 shows the results obtained for the in-cavity dimensions. The

results showed that in most cases computer models over-predicted the change in

dimensions, resulting in smaller distances when compared to experimental

results. The largest differences were consistently obtained for distance D2,

reaching a difference of up to 0.5% in run 7. Computer model and experimental

predictions came in closer agreement for distances D1 and D3, with differences

averaging 0.14% and 0.15%, respectively. On average, the differences between

both sets of results across all distances were about 0.2%.

Over prediction of casting warpage by the computer models might be a

likely explanation for the consistent under estimation given by the computer

205

models. It is conjectured that the weak resistance of casting to bending loading

is the artifact result of the linear reduced integration elements used to represent

it. Linear reduced integration elements are prone to hourglassing and tend to be

too flexible under bending loads, leading to greater than expected warpage on

the casting.

Fig. 5.16 Comparison of results for in-cavity dimensions

The results obtained for the across parting plane dimensions are shown in

Fig. 5.17. In general, the difference between the computer model predictions

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and the experimental measurements were larger than for the in-cavity

dimensions. The differences were as large as 1.7%, which might be considered

rather noticeable. The results showed that in most cases the differences were

larger for distance H4, located the closest to the gate. Except for one case, the

results showed that the obtained differences for distances H2 and H3 were

smaller than 0.5%, averaging 0.03% and 0.20% across all runs respectively.

These results may also show that the computer model is able to predict across

parting plane dimensions within acceptable levels for some locations and may

argue that the inability to predict in other locations might be attributed to an

inadequate modeling assumption such as a boundary condition.

Fig. 5.17 Comparisons of results for across parting plane dimensions

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A likely explanation to help describing the observed differences in

distortion predictions for across parting plane dimensions might be obtained by

looking at Fig. 5.18 and Fig. 5.19. Fig. 5.18 shows the different across parting

plane dimensions obtained for all the experimental cases. Distance H2 is the

farthest from the gate, whereas distance H4 is at the gate. From this plot it can

be seen that the across parting plane dimensions near the gate are larger. This

might be explained due to the vertical off-center positioning of the die, whose

geometric center is located 76.2mm (3in) down relative to the geometric center of

the platen. Additionally, the center of pressure of the cavity is located 10mm

(0.4in) down, relative to the geometric center of the die. This off-center

positioning of the die and cavity translates into an off-center loading during

operation and causes the die to open up at the bottom, leading to larger across

parting plane dimensions near the gate. Additionally, since the casting is gated

from the bottom, solidification progresses from the top-down and any premature

solidification of the top regions before the intensification pressure takes effect

would concentrate this pressure loading near the gate, contributing to lager

across parting plane dimensions.

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Fig. 5.18 Distribution of across parting plane experimental dimensions

Fig. 5.19 shows the across parting plane dimensions obtained from the

computer model results. These results predicted different trends when compared

to the experimental results, suggesting larger dimensions far from the gate. This

might be explained due to the way the clamping force was modeled. In this

research work clamping force was modeled by applying a constant pressure load

at the back of the ejector and cover platens. The load was assumed to be of

equal magnitudes at the top and bottom and no account for load imbalance due

to off-center positioning of the die was taken. Moreover, having a constant

pressure load prevents from capturing any load redistribution that takes place

after the intensification pressure is applied. The limitations of this approach may

29.7

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Distance (m

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209

cause the machine to behave too stiff preventing any cavity enlargement,

especially across parting plane, leading to smaller dimensions.

Additionally, the observed differences may also be due to the inability to

model the intensification pressure adequately. As described, the intensification

pressure is the result of the loading action of the pressurized casting acting on

the die. These pressurized solidification conditions may guarantee a tight

contact between the casting and the die at all times, preventing the casting from

freely shrinking across the parting plane. Because the solid elements used to

represent the casting cannot carry a hydrostatic pressure, the described

solidification conditions cannot be modeled. In this modeling work the

intensification pressure is modeled as a pressure load acting on the die cavity

surface and a de-coupling at the casting/die interface exists that allows the

casting to shrink freely across the parting plane. This modeling limitation may

also lead to smaller across parting plane dimensions, especially at locations

where the intensification pressure effects are larger, particularly near the gate.

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Fig. 5.19 Distribution of across parting plane simulation dimensions

5.5 Testing the adequacy of part distortion modeling assumptions

Because of the high degree of complexity commonly found in nature, the

use of mathematical models for simulating a particular system always requires

the use of assumptions to be able to solve the given problem. Developing

computer models in die casting has always represented a real challenge

because of the many interrelated physical phenomena that take place during the

process of making a casting. Currently, all of the available codes used for part

distortion modeling in die casting rely on a set of assumptions limiting to varying

degrees the accuracy of the numerical predictions.

29.429.529.629.729.829.930

30.130.230.3

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211

It was decided to investigate the adequacy of the different sets of

modeling assumptions most commonly found. Computer models were generated

to reproduce the different sets of assumptions taken. Table 5.8 shows six

different sets of modeling assumptions analyzed. Case 1 represents the

methodology presented in this dissertation work. Case 2 was designed to

investigate the effect of having an elastic constitutive model representing the

casting and represents the same set of assumptions as Case 1, but the casting

constitutive model was taken to be purely elastic. Case 3 was designed to

investigate the contributions of the elastic deflections of the tooling in casting final

dimensions. In this case the elastic deflections of the die are not accounted for

and the casting is assumed to be ejected with the same shape as the room

temperature cavity and only having a characteristic ejection thermal profile.

In case 4 the casting is assumed to be ejected with the same shape as the

operating cavity at the point of ejection and with a characteristic thermal profile.

Case 5 represents the same set of assumptions as case 1, but the casting is

assumed to be ejected stress-free. This case was designed to investigate the

effect of the residual stress predictions at ejection obtained in this research work

in casting final dimensions. Case 6 represents a slight variation of case 4 in

which the casting is represented using an elastic constitutive model. This case

was designed to investigate the validity of assuming an elastic model for the

casting after ejection.

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Case Modeling assumptions

1

• Elastic deflections of die • Contact interactions casting/die • Casting is ejected with the predicted shape, stress and thermal profile • Elastic-perfectly-plastic constitutive model

2

• Elastic deflections of die • Contact interactions casting/die • Casting is ejected with the predicted shape, stress and thermal profile • Elastic constitutive model

3 • Casting shape matches room temperature cavity shape • Casting ejected stress-free and with predicted thermal profile • Elastic-perfectly-plastic constitutive model

4 • Casting shape matches distorted cavity shape • Casting ejected stress-free and with predicted thermal profile • Elastic-perfectly-plastic constitutive model

5

• Elastic deflections of die • Contact interactions casting/die • Casting is ejected stress-free, with the predicted shape and thermal

profile • Elastic-perfectly-plastic constitutive model

6 • Casting shape matches distorted cavity shape • Casting ejected stress-free and with predicted thermal profile • Elastic constitutive model

Table 5.8 Proposed sets of modeling assumptions

The computer models were prepared using the described modeling

assumptions and the combination of process conditions of run 5 from the DOE as

shown in Table 5.1. This run was selected because it represents the nominal

combination of process conditions designed to produce the casting under study.

The numerical predictions of the computer models were compared against the

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experimental data obtained for run 5. As with the previous analysis, the median

of the experimental results was taken as the single numerical value to be used

for comparing the experimental data with the computer model predictions.

The computer model results were expressed in the fixture coordinate

system using the same affine transformation as described in section 5.3.2.1. Fig.

5.20 shows the results obtained for the in-cavity dimensions. The results show

that the best predictions were obtained in case 4, where the casting is assumed

to be ejected stress-free and with the same shape as the distorted die cavity at

the point of injection. Under these set of assumptions, assuming an elastic

model for the casting material yields results as good as assuming an elastic-

plastic model. The predictions for case 1, which uses the methodology proposed

in this research work, were substantially improved when the residual stresses at

ejection are not accounted for. This observation shows the effect the residual

stresses at ejection can have in driving the casting distortion after ejection.

The results showed that the worst predictions were obtained when an

elastic constitutive model is used to represent the casting throughout the

analysis. These results clearly show that the use of an elastic model is not

adequate for modeling casting distortion. The second worst predictions were

obtained when the elastic deflections of the die are not accounted for. These

results show the significant contributions of the die deflections in the in-cavity

casting dimensions and suggest the die and machine elements must be included

in the analysis.

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Fig. 5.20 Comparisons of results for in-cavity dimensions predicted by the different sets of modeling assumptions

Fig. 5.21 shows the results obtained for the across parting plane

dimensions. The predictions for these distances followed a different pattern as

the in-cavity dimensions, and the set of modeling assumptions that were not able

to provide good predictions previously resulted to be reasonably good for across

parting plane dimensions. A clear example of this situation can be seen from the

across parting plane predictions given by case 2, which uses an elastic model for

the casting. The predictions given by this case were the closest to the

experimental results for all distances, showing differences of 0.07 and 0.11% for

distances H2 and H3, respectively. The results of case 3 also showed the close

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agreement obtained for the across parting plane distances, where before they

were a poor predictor for in-cavity dimensions.

In consistency with the results of Fig. 5.17, the obtained data also shows

that all cases over estimate the shrinkage of distance H4. This observation

seems to suggest that the modeling methodology relies on an assumption that

consistently affects the across parting plane dimensions near the gate. As was

mentioned before, the overly stiff behavior of the machine might be attributed to

the constant pressure used to model the clamping force effects. Additionally, the

inability to model the intensification pressure adequately may lead to smaller

dimensions as well.

Fig. 5.21 Comparison of results for across parting plane dimensions predicted by the different sets of modeling assumptions

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5.6 Stress results

The ability to predict the deformation and residual stresses in a casting is

of vital importance for casting design and performance. It is well known that the

distribution and magnitude of the stresses on a casting can severely limit its

function. Moreover, the residual stresses can have a significant impact on the

casting final dimensions if their magnitudes and distributions are such that

twisting and warping result from the state of stress. Because of their importance,

it was decided to study the behavior of the stress evolution on the castings using

the computer model results.

To analyze the behavior of the residual stresses on the casting, a series of

points of interest were selected and their stress, strain and temperature profiles

were studied. The plots were generated from the simulation results of run 5,

corresponding to a 9s dwell time and an intensification pressure of 680bars as

listed in Table 5.1. Although the magnitudes of the stresses and strains vary

depending on the process conditions used to produce the casting, run 5 was

selected because it represents the nominal process conditions designed to

produce the casting under study.

Fig. 5.22 shows the selected locations for this analysis. The points were

all located at the casting symmetry plane. The von Mises, maximum principal,

medium principal and minimum principal stresses were all extracted from the

finite element’s centroid at the locations shown. The cooling profile of the

selected elements is plotted along with the stresses to help interpreting the

results.

217

To gain more insight into the stress behavior, the maximum principal,

medium principal and minimum principal strains at the centroid of the selected

elements were plotted as well. It was decided to analyze the strains because

they may provide more indication into the nature of the cooling conditions the

casting experiences, since thermal strains as such do not cause stresses,

stresses develop only if the shrinkage is constrained either due to a die wall or

due to differential thermal cooling of different regions of the casting.

Fig. 5.22 Analyzed stress locations at the symmetry plane

Fig. 5.23 to Fig. 5.26 show the stress and temperature profiles at the

selected regions. The results depict a rather consistent trend at these locations,

suggesting a stress built-up while the casting cools in the die followed by stress

relaxation after ejection. The stress built-up may be explained by the fairly rigid

constrained cooling conditions the casting experiences while it remains in the die.

218

At ejection, when this restraint is removed and the casting is free to shrink, some

degree of relaxation takes place as reflected by the observed sudden drop in the

stress plots.

The stress plots suggest that the degree of relaxation after ejection varies

depending on location. Locations 1 and 2 show that after ejection the stresses

relax and remain low for the rest of the cooling time. On the other hand, the

stress plots for locations 3 and 4 suggest that although relaxation takes place

after ejection, the stresses build up after ejection. A likely explanation for this

behavior might be the fact that these locations are situated near a hot spot and

deformation might be still taking place, leading to an increase of the stresses at

these regions. These results also show that the magnitudes of the computer

model stress predictions are within an expected range. At the selected locations,

the stress results vary between 30 to 60 MPa, considered within acceptable

levels.

219

Fig. 5.23 Stress and temperature profiles at location 1

220


221


222


A useful piece of information regarding the stress behavior in the casting

can be obtained by looking at the strain profiles. As mentioned, the stresses on

a casting develop due to the constrained shrinkage conditions that can be

developed due to differential thermal cooling between different regions of the

casting and also due to contact between the casting and the die walls. Thus,

looking at the strains can provide insights into the nature of the cooling conditions

223

because the thermal strains just provide a “thermal load” or mismatch that is

mapped into stresses if some restrainting is experienced.

Fig. 5.27 to Fig. 5.30 show the profiles for the maximum, medium and

minimum principal strains at the same locations shown in Fig. 5.22. In

agreement with the stress plots, the strain curves show that strains are indeed

developed while the casting cools in the die. The development of these strains

might be due to plastic behavior of the rather weak casting, unable to resist the

restraint of the stiff die walls. The weak behavior of the casting at these

temperatures can be well explained by looking at material properties of the

casting shown in Fig. 4.43 and Fig. 4.44. These curves show that casting yields

under perfect plasticity at the temperature range observed while cooling in the

die and is therefore rather weak to resist any constrained shrinkage.

After ejection, the strain plots show a rather interesting trend. When the

casting is ejected, the strains experienced a sudden increase, followed by an

asymptotic downward trend while the casting cools to room temperature. The

sudden rise in magnitude might be explained by the stress relaxation taking

place, since at this point the casting can shrink freely potentially causing a rise in

the thermal strains that are accommodated as plastic stresses leading to

relaxation. The downward asymptotic trend may correspond to the ability of the

casting to shrink freely as it cools to room temperature.

224

Fig. 5.27 Strain profiles at location 1

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5.7 Summary

The statistical analysis done using the CMM measurements predicted that

the pressure, dwell time and their interaction were all statistically significant for

both, the in-cavity and across parting plane dimensions. The interaction plots for

the in-cavity distances showed the degree to which complex relationship

between the ejection temperature profile and the residual stresses at ejection can

drive the dimensional behavior of the casting. The across parting plane

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interaction plots showed more consistent trends across all three distances.

These results also suggested larger dimensions for distances near the gate.

Computer model interaction predictions correlated well with only a few

experimental cases. The dimensional predictions for in-cavity dimensions

showed close agreement with the experimental results, with differences

averaging 0.18% across all analyzed distances. On the other hand, the

predictions for across parting plane dimensions showed larger differences,

especially for the dimensions near the gate. The observed differences in results

for the distance at the gate were as large as 1.79%. However, the results for the

other two dimensions were much closer to the experimental measurements with

differences averaging 0.03% and 0.20%, respectively.

It is conjectured that the larger differences in results for across parting

plane dimensions might be attributed to modeling aspects. The constant

pressure load used to model clamping force effects is thought to cause an overly

stiff behavior of the machine, leading to smaller deflections and potentially to

smaller across parting plane dimensions. Additionally, the inability to model

pressurized solidification for the casting prevents from having a tight contact

between the casting and the die, which may also lead to a larger shrinkage

across parting plane.

The results obtained from testing the adequacy of different sets of

modeling assumptions yielded different trends for in-cavity and across parting

plane dimensions. On the one hand, for in-cavity dimensions the best results

were obtained when the casting is assumed to be ejected stress-free and with

229

the same shape of distorted die cavity shape at ejection. The use of an elastic or

an elastic-plastic model for the casting material did not have any impact on the

results. The worst predictions were obtained when an elastic model is used for

the casting material. On the other hand, for across parting plane dimensions the

best predictions were given when the casting material is assumed to be elastic.

Nonetheless, in all cases the shrinkage for across parting plane dimensions were

over predicted.

The analysis of the stress profiles provided insights into the evolution of

the stresses during the whole cooling period. A stress built-up was observed

while the casting cools in the die, followed by a stress relaxation at the point of

ejection. The obtained plots suggested that the evolution of the stresses after

ejection is location dependent, showing some degree of stress built-up for

regions undergoing distortion after ejection. The analysis of the strain plots

correlated well with the observed stress profiles, clearly depicting the built-up and

relaxation that take place in the casting as it cools.

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6 CHAPTER 6

CONCLUSIONS

6.1 Introduction

The motivation for this research work was the need to develop a computer

modeling methodology to predict final dimensions and residual stresses in a die

casting. The difficulty of the problem consisted in accounting for the elastic

deflections experienced by the die, which bring about dimensional changes in the

cavity during operation. Thus, a methodology aimed at predicting final casting

dimensions must account for these dimensional changes, since the initial casting

shape at the end of filling is determined by the distorted die cavity.

Accounting for the dimensional changes taking place in the die cavity

requires tracking its distortion during the loading. Clamping, temperature and

intensification pressure are the main static loads considered when modeling die

distortion. The machine components must be included in the model to properly

simulate the loading conditions. The main complexity lies in how the cavity

distortions are tracked, since the method for tracking the distortion must be

accurate to within tenths of a millimeter to be of any use.

231

Additionally, stresses in a casting have a significant impact in driving its

distortion and the casting material must be accurately represented in the model.

This requires accurate temperature and strain rate dependent constitutive

models for representing the casting material, since the mapping of the strains to

stresses relies on the assumed constitutive model.

A modeling methodology for predicting final dimensions and residual

stresses in a die casting was developed using Abaqus, a commercially available

FEA package. The material properties needed to represent the casting material

were determined as part of this research work as well. The predicted dimensions

given by the model were in relatively close agreement with experimental

measurements. The methodology over predicts shrinkage for across parting

plane dimensions located near the gate.

The method presented as part of this research work represents a starting

point towards developing accurate models for the study of dimensions in die

castings. Additionally, residual stress predictions can be obtained from the

results, providing a useful piece of information for casting design. The use of this

procedure may position casting designers and die builders in a better situation by

providing them with useful data available at their computer desks before any

casting is produced or any die is machined.

6.2 Research contributions

A computer model methodology to predict final casting dimensions and

residual stresses for the die casting process has been developed. The

232

procedure presented is able to account for the elastic deflections experienced by

the die. The die and machine components are part of the model and the main

process loads causing the elastic deflections of the die are modeled. The

method relies on using a shell mesh tied to the die cavity that tracks its distortion

during the application the clamping, temperature and intensification pressure

loads. Furthermore, accurate representation of the casting material was done by

using temperature and strain rate dependent mechanical properties determined

as part of this research work.

The computer model predictions were evaluated with experimental

measurements taken on production castings. Casting distortion characterized by

a set of in-cavity and across parting plane dimensions was determined for a

series of cases simulating different process conditions. The computer model

results were able to predict dimensions with close agreement, showing the

largest differences for the across parting plane dimensions near the gate.

6.3 Conclusions

6.3.1 Finite element modeling

The finite element methodology presented in this research work was the

end result of a series of modeling efforts done to develop a reliable method to

predict the distortion and residual stresses on a die casting. During the course of

this development a series of modeling difficulties were experienced, mainly

dealing with tracking the distortion of the die cavity. This section describes the

233

conclusions drawn from the challenges faced and the lessons learned while

overcoming such obstacles.

Tracking the distortion in the cavity was initially attempted by modeling the

interaction of fluid and solid elements in ADINA. The FSI model provided an

adequate way of describing the distorted die cavity shape, since the fluid casting

followed the cavity at all times. Moreover, the intensification pressure could be

modeled as load coming from the fluid casting, as supposed to be modeled as a

load acting onto the die. However, since liquid elements cannot turn into solids

modeling casting distortion required a coupled thermal-mechanical analysis,

which used the FSI predictions as initial conditions. Differences in displacement

predictions between the FSI and the thermal-mechanical model caused contact

divergence that were not resolved, eventually leading to abandoning this

modeling procedure.

Tracking of the distorted die cavity was successfully done by using a shell

mesh tied to the die cavity. The shell mesh was formed using the surface

elements of the three dimensional casting mesh, both meshes sharing the same

surface nodes. The shell was tied to the cavity and the die distortion was

modeled by applying the clamping, temperature and intensification pressure

loads in a static analysis. The displacement predictions given by the shell were

mapped onto the casting mesh, obtaining this way a description of the distorted

die cavity but now on a three dimensional mesh. The assumption behind this

methodology was that the distortions in the die are small enough that they can be

mapped only to the surface of the casting without affecting its interior structure.

234

Semi-rigid body motions were observed to cause contact divergence

during this modeling work. In Abaqus/Standard, divergence can be experienced

during the contact iterations if rigid body motions are experienced between

surfaces that are expected to come into contact. Modeling the effects of

clamping force produced semi-rigid body motion due to tie bar stretching and

difficulties were experienced when contact was being established between the

casting and die cavity surfaces at the onset of cooling. To overcome this

modeling issue, the tie bars were excluded from the model and the clamping

force effects were modeled by applying a pressure load on both platens. The

total force of equal magnitude on both sides suppressed any semi-rigid body

motion and allowed the contact between the casting and the die to be

established without any divergence issues. It is worth mentioning that this

modeling limitation might be overcome if Abaqus/Explicit is used to solve this

model since the contact formulations can readily accommodate rigid body

motions between contacting surfaces during the course of the analysis.

The use of reduced integration finite elements was required to represent

the casting in the computer models. Reduced integration elements were

employed because volumetric locking was observed when fully integrated

elements were used. Volumetric locking on fully integrated finite elements

results because modeling the incompressible material response characteristic of

plasticity adds kinematic constraints to an element, which for this case requires

the volume at the element’s integration point to remain constant. In some

circumstances, modeling this material response may over constrain the element,

235

causing an overly stiff behavior that locks it. Reduced integration elements use

fewer integration points that help satisfying the kinematic constraints more easily.

Modeling the hydrostatic pressure on the casting still remains as a

modeling opportunity for further research. The ability to have this degree of

freedom will allow modeling the pressurized solidification conditions in the

casting more adequately, since the hydrostatic state of stress represents the

initial conditions for the stress calculations. Having this modeling capability will

also provide a better representation of the interaction conditions between the

casting and the die, since the intensification pressure could be transferred from

the casting to the die removing de-coupling between them.

6.3.2 Determination of mechanical properties for casting material

The temperature and strain rate dependent mechanical properties for the

die casting alloy A380.0 were determined as part of this research work. This task

required the design of testing procedures, including the selection of testing

temperatures and strain rates, selection of machine, determination of machine

operational procedures, specimen selection, production and preparation. The

conclusions drawn from the different stages of this project are described in this

section.

An experimental design matrix was used to select the temperatures and

strain rates for conducting the tests. The temperatures were selected to span the

whole solidification range, while the selected strain rates were determined to be

characteristic of the cooling conditions in the die casting process. Temperatures

ranged from 25 ºC to 500 ºC, while the strain rates varied from 1x10-2 to 1x10-6 s-

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1. Because the two design variables had different levels, an hexagonal design

was used for designing the matrix.

The selected specimen design used for the tensile tests corresponded to

the fatigue design recommended for die casting alloys. This design was selected

instead of the tensile test design because of space restrictions in the Gleeble

chamber. The production of the specimens was carried out by an industrial

partner using an existing tooling. Due to large porosity content found in the test

bars, 360 out of the 800 specimens produced were selected for the tests. It is

worth mentioning that the large porosity content found in the bars is the result of

the gating design in the insert.

The tensile tests were conducted in a Gleeble thermo-mechanical

simulator. This machine has been used extensively to determine the mechanical

properties for direct chill aluminum alloys and was selected for this project.

Preliminary tests were initially conducted using a Gleeble 3800. However,

limitations in the minimum displacement rates that can be achieved with this

model prevented from reaching the smallest strain rate magnitudes required in

this project. This displacement rate limitation resulted from the 200GPM

hydraulic valve installed in the machine model, allowing a minimum displacement

rate of 0.1mm/s. The tests were finally conducted in a Gleeble 1500, whose

60GPM hydraulic valve allowed achieving the smallest required strain rates.

Although the minimum strain rates were achieved, a 10-15GPM hydraulic valve

would have been recommended to guarantee accurate control of the machine

jaws.

237

Achieving the intended forces for the constant force tests posed some

difficulties during testing. Since the load cell installed in the Gleeble 1500 is

rated for 80KN, the applied forces for the tests fell within 1-4% of the load cell

output. The control system of the machine had difficulties interpreting the

readings appropriately due to excessive noise in the data. Although the tests

were carried out successfully, a smaller cell of about 10KN would have provided

a better feedback to the machine control system and allowed more accurate

controlled tests.

Among the twenty different runs carried out the tests performed at the

highest temperatures and smallest strain rates posed the biggest difficulties. The

smallest strain rates were difficult to achieve because they required small

displacement rates or small forces, which both were difficult to control for the

machine due to the reasons described above. At the highest temperatures,

another difficulty was faced due to surface oxidation of the test bars. This

oxidation repeatedly caused the thermocouple wires to detach from the surface

and the test had to be redone. A controlled testing environment using Argon had

to be employed to prevent surface oxidation and guarantee proper thermocouple

placement during the test.

From the results obtained, several conclusions can be drawn. At

temperatures below 100 ºC, the material did not exhibit strain rate dependency.

The flow stress was not observed to change under the tested strain rates. Strain

rate dependency was observed for temperatures above 145 ºC. Perfect plasticity

was obtained for temperatures above 220 ºC at strain rates ranging about 1x10-6

238

s-1. The yield strength of the material showed extremely high strain-rate

dependency for temperatures above 300 ºC, with magnitudes differing by as

much as 40MPa at a given temperature under different strain rates. An

interesting observation was noticed in which a higher flow stress was obtained at

486 ºC and a strain rate of 1x10-3 s-1 as compared with a case at 444 ºC and

1x10-6 s-1. This result shows the effect the strain rate can have on the flow

stress.

6.3.3 Design of Experiments

6.3.3.1 Experimental results

Several conclusions can be drawn from the statistical analysis done using

the results of the experimental measurements taken on production castings. The

ANOVA results showed that the pressure, dwell time and their interaction were

all statistically significant for both, in-cavity and across parting plane dimensions.

The obtained results strongly suggest that dimensional stability of castings is

tightly related to the interaction of these typical process control variables.

Generally speaking, the interaction plots for in-cavity dimensions did not

seem to suggest that longer dwell times lead to lesser distortion. On the

contrary, the plots showed the complex interplay of residual stresses and

temperature profiles at ejection and its impact in driving in-cavity dimensions.

The variability of the results across the three analyzed distances also showed

that different dimensions may respond differently to the process control variables.

This observation may argue that adjusting process control variables to bring one

239

dimension within tolerances may result in placing another dimension out of

tolerances. Therefore, care must be exercised when adjusting process variables.

The interaction plots for the across parting plane dimensions showed

more consistency than the in-cavity results across all distances. The pressure

plots at all levels followed similar trends and the obtained results suggested that

longer dwell times lead to greater shrinkage only when 340bars where used for

intensification pressure. On the other hand, when 170 or 680bars were used for

intensification pressure, the plots showed a minimum and a maximum at the 9s

dwell time, respectively. The experimental measurements showed that the

across parting plane dimensions were consistently larger near the gate. This

result may argue for the presence of pressure gradients within the casting and

may also suggest premature solidification in the casting before intensification

pressure takes effect.

6.3.3.2 Computer model results

To test the adequacy of the developed modeling methodology, computer

models simulating the same combination of process conditions as in the

experimental design were prepared. For all the simulated cases, the results

given by the computer models consistently over predicted the casting distortion

and the shrinkage across the parting plane.

The computer model predictions for in-cavity dimensions came in

relatively close agreement with the experimental results. The largest differences

were observed for only one distance, where the differences were as large as

0.5%. Nonetheless, it was observed that on average, the difference between the

240

computer model predictions and the experimental measurements for these in-

cavity distances was about 0.18%.

The predictions for the across parting plane dimensions showed a greater

discrepancy with the experimental results. The largest differences were

observed for the distance near the gate, where contrary to the experimental

measurements the computer model predicted smaller distances. The numerical

differences for this dimension were as large as 1.79%. However, the predictions

for the other two distances showed a closer agreement, averaging a difference of

0.12%.

As was mentioned, the differences between the computer model

predictions and the experimental measurements might be attributed to several

factors. For in-cavity dimensions, the consistent under prediction of the

dimensions might be explained by excessive casting warpage. Excessive

warpage in the casting might have been obtained due to the overly weak

behavior of the reduced integration elements used. These types of elements are

known to over-predict distortion under bending loads due to hourglassing, and a

minimum of four elements through the thickness must be used. For the casting

mesh only three elements through the thickness were used, which may have

potentially led to greater distortion.

For the across parting plane dimensions, clamping force modeling was

identified as one of the factors leading to smaller distances. Since clamping

force was modeled by applying equal pressure loads on the cover and ejector

platens, tie bar imbalance due to off center positioning of the die was not

241

accounted for. Moreover, this method is unable to accommodate any load

redistribution in the tie bars that may result when the intensification pressure is

applied. Thus, it is conjectured that modeling clamping force by using a constant

pressure load causes the machine to behave too stiff and may lead to smaller

across parting plane dimensions.

Additionally, the inability to model the intensification pressure adequately

may also lead to smaller across parting plane dimensions. It is conjectured that

the lack of pressurized solidification conditions in the casting allow it to freely

shrink away from the die, where in reality a tight contact exists due to the high

internal pressure present in the liquid metal. This situation may well explain the

largest differences between the computer model predictions and the

experimental results for the distance near the gate, where the intensification

pressure effects are larger.

6.3.4 Adequacy of different sets of modeling assumptions

As part of this research work, the adequacy of several sets of modeling

assumptions in predicting in-cavity and across parting plane dimensions was

evaluated. The results of this exercise provided valuable insights into the validity

of the existing modeling assumptions and exhibited their respective limitations.

For the in-cavity dimensions, the best predictions were obtained when the

casting was assumed to be ejected stress-free and with the same shape as the

distorted die cavity at the point of injection. Under these assumptions, using an

elastic or an elastic-plastic material model for the casting did not have any impact

in the final dimensions. The predictions of the modeling methodology developed

242

in this research work improved substantially when the residual stresses at

ejection were not accounted for. The worst predictions were obtained when the

casting material is assumed to be elastic throughout the analysis. Ignoring the

contributions of the elastic deflections of the die yielded the second worst

predictions

The across parting plane predictions followed different trends. Contrary to

the predictions for the in-cavity dimensions, the best across parting plane

predictions were observed when the casting material was assumed to be elastic.

Assuming the casting is ejected stress-free and with the same shape as the

distorted cavity predicted relatively good results as well. The predictions of the

proposed methodology developed in this research work did not improve when the

residual stresses at ejection were not accounted for.

6.3.5 Analysis of residual stress profiles

Because of the difficulty in measuring residual stresses in production

castings, the computer model results were used to analyze the evolution of

stresses at several locations within the casting. For different locations at the

parting plane, the stress profiles depicted a clear stress built-up while the casting

is in the die, followed by a stress relaxation at the point of ejection. The stress

built-up was explained by the restrainted shrinkage conditions in the die, while

the relaxation was explained by the ability of the casting to shrink freely after

ejection. The obtained plots showed that the stresses followed different trends

depending on the location and a built-up may be experienced at locations such

as a hot spot undergoing constrained shrinkage while cooling on air.

243

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250

APPENDIX A

TEST BARS INSERT

251

252

APPENDIX B

TENSILE BARS MATERIAL CHEMICAL COMPOSITION AND DIE CASTING PROCESS CONTROL AND CYCLE PARAMETERS

253

254

255

APPENDIX C

COLLECTED DATA DURING TENSILE TESTS

256

Run order

Run Replication Temperature (ºC)

Strain rate (s-1)

Date Specimen #

1 12 1 39 1x10-5 5/10/07 410 2 8 1 81 1x10-2 5/10/07 29 3 6 1 221 1x10-2 5/10/07 677 4 2 1 486 1x10-3 5/10/07 686 5 2 2 486 1x10-3 5/10/07 326 6 3 1 444 1x10-2 5/10/07 435 7 2 3 486 1x10-3 5/10/07 468 8 4 1 381 1x10-2 5/10/07 480 9 10 1 25 1x10-4 5/25/07 99 10 18 1 444 1x10-6 5/10/07 829 11 18 2 444 1x10-6 5/11/07 88 12 15 1 221 1x10-6 5/11/07 541 13 16 1 304 1x10-6 5/26/07 835 14 2 4 486 1x10-3 5/26/07 795 15 1 1 500 1x10-4 5/26/07 239 16 4 2 381 1x10-2 5/26/07 346 17 16 2 304 1x10-6 5/26/07 24 18 16 3 304 1x10-6 5/26/07 574 19 19 1 486 1x10-6 5/26/07 90 20 13 1 81 1x10-6 5/29/07 257 21 8 2 81 1x10-2 5/29/07 399 22 12 2 39 1x10-5 5/29/07 717 23 5 1 304 1x10-2 5/29/07 84 24 14 1 144 1x10-6 5/30/07 366 25 7 1 144 1x10-2 5/30/07 403 26 11 1 25 1x10-4 5/30/07 106 27 16 4 304 1x10-6 5/30/07 135 28 10 2 25 1x10-4 5/30/07 641 29 19 2 486 1x10-5 5/30/07 240 30 9 1 39 1x10-3 5/31/07 747 31 1 2 500 1x10-4 5/31/07 286 32 11 2 25 1x10-4 5/31/07 316 33 13 2 81 1x10-6 6/1/07 799 34 6 2 221 1x10-2 5/31/07 96 35 17 1 381 1x10-6 5/31/07 405 36 5 2 304 1x10-2 5/31/07 177 37 12 3 39 1x10-5 5/31/07 264 38 12 4 39 1x10-5 5/31/07 381 39 20 1 263 1x10-4 5/31/07 56 40 1 3 500 1x10-4 5/31/07 712 41 2 5 486 1x10-3 5/31/07 791 42 17 2 381 1x10-6 6/2/07 628 43 1 4 500 1x10-4 5/31/07 485 44 18 3 444 1x10-6 6/1/07 127 45 17 3 381 1x10-6 6/3/07 646 46 4 3 381 1x10-2 6/1/07 37 47 17 4 381 1x10-6 6/5/07 285

Table C.1 Data collection for tensile tests

257

Table C.1 continued

48 3 2 444 1x10-2 6/1/07 651 49 10 3 25 1x10-4 6/1/07 472 50 1 5 500 1x10-4 6/1/07 154 51 14 2 144 1x10-6 6/4/07 31 52 20 2 263 1x10-4 6/5/07 335 53 7 2 144 1x10-2 6/5/07 381 54 18 4 444 1x10-6 6/5/07 689 55 7 3 144 1x10-2 6/5/07 206 56 5 3 304 1x10-2 6/5/07 33 57 14 3 144 1x10-6 6/6/07 383 58 19 3 486 1x10-5 6/5/07 741 59 4 4 381 1x10-2 6/5/07 521 60 19 4 486 1x10-5 6/5/07 191 61 9 2 39 1x10-3 6/5/07 256 62 6 3 221 1x10-2 6/5/07 513 63 13 3 81 1x10-6 6/6/07 656 64 11 3 25 1x10-4 6/5/07 267 65 19 5 486 1x10-5 6/5/07 59 66 3 3 444 1x10-2 6/5/07 408 67 9 3 39 1x10-3 6/5/07 364 68 8 3 81 1x10-2 6/5/07 698 69 20 3 263 1x10-4 6/5/07 2 70 17 5 381 1x10-6 6/7/07 483 71 20 4 263 1x10-4 6/5/07 247 72 4 5 381 1x10-2 6/7/07 761 73 5 4 304 1x10-2 6/7/07 58 74 11 4 25 1x10-4 6/7/07 620 75 10 4 25 1x10-4 6/7/07 30 76 11 5 25 1x10-4 6/7/07 535 77 6 4 221 1x10-2 6/7/07 130 78 10 5 25 1x10-4 6/7/07 786 79 6 5 221 1x10-2 6/7/07 694 80 14 4 144 1x10-6 6/7/07 831 81 18 5 444 1x10-6 6/7/07 710 82 12 5 39 1x10-5 6/9/07 280 83 15 2 221 1x10-6 6/7/07 258 84 15 3 221 1x10-6 6/8/07 503 85 9 4 39 1x10-3 6/8/07 753 86 7 4 144 1x10-2 6/8/07 464 87 14 5 144 1x10-6 6/9/07 484 88 15 4 221 1x10-6 6/10/07 697 89 8 4 81 1x10-2 6/8/07 283 90 3 4 444 1x10-2 6/8/07 740 91 20 5 263 1x10-4 6/9/07 349 92 13 4 81 1x10-6 6/11/07 255 93 9 5 39 1x10-3 6/8/07 265 94 13 5 81 1x10-6 6/11/07 104 95 5 5 304 1x10-2 6/8/07 23 96 8 5 81 1x10-2 6/8/07 757

258

Table C.1 continued

97 16 5 304 1x10-6 6/11/07 416 98 13 5 444 1x10-2 6/8/07 642 99 15 5 221 1x10-6 6/10/07 107 100 7 5 144 1x10-2 6/8/07 450

259

APPENDIX D

SAMPLE MEASUREMENTS OBTAINED FROM COORDINATE MEASUREMENT MACHINE

260

261

262

263

264

265

266

267

APPENDIX E

RESEARCH CASTING INSERT DIMENSIONS

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