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Mathematical modeling of microstructure and residual stress evolution in cast iron calender rolls Maijer, Daan Michiel 1998

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MATHEMATICAL MODELING OF MICROSTRUCTURE AND RESIDUAL STRESS EVOLUTION IN CAST IRON CALENDER ROLLS by DAAN MICHIEL MAIJER B.ASc, The University of British Columbia, 1994 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDffiS (Department of Metals and Materials Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December 1998 © Daan Michiel Maijer, 1998 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, 1 agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of (HcWs aw) lr1a,kr]diU F , ^ \ X A ^ C \ The University of British Columbia Vancouver, Canada Date hzjLPpJ^J- L£ , DE-6 (2/88) ii Abstract Mathematical models have been developed for the purpose of predicting the evolution of microstructure and residual stress that develops during the manufacture of hypo-eutectic cast iron rolls for the paper industry. These include, a finite element based heat flow model to predict the evolution of temperature and microstructure, and a preliminary finite element based stress model to estimate the evolution of residual stress. Both models have been implemented in the commercial finite element code ABAQUS. To characterize the evolution of solidified microstructure, specialized routines, employing relationships describing nucleation and growth of equiaxed primary austenite, gray iron and white iron, were formulated and incorporated into an ABAQUS thermal model through user-defined subroutines. These relationships have been adapted and extended from a number of investigations describing equiaxed cast iron solidification presented in the literature. In addition, a preliminary columnar growth model was also implemented and tested to describe white iron growth. To describe the evolution of residual stress, a preliminary thermal stress model was developed to describe the evolution of stress and strain throughout the casting process. The preliminary stress model utilizes the temperature and microstructure predictions of the thermal model as input and incorporates phase and temperature dependent properties describing the elastic modulus, thermal dilatation, as well as, the strain rate independent, Von Mises, plastic deformation of the various phases. In order to validate the various components of the overall model, a series of measurements involving two industrial castings - a Quik-cup casting and a reduced scale Ill roll casting - were undertaken to provide thermal, microstructural, and residual stress data for use in 'fine-tuning' and validating the mathematical models. Comparison of the predicted and measured temperatures show good overall agreement for both casting geometries. Microstructure predictions for the Quik-cup casting agree with the observed microstructure. However, in the reduced scale roll casting, key microstructure phenomena were inaccurately predicted with the equiaxed microstructure model. Improvements in the microstructure predictions were observed after the white iron columnar growth model was implemented. Residual stress predictions for the reduced scale roll casting show good overall agreement with the OD surface measurements. The results of this analysis show the importance and usefulness of developing the ability to predict residual stress and microstructure evolution in cast iron rolls. iv Table of Contents Abstract ii Table of Contents iv List of Tables . viii List of Figures ix List of Symbols xvi Acknowledgements xviii Chapter 1 Introduction 1 1.1 Thermoroll Manufacturing 2 1.2 General Description of Roll Failures 4 1.3 General Discussion of Residual Stress 5 Chapter 2 Literature Review 12 2.1 Cast Iron Microstructure Models 12 2.1.1 Growth Morphology 13 2.1.2 Phase Stability 14 2.1.3 Primary Austenite Growth 17 2.1.4 Nucleation and Growth Model 18 2.1.4.1 Evolution of Fraction Solidified 18 2.1.4.2 Nucleation Kinetics 21 2.1.4.3 Growth Kinetics 23 2.1.5 Microsegregation 25 2.1.6 Gray and White Cast Iron Modeling 27 2.2 Thermal Models 29 2.3 Residual Stress Models .....31 2.3.1 Constitutive Material Behaviour 33 2.3.2 Transformation Plasticity 34 Chapter 3 Scope and Objectives 45 3.1 Scope of Research Programme 45 3.2 Objectives of Research Programme 48 V Chapter 4 Experimental Measurements 49 4.1 Reduced Scale Roll Casting 49 4.1.1 Casting Instrumentation , 51 4.1.2 Casting Procedure 53 4.1.3 Thermal Response 54 4.1.3.1 Mould Preheat 55 4.1.3.2 Casting Solidification 55 4.1.3.3 Long Term Cooling 58 4.1.4 Residual Stress Measurements 58 4.1.5 Microstructural Characterization... 59 4.2 Quik-cup 62 4.2.1 Procedures 63 4.2.2 Thermal Response 64 4.2.3 Microstructure Characterization 65 4.3 Summary 67 Chapter 5 Model Development 85 5.1 FEM Background 86 5.1.1 Element Interpolation 87 5.1.2 Numerical Integration 88 5.1.3 ABAQUS Solution Method 88 5.2 Thermal Model .90 5.2.1 General Thermal Model Formulation 90 5.2.2 Implementation of the Microstructure Model 91 5.2.2.1 Micro-mode Time Integration 93 5.2.2.2 Multi-component Segregation 94 5.2.2.3 Phase Impingement 95 5.2.2.4 Nucleation Attenuation 96 5.2.2.5 Phase Normalization 97 5.2.3 Latent Heat 98 5.2.4 Temperature and Phase Dependent Thermal Properties 100 5.3 Stress Model 101 5.3.1 Genera] Stress Model Formulation 101 5.3.2 Elastic Mechanical Properties 102 5.3.3 Dilatation Effects 102 5.3.4 Constitutive Behaviour 104 Chapter 6 Microstructure and Thermal Model Application I l l 6.1 Analysis of the Quik-cup Casting 112 6.1.1 Mesh 112 6.1.2 Boundary Conditions 113 6.1.3 Initial Conditions.. 116 6.1.4 Microstructure Model 'Fine-Tuning' 116 6.1.4.1 Phase Stability Temperatures 116 6.1.4.2 Nucleation and Growth Kinetics Coefficients 117 6.1.5 Predictions and Comparisons to Measured Data 118 6.1.5.1 Temperature and Microstructure Results and Comparisons 118 6.1.5.2 Effects of Formation Temperature 722 6.1.5.3 Solute Segregation 123 6.1.5.4 Nucleation Effects 123 6.1.6 Sensitivity 125 6.1.6.1 Microstructure Parameter Sensitivity 126 6.1.6.2 Thermal Parameter Sensitivity 128 6.2 Analysis of Reduced Scale Roll Casting 129 6.2A Mesh 129 6.2.2 Boundary Conditions 131 6.2.2.1 Preheat Boundary Conditions 131 6.2.2.2 Solidification and Cooling Boundary Conditions 132 6.2.3 Initial Conditions 133 6.2.4 Comparison of Predictions and Measured Data 134 6.2.4.1 Temperature Prediction for the Preheat Stage 134 6.2.4.2 Thermal Predictions of the Solidification Stage 135 6.2.4.3 Thermal Predictions of Long Term Cooling 139 6.2.4.4 Microstructure Predictions 140 6.3 Microstructure Model Extension for Columnar Growth 142 6.3.1 Columnar Model Formulation 144 6.3.2 Application of Columnar Model for White Iron Solidification 148 6.3.2.1 Columnar and Equiaxed Predictions in the Reduced Scale Roll Casting 148 6.3.2.2 Columnar and Equiaxed Quik-cup Casting Predictions 150 6.4 Summary 150 Vll Chapter 7 Preliminary Residual Stress Analysis of the reduced scale roll casting 173 7.1 Mesh 176 7.2 Boundary Conditions 276' 7.3 Initial Conditions 177 7.4 Residual Stress Predictions 177 7.4.1 Residual Stress Evolution 177 7.4.2 Residual Stress Profiles 181 7.5 Sensitivity Analysis of Stress Model 183 7.5.1 Microstructure Sensitivity 183 7.5.2 Material Constitutive Behaviour Sensitivity 185 Chapter 8 Summary and Conclusions 200 8.1 Recommendations for Future Work 204 Bibliography 200 Appendix A Thermocouple Calibration Data For Reduced Scale Roll Casting 210 Appendix B Quik-cup Casting Thermal Analysis 211 Appendix C Governing Equations and Conditions 212 Cl Thermal Equations 212 C.l . l Quik-cup Thermal Equations 212 C. 1.1.1 Initial Conditions 213 C.l.l.2 Boundary Conditions 213 C. 1.2 Reduced Scale Roll Thermal Equations 213 C. 1.2.1 Initial Conditions 214 C.l.2.2 Boundary Conditions 214 C.2 Stress Equations 215 Appendix D Quik-cup Casting Sensitivity Analysis Plots 216 Vlll List of Tables Page TABLE 1.1- NEXGEN THERMOROLL CAST IRON COMPOSITION[2] 8 TABLE 1.2 - DIMENSIONS OF FINISHED NEXGEN THERMOROLLS[2] 8 TABLE 2.1 - MICROSTRUCTURE SPECIFIC VOLUMETRIC LATENT HEAT VALUES[14] 36 TABLE 4.1 - COMPOSITION OF THE CAST IRON ALLOY, K T , FOR THE REDUCED SCALE ROLL CASTING AND THE QUIKCUP 69 TABLE 4.2 - RESULTS OF THE RESIDUAL STRESS MEASUREMENTS PERFORMED ON REDUCED SCALE ROLL CASTING; (JA IS THE AXIAL STRESS AND <7tIS THE TANGENTIAL STRESS 69 TABLE 5.1- THERMO-PHYSICAL PROPERTIES OF HYPO-EUTECTIC CAST IRON AND A BONDED SILICA SAND MOULD MATERIAL[61 ] 106 TABLE 6.1 - NUCLEATTON AND GROWTH KINETICS PARAMETERS FOR MICROSTRUCTURE MODEL 152 TABLE 6.2 - MESH DESCRITION FOR THE REDUCED SCALE ROLL CASTING MODEL 152 TABLE 6.3 - SUMMARY OF THE MICROSTRUCTURE AND THERMAL PARAMETERS INVESTIGATED IN THE SENSITIVITY ANALSYIS AND THEIR EFFECTS ON THE FRACTION GRAY AND THE DURATION OF THERMAL RECALESCENCE AT THE QUIK-CUP CASTING CENTER 153 TABLE A . 1 - THERMOCOUPLE CALIBRATION FOR THE SHEATHED TYPE-K THERMOCOUPLES USED IN THE REDUCED SCALE ROLL CASTING 210 ix List of Figures Page F I G U R E 1.1- S C H E M A T I C O F A M O D E R N S O F T NIP C A L E N D E R . THIS E X A M P L E IS F O R A N 8 M W I D E M A C H I N E R U N N I N G A T O V E R 1000M/MIN A N D E Q U I P P E D W I T H H E A T E D C H I L L E D I R O N R O L L S ( U P P E R L E F T , L O W E R R I G H T ) A N D Z O N E D E F L E C T I O N C O N T R O L L E D C O V E R E D R O L L S [ 1 ] 9 F I G U R E 1.2 - M O U L D C O N F I G U R A T I O N F O R T H E R M O R O L L C A S T I N G ( R E P R O D U C E D I N - P A R T FR0M[4]) 9 F I G U R E 1.3 - T Y P I C A L SOLIDIFICATION C O O L I N G C U R V E S F O R A ) G R A Y I R O N , B ) W H I T E I R O N , A N D C ) M O T T L E D IRON[8] 10 F I G U R E 1.4 - I L L U S T R A T I O N O F T H E 1ST, 2ND, A N D 3RD C A T E G O R Y O F R E S I D U A L S T R E S S E S [9] 11 F I G U R E 1.5- T A K I N G T H E R E S I D U A L S T R E S S INTO A C C O U N T U S I N G A H A I G H D I A G R A M (aA = A L T E R N A T E A M P L I T U D E O F T H E A P P L I E D D Y N A M I C S T R E S S , aM = M E A N S T R E S S O F T H E A P P L I E D S T R E S S , A F I = R E S I D U A L STRESS)[7] 11 F I G U R E 2.1 - DIVISION O F C O M P U T A T I O N A L S P A C E F O R D E T E R M I N I S T I C A N D P R O B A B I L I S T I C M O D E L I N G O F C O U P L E D M A C R O - T R A N S P O R T A N D T R A N S F O R M A T I O N KINETICS[11] 37 F I G U R E 2.2 - S C H E M A T I C R E P R E S E N T A T I O N O F A , B C O L U M N A R A N D C , D E Q U I A X E D G R O W T H M O R P H O L O G I E S F O R A , C D E N D R I T I C A N D B , D E U T E C T I C T Y P E A L L O Y S [10] 37 F I G U R E 2.3 - A U S T E N I T E L I Q U I D U S T E M P E R A T U R E S P L O T T E D A S A F U N C T I O N O F C A R B O N E Q U I V A L E N T A F T E R A L A G A R S A M Y ETAL. [25] 38 F I G U R E 2.4 - E U T E C T I C S O L I D I H C A T I O N R A N G E S A S A F U N C T I O N O F SILICON[28] 39 F I G U R E 2.5 - R E L A T I O N S H I P S B E T W E E N F R A C T I O N O F S O L I D , FS, A N D A V E R A G E G R A I N R A D I U S , R, C A L C U L A T E D U S I N G V A R I O U S I M P I N G E M E N T M O D E L S : R A D I U S H A S B E E N N O R M A L I Z E D B Y C R I T I C A L R A D I U S O F C U B I C A R R A N G E M E N T O F S P H E R E S [31] 39 F I G U R E 2.6 - E F F E C T S O F V A R Y I N G T H E I M P I N G E M E N T F A C T O R O N T H E C O O L I N G C U R V E O F A 2 5 . 4 M M - D I A M E T E R CASTING[14] 40 F I G U R E 2.7 - S C H E M A T I C C O M P A R I S O N B E T W E E N A S S U M P T I O N S O F I N S T A N T A N E O U S A N D C O N T I N U O U S N U C L E A T I O N M O D E L S [11] 40 F I G U R E 2.8 - C O N T I N U O U S N U C L E A T I O N M O D E L W H E R E U N D E R C O O L I N G , ATL, D E T E R M I N E S T H E N U M B E R O F N U C L E I , NL, W I T H I N T H E L I Q U I D B Y E V A L U A T I N G T H E I N T E G R A L O F T H E N U C L E I DISTRIBUTION[32] 41 F I G U R E 2.9 - S C H E M A T I C O F S O L I D I H C A T I O N C O O L I N G C U R V E I L L U S T R A T I N G T H E I N F L U E N C E O F A L L O Y S E G R E G A T I O N O N E U T E C T I C S O L I D I H C A T I O N A N D T H E C O N D I T I O N S L E A D I N G T O I N T E R C E L L U L A R C A R B I D E F O R M A T I O N [ 8 ] 41 X FIGURE 2.10 - COMPARISON OF SEGREGATION RELATIONSHIPS AS A FUNCTION OF FRACTION SOLIDIFIED FOR P IN 5-FE[36] 42 FIGURE 2.11- COMPARISON OF SEGREGATION RELATIONSHIPS AS A FUNCTION OF FRACTION SOLIDIFIED FOR C IN Y-FE[36] 42 FIGURE 2.12- UNIAXIAL VISCO-PLASTIC RHEOLOGICAL MODEL (REPRODUCED IN-PART FROM [49]) 43 FIGURE 2.13- YIELD SURFACE FOR GRAY IRON WITH AXES CORRESPONDING TO THE PRINCIPAL STRESS DIRECTIONS[51] 43 FIGURE 2.14 - COMPARISON OF THE STRESS RATIO VS. TIME RESULTS COMPUTED WITH A STANDARD YIELD SURFACE (VON MLSES) AND A NEW YIELD SURFACE. STRESS RATIO IS THE MAXIMUM COMPUTED PRINCIPAL TENSILE STRESS DIVIDED BY YIELD STRESS[51] 44 FIGURE 2.15- RESIDUAL STRESS COMPARISON BETWEEN CALCULATIONS WITH AND WITHOUT TRANSFORMATION PLASTICITY, AND EXPERIMENTALLY MEASURED RESIDUAL STRESS IN AN OIL QUENCHED STEEL PLATE[45] 44 FIGURE 4.1 - REDUCED SCALE ROLL CASTING GEOMETRIC CONFIGURATION AND DIMENSIONS WITH DIFFERENT SHADINGS FOR CHILL MOULD, CASTING, AND BONDED SAND CORE 70 FIGURE 4.2 - PHOTOGRAPH OF THE ASSEMBLED REDUCED SCALE ROLL CASTING MOULD. . 71 FIGURE 4.3 - SCHEMATIC OF A TYPE-K THERMOCOUPLE AND THE SURROUNDING MATERIALS TO PROVIDE PROTECTION FROM THE MELT IN THE REDUCED SCALE ROLL CASTING 71 FIGURE 4.4 - PHOTOGRAPH OF THERMOCOUPLES MOUNTED IN THE BONDED SAND CORE. .. 72 FIGURE 4.5 - PHOTOGRAPH OF THE REDUCED SCALE ROLL CASTING BEING POURED AT WALZENIRLE 72 FIGURE 4.6 - PHOTOGRAPH OF THE REDUCED SCALE ROLL CASTING AFTER BREAKOUT FROM THE MOULD 73 FIGURE 4.7 - PREHEAT TEMPERATURES PRIOR TO CASTING THE REDUCED SCALE ROLL CASTING PLOTTED ON A MOULD SCHEMATIC 73 FIGURE 4.8 - THERMAL HISTORY OF REDUCED SCALE ROLL CASTING DURING SOLIDIFICATION FOR: A) MID-HEIGHT AND B) BOTTOM HEIGHT THERMOCOUPLES 74 FIGURE 4.9 - THERMAL HISTORY OF REDUCED SCALE ROLL CASTING CHILL MOULD DURING SOLIDIFICATION FOR THE MID- AND BOTTOM HEIGHT THERMOCOUPLES 75 FIGURE 4 .10 - EXTENDED COOLING THERMAL RESPONSE OF THE REDUCED SCALE ROLL CASTING FOR: A) MID-HEIGHT AND B) BOTTOM HEIGHT 76 FIGURE 4.11- SCHEMATIC OF RING-CORE RESIDUAL STRESS MEASUREMENT METHOD[56] . '. 77 FIGURE 4.12- RADIAL HARDNESS PROFILES MEASURED AT WALZEN IRLE AND U B C ON SECTIONS FROM THE REDUCED SCALE ROLL CASTING 77 xi FIGURE 4.13 - MICROSTRUCTURE AT THE O D SURFACE OF THE REDUCED SCALE ROLL CASTING SHOWING DARK AREAS OF PEARLITE AND LIGHT AREAS OF IRON-CARBIDE (100X, 2% N l T A L ETCH) 78 FIGURE 4.14 - MICROSTRUCTURE 10 MM FROM THE O D SURFACE OF THE REDUCED SCALE ROLL CASTING SHOWING LIGHT AREAS OF IRON-CARBIDE, GRAY AREAS OF PEARLITE AND B L A C K FLAKES OF GRAPHITE (50x, 2% N l T A L ETCH) 78 FIGURE 4.15- MICROSTRUCTURE 35 MM FROM THE O D SURFACE OF THE REDUCED SCALE ROLL CASTING SHOWING LIGHT AREAS OF IRON-CARBIDE, GRAY AREAS OF PEARLITE AND B L A C K FLAKES OF GRAPHITE (50X, 2% N l T A L ETCH) 79 FIGURE 4.16- MICROSTRUCTURE 75 M M FROM THE O D SURFACE OF THE REDUCED SCALE ROLL CASTING SHOWING LIGHT AREAS OF IRON-CARBIDE / IRON-PHOSPHIDE, GRAY AREAS OF PEARLITE AND B L A C K FLAKES OF GRAPHITE (50X, 2% N l T A L ETCH) 79 FIGURE 4.17 - FRACTION GRAY AND WHITE IRON PROFILES FOR THE REDUCED SCALE ROLL CASTING, AS CONVERTED FROM THE WALZEN IRLE HARDNESS DATA 80 FIGURE 4.18- ENLARGED INTERIOR MICROSTRUCTURE OF THE REDUCED CAST ROLL CASTING ETCHED USING: A) 2% NITAL SHOWING IRON-CARBIDE AND IRON-PHOSPHIDE AS LIGHT PHASE AND B) MURAKAMI'S ETCH SHOWING IRON-PHOSPHIDE AS GRAY PHASES IN AREAS UNETCHED IN A) (200X) 81 FIGURE 4 .19- GEOMETRY AND DIMENSIONS OF THE QUIK-CUP CASTING MOULD; FRONT AND TOP VIEWS SHOWN 82 FIGURE 4.20 - MEASURED THERMAL RESPONSE AT THE QUIK-CUP CASTING CENTER 82 FIGURE 4.21 - PHOTOGRAPH OF MACRO-SHRINKAGE POROSITY IN THE SECTIONED QUIKCUP CASTING 83 FIGURE 4.22 - MICROSTRUCTURE 5 MM FROM THE QUIK-CUP CASTING CENTER WITH IRON-CARBIDE AS A LIGHT PHASE, PEARLITE AS A GRAY PHASE, GRAPHITE AS BLACK FLAKES, AND POROSITY AS LARGE BLACK AREAS (50X, 2% NLTAL ETCH) 83 FIGURE 4.23 - MICROSTRUCTURE AT THE BOTTOM QUIK-CUP CASTING CORNER WITH IRON-CARBIDE AS A LIGHT PHASE AND PEARLITE AS A GRAY PHASE (100X, 2% N l T A L ETCH) 84 FIGURE 4.24 - VARIATION OF PHASE FRACTION GRAY WITH DISTANCE FROM THE QUIK-CUP CASTING CENTER. RESULTS CONVERTED FROM HARDNESS MEASUREMENTS.. 84 FIGURE 5.1- SCHEMATIC OF PROBLEM DOMAINS AS THEY APPLY TO FEM.[57] 107 FIGURE 5.2 - THREE DIMENSIONAL MAPPING OF DIFFERENT ELEMENT TYPES FROM THE CARTESIAN (GLOBAL) SPACE TO A LOCAL COORDINATE SYSTEM. [57] 107 FIGURE 5.3 - FLOW CHART OF USER SUBROUTINES DEVELOPED TO CALCULATE MICROSTRUCTURE WITHIN A B A Q U S 108 FIGURE 5.4 - VARIATION OF ELASTIC MODULUS WITH TEMPERATURE FOR GRAY AND WHITE IRON [62] 109 FIGURE 5.5 - COEFFICIENTS OF THERMAL EXPANSION VERSUS TEMPERATURE FOR GRAY AND WHITE IRON, AFTER[3,62] 109 FIGURE 5.6 - THERMAL STRAINS WITH AND WITHOUT PEARLITE EXPANSION FOR GRAY AND WHITE IRON 110 xu F I G U R E 5.7 - T E N S I L E S T R E S S - S T R A I N R E S P O N S E O F G R A Y A N D W H I T E I R O N A T V A R I O U S T E M P E R A T U R E S [62] 110 F I G U R E 6.1- Q U I K C U P C A S T I N G ( L I G H T P O R T I O N ) A N D M O U L D ( D A R K P O R T I O N ) SIMPLIFIED T O A 1-8TH S E C T I O N A N D M E S H E D W I T H 8 - N O D E B R I C K E L E M E N T S 154 F I G U R E 6.2 - M E A S U R E D T E M P E R A T U R E R E S P O N S E N E A R T H E Q U I K - C U P C A S T I N G C E N T E R A N D V A R I O U S C A R B E ) E E U T E C T I C T E M P E R A T U R E S 154 F I G U R E 6.3 - M E A S U R E D A N D P R E D I C T E D T E M P E R A T U R E S A T T H E Q U I K - C U P C A S T I N G C E N T E R 155 F I G U R E 6.4 - P R E D I C T E D P H A S E E V O L U T I O N W I T H T I M E A T T H E Q U I K - C U P C A S T I N G C E N T E R . 155 F I G U R E 6.5 - P R E D I C T E D T E M P E R A T U R E R E S P O N S E A T T H E B O T T O M C O R N E R O F T H E Q U I K -C U P C A S T I N G 156 F I G U R E 6.6 - P R E D I C T E D P H A S E E V O L U T I O N W I T H T I M E A T T H E B O T T O M C O R N E R O F T H E Q U I K - C U P C A S T I N G 156 F I G U R E 6.7 - C O M P A R I S O N O F P R E D I C T E D A N D M E A S U R E D F R A C T I O N S O F G R A Y A L O N G T H E D I A G O N A L F R O M T H E Q U I K - C U P C A S T I N G C E N T E R T O T H E B O T T O M C O R N E R . N O T E : M E A S U R E D R E S U L T S W E R E C O N V E R T E D F R O M H A R D N E S S M E A S U R E M E N T S 157 F I G U R E 6.8 - A U S T E N I T E , G R A Y I R O N E U T E C T I C , A N D W H I T E I R O N E U T E C T I C P H A S E F O R M A T I O N T E M P E R A T U R E S V A R I A T I O N W I T H T I M E A N D T H E P R E D I C T E D T H E R M A L R E S P O N S E A T T H E C E N T E R O F T H E Q U I K - C U P C A S T I N G 157 F I G U R E 6.9 - V A R I A T I O N O F L I Q U I D C O N C E N T R A T I O N ( C , SI, P) IN W T % W I T H T I M E A T T H E Q U I K - C U P C A S T I N G C E N T E R 158 F I G U R E 6.10 - P R E D I C T E D G R A Y A N D W H I T E I R O N N U C L E A T I O N P R O F I L E S W I T H A N D W I T H O U T C O R R E C T I O N 158 F I G U R E 6.11- SENSITIVITY A N A L Y S I S O F T H E G R A Y G R O W T H C O E F F I C I E N T S H O W I N G : A ) T H E R E S U L T I N G V A R I A T I O N S IN T H E P R O F I L E S O F V O L U M E F R A C T I O N G R A Y I R O N A N D B ) T H E T E M P E R A T U R E E V O L U T I O N S A T T H E C E N T E R A N D C O R N E R O F T H E Q U I K - C U P C A S T I N G . M A X / M I N V A R I A T I O N S A R E S H O W N R E L A T I V E T O T H E R E F E R E N C E R E S U L T 159 F I G U R E 6.12 - SENSITIVITY A N A L Y S I S O F T H E N O M I N A L G A P H E A T T R A N S F E R C O E F F I C I E N T S H O W I N G : A ) T H E R E S U L T I N G V A R I A I T O N S IN T H E P R O F I L E S O F V O L U M E F R A C T I O N G R A Y I R O N A N D B ) T H E T E M P E R A T U R E E V O L U T I O N S A T T H E C E N T E R A N D C O R N E R O F T H E Q U I K - C U P C A S T I N G . M A X / M I N V A R I A T I O N S A R E S H O W N R E L A T I V E T O T H E R E F E R E N C E R E S U L T 160 F I G U R E 6.13- A X I S Y M M E T R I C G E O M E T R Y A N D M E S H O F T H E R E D U C E D S C A L E R O L L C A S T I N G 161 F I G U R E 6.14 - T E M P E R A T U R E P R E D I C T I O N F O R T H E P R E H E A T P L O T T E D A S A ) S C H E M A T I C S H O W I N G P R E D I C T E D A N D ( M E A S U R E D ) T E M P E R A T U R E S IN ° C A N D B ) A C O N T O U R P L O T O F T H E P R E D I C T E D T E M P E R A T U R E S 162 F I G U R E 6.15 - T E M P E R A T U R E E V O L U T I O N S D U R I N G P O U R I N G S T A G E A T V A R I O U S A X I A L L O C A T I O N S A L O N G T H E R E D U C E D S C A L E R O L L C A S T I N G O D S U R F A C E 162 Xlll F I G U R E 6.16- C O N T O U R S O F : A ) T E M P E R A T U R E A N D B ) P H A S E F R A C T I O N SOLIDIFIED P R E D I C T I O N S IN T H E R E D U C E D S C A L E R O L L C A S T I N G A F T E R 1800S O F S O L I D I F I C A T I O N T I M E 163 F I G U R E 6.17 - C O M P A R I S O N O F T H E M E A S U R E D A N D P R E D I C T E D T E M P E R A T U R E E V O L U T I O N A T V A R I O U S L O C A T I O N S W I T H I N T H E R E D U C E D S C A L E R O L L C A S T I N G F O R : A ) M I D - H E I G H T A N D B ) B O T T O M H E I G H T L O C A T I O N S 164 F I G U R E 6 .18- C O M P A R I S O N O F T H E C H I L L M O U L D M E A S U R E D A N D P R E D I C T E D T E M P E R A T U R E E V O L U T I O N S F O R T H E R E D U C E D S C A L E R O L L C A S T I N G A T : A ) M I D - H E I G H T A N D B ) B O T T O M H E I G H T L O C A T I O N S 165 F I G U R E 6 .19- L O N G T E R M C O O L I N G C O M P A R I S O N O F T H E M E A S U R E D A N D P R E D I C T E D T E M P E R A T U R E E V O L U T I O N A T V A R I O U S L O C A T I O N S W I T H I N T H E R E D U C E D S C A L E R O L L C A S T I N G F O R : A ) M I D - H E I G H T A N D B ) B O T T O M H E I G H T L O C A T I O N S 166 F I G U R E 6.20 - L O N G T E R M C O O L I N G C O M P A R I S O N O F T H E C H I L L M O U L D M E A S U R E D A N D P R E D I C T E D T E M P E R A T U R E E V O L U T I O N S F O R T H E R E D U C E D S C A L E R O L L C A S T I N G A T : A ) M I D - H E I G H T A N D B ) B O T T O M H E I G H T L O C A T I O N S 167 F I G U R E 6.21 - M E A S U R E D A N D P R E D I C T E D M I C R O S T R U C T U R E P R O F I L E S O F T H E R E D U C E D S C A L E R O L L C A S T I N G 168 F I G U R E 6.22 - S C H E M A T I C O F C O L U M N A R G R O W T H M O D E L S H O W I N G D I F F E R E N T S T A G E S IN T H E G R O W T H O F A C O L U M N A R F R O N T 168 F I G U R E 6.23 - S C H E M A T I C O F T H E E F F E C T S O F C O L U M N A R F R O N T G R O W T H D I R E C T I O N . . 169 F I G U R E 6.24 - M I C R O S T R U C T U R E P R O F I L E S O F T H E R E D U C E D S C A L E R O L L C A S T I N G C O M P A R I N G T H E O R I G I N A L E Q U I A X E D A N D E X T E N D E D E Q U I A X E D / C O L U M N A R M O D E L P R E D I C T I O N S T O T H E M E A S U R E D D A T A , A S : A ) G R A Y I R O N M I C R O S T R U C T U R E P R O F I L E S A N D B ) W H I T E I R O N M I C R O S T R U C T U R E P R O F I L E S . 170 F I G U R E 6.25 - M I C R O S T R U C T U R E P R O F I L E S O F T H E Q U I K - C U P C A S T I N G C O M P A R I N G T H E O R I G I N A L E Q U I A X E D A N D E X T E N D E D E Q U I A X E D / C O L U M N A R M O D E L P R E D I C T I O N S T O T H E M E A S U R E D D A T A , A S : A ) G R A Y I R O N M I C R O S T R U C T U R E P R O F I L E S A N D B ) W H I T E I R O N M I C R O S T R U C T U R E P R O F I L E S 171 F I G U R E 7.1- S C H E M A T I C O F G E O M E T R Y A N D M E S H U S E D IN T H E S T R E S S A N A L Y S I S O F T H E R E D U C E D S C A L E R O L L C A S T I N G 187 F I G U R E 7.2 - P R E D I C T E D E V O L U T I O N O F T H E A ) T E M P E R A T U R E A N D B ) T H E R M A L S T R A I N IN T H E A X I A L D I R E C T I O N A T T H E MUD H E I G H T F O R T H E ID S U R F A C E , M L D R A D I U S , A N D O D S U R F A C E O F T H E R E D U C E D S C A L E R O L L C A S T I N G 188 F I G U R E 7.3 - P R E D I C T E D E V O L U T I O N O F T H E A ) STRESS A N D B ) E L A S T I C S T R A I N I N T H E A X I A L D I R E C T I O N A T T H E M I D H E I G H T F O R T H E ID S U R F A C E , M L D R A D I U S , A N D O D S U R F A C E O F T H E R E D U C E D S C A L E R O L L C A S T I N G 189 F I G U R E 7.4 - P R E D I C T E D E V O L U T I O N O F P L A S T I C S T R A I N IN T H E A X I A L D I R E C T I O N A T T H E M I D H E I G H T F O R T H E ID S U R F A C E , M I D - R A D I U S , A N D O D S U R F A C E O F T H E R E D U C E D S C A L E R O L L C A S T I N G 190 F I G U R E 7.5 - P R E D I C T E D R A D I A L P R O F I L E S O F T H E A ) S T R E S S A N D B ) P L A S T I C S T R A I N C O M P O N E N T S A T T H E M I D H E I G H T O F T H E R E D U C E D S C A L E R O L L C A S T I N G . ..191 xiv F I G U R E 7.6 - P R E D I C T E D A X I A L P R O F I L E S O F T H E A X I A L A N D T A N G E N T I A L S T R E S S E S A L O N G T H E ID A N D O D S U R F A C E S O F T H E R E D U C E D S C A L E R O L L C A S T I N G . T H E M E A S U R E D O D A X I A L A N D T A N G E N T I A L R E S I D U A L S T R E S S D A T A H A S B E E N P L O T T E D A S S Y M B O L S 192 F I G U R E 7.7 - M I C R O S T R U C T U R A L SENSITIVITY O F T H E P R E D I C T E D R A D I A L P R O F I L E S O F A ) S T R E S S A N D B ) P L A S T I C S T R A I N C O M P O N E N T S A T T H E M I D H E I G H T O F T H E R E D U C E D S C A L E R O L L C A S T I N G . B A S E O N T H E S T R E S S M O D E L P R E D I C T I O N S W I T H M I C R O S T R U C T U R E I N P U T F R O M T H E E Q U I A X E D A N D C O L U M N A R A N D E Q U I A X E D M O D E L S 193 F I G U R E 7.8 - M O D I F I E D M I C R O S T R U C T U R E P R O F I L E E M P L O Y E D IN T H E S T R E S S M O D E L SENSITIVITY A N A L Y S I S . T H E M O D I F I E D M I C R O S T R U C T U R E P R O F I L E IS C O M P A R E D W I T H T H E C O L U M N A R - E Q U I A X E D M O D E L A N D T H E M E A S U R E D M I C R O S T R U C T U R E P R O F I L E S 194 F I G U R E 7.9 - I N F L U E N C E O F T H E W H I T E I R O N O D S U R F A C E M I C R O S T R U C T U R E O N T H E P R E D I C T E D R A D I A L P R O F I L E S O F A ) STRESS A N D B ) P L A S T I C S T R A I N C O M P O N E N T S A T T H E M I D H E I G H T O F T H E R E D U C E D S C A L E R O L L C A S T I N G . B A S E D O N T H E S T R E S S M O D E L P R E D I C T I O N S W I T H M I C R O S T R U C T U R E I N P U T F R O M T H E C O L U M N A R - E Q U I A X E D M O D E L A N D A M O D I F I E D P R O F I L E 195 F I G U R E 7 .10- I N E L A S T I C D E F O R M A T I O N SENSITIVITY O F T H E P R E D I C T E D R A D I A L P R O F I L E S O F A ) S T R E S S A N D B ) P L A S T I C S T R A I N C O M P O N E N T S A T T H E M I D H E I G H T O F T H E R E D U C E D S C A L E R O L L C A S T I N G . B A S E D O N T H E S T R E S S M O D E L P R E D I C T I O N W I T H A N D W I T H O U T T H E I N F L U E N C E O F I N E L A S T I C D E F O R M A T I O N 196 F I G U R E 7 .11- SENSITIVITY P E R T U R B A T I O N O F G Y S I N F L U E N C I N G T H E P R E D I C T E D R A D I A L P R O F I L E S O F A ) S T R E S S A N D B ) P L A S T I C S T R A I N C O M P O N E N T S IN T H E A X I A L D I R E C T I O N A T T H E M I D H E I G H T O F T H E R E D U C E D S C A L E R O L L C A S T I N G . M A X / M I N V A R I A T I O N S A R E S H O W N R E L A T I V E T O T H E L I T E R A T U R E B A S E D OYS 197 F I G U R E 7.12- SENSITIVITY O F A X I A L P L A S T I C S T R A I N E V O L U T I O N A T T H E O D S U R F A C E L O C A T I O N T O P E R T U R B A T I O N S IN fJYs- M A X / M I N V A R I A T I O N S A R E S H O W N R E L A T I V E T O T H E L I T E R A T U R E B A S E D CTys 198 F I G U R E B . 1 - T H E R M O C O U P L E R E S P O N S E O F Q U I K - C U P C A S T I N G S A M P L E S 211 F I G U R E D . 1 - SENSITIVITY A N A L Y S I S O F T H E G R A Y A N D W H I T E I R O N N U C L E A T I O N C O E F F I C I E N T S H O W I N G : A ) T H E R E S U L T I N G V A R I A T I O N S IN T H E P R O F I L E S O F V O L U M E F R A C T I O N G R A Y I R O N A N D B ) T H E T E M P E R A T U R E E V O L U T I O N S A T T H E C E N T E R A N D C O R N E R O F T H E Q U I K - C U P C A S T I N G . M A X / M I N V A R I A T I O N S A R E S H O W N R E L A T I V E T O T H E R E F E R E N C E R E S U L T 216 F I G U R E D.2 - SENSITIVITY A N A L Y S I S O F T H E W H I T E G R O W T H C O E F F I C I E N T S H O W I N G : A ) T H E R E S U L T I N G V A R I A T I O N S IN T H E P R O F I L E S O F V O L U M E F R A C T I O N G R A Y I R O N A N D B ) T H E T E M P E R A T U R E E V O L U T I O N S A T T H E C E N T E R A N D C O R N E R O F T H E Q U I K - C U P C A S T I N G . M A X / M I N V A R I A T I O N S A R E S H O W N R E L A T I V E T O T H E R E F E R E N C E R E S U L T 217 XV FIGURE D.3 - SENSITIVITY ANALYSIS OF THE CARBIDE EUTECTIC TEMPERATURE SHOWING: A) THE RESULTING VARIATIONS IN THE PROFILES OF VOLUME FRACTION GRAY IRON AND B) THE TEMPERATURE EVOLUTIONS AT THE CENTER AND CORNER OF THE QUIK-CUP CASTING. M A X / MIN VARIATIONS ARE SHOWN RELATIVE TO THE REFERENCE RESULT 218 FIGURE D.4 - SENSITIVITY ANALYSIS OF THE INITIAL POUR TEMPERATURE SHOWING: A) THE RESULTING VARIATIONS IN THE PROFILES OF VOLUME FRACTION GRAY IRON AND B) THE TEMPERATURE EVOLUTIONS AT THE CENTER AND CORNER OF THE QUIK-CUP CASTING. M A X / MIN VARIATIONS ARE SHOWN RELATIVE TO THE REFERENCE RESULT 219 FIGURE D.5 - SENSITIVITY ANALYSIS OF THE MINIMUM GAP CONDUCTANCE TEMPERATURE SHOWING: A) THE RESULTING VARIATIONS IN THE PROFILES OF VOLUME FRACTION GRAY IRON AND B) THE TEMPERATURE EVOLUTIONS AT THE CENTER AND CORNER OF THE QUIK-CUP CASTING. M A X / MIN VARIATIONS ARE SHOWN RELATIVE TO THE REFERENCE RESULT 220 FIGURE D.6 - SENSITIVITY ANALYSIS OF THE EMISSIVITY SHOWING: A) THE RESULTING VARIATIONS IN THE PROFILES OF VOLUME FRACTION GRAY IRON AND B) THE TEMPERATURE EVOLUTIONS AT THE CENTER AND CORNER OF THE QUIK-CUP CASTING. M A X / MIN VARIATIONS ARE SHOWN RELATIVE TO THE REFERENCE RESULT 221 List of Symbols Latin Symbols Description Units a,- solution values A nucleation coefficient m"3 K"2 %C, %Si, %P concentration carbon, silicon, and phosphorus in liquid wt% CL liquid composition wt% C0 initial liquid composition wt% CP specific heat J kg"1 K"1 E Young's modulus Pa fcomi fraction of gap heat transfer via conduction fum fraction limit of gap heat transfer via conduction fs volume fraction solidified feq volume fraction solidified equiaxed fC(,i volume fraction solidified columnar f"s' f"w normalized gray and white iron volume fraction solidified fs rate of solidification s"1 h film coefficient for free convection W m"2 K" heff effective heat transfer coefficient Wm"2 K" hcond conductive component of heg W m"2 K" hrad_ radiative component of heff Wm"2 K" Jc, Jc segregation coefficient or conductivity Wm"1 K" L volumetric latent heat T -3 J m h nucleation rate rrf3 s"1 N number of grains per unit volume -3 m N shape functions q" heat flux Wm"2 Q volumetric heat source term Wm"3 Q activation energy J mol"1 R grain radius m R Gas constant (8.3144) J K"1 mol Sij deviatoric stress Pa T temperature °C TL liquidus temperature °c Teut graphite eutectic temperature °c Tcarb iron carbide eutectic temperature °c Tcast temperature of the casting surface °c Tmould temperature of mould surface °c Tsurft Too surface and ambient temperature °C AT liquid undercooling K t time s casting time s xvii u trial function vc„i columnar growth velocity m s"1 V growth velocity m s"1 Wi, Wj, Wk weight coefficients x characteristic length m Greek Symbols a coefficient of thermal expansion d^, 5 Kronnecker delta or Dirac delta £eff effective radiation emissivity £cast, £int/uld emissivity of cast and mould 4 total strain elastic strain 4 thermal strain u transformation strain £"' >j transformational plasticity strain £in •j in-elastic strain creep strain rate total extended volume fraction extended volume fraction of phase j r boundary of Q u growth coefficient V Poisson's Ratio Q problem domain a sub-domain of Q, p density O'rad Stefan-Boltzmann constant (5.6696x K-1 s-1 m2 3 m 3 m kg m"3 W nf2 K"4 r/ 5' 7, (fSJ1, c/ 5 ' 7 7 7 1 s t , 2 n d , and 3 r d categories of residual stress Pa or residual stress Pa <ja alternating stress Pa am mean stress Pa Uij, <Jkk components of stress Pa r local solidification time s y/ nucleation attenuation coefficient xvm Acknowledgements I would like to thank my research supervisor, Dr. Steve Cockcroft, for his guidance and support during the course of my research programme. He is a true mentor and friend whose opinions and insights are highly valued and often sought. I also gratefully acknowledge Dr. Bruce Hawbolt who acted as a sounding board for microstructure evolution theories and who critiqued my thesis, often on short notice. The strong theoretical background of Dr. Alain Jacot was a significant knowledge base from which I often drew and informal discussions with Dr. Michel Rappaz were beneficial at a critical time in the programme. I am grateful to Craig Reid, formerly with MacMillan Bloedel Research, for committing to the academic research process to investigate this industrial problem. The in-kind and financial support of Walzen hie GmbH and specifically the assistance of Walter Patt and Ludwig Hellenthal were instrumental to the success of this programme. The financial support of the Science Council of British Columbia was also greatly appreciated. I am thankful for the invaluable encouragement of my family and friends. Special thanks to Jen and the space kitties, Lucy and Linus, for their love and support. 1 CHAPTER 1 INTRODUCTION The formation of residual stresses can have an important influence on the stress state of components under operational conditions. Specifically for thermorolls used in paper calendering, a level of residual stress is required for safe operation in this cyclic loading application. The recent failure of two thermorolls on the MacMillan Bloedel's Alberni Specialties Nexgen soft nip calender prompted MacMillan Bloedel to question the safety of the manufacturer's roll design. As the residual stresses are an important consideration in design, the prediction of residual stress formation in thermoroll manufacturing would provide a tool to assess and improve thermoroll design. Calendering is the final stage in the paper manufacturing process before winding paper on the reel. During calendering, paper passes through a series of rolls where it is heated and subjected to pressure along the length of the nip (zone of contact between two rolls) to improve surface finish and reduce cross machine direction (CD) variations in paper thickness[l]. When manufacturing coated paper, as in the Nexgen paper machine at MacMillan Bloedel's Alberni Specialties mill, calendering has the added responsibility of developing paper gloss[l]. Calender performance is regulated by controlling the roll operating temperatures and nip loading. The original calenders employed nips formed through mating two hard cast iron rolls. Recent trends in the industry are to adopt instead the use of soft calender nips where a hard cast iron roll mates with an elastomer covered soft roll[l]. This type of soft nip calender, shown schematically in Figure 1.1, is used in the Nexgen paper machine. 2 The soft rolls consist of a cast iron shell with an elastomer covering, while the hard rolls have a gray cast iron shell with a chilled iron surface layer. The hard rolls are referred to as thermorolls as they have heated oil circulating in sub-surface axial bores in order to heat the roll to operating surface temperatures of approximately 200°C. Hydraulic actuators on the thermoroll journal bearings are employed to develop nip line-loads of 350kN/m. The present thermoroll design criteria takes advantage of a high compressive residual stress which develops at the roll outer diameter (OD) during the manufacturing process. The OD compressive stress acts to offset the tensile stress resulting from roll heating to reduce the overall OD stress to within the desired safety envelope. Consequently, this design and its safety hinge on the presence of compressive residual stress at the OD surface. As part of an effort to assess roll design safety and improve manufacturing, this project will develop the capability to quantitatively predict residual stress formation during . manufacturing of thermorolls. In order to understand some of the issues surrounding the residual stress formation in thermorolls, some background on thermoroll manufacturing, recent thermoroll failures, and residual stress formation is presented in the following sections. 1.1 Thermoroll Manufacturing The majority of oil heated calender rolls are manufactured by two companies, Walzen Irle and SHW, both located in Germany. These companies use essentially the same process and techniques to produce the rolls apart from some proprietary differences. Roll 3 shells are cast in a vertical mould constructed of circular cast iron chill blocks placed on top of each other in a casting pit (refer to Figure 1.2). Bonded-sand cores are used in combination with the chill sections during thermoroll casting to achieve the proper residual stress state and reduce costs associated with machining the center bore of the shell. The melt is poured into a side mounted gating system that enters the bottom of the mould at an angle tangential to the circumference. For the Nexgen thermorolls, the melt weighed 75 tonnes and was poured in times ranging between 90 and 100 seconds[2]. The side gating entry into the mould gives the liquid metal a high tangential velocity that forces it against the mould chill blocks. The centrifugal force experienced by the melt causes light impurities such as entrained refractory to be displaced toward the center of the casting. The cast iron used in the Nexgen thermorolls, is a gray cast iron with the composition shown in Table 1.1. The roll shell solidifies with a clear chilled iron OD surface layer which has a microstructure consisting of iron carbide (cementite) embedded in a matrix of pearlite[3]. Moving toward the inner diameter (ID), the microstructure then shifts to that of an equilibrium gray iron microstructure, comprised of flake graphite particles in a pearlite matrix. The microstructure in the transition zone is referred to as being mottled. The relative thicknesses of chill, mottle, and gray microstructure phases for the Nexgen rolls were approximately 10 mm, 30 mm, and 150 mm, respectively[4]. The conditions required to form these distinct zones of cast iron microstructure are presented schematically in Figure 1.3. Under all conditions, the first solid phase to form is primary austenite followed by one or a combination of the two eutectic phases depending on the cooling rate - i.e. gray iron, consisting of austenite and graphite and/or 4 white iron consisting of austenite and iron carbide. For example, gray iron forms at slow cooling rates, when solidification occurs below the stable graphite eutectic temperature and above the metastable iron carbide eutectic temperature, as show in Figure 1.3a. White iron forms at fast cooling rates, when solidification occurs completely below the metastable iron carbide eutectic temperature, as shown in Figure 1.3b, and mottled iron (a combination of gray and white) forms at intermediate cooling rates, when solidification occurs in both regimes (refer to Figure 1.3c). Following solidification, an increasing amount of eutectic austenite combines with the proeutectic austenite in gray and/or white iron prior to the eutectoid transformation to pearlite. At this time, the casting has achieved the final microstructure distribution. After pouring and cooling for one week, the roll shell is removed from the mould for machining. Machining entails removal of the material from the OD and ID to attain the approximate finished dimensions. In addition, gun drilling is used to bore the axial oil alleyways around the periphery of the roll at a depth of approximately 65 mm below the OD face. Finally, the OD crown and surface are finish ground to the tolerances set by purchaser. Table 1.2 provides the finished roll and oil bore dimensions of the Nexgen thermorolls. 1.2 General Description of Roll Failures In-service failures of two Nexgen thermorolls in August and September of 1996 were first identified by oil streaks on the paper following Stack No. 1 in the soft nip calender. In both cases, the failed thermorolls had a large circumferential crack (-875 mm) that intersected several oil bores which had lead to the observed streaks. Craze cracks typical 5 of incidental water contact were present on the first roll to fail at several axial locations, including in the vicinity of the large circumferential crack. This craze cracking was not present on the second failed roll, but a colony of small circumferential cracks (25-100 mm) was located approximately 100 mm from the main crack[5]. The resulting failure investigation by MacMillan Bloedel, the machine builder (Valmet), and the roll manufacturer (SHW) has not conclusively determined the cause of the failures. 1.3 General Discussion of Residual Stress Residual stresses, sometimes called internal stresses, are stresses present in a component in the absence of external loading (including gravity) or thermal gradients[6]. Since residual stress is an equilibrium stress state, the sum of resultant forces and moments within a component must necessarily be zero. There are three classifications of residual stress based on the distance or length scale over which they are observed[6]. The different categories of residual stress and their relation to each other are illustrated schematically in Figure 1.4. Macroscopic stress, the first category of residual stress, is a long range definition (from fim to cm) extending over many grains. Macroscopic stresses, denoted cfs' ' in Figure 1.4, are a bulk measure of the stress state in a component. The second category, microstress (o*5, " in Figure 1.4), is a short range definition (order |im) with observation dimensions on the order of single grains. This category of residual stress is observed between grains with differing physical characteristics, or microstructural phase, and between precipitate particles and their matrix. The third category, residual stresses observed on a length scale 6 between 100 and 1000 A, (refer to rj R S ' 1 1 1 in Figure 1.4) arise due to differences across dislocations in a grain or around an embedded particle. Irrespective of the scale, residual stresses are caused by differential in-elastic deformation and/or dilatation. Typically, residual stresses are generated during component manufacturing, but they can also be generated or modified during operational use. Causes of residual stress in manufacturing include casting, heat treatment, surface treatments, and machining. In fatigue applications, residual stresses, <7r, can increase the limit of safe alternating stresses, oa, by reducing the mean stress, <7m, in the component (referring to Figure 1.5)[7]. This effect is clearly shown in the Haigh diagram of Figure 1.5 where the Goodman line is used to define a safe operating envelope for applications involving a combination of alternating and mean stress. In Figure 1.5, the permissible stress amplitude increases as the compressive residual stress in the component increases. This example applies directly to the operational conditions in a soft nip calender thermoroll. Alternating bending stresses result from the combination of gravity forces and nip loading, while mean bending stresses are defined by thermal and residual stresses. The use of a Haigh diagram like Figure 1.5 thus necessarily requires accurate knowledge of residual stress at the OD surface of a thermoroll. Designs based on the Haigh diagram (Figure 1.5) employ the bulk average residual stress state or macroscopic stresses. Resolution of residual stresses on the micro-scale is potentially useful in analysis of crack nucleation and growth, but typically cannot be incorporated into large scale engineering designs because of the resolution required. In view of this, only macroscopic residual stresses and their development will be considered in this project. Thus, the literature review presented in the following Chapter, considers only macroscopic stress in discussions of residual stress prediction. Table 1.1 - Nexgen thermoroll cast iron composition[2]. Composition (wt%) C Si Mn P S 3.50-3.90 0.32-0.66 0.20-0.40 0.40-0.80 0.08 -0.16 Table 1.2 - Dimensions of finished Nexgen thermorolls[2]. OD Oil bore ID Oil bore Number of Face Length Bearing Center circle diameter oil bores (mm) (mm) (mm) (mm) (mm) (mm) 1500 1370 1120 32 45 7990 9000 Stack #1 Stack #2 Figure 1.1- Schematic of a modern soft nip calender. This example is for an 8m wide machine running at over lOOOm/min and equipped with heated chilled iron rolls (upper left, lower right) and zone deflection controlled covered rolls[ 1 ]. Bonded Sand Core Cast Iron Mould " Tundish -Gating System Figure 1.2 - Mould configuration for thermoroll casting (reproduced in-part from[4]). 10 UJ <E 3 t -< ee u i C L 2 V^-y LIQUIDUS GRAPHITE \ EUTECTIC \ yS" v \ END OF IRON CARBIDE ^SOLIDIFICATION EUTECTIC \ (a) TIME y LIQUIDUS GRAPHITE EUTECTIC IRON CARBIDE\ EUTECTIC (b) TIME Ul cc CC Ul CL z u l y-GRAPHITE \ EUTECTIC \ . IRON CARBIDE \ / " \ END OF X/SOLIDIFICATION EUTECTIC \J ^CHILL (c) TIME Figure 1.3 - Typical solidification cooling curves for a) gray iron, b) white iron, and c) mottled iron[8]. Figure 1.5- Taking the residual stress into account using a Haigh diagram (aa = alternate amplitude of the applied dynamic stress, <5m = mean stress of the applied stress, 07 = residual stress)[7]. 12 CHAPTER 2 LITERATURE REVIEW A review of the literature relevant to the prediction of microstructure and residual stress evolution in cast iron is presented in this chapter. The key to predicting the microstructural evolution, rests on an ability to quantify the nucleation and growth kinetics of primary austenite, gray iron, and white iron and how they are affected by the melt chemistry and prevailing macroscopic heat flow conditions. The microstructure phase evolution and the macroscopic heat flow will influence the residual stress evolution through physical properties and differential thermal contraction. Thus, the complete analysis requires solution of phenomena occurring at both the scale of the microstructure (micro-scale) and at the scale of the component (macro-scale). The relevant subject material will be discussed in order from micro to macro-scale. 2.1 Cast Iron Microstructure Models Heat transfer is coupled with microstructure evolution through the release of latent heat during phase transformations. Thus, the fraction solidified or transformed must be accounted for as a function of temperature and/or time. Methodologies for modeling solidification microstructure were reviewed by Rappaz[10] and more recently by Stefanescufl 1] and the reader is referred to these articles for a complete discussion of the available modeling methods. The two techniques employed in microstructure modeling are termed the 'deterministic' and 'probabilistic' methods. In the deterministic approach, macro-volume 13 elements are divided into micro-volumes, usually spheres, of microstructural phases (refer to Figure 2.1). This method gives a measure of the fraction of microstructure in a volume element, but does not differentiate between individual grains because the position and size of the individual micro-volume elements are not tracked[ll]. In contrast, the probabilistic method tracks development of individual microstructural features in a large macro-volume element (refer to Figure 2.1). This method gives a graphical representation of the microstructure, but it is more computationally intensive than the deterministic approach. For Fe-C melts with eutectic compositions, prediction of the transformation of liquid to eutectic phase can be calculated with either the deterministic or probabilistic technique. However, since the majority of the published investigations into cast iron solidification use the deterministic technique, the following discussion will be limited to that method. 2.1.1 Growth Morphology To correctly predict the evolution of microstructure during the solidification process, it is important to understand the possible growth morphologies. Figure 2.2 illustrates the two primary forms of microstructure morphologies, columnar and equiaxed, for dendritic and eutectic solidification. In columnar microstructures, the growth rate is determined by the speed of the eutectic or liquidus isotherm. There is an initial nucleation event which determines the density of columnar grains, but subsequent solidification is independent of a nucleation event. Undercooling below the eutectic or liquidus isotherm is directly proportional to the speed of solute redistribution. For equiaxed microstructures, nucleation is a precursor to growth and these phenomena are a function of the undercooling. 14 Cast iron solidification has exclusively been modeled using equiaxed solidification models. For the most part, these models focused on gray iron solidification where cooling rates are slower and microstructural analysis clearly illustrates that the solidified microstructure morphology is equiaxed[12,13,14,15]. The studies that include chill or white iron formation have also utilized equiaxed solidification models, but there have been no reported assessments of the microstructure[16,17,18,19,20,21,22]. The overall goal of these studies to predict white iron formation was to reduce or identify areas within a casting where white iron forms. Thus, accurate prediction of the white iron morphology was not assessed. In order to accurately predict the evolution of solidified microstructure, be it either columnar or equiaxed, an accurate assessment of the growth temperature is first required. 2.1.2 Phase Stability The ability of a particular phase to nucleate and grow is contingent on it being thermodynamically stable. To begin, therefore, it is important to establish the temperature range(s) for stability of the various solid phases starting first with primary austenite and following with gray and white iron. This information is necessary for equiaxed solidification to determine first nucleation kinetics and second growth kinetics, and for columnar growth to determine growth temperature. Heine investigated the compositional effect of carbon and silicon on the austenite liquidus temperature under various melting conditions[23,24]. The cast iron melt compositions in these experiments had a range of 0.5 - 3.35 wt% for carbon, 1.25 - 3.5 wt% for silicon and <0.1 wt% phosphorus. The different melting conditions consisted of 15 melting under deoxidized argon below 1454.4°C (TAL), melting under an oxidizing atmosphere of C 0 2 (OTAL), and melting under deoxidized argon at temperatures above 1454.4°C to reduce A1203 and retain 0.01% Al in the melt (HTAL). The austenite liquidus temperature equations describing these different melting conditions were determined to be a function of the carbon equivalent and extended to include phosphorus. These 3 temperature relationships are plotted in Figure 2.3 as a function of carbon equivalent and provided below as Equations 2.1, 2.2, and 2.3: TAL(°C) = 1569- 97.3(%C + 0.25%5i + 0.5%P) (2.1) OTALCQ = 1594.4 -102.2(%C + 0.25%5i + 0.5% P) (2.2) HTAL (°Q= 1540 - 92.06(%C + 0.25%5i + 0.5%P) (2.3) where %C, %Si, and %P are the weight percent of carbon, silicon, and phosphorous in the liquid, respectively. Alagarsamy et al. [25] compared Equations 2.1, 2.2, and 2.3 with another austenite liquidus relationship, Equation 2.4, that was described in a BVCIRA report as the "most representative equation[26]." In Figure 2.3, the austenite liquidus temperatures described by the TAL, OTAL, and HTAL equations decrease as the carbon equivalent increases. The similar slopes of these lines give the relationships an almost constant offset from each other. However, the BVCIRA austenite liquidus temperature traverses these lines because of a larger variation with carbon equivalent. Alagarsamy et al. concluded that the austenite liquidus temperature is not just a function of the composition, but also depends on the melting process variables[25]. TL (°C) = 1669 -124(%C + 0.25%5i + 0.5%P) (2.4) Eutectic arrest temperatures were measured by Heine and used to estimate extremes for the graphite and carbide eutectic temperatures[23,27]. Glover et al. have 16 furthered this work with a series of casting measurements to determine the graphite eutectic, Teut, and carbide eutectic, Tcaru, temperatures (refer to Equations 2.5 and 2.6) as functions of silicon and phosphorus composition[28]. The effectiveness of silicon as a carbide or chill reducer can be seen in Figure 2.4 where the Teut and Tcaro have been plotted as functions of silicon content. Thus, when white iron is desired in a casting, lower silicon contents will enhance the ability to form substantial amounts of chilled cast iron. r e H ((°C)= 1135.06 + 13.89%SJ - 2.05%Sz2 (2.5) Tcarh (°C)= 1138.2 - 6.93(%5i + 2.5% P) -1.717(%5i + 2.5%P)2 (2.6) Although the work by Glover et al. includes the influence of phosphorus on the carbide eutectic, there is no mention of the experimental phosphorus contents in the publication. Other relationships for the graphite and carbide eutectic which include the influence of phosphorus on the eutectic temperatures have been published. In a series of publications on cast iron chilling susceptibility, Fras and Lopez report equations for the austenite liquidus, graphite eutectic, and carbide eutectic temperatures as[16,17]: Ty (°C) = 1636 -113(%C + 0.25Si + 0.5%P) (2.7) Teul (°C)= 1153.9 + 5.25%Si - 14.88%P (2.8) Tcarh ( ° c ) = 1 1 3 4 - 9 - 1 1 -02%5i - 39.7%P (2.9) In a publication describing a model of the gray to white transition, Nastac and Stefanescu report the graphite and carbide eutectic temperatures as[18]: Teul (°C)= 1154 + 4(SiL)L-2{MnL)L-30{PL)L (2.10) Tea* (°C) = 1148 -\5{SiL)L+3{MnL)L-3l{PL)L (2.11) 17 where (x^1 is the volume average concentration of x in the liquid phase. Comparing the carbide eutectic temperature equations, it is clear that phosphorus has the largest impact on temperature. A similar impact is also seen in the graphite eutectic Equations 2.8 and 2.10. It should be noted that typically in cast irons the silicon content is greater than 1.5wt% while the phosphorus content is less than 0.1wt%[8]. Thus, at low concentrations (<0.1wt%) these temperatures are extremely sensitive to phosphorus, but it is not clear whether this sensitivity extends beyond the range of phosphorus compositions for which the eutectic temperatures have been experimentally measured. Clearly, from these temperature relationships, the composition of the liquid will play an important roll in determining the stability regions for microstructural evolution in addition to cooling rate. These temperature relationships are used to calculate the liquid undercooling which defines the nucleation and growth kinetics for the growing phases. 2.1.3 Primary Austenite Growth For cast iron melts of hypo-eutectic compositions, primary austenite forms prior to the eutectic transformation. At temperatures below the liquidus temperature and above the eutectic transformation temperature, the fraction of liquid transformed to primary austenite faus can be calculated as a function of temperature T, assuming an inverse lever rule dependence (Equation 2.12)[14]: 1 T-T, f = (2.12) Jaux 1-k'T-T^ K J where TL is the liquidus temperature, Ty is intersection point of the liquidus and solidus curves, and k' is the ratio of slope of the liquidus curve to the slope of the solidus curve. 18 As previously discussed, Heine[23] developed equations to describe the austenite liquidus temperature as a function of carbon, silicon and phosphorus content. Unfortunately, the equation for the liquidus / solidus intersection point was not developed as a function of phosphorus. In a study of solidification kinetics modeling, Fras et al. [15] detailed a methodology to predict primary austenite and eutectic growth using a nucleation and growth kinetics model. The ability of this model to predict primary austenite formation was not validated with measurements. However, the method of prediction is of particular use because it is similar to those employed for eutectic predictions. 2.1.4 Nucleation and Growth M odel The primary goal of a nucleation and growth kinetics model is to predict the evolution in fraction solidified as a function of time or undercooling. Coupled with the deterministic approach, the nucleation and growth model is applied to a domain representing the casting that has been discretized into macro-volume elements in which the various microstructural phases are allowed to evolve. In order to couple the thermal and microstructure evolution, the rate of fraction solidified must be known and incorporated with latent heat release. 2.1.4.1 Evolution of Fraction Solidified Below the liquidus and/or respective eutectic temperature, nucleation and growth of a particular equiaxed phase can occur. For single phase growth, the unimpeded growth of 19 spherical grains, described as the extended volume fraction of the phase transformed, <pe, can be calculated as[10]: f 4 , (j)=\h{T)-7tR\T,t)dT (2.13) o 3 where n(z) is the nucleation rate at timer and R(%t) is the grain radius nucleated at time 1 and grown until time t. However, the use of Equation 2.13 requires that each growth step be tracked and stored to facilitate the integral. In a study of the columnar to equiaxed transition, Hunt developed a series approximation to Equation 2.13, which is presented below as Equation 2.14[29]. ^Y,NkR] (2.14) k In Equation 2.14, N is the volumetric grain density and R is the grain radius assuming a spherical grain geometry. The series approximation term, ^JNkRl, accounts for the k nucleation and growth of each Jch crystal or grain. In this manner, it is possible to account for the effect of individual grain nucleation and growth while not actually tracking and storing individual grain growth information. For this calculation, temperature is necessarily assumed constant across the volume occupied by the growing microstructure. The extended volume fraction is equivalent to the volume fraction solidified during the initial stages of solidification when impingement of grains is unlikely. As solidification proceeds, the extended volume fraction can be corrected to account for grain impingement and to determine the actual volume fraction transformed,^ , using the 20 well know Avrami correction factor (1 - /v)[30]. In incremental form the Avrami relationship is expressed as: *M-fM (2.15) Equation 2.15 assumes there are increasing amounts of impingement from the beginning of solidification. Near the completion of solidification, the limitation of Equation 2.15 is that the volume fraction transformed equals one only as the radius equals infinity. Some researchers have developed new approaches to account for this impingement. Rappaz reported on the new approach developed by Zou Jie[31], where impingement was calculated for uniformly sized spheres centered at the nodal points of two common lattice structures, simple cubic and close-packed[10]. The different fraction solidified calculated using these impingement factors and the Avrami correction have been plotted in Figure 2.5 as a function of normalized radius. Contrary to the Avrami correction factor, impingement correction begins at 52 and 74% fraction solidified for the simple cubic and close-packed corrections, respectively. The Avrami impingement factor is the most widely used extended volume correction in cast iron research publications. One notable exception to this trend was reported by Goettsch and Dantzig[14]. In addition to applying the standard (l-/v) correction factor, correction factors of 1, (l-fs)2 and (1-jQ3 were implemented. The effects of varying the impingement factors on predicted temperature for a cylindrical casting are shown in Figure 2.6. As the impingement factor exponent is increased from 0 to 3, the rate of solidification decreases causing a gradual release of latent heat until the end of solidification. It should be noted that instead of applying the Avrami impingement correction to the extended volume, Goettsch and Dantzig applied the impingement 21 correction to the growth velocity. However, the effects of applying the impingement factor to the growth velocity were not investigated. The Hunt extended volume model and some form of impingement provides a general methodology for tracking the evolution of primary austenite, gray iron, and white iron providing the evolution in number of grains and grain radius can be quantified. 2.1.4.2 Nucleation Kinetics Nucleation kinetics provide a measure of the grain density in a casting. In cast irons, nucleation kinetics have been described using both instantaneous and continuous nucleation models, shown in Figure 2.7. Instantaneous nucleation models assume that all grain nuclei, N, are generated at a single temperature, TN. While investigating cast iron solidification, Stefanescu used an instantaneous nucleation model of the form[12]: ^-=NSS{T-TN) (2.16) ol where dN/dT is a function representing the number of nuclei versus undercooling, Ns is the total number of nuclei at TN, and 8 is the Dirac delta function. The consequence of assuming all nuclei form at a single temperature is that, numerically, all grains will have the same size and this is true for very few conditions. In contrast, continuous nucleation models assume that the number of nuclei formed is a function of temperature. In pioneering work by Oldfield, the number of nuclei in cast iron was found to obey the relationship[13]: N = A-AT2 (2.17) 22 where TV is the number of nuclei per unit volume, AT is the local undercooling of the liquid (temperature below the respective liquidus or eutectic temperature), and A is a nucleation coefficient which can be experimentally determined. Although further work, such as that by Thevoz et al.[32], has been done to describe continuous nucleation, the relationship developed by Oldfield, Equation 2.17, continues to be used extensively because of its simplicity and accuracy. For comparison, the statistical distribution function used by Thevoz to describe continuous nucleation is given in Equation 2.18[32]: where ATN is the nucleation distribution center, ATa is the standard deviation of the distribution, and Ns is the maximum number of nuclei that can be formed. A graphical representation of this relationship is shown in Figure 2.8. For a given undercooling AT/, the number of nuclei, «/, can be determined by integrating the nucleation distribution from AT= 0 to AT}. Nucleation is complete at the maximum undercooling ATmax, and the total number of nuclei, n2, can be calculated by integrating the nucleation distribution between 0 and ATinax. Equation 2.18 accurately represents nucleation as a function of undercooling, but with the number of parameters to be experimentally determined, it is difficult to implement. For gray iron solidification, 01dfield[13] reported experimentally determined values for A varying from 0.91 - 7.12 cm"3K"2 (0.91xl06 - 7.12xl06 nf3K"2). In a study of gray iron solidification, Goettsch and Dantzig[14] found that A equal to 36.63x103 dN _ Ns •exp (AT-ATj -2{ATj (2.18) 23 mm' 3K" 2 (36.63xl06 m"3K"2) for gray iron nucleation gave the best thermal prediction results. In a more recent publication, Nastac and Stefanescu[18] used separate instantaneous nucleation relations for gray and white iron based on cooling rate where the number of white iron nuclei is 5 times greater than the gray iron nuclei for the same cooling rate. For primary austenite dendrites, Fras et al.[15] used 500 cm"3K~2 (500xl0 6 m" 3K" 2)forA r . Goettsch and Dantzig[14] defined a criteria for employing Equation 2.17 to ensure that the number of nucleation sites does not decrease. After nucleation initiates at the liquidus or eutectic temperature, nuclei will continue to form until either solidification ends or there is thermal recalescence and the cooling rate becomes positive - i.e. the maximum undercooling is reached. In the cases were thermal recalescence occurs, nucleation is halted and the number of nuclei is constant until solidification is finished. 2.1.4.3 Growth Kinetics The growth of dendrites and eutectic phases in multi-component alloys is complex. Presently, most of the approaches adopted for solidification analysis codes require significant simplification, particularly with respect to growth morphology. Despite more and more rigorous attempts to characterize growth kinetics, most relationships generally have the form of Equation 2.19, which relates the growth velocity, V, of a grain to the local undercooling, Zt7T[ 11]: dR V= — = [lAT2 (2.19) 24 In Equation 2.19, R is the radius of a simple spherical grain and ji is a growth coefficient. Stefanescu presented a number of different equations for growth constants of different equiaxed grain types in his review[l 1]. However, for cooperative eutectics such as gray and white iron, the growth coefficient is typically a constant fit to measured data. 01dfield[13] calculated a range of gray iron growth coefficients of 2.5x10"8 -34.5xl0"8 m-s"'-K"2 for a cast iron of 4.1 carbon equivalent. In a two part paper on competitive growth of gray and white iron eutectics, Magnin and Kurz[33,34] measured the gray and white iron growth rates of cast irons with a number of different compositions. For alloy compositions of the Fe-C eutectic, Fe-C-0.5wt%Si, and Fe-C-0. lwt%P, Magnin and Kurz[33] measured gray iron growth coefficients of 9.18xl0"8, 5.95xl0~8, and 0.694xl0"9 m-s_1K"2, respectively. For a model of the gray to white transition, Nastac and Stefanescu[18] used values of 8.0xl0"7 - 2.4xl0"6 and 3.0x10" m-s" KT for the growth coefficients of white and gray iron, respectively. In the case of primary austenite, the standard approach is to assume that solidification obeys the equilibrium phase diagram and calculate the fraction transformed using the inverse Lever rule[14]. However, in one study by Fras et al.[\5], an attempt to account for austenite growth was made based on an alternative theoretical method proposed by Lipton et al. [35]. In this model, the primary austenite growth coefficient is presented as a function of the thermal and solute diffusion parameters for a binary alloy. The application of this method to multi-component alloys such as cast iron requires a knowledge of the alloy specific equilibrium phase diagrams which are typically not avail able [11]. 25 2.1.5 Microsegregation During solidification, solute segregation will influence nucleation and growth kinetics through its influence on undercooling, (the undercooling that a particular phase sees is dependent on the liquidus and/or eutectic temperature, which in turn is dependent on the local liquid composition as described previously in the Phase Stability section). Some of the different models developed to account for microsegregation are presented in the review papers by Rappaz[10] and Stefanescu[l 1]. The influence of segregation in cast irons, shown schematically in Figure 2.9, can lead to inter-granular carbide formation when white iron growth initiates near the end of solidification; hence, it is important to include segregation[8]. In a theoretical analysis, Kobayashi compared the results of various models with the exact solution for the segregation of solutes, such as carbon, phosphorus, and sulphur in iron[36]. Kobayashi evaluated the extremes of segregation — equilibrium segregation where there is complete diffusion in the solid and liquid, and the Scheil equation which assumes no diffusion in the solid and complete diffusion in the liquid. These extremes were compared with the Brody-Flemings[37] and Clyne-Kurz[38] segregation solutions which assume limited diffusion in the solid. The comparison of solute models for phosphorus segregation in 8-iron (refer to Figure 2.10) shows that the Scheil[39] equation provides an adequate description of segregation for low diffusivity solutes like phosphorus. For high diffusivity solutes, such as carbon in austenite, the Scheil equation deviates significantly from the exact solution for fractions solid greater than 50% as shown by Figure 2.11. 26 The Scheil equation, written in terms of the solute composition in the inter-granular liquid, is given by the expression[40]: CL=C0{l-f.T] (2.20) where CL is the concentration of solute in wt%, C() is the initial liquid concentration and k is the segregation coefficient. For materials such as cast iron, no general methodology for describing individual component segregation in multi-component systems involving multiple phases exists. Hence, binary interaction behavior on a component by component basis is usually adopted. The segregation coefficient for carbon during primary austenite solidification, kc, is 0.48. During eutectic solidification of gray iron and white iron, there is negligible carbon segregation because there is a net carbon redistribution between the eutectic austenite and the graphite or carbide. Kagawa and Okamoto[41] have shown thermodynamically that the silicon segregation coefficient is dependent on the microstructural phase solidifying and the concentration of silicon in the liquid. The reported silicon segregation relationships are[41]: *Si-=1.71 kSieu=\ .70-0.31 %5/+0.05%5/2 (2.21) *fi^=0.88-O.05%Si The primary focus in cast iron segregation modeling has been on carbon and silicon segregation because the concentrations of these solutes have the most significant effects on solidification based on published experimental data. The roll of phosphorus segregation and a method of describing this segregation has not been clearly defined in the literature. For the most part, phosphorus segregation has been overlooked because 27 the cast iron alloy systems investigated to-date have been low in phosphorus and the resulting impact of concentration limited phosphorus segregation is minor[8]. 2.1.6 Gray and White Cast Iron Modeling A number of investigations have focussed on the prediction of white iron formation during cast iron solidificationf 14,18-22]. Within these investigations, a wide variety of approaches, centering around the deterministic microstructure prediction method, have been developed to describe the evolution of white and gray iron. The early attempts by Krivanek and Mobley[19], and Stefanescu and Kanetkar[20] employed a finite difference solution method to solve the heat transfer and microstructural equations. These two investigations focused on predicting the evolution of microstructure in the simple geometries of a chill block and a bar casting. The results of these investigations were limited to the prediction of completely gray or white iron microstructure at each nodal location. Following these early investigations, Stefanescu et al. [18,21,22] and Goettsch and Dantzig[14] have employed finite element analysis to solve the combined gray and white iron microstructure evolution in a more complex step casting geometry. The results of these investigations show good agreement with the experimental measurements and illustrate the applicability of this modeling approach to gray and white iron microstructure predictions. Although each of the studies mentioned here employs the deterministic microstructure prediction method, different sub-components were employed in each model. Specifically, the most recent study of gray and white iron prediction by Nastac and Stefanescu[18] does not include the formation of primary austenite. The evolution of 28 primary austenite influences the thermal and microstructure evolution through the latent heat release associated with this transformation. Goettsch and Dantzig[14] accounted for primary austenite formation through the inverse lever law, while Banerjee et al.[22] calculated the austenite dendrite growth velocity and employed the previously described equiaxed growth model to determine the volume fraction of primary austenite formed. In another variation, the influence of microsegregation on the transformation temperatures was neglected by Goettsch and Dantzig[14], while Nastac and Stefanescu[18] and Banerjee et al. [22] employ differing solute segregation methods to include the influence of microsegregation. Other differences center around nucleation and growth kinetics laws which are alloy specific. In each of these investigations, the extension of the single phase equiaxed deterministic approach to the multi-phase regime has been employed, but inadequate discussion of this extension has been provided. The equations describing primary austenite, gray iron, and white iron solidification are established with general criteria for applying them, but a method of assessing the combined growth of each phase is left out. In the case of each phase, the Avrami correction factor is applied to the transformation equations. A clear method of assessing the independent and combined influence of each cast iron phase must be defined in a comprehensive model of cast iron solidification. With the nucleation and growth kinetics defined for the liquid/solid transformations, as well as the influence of segregation during cast iron solidification, it is possible to predict the solidified microstructure in cast iron components. The results of these calculations should in principle enable the accurate prediction of temperature 29 distribution in cast components and provide a mechanism for assigning physical properties dependent on microstructure during a stress analysis. 2.2 Thermal Models A full assessment of heat transport in casting processes where liquid is present for part of the time requires solution of heat, momentum and mass transport equations. However, in many applications satisfactory accuracy can be obtained by considering only heat transport through conduction and assuming that the liquid and solid regimes can be modeled as one domain[10]. The governing partial differential equation for the simplified system is given as Equation 2.22. pCp(T)^ = V[k(T)VT] + Q with Q=L-fs (2.22) where T is the temperature, k is the thermal conductivity, p is the density, Cp is the specific heat and Q is a volumetric source term associated with latent heat evolution. Apart from the characterization of the necessary boundary and initial conditions, one of the most challenging aspects of model formulation is in the evaluation of the volumetric source term, Q, as it requires quantification of the rate of change in fraction solidified and/or transformed, fs, and a knowledge of the volumetric latent heat of transformation, L[10]. Incorporation of the microstructure dependent latent heat term couples the macro heat transfer and microstructure analyses explicitly, yielding a non-linear analysis of considerable complexity[42]. Published values for the heats associated with primary austenite, gray eutectic and white eutectic transformations are presented in Table 2.1. 30 One problem that arises with the linking of the micro and macro-scale models is related to the time integration of the two models. Numerous methodologies have been developed to ensure sufficient resolution in the micro-scale integration, while at the same time, maintaining reasonable overall macro model execution times. Two of these methods, the micro-latent heat method and the micro-enthalpy method, have recently been discussed in the literature[43]. Although it is somewhat unclear, both methods appear to initially neglect latent heat evolution in the macro model integration step then correct for latent heat omission in a series of smaller integration steps conducted within the microstructural model. The alternative is to include the latent heat source term within the macro model, but employ a series of sub-steps within the microstructure procedure. For example, 10 integration steps within the micro model may be employed for every macro model integration step. The rate of latent heat generation is generally evaluated based on the rates of evolution in microstructures calculated from the predicted macro-scale temperature solution. The disadvantage of this method is that it makes the problem highly non-linear requiring an iterative method of solution such as the Newton-Raphson method[44]. The ability to characterize the initial and boundary conditions is important for accurate solutions of the heat conduction equation. Temperatures used as initial conditions for the casting and the mould are the pouring temperature and room temperature, or possibly a preheat temperature, respectively. Boundary conditions representing the interactions between a casting and the surrounding environment are difficult to quantify because of complex heat transfer conditions that arise as a result of the formation of air gaps between the casting and the mould[10]. A method for 31 characterizing these boundary conditions is to measure the cooling conditions during casting and relate this to heat flux or heat transfer coefficient boundary conditions. Heat loss from external surfaces is typically more straight-forward providing radiation does not play a significant role. The thermophysical properties (k, C p , p) used in these models also add complexity since they are commonly temperature and/or microstructure dependent. Precise description of the latent heat released and the physical properties is essential to accurately quantify the thermal field in a casting. Knowledge of the thermal and microstructure fields in a casting provides the necessary input to perform a residual stress analysis. 2.3 Residual Stress Models To analyze the stress development in a component undergoing a phase transformation, the thermomechanical behavior must be characterized. Denis considered the components of thermomechanical behavior as: 1) metallurgical factors: the development of microstructure and its impact on the thermal field, 2) mechanical factors: the elastic/plastic behavior of the multiple phases developed in a component , and 3) the interactions between stress/strain and phase transformations[45]. Although this work involved characterizing stress/phase transformation interactions in quenched steel components, the approach is applicable to cast iron components. For a material undergoing a phase transformation, the total macroscopic incremental strain, de',,, can be written in the form: 32 d^^d^+defll+de^+de'l'+de:" (2.23) where de*. is the elastic strain, ofe^ 'is the thermal strain, de'^is the transformation strain, dejjis the transformation plasticity strain, and Je'" is the plastic strain[45,46]. In describing the relations for their computer code "Hearts", Ju et al. [47] expressed the relations for elastic and thermal strains as: e l+V U £. e\-=—Vi--°kkS, (2.24) e$=a(T-To)S0 (2.25) where v is Poisson's ratio, E is the Young's modulus, a is the coefficient of thermal expansion, tris stress, T„ is the reference temperature, and Sis the Kronnecker delta. The physical properties v, E, and a are material or phase dependent, as well as temperature dependent. For the multiphase cases, the physical properties are described by the mixture law. The mixture law calculates an average property based on the fractions of a phase present. Strains resulting from transformation induced dilatation are a function of the fraction transformed. A typical formulation for these strains is: e£=8,Xy 4eiT (2.26) where e" is the measured transformation strain resulting from the fraction, yk, formation of phase, fc[48]. 33 2.3.1 Constitutive Material Behaviour At temperatures below half the melting point, classical plasticity theory with an associative hardening rule is the standard method used to calculate plastic strains in metals[45]. This method calculates plastic strains independently of strain rate based on a constitutive equation. At temperatures above half the melting point, in-elastic deformation becomes strain rate dependent necessitating the use of visco-plastic or creep theories. In a review of visco-plasticity, Zienkiewicz and Cormeau define visco-plasticity as strain rate dependent inelastic deformation occurring above a threshold or yield stress[49]. Creep is defined as a special case of visco-plasticity where the yield stress is zero and strain rate dependent inelastic deformation occurs at any applied load. The visco-plastic relationship and its simplification to creep are illustrated in rheological form in Figure 2.12. Commercial finite element packages can model creep with their standard constitutive relations, but they must be adapted to implement visco-plasticity in the rheological manner shown in Figure 2.12[50,51]. There are many relations used to describe the steady-state creep strain rate, ev, at high temperatures, one such empirical relation being the Hyperbolic-Sine Law Model [52]: 8, = A{sinhao)n exp\ — ^ I (2.27) \RTj where A, a, and n'are fitting parameters, Q is the activation energy for creep, and R is the Gas Constant. In pure metals at high temperatures, Q is equivalent to the energy for self diffusion, but for multi-component alloys, it is easier to determine Q from 34 experimental measurements[52]. At low stresses, Equation 2.27 reduces to a power law relation: f_r>\ e, = A,a"' exp Q (2.28) This type of creep relation was used by Wiese and Dantzig during their investigation of residual stress development in a gray iron casting[51]. Wiese and Dantzig also characterized gray iron's asymmetric yielding behavior, shown in Figure 2.13[51,53]. In their study, the yield behavior was determined for the tensile and compressive domains at five different temperatures and incorporated with visco-plasticity into the commercial finite element package ANSYS for thermal stress calculations. Wiese and Dantzig employed this cast iron constitutive model in an investigation of the residual stress evolution in a small gray iron casting. Figure 2.14 shows the importance of yield criteria by comparing the stress ratio results using the Wiese and Dantzig yield criteria with those produced with the standard von Mises yield criteria. 2.3.2 Transformation Plasticity Denis and colleagues investigated the interactions between phase transformations and stress generation in quenched and heated steel components[45,46,54]. Besides the effects of stress on transformation kinetics, they also observed transformation plasticity, a 'mechanical interaction', where material undergoing a phase transformation plastically deforms at stresses below yield[45]. The two contributions to this phenomenon are anisotropic plastic accommodation of the transformation strain and product phase 35 orientation by the stress state[45]. Product phase orientation is only observed when phase transformation strains have shear components such as in martensite formation. Thus for transformations in cast irons, only the anisotropic plastic accommodation of the pearlite phase must be considered. Denis et al. reviewed the different models used to predict transformation plasticity and subsequently proposed a phenomenological model to relate transformational plasticity to experimental measurements[46]. The formulations used by Denis and her colleagues for transformation plasticity strain, d'\ and strain rate, eV ,are: where K and/* are experimentally determined constants, s,y is the deviatoric stress, and yk is the fraction transformed[45]. Referring to Figure 2.15, Denis presented a clear example of the impact that transformation plasticity can have on the accuracy of residual stress calculations. Figure 2.15 compares the results of calculations with and without transformation plasticity to experimentally measured residual stress in an oil quenched steel plate. The impact of transformation plasticity for applications involving slow cooling rates, such as in those found in large section castings, has not been determined and may be minimal. In summary, the stress model required to predict residual stress evolution in cast components must account for the five strain relations discussed. Accurate description of these strain phenomena requires the results of thermal and microstructural simulations as E'» = Kofk(yk) (2.29) i'!;=^Kfk(yk )ykSjj (2.30) 36 input. Thus, the thermal, microstructural, and stress evolution must be predicted to study the formation of residual stress in cast components. Table 2.1 - Microstructure specific volumetric latent heat values[14]. L (MJm"3) Austenite 1841 Gray Iron 1615 White Iron 1491 37 Figure 2.1 - Division of computational space for deterministic and probabilistic modeling of coupled macro-transport and transformation kinetics[l 1]. Figure 2.2 - Schematic representation of a, b columnar and c, d equiaxed growth morphologies for a, c dendritic and b, d eutectic type alloys[10]. 38 Carbon Equivalent (%C +0.25%Si+0.5%P) Figure 2.3 - Austenite liquidus temperatures plotted as a function of carbon equivalent after Alagarsamy et a/.[25]. 39 Figure 2.4 - Eutectic solidification ranges as a function of silicon[28]. Dimension-less radius Figure 2.5 - Relationships between fraction of solid, fs, and average grain radius, R, calculated using various impingement models: radius has been normalized by critical radius of cubic arrangement of spheres[31]. 40 Time (s) Figure 2.6 - Effects of varying the impingement factor on the cooling curve of a 25.4mm - diameter casting[14]. AT„ AT AT„ AT a) instantaneous nucleation (site b) continuous nucleation saturation) Figure 2.7 - Schematic comparison between assumptions of instantaneous and continuous nucleation models[11]. 41 grain density nucleation d is t r i button cooling curve Figure 2.8 - Continuous nucleation model where undercooling, ATi, determines the number of nuclei, ni, within the liquid by evaluating the integral of the nuclei distribution[32]. TIME Figure 2.9 - Schematic of solidification cooling curve illustrating the influence of alloy segregation on eutectic solidification and the conditions leading to intercellular carbide formation[8]. 42 SCHEIL 4.48 I i i i i — I — i — i — i — i — > 0 0 . 5 1 . 0 Figure 2.10 - Comparison of segregation relationships as a function of fraction solidified for Pin 5-Fe[36]. SCHEIL I / / / / EXACT / \ I I I I I I I L — < 1 — J 0 0 . 5 1 . 0 Figure 2.11 - Comparison of segregation relationships as a function of fraction solidified for C in y-Fe[36]. 43 Visco-plastic element rate-dependent plasticity Elastic element i / W W \ x Plasticity element Allows deformation when a > yield Figure 2.12 - Uniaxial visco-plastic rheological model (reproduced.in-part from[49]). Figure 2.13 - Yield surface for gray iron with axes corresponding to the principal stress directions[51]. 44 Figure 2.14 - Comparison of the stress ratio vs. time results computed with a standard yield surface (von Mises) and a new yield surface. Stress ratio is the maximum computed principal tensile stress divided by yield stress[51]. 200J--3001 1 r ~i r -JZ lOOr »V -calculated - experimental O 1 5 3 0 4 5 6 0 75 6 0 4 5 3 0 15 O DEPTH BENEATH S U R F A C E , m m 15 3 0 4 5 6 0 7 5 6 0 4 5 3 0 15 DEPTH BENEATH SURFACE, mm a) Without transformation plasticity b) With transformation plasticity Figure 2.15 - Residual stress comparison between calculations with and without transformation plasticity, and experimentally measured residual stress in an oil quenched steel plate[45]. 45 CHAPTER 3 SCOPE AND OBJECTIVES 3.1 Scope of Research Programme The goal of this research programme was to develop a method of predicting the residual stress distribution in cast iron thermorolls to aid in their design and safe operation. As this capability requires a knowledge of the distribution of phase content and the evolution of temperature during solidification and subsequent cooling, the research programme necessarily was also concerned with the microstructural evolution and heat transport during casting. To achieve this goal, fundamentally based mathematical models were developed to predict the evolution in temperature, microstructure, and residual stress during the solidification and cooling of a casting using the commercial finite element code, ABAQUS*. ABAQUS excels in non-linear heat transfer and stress analyses, but does not have the built-in capability to predict microstructure. This capability was added through "user subroutines" which employ relationship describing nucleation and growth kinetics to predict the evolution of microstructure. The influence of microstructure was incorporated into the thermal and stress analyses through the use of microstructure phase dependent thermal and mechanical properties. A B A Q U S is a registered trademark of Hibbitt, Karlsson & Sorensen, Inc. 46 Experimental measurements from a casting similar in geometry to a thermoroll were required for validation of the mathematical models and to ensure industrial relevance. The German roll manufacturer, Walzen hie, instrumented, cast, and analyzed a reduced scale shell casting similar to a thermoroll. As the time-scale resolution of the temperature data was not sufficient to validate the coupling between the thermal and microstructure models in the reduced scale roll casting, a small casting, employed in many foundries to assess the carbon equivalent of cast iron melts, was also employed for model validation. High resolution temperature data, obtained from a thermocouple embedded in this small casting, was used to fit the nucleation and growth kinetics parameters and validate the thermal/microstructure model. After casting, Walzen Lie performed residual stress measurements on the reduced scale shell casting prior to sectioning the casting. Sections of the reduced scale shell casting and a number of the small castings were sent to UBC for microstructure and hardness characterization. Thermal and mechanical property data, necessary for the thermal and stress models, was supplied by the manufacturer, Walzen hie. Some of this data has been measured by Walzen hie, while a large portion of it has been adapted from literature sources. This data was used without prejudice in the analysis of the two castings. An assessment of the sensitivity of the models to this data was performed to ensure there were no limiting parameters within the data. Other parameters used in the models, needed for description of the nucleation and growth kinetics were based on values published in the literature, but were 'fine-tuned' by fitting the measured variation in microstructure. 47 Thermal and microstructure analysis of the small scale casting was used to 'fine-tune' the nucleation and growth kinetics parameters through comparison of predictions with the measured temperatures and microstructure distribution. After 'fine-tuning', a thermal and microstructure analysis was performed on the reduced scale roll casting. The results of this analysis indicated that an extension of the equiaxed microstructure model was warranted. A preliminary extension to predict columnar growth was implemented in the existing microstructure model and used to predict the evolution of temperature and microstructure in the small scale casting and the reduced scale roll casting. The microstructure and thermal analysis results of the reduced scale roll casting were employed as input to the stress model to predict the evolution of residual stress. The final stress distribution was compared with measured results. 48 3.2 Objectives of Research Programme The primary objective of the present study is: • To formulate, develop, and verify mathematical models capable of predicting the temperature, microstructure, and stress field evolution in cast iron shells during manufacturing. In accomplishing the primary objective, the following sub-objectives were formulated: • To develop and implement, within ABAQUS, equations describing the microstructure nucleation and growth kinetics of gray and white iron based on temperature and microstructure data obtained from Quik-cup samples. • To measure the thermal history and final microstructure in a reduced scale shell casting for comparison with model predictions. • To measure the residual stress in a reduced scale shell casting for comparison with model predictions. 49 CHAPTER4 EXPERIMENTAL MEASUREMENTS To retain an industrial focus and provide data essential for model validation, a campaign of experimental measurements was undertaken in collaboration with the German roll manufacturer, Walzen Irle. In order to avoid problems associated with the disclosure of proprietary technology, the experiments were conducted on a reduced scale roll casting employing a variant on the nominal casting process. Unfortunately, as will be discussed in the next section, the time-scale resolution of the measured temperatures in the reduced scale roll casting was not sufficient to fit and validate the microstructure model parameters. Consequently, a second set of small scale castings were made that provided continuous temperature data throughout solidification. These castings, called Quik-cups*, are a commercial product used in foundries for thermal analysis to assess carbon equivalents in cast irons. The next two sections discuss the experimental method and results of the two castings produced in collaboration with Walzen Irle. 4.1 Reduced Scale Roll Casting The reduced scale roll casting (refer to Figure 4.1) was designed based on the proportions of a full sized thermoroll with the goal of attaining a similar distribution of microstructure. The desired distribution includes a depth of clear chilled white iron on 50 the casting OD and a gradual shift in microstructure to gray iron with increasing depth into the casting. The scale of this casting relative to a full sized thermoroll is approximately 1/4 of the radial dimensions and 1/8 of the axial length. Based on the casting dimensions given in Figure 4.1, the weight of the roll casting was approximately 850kg. The mould, as shown in Figure 4.1, is comprised of a chill mould, and bonded sand core and base sections. The chill mould, manufactured from gray cast iron, has a high thermal conductivity relative to the sand sections of the mould. The high cooling rates induced in the casting during solidification by the chill mould promote white iron formation at the casting surface. The chill depth in these castings can be adjusted by changing the chill mould thickness. The base sand section of the mould, made of a bonded sand material, insulates the bottom end of the casting to ensure proper mould filling prior to solidification. The core of the mould was constructed of a commercially available bonded sand by cold-pressing the sand into a Styrofoam moulding. The core was mounted on the sand base section using a mould adhesive and vertically stabilized at the top of the chill mould using wooden blocks. A photograph of the assembled mould is shown as Figure 4.2. In Figure 4.2, the pour cup, located to the right of the chill mould, is constructed of sections of cast iron pipe lined with refractory tubes. The mould system described above has the same components as used by Walzen Irle for their commercial castings. Using this casting technique, industrial measurements of the thermal history, evolved microstructure, and residual stress state of the reduced * Quik-cup is a registered trademark of Heraeus Electro-Nite Canada Ltd. 51 scale roll casting were performed. The method of acquiring this data as well as the results obtained are presented in the following sections. 4.1.1 Casting Instrumentation The instrumentation system and thermocouples employed to monitor the thermal history of the reduced scale roll casting were provided by Walzen Irle at their casting facility in Germany. Thermocouples were mounted in sets of six at two vertical heights - mid and bottom. Each set of thermocouples was divided amongst radial positions in the chill mould, casting shell, and bonded sand core. The thermocouple locations have been plotted in Figure 4.1 as solid dots. The interior thermocouples were molded into the bonded sand core during its fabrication. The chill mould thermocouples were positioned in their respective locations through holes drilled in the chill mould. The thermocouples employed by Walzen Irle were a type-K (nickel chrome -nickel) thermocouple with a surrounding Inconel sheath for protection from oxidative or corrosive environments. These thermocouples, manufactured by Heraeus Sensor, are reusable and normally used in heat treatment applications. In this application, positioning the thermocouples in the center of the solidifying shell posed a logistical problem because the Inconel sheathing offers little stability at the solidification temperatures of cast iron (~1250°C). Also, without some method of protecting the thermocouples, they would be solidified into the casting resulting in their loss at a significant cost. Walzen Irle worked with their thermocouple supplier to develop a method of mounting the thermocouples within the casting to ensure proper location and that they could be removed for subsequent use. The solution was to mount each thermocouple in a 52 protective quartz tube. The quartz tube and thermocouple assembly, shown schematically in Figure 4.3, were mounted at each thermocouple position in the casting. Figure 4.4 is a picture of the bonded sand core showing the method of mounting and the placement of the thermocouples. Extensions of sheathed thermocouple wire can be seen in the foreground of Figure 4.4. The monitoring and control system for a heat treatment furnace, manufactured by Honeywell, was used to acquire data from the thermocouples mounted in the casting and mould. This method of recording the data was employed because an integrated computer based data acquisition was unavailable at Walzen Irle. The monitoring station was configured to supply a text print out of the temperature at each thermocouple at set time increments. The maximum data rate that could be obtained was one data set per minute -i.e. 12 thermocouple readings every minute. During the initial stages of solidification, when the rates of temperature change are high, the maximum data rate was employed. One hour after casting, the time increment between data sets was increased to five minutes because the rates of temperature change had decreased. Prior to casting, the thermocouples were calibrated with a known heat source, at a temperature of 500°C. The heat source was, in-turn, placed in contact with each thermocouple and the measured temperature was recorded. The measured temperatures were from 3 to 7°C above the known heat source temperature. The calibration data has been included as Appendix A. This offset temperature data was used as calibration for each thermocouple and the corresponding amount was subtracted from each signal measured during solidification and cooling of the reduced scale roll casting. 53 The thermocouple assemblies, shown in Figure 4.3, were comprised of a type-K thermocouple sheathed in Inconel and placed in a quartz tube. Likely sources of error in the thermocouple assemblies includes contact resistance, response time, and movement of the thermocouples during casting. The calibration of the thermocouple assemblies with a known heat source minimizes the temperature uncertainties related to contact resistance, while post casting assessment of the thermocouple locations reduces the errors associated with thermocouple movement. No attempt was made to quantify the response time which is likely to be significant during rapid changes in temperature. The anticipated slow response time will impact temperature measurements during pouring and initial solidification. Following these initial stages, the large thermal mass of the casting relative to the thermocouple assemblies will result in gradual changes in temperature with little temperature lag expected in the thermocouple assemblies. 4.1.2 Casting Procedure Following Walzen Irle's industrial practice, the chill mould section was preheated prior to pouring the reduced scale roll casting. Preheating was performed with a gas burner positioned inside the chill mould section. After heating the chill mould to approximately 140°C, the chill mould was lifted by crane and assembled together with the unheated sections of the mould which include the bonded sand core and a bonded sand lined cast iron base section. One and a half hours elapsed between final mould assembly and the pouring of the roll casting, giving time for the preheated chill mould to cool and the other sections of the mould to be heated by heat transfer from the preheated chill mould. The 54 preheat temperature and the total time the mould was allowed equilibrate before pouring were found to be important factors in assessing the heat flow conditions during this stage. The liquid cast iron was prepared in Walzen Irle's melt shop according to one of their proprietary material grades, designated KT. A material sample was taken from the melt and analysis revealed the composition presented in Table 4.1. After transferring the melt to a transport ladle, a melt temperature of 1280°C was measured. Following standard Walzen Irle foundry practice, a thermal analysis sample and a keel block were cast to assess carbon content, the eutectic arrest temperature, and the chilling susceptibility. The melt was then transferred to the casting shop in preparation for pouring. The castings were poured by Walzen Irle employees, as shown in Figure 4.5. The time required to pour the casting was approximately 1.5min. After pouring, the casting was allowed to cool to room temperature before breaking it from the mould. Figure 4.6 shows a photograph of the casting after breakout from the mould. Residual stress analysis was performed by Walzen Irle on the as cast section prior to sectioning the casting for microstructure evaluation at Walzen Irle and UBC. 4.1.3 Thermal Response Based on the casting procedure, the thermal response of the reduce scale roll casting can be broken into three regimes - preheat, solidification, and cooling. 55 4.1.3.1 Mould Preheat Within the preheat regime, the chill mould, at a temperature of ~140°C, was assembled with the room temperature, 20°C, bonded sand core and mould base and allowed to equilibrate for 1.5 hours. In Figure 4.7, the temperatures prior to casting, plotted on a schematic of the roll casting mould, show that the chill mould had cooled to a temperature below 90°C and the bonded sand core had heated to temperatures above 60°C. There was a 9°C difference between the mid- and bottom height temperatures at the chill mould surface. The difference between the ID and OD chill mould temperatures was 8°C at the mid-height and 12°C at the bottom level. This increase in bottom level temperature difference is likely due to the longer radial conduction path at this position and conduction to the mould base. Between the bonded sand core and the ID of the chill mould, there was approximately a 20°C temperature difference. A model of the casting thermal history must include the preheat segment of the casting process to incorporate the impact of mould temperature gradients developed prior to casting. 4.1.3.2 Casting Solidification The temperature profiles of the casting, recorded during solidification, are presented in Figure 4.8. The mid-height thermocouple data are given in Figure 4.8a and the bottom height data in Figure 4.8b. Each plot in Figure 4.8 shows the temperature profiles of the casting with time at the OD surface, the mottle zone (20mm from the OD surface), the 56 gray zone (55 mm from the OD surface), and the bonded sand core OD 1. In both plots, the casting surface shows a high initial cooling rate which decreases after 3 min. Comparing the thermal profiles of the surface, mottle zone and gray zone shows that the cooling rate decreases with increasing distance from the surface. The bonded sand core OD thermal response shows a rapid increase in temperature after pouring which slows to a moderate rate of temperature increase until solidification is complete. There is a continued increase in the bonded sand core OD temperature until the casting ID temperature is lower than that of the core and heat removal from the core begins. At this point, the core temperature begins to decrease. The initial melt temperature after pouring is difficult to assess from Figure 4.8. The first temperature point on each of the plots presented in Figure 4.8 is obtained 1 min after the start of pouring and the corresponding maximum temperature measured by the thermocouples at this time was 1206°C at the bottom height mottle zone thermocouple. The maximum temperature at the mid height was measured at the next recording time as 1187°C in the mottle zone. The time offset of the maximum initial temperatures is due to the 1.5 min pouring time. As previously discussed, the last recorded melt temperature prior to pouring was 1280°C. Thus, the initial melt temperature of the casting after pouring must be between this temperature and the maximum recorded temperature. Other than moderate changes in cooling rates, the effects of microstructure evolution cannot be seen in the temperature responses of the casting OD and the mottle The thermocouple positions in the reduced scale roll casting were determined during post-casting examination by measuring the depth of the holes left by the protective quartz tubes. This procedure could only be preformed for the mid-height location, as the bottom height casting section was not available. 57 zone. The gray zone temperature data has thermal plateaus associated with the formation of proeutectic austenite at 1172°C and eutectic solidification at 1125°C. However, with such coarse time resolution of the temperature data, it is not possible to identify the nucleation and growth of austenite, gray iron and white iron phases from these thermal profiles. These temperature profiles provide a means of assessing the basic macroscopic heat flow capabilities of a thermal model, but lack the time-scale resolution required to fit parameters associated with a microstructure model. The thermal response of the chill mould at the mid- and bottom heights is presented in Figure 4.9. After an initial rapid increase in temperature, the chill mould ID temperatures, plotted with plus symbols in Figure 4.9, decrease to ~430°C at the mid-height and ~530°C at the bottom height. This temperature decrease is likely a result of reduced heat transfer associated with gap formation between the casting and the mould. In Figure 4.9, the OD temperatures at both heights, designated with circle symbols, showed a gradual increase in temperature over the 30 minute interval examined in Figure 4.9. The thermal response of the mottle zone thermocouples at mid- and bottom heights appear to follow the same trends during the initial stages of solidification. However, the mid-height mottle zone thermocouple appears to cool too rapidly as compared to the bottom height mottle zone thermocouple. Near the end of solidification, the mid-height mottle zone thermocouple was reporting the same temperature as the surface of the casting. This situation is not possible for the cooling conditions present in this application and consequently, the data from this thermocouple was not used for model calibration and validation. 58 4.1.3.3 Long Term Cooling The thermal responses of the casting and mould for 5 hours after casting are presented in Figure 4.10; subsequent cooling continued until the casting was at room temperature. As shown in Figure 4.10, the temperature gradients in the casting and chill mould, which developed during solidification, dissipate as cooling proceeds. The austenite decomposition temperature, 724°C, can be estimated from the change in slope of the temperature responses. The thermal response of the bottom height bonded sand core OD thermocouple indicates a shift in the thermocouple position after -3.5 hrs (refer to Figure 4.10b). The cooling response of this thermocouple after the temperature shift appears to be consistent with the other thermocouples, albeit shifted. 4.1.4 Residual Stress Measurements Once the casting was removed from the mould, a residual stress analysis was performed by Walzen Irle using their standard residual stress measurement technique. Walzen Irle employs the ring-core method of residual stress characterization which is illustrated in Figure 4.11. In the ring-core method, a strain gauge rosette is mounted on the surface of the piece to be tested and material surrounding the section with the strain gauge is removed. The removal of material relaxes the elastic strains at the surface of the section and can be related to the residual stress prior to testing. This method provides a measure of the surface residual stress, but gives no indication of the residual stress variation through the thickness of the casting. 59 Residual stress measurements were conducted on the casting OD surface at mid-height and at 240mm from each barrel length edge as indicated in Figure 4.1. At each height, two residual stress measurements were made on the surface at the 0° and 180° positions around the circumference of the casting. The results of these measurements, presented in Table 4.2, provide a measure of the residual stress in the axial and tangential directions. The measurements indicate there was little variation in residual stress between the measurement points at each height. The maximum axial residual stress, approximately -150 MPa, was measured at the top position in the casting. The axial residual stress decreased with axial distance towards the bottom of the roll. The tangential residual stress values show the same trend, but were consistently lower than the axial residual stresses. The development of an axial residual stress distribution is indicative of either a variation in mechanical properties resulting from microstructural phase variation in the casting or a variation in the amount of in-elastic deformation occurring during solidification. 4.1.5 Microstructural Characterization At the positions shown in Figure 4.1, cross-sections, 4 cm thick, were cut from the reduced scale roll casting following the residual stress measurements. Walzen Irle cut full radial sections from each of the cross-sections before shipping the remaining cross-section pieces to UBC. On the radial sections, Walzen Irle examined the microstructure and measured the hardness at various radial distances. The hardness measurements conducted at Walzen Irle for the two heights are presented in Figure 4.12. Walzen Irle 60 measured hardness in the first 20 mm from the casting OD surface using a Vickers hardness tester and the remaining interior measurements were conducted using a Brinell hardness tester. Based on the hardness data presented in Figure 4.12 and the associated microscopic examinations, Walzen Irle reported that there was no axial variation in the microstructure. This would seem to suggest that the axial variation in residual stress is due to a variation in the amount of in-elastic deformation experienced during solidification. ; For the upper height cross-section, a similar series of examinations were conducted at UBC. Duplicate radial hardness profiles were measured at two different locations on the upper cross-section using a Rockwell C hardness scale and the results are plotted as triangle and circle symbols in Figure 4.12. The large scatter present in the measurements results from the coarse microstructure and small diamond hardness indentor. Allowing for the scatter, the UBC measurements show the same variation in hardness as the Walzen Irle measurements. These measurements were then correlated with microstructure observations. Microstructure analysis was performed on samples cut from various radial positions in the upper cross-section. These samples were rough polished on a series of rotating disc polishing wheels ranging from 80 grit to 600 grit. Final polishing was performed with 5 pm diamond paste. After the final polish, the samples were etched with a 2% Nital solution before viewing and photographing on a microscope. The microstructure at the surface of the casting, shown in Figure 4.13, is completely white iron. In Figure 4.13, the light areas of iron carbide and dark pearlite areas are characteristic of a white iron microstructure. Although the magnification of Figure 4.13 61 is too high, there is some preferential orientation of the microstructure that may indicate columnar growth. The dendritic appearance of the white iron is consistent with a divorced eutectic microstructure where the eutectic austenite forms on the already present proeutectic austenite dendrites and the iron-carbide forms interdendritically as a semi-coherent phase. Moving to a position 10 mm from the casting OD surface (refer to Figure 4.14), isolated areas of gray iron begin to appear. The gray iron microstructure is characterized by flakes of graphite (black phase) radiating through areas of pearlite. Figure 4.15 shows the microstructure at 35 mm from the casting OD surface. At this position, individual nodules of gray iron have coalesced and formed bands of gray iron with isolated areas of white iron. Increasing amounts of gray iron microstructure are observed as the depth from the casting OD surface increases. At 75mm from the casting OD surface, the microstructure is almost completely gray iron (refer to Figure 4.16). Correlating the hardness data presented in Figure 4.12 with the observed microstructure, it was possible to convert the hardness measurements to a fraction of gray or white iron. Assuming that the gray and white iron are the major microstructural phases present, the fraction of gray and white iron can be obtained by solving Equation 4.1: Hm=HJw+Hgfg ' (4.1) where H m is the measured hardness, H w is the hardness of white iron, H g is the hardness of gray iron, and fw and fg are the fraction of white and gray iron. Using the min and max hardness values from Figure 4.12, white iron has a hardness of 543 VHN (53 Rc) and gray iron has a hardness of 220 VHN (12 Rc). The resulting variation of fraction gray and white iron with radial distance from the surface is presented in Figure 4.17. This 62 radial variation of microstructure will impact the residual stress development in the casting through the variation in mechanical properties of gray and white iron. In Figure 4.16, the small areas of white phase were thought to be white iron. However, white iron this far from the casting surface is unlikely because the cooling rates are low. Figure 4.18a is a higher magnification view of the microstructure near that of Figure 4.16. In this photo, a number of white phase areas are visible. This sample was re-polished and etched with Murakami's reagent which was reported to attack iron-phosphides[55]. The resulting photomicrograph at the same location is shown in Figure 4.18b. For the most part, the white phase observed in Figure 4.18a has been turned dark by the Murakami etch in Figure 4.18b. This indicates that the small areas of white phase in the interior of the casting are mostly iron-phosphide. 4.2 Quik-cup In lieu of the poor time resolution in thermocouple data from the reduced scale roll casting, an additional small scale commercial casting, called Quik-cup, was chosen to provide high resolution temperature measurements, as well as microstructure data. The Quik-cup thermal analysis system, manufactured by Heraeus Electro-nite, is used by Walzen Irle to analyze melt compositions prior to pouring each production casting. The Quik-cup casting mould, shown in Figure 4.19, produces a small rectangular casting with a slight cross-sectional taper. The Quik-cup mould is made of a phenolic resin bonded sand with the trade name Plastisand. A type-K thermocouple, sheathed in a quartz tube, is mounted horizontally through the center of the mould. This thermocouple is monitored 63 with a chart recorder which provides continuous output during the solidification temperature regime. During the production of the reduced scale roll casting, a Quik-cup casting was poured according to standard operating procedure and the thermal data was used to confirm the carbon equivalent and eutectic arrest temperatures. At the time, it was still hoped that the thermal data from the reduced scale roll casting would have sufficient resolution to calibrate and validate the microstructure model. Although the Quik-cup thermal data was obtained, the Quik-cup casting was scrapped before its usefulness was realized. Thus, Walzen Irle produced a series of Quik-cup castings, similar in composition to the reduced scale roll casting, and sent these castings with the recorded thermal data to UBC for analysis. 4.2.1 Procedures In the Walzen Irle casting production process, Quik-cup castings are prepared after the melt has been transferred to the pouring ladle. The Quik-cup mould is placed on a steel stand with an integrated contact point for the embedded thermocouple. A sample cup is used to remove melt from the ladle and fill the Quik-cup. As the sample solidifies, thermal data is recorded. At this stage, the Quik-cup casting would typically be discarded. For this study, Walzen Irle poured 7 Quik-cup castings in succession from the same melt. The composition of this melt, presented in Table 4.1, was similar to the composition of the reduced scale roll casting. The melt temperature, 1250°C, was 64 measured in the ladle prior to sampling. The castings were broken from their Quik-cup moulds before shipment to UBC for analysis. 4.2.2 Thermal Response The measured temperature at the center of the Quik-cup casting has been plotted versus time in Figure 4.20 for one of the castings. As can be seen, the temperature response in Figure 4.20 clearly reflects the various phase transformations occurring during solidification similar to the schematic cooling curves presented in Figure 1.3 of Chapter 1. Upon reaching the maximum temperature of 1197°C, 3.75s after pouring, the casting cooled to the proeutectic austenite formation temperature of 1172°C. Associated with the austenite formation, there was a plateau in the cooling curve which lasted until this solidification slowed. Cooling continued until the eutectic solidification began and the heat released caused a thermal recalescence at the eutectic arrest temperature. From this thermal profile, it was not possible to identify the eutectic phases that were forming at each time. The temperature response shown in Figure 4.20 was typical of the other Quik-cup castings. The data from these castings show ±3°C differences in the maximum and eutectic arrest temperatures. These small differences illustrate the reproducibility of this type of casting. The remaining Quik-cup casting temperature response curves have been included in Appendix B. 65 4.2.3 Microstructure Characterization Microstructure analysis was performed on 3 of the 7 Quik-cup castings. The analyzed castings were sectioned vertically along the horizontal cross-section diagonal to provide the largest surface area for analysis; also consistent with the largest variation in cooling conditions. Through this sectioning method, the microstructures corresponding to the position of maximum cooling rate, the bottom corner, and the position of minimum cooling rate at the casting center were examined. After sectioning, the samples were polished and etched following the same procedure used for the sample of the reduced scale roll casting. Considerable macro-shrinkage porosity was observed at the center of the castings surrounding the quartz tube shielded thermocouple. Figure 4.21 is a macro-scale photograph of the sectioned Quik-cup casting. In Figure 4.21, a large macro-shrinkage pore is observed at the center of the casting. Smaller macro-shrinkage pores radiate out from the large pore in all directions. During sectioning, it was noted that the thermocouple, mounted in the center of the large pore, was not in contact with the bulk of the casting. Figure 4.22 is a photomicrograph of the microstructure at 5 mm from the casting center along the diagonal towards the bottom corner. The microstructure observed at positions closer to the center than Figure 4.22 were predominately porosity. In Figure 4.22, the microstructure is white iron with a few areas of gray iron. Figure 4.22, also, shows areas of macro-shrinkage porosity. Estimating the phase areas in Figure 4.22, the phase fractions of white and gray iron are 60% and 40%, respectively. 66 Moving towards the bottom corner of the Quik-cup, the cooling rates increase and as a consequence, increasing amounts of white iron develop. The microstructure at the bottom corner of the Quik-cup, presented in Figure 4.23, is entirely white iron. It should be noted that the microstructure in this region appears to be in transition towards a directionally orientated microstructure typical of a columnar dendritic growth morphology. Given the opportunity, it is likely that an oriented microstructure could develop a small distance from the casting surface. Hardness measurements were performed at positions along the diagonal from the bottom corner to the center of the Quik-cup casting and at positions offset 5 mm to the left and right of the diagonal. These measurements were converted to gray and white iron phase fractions following the previously described technique. The variation of gray iron phase fraction with distance from the Quik-cup center is plotted in Figure 4.24. At the bottom corner of the Quik-cup, Figure 4.24 shows the phase fraction of gray equal to 0. The gray iron phase fraction increases gradually as the distance from the center decreases. The gray iron phase fraction for positions less than 5 mm from the casting center show a deceiving increase. As previously indicated, 40% gray iron was estimated by employing an area based phase fraction average 5 mm from the center of the Quikcup casting. Thus, the observed gray iron phase fraction is considerable less than the 67% gray iron phase fraction calculated from the hardness measurements. It is likely that at these positions in the casting, the small areas of macro-shrinkage porosity cause a decrease in the measured hardness of the cast iron and a corresponding erroneous increase in the calculated gray iron phase fraction. 67 The Quik-cup casting exhibits microstructure ranging from clear white iron to mottled white and gray iron. When compared with the full range of microstructure observed in the reduced scale roll casting, it is clear that the Quik-cup casting represents only a small portion of the casting conditions present in the reduced scale roll casting. Qualitatively, the measured temperature data can be used to assess the differences in casting conditions. At the center of the Quik-cup casting, the casting cools 133°C in 163s from a maximum measured temperature of 1197°C. The overall cooling rate at the center location in the Quik-cup casting is 0.8°C/s. In the reduced scale roll casting, the overall cooling rates, calculated following a similar procedure, were l.l°C/s at the O D surface, 0.4°C/s at the mottle zone, and 0.1°C/s at the gray zone thermocouple locations. Thus, the overall cooling rate in the reduced scale roll casting is lower than the cooling rate in the Quik-cup within the first 20mm from the O D surface. This confirms that the Quik-cup represents only a small portion of the solidification conditions experience in the reduced scale roll casting. 4.3 Summary Reduced scale roll casting results provided thermal data suitable for validating bulk heat flow in the casting. To validate micro-scale calculations, the Quik-cup was needed because it provided higher time-scale resolution temperature data. Examining the Quik-cup temperature data allows the identification of specific microstructure transformation events such as the austenite formation temperature and the eutectic arrest temperature. The residual stress measurements of the reduced scale roll casting showed an axial variation with the highest values at the top of the casting. This variation in residual 68 stress was not mirrored in the observed microstructure. A possible explanation could be an axial variation in permanent deformation occurring during solidification. Microstructure analysis of the reduced scale roll casting revealed a distribution of microstructure of white iron at the casting surface with a gradual transition to gray iron at the casting interior. The solidification conditions in the Quik-cup casting resulted in a white iron surface layer with a transition to approximately 40% gray iron at the casting center. Thus, the solidification conditions in the Quik-cup casting only match a small portion of those observed in the reduced scale roll casting. 69 Table 4.1 - Composition of the cast iron alloy, KT, for the reduced scale roll casting and the Quikcup. Casting Composition (wt%) C Si P S Mn Reduced Scale 3.63 0.61 0.53 0.15 0.28 Quikcup 3.60 0.51 0.55 0.15 0.20 Table 4.2 - Results of the residual stress measurements performed on reduced scale roll casting; aa is the axial stress and <r, is the tangential stress. Circumferential Position Stress Orientation 240mm from Top End Mid-Height 240mm from Bottom End 0° -81 -104 -151 S t -92 -100 -126 180° -92 -109 -148 0-, -80 -84 -120 70 670mm 640mm 400mm 200mm Material Legend Cast Iron [ ,J Sand Core/Base ^ Chill Mould Residual Stress Measurements Thermocouples 6 radial positions for each height: Chil l mold OD (surface) Chi l l mold ID Casting O D Mottle zone (20 mm from casting OD) Gray zone (55 mm from casting OD) Core OD / Casting I D Sections cut for Microstructure / Hardness Profiles 420mm Casting Inlet Figure 4.1 - Reduced scale roll casting geometric configuration and dimensions with different shadings for chill mould, casting, and bonded sand core. Figure 4.2 - Photograph of the assembled reduced scale roll casting mould. ?/////////////////////////////////^^^^^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ Quartz Tube Inconel Sheath Thermocouple Figure 4.3 - Schematic of a type-K thermocouple and the surrounding materials to provide protection from the melt in the reduced scale roll casting. Figure 4.5 - Photograph of the reduced scale roll casting being poured at Walzen Irle. 73 Figure 4.6 - Photograph of the reduced scale roll casting after breakout from the mould. h n i 1 71 59+ +82 8 9 81 Temperatures in degrees C 70 3 Figure 4.7 - Preheat temperatures prior to casting the reduced scale roll casting plotted on a mould schematic. 74 a) Time (min) Figure 4.8 - Thermal history of reduced scale roll casting during solidification for: a) mid-height and b) bottom height thermocouples. 75 700 0 A , , , , , 1 0 5 10 15 20 25 30 Time (min) Figure 4.9 - Thermal history of reduced scale roll casting chill mould during solidification for the mid- and bottom height thermocouples. 76 1200 -1000 w 800 D U 3 « 600 a S H 400 H 200 0.00 1.00 Mid-Height Temperature Data at: Casting OD Mottle Zone (20mm from casting OD) Gray Zone (55mm from casting OD) Bonded Sand Core OD Chill Mould OD Chill Mould ID + + + + + + + + + + + + + M l i M i » * « H l i i B i iii a) 2.00 3.00 Time (hrs) 4.00 5.00 1200 1000 •-3 ca -3 . 3 H 800 600 -t 400 200 0.00 1.00 b) 2.00 3.00 Time (hrs) 4.00 5.00 Figure 4.10 - Extended cooling thermal response of the reduced scale roll casting for: a) mid-height and b) bottom height. 77 strain gauge Figure 4.11 - Schematic of ring-core residual stress measurement method[56]. 600 500 £ 400 a TJ « 300 HM c 200 £ 100 •— Walzen Irle Lower Profile B — Walzen Irle Upper Profile A UBC Upper Profile o UBC Upper Profile 2 o lb o T 60 50 4- 40 e 30 T» M a u M 4- 20 o 10 + 0 20 40 60 80 Radial Depth From Casting OD (mm) 100 Figure 4.12 - Radial hardness profiles measured at Walzen Irle and UBC on sections from the reduced scale roll casting. 78 100pm Figure 4.13 - Microstructure at the OD surface of the reduced scale roll casting showing dark areas of pearlite and light areas of iron-carbide (lOOx, 2% Nital etch). 200pm Figure 4.14 - Microstructure 10 mm from the OD surface of the reduced scale roll casting showing light areas of iron-carbide, gray areas of pearlite and black flakes of graphite (50x, 2% Nital etch). 79 200|jm Figure 4.15 - Microstructure 35 mm from the OD surface of the reduced scale roll casting showing light areas of iron-carbide, gray areas of peaiiite and black flakes of graphite (50x, 2% Nital etch). 200|jm Figure 4.16 - Microstructure 75 mm from the OD surface of the reduced scale roll casting showing light areas of iron-carbide / iron-phosphide, gray areas of pearlite and black flakes of graphite (50x, 2% Nital etch). 0 10 20 30 40 50 60 70 80 Radial Distance from the Casting Surface (mm) Figure 4.17 - Fraction gray and white iron profiles for the reduced scale roll casting converted from the Walzen Irle hardness data. 81 a) 50|jm Figure 4.18 - Enlarged interior microstructure of the reduced cast roll casting etched using: a) 2% Nital showing iron-carbide and iron-phosphide as light phase and b) Murakami's Etch showing iron-phosphide as gray phases in areas unetched in a) (200x). 82 52mm 38mm Type-K Thermocouple— Shielded in Quartz Tube 24mm 44mm Figure 4.19 - Geometry and dimensions of the Quik-cup casting mould; front and top views shown. Austenite Temperature Eutectic Arrest Temperature 25 50 75 100 Time (sec) 125 150 175 Figure 4.20 - Measured thermal response at the Quik-cup casting center. 83 Figure 4.21 - Photograph of macro-shrinkage porosity in the sectioned Quikcup casting. 200|im Figure 4.22 - Microstructure 5 mm from the Quik-cup casting center with iron-carbide as a light phase, pearlite as a gray phase, graphite as black flakes, and porosity as large black areas (50x, 2% Nital etch). a o -<a u c o "3 ss -0.7 0.6 0.5 0.4 0.3 0.2 0.1 Casting Center Shrinkage Porosity; a Right • Center Left 4* 0 5 10 15 20 25 30 35 Distance from Quikcup casting center (mm) Figure 4.24 - Variation of phase fraction gray with distance from the Quik-cup casting center. Results converted from hardness measurements. 85 CHAPTER 5 MODEL DEVELOPMENT The problem of residual stress development in castings is fully coupled and highly non-linear because of latent heat evolution, stress-phase transformation interactions, and non-linear visco-plastic deformation. Based on the modeling work reviewed, the finite element method (FEM) provides a convenient procedure to mathematically model the development of the thermal, microstructural, and strain fields that drive the development of residual stress[l 1,32,42,45,51]. The commercial finite element software, ABAQUS, was employed as a solution platform for this problem. ABAQUS was chosen because it provides highly developed non-linear solution capabilities and a well-documented method for extending the program's capabilities. The thermal stress model, employed in this study to predict the evolution of residual stress, is composed of two components - thermal and stress. The thermal model solves the non-linear heat transfer problem that results from the evolution of latent heat and the inclusion of temperature and phase dependent material properties. A microstructure model has been constructed from the techniques described in Chapter 2 and implemented in the thermal model to predict the evolution of microstructure and its influence on temperature. The stress model employs temperature and phase dependent thermal contraction, elastic modulus and in-elastic deformation behaviour to calculate the evolution of residual stress. The temperature and microstructure predictions from the thermal model act as input for this stress model. Thus, through temperature and phase dependent physical properties, as well as thermal contraction, the stress model is coupled 86 to the thermal and microstructure predictions. This formulation of the thermal stress model couples the stress results to the thermal and microstructure results, but does not include the influence of stress on the thermal and microstructural solution. The reader is referred to the text by Zienkiewicz and Taylor for a complete explanation and discussion of the issues relating to the finite element solution procedure[57]. Discussion in the following sections includes an overview of the FEM concepts relevant to the formulation of thermal stress models, the implementation of a microstructure model, and overviews of the important factors in thermal and stress models. A detailed description of the application of the FEM to thermal and stress analysis will not be presented because these algorithms were not developed during the course of this research programme. 5.1 FEM Background The generalized concept of FEM analysis is to solve a problem in domain, Q, with boundary conditions on the boundary, T, using a series of differential equations developed for the sub-domain regions, Q., (referring to Figure 5.1). Using this approach, it is possible to solve a wide variety of engineering problems dealing with phenomena such as heat transfer and deformation. In heat transfer problems, the desired solution is usually the temperature throughout the domain, Q., resulting from heat flux or prescribed temperature boundary conditions along the boundary, T. In a stress model, the deformations or displacements in domain, Q., are determined from a knowledge of the external forces or displacements on the boundary, T. Following this generalized FEM concept, discussion in this section will focus on the FEM knowledge required to 87 understand the formulation of the microstructure model and how this model impacts the thermal and stress models. 5.1.1 Element Interpolation Within each sub-domain region or element, the trial function, u, which satisfies the governing differential equations, is calculated at various locations with the approximate form: where A/, are shape functions defined in terms of the independent variables such as the spatial coordinates, the parameters, a,-, are the solution values which are all or partially unknown, and n is the number of positions or nodes within the element. In standard formulations, the shape functions, also called interpolation functions, are linear or quadratic polynomials, depending on the number of nodes per element. The interpolation functions are chosen such that they equal one at a single node within the element and zero at all other nodes. Thus, at any nodal position, the corresponding trial function will equal the nodal solution value, i.e. w, = a,. Interpolation functions are also used to map elements from arbitrary shapes in global coordinate space to regular shapes in local coordinate space. Figure 5.2 illustrates elemental mapping for a few element shapes. Elements are called isoparametric when the same interpolation functions are used for the coordinate transformation and the trial solution function. In this study, 4 node, 2-dimensional axisymmetric and 8 node, 3-dimensional hexagonal isoparametric linear elements were employed. These element n (5.1) 88 types and the geometric reasons for their choice will be discussed when the model applications are presented in Chapters 6 and 7. 5.1.2 Numerical Integration In order to assemble the matrix of differential equations describing the global problem of Figure 5.1, numerical integration is employed on an elemental basis. Of the techniques available for numerical integration, the Gauss quadrature method is particularly well suited to FEM because it requires the least number of function evaluations[57]. In 3-dimensions, the Gauss quadrature integration of a function f(u,v,w) is described by the following expression: ' ' ^ m m m J J\f(u,v,w)du dv dw = £ X X Wi WJ W* f^>vrwk) (5-2) where W,, Wj, and Wk are weighting coefficients at locations i,j, and k, respectively and m is the number of integration or gauss points within an element. Employing this integration formulation, the 2-dimensional axisymmetric elements and the 3-dimensional hexagonal elements utilized in this study are defined with 2x2 and 2x2x2 integration points, respectively. The integration points provide one of the means by which spatially dependent variables can be addressed in finite element models. For example, the material properties can be made a function of microstructure quantities which are calculated at the integration points. 5.1.3 ABAQUS Solution Method A feature that makes ABAQUS particularly well suited to solving the microstructure, heat transfer and stress problems is that its non-linear solution capabilities are highly 89 developed and robust. ABAQUS uses a modified Newton solution method to solve the nonlinear equilibrium equations and ensure overall equilibrium of the solution[58]. The modified Newton method is an incremental solution algorithm that employs a series of piecewise linear FEM approximations to approach a solution. An acceptable solution is obtained when the change in incremental solutions becomes small relative to some tolerance. Within a time increment, ABAQUS predicts a solution based on the material properties and loads (heat fluxes or forces) at the increment start. After updating the properties and loads based on the predicted solution, a second solution is calculated to check convergence. The convergence check ensures that the change in solution values and load levels are within their prescribed tolerances. If equilibrium has been satisfied, a new increment is started; otherwise, another 'equilibrium iteration' is performed to find a solution based on the updated data. In extreme cases where a solution cannot be found within a reasonable number of iterations, the increment is started over with a decreased time step. The ability of this method to manage highly non-linear latent heat release analyses make it well-suited to the microstructure problem. Since the modified Newton solution method relies on a series of incremental solutions, the resulting solution in a transient analysis will be sensitive to the time step. Thus, time step selection is an important issue and ABAQUS has addressed this through the implementation of an adaptive time step algorithm. When setting up the problem, the user supplies a maximum and minimum time step, as well as a tolerance for the maximum solution change within a time increment. Based on the solution change 90 tolerance and the rates of convergence of the solution, ABAQUS adapts the time step to ensure solution accuracy while maintaining the largest acceptable time step. 5.2 Thermal Model A transient heat conduction model was developed using ABAQUS to predict the temperature evolution in the two sample castings. A microstructure subroutine is used in conjunction with this model to determine the evolution of latent heat. The boundary conditions, including gap heat transfer and radiation/natural convection, were formulated from a knowledge of the casting processes and their geometry. The final components of this model are temperature and phase dependent thermal properties. In this section, general issues relating to the thermal model and the implementation of the microstructure model will be presented. Specific details such as the mesh development and the boundary conditions for each casting will be presented in Chapter 6, while describing the specific applications of the thermal model. 5.2.1 General Thermal Model Formulation Within the FEM formulation of a heat transfer model, temperatures are stored at the nodal positions in an element. During a solution increment, the temperatures are interpolated to the integration point locations before solving the heat transfer equations. Material properties that are temperature dependent are evaluated at these integration point locations based on the interpolated temperatures. Loads in the forms of heat fluxes and prescribed temperatures are applied at prescribed positions within the model. The result of these calculations is a new set of temperatures at each node in the mesh. Within the 91 context of ABAQUS, multiple temperature solutions are found in each time increment to ensure equilibrium is satisfied. The first-order heat transfer elements provided by ABAQUS, such as the 4-node axisymmetric quadrilateral and 8-node brick elements used in this study, have a non-standard formulation. The integration points in these elements are located at the corners of the element rather than within the element as in standard formulations. This modification improves the performance of these elements when strong latent heat effects are present[58]. ABAQUS recommends that these 'first-order' elements be employed in problems where there are latent heat effects. 5.2.2 Implementation of the Microstructure Model Although ABAQUS is capable of solving a variety of non-linear problems including transient heat conduction and thermal stress analysis, the basic suite of routines provided do not enable solution of microstructure evolution directly. However, ABAQUS provides a series of well documented subroutines that can be developed by the user to extend the code's capabilities[59]. A custom routine was developed to evaluate the evolution of microstructure based on the cast iron equiaxed modeling discussed in the Literature Review. The evolution of microstructure is coupled to the thermal analysis through the volumetric latent heat source term, Q. Secondary coupling exists through the influence microstructure has on thermo-physical properties. Within ABAQUS, this source term and the associated evolution in microstructure is calculated in the 'user subroutine' 92 HETVAL. This subroutine is called at the integration points of each element for each time increment of the thermal model. The parameters relating to microstructure, such as fraction solidified to austenite, gray iron and white iron, are stored in user-defined "state-dependent variables" at each integration point. In tracking microstructure evolution, a total of 40 parameters are stored at each integration point. The results of these microstructure calculations are accessible from other subroutines and can be visualized through ABAQUS-Post, the post processing utility. The microstructure subroutine is called, during each time step, for each trial temperature solution calculated during the modified Newton solution procedure of ABAQUS. For each trial temperature solution, nucleation and growth kinetics relations are applied according to the flow chart in Figure 5.3. During each microstructure evaluation, nucleation and growth calculations for austenite, gray iron and white iron are performed if the current temperature is below the specific transformation temperature and, in the case of austenite, if the temperature is above the graphite and iron carbide eutectic temperatures. In this manner, it is possible for the gray and white iron phases to be forming at the same time. The application shown schematically in Figure 5.3, is unique to others presented in the literature in that the evolution of all three phases, primary austenite, gray and white eutectic are accounted for. In addition, the model uses nucleation and growth kinetics to predict the evolution of primary austenite instead of simply calculating the fraction transformed at a given temperature from the lever-rule applied to the Fe-C phase diagram. 93 5.2.2.1 Micro-mode Time Integration As discussed in Chapter 2, problems arise when linking a macro-scale thermal model and a micro-scale microstructural model with regards to time integration. The challenge of linking these two models is to maintain a reasonable macro-scale time step while ensuring a sufficiently small micro-scale time step to provide adequate integration of the micro-scale phenomena. Numerous methodologies have been developed to avoid these problems by omitting latent heat from the thermal solution and post-correcting the solution. These methods are impractical within the context of an ABAQUS thermal model because equilibrium would not be satisfied with post-corrections of temperature. Further, these approaches would neglect the strength of ABAQUS, in ignoring its ability to handle highly nonlinear problems. In an effort to address these problems for an ABAQUS thermal model, the microstructural procedures are integrated in a series of n micro-scale time steps within each macro-scale time step. An adaptive method for determining the optimum number of micro-scale time steps has been implemented based on the change in extended volume fraction. Initially, the microstructure calculations are performed in one micro-scale time step. The resulting increment in the extend volume fraction, 8(j)e, is compared to an extended volume fraction increment limit, S</>e m a x . When 8<pe is greater than <50 e t m a x, the microstructure calculations are performed again with enough micro-scale time integration steps to ensure that 6<j>e is less that S@ema during each micro-scale time step. The extended volume fraction increment limit chosen was 0.1. This value ensures there will be few iterations during the initial stages of solidification when increments in extend 94 volume fraction are low and increased incrementation in the later stages of solidification when increments in growth are large. The number of micro-scale time integration steps was limited to 500 during each macro-scale step to ensure a reasonable efficiency within the incrementation. This limit corrects the tendency of the adaptive micro-scale algorithm to perform excessive numbers of increments ( > 1,000) in the final stages of solidification. After finishing the specified number of micro-scale time steps, the resulting microstructure solidification rates, fs, of each fh phase are used to calculate the new rate of volumetric latent heat released according to Equation 5.3. ABAQUS calculates a second trial solution and checks equilibrium before moving on to the next macro-scale time increment. 5.2.2.2 Multi-component Segregation The relationships presented in Figure 5.3, describing the nucleation and growth routines, are for the most part adapted from a variety of literature sources as described in the Literature Review. There is one notable exception, however. As previously indicated, there is no general method reported for accounting for segregation in multi-component, multi-phase systems, such as hypo-eutectic cast iron. The approach adopted in the present study was to assume that the Scheil Equation for binary segregation holds, but it is applied on a component by component basis to each phase. The resulting incremental relationship is presented in Equation 5.4: (5.3) 95 ^ • = 1 ^ ^ ^ , ; (5-4) where SCLi is the change in inter-granular liquid composition with respect to component i, (i being C, P or Si), CLJ is the composition of the inter-granular liquid with respect to component i at the beginning of the increment, kjj is the segregation coefficient for the ith component of the j"1 phase, 6fSij is the previous increment in fraction transformed for the j'h phase, and fs is the total fraction solidified at the beginning of the increment. For carbon segregation, the fraction transformed, fx, is the fraction of austenite formed because no net segregation of carbon is assumed to take place in either the gray or white iron eutectics. The segregation coefficient, kjj, in the case of carbon is taken from the Fe-C binary phase diagram. For silicon segregation, the fraction transformed, fs, is equal to the total fraction solid formed (sum of fraction formed of austenite, gray and white iron). In this way, all three phases contribute to the component balance of silicon in the inter-granular liquid. For silicon, the segregation coefficients were taken from Kagawa and Okamoto[41] as described in the Background and Literature section. Phosphorus segregation was omitted because there was no clear method for accounting for its effect on the eutectic temperatures. The problems associated with phosphorus and the eutectic temperatures will be discussed in more detail in Chapter 6. 5.2.2.3 Phase Impingement The well known Avrami correction factor can be used to account for grain impingement in extended volume predictions of single phase growth. In order to describe transformations involving multiple phases, a two step calculation was performed. The 96 Avrami correction factor was applied to the total increment in extended volume fraction growth, 8(p(,, to obtain an increment in total fraction transformed, 6fs. To distribute the increment in fraction transformed to the increments in the phase transformed, a normalizing relationship was written as: Thus, the total extended volume fraction is corrected based on the overall fraction transformed and this is further partitioned to separate the increments of volume fraction formed of austenite, gray iron, and white iron during each growth increment. 5.2.2.4 Nucleation Attenuation The majority of investigations discussed in the Literature Review dealt with the nucleation and growth of a single cast iron phase, i.e. gray iron. For the multi-component case of cast iron, the growth of a single phase will likely impact the ability of other phases to nucleate and grow. To account for the influence of one phase on the capacity of other phases to form, a correction factor was applied to the relationship describing nucleation kinetics as per Equation 5.6. 5Ki (5.5) Nj = AATny¥j (5.6) Where iV. is the number of nuclei per unit volume of the j phase - e.g. austenite, white or gray iron - and *P. is the attenuation factor for nucleation of phase j, evaluated according to the following expression: (5.7) 97 The sum in Equation 5.7 is that of the volume fractions of all other competing phases excluding the jtn phase. For low cooling rate conditions, when appreciable amounts of austenite and gray iron have formed before the white iron eutectic temperature is reached, the attenuation factor will decrease the nucleation kinetics of white iron. Conversely, for high cooling rate conditions, the slow solidification rates of austenite and gray iron allow white iron to nucleate uninhibited. Although, under these conditions, white iron forms the majority of the solidified phase, it has no effect on the nucleation of austenite and gray iron as they will have nucleated prior to the white iron eutectic temperature. In the limiting case, under the conditions where a single phase nucleates and grows, Equation 5.6 defaults to the original relationship, Equation 2.17, developed by 01dfield[13]. 5.2.2.5 Phase Normalization Using the nucleation and growth kinetics model, phase evolution is calculated as the volume fractions of primary austenite, gray iron, and white iron. Unfortunately, through examination of the solidified microstructure, it is not possible to determine the amount of primary austenite. Microstructure examination provides only an estimate of the gray and white iron phase fractions. Also, material property data have been measured for gray and white iron and not their constituents of austenite, graphite, and iron carbide. Consequently, once solidification was complete, the calculated phase fractions of gray and white iron were normalized with a re-distribution of the primary austenite. The normalized phase fractions of gray and white iron were calculated with Equations 5.8 and 5.9. 98 s,w (5.8) f" =- (5.9) As defined, the microstructure subroutine provides a model for nucleation and growth of multi-phase equiaxed alloys such as cast iron. The evolution in microstructure will impact the thermal and stress evolutions through it's influence on thermal and mechanical properties as well as through latent heat release. 5.2.3 Latent Heat The release of latent heat is a complex non-linear phenomena which can cause considerable problems in heat transfer analysis especially in those cases where latent heat is released over a small temperature range. The different methods of incorporating latent heat release in thermal models can be separated into two categories: 1) methods involving enhanced specific heat or 2) methods involving a volumetric heat release. In the first case, latent heat is released linearly between the transformation start and finish temperatures by enhancing the specific heat of the transforming material. This method is adequate for those problems where the transformation occurs over the same temperature range regardless of the local cooling conditions - e.g. transformations that do not depend on phase transformation kinetics. The second method of latent heat release employs a volumetric heat source term evaluated at each integration point which is more flexible and hence, provides a means to couple the heat released to a phase transformation kinetics relationship. Employing ABAQUS, both of these methods were utilized for the problem at hand. 99 The latent heat release associated with cast iron solidification is defined by the kinetics of transformation. Consequently, the second method of incorporating latent heat release was employed for cast iron solidification. To facilitate this kinetics-base latent heat approach, the user-subroutine for internal volumetric heat release, HETVAL, is invoked within the ABAQUS thermal models for elements associated with the casting. At each material integration point within the casting, this subroutine calculates the volumetric latent heat flux as per Equation 5.3 using the latent heat values provided in Table 2.1. The latent heat released during the pearlite transformation in cast iron must be considered when predicting the temperature evolution in castings under long term cooling. Under the slow cooling conditions experienced in the roll casting (refer to Figure 4.10), this transformation does not appear to be influenced by kinetics since it occurs at the same temperature throughout the casting. The latent heat option in ABAQUS was utilized for this transformation in long term simulations. The parameters describing the transformation are start and finish temperatures of 720°C and 680°C, respectively and 82.0xl03 Jkg"1 as the latent heat of pearlite transformation[60]. Based on the Fe-C phase diagram and the composition of the reduced scale roll casting, the equilibrium fraction of austenite in gray and white iron at the pearlite transformation is 0.97 and 0.52, respectively. Thus, this method of accounting for the latent heat associated with the austenite to pearlite transformation will tend to over estimate the latent heat release in areas of white iron because the actual fraction of austenite transforming to pearlite is considerably lower in these regions. However, the latent heat 100 release associated with this transformation is considerably smaller than that of solidification and the impact should be minimal. 5.2.4 Temperature and Phase Dependent Thermal Properties The large temperature range experienced in casting processes necessitates the use of temperature dependent thermo-physical properties for the casting and mould materials. This temperature dependence adds to the overall non-linearity of the thermal problem. In multi-phase cast iron solidification, the non-linearity is further complicated by the need for phase dependent properties to account for the behaviour differences between white and gray iron. ABAQUS allows the user to define a table of material properties, incorporating multiple dependencies such as temperature and user-defined variables. ABAQUS linearly interpolates between the tabular material property data when updating the material properties at an integration point. Pehlke et al. reviewed the thermal properties of casting alloys and mould materials[61]. The generic thermo-physical property data for hypo-eutectic gray and white cast iron, presented in Table 5.1, were used as input for the thermal models. The same density was specified for each cast iron phase to preserve mass within the model since the dimensions of the mesh will not be changing with temperature or phase content. Also, the specific heat of each cast iron phase was the same since the casting is the same material on a macro-scale. As show in Table 5.1, thermal conductivity is the only property exhibiting a variation between the different cast iron phases. The liquid thermal conductivity has been elevated to approximate the effects of convection in the melt. 101 Phelke et al. presented thermo-physical properties for a number of different bonded sands. The bonded sand portions of the sample castings were assigned the properties of the silica bonded sand presented Table 5.1. As defined, the thermal model incorporates the influence of latent heat release and the effects of temperature and phase dependent material properties on temperature evolution. The results of a thermal/microstructure analysis could then be used as input for a stress analysis to determine the distribution of residual stress in a casting. 5.3 Stress Model A stress model was formulated to analyze the residual stress development in a component as it solidifies and cools. This model characterizes the majority of the thermomechanical phenomena discussed in the Literature Review. Specifically, the model describes strains due to elastic and in-elastic deformation, as well as thermal dilatational strains. No attempt was made to characterize the effects of transformation plasticity. Transformation plasticity is usually observed in samples that are quenched through the pearlite transformation. In the reduced scale roll casting, the slow cooling through this transformation ensures that transformation plasticity effects will be minimized. 5.3.1 General Stress Model Formulation In an FEM stress analysis, the solution variables, stored as a vector at the nodes, are the components of displacement. Thus, in a 3-dimensional problem, there are 3 degrees of freedom at each node. In stress analysis, loads are defined as prescribed displacements and forces. Employing the interpolation functions, it is possible to calculate the strain 102 and stress at any position within an element. From these quantities, the FEM equations may be developed through the imposition of the principle of virtual work within the element. This formulation is further extended to include in-elastic deformation. These concepts have been overly simplified here and the reader is directed to Zienkiewicz and Taylor[57] for a more complete description. The model developed to track the stress evolution in the reduced scale roll casting requires the thermal and microstructure model results as input. These results provide the evolution of temperature and microstructure of each material integration point in the casting. As in the thermal model, this data is then used to evaluate the temperature and phase dependent physical properties. 5.3.2 Elastic Mechanical Properties Increments in the elastic strain in a material can be calculated using Equation 2.24 providing the elastic modulus and Poisson's ratio are known. In their investigation of residual stress in gray iron castings, Wiese and Dantzig presented a temperature dependent elastic modulus[51]. In an investigation of roll casting, Beer utilized this data to characterize the elastic modulus of white iron[62]. The elastic moduli of gray and white iron are plotted in Figure 5.4 versus temperature. Beer also reported the Poisson's ratio of gray and white iron as 0.26 and 0.3 respectively. 5.3.3 Dilatation Effects Based on the temperatures in a casting at each time increment, the thermal strain at any position can be evaluated using the coefficient of thermal expansion. The coefficients of 103 thermal expansion for gray and white iron were measured by Walzen Irle for the operational temperature range of their rolls, 20°C to 400°C[3]. The data was extended to the temperature range experienced in a solidifying casting by Beer[62]. The combined coefficients of thermal expansion from these sources are plotted versus temperature in Figure 5.5. As Figure 5.5 shows, the thermal expansion coefficient of white iron measured by Walzen Irle is anisotropic. The orientational dependency in the white iron arises from directional solidification caused by high cooling rates at the casting OD surface. These differences in thermal dilatation behaviour are incorporated into ABAQUS using an anisotropic thermal expansion material definition. During extended cooling, the decomposition of austenite into pearlite results in a volumetric expansion and a change in the thermal dilatation behaviour of the cast iron phases (refer to Figure 5.6). Denis et al. reported the strain associated with the pearlite transformation as 5.0xl0"3[63]. Employing the same approach as described in the Thermal model section, the proportions of austenite in gray and white iron were used to calculate the phase specific volumetric expansion associated with the pearlite transformation. The thermal strains with and without the pearlite transformation, plotted in Figure 5.6, were calculated with Equations 2.25 and 2.26 and the data of Figure 5.5. Within ABAQUS, thermal expansion is defined as the total thermal expansion, ex, relative to a reference temperature, T. Equation 5.10 was used to convert the thermal strains, dh, of Figure 5.5 to total thermal expansions. When entered into ABAQUS, the anisotropic thermal strain is calculated based on the thermal model temperature, T, as well as the evolved microstructure. 104 a = e'"/(T-T") (5.10) 5.3.4 Constitutive Behaviour As Wiese and Dantzig described, the in-elastic behaviour of gray cast iron can be approximated by a complex series of yield surfaces[51]. However, this investigation did not include the in-elastic behaviour of white iron. The photomicrographs of white iron in the Quik-cup and reduced scale roll castings reveal that the iron carbide in white iron forms a large portion of the microstructure and appears to have a regular spaced phase distribution (refer to Figure 4.13). Thus, the in-elastic behaviour of white iron is likely much different than that exhibited by gray iron which has an irregular distribution of graphite flakes. Regardless of microstructure, the in-elastic behaviour of cast iron will increase in complexity at high temperatures as creep deformation becomes active. A relatively new feature has been implemented in ABAQUS to simulate the specific constitutive behaviour of gray iron. This constitutive model employs the Von Mises yield criterion for compressive yielding and the Rankine Criterion for tensile yielding. These criteria assume yielding is pressure-independent and governed by the deviatoric stresses alone in compression and in tension, yielding is governed by the maximum principal stress. Unfortunately, this model is still under development and at present is only applicable to monotonic loading conditions. Also, this model cannot be used in conjunction with the anisotropic thermal contraction behaviour previously discussed. It is essential to have proper mechanical data measured for a broad range of temperatures in order to predict in-elastic cast iron behaviour. Wiese and Dantzig 105 employ tensile and compressive stress-strain data measured for gray cast iron at a range of temperatures from an investigation by Ozgu[64]. Beer extended this data to describe the tensile behaviour of white iron[62]. This stress-strain behaviour for gray and white iron, presented in Figure 5.7, was employed as input for a stress model in ABAQUS. Without the proper characterization of white iron behaviour in compression, the stress model was limited to the standard plasticity approach. This approach assumes symmetric tensile and compressive behaviour with the yield surfaces defined by Figure 5.7. Following this approach, creep strains were neglected. 106 Table 5.1 - Thermo-physical properties of hypo-eutectic cast iron and a bonded silica sand mould material[61]. T K T C p P (°C) (Wm"'K"') (°C) (Jkg') (kgm"3) Liquid Iron 1175 18.8 Same as 7300 1400 270 Gray Iron Austenite 760 32.0 Same as 7300 1175 18.8 Gray Iron Gray Iron 140 60.0 100 548 7300 280 44.1 662 622 420 40.9 803 705 560 37.1 1100 746 700 33.6 1200 916 840 28.1 980 22.5 1120 18.8 1175 18.8 White Iron 140 25.3 Same as 7300 280 26.3 Gray Iron 420 24.4 560 20.6 700 16.9 980 16.9 1120 18.8 1175 18.8 Mould Sand 100 0.90 100 827 1500 300 0.74 300 980 500 0.67 500 1060 700 0.69 700 1116 900 0.80 900 1161 1100 1.00 1100 1197 1300 1.28 1300 1228 Figure 5.2 - Three dimensional mapping of different element types from the Cartesian (global) space to a local coordinate system. [57] 108 (Start Micro Calc.) IX r+A,<r, Restart Micro Calc with: f n=int S(t>e •+i n= 1 z r : ->| Do n micro-scale steps CLC,CLSi viaScheil I At _ T'+A'-T' 5r=— ST=- — n (Start NucGro) ^(CLCC^UT+OT) N=AAT2¥ V=juAT2 J t, (Stop NucGro) f Call NucGro Call NucGro f o r ^ e , e W ^ / Call NucGro A forC (Stop Micro Calc.) Figure 5.3 - Flow chart of user subroutines developed to calculate microstructure within A B A Q U S . 109 Figure 5.5 - Coefficients of thermal expansion versus temperature for gray and white iron, after[3,62]. 110 0.000 -0.002 -0.004 a -0.006 '8 -0.008 "8 S -0.010 <u J: H -0.012 -0.014 -0.016 -0.018 Gray Iron White Iron (radial) White Iron (tangential/axial) With Pearlite Expansion Without Pearlite Expansion 200 400 600 800 Temperature (°C) 1000 1200 Figure 5.6 - Thermal strains with and without pearlite expansion for gray and white iron. Figure 5.7 - Tensile stress - strain response of gray and white iron at various temperatures [62]. I l l CHAPTER 6 MICROSTRUCTURE AND THERMAL MODEL APPLICATION Prior to analyzing the residual stress distribution in the reduced scale roll casting, the evolution of temperature and microstructure in the Quik-cup and reduced scale roll castings were analyzed with the aid of the coupled microstructure and thermal model. Comparisons of the Quik-cup casting thermal and microstructure predictions with the measured data were used to 'fine-tune' the analysis via the microstructure model parameters and the thermal boundary conditions. The 'fine-tuned' microstructure parameters were employed in the analysis of the reduced scale roll casting with comparisons between measured predicted temperature used to quantify the boundary conditions. Discussion in the next section will indicate that the equiaxed microstructure model did not fully describe the evolution of microstructure for the broad range of cooling conditions experienced in the reduced scale roll casting. An extension of the microstructure model has been proposed to predict columnar white iron growth as observed at the surface of the roll. This extended microstructure model was applied to the Quik-cup and reduced scale roll castings to update the microstructure predictions. The application of the microstructure and thermal models to the Quik-cup and reduced scale castings is now discussed. The discussion will include a description of the element meshes, boundary and initial conditions, and results of the analyses. The extension of the equiaxed microstructure model to predict columnar growth is presented 112 and the results are compared with the original microstructure results and the measured data. 6.1 Analysis of the Quik-cup Casting As previously discussed, the Quik-cup casting was chosen as the test piece for validation of the heat transfer and microstructure models. Prior to presenting the thermal and microstructure predictions, the finite element mesh, boundary and initial conditions, and the microstructure model 'fine-tuning' are discussed. A statement of the governing partial differential equation, as well as, the relevant boundary and initial conditions is presented in Appendix C section C.l . l . 6.1.1 Mesh The small size and the close proximity of the mould sides to each other result in 3-dimensional heat flow within the Quik-cup casting. Applying planes of symmetry vertically across the centerline and diagonally from the corner, the geometry of the Quik-cup casting and mould were approximated with the l/8th section shown in Figure 6.1. The small lip of bonded sand mould that extends from the mould base (referring to Figure 4.19) was neglected since it has little influence on the mould heat transfer. The heat transfer implications of each of these simplifications were tested and the geometry, shown in Figure 6.1, represents the final solution arrived at through a series of simplifications. The geometry was discretized into 8-node hexagonal (brick) elements using a commercial software package, PATRAN, as a geometry pre-processing program. The 113 mesh consists of 594 elements and 977 nodes. The maximum horizontal element dimension in the Quikcup casting is ~6mm and the vertical element dimensions are uniform at 2.5 mm. 6.1.2 Boundary Conditions The boundary conditions required for solution of the thermal problem are adapted from well known equations describing free convective, q".onv, and radiative, q"md, heat transport. The standard equations describing these heat transfer phenomena follow as Equations 6.1 and 6.2[65]: where hamv is a heat transfer coefficient, <7rad is the Stefan-Boltzmann constant (5.6696x10"8Wm_2K'4), esurf is the emissivity of the radiating surface, Tsurf is the surface temperature, and T„ is the ambient environment temperature. These boundary conditions were used to describe heat transfer from the external surface of the mould and casting to the surrounding environment. The film coefficient for free convection, hconv, was set equal to 10Wm"2K"'. Average emissivities for the bonded sand and cast iron were adapted from a radiative heat transfer text as esand = 0.46 and ecas, = 0.50[66]. The complex interaction occurring along the internal interface between the casting and the mould is a difficult boundary condition to quantify. In the absence of the thermocouple, a casting of this geometry would develop a gap along the vertical mould/casting interfaces during solidification because of thermal contraction. The thermocouple wire in this casting is shielded by a quartz tube mounted horizontally ) (6.1) (6.2) 114 through the center of the casting (refer to Figure 4.19). This quartz tube fixes the casting vertically and leads to an additional gap forming horizontally across the mould/casting interface at the base of the casting. Dominik reviewed the theories used to describe gap heat transfer in casting processes[67]. The heat transfer across a gap is typically characterized by a combination of conduction and radiation. After pouring, a mould and casting are in good contact and the heat transfer across their interface is not limited by a gap. As the casting solidifies and contracts, heat transfer across the mould casting interface will decrease as the area of contact between the casting and mould decreases during gap formation. In the limiting case when there is no longer any contact, heat transfer across a gap is the result of conduction through air in the gap and radiation between the two constituent surfaces. Within the Quik-cup casting and mould, there was no means of measuring the gap distance or the heat transfer across the gap. Consequently, a generalized form of the gap conduction equation was employed in which the effective heat transfer coefficient, hejf, was assumed to vary with temperature of the casting: %,p=Kff(Jca.srTml,U!d) (6-3) where Tcast and Tmmid are the casting and mould surface temperatures, respectively. Prior to solidification, the heat transfer is high because of contact conduction. Thus, above 1100°C, the effective heat transfer coefficient was equal to a nominal value for contact conductive heat transfer, hamd , assumed to be equal to 1500Wm"2K"'. Below the solidification temperature, heat transfer due to contact conduction will decrease and radiation heat transfer will increase as a gap forms. Between a temperature 115 below the solidification temperature, Tmux, and a minimum temperature, Tmin, the effective heat transfer coefficient was calculated with Equations 6.4 and 6.5: Kff = KonJcond + (1 - fcond )h«ul (6-4) famd~~Z ( 1 — / l i m ^ / l i m (6-5) max min where fcond is the fraction of heat transfer across the gap occurring via contact conduction, and fum is the limiting fraction of contact conductive heat transfer. The lower limit of heat transfer via conduction, fum equal to 0.005, was used to simulate heat transfer across a gap via conduction through a gas. Through a series of trial and error calculations, the gap formation temperatures, 7 ^ and Tmin were set equal to 1100°C and 950°C, respectively. In Equation 6.4, the radiation heat transfer between the casting and mould surface, hra(i, was calculated according to Equations 6.6 and 6.7. The effective emissivity, calculated according to Equation 6.7, assumes that the gap is formed by two parallel surfaces that are in close proximity to each other[68]. This technique is a common method of calculating the radiation heat transfer across a gap[67]. ^rad ^^eff ^Fcast mould )(Tca,+Tmmld) (6.6) 1 1 1 £ e f f = ~ i (6-7) £ £ cast mould The gap interface conditions were specified in the thermal model as gap conduction surfaces within ABAQUS. The materials whose elements form the surface interfaces were defined with coincident nodes. The gap heat transfer algorithm described by Equations 6.3 to 6.7, was implemented in ABAQUS using the user-subroutine 116 GAPCON. For each element along the interfaces, GAPCON is called to calculate the effective heat transfer coefficient. 6.1.3 Initial Conditions The initial temperature of the Quik-cup casting was estimated to be 1225°C based on the measured temperature data (refer to Figure 4.20) and the reported pouring temperature of 1250°C. The mould was assumed to be at the ambient temperature of 20°C. 6.1.4 Microstructure Model 'Fine-Tuning' The preceding discussion has fully defined the thermal conditions required to solve the heat transfer problem for the Quik-cup casting. Prior to providing the results of the Quik-cup analysis, some discussion is presented on tuning the microstructure model. 6.1.4.1 Phase Stability Temperatures Based on the Quik-cup casting composition (refer to Table 4.1), the phase stability temperatures described by Equations 2.4, 2.5, and 2.6, were applied in the microstructure model. The phase transformation temperatures calculated with these relationships are: TL = 1172.7°C, Teut = 1141.6°C, and Tcarb = 1119°C. Utilizing nucleation and growth kinetics parameters adapted from literature sources, initial attempts to apply the microstructure model produced results that were inconsistent with the observed microstructure. Near the Quik-cup casting center, the predictions yielded a completely gray iron microstructure, whereas the observed microstructure was approximately 41% gray iron and 59% white iron. 117 The inability of the microstructure/heat transfer model to predict white iron growth near the Quik-cup casting center was traced to the carbide eutectic phase stability relationship. Figure 6.2 is a plot of the measured temperature response near the Quik-cup casting center (solid line) together with various carbide eutectic phase stability temperatures (broken lines). The original carbide eutectic temperature (Equation 2.6), calculated for the composition in Table 4.1 and plotted as a dotted line, is considerably lower than the eutectic arrest temperature of the measured data. Employing this temperature, the model predicts no carbide growth near the Quik-cup center. As discussed in the Literature Review, Equation 2.6 was validated against phosphorus compositions limited to a maximum of 0.1 wt% which is considerably lower than the 0.55wt% of this study. The dashed line plotted in Figure 6.2 was calculated by omitting the influence of phosphorus in Equation 2.6. This elevated transition temperature would provide a large undercooling for white iron growth throughout the casting, resulting in a completely white iron structure, which is also incorrect. The approach opted for was to modify Equation 2.6 so as to shift the carbide eutectic temperature up 9°C to a value consistent with the observed temperature response. The result, Equation 6.8, has been plotted as a dotted and dashed line in Figure 6.2. Tcarh(°C)= 1147.2 - 6.93(%Si + 2.5% P) - 1.717(%Si + 2.5%P)2 (6.8) 6.1.4.2 Nucleation and Growth Kinetics Coefficients A series of simulations were performed with the microstructure and thermal model to determine the optimum nucleation and growth kinetics parameters. Trial and error was used to assess which parameters provided the optimal solution. The nucleation and growth parameters that best describe the microstructure evolution in the 118 Quik-cup casting are presented in Table 6.1. This set of parameters for the Quik-cup casting falls within the range of parameters presented in the Literature Review. The following section presents the results obtained with these parameters. 6.1.5 Predictions and Comparisons to Measured Data Focus is now shifted to the temperature and microstructure results of the Quik-cup analysis. The comparison of the experimental data with the predicted temperature and microstructure provides a means for assessing the capabilities of the models. 6.1.5.1 Temperature and Microstructure Results and Comparisons The measured temperature response obtained from the embedded thermocouple (crosses) are plotted in Figure 6.3 together with the predicted temperature response at several locations close to the Quik-cup center (continuous lines) for a typical Quik-cup casting. As can be seen, the predicted temperature responses in Figure 6.3 clearly reflect the various phase transformations occurring during solidification similar to the measured response. Quantitatively, however, it is evident that the best agreement between the measured and the predicted data is obtained for a location between the nodes 5mm and 10mm from the casting center. At the 5mm distance from the casting center, the duration of the plateau or the thermal arrest is longer than that observed in the measured response. Near the end of solidification at the 5mm location, the prediction and measured data agree because of the rapid cooling which ensues after solidification is complete. The predictions at 10mm from the casting center show the correct thermal arrest duration, but agreement with the measured response worsens after solidification is complete at this location. 119 The discrepancy in the model predictions at the center may be due in part to uncertainty in the exact location of the thermocouple. In addition, some macro-shrinkage porosity was typically observed at the center of each Quik-cup casting around the quartz tube, as discussed in Chapter 4. The majority of macro-shrinkage porosity was observed to end 5mm from the casting center. The effect of the shrinkage porosity near the center and the effect of the quartz tube surrounding the thermocouple would be to decrease the thermal mass of the sample relative to the model, resulting in a thermal arrest of shorter duration. Moreover, the quartz tube may also act to radiatively transport heat from the casting center, which would also enhance cooling in the actual casting relative to the model. For simplicity, the model predictions at a location 5mm from the casting center will be taken to be representative of the behaviour at the center for the remainder of the Quik-cup casting discussion. The evolution of each phase with time has been plotted in Figure 6.4 for the Quik-cup center. Comparing the phase evolution in Figure 6.4 with the predicted temperature response in Figure 6.3 explains the characteristic temperature response predicted by the model. Following the predicted temperature response in Figure 6.3, there is a rapid cooling regime starting at the initial pour temperature of 1225°C and continuing until the austenite formation temperature, ~1172°C. The latent heat released during the austenite formation results in an offset in the temperature response before resuming moderate cooling. As cooling continues past the gray iron eutectic temperature, the austenite formation stops and the formation of gray iron begins. The slow kinetics of the gray iron phase development, relative to austenite and white iron, result in a gradual increase in the fraction of gray and a slow release of latent heat which produces the 120 characteristic thermal recalescence at 1125°C, shown in Figure 6.3. As the gray iron formation slows, the temperature begins to decrease and white iron formation begins. The rapid formation of white iron completes solidification in a comparatively small amount of time. The simulation is terminated prior to reaching the austenite eutectoid decomposition temperature. (In the current model, austenite decomposition kinetics are not considered, however, the latent heat associated with this transformation is included.) As can be seen in Figure 6.3, there is good agreement between the predicted and measured temperatures at the Quik-cup casting center. This serves as validation for the thermal model and the boundary conditions, but only partially validates the microstructure model in so far as the sequence and amounts of latent heat evolution associated with the various transformations is consistent with the measured temperature response. To fully validate the microstructure model, direct comparison of the final phase fraction content is required. In Figure 6.4, the solidified phase volume fractions predicted at the center of the casting are 0.13 austenite, 0.35 gray iron, and 0.52 white iron. Applying the normalization scheme of Equations 5.8 and 5.9, the final volume fractions of gray and white iron are 0.40 and 0.60, respectively. Thus, the microstructure predictions are in good agreement with the 40% gray iron and 60% white iron estimated at the Quik-cup center through phase area estimation in Chapter 4. The maximum cooling rate in the Quik-cup casting is experienced at the bottom corners where heat extraction to the mould is occurring in three directions. At this position in the casting, the temperature response (refer to Figure 6.5) shows a rapid cooling upon pouring and a subsequent rebound due to latent heat release. (Note: Figure 6.5 has been plotted with a smaller time scale in order to better resolve the temperature 121 evolution during solidification.) The evolution of phase fractions at this location has been plotted in Figure 6.6 with the smaller time scale. The extreme initial cooling conditions at this position in the casting allow the temperature to drop below the white iron eutectic temperature before any significant amounts of gray iron form. The speed of the white iron formation, dictated by the high growth rate coefficient and the nucleation kinetics of white iron (refer to Table 6.1), results in white iron forming the majority of the phase at this position in the Quik-cup casting. Figure 6.6 shows that 0.1 volume fraction pro-eutectic austenite forms along with 0.9 volume fraction white iron. The short solidification time, 1.5 seconds, at this position prevents gray iron from forming before complete solidification. Thus, the predicted microstructure, 100% white iron after normalization, agrees with the observed microstructure at the casting corner (refer to Figure 4.23), where white iron is the only phase present. A more comprehensive comparison between the measured and predicted microstructures is presented in Figure 6.7. In Figure 6.7, the volume fraction of gray iron converted from the measured hardness data has been plotted, as a function of distance from the Quik-cup center, together with the normalized gray iron model predictions. The large degree of scatter is attributed to the coarse nature of the hardness measurements. The resulting error in the volume fraction gray iron may be as high as ±10%. Excluding the region of porosity near the center, the results show satisfactory agreement for the center diagonal, given the scatter in the measured data. Moreover, the results for the center diagonal are generally higher than the off-diagonal measurements as would be expected. The cooling conditions experienced at each location along the center to corner traverse represent the full range of conditions experienced in the Quik-cup casting. The 122 agreement between model predictions and measured values along this traverse validate the microstructure model for the cooling rates experienced within the Quik-cup casting. 6.1.5.2 Effects of Formation Temperature In Figure 6.4, it is apparent that in the final stage of solidification, the gray iron forms slowly while the remaining liquid volume fraction solidifies as white iron. To examine this result, the variation in the formation temperatures for austenite, gray iron and white iron throughout solidification have been plotted in Figure 6.8 for the center of the Quik-cup casting. The predicted temperature response at this position in the casting is also included in Figure 6.8. The formation temperatures vary throughout the solidification period because of segregation. For each phase, the driving force for growth can be estimated by the difference between the predicted temperature and the appropriate phase formation temperature. For phases with negative undercoolings, where Tpredicted > Tformation, there is no driving force for growth. However, for all positive undercoolings, where Tpredicted > Tf o r m a t i on , growth will proceed proportional to the magnitude of the undercooling squared (refer to Equation 2.19). Thus, from Figure 6.8, it can be seen that the gray iron solidification has a significant driving force for growth throughout the period of solidification. In Figure 6.4, it is apparent that the low growth coefficient of gray iron results in a gradual formation of gray iron throughout solidification. Although white iron has a positive undercooling at the recalescence temperature, the driving force is not large enough to facilitate white iron growth. As the thermal recalescence ends, the driving force for white iron growth increases and the solidification rate of white iron increases. The remaining liquid volume faction solidifies primarily as white iron because of the comparatively high growth coefficient of white iron. 123 6.1.5.3 Solute Segregation As previously mentioned, the transformation temperatures vary with fraction solidified due to solute segregation during solidification. The time variations of solute (carbon, silicon, phosphorus) concentrations in the liquid are presented in Figure 6.9 for the center location in the Quik-cup casting. Austenite solidification results in carbon enrichment in the local liquid because the carbon segregation coefficient is less than one. This carbon segregation leads to the decrease in austenite growth driving force shown in Figure 6.8 by the corresponding decreases in the predicted temperature response and the austenite formation temperature. Prior to white iron formation, the variations in gray and white iron formation temperatures are caused by the inverse segregation of silicon during the austenite and gray iron formations. The silicon segregation coefficient during white iron formation is less than one. This silicon segregation coefficient for white iron and the quick solidification rate of white iron lead to the extreme variations of the formation temperatures near the end of solidification, shown in Figure 6.8. 6.1.5.4 Nucleation Effects At the scale of the microstructural phases, the microstructure model can accurately predict the evolution of gray and white iron. However, the predictions of an underlying parameter, such as nucleation, illustrate the limitations of the microstructure model. In Figure 6.10, profiles of gray and white iron nucleation densities have been plotted versus the distance from the center of the Quik-cup casting. The nucleation density values predicted by and used in the microstructure model to calculate the extended volume fraction have been labeled in Figure 6.10 as uncorrected nucleation. At the surface of the 124 casting, the white and gray iron nucleation profiles show a high nucleation density 10 3 (-3.0x10 nv). The nucleation densities decrease toward the center. This behaviour is expected since the nucleation density is a function of the undercooling squared at each position in the casting (refer to Equation 2.19). However, when compared with the observed microstructure, there is a discrepancy between the predicted nucleation density and the expected grain density. This effect is most pronounced at the corner of the casting where no grains of gray iron are observed, yet the model predicts the gray grain density to be a maximum, JVA, = 3.0x10 m"". In an attempt to deal with this obvious shortcoming, a corrected nucleation density can be calculated by multiplying the predicted nucleation density of phase j by the / h phase volume fraction. This assumes the predicted nucleation densities to be the local density of nuclei within the volume fraction of phase j. The correction calculation, described by Equation 6.9, was performed at each material integration point and the resulting nucleation profiles employing this correction have been plotted in Figure 6.10. The corrected gray iron nucleation profile increases as the distance to the center decreases. Although this result is inline with qualitative expectations, there remains a substantial gray iron nucleation density at the surface of the casting (Ng =108m"3) where no gray iron grains were observed. NhCor=Nhimd-f^ (6.9) Equation 2.17 was developed by 01dfield[13] to describe nucleation in cast irons under single phase solidification conditions. From the reported results, the extension of this relationship to multi-phase nucleation does not qualitatively or quantitatively 125 describe the nucleation observed in a casting. The problems of extending this relationship to multi-phase growth result from the dependence of nucleation on the undercooling temperature. A nucleation relationship with time and temperature dependencies might be better suited to this problem. Using such a relationship, it could be possible to predict the absence of gray iron nuclei at the Quik-cup casting surface. The development of a nucleation relationship better suited to multi-phase growth conditions presents a problem which will be left for future work. The predictions discussed in the preceding sections illustrate the ability of the thermal and microstructure models to accurately predict the evolution of temperature and microstructure phases in the Quik-cup casting. This small casting provided a means of establishing the microstructure parameters prior to applying the models to the reduced scale roll casting. Before analyzing the reduced scale casting, the sensitivity of the thermal and microstructure models to variations in key parameters will be assessed to ascertain potential sources of error. 6.1.6 Sensitivity A sensitivity analysis was performed to assess the influence of key parameters on the predictions of temperature and microstructure. Although each parameter in the microstructure and thermal models could potentially alter the predictions of the models, parameters with the most uncertainty were chosen for the sensitivity analysis. Thus, the sensitivity of the predictions to parameters such as the material properties of the cast iron and the sand were not included because they were taken from documented sources. The 126 analysis was performed using the Quik-cup casting geometry because of the faster execution time compared to that of the reduced scale roll casting case. The parameters investigated in the sensitivity analysis were separated into two categories: microstructure and thermal. These parameters, summarized in Table 6.3, were perturbed from their fitted values to maximum and minimum values and the results were compared with the reference result. Each perturbation presented in Table 6.3, represents a separate run of the Quik-cup casting model; in total, there were 16 sensitivity runs. In Table 6.3, the relative change of volume fraction gray and duration of thermal recalescence at the Quik-cup casting center, were used to assess the sensitivity of the model2. These results are discussed in the next two sections along with a baseline result for the microstructure and thermal parameters. 6.1.6.1 Microstructure Parameter Sensitivity The sensitivity of microstructure and temperature predictions to variation in the gray growth coefficient are presented in Figure 6.11. In Figure 6.11, the reference results (solid line) are compared with predictions based on maximum and minimum variations of the reference value. The profiles of volume fraction gray iron have been plotted versus distance from the Quik-cup casting center in Figure 6.1 la. The volume fraction gray iron profiles are low at the casting corner (furthest distance from center) where the cooling 2 The Quik-cup casting center results in Table 6.3 are for the same 5mm diagonal location discussed in the previous section on Quik-cup casting predictions. 127 rates are highest and increase as the distance to the center decreases. The perturbations in gray iron growth coefficient produce large variations in the gray iron profiles. As expected, increasing the growth coefficient leads to a larger volume fraction of gray iron, while decreasing the growth coefficient leads to smaller volume fraction of gray iron. The temperature evolutions at the center and corner of the Quik-cup casting, plotted in Figure 6.1 lb, show that the gray growth coefficient has little effect on temperature. The only notable effect of this parameter on the thermal history is a variation in the recalescence temperature. This effect can be used to explain the off-center gray volume fraction prediction for the case where the gray growth coefficient was increased (refer to Figure 6.11a). In Figure 6.11a, the profile corresponding to the increased gray growth coefficient reaches a maximum at 10mm from the casting center, while the two other profiles have maximum gray volume fractions at the center of the casting. The increased thermal recalescence temperature results in a lower undercooling for growth. Thus, although the growth coefficient was higher, the driving force for growth was less at the center of the casting and a decreased volume fraction of gray iron resulted. In Table 6.3, the microstructure parameter sensitivity summary illustrates the sensitivity of the remaining microstructure parameters. The gray growth coefficient shows the largest microstructural sensitivity followed, in order of sensitivity, by the carbide eutectic temperature, the white growth coefficient, and the nucleation coefficient. The thermal sensitivities, summarized by the duration of the thermal recalescence plateau, illustrate the limited effect (Afrec<3s) that the microstructure parameters have on the thermal results. The notable exception is the carbide eutectic temperature which has a slightly larger impact on the thermal recalescence duration (Atrec+~4s). Plots of the 128 sensitivity effects, in the form of Figure 6.11, for remaining microstructure parameters have been included in Appendix D. 6.1.6.2 Thermal Parameter Sensitivity With regards to the model sensitivity to variations in the thermal parameters, the thermal and microstructure results are most sensitive to the nominal gap heat transfer coefficient, hgap (refer to Table 6.3). The effects of this parameter on the volume fraction gray iron profile and the temperature evolution are presented in Figure 6.12. Compared with the gray growth coefficient (refer to Figure 6.1 la), the nominal gap heat transfer coefficient has a moderate influence on the profile of volume fraction gray iron, presented in Figure 6.12a. When the gap heat transfer is decreased, cooling rates throughout the casting decrease which results in increased fractions of gray iron. This effect is most pronounced at the center of the casting. In Figure 6.12b, the temperature evolution at the corner of the casting shows a large sensitivity to variations in the gap heat transfer. This sensitivity is a direct result of the impact the gap heat transfer coefficient has on heat transfer at the casting surface. The effects of the nominal gap heat transfer coefficient at the casting center are not as pronounced because of the distance from the casting / mould interface. At the center, the thermal recalescence duration increases (Atrec+\6s) as the gap heat transfer decreases. This longer thermal recalescence is responsible for the sharp increase in gray iron volume fraction at the casting center, shown in Figure 6.12a. The sensitivity of the thermal predictions to the remaining thermal parameters are presented in Table 6.3. The summary clearly illustrates the direct impact the thermal 129 parameters have on the thermal recalescence duration. Compared to the microstructure parameter sensitivities, the gray iron volume fraction at the Quik-cup casting center is not as sensitive to the thermal parameters. The gray iron volume fraction was most influenced by the lower gap heat transfer coefficient. As discussed, the lower cooling rates in this case allow more time for gray iron to transform. The thermal recalescence duration shows a corresponding increase under these conditions. Plots of the remaining thermal parameter sensitivity results are included in Appendix D. 6.2 Analysis of Reduced Scale Roll Casting The reduced scale roll casting was analyzed using the 'fine tuned' nucleation and growth parameters established from analysis of the Quik-cup. Although, the same microstructure model was applied in this analysis, the geometry and casting process, such as preheating and long term cooling, required a thermal model with a considerable increase in complexity. A statement of the governing partial differential equation, as well as, the relevant boundary and initial conditions is presented in Appendix C section C.1.2. 6.2.1 Mesh The geometry of the reduced scale roll casting was simplified to the 2-dimensional axisymmetric section shown in Figure 6.13a. This geometry does not include the casting inlet gate because it cannot be described by an axisymmetric analysis. The influence of this geometric feature on the temperature solution has not been formally assessed. A 3-dimensional analysis would be required to describe the influence of this geometric feature on the casting. The benefits of simplification from a 3-dimensional to a 2-130 dimensional axisymmetric analysis are reductions in model complexity and size which correspond to a reduction in the solution time. The reduced scale roll casting model employs three material types to describe the roll, permanent mould and sand mould sections (refer to Figure 6.13a). The cast iron thermo-physical property data, presented in Table 5.1, was assigned to the roll material section. The permanent mould sections, consisting of the chill mould and the mould base, were assigned the gray iron thermo-physical properties of Table 5.1. The thermo-physical properties of bonded sand were assigned to the core and base sand sections. The mesh generated for the reduced scale roll casting is presented in Figure 6.13b and summarized in Table 6.2. The casting mesh consists of 4-node linear axisymmetric elements with a fine mesh resolution of 5mm radial and 10mm axial element lengths. These linear temperature elements required a high mesh density to ensure proper accounting of the latent heat released during solidification. In the cast iron and bonded sand mould sections, 8-node quadratic axisymmetric elements were used to allow a decrease in the mesh resolution. The mesh resolution within these sections corresponds to ~20mm element lengths in the radial and axial dimensions. The mesh corresponding to each material was generated with coincident nodes to form interfaces between each material contact area. This formulation facilitates the use of different element types in the same model, as well as, providing the flexibility need to define boundary conditions that describe the different stages of the casting process. 131 6.2.2 Boundary Conditions Although some variations were required, the boundary conditions applied to the reduced scale roll casting are of the same form as those applied to the Quik-cup casting. Heat transfer from the cast iron and bonded sand mould surfaces to the surrounding environment was calculated via Equations 6.1 and 6.2, with the same convective heat transfer coefficient and emissivities as applied to the Quik-cup casting. As in the Quik-cup casting, the interface areas between the different materials represent the most complex boundary conditions in the reduced scale roll casting. The boundary conditions that are modified for specific casting stages will be described in the following sub-sections. However, there are a number of contact interfaces that were defined with constant gap heat transfer coefficients of 2500Wm"2K"1. These interfaces have been marked in Figure 6.13a with "C HTC". The high gap heat transfer coefficient is intended to provide low thermal resistance across these interfaces and results from the assumption that these areas of the mould assembly are in good thermal contact. 6.2.2.1 Preheat Boundary Conditions During the casting trial, the preheated chill mould was assembled with the other mould sections prior to pouring the casting. In the interval of 1.5 hours that elapsed prior to casting, some heat was transferred from the chill mould section to the environment and to the other sections of the mould. As previously discussed in Chapter 4, temperature gradients developed in the mould sections during this period. The resulting temperature distribution acts as the initial condition for the casting stage of the process. 132 Consequently, a separate set of boundary conditions were needed to simulate events after mould assembly and prior to mould filling. A model to fully describe heat transfer during the preheat stage must include heat conduction through the solid sections, convective cooling caused by the buoyancy of air in and around the mould (fluid flow calculations), and a radiation analysis with view factors. However, this type of model is considerably more complex than required for this investigation. Instead, a model which simulates the redistribution of heat from the chill mould to the other mould sections and the surrounding environment was found to be sufficient. Following this approach, convective and radiative heat transfer within the mould cavity were lumped together and approximated with a gap conductance boundary condition. This gap conductance boundary condition, similar to Equation 6.3, was defined between the chill mould and base sand ID surfaces and the bonded sand core OD surface (refer to Figure 6.13a). The gap conductance coefficient was set equal to 5Wm'2K"' following a series of trial and error simulations of the preheat stage. The gap boundary condition is typically applied to interfaces that are in close proximity to each other. Although there is a 10cm gap between the chill mould and the core, the gap boundary condition provides a convenient means of directly redistributing the heat from the chill mould to the bonded sand core. 6.2.2.2 Solidification and Cooling Boundary Conditions In the second stage of the thermal analysis of the reduced scale roll casting process, the liquid cast iron is introduced and allowed to cool to room temperature. The boundary conditions associated with the preheat stage are removed and the casting and its boundary 133 conditions are introduced. The interface boundary conditions between the chill and sand ID surfaces of the mould and the casting OD surface are similar to those applied in the Quik-cup casting simulation. The temperature range defining the gap heat transfer ramp from conduction to radiation has been widened with TmaX equal to 1100°C and Tmin equal to 850°C. The upper gap heat transfer coefficient for conduction is unchanged at ISOOWm'V. The gap heat transfer coefficient between the casting ID surface and the bonded sand core was equal to 500Wm"zK"'. The heat transport associated with mould filling and liquid convection cannot be modeled within the constraints of an ABAQUS heat transfer model. However, a method was devised to approximate the effect of mould filling on heat transfer. In this method, the casting is instantaneously introduced into the model, however, interfacial heat transfer between the casting and the mould is left 'turned off. Over the interval of time taken to pour the casting, the gap heat transfer between the casting and the mould sections is activated as a function of time and the axial height along the casting surface. Thus, heat transfer to the mould is activated along the axial length of the casting surface gradually from the bottom of the casting to the top consistent with the manner in which molten metal is introduced into the mould. The pouring time, 1.5min, was used as the total time to initiate gap heat transfer along the length of the casting. 6.2.3 Initial Conditions The initial temperature of the chill mould was reported as 140°C after preheating with a torch. It is unlikely that a uniform temperature distribution resulted from this method of heating. However, in the absence of thermocouple data for the chill mould preheating 134 stage, it was necessary to assume that the chill mould initial temperature was uniform at 140°C. The bonded sand sections and the cast iron base section of the mould were at room temperature, 20°C, prior to final assembly with the preheated chill mould. The casting temperature was estimated at 1250°C from a combination of the reported ladle temperature and the maximum measured temperature. 6.2.4 Comparison of Predictions and Measured Data The analysis of the reduced scale roll casting employed the 'best fit' microstructure parameters found in the Quik-cup analysis. With regard to the microstructural model, the only difference between the Quik-cup and reduced scale roll casting analysis is the compositional difference shown in Table 4.1. Similar to the experimental discussion of the reduced scale roll casting, the analysis was separated into preheat, solidification, and long term cooling stages. 6.2.4.1 Temperature Prediction fo r the Preheat Stage As described previously, prior to pouring the casting, the mould was allowed to equilibrate for 1.5 hours after the preheated chill mould section was assembled with the other mould sections. The predicted and measured (in brackets) temperatures of the chill mould section are plotted on a schematic of the mould in Figure 6.14a for comparison purposes. The temperature contours that develop in the mould base sections and the bonded sand core are plotted in Figure 6.14b. As can be seen, the agreement between the predicted and measured temperatures is satisfactory overall. However, there is a tendency for the radial gradient between the 135 RO and OD surfaces of the chill mould to be underestimated by the model. The absence of this gradient is likely due to the initial temperature distribution employed in the model for the chill mould section. Preheating the chill mould with a torch would not result in a uniform temperature distribution as assumed. Instead, there would likely be an axial and a radial temperature gradient in the chill mould section consistent with the uneven heating expected from a torch and cooling during mould assembly. However, as was previously mentioned, in the absence of thermocouple data for the chill mould preheating, an initial condition of 140°C was imposed throughout the chill mould. 6,2.4.2 Thermal Predictions of the Solidification Stage The reduced scale roll casting was poured in a 1.5min period following the preheat stage. As discussed above, pouring the casting was simulated with a time-dependent internal gap boundary condition. The temperature evolution at a number of nodes along the OD surface of the casting are presented in Figure 6.15. In Figure 6.15, the time delay in the temperature response at each axial height simulates the variation of the metal level height during pouring. At each axial location, the casting remains at the pour temperature until the internal gap boundary condition is activated and heat transfer to the mould begins. Initially, a high gap heat transfer coefficient causes a rapid temperature decrease at the casting surface as the molten metal looses superheat. The reduction in cooling rate at each location is the result of several factors including a decreasing gap heat transfer coefficient and the heat-up of the mould materials. The gap heat transfer coefficient decreases from conduction conditions at surface temperatures above 1100°C to a combination of radiation and limited conduction conditions at surface temperatures below 136 850°C. The two characteristic cooling curves observed in Figure 6.15 correspond to the different mould materials - the bonded sand base and the chill mould - in contact with the casting OD surface. An assessment of accuracy of the mould filling simulation technique cannot be made because of the lack of thermal data with good time-scale resolution at a number of axial heights. Also, it should be noted that there was a large computational penalty associated with this pouring simulation. The number of increments required to solve the 90s pouring simulation and the subsequent solidification stage was -8600. The alternative approach, assuming instantaneous pouring, reduces the number of increments required to solve the solidification stage to -1650 or a factor of 5 in computation time. This is a significant computational penalty, but the predicted results agree better with the time delay exhibited by the experimental data. Following mould filling, focus can now be shifted to the thermal results of the solidification stage, which is 1800s in duration. The temperature distribution of the roll and mould after 1800s is shown in Figure 6.16a. In Figure 6.16a, heat from the roll has been conducted into the surrounding material sections. However, the outer surfaces of the mould base remain close to ambient temperature, 20°C. The roll temperature along the OD and top surfaces has cooled considerably from the pour temperature of 1250°C. The portion of the roll below the chill mold is insulated by sand on three side and thus, is at the highest temperature in the roll. Contours of the solidified phase fractions of austenite, gray iron, white iron and the total amounts formed are presented in Figure 6.16b. The total fraction transformed plot shows that the lower portion of the roll has not 137 solidified. The gray and white iron phase fraction plots indicate that a large fraction of white iron has developed at the OD surface of the roll. The white iron phase fraction decreases with distance to the ID surface of the roll. A discussion of the final solidified microstructure is presented in a following section. Further discussion in this section will focus on the evolution in temperature at the thermocouple locations. In Figure 6.17, the thermal predictions (lines) of the solidification stage are plotted with the measured temperature data (symbols) for the mid- and bottom height positions in the reduced scale roll casting. At both heights in the casting, the measured and predicted temperature data exhibit similar trends. At the casting OD surface, there is an initial rapid decrease in temperature which slows as the gap heat transfer coefficient decreases and the mould ID surface temperature increases. At the thermocouple locations within the casting, the impact of the rapid surface cooling decreases with increasing distance from the surface. The maximum difference between the measured and the predicted temperature response occurs at the casting OD locations. At these OD locations, small differences in thermocouple location will result in large temperature differences because of high temperature gradients. In Figure 6.17a, the temperature difference between the 20mm sub-surface location and the casting OD is 160°C at a total time of 5600s. The thermocouples employed to measure the data were sheathed and mounted in quartz tubes prior to embedding in the bonded sand mould. This introduces some amount of uncertainty as to the exact location of the thermocouple bead within the thermocouple assembly. As discussed in Chapter 4, an attempt was made to measure the depths of the holes left by the quartz tubes in the center section which was shipped to UBC. However, 138 it was not possible to measure the mould thermocouple locations. Thus, considering the uncertainty in the thermocouple positions, the overall agreement between the predicted and measured thermal responses in the casting is reasonable. The predicted and measured temperature responses of the bonded sand core OD surface are also presented in Figure 6.17. The agreement between the measured and predicted temperature responses is good during the initial and intermediate portions of the solidification stage. However, near the end of solidification, the measured temperature response begins to cool at 6700s while, in the predicted temperature response, cooling is delayed until 7000s. The cause of this error in the predicted response is unclear. Modification of the gap heat transfer coefficient between the casting and the core has little effect on the time at which the temperature drops. The most likely cause of the error is uncertainty in the bonded sand thermo-physical properties. The thermo-physical properties of this bonded sand were taken from typical mould sand properties reported by Phelke et al. [61] because the specific properties of the bonded sand employed in the mould were unavailable from the manufacturer. The temperature responses in the mould at the mid- and bottom heights are plotted in Figure 6.18. The model temperature predictions show good agreement at the mid-height locations. However, the model predictions for the chill mould ID location do not agree with the measured temperature data at the bottom height. The chill mould ID predicted temperature responses at the mid- and bottom heights are comparable to each other. Whereas, the measured temperature response at the bottom height is consistently higher than the mid-height measured temperature response. Numerous attempts with the thermal model were made to extend the temperature difference between the chill mould 139 ID and OD surfaces at the bottom height. However, the high conductivity of the cast iron chill mould (refer to Table 5.1) prevents the temperature difference from increasing any further than is presented in Figure 6.18. It is likely that the chill mould ID thermocouple at the bottom height was in partial contact with the casting OD surface causing the elevated temperature measurements at this location. 6.2.4.3 Thermal Predictions of Long Term Cooling The predicted and measured temperature responses for long term cooling in the reduced scale roll casting are presented in Figure 6.19 and Figure 6.20. In Figure 6.19, the predicted temperature evolution in the casting at the mid- and bottom heights illustrate the decrease in temperature gradients within the casting as cooling proceeds. Prior to the pearlite transformation, beginning at 725°C, the predicted temperature responses show satisfactory agreement with the measured data. After the pearlite transformation, the predicted and measured temperature responses converge and show good agreement for the duration of cooling shown in Figure 6.19. In Figure 6.20, the predicted mid-height chill mould temperature evolution is slightly cooler than the measured response at the chill mould ID and hotter along at the OD surface of the chill mould. While at the bottom height, the predicted chill mould OD surface response is in good agreement with the measured response. As discussed in the previous section, the chill mould ID thermocouple at the bottom height is consistently higher in temperature than the predicted response, likely due to partial contact between the casting and the thermocouple assembly. 140 6.2.4.4 Microstructure Predictions The predicted and measured microstructure profiles of gray and white iron are compared in Figure 6.21. Although the same general trends are exhibited by the predicted and measured data, the overall agreement between them is poor. Specifically, the measured microstructure profiles at the OD surface show a plateau of white iron extending 20mm into the casting. Following this white iron plateau, there is a rapid decrease and extinction of the white iron phase, and a corresponding increase in the gray iron phase fraction. Contrary to these measured results, the predicted microstructure profile shows a steady decline in the white iron phase fraction from the OD surface to 40 mm from the ID surface. At this position, 20% white iron is predicted and the volume fraction of white iron does not decrease until near the ID surface. Errors, resulting from the bulk nature of the hardness tests, are expected in the measured microstructure fractions and may be as high as ±10%. This might explain the lack of agreement at the OD of the casting where the cooling rates are as high as those experienced in the Quik-cup casting. However, outside this region of high cooling rates, the model consistently over predicts the amount of white iron. Thus, the application of the microstructural model to a broader range of cooling conditions than present in the Quik-cup casting reveals some limitations in the model. Attempts to modify the phase fraction profiles through the microstructure parameters were unsuccessful. Enhancing the gray iron nucleation and kinetics parameters caused the desired decrease in white iron near the casting ID, but also increased the gray iron phase fractions at the casting surface. The same effects were observed when depressing the white iron nucleation and kinetics parameters. The 141 coupled nature of the microstructural phase nucleation and growth prevents a better fit to the measured microstructure profiles. The lack of a predicted white iron plateau at the OD surface of the casting can be attributed to the rapid evolution of gray iron. In the previous discussion of the Quik-cup casting results, it was noted that a nucleation law that was a function of time and temperature would be better suited for describing the surface solidification conditions in the Quik-cup casting. In this case, it is likely that a nucleation relationship of this type would improve the microstructure predictions at the casting surface to some extent. A central assumption of the microstructure model developed in this study was that the growth morphologies of gray and white iron are equiaxed. Throughout the literature, this assumption has been made in cases of cast iron eutectic growth. However, the photomicrographs of areas near the surface of the reduced scale roll casting and the Quik-cup casting (refer to Figure 4.13 and Figure 4.23) show an oriented white iron phase interspersed with equiaxed grains of gray iron. This directionality of the white iron phase is also supported by the thermal expansion properties measured by Walzen Irle in roll castings (refer to Figure 5.5). The thermal expansion data exhibits a preferential contraction in the radial direction caused by a radial orientation of the white iron phase in these roll castings. The assumption of an equiaxed white iron growth morphology in this study may have limited the capabilities of the microstructure model and prevented accurate microstructure predictions. Consequently, an attempt has been made to extend the present microstructure model to predict columnar white iron growth. 142 6.3 Microstructure Model Extension for Columnar Growth The microstructure profiles predicted by the equiaxed microstructure model illustrate the inability of this type of model to accurately predict the evolution of the white iron phase in roll castings over a broad range of cooling conditions (refer to Figure 6.21). Microstructure analysis of the Quik-cup casting and the reduced scale roll casting reveals that the white iron phases, initiating at the casting surfaces, have columnar growth morphologies oriented towards the casting centers. To account for this, a columnar growth morphology extension has been proposed and implemented within the microstructure model. The pioneering investigation by Hunt[29], to analytically model columnar and equiaxed growth, has been built upon by a number of authors[10,69,70,71]. Although, none of these studies have dealt with columnar growth in cast iron alloys, their work can be extended to this area. An underlying assumption in most of these studies is that the growth rate of the columnar front is equal to the velocity of the eutectic isotherm. To apply this technique, the temperature field and the corresponding gradient within the solidifying region must be know. For this case, the velocity of the eutectic isotherm, vs, may be described as: v = dT/dt (6.10) where VT is the temperature gradient. The columnar front velocity is subsequently used to determine the position of the columnar front at the end of a time increment. This approach is valid for solidification conditions where the columnar front exhibits small undercooling. 143 Implementation of a columnar front tracking algorithm within ABAQUS presents a number of challenges both from a practical and numerical viewpoint. Firstly, the suite of user subroutines available within ABAQUS, do not provide access to the necessary information to calculate the velocity of the eutectic isotherm at a given time and position in a straightforward manner. Consequently, the columnar front undercooling cannot be readily determined in order to establish at what temperature the latent heat associated with the front should be released. Secondly, even if the first problem is overcome, ABAQUS will not converge if presented with the situation where there is step change in latent heat evolution associated with the arrival of the columnar front at a particular material integration point. In the present equiaxed model, the discrete volume of material associated with each material integration point is assumed to be of uniform temperature, or isothermal. Thus, the volume fraction of a particular phase, and more importantly, the rate of change in volume fraction, is also assumed uniform over the volume of material associated with the material integration point. In the context of the equiaxed model, this approach is satisfactory as there is a gradual release of heat associated with the process of nucleation and growth of a particular phase. In a columnar model, such approach would result in the instantaneous transformation of liquid to solid within the volume associated with the material integration point upon reaching the front temperature (formation temperature less the undercooling). The consequence of this transformation would be an instantaneous release of all the latent heat associated with the volume fraction liquid, making convergence impossible. Thus, an alternate approach had to be developed that 144 better reflects the latent heat effects of a columnar front with a finite velocity moving through a fixed non-isothermal mesh. 6.3.1 Columnar Model Formulation The columnar growth model developed in this investigation draws on the previously described work by other investigators[ 10,29,69,70] and incorporates some significant simplifications to satisfy the rigorous equilibrium constraints of ABAQUS. Although it will be clear that the results of this columnar growth model are a step forward in the predictive capability of the microstructure model presented in this study, it should be noted that the columnar growth model lacks the sophistication of some of the previous work[ 10,70]. Nevertheless, its simplicity makes this approach useful for FEM-type calculations in commercial codes. In this investigation, columnar white iron growth begins when the material integration point cools below the carbide eutectic temperature as illustrated schematically in Figure 6.22. As in the equiaxed growth model, the carbide eutectic temperature will vary as columnar and equiaxed solidification proceeds because of solute segregation to the liquid. Columnar growth proceeds as a front, shown in Figure 6.22, oriented perpendicular to and growing in the opposite direction of the direction of heat flow. As in equiaxed solidification, the growth velocity of the white iron columnar front is assumed to be proportional to the square of the undercooling below the carbide eutectic temperature based on the reference temperature at the material integration point (refer to Equation 2.19). The same white iron growth coefficient employed for equiaxed solidification has been applied to columnar solidification. 145 The increment in fraction transformed of columnar white iron phase, 6fcnt, is calculated as: St (6.11) x where x is the characteristic length of the element parallel to the growth direction of the columnar front and vc„t is the columnar front velocity. The columnar front, described by Equation 6.11, is a solidification front oriented perpendicular to x. The impact of x on the predicted front growth is shown schematically in Figure 6.23. When the columnar front is oriented perpendicular to x, the predicted front is accurately predicted (refer to Case 1 in Figure 6.23). However, in Case 2 of Figure 6.23 the actual characteristic length, x', is larger than x which results in an over prediction of the fraction transformed (refer to Equation 6.11). The volume fraction columnar is tracked as the sum of the increments in columnar growth and the total fraction transformed is now the sum of equiaxed and columnar phase fractions. Coupling this columnar growth model to the equiaxed model previously described allows both phases to grow in competition with the other. For casting conditions that produce large undercoolings, equiaxed phase growth will dominate because of the large numbers of equiaxed nuclei combined with high growth rates that result from large undercoolings. As cooling rates decrease, moderate undercoolings provide a lower driving force for equiaxed solidification and allow columnar growth to dominant solidification. Finally, if the undercoolings decrease further, a transition will occur from columnar to equiaxed growth. This transition from columnar to equiaxed growth is typically referred to as the CET[29]. 146 Equiaxed solidification, occurring prior to the arrival of the columnar front, will end when the columnar front passes. The equiaxed grains that have grown when the columnar front passes will disrupt the front's growth as the front attempts to accommodate them. The critical volume fraction of equiaxed grains that completely disrupts a columnar front was defined by Hunt[29] as feq equal to 0.49. If the volume fraction of equiaxed phase reaches this critical level before the columnar front arrives, columnar growth at this position will end. There are a number of issues of interest that follow from the formulation of the present columnar model and its incorporation into ABAQUS. These issues and their implications have been summarized below: • The standard analytical method of locating the columnar front during steady state growth is via an undercooling calculated from the isotherm velocity. In the model presented, a columnar front begins to grow when the eutectic temperature is reached and growth will accelerate as undercooling increases (refer to Figure 6.22). The predicted location of the columnar front will fall between the eutectic isotherm and the columnar isotherm predicted using the analytical method. For cooling conditions where columnar growth occurs with minimal undercooling, the differences between the method used in this study and the standard approach will be minimal. • As the mesh size in the present model decreases, the isotherm of the predicted columnar front will approach the eutectic isotherm because the characteristic length of the element is decreasing (refer to Equation 6.11). For cooling conditions that result in high temperature gradients, the error associated with the predictions will 147 increase as the mesh size decreases. Thus, contrary to most FEM mesh size relationships, a coarse mesh will improve the predictive ability of this model. Without explicit tracking of the columnar front, equiaxed growth may occur in regions of an element that have already solidified. With a uniform temperature assumed within each volume region, the proportions of equiaxed growth behind the columnar growth front cannot be assessed. One of the key parameters in the columnar model is the characteristic length over which the columnar front is growing. Within the user-subroutine HETVAL, referenced to calculate the microstructure evolution in ABAQUS, a number of key model and solution variables are missing, specifically the nodal coordinates and the temperature gradients. The nodal coordinates are needed to determine the characteristic length of an element. In the absence of the coordinates, the model cannot adapt to changes in mesh size. The temperature gradient is required to determine the growth direction of the columnar front. Without the temperature gradient, columnar phase fractions will be over predicted when growth occurs in directions which are not perpendicular to element faces (refer to Figure 6.23). This error will mainly affect corner geometry sections of mesh. The errors associated with the missing nodal coordinates and temperature gradient can be minimized by using a uniform mesh size and constant characteristic distance. For the present castings, these errors will have limited impact in the reduced scale roll casting since the primary direction of heat flow is radial. The small size and high heat transfer in the corner of the Quik-cup casting will make it more susceptible to this error. 148 For the casting geometries investigated in this study, the majority of these issues will have little impact on the white iron columnar predictions. The errors associated with this columnar model are outweighed by the simplicity and general applicability that this approach provides. 6.3.2 Application of Columnar Model for White Iron Solidification The predicted microstructure in the Quik-cup and reduced scale roll castings was updated employing a model with the white iron columnar growth extension. Since the nucleation and growth parameters employed with the equiaxed model over-predict the white iron phase fraction in the interior of the reduced scale roll casting, changes to the parameters were required before the columnar growth model could be incorporated. To preferentially decrease the phase fraction of equiaxed white iron relative to that of columnar white iron, the nucleation kinetics coefficient of white iron was reduced two orders of magnitude to 12.86xl04 m"3. The microstructure results for the reduced scale roll casting are presented next, with the Quik-cup casting results following for comparative purposes. The variations in microstructure predictions had little effect on the thermal predictions. For this reason, they will not be represented. 6.3.2.1 Columnar and Equiaxed P redictions in the Reduced Scale Roll Casting The microstructure profiles for the reduced scale roll casting predicted with the combined columnar and equiaxed microstructure model are presented in Figure 6.24. The original equiaxed microstructure profile predictions, as well as, the measured profiles have also been included in Figure 6.24. The gray iron profiles in Figure 6.24a show the columnar and equiaxed model formulation has increased the predicted gray iron volume fraction 149 across the entire radial traverse. Near the ID, the increase in gray iron volume fraction brings the extended model closer to the measured result. However, in the region 80 - 100 mm from the ID surface, the increase in gray iron volume fraction weakens the agreement with the measured result when compared with the equiaxed model. In Figure 6.24b, the white iron profiles of the columnar, equiaxed, and their combined results have been plotted for the columnar and equiaxed model. As in the equiaxed model, the combined columnar and equiaxed model over predicts the growth of gray iron at the casting surface which leads to a rapid decrease in the white iron volume fraction near the casting OD surface. Following the separate components of the columnar and equiaxed model white iron profile shows a sub-surface maximum in the columnar profile near the OD surface. The evolution of equiaxed white iron at the OD surface prevents the maximum in the columnar profile from occurring at the surface. From the sub-surface maximum, the columnar profile decreases with distance from the ID surface. The equiaxed profile decreases rapidly over a small distance from the OD surface and rebounds to an almost uniform 0.05 volume fraction in the 10mm to 75mm region from the ID surface. The transition region of equiaxed grains at the casting surface to columnar grains just inside the surface is a typical phenomena observed in castings. However, in this case, the rebound of the equiaxed white iron phase is undesirable. In fact, suppressing equiaxed white iron growth would result in a microstructure profile that was considerably closer to the measured result in Figure 6.24b. The results shown in Figure 6.24 clearly indicate that the extension of the microstructure model to predict columnar growth has brought the model closer to predicting the observed microstructure. 150 6.3.2.2 Columnar and Equiaxed Quik-cup Casting Predictions Application of the columnar and equiaxed model to the l/8th section of the Quik-cup geometry was not appropriate because of the variations in the mesh, shown in Figure 6.1. A XA section mesh of the Quik-cup geometry was developed with element lengths approximately 2.5mm in each direction. The microstructure profiles of the equiaxed and columnar model predictions for the Quik-cup casting are presented in Figure 6.25. The profiles plotted in Figure 6.25 were generated along the diagonal from the center to the corner of the casting. Based on the previous discussion of the columnar formulation, the errors in these microstructure predictions are likely great because of the multi-dimensional heat flow in this small casting. In Figure 6.25a, the gray iron profiles indicate that the columnar and equiaxed model predicts a lower volume fraction of gray iron. The low gray iron profile is more consistent with the measured profile. The white iron profiles in Figure 6.25b show a similar trend. As in the reduced scale roll casting columnar predictions, there is a region of equiaxed white iron at the surface of the Quik-cup which rapidly decreases and is replaced by columnar white iron growth. The phenomena is consistent with the observed region of random oriented growth at the surface of the Quik-cup casting (refer to Figure 4.23). 6.4 Summary The models formulated in Chapter 5 were used to predict the evolution of temperature and microstructure in the Quik-cup and reduced scale roll castings. Prior to presenting 151 the results, the geometry, boundary conditions and initial conditions employed for each casting were summarized. The Quik-cup casting was used to fine-tune the nucleation and growth parameters needed to describe the microstructure evolution in the present cast iron system. The carbide eutectic temperature was raised 9°C to facilitate the growth and nucleation of white iron. The Quik-cup casting thermal and microstructure predictions agreed with the measured data. These predictions were used as a baseline for a sensitivity analysis of key thermal and microstructural parameters. As expected, the microstructure parameters had the greatest effect on microstructure predictions, while variations in the thermal parameters dominate the temperature predictions. The temperature predictions for the reduced scale roll casting showed satisfactory overall agreement with the measured temperature evolutions. The preheat temperatures were predicted in an initial stage without the casting active in the model. The results of this section were satisfactory, although the observed temperature gradient in the chill mould was under-predicted. An incremental method of initiating heat transfer from the casting to the mould was used to simulate mould filling. Following mould filling, the solidification and long term cooling stages showed satisfactory agreement with the measured temperature data. The microstructure model in the reduced scale roll casting over-predicted the evolution of white iron for a large region extending from the ID surface. At the OD surface, the measured white iron plateau was not predicted. Based on these results, an extension to the microstructure model was proposed to described columnar white iron 152 growth. Once formulated, the columnar growth model provided better agreement with the measured results. However, the white iron plateau could not be predicted. The columnar and equiaxed model also improved the microstructure predictions in the Quik-cup casting. The following chapter deals with the calculation of the stress field based on the thermal and microstructure predictions for the reduced scale roll casting. Table 6.1 - Nucleation and growth kinetics parameters for microstructure model. A (m-3K-2) (m s"'K"2) Austenite 5.00x10* 9.0xl0"7 Gray Iron 12.86xl06 1.6xl0"8 White Iron 12.86106 3.0xl0"7 Table 6.2 - Mesh description for the reduced scale roll casting model. Material Section Number of Number of Elements Nodes Casting 3084 7886 Cast Iron Mould 632 2132 Bonded Sand 558 1905 153 Table 6.3 - Summary of the microstructure and thermal parameters investigated in the sensitivity analsyis and their effects on the fraction gray and the duration of thermal recalescence at the Quik-cup casting center. Parameter Description Max / Min Max / Min Relative Change (reference value) Variations Values Af g(%) At r e c (s) Microstructure Parameters Nucleation Coefficient, A 2A 25.7E+06 7 1 (A = 12.8E+06m'3K"2) 0.5A 6.4E+06 -8 -1 Gray Growth Coefficient, u.g 2-fig 3.2E-08 28 2 (H g= 1.6E-08 ms"'K"2) 0.5-u,g 0.8E-08 -32 -1 White Growth Coefficient, u.w 2-llw 0.6E-06 -14 -1 (u.w = 0.3E-06 ms"'K"2) 0.5-p:w 0.15E-06 17 3 Carbide Eutectic Temperature, T c a r b T curb + 3 ° C 1131 -27 -4 (7/c.urf, = 1128°C) T curb ^ 1125 22 5 Thermal Parameters Initial Temperature, Tinit Tint, +25°C 1250 5 8 (Tmit = 1225°C) T -25°C 1200 -3 -10 Nominal Gap HTC, h Rap 2-h n gap 3000 -2 -9 (hgap = 1500Wm"2K"') 0.5-hgap 750 15 16 Gap Formation Temp., Tmin T + 50°C -* min ' J V J v-1000 2 4 (Tmn =950°C) T -50°C x mm ^ 900 -1 0 Radiative Emissivities, f 2-£ 0.75, 0.69 -3 -7 (£ c a s t = 0.5, esand = 0.46) 0.5-£ 0.25, 0.23 3 9 154 Figure 6.1 - Quikcup casting (light portion) and mould (dark portion) simplified to a 1-8th section and meshed with 8-node brick elements. Figure 6.2 - Measured temperature response near the Quik-cup casting center and various carbide eutectic temperatures. 155 u Center 5 mm from Center Omm from Center Measured 50 100 Total Time (sec) Figure 6.3 - Measured and predicted temperatures at the Quik-cup casting center. -a o oo e o 1 r -0.9 ' -0.8 7 0.7 | -0.6 ' -0.5 ~-0.4 7 0.3 ' -0.2 r 0.1 ° o L Austenite Gray Iron White Iron Total / / / / - - I 50 100 Total Time (sec) 150 Figure 6.4 - Predicted phase evolution with time at the Quik-cup casting center. 156 1200 y 1150 C3 I" 1100 H 1050 Corner Predicted Temperature 1000 I I I I I I I I I I I I I I I I I I l _ l I i i i i I 0 1 2 3 4 5 Total Time (sec) Figure 6.5 - Predicted temperature response at the bottom corner of the Quik-cup casting. o c o PH 0.9 7 0.8 7 0.7 7 0.6 r 0.5 7 0.4 7 0.3 7 0.2 7 0.1 7 0 t— 0 / /' / / Austenite — — — - Gray Iron White Iron Total / / / ' / / / / / / / i I i i i i I i i i ' I ' ' ' ' I ' ' ' ' ' 1 2 3 4 5 Total Time (sec) Figure 6.6 - Predicted phase evolution with, time at the bottom corner of the Quik-cup casting. 157 Shrinkage Porosity Distance from center (mm) Figure 6.7 - Comparison of predicted and measured fractions of gray along the diagonal from the Quik-cup casting center to the bottom corner. Note: Measured results were converted from hardness measurements. 1240 -1220 1200 - \ 1180 1-1160 H 1140 | 1120 |— <u g- 1100 | H 1080 h 1060 -1040 -1020 -1000 0 Austenite Formation Temperature Gray Iron Formation Temperature White Iron Formation Temperature Predicted Temperature J I I I L. 50 100 Total Time (sec) 150 Figure 6.8 - Austenite, gray iron eutectic, and white iron eutectic phase formation temperatures variation with time and the predicted thermal response at the center of the Quik-cup casting. 158 c o c u o c o U 4 h 2 h 1 h Carbon Concentration Silicon Concentration Phosphorus Concentration _L -I I I L _L 50 100 Total Time (sec) 150 Figure 6.9 - Variation of liquid concentration (C, Si, P) in wt% with time at the Quik-cup casting center. c u Q c o o 3 10"' H 10" h 10 Gray Nuclei White Nuclei Corrected Nucleation Profiles Uncorrected Nucleation Profiles _L 10 20 Distance from Center (mm) 30 Figure 6.10 - Predicted Gray and white iron nucleation profiles with and without correction. 159 90Q t I I I I I I I I ! I I I I I I I 1 0 50 100 150 Total Time (sec) b) Figure 6.11 - Sensitivity analysis of the gray growth coefficient showing: a) the resulting variations in the profiles of volume fraction gray iron and b) the temperature evolutions at the center and corner of the Quik-cup casting. Max / min variations are shown relative to the reference result. 160 Total Time (sec) b) Figure 6.12 - Sensitivity analysis of the nominal gap heat transfer coefficient showing: a) the resulting variaitons in the profiles of volume fraction gray iron and b) the temperature evolutions at the center and corner of the Quik-cup casting. Max / min variations are shown relative to the reference result. 162 Figure 6.14 - Temperature prediction for the preheat plotted as a) schematic showing predicted and (measured) temperatures in °C and b) a contour plot of the predicted temperatures. o. E 1 100 Axial Height Relative to Base 0.2m 0.3m Bottom TC's Mid T C s - 1.25m I _i i I 5420 5440 5460 5480 Total Time (sec) Figure 6.15 - Temperature evolutions during pouring stage at various axial locations along the reduced scale roll casting OD surface. Figure 6.16 - Contours of: a) temperature and b) phase fraction solidified predictions in the reduced scale roll casting after 1800s of solidification time. 164 Predicted Measured Location , Total Time (sec) b) Figure 6.17 - Comparison of the measured and predicted temperature evolution at various locations within the reduced scale roll casting for: a) mid-height and b) bottom height locations. 165 Figure 6.18 - Comparison of the chill mould measured and predicted temperature evolutions for the reduced scale roll casting at: a) mid-height and b) bottom height locations. 166 Predicted b) u 6 Measured Location o Casting OD • 20mm from Casting OD A 55 mm from Casting O D X Bonded Sand Core OD 10000 15000 Total Time (sec) 20000 25000 Figure 6.19 - Long term cooling comparison of the measured and predicted temperature evolution at various locations within the reduced scale roll casting for: a) mid-height and b) bottom height locations. 167 o. S V Q Q O O O Q Q Predicted Measured Location V O Chill Mould ID Chill Mould OD j i I i_ _j i i i_ 10000 b) 15000 20000 Total Time (sec) 25000 Figure 6.20 - Long term cooling comparison of the chill mould measured and predicted temperature evolutions for the reduced scale roll casting at: a) mid-height and b) bottom height locations. 168 Radial Distance from ID Surface (mm) Figure 6.21 - Measured and predicted microstructure profiles of the reduced scale roll casting. Stage 0 Tref > Tcarb No Columnar Growth Stage 1 Tref < Tcarb Columnar Growth Starts Stage 2 Tref < Tcarb Columnar Front Reaches Reference Point Reference Point Stage 3 Tref < Tcarb Columnar Growth Finishes Direction of Heat Flow Direction of Columnar Growth Figure 6.22 - Schematic of columnar growth model showing different stages in the growth of a columnar front. 169 Case 1 Columnar front growing perpendicular to the element face. >} >) >) T, >) Case 2 Columnar front growing at an angle to the element face. >) >; >} >) Growth Direction Growth Direction Increments in fraction transformed calculated correctly. Increments in fraction transformed over predicted. Figure 6.23 - Schematic of the effects of columnar front growth direction. 170 Figure 6.24 - Microstructure profiles of the reduced scale roll casting comparing the original equiaxed and extended equiaxed / columnar model predictions to the measured data, as: a) gray iron microstructure profiles and b) white iron microstructure profiles. 171 Figure 6.25 - Microstructure profiles of the Quik-cup casting comparing the original equiaxed and extended equiaxed / columnar model predictions to the measured data, as: a) gray iron microstructure profiles and b) white iron microstructure profiles. 172 CHAPTER 7 PRELIMINARY RESIDUAL STRESS ANALYSIS OF THE REDUCED SCALE ROLL CASTING Significant effort in this work has been directed toward developing a model capable of predicting the evolution of microstructure in cast iron rolls. In this particular application, one of the key issues is that microstructure has a bearing on the residual stress distribution. For this reason, a preliminary stress analysis of the reduced scale roll casting has been completed and is presented in this Chapter. The analysis is preliminary in the sense that constitutive behaviour for the material used in this analysis is based on crude data available in the literature. Moreover, only a limited amount of residual stress measurements are available for validation. Further study and quantification of cast iron material behaviour and residual stress measurements would be required to complete this topic, which is beyond the scope of the present research. The completion of the thermal and microstructure analysis provides the input required to perform a stress analysis and predict the evolution of residual stress in the reduced scale roll casting which develop during solidification and subsequent cooling. The thermal model, presented in Chapter 5, included the influences of the chill.mould, bonded sand core and mould base on the evolution of temperature and microstructure in the reduced scale roll casting. In the stress model, the influence of the mould and core have been assumed to be negligible. Thus, the stress analysis need only consider the cast iron roll, greatly simplifying the problem. 173 The development of stresses and strains in the reduced-scale roll is complex owing to a number of factors including the large range of temperature experienced in the casting process, as well as, the positional variation in phase content and morphology that develops in the cast iron. Initially, the cast iron starts as a liquid, incapable of supporting shear stresses. As it begins to solidify and become coherent, stresses can begin to develop starting first in the solidified shell. Ideally the stress analysis should involve only the solid material and should neglect the liquid as it is incapable of supporting a load. However, in the present analysis methodology, the entire domain is analyzed (complete mesh) for each increment in time. Thus, at the beginning of the casting process the liquid is included in the equilibrium force balance. The approach adopted in the present work is to try and minimize the material 'stiffness' and therefore, the load carried by the liquid by using a temperature dependent elastic modulus as described in Chapter 5 (refer to Figure 5.4). Implementation, yields an elastic modulus of O.lMPa for the liquid which is ramped up to phase dependent value at the solidus temperature. In addition to the liquid to solid transformation, large variations in the constitutive behaviour of the solid material will be experienced as it cools from the solidus temperature to room temperature. At temperatures above approximately half the melting temperature, cast iron will exhibit the ability to creep when under load and will flow at a strain rate dependent on the applied load. Within the roll, the cast iron will be subject to a large variation in strain rate from OD to ID due to the associated variation in cooling rate. Thus, it may be anticipated that the OD material will tend to be relatively 'stiff owing to the high strain rates it experiences during initial quench and the ID material will tend to be relatively 'soft' owing to the low strain rates and cooling rates at the center. In 174 the absence of the necessary data, the approach adopted in the present work is to assume that the material behaves as an idealized elastic-plastic material with temperature dependent elastic modulus, and temperature dependent yield stress and hardening curves. This approach ignores any effect that strain rate will have on the 'stiffness' of the material and, for this reason, the present work should be considered qualitative. As discussed in the preceding chapter, the cast iron microstructure within the roll varies from predominantly columnar white iron at the OD surface to predominantly equiaxed gray iron in the interior. The variation in microstructure is input directly from the thermal model. The morphology of the solidified microstructure directly influences residual stress through the orientational variations in the components of thermal strain. As discussed in Chapter 5, the columnar white iron that solidifies in these roll castings exhibits anisotropic thermal expansion behaviour (refer to Figure 5.5). Consequently, the white iron microstructure contracts preferentially in the radial direction. The equiaxed gray iron microstructure, exhibiting no thermal strain anisotropy, will contract evenly in each direction. The predicted evolution of residual stress in the reduced scale roll casting will be discussed and compared to measurements following a description of the mesh, boundary conditions and initial conditions. Also, a sensitivity assessment was performed to determine the influence of the microstructure and the inelastic deformation on the resulting residual stress and strain distributions. A statement of the incremental strain components and their dependencies as implemented in the stress model for this study is presented in Appendix C section C.2. 175 7.7 Mesh The same reduced scale roll casting mesh employed in the thermal and microstructural analysis, shown in Figure 7.1, was utilized in the stress analysis to facilitate direct input of the predicted temperatures and microstructure distribution. The mesh of the roll, consisting of 4-node linear axisymmetric elements, was assigned cast iron mechanical properties based on those presented by Beer[62] (refer to Figures 5.4 to 5.7). The associated variation in thermomechanical properties (a, E, Oy.s.) have been input to ABAQUS through a series of tabulated data sets. For temperatures and microstructure contents not expressly given in the tabulated data, ABAQUS linearly interpolates to determine the properties based on the temperature and microstructure at the integration points. 7.2 Boundary Conditions Strictly speaking, there are no boundary conditions required for the axisymmetric analysis of the reduced scale roll casting. Practically, however, without a single boundary condition, ABAQUS reports warnings of problems associated with an ill-defined problem. To avoid these warnings, the ID surface node at the base of the roll was fixed in the axial direction, while being free to move in the radial direction (refer to Figure 7.1). Further boundary conditions were not required since the roll is allowed to contract freely, neglecting any possible contraction resistance caused by the bonded sand core. The contraction of the roll throughout cooling ensures that the OD surface of the roll will not come in contact with the mould ID surfaces. It should be noted that there is a 176 slight expansion during the pearlite transformation, but this local expansion is not large enough to counter the previous thermal contraction (refer to Figure 5.6). 7.3 Initial Conditions The reduced scale roll casting was assumed to be strain free at the beginning of the stress analysis. This is a good assumption since the stress analysis commences with the reduced scale roll casting as liquid cast iron. The pour temperature, 1250°C, of the thermal model was utilized as the initial temperature condition. The white iron phase fraction, at each node in the mesh, is also supplied as an initial condition. 7.4 Residual Stress Predictions The columnar-equiaxed model microstructure results were employed as input for the residual stress analysis of the reduced scale roll casting because these represent the best microstructure predictions when compared with measured results. Initial discussion of the predicted residual stress focuses on the evolution of the axial components of stress and strain. After understanding the evolution of residual stress in the roll, the final radial profiles of radial, tangential and axial components of stress and permanent strain are compared. 7.4.1 Residual Stress Evolution In this section, the evolution of axial residual stress will be examined at the nodes located at the JD surface, mid-radius, and OD surface at the mid-height in the reduced scale roll casting. In order to understand the complex evolution of the axial residual stress at these 177 locations, it is important to first understand the evolution of temperature and thermal strain plotted in Figure 7.2. This discussion provides the insight required to understand the evolution of stress and elastic strain, plotted in Figure 7.3. Finally, the accumulated plastic strain is examined in Figure 7.4. Combined analysis of these plots, in this order, provides insights into the evolution of residual stress during the solidification and extended cooling of the reduced scale roll casting. Prior to discussing the evolution of residual stress, it is important to reiterate the factors which contribute to its formation. Microstructural variations contribute to residual stress because of differential thermal strains that evolve due to differences in individual phase thermal contraction behaviour, as well as, differences in the amount of expansion associated with the austenite to pearlite transformation. The accumulation of differential amounts of in-elastic strain has a significant influence on residual stress evolution. It is also important to note that, since yield stress is microstructure dependent, microstructure will also influence in-elastic strain accumulation. In the absence of these variations in microstructure and in-elastic strain there would be no residual stress developed in the roll. However, in the reduced scale roll casting, the evolution of variations in microstructure and in-elastic deformation result in complex residual stress distributions. The driving force for residual stress formation during the solidification of the reduced scale roll casting is the evolution of temperature, shown in Figure 7.2a. Two distinct stages of temperature evolution are distinguishable in Figure 7.2a. The first stage, including the initial OD surface quench at time equal to Os and the final roll solidification at time equal to ~2000s, is characterized by high thermal gradients. 178 Thermal contraction, at each position in the roll, begins after the temperature cools below the solidus temperature, assumed to be 1125°C. This material behaviour combined with high thermal gradients in the roll result in a time-delay in thermal contraction at the radial locations, as shown in Figure 7.2b. In the second stage of cooling in Figure 7.2a, the thermal gradients decrease and the austenite to pearlite transformation occurs as long term cooling continues. In Figure 7.2b prior to the austenite transformation, the decrease in thermal gradients causes a decrease in difference between OD and ID surface thermal contraction. The austenite decomposition temperature of 725°C is reached by the OD surface at approximately 3900s, followed by the mid-radius at 5100s and the ID surface at 5350s. Figure 7.2b shows the corresponding expansion of the roll at each of these times as well as the differential amounts of expansion resulting from the variation in microstructure at these locations. As previously discussed in Chapter 5, the lower fraction of austenite in white iron compared to gray iron results in smaller expansion during the pearlite transformation. Thus, the amount of thermal expansion at each radial location increases with decreasing radius from the OD surface. Following the austenite transformation, thermal contraction continues during long term cooling. Although Figure 7.2a indicates low thermal gradients along the roll radius, the austenite to pearlite transformation has resulted in the large strain gradient shown in Figure 7.2b at 15000s. This large differential strain is a long term effect associated with the microstructure variation throughout the roll and is an important factor in the evolution of residual stress. The evolution of stress and elastic strain in the axial direction are compared in Figure 7.3. Overall, Figure 7.3 shows the direct relationship between stress and elastic strain. The strain curves, scaled by the elastic modulus, have the same characteristic 179 variations as the stress curves. The specifics of these characteristic variations will now be discussed. Considering the initial stage of solidification (the first 400s), the OD surface of the casting is quenched causing it to contract relative to the inner section of the roll. This contraction leads to peak tensile stresses at the OD surface (refer to Figure 7.3a) which result in yielding and the accumulation of tensile plastic strains, as shown in Figure 7.4. In Figure 7.3a, the OD surface tensile stress result in compressive stresses at the mid-radius and ID surface of the roll. Liquid cast iron, defined for model temperatures above the austenite liquidus temperature, was formulated as an elastic material with a low modulus of elasticity (~lMPa). Thus, prior to the liquidus temperature, no plastic strains are generated (refer to Figure 7.4) and the low elastic modulus results in low levels of compressive stress (refer to Figure 7.3a) at the mid-radius and ID surface locations. As discussed previously, the liquid should remain in a zero state of stress. The low elastic modulus used in the model for the liquid is meant to approximate this behaviour and it appears to provide an adequate result. During the initial stage of solidification, the mid-radius and ID surface begin to solidify at the austenite liquidus temperature of 1172°C. Between the liquidus temperature and the solidus temperature of 1125°C, the material properties are linearly ramped from the elastic liquid properties to the temperature and phase dependent properties presented in Chapter 5. As shown in Figure 7.4, the liquid to solid transition coincides with the ability of the solid to accumulate in-elastic strains. 180 The incremental contraction of the reduced scale roll casting from the OD to the ID surfaces cause compressive stresses throughout the interior molten portions during solidification. As solidification proceeds towards the ID surface, the differential thermal contraction of the roll gradually releases the tensile stress built up in the OD portions of the casting (refer to Figure 7.3a). Once solidification is complete at ~2000s, the stress state is near OMPa. The decrease in thermal gradients, following solidification, result in the reversal of the stress states placing the OD surface in compression and the LD surface in tension. The beginning of the austenite to pearlite transformation, resulting in the expansion of the OD surface prior to the roll interior, causes an increase in compressive stress at the OD surface and a corresponding yielding in compression, as shown in Figure 7.4. The time-delay in the austenite decomposition expansion, as well as the.increase in the amount of expansion throughout the interior of the roll, releases the majority of the stress built up in the roll (refer to Figure 7.3). At 7700s, the end of the austenite decomposition, the axial stress state in the roll is 23MPa at the OD surface, -19MPa at the mid-radius, and lOMPa at the ID surface. Subsequent cooling from this time, results in increased compressive stresses at the surface of the casting because of the thermal contraction differences between gray and white iron (refer to Figure 5.6). At room temperature, the final residual stress at each of the locations followed in this discussion are: -94MPa at the OD surface, -5.5MPa at the mid-radius, and 66.5MPa at the ID surface. 7.4.2 Residual Stress Profiles The final radial profiles of stress and plastic strain are presented in Figure 7.5 for the mid-height location. In Figure 7.5a, the tangential and axial stress profiles show a 181 maximum compressive stress at the OD surface of the roll. This compressive peak at the surface is directly related to the tensile plastic strains that developed at this location during the initial cooling of the roll. The compressive axial and tangential stresses at the OD surface decrease with distance from the OD surface. At 25mm from the OD surface, the axial and tangential stress profiles exhibit a sub-surface tensile peak. A decrease in the axial and tangential stress profiles follows the tensile peaks and leads into a gradual increase in tensile stress to the nominal ID surface stress. The plastic strain profiles in Figure 7.5b show a continuous change in axial and tangential plastic strains across the radius of the roll. Current design practices for cast iron rolls in the paper industry assume a gradual variation in the axial and tangential stress changing from compressive stress at the OD surface to tensile stress at the ID surface. These profiles do not include the sub-surface tensile peak found in this study. It is likely that the presence of a sub-surface tensile peak would decrease the margin of safety designed into these rolls. The predicted axial profiles of tangential and axial stress along the OD and ID surface of the reduced scale roll casting are shown in Figure 7.6. The large stress variations near the roll ends are the result of effects associated with the complex heat flow conditions and the reduction of axial stresses at the upper and lower edges of the roll. The OD surface residual stress measurements, described in Chapter 4, have been included in Figure 7.6 for comparison with the predicted stress profiles. The predicted axial and tangential stresses show good agreement with the measured data at the mid- and 182 upper height locations. The stresses predicted at the lower height location appear to be under-predicted when compared with the measurements. Causes of this under-prediction are unclear. It is worth noting, that these results were obtained with unmodified constitutive behaviour taken from the literature. It is difficult to say whether this good agreement follows from a combination of offsetting errors or from the ability of the strain rate independent yield behaviour to accurately describe the in-elastic deformation occurring in this casting. The overall comparison between predicted and measured OD surface stresses validates the stress model predictions along the roll OD surface. Validation of the sub-surface peak predicted by the model can only be accomplished by through-thickness residual stress measurements, which were not completed within the scope of the present study. 7.5 Sensitivity Analysis of Stress Model A limited sensitivity analysis of the stress model was undertaken to elucidate the impact of microstructure and constitutive behaviour on the predicted residual stress. While there are certainly more factors which could be investigated, the microstructure and constitutive behaviour represent the most likely sources of uncertainty in the stress model. 7.5.1 Microstructure Sensitivity The sensitivity of the stress model to microstructure was tested by utilizing the equiaxed microstructure model predictions as input for the stress model. The main difference 183 between this input microstructure profile and the microstructure profile from the columnar-equiaxed model is an increased white iron content within the region extending 75mm out from the ID surface. The predicted stress profiles at the mid-height location are plotted in Figure 7.7a (thick lines) along with the previously presented predictions (thin lines) for comparison. The overall effect of the equiaxed model microstructure was to increase the peak stress levels at the OD and ID surfaces. The plastic strain profiles at the mid-height location, shown in Figure 7.7b, indicated that the change in microstructure had no effect on the accumulated plastic strain. Thus, the stress differences observed in Figure 7.7a are caused by changes in the state of elastic strain. The differences in elastic strain result from material property differences along the radial profile associated with microstructure variations. The experimentally observed microstructure profiles in the reduced scale roll casting exhibited a plateau of white iron extending 10mm inward from the OD surface. Neither of the predicted microstructure results employed in the stress analyses exhibited this observed behaviour (refer to Figure 6.24). To assess the impact of the OD white iron plateau, the stress model was run with microstructure input modified from the columnar-equiaxed model results to exhibit the observed plateau. The modified microstructure profile is shown in Figure 7.8 with the measured microstructure profile provided for comparison. The predicted radial profiles of stress for this case are presented in Figure 7.9, with original columnar-equiaxed microstructure input stress predictions as reference. There is little difference in the compressive OD surface stresses of the two cases shown in Figure 7.9a. However, the modified microstructure profile with the white iron plateau has substantially increased the sub-surface tensile peak. Also, the high sub-surface 184 tensile peak results in lower ID surface stress levels. Similar to the previous microstructure sensitivity case, little change in the plastic strain state were observed. As before, this indicates that the change in stress state is a direct result of the material property variations across the microstructure profile. Clearly the residual stress predictions are sensitive to the input microstructure predictions. However, it is interesting to note that in each of the different microstructure sensitivity cases presented, the same OD surface compressive stress was predicted. This bolsters confidence in residual stress predictions, but also points to the need for a more complete validation of the model using through-thickness residual stress measurement techniques. 7.5.2 Material Constitutive Beh aviour Sensitivity The constitutive behaviour of gray and white cast iron is based on stress-strain behaviour of gray and white cast iron reported in the literature[62]. As discussed in Chapter 5, the reported behaviour is a series of plasticity curves for gray and white iron at different temperatures. The impact of in-elastic deformation on the development of residual stress in the reduced scale roll casting is best determined by the limiting case where the cast iron behaves elastically. In Figure 7.10, the stress and elastic strain predictions of the analysis suppressing in-elastic deformation are compared to the original predictions with in-elastic deformation. The axial and tangential residual stress profiles, in Figure 7.10a, predicted without in-elastic deformation show high tensile stresses at the OD surface which decrease with distance to high compressive stresses at the ID surface. This result is almost completely opposite to the predictions with in-elastic deformation. The 185 predicted elastic strains of the model without in-elastic deformation, presented in Figure 7.10b, indicate a significant increase in the level of stored elastic strain throughout the roll profile. As in the case of stress, this is contrary to the predictions of the model with in-elastic deformation. Overall, this comparison shows that the model predictions are sensitive to the constitutive behaviour definition employed for cast iron in the stress model. Current industrial design practices assume that the residual stress state in a roll is a result of the differential contraction behaviour between gray and white iron. This is based solely on the thermal contraction behaviour of cast iron - i.e. gray iron contracts more than white iron and therefore, the OD surface is in compression and the ID in tension (refer to Figure 5.6 without pearlite expansion). This view neglects the impact of the austenite to pearlite transformation expansion on the accumulated thermal-elastic strain. From Figure 7.10, it is clear that in-elastic strain accumulation has a significant effect on the residual stress distribution. To better assess the impact of in-elastic deformation on the residual stress state in the roll, the stress-strain data defining in-elastic deformation was perturbed from the baseline literature values by ± 20%. The radial profiles of stress and plastic strain in the axial direction for the max and min perturbations are compared with the baseline predictions in Figure 7.11. At the OD surface, the results shown in Figure 7.11 are counterintuitive. The compressive stress at the OD surface increased with yield stress because there was an increase in plastic strain. With an increase in yield stress, however, it was expected that the plastic strain would decrease. Conversely, at the LD surface, the expected results - lower tensile stress and less plastic strain - are observed. 186 In order to understand the counterintuitive OD surface stress prediction, the evolution of plastic strain at this location has been plotted in Figure 7.12. In Figure 7.12, the modification of Cys affects the initial in-elastic deformation on quenching as expected. Increases in the yield strength decrease the accumulated plastic strain, while decrease in the yield strength increase it. The austenite to pearlite transformation causes a large correction or undoing of the tensile in-elastic strain in the baseline and reduced ays cases. However, the increased Oys case exhibits no correction. The result of these changes is an increase in plastic strain for the max ays case. Although the material may be stronger, it is possible for this condition to result in more plastic strain and thus, residual stress. Figure 7.1 - Schematic of geometry and mesh used in the stress analysis of the reduced scale roll casting. 188 Figure 7.2 - Predicted evolution of the a) temperature and b) thermal strain in the axial direction at the mid height for the ID surface, Mid radius, and OD surface of the reduced scale roll casting. 189 Figure 7.3 - Predicted evolution of the a) stress and b) elastic strain in the axial direction at the mid height for the ID surface, Mid radius, and OD surface of the reduced scale roll casting. 190 0.002 i 0.0015 0.001 ' § 0.0005 | 3 01 •£ -0.0005 | < -0.001 -0.0015 h ID Surface Mid-Radius OD Surface _l_ J 5000 10000 Time Since Start of Mould Filling (sec) 15000 Figure 7.4 - Predicted evolution of plastic strain in the axial direction at the mid height for the ID surface, mid-radius, and OD surface of the reduced scale roll casting. 191 0.005 r-0.004 7 0.003 7 " 0.002 -.5 0.001 o 0 E -0.001 -0.002 -0.003 -0.004 -0.005 b) Radial Strain Axial Strain Tangential Strain _j I 1 i_ J i_ J L j 1 1 i_ 25 50 75 Radial Distance from ID Surface (mm) 100 Figure 7.5 - Predicted radial profiles of the a) stress and b) plastic strain components at the mid height of the reduced scale roll casting. 192 Figure 7.6 - Predicted axial profiles of the axial and tangential stresses along the ID and OD surfaces of the reduced scale roll casting. The measured OD axial and tangential residual stress data has been plotted as symbols. 193 Columnar + Equiaxed Equiaxed Model Model Radial Stress Axial Stress Tangential Stress -150 a) Radial Distance from ID Surface (mm) Columnar + Equiaxed Equiaxed Model Model Radial Strain Axial Strain Tangential Strain b) 25 50 75 Radial Distance from ID Surface (mm) 100 Figure 7.7 - Microstructural sensitivity of the predicted radial profiles of a) stress and b) plastic strain components at the mid height of the reduced scale roll casting. Base on the stress model predictions with microstructure input from the equiaxed and columnar and equiaxed models. 194 Radial Distance from ID Surface (mm) Figure 7.8 - Modified microstructure profile employed in the stress model sensitivity analysis. The modified microstructure profile is compared with the columnar-equiaxed model and the measured microstructure profiles. 195 150 h-1 0 0 h a* S -50 h -ioo h -150 a) Columnar Modified + Equiaxed Microstructure Model Profile Radial Stress Axial Stress Tangential Stress 7 / ; / ^ \ » w w 25 50 75 100 Radial Distance from ID Surface (mm) 0.005 r 0.004 0.003 K" 0.002 F-o.ooi o h -o.ooi --0.002 -0.003 --0.004 -0.005 b) 0 Columnar Modified + Equiaxed Microstructure Model Profile Radial Strain Axial Strain Tangential Strain J I I L . j i i _l_ 25 50 75 Radial Distance from ID Surface (mm) 100 Figure 7.9 - Influence of the white iron OD surface microstructure on the predicted radial profiles of a) stress and b) plastic strain components at the mid height of the reduced scale roll casting. Based on the stress model predictions with microstructure input from the columnar-equiaxed model and a modified profile. 196 With Without Inelastic In-elastic Deformation Deformation Radial Distance from ID Surface (mm) 0.002 With Without Inelastic In-elastic Deformation Deformation 0.001 h oh -0.001 Radial Strain • Axial Strain Tangential Strain _] I I L. _l_ _!_ _l I I L. _L 1 '• 1 •— b) 25 50 75 Radial Distance from ID Surface (mm) 100 Figure 7.10 - Inelastic deformation sensitivity of the predicted radial profiles of a) stress and b) plastic strain components at the mid height of the reduced scale roll casting. Based on the stress model prediction with and without the influence of inelastic deformation. 197 Figure 7.11 - Sensitivity perturbation of Oys influencing the predicted radial profiles of a) stress and b) plastic strain components in the axial direction at the mid height of the reduced scale roll casting. Max / min variations are shown relative to the literature based C?YS-198 Literature based 0 V Ov < !-0.8 _L J i i i_ 5000 10000 Time Since Start of Mould Filling (sec) 15000 Figure 7.12 - Sensitivity of axial plastic strain evolution at the OD surface location to perturbations in ays- Max / min variations are shown relative to the literature based ays-199 CHAPTER 8 SUMMARY AND CONCLUSIONS The research programme has focussed on developing mathematical models capable of predicting the evolution of temperature, microstructure and residual stress during the manufacturing of cast iron rolls. A series of measurements involving two industrial castings - a Quik-cup casting and a reduced scale roll casting - were undertaken to provide thermal, microstructural, and residual stress data for use in 'fine-tuning' and validating the mathematical models. A two part thermal stress model based on the commercial finite element code ABAQUS was developed to predict: 1) the evolutions of temperature and microstructure, and 2) the formation of residual stress during the solidification of hypo-eutectic cast iron. Specialized routines, employing relationships describing nucleation and growth of equiaxed primary austenite, gray iron and white iron, were formulated and incorporated into an ABAQUS thermal model through user written subroutines. The relationships used to describe microstructure evolution have been adapted and extended from a number of investigations describing equiaxed cast iron solidification in the literature. A general methodology has been proposed and implemented for dealing with multiphase growth through increments in extended volume fraction of the various phases. Further, a method of adaptive time-stepping was implemented to improve computational efficiency. The thermal and microstructure model was 'fine-tuned' and validated against the temperature and microstructure data collected from the Quik-cup castings, which contained a thermocouple embedded approximately in the center of the casting. The 200 'fine-tuning' involved systematic adjustment of the parameters describing the thermal boundary conditions, as well as the parameters needed to quantify microstructural evolution. The magnitude of the parameters resulting from this exercise were found to fall within the range of previously published values except for the relationship describing the compositional dependence of the white iron eutectic temperature. In order to fit the microstructure predictions to observed microstructure, the white iron eutectic temperature had to be adjusted upwards in temperature, presumably to account for the effect of high phosphorus levels in the melt examined. The predicted phase distribution has been compared with the measured phase distribution inferred from hardness measurements preformed on a sectioned Quik-cup casting. The overall agreement between the model predictions and the measured microstructure is good. At the center of the Quik-cup casting, the model predicts 0.40 normalized volume fraction gray iron and 0.60 white iron, which compares favorably with 0.40 gray iron and 0.60 white iron measured at the center. At the corner of the casting, both the model and measurements show 100% white iron. Analysis of the reduced scale roll casting required a significant increase in the complexity of the thermal model to describe the various stages in the casting process. Thermal boundary conditions simulating the preheat and mould filling stage were developed and implemented. Overall, the agreement between the measured and predicted temperatures during solidification and long term cooling of the reduced scale roll casting is acceptable. Likely sources for error in this area revolve around uncertainties in thermocouple location, as well as the influence of the sheathing used to protect the thermocouples. 201 The 'fine tuned' microstructure parameters, validated for the Quik-cup casting, were utilized in the analysis of the reduced scale roll casting. Comparison of the microstructure predictions with the measured microstructure profile revealed the inability of the equiaxed microstructure model to predict the evolution of microstructure under the broad range of cooling conditions present in the reduced scale roll casting. Specifically, the equiaxed microstructure model over-predicted the evolution of white iron in the interior of the roll (20% white iron predicted, but none was measured). As well, the equiaxed microstructure model failed to predict the white iron plateau observed at the OD surface of the roll. Based on the results of the equiaxed microstructure analysis of the reduced scale roll casting, the need to also predict the growth of columnar white iron in addition to equiaxed white iron was identified and a preliminary columnar growth model has been implemented in the microstructure model. A number of simplifications were necessary for this columnar growth extension, but the result is a model that is easy to implement within specific conditions of applicability. The reduced scale roll casting and the Quik-cup casting were re-analyzed with the equiaxed and columnar microstructure model. In the reduced scale roll casting, the white iron predicted in the interior of the roll decreased to 5%, representing a significant improvement. There was no improvement in the white iron predictions near the OD surface. The inability of the microstructure model to predict a white iron plateau near the OD surface is likely a result of the time-independent nucleation law employed. These microstructure predictions are an improvement over those predicted by the equiaxed model and, as a whole, satisfactory agreement with measured results has been obtained. 202 The columnar and equiaxed model predictions for the Quik-cup casting showed an increase in the white iron phase fraction at each location in the casting. The analysis of this small scale casting exemplified the limitations of the columnar model. The multi-dimensional heat flow in this casting results in an off angle growth front and consequently, white iron is over predicted by the columnar model formulated in this study. The residual stress analysis of the reduced scale roll casting utilized the temperature and microstructure predictions as input. Evolution of residual stress revealed that tensile plastic strains, formed at the OD surface of the roll during the initial stage of cooling, are a driving force in the development of OD surface compressive stresses. The other key factor, critical to the evolution of residual stress, is the distribution of microstructure because of the variation of material properties it causes. The final OD surface residual stress profile showed good agreement with residual stress measurements. The final results of this stress analysis should be considered qualitative, since the influence of strain rate dependent plasticity has not been investigated. The predicted radial profile of residual stress showed sub-surface axial and tangential tensile peaks. Not currently incorporated into current design practice, the presence of sub-surface tensile peaks will impact the operational stress state of a roll. This final result of the study shows the importance and usefulness of developing the ability to predict residual stress and microstructure evolution in cast iron rolls. 203 8.1 Recommendations for Future Work The initial application of the microstructure model to the Quik-cup casting illustrated the need for improved understanding of the influence of phosphorus on the eutectic and carbide transformation temperatures. Moreover, eutectic and transformation temperatures developed for high phosphorus cast irons would allow the influence of phosphorus segregation on the transformation temperatures to be included in the microstructure model. Furthermore, the temperature dependent nucleation law for cast iron should be expanded to be both temperature and time dependent. The inclusion of a nucleation relationship with these dependencies will improve the ability of the microstructure model to predict preferential nucleation of white iron over gray iron at the surface of a casting. The columnar microstructure extension developed for this study improved the capabilities of the microstructure model. However, the simplifications required to implement the columnar extension in ABAQUS, limit the general applicability of this model. The introduction of the thermal gradient and mesh dimensions would eliminate many of these limits. In the stress analysis, further investigation into time dependent plasticity/creep relations is needed. Although temperature dependent plasticity curves were implemented in this study, rate dependent plasticity is likely to have an influence on the in-elastic deformation occurring in the initial stage of cooling. 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"Summary of Thermal Properties for Casting Alloys and Mold Materials." NTIS Bulletin, NTIS-PB83-211003, 1982. 62. R. Beer. Multidisziplindre Optimierung von Zylinderfbrmigen Gufibauteilen mit Mehreren Zielvorstellungen. University of Siegen (1996). 63. S. Denis, S. Sjostrom, A. Simon. "Coupled Temperature, Stress, Phase Transformation Calculation Model Illustration of the Internal Stresses Evolution during Cooling of a Eutectoid Carbon Steel Cylinder." Metall. Trans., 18A (1987): 1203-1212. 64. M.R. Ozgu. "Thinner-Walled Ingot Molds: Mathematical Modeling of Thermal Stresses and Plant Trials With Lighter Big-End-Up Molds." In Steelmaking Conference, 66"', Atlanta, Ga., (1982): 27-39. Iron and Steel Society. 65. F.M.White. Heat and Mass Transfer. Addison-Wesley, 1991. 66. M.F. Modest. Radiative Heat Transfer. McGraw-Hill Series in Mechanical Engineering. McGraw-Hill Inc., 1993. 209 67. B. Dominik. Mathematical Modeling of Heat Transfer in the Vacuum Investment Casting of Superalloy IN718. M.A.Sc. diss., University of British Columbia. 68. ABAQUS/Standard User's Manual, vol. II. Hibbitt, Karlsson, and Sorenson, Inc., 1997. 69. T.W. Clyne. "Numerical Treatment of Rapid Solidification." Met. Trans. 15B (1984): 369-381. 70. C.Y. Wang and C. Beckermann. "Prediction of Columnar to Equiaxed Transition during Diffusion-Controlled Dendritic Alloy Solidification." Met. Trans. 25A(1994): 1081-1093. 71. M . M . Hamdi, M . Bobadilla, H. Combeau, G. Lesoult. "Numerical Modeling of the Columnar to Equiaxed Transition in Continuous Casting of Steel." In Modeling of Casting, Welding and Advanced Solidification Processes VIII (1998): 375-382. TMS. 210 Appendix A Thermocouple Calibration Data For Reduced Scale Roll Casting Table A. 1 - Thermocouple calibration for the sheathed type-k thermocouples used in the reduced scale roll casting. Channel Location Temperature Measured Temperature No. of Heat Source Temperature Offset 9 Bot Mid OD 500°C 505°C 5°C 10 Mid Mid OD 500°C 504°C 4°C 11 Bot Mid ID 500°C 505°C 5°C 12 Mid Mid ID 500°C 506°C 6°C 13 Bot Cast OD 500°C 507°C 7°C 14 Mid Cast OD 500°C 506°C 6°C 15 Bot Cast 20mm 500°C 507°C 7°C 16 Mid Cast 20mm 500°C 503°C 3°C 17 Bot Cast 55mm 500°C 507°C 7°C 18 Mid Cast 55mm 500°C 504°C 4°C 19 Bot Core OD 500°C 505°C 5°C 20 Mid Core OD 500°C 504°C 4°C 211 Appendix B Quik-cup Casting Thermal Analysis Figure B.l - Thermocouple response of Quik-cup casting samples. 212 Appendix C Governing Equations and Conditions Cl Thermal Equations The governing partial differential equation describing heat transfer in a casting in the presence of liquid and/or solid is: PC, dt dx dy dz ii -It, 'Ki i Ki + + Ki ' dt dx dy dz dx k + -f dT^ dy + • ^ dy) dz k— V 5 z v ( C l ) where: Cp = f(T), k = f(TfsJ) Convective heat transport was neglected in the thermal models of the Quik-cup and the reduced scale roll casting. Instead, the high temperature thermal conductivity was enhanced to simulate the effects of convection. C.l. l Quik-cup Thermal Equations Neglecting heat transport through convection Equation C. 1 reduces to: f dT^ dy d(,dT} ,+ — k— v dy 1 dz\ dz J where Q = ^ L ; -j=i KL dt where: Cp = f(T), k = f(T,fSJ) 213 C. 1.1.1 Initial Conditions :1225°C fort = Os C . l . 1 . 2 Boundary Conditions Heat transfer to the environment, applied to each external element face: ff 7 I ' 7 ' 'I' \ tfconv cnnv \ .vu// <» / Qmd^^xurf ^Fsurf~^= ) Heat transfer between the casting and mould, applied to each contacting surface: Kff = KondJcond + 0 — fcond^Kad T - T f =.J-cast ^nun / I j r y , j r J cond rp T J\im^d\\m max min Kad=<7£ejf ^FcustJrTmmi(i X^avz+^mwHW ) 1 % ~ 1 1 _ _1 cast mould C.1.2 Reduced Scale Roll Thermal Equations Neglecting heat transport through convection and expressing Equation C l in 2-dimensional cylindrical coordinates: P " dt ^ r d r v d r J + — dz where Q = ^ L y j=l tit where: C r = f(T), k = f(T,fsJ) 214 C. 1.2.1 Initial Conditions T M l m i l u l d ^ O ° C foxt = Qs Tm„ulclhuxe=20°C fovt = Os Tcasl =1250°C for t =5400* C. 1.2.2 Boundary Conditions Heat transfer to the environment, applied to each external element face: Qcanv conv \ surf «> / qrad^xirffarf-T-) Heat transfer between the OD casting surface and the chill mould, applied to each contacting surface: 9gap~^ e]j' ^Fcast^rmnld ) Kff = Kimilfcond + 0 ~ famd)Kud T -T • f -least m n w i f \ + f J cond rp T J l i m l i m max min Kad~^£eS ^Fcast^mould )(7«i.v/mould ) 1 1 cast mould 215 C.2 Stress Equations The total incremental strain is: with: . e l+V V „ d£\. = do, do^8, V J7 E de*=aildTSfl Jcin _ r i df flow d£» - d l i f l o ~ where: E = f(T, fsJ), ay = f(J, fsJ), ffll)W = f(T, i\ fsJ) Transformation strains, d£y , are incorporated as a thermal dilatation effect and the influence of transformation plasticity, d£? , has been neglected. Appendix D Quik-cup Casting Sensitivity Analysis Plots 216 0.9 '-0.8 '-O 2 0.5 tt. o > 0.3 b ) 0.2 0.1 2-A A = 12.86E6 m"3K"2 0.5*A 10 20 Distance from Center (mm) Figure D.l - Sensitivity analysis of the gray and white iron nucleation coefficient showing: a) the resulting variations in the profiles of volume fraction gray iron and b) the temperature evolutions at the center and corner of the Quik-cup casting. Max / min variations are shown relative to the reference result. 217 i r Distance from Center (mm) b ) Figure D.2 - Sensitivity analysis of the white growth coefficient showing: a) the resulting variations in the profiles of volume fraction gray iron and b) the temperature evolutions at the center and corner of the Quik-cup casting. Max / min variations are shown relative to the reference result. 218 i r T 4. + 3°C Distance from Center (mm) b) Figure D.3 - Sensitivity analysis of the carbide eutectic temperature showing: a) the resulting variations in the profiles of volume fraction gray iron and b) the temperature evolutions at the center and corner of the Quik-cup casting. Max / min variations are shown relative to the reference result. 219 i r o.9 F- T. .. + 25°C Figure D.4 - Sensitivity analysis of the initial pour temperature showing: a) the resulting variations in the profiles of volume fraction gray iron and b) the temperature evolutions at the center and corner of the Quik-cup casting. Max / min variations are shown relative to the reference result. 220 11-b) T + 50°C gap T = 950°C gap T -50°C gap 10 20 Distance from Center (mm) Figure D.5 - Sensitivity analysis of the minimum gap conductance temperature showing: a) the resulting variations in the profiles of volume fraction gray iron and b) the temperature evolutions at the center and corner of the Quik-cup casting. Max / min variations are shown relative to the reference result. 221 i r-b) 0.9 0.8 0.7 o c 0.6 .0 o a 0.5 6 0.4 > 0.3 0.2 0.1 0 2-e e , cast 0.5-e  =0.5,8 =0.46 cast ' sand 10 20 Distance from Center (mm) Figure D.6 - Sensitivity analysis of the emissivity showing: a) the resulting variations in the profiles of volume fraction gray iron and b) the temperature evolutions at the center and corner of the Quik-cup casting. Max / min variations are shown relative to the reference result. 

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