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Hot tearing and constitutive behaviour of semi-solid aluminum alloys 2007

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  HOT TEARING AND CONSTITUTIVE BEHAVIOUR OF SEMI-SOLID ALUMINUM ALLOYS by ANDRÉ BERNARD PHILLION B.Eng., McMaster University, 2002 M.A.Sc., The University of British Columbia, 2004   A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF   DOCTOR OF PHILOSOPHY  in  THE FACULTY OF GRADUATE STUDIES   (Materials Engineering)   THE UNIVERSITY OF BRITISH COLUMBIA   December, 2007 © André Bernard Phillion     ii Abstract The occurrence of hot tearing during solidification is one of the major factors influencing both the quality and productivity of aluminum castings. In order to reduce the formation of hot tears, quantitative information regarding both hot tearing formation and semi-solid deformation is essential. In this study, the mechanisms of hot tearing and semi-solid deformation have been investigated via two novel techniques: x-ray micro-tomography on material deformed in the semi-solid region, and development of a three phase microstructural model based on a geometry derived from a Voronoi diagram with rounded corners and porosity. Numerical techniques were utilized to quantify both the size evolution and orientation of internal damage relative to void growth. In order to conduct the above research, a new semi-solid tensile deformation methodology was devised which uses a two thermocouple control technique to enable accurate measurement of semi-solid tensile strength and ductility. The experimental work was conducted on the aluminum – magnesium alloy AA5182 in the as-cast and hot isostatic pressing (HIP) states. The x-ray micro-tomography technique was used to observe that semi-solid deformation is accommodated by internal damage via growth of as-cast porosity and the nucleation of new damage-based voids. As the volume fraction of damage increases, the growth of voids occurs in an orientation perpendicular to the loading direction, both through expansion within the grain boundary liquid and via coalescence between voids. The damage then localizes, causing failure. The finite element semi-solid microstructural model was used to explore the effects of fraction solid, fraction porosity, and grain size on semi-solid constitutive behaviour. The simulations revealed that increased grain size and fraction porosity lead to a reduction in flow stress for a given fraction solid. Furthermore, local strain accumulation was linked to hot tearing, since strain localizes in the liquid very early in the deformation process. Based on the model predictions, a new constitutive relationship was developed over the range 0.75 < fs < 0.95. Together, these two techniques have provided powerful new insight, such as the critical role played by as-cast porosity, on the phenomena of hot tearing and semi-solid deformation in aluminum alloys.     iii Table of Contents Abstract.................................................................................................................... ii Table of Contents ................................................................................................... iii List of Tables ............................................................................................................v List of Figures......................................................................................................... vi Acknowledgements................................................................................................. ix Co-Authorship Statement .......................................................................................x Chapter 1: Introduction ..........................................................................................1 1.1 Primary Aluminum Consolidation Processes ......................................................................... 1 1.2 Hot Tearing............................................................................................................................. 5 1.3 Semi-Solid Constitutive Behaviour ...................................................................................... 11 1.4 X-ray Micro-Tomography .................................................................................................... 20 1.5 Scope and Objective of Thesis ............................................................................................. 21 1.6 References ............................................................................................................................ 23 Chapter 2: X-ray Micro-Tomographic Observations of Hot Tear Damage in an Al–Mg Commercial Alloy ................................................................................26 2.1 Introduction and Background ............................................................................................... 26 2.2 Experimental Methodology .................................................................................................. 27 2.3 Results and Discussion ......................................................................................................... 29 2.4 Conclusions .......................................................................................................................... 36 2.5 References ............................................................................................................................ 37 Chapter 3: Quantitative Assessment of Deformation-Induced Damage in a Semi-Solid Aluminum Alloy via X-ray Micro-Tomography .............................38 3.1 Introduction and Background ............................................................................................... 38 3.2 Experimental Methodology .................................................................................................. 40 3.3 Results and Discussion ......................................................................................................... 44 3.4 Conclusions .......................................................................................................................... 61 3.5 References ............................................................................................................................ 62       iv Chapter 4: A New Methodology for Measurement of Semi-Solid Constitutive Behaviour and its Application to Examination of As-Cast Porosity and Hot Tearing in Aluminum Alloys.................................................................................63 4.1 Introduction and Background ............................................................................................... 63 4.2 Experimental Methodology .................................................................................................. 66 4.3 Results and Analysis............................................................................................................. 73 4.4 Discussion............................................................................................................................. 81 4.5 Conclusions .......................................................................................................................... 86 4.6 References ............................................................................................................................ 87 Chapter 5: Three-Phase Microstructural Model for Prediction of Semi-Solid Constitutive Behaviour in Aluminum Alloys ......................................................89 5.1 Introduction .......................................................................................................................... 89 5.2 Semi-Solid Equiaxed-Globular Grain Geometry ................................................................. 91 5.3 Finite Element Model ........................................................................................................... 97 5.4 Results and Discussion ....................................................................................................... 103 5.5 Conclusions ........................................................................................................................ 115 5.6 References .......................................................................................................................... 116 Chapter 6: A New Semi-Solid Constitutive Equation for AA5182 ................ 117 6.1 Introduction ........................................................................................................................ 117 6.2 Three-Phase Semi-Solid Microstructural Model................................................................ 119 6.3 Semi-Solid Empirical Constitutive Relationship................................................................ 126 6.4 Implementation within Industrial Process Models and Application to Other Alloys......... 132 6.5 Conclusions ........................................................................................................................ 133 6.6 References .......................................................................................................................... 134 Chapter 7: Summary and Conclusions............................................................. 135 7.1 Summary and Conclusions ................................................................................................. 135 7.2 Future Work........................................................................................................................ 137      v List of Tables Table 2A: Summary of analysis of XMT data.............................................................................. 27 Table 3A: Specimen text matrix and results of x-ray tomographic analysis. ............................... 42 Table 4A: Advantages and disadvantages of the two testing approaches. ................................... 65 Table 4B: Experimental conditions and results. ........................................................................... 67 Table 4C: Comparison of damage-related measurements via XMT in select as-cast and HIP specimens. ................................................................................................................................ 84 Table 5A: The relationship between fraction solid and temperature for AA5182 after Thompson [15]. ........................................................................................................................ 93 Table 5B: AA5182 Ludwik parameter correlations for K, n, and m [17]. .................................... 99 Table 5C: The variation in Δσ as a function of Ng, calculated from 25 different geometric configurations (fs = 0.90, d  = 150 μm). ................................................................................ 106 Table 6A: Comprehensive series of model runs conducted to predict the semi-solid constitutive behaviour of AA5182. ........................................................................................ 123 Table 6B: Variation in fs(T) for AA5182 after Thompson [24].................................................. 123 Table 6C: Results showing the effect of grain geometric configuration at ε = 0.01 on model predictions. ............................................................................................................................. 124 Table 6D: Fitting parameters for the empirically-based constitutive relationship. .................... 128      vi List of Figures Figure 1.1: Process related defects in DC casting........................................................................... 4 Figure 1.2: A typical hot tear from a rectangular ingot. ................................................................. 6 Figure 1.3: In situ observations of hot tearing in (a) succinonitrile acetone (after [42]), and (b) Al–0.5Cu (after [63]). In (a), ‘A’ is a pulling stick, ‘B’ is a grain boundary, and ‘C’ is a bridge between the two grains. In (b), the surface strain contours are shown. ........................ 10 Figure 1.4: The ductility in the semi-solid state for an Al–4.5wt% Cu alloy (after [65]). ........... 12 Figure 1.5: Comparison of semi-solid failure stress for AA5182 measured by Colley [39] and Van Haaften [70]...................................................................................................................... 17 Figure 2.1: Diagram of specimen geometry, including location of tomography and radiography scans..................................................................................................................... 28 Figure 2.2: SEM image of specimen 4, showing the presence of liquid at the crack  interface. The marker ‘A’ highlights one area where liquid was present. ............................................... 31 Figure 2.3: Radiograph images showing the axial extent of damage in specimens 3 and 4, strained to values of: (a) 0.16 and (b) 0.20. ............................................................................. 31 Figure 2.4: 3D micro-tomographic reconstruction of the damage and porosity in the specimens strained to values of: (a) as-cast, (b) 0.02, (c) 0.06, (d) 0.16, and (e) 0.20. The marker ‘A’ highlights the large hot tear from the radiograph in Fig.  2.3(a). ........................... 32 Figure 2.5: Void density distribution. ........................................................................................... 34 Figure 3.1: Interrupted tensile test specimen geometry, showing the reduced area region. ......... 42 Figure 3.2: Transverse sections from the tomographic reconstruction of specimen A showing the initial porosity in the specimen, and then the development of further damage with application of strain: (a) ε = 0, %P = 0.52 (A0), (b) ε = 0.09, %P = 3.22 (A1), and (c) ε = 0.39, %P = 16.49 (A2). ....................................................................................................... 47 Figure 3.3: 3D morphology of the internal damage in specimen A observed by tomography at various levels of strain in a quarter-section of the deformed region at strain levels of: (a) ε = 0, %P = 0.52 (A0), (b) ε = 0.09, %P = 3.22 (A1), and (c) ε = 0.39, %P = 16.49 (A2) (next page)................................................................................................................................ 48 Figure 3.4: Transverse sections from the high resolution x-ray micro-tomography scans of (a) as-cast AA5182, and (b) specimen C1, ε = 0.32....................................................................... 49 Figure.3.5: The influence of tomographic spatial resolution on the void number density as a function of equivalent radius. The resolution for diameters 2 and 7 mm is 2.5 and 9 μm. ..... 53 Figure 3.6: The effect of total strain on the void number density distribution in specimen A. .... 53     vii Figure 3.7: The effect of total strain on the void number density distribution in specimen B. .... 54 Figure 3.8: Comparison of the void number density distribution for specimens A, B, and C after significant semi-solid deformation. ................................................................................. 54 Figure 3.9: Pole figures showing the evolution of the morphological texture of the voids in specimen A as a function of strain: (a) ε = 0 (A0), (b) ε = 0.09 (A1), and (c) ε = 0.39 (A2)...... 56 Figure. 3.10: The effect of strain on the orientation of the voids relative to the loading direction for specimen A. ......................................................................................................... 57 Figure 3.11: 3D morphology of the internal damage in the specimen A1 which was used for the FE analysis. Two separate voids are marked – V1 and V2................................................. 59 Figure 3.12: Contour plots of the von Mises stress at different relative depths inside specimen A1 at the onset of plastic yielding: (a) 0 μm, (b) +40 μm in y-dir, and (c) +100 μm in y-dir. The locations of voids V1 and V2 from Fig. 3.11 are also marked in (b). ............................... 60 Figure 4.1: Specimen geometry showing dimensions and locations of control thermocouples... 67 Figure 4.2: Flowchart outlining the Two Thermocouple Control Technique for conducting semi-solid deformation tests in the Gleeble 3500 thermomechanical simulator. .................... 71 Figure 4.3: An example of the temperature-time profile obtained using the 2TC technique on HIP material. The arrow at ‘A’ marks the start of deformation............................................... 72 Figure 4.4: An example of the force and dimetral change during semi-solid deformation on HIP material. The arrow at ‘A’ marks the start of deformation............................................... 72 Figure 4.5: 2D cross-sectional slices from 3D tomography scans of the undeformed material: (a) as-cast AA5182, and (b) HIP AA5182. .............................................................................. 75 Figure 4.6: Solute profiles of Mg content in AA5182 in both the as-cast and HIP conditions. ... 75 Figure 4.7: SEM Images of the fracture surface of as-cast AA5182 specimens tested using the 2TC methodology: (a) 500°C, (b) 520°C, (c) 540°C, (d) 560°C. ............................................ 76 Figure 4.8: Stress-strain curve for as-cast AA5182 at 520, 527, 535 and 545°C. ........................ 78 Figure 4.9: Variation in (a) flow stress and (b) true strain with temperature for as-cast AA5182 and HIP AA5182. ...................................................................................................... 78 Figure 4.10: 2D cross-sectional slices from 3D tomography scans of deformed specimens: (a) as-cast AA5182 (T = 560°C, ε = 0.0008), (b) HIP AA5182 (T = 560°C, ε = 0.09), and (c) as-cast AA5182 (T = 500°C, ε = 0.08). ................................................................................... 80 Figure 5.1: Schematic showing the stages of the semi-solid microstructural model; (a) Voronoi diagram, (b) fs = 0.90 with sharp corners, (c) round corners and (d) porosity........... 92 Figure 5.2: A comparison of the grain geometry with sharp edges vs. round corners. The grain nuclei, rays to the vertices, variable α and marker ‘A’ are also shown.......................... 96     viii Figure 5.3: Variation in flow stress with temperature for the experimental data from Chapter 4, and the base-line [17] and enhanced Ludwik equations; ε&  = 0.0015 s-1. .......................... 100 Figure 5.4: An example model domain, constructed using the geometric methodology outlined in Section 5.3; fs = 0.90, d = 150 μm, and Ng = 56 ................................................. 102 Figure 5.5: Effect of fraction solid on the predicted semi-solid constitutive behaviour in the range 0.75 < fs < 0.95 ( d  = 225 μm, fp = 0). ......................................................................... 104 Figure 5.6: The effect of σl on the bulk stress-strain curve predicted by the semi-solid microstructural model; fs = 0.95, T = 570°C, and d  = 150 μm............................................. 106 Figure 5.7: The effect of solid element size on the bulk stress-strain curve predicted by the model; fs = 0.90, T = 580°C, d  = 150 μm, and Ng = 56........................................................ 107 Figure 5.8: Variation in predicted tensile response of the semi-solid material as a function of geometric configuration; fs = 0.90, d  = 150 μm, and Ng = 56. ............................................. 107 Figure 5.9: Comparison of the predicted and experimentally measured semi-solid constitutive behaviour of AA5182; T = 570°C, fp = 0, and d  = 225 μm.................................................. 110 Figure 5.10: Local Strain vs. Distance (a) on the surface of a solidifying AA6111 alloy (after [19]) and (b) as calculated from the microstructure model; fs = 0.90, fp = 0, and d  = 150 μm. ......................................................................................................................................... 111 Figure 5.11: Effect of (a) grain size and (b) void area fraction on the predicted semi-solid constitutive behaviour. ........................................................................................................... 113 Figure 5.12: Finite element simulation showing strain localization between voids A and B; fs = 0.95, fp = 0.004, and d  = 150 μm. ..................................................................................... 114 Figure 6.1: Critical features of the three phase semi-solid microstructural model..................... 120 Figure 6.2: Effect of fraction solid on the predicted semi-solid constitutive behaviour in the range 0.75 < fs < 0.95 ( d  = 150 μm, fp = 0). ......................................................................... 124 Figure 6.3: Effect of grain size on the predicted semi-solid constitutive behaviour in the range 75 < d  < 300 μm (fs = 0.90, fp = 0). ...................................................................................... 125 Figure 6.4: Effect of fraction porosity on the predicted semi-solid constitutive behaviour at fs = 0.80 and fs = 0.95 ( d  = 150 μm)..................................................................................... 125 Figure 6.5: Validation of the empirical constitutive relationship via the model predictions for the case without porosity: (a) d  = 300 mm, (b) d  = 75 mm................................................ 130 Figure 6.6: Validation of the empirical constitutive relationship for the case where porosity is included at fs = 0.85 and fs = 0.95 ( d  = 150 μm). ................................................................. 131 Figure 6.7: Validation of the semi-solid AA5182 empirical constitutive via the experimental measurements from Chapter 4. .............................................................................................. 131     ix Acknowledgements This thesis was made possible thanks to the help of many colleagues, and I would like to give special thanks to the following: My supervisors, Professor Steven Cockcroft and Professor Peter Lee, for their advice and insight with respect to the problems of semi-solid constitutive behaviour and hot tearing. Furthermore, their support for conducting this project at both The University of British Columbia, and Imperial College, London, and their support in the writing papers and presenting at major conferences greatly enriched my PhD experience. Professor Daan Maijer for fruitful modeling discussions and many evenings of being entertained by the Vancouver Giants Hockey Club. My colleagues, Sujay Sarkar, Babak Raeisinia, Devashish Fuloria and Junsheng Wang, to name a few, for their hours spent on experimental help and discussions on the (often controversial) topics of materials science, sporting events, and current affairs. My friends and family, in particular my parents, for their continual wholehearted support and encouragement. My wife, Suzanne, for her loving participation with me in this great adventure called a PhD project, along paths that have taken us beyond the furthest hills and the highest mountains. Finally, I would like to acknowledge the Natural Sciences and Engineering Research Council of Canada – Canada Graduate Scholarship, the Alcan Fellowship, and the Killam Predoctoral Fellowship, for financial support during my course of study.      x Co-Authorship Statement This thesis project has been conducted with colleagues from the University of British Columbia and Imperial College, London. With the exception of my supervisors, who offered detailed advice on experimental procedures, analysis methods, and results interpretation, I am the primary contributor to the work assembled below. In Chapters 2 and 3, I performed the key deformation experiments, tomographic characterization of the samples and developed the software tools for analysis of the tomographic images. In Chapter 4, I developed the new methodology enabling accurate measurement of semi-solid tensile strength and ductility. I wrote a program allowing these tests to be conducted and I assembled the data for analysis in collaboration with my supervisors. The rank sort analysis to characterize the extent of homogenization in the as-cast and HIP materials was performed by Robert Atwood at Imperial College, London. In Chapters 5 and 6, I devised the specific technique for generation of semi-solid microstructure geometry for finite element analysis, I conducted the virtual experiments and I created the form of the new constitutive relationship. In all cases, I was the principal author of the text and the figures, and incorporated the suggestions made by my supervisors into each Chapter’s final version, which was / will be submitted for publication.   1 Chapter 1: Introduction 1.1 Primary Aluminum Consolidation Processes 1.1.1 Introduction With increased emphasis being placed on decreasing hydrocarbon-based fuel consumption and improved product performance worldwide, replacement of ferrous-based components with aluminum-based components is often proposed. While aluminum may provide significant fuel savings through its light weight, increased usage will only occur through improvements in the upstream processing stages. One area of particular concern for the aluminum industry is the casting process, since the majority of refining and recycling operations occur above the melting temperature of the base metal. This process also has a number of important technological challenges with respect to improving productivity and quality. Firstly, solidification defects such as porosity, inclusions, and hot tearing, are detrimental to quality and result in decreased product yield. These defects have proved difficult to eliminate because they form due to a myriad of metallurgical and processing factors. Secondly, the amount of downstream effort required to optimize the final material properties, such as strength and ductility, depends much on the as- solidified grain structure. Thus, the challenge is to reliably manufacture high quality aluminum components at a minimum cost. Over the past few years, there has been a number of casting process improvements, such as continuous casting via the horizontal direct chill method, simultaneously casting aluminum ingots with multiple alloy layers, and spray water cooling systems for improved dimensional control. Much of this recent advancement was aided by the development of increased computational capability. Sophisticated solidification models have been developed covering fundamental issues such as microstructure nucleation and growth [1], important phenomena such as porosity [2] and hot tearing [3], and also numerical simulations of the temperatures (e.g. [4]) and stresses (e.g. [5, 6]) occurring during the casting of commercial products. These modeling efforts, along with excellent control over process parameters afforded by computers, have improved the reliability of the casting process and also reduced costs. While computer simulations have become sophisticated and established methods of research and development, solidification experimentation has begun to lag behind. In particular, there has Introduction    2 been little experimental work aimed at in situ observation of the developing solidification microstructure of alloys, although this area is growing due to new equipment such as x-ray tomography at the European Synchrotron Radiation Facility (ESRF) in France (e.g. [7, 8]). (In situ solidification experiments can be categorized by three different techniques: organic analogues [9-14], in which direct observation is made on organic systems that seem to behave in a similar fashion to metals, confocal scanning laser microscopy [15-21], and x-ray radiography and tomography [22-28]). 1.1.2 Direct Chill Casting Process for Aluminum Ingot Production Although it has only been produced commercially since the late 1880's, aluminum has become the world's generic lightweight metal. In 2005, nearly 32 million tonnes of aluminum were produced. Annual volumetric production now exceeds all other metals combined with the exception of steel [29]. Due to its low density, high strength, and corrosion resistance, aluminum has found use in most areas of manufacturing, including transportation, electrical systems, and construction. The most common aluminum ore, bauxite, is plentiful and occurs mainly in tropical and sub- tropical areas: Africa, West Indies, South American and Australia. Bauxite is refined to the oxide alumina and is then electrolytically reduced to metallic aluminum via the Hall-Héroult process. Approximately ten percent of the world’s primary aluminum production is in Canada, owing to this country’s abundant supply of inexpensive, hydro-electric energy. For aluminum alloys, the dominant solidification technique is the semi-continuous process known as Direct Chill (DC) casting, due to its relative simplicity, low capital cost of equipment and relatively high quality level [30]. The two basic formats are: ingots for rolling applications and billets for extrusion applications, and are generally cast four to ten meters in length. The DC casting process consists of an open, water-cooled mold together with systems for metal delivery and ingot withdrawal. At the beginning of the process, the bottom of the mold is plugged by the bottom block. Superheated liquid metal is poured into the mold at a predetermined filling rate and begins to cool as the bottom block and mold conduct heat away from the liquid metal. Once the metal reaches a certain height within the mold, the bottom block and the ingot exit the base of the mold and are lowered into the casting pit. Water is sprayed from a series of holes located around the circumference at the base of the mold directly onto the cast surface to continue the Introduction    3 cooling process and to ensure that solidification occurs rather quickly. Once the ingot has reached a predetermined length, it is removed from the casting pit and sent for downstream processing. 1.1.3 Process Related Defects in Direct Chill Casting There are three major process-related defects in DC casting: hot tearing, cold cracking, and dimensional control [31]. Common locations of these defects are shown in Fig. 1.1. Hot tears and cold cracks refer to cracks that have formed in the ingot during the casting process. Hot tears occur at temperatures above the solidus. These defects are generally caused by the application of thermomechanical loading on semi-solid material exhibiting little ductility, and may be exacerbated by a lack of metal flow to feed the shrinkage associated solidification [32]. Hot tears have also been linked to high casting speeds, bottom block design, thermal gradients, alloy chemistry, and variability in cooling conditions during the start-up phase. In contrast, cold cracks form below the solidus temperature, in the typical manner either through high temperature ductile yielding or fracture [33]. Dimensional control defects include the phenomena known as butt curl, butt swell, and lateral pull-in [34]. These defects form due to differential thermal contraction between the surface and the centre of the casting, and result in geometry that is outside the allowable specifications. The current work deals with the hot tearing defect. This defect is of great concern since it usually results in rejection of the entire casting. It has been found that reducing the casting velocity decreases the risk of hot tear formation because it directly governs the strain applied to the mushy zone, and the pressure decrease in the liquid [35]. Unfortunately, low casting speeds will also increase the likelihood of cold cracks to form, and reduce productivity. Introduction    4 Butt Curl INGOT BOTTOM BLOCK CL J Cold Crack Trouser Cold Crack Centreline Hot Crack  Figure 1.1: Process related defects in DC casting. Introduction    5 1.2 Hot Tearing 1.2.1 Solidification Behaviour The solidification of alloys is a complicated phenomenon involving processes such as compound formation, phase changes, heat transfer and mass transfer. At the beginning of the solidification process, the primary solid phase nucleates, and grows in the form of grains. Once a critical fraction of solid is reached, the grains begin to interact with each other – firstly chemically and then physically by contacting and bridging to form a solid skeleton. The point of first contact has often been called the coherency point. At fraction solids below the coherency point, the solid-liquid mixture is properly called a slurry and behaves as a highly viscous liquid. At fraction solids above the coherency point, the solid-liquid mixture is usually called a mush or semi-solid and is characterized by a developing solid network surrounded by liquid. The response to load of a semi-solid is complex, and may include both creep and asymmetric constitutive behaviour [36, 37], and large ductility changes with temperature [38, 39]. Semi-solid constitutive behaviour is highly dependent on both the fraction of solid present in the mushy zone, and also the shape of the solidifying network. This behaviour can be broadly divided into four stages, described in Section 1.3, which correspond to the decreasing ability of liquid to move within the solid structure. At high fraction solid, there is a continuous and interconnected liquid film surrounding the dendrites yet the permeability of the solid network is too small for liquid to flow. Application of load may result in hot tearing if the ductility is insufficient to accommodate strain. In the final stages of solidification, considerable alloy strength and ductility develops because solid bridging has occurred between most of the grains. 1.2.2 Hot Tearing Theories Hot tears, solidification cracks, or hot cracks are names given to cracks in castings that appear to have formed at temperatures above the solidus of the alloy [40]. These defects are easily identified since their form is often that of a ragged, branching fracture with a morphology resembling semi-solid dendrites [41]. A typical hot tear found on the surface of an aluminum ingot is provided in Fig. 1.2. From previous work, it has been clearly shown that hot tears form as interdendritic openings in the last stages of solidification [42], and that the permeability of the solid network plays a large role in their initial formation [43]. Introduction    6  Figure 1.2: A typical hot tear from a rectangular ingot.  Industrially, hot tearing is commonly encountered during the casting of alloys with large freezing ranges. These alloys are particularly susceptible to hot tearing since a comparatively long time is spent in the solidification window vulnerable to hot tearing, when thin liquid films surround the dendrites. The effect of large freezing ranges on hot tearing was recently examined by Rappaz et al. [44] in a model for prediction of dendrite-arm and grain coalescence during the last stages of solidification. This work showed that large-misorientation grain boundaries, with an interfacial energy γgb greater than twice the solid-liquid interfacial energy γsl, require an undercooling for grain coalescence. In contrast, small misorientation boundaries have a much smaller interfacial energy and are able to coalescence immediately upon impingement. Grain coalescence in large freezing range alloys, with equiaxed and randomly oriented grains, will thus occur at a depressed temperature and will further lengthen the temperature range vulnerable to hot tearing. The combination of a lengthy solidification window, thermal contraction, and mechanical loading of the solid network leads to the formation of hot tears. To reduce the occurrence of hot tearing, a number of researchers have investigated the underlying theories and mechanisms. Singer and Jennings [45] performed a series of ring castings on aluminum-silicon alloys and proposed that cracking occurs under conditions of high thermal contraction in the semi-solid regime. Pellini [46] proposed a strain-controlled mechanism based on observations of hot tearing in aluminum-copper alloys. Rappaz et al. [47] Introduction    7 proposed that hot tearing is strain rate controlled since low strain rates in the mushy zone could be offset by both solid deformation and liquid flow. Feurer [48] examined the influence of alloy composition and solidification conditions on hot tearing, and proposed that this defect was a result of the inability of liquid to feed the solidification shrinkage. Warrington and McCartney [49] examined the effect of grain refining on hot tearing, and found that both columnar and equiaxed-globular grain structures lead to high hot tearing susceptibility, while equiaxed- dendritic structures were resistant to hot tears. Sigworth [50] approached hot tearing by considering that liquid surrounding grains was a stress riser on the solid network, and acts as a crack initiation site. Using experimental data from ring castings, Guven and Hunt [51] showed that hot tears initiate in thin films of liquid between grains. Nagaumi et al. [52] performed hot tensile tests on a laboratory-cast high strength Al–Mg–Si alloy with varying Fe content. The results showed that small changes in trace alloy elements will greatly affect hot tearing by altering the solubility, shape, and volume fraction of the last-to-solidify intermetallic phases. Kim et al. [53] investigated hot tearing in nickel alloys, and observed that the mechanism of hot tear formation was cooling-rate dependent with fast cooling rates promoting fracture between the primary dendrites and slow cooling rates promoting fracture along the grain boundaries. Zhou and Volek’s recent work [54], also on nickel alloys, seem to corroborate the findings by Kim et al. These authors observed that cohesion of the solid phase occurs at lower fraction solids with higher cooling rates, due to the finer dendritic structure. Suyitno et al. [55] studied the effect of casting speed and alloy composition on microstructure formation and hot tearing in an industrial setting by DC casting aluminum billets of various copper content. This work resulted in the development of a casting speed–copper concentration–hot tearing susceptibility diagram to improve DC casting shop practices. While these studies have proved insightful with respect to understanding the growth of hot tearing, and also the development of processes which are less susceptible to hot tearing, they lack insight into the underlying mechanisms controlling hot tear formation such as nucleation and early-stage growth. One recent study designed to identify hot tearing nucleation was performed by Farup et al. [42], who used organic analogues to directly observe the formation of hot tears under applied transverse load. They concluded that hot tears always occur at grain boundaries and found three nucleation mechanisms including tearing between grains, on pores caused by solidification shrinkage, and as round pores nucleating in the liquid. In a second recent study, Fredriksson et al. Introduction    8 [56] presented a thermodynamic description of hot tear nucleation. In this work, it was proposed that hot tear nucleation and growth is enhanced by the supersaturation of vacancies since these vacancies will cluster to form voids at grain boundaries or at solid-liquid interfaces. Furthermore, higher cooling rates promote vacancy coalescence due to the smaller dendrite arm spacing. After nucleation, hot tear growth would occur by a combination of vacancy diffusion and also by the decrease in free energy when stored elastic energy from thermal and mechanical stress is released as crack growth. Based on the above theories and associated experiments, a number of computational models have been developed to predict the occurrence of hot tears. These criteria generally fall into three classifications: those relating to the transport of liquid at high fraction solid (e.g. [57]); those relating to the mechanical aspects of the problem (e.g. [46]); and those which combine these features (e.g. [58]). Unfortunately, as shown quantitatively by Suyitno [59] and Phillion [60], the majority of these criteria agree poorly with industrial experience. The reason for this is twofold. Firstly, many of the criteria assume that the critical stage for hot tearing is the initial void formation and do not take into account void growth and coalescence processes. Secondly, the relationship between semi-solid microstructures and crack development is not well understood. 1.2.3 In situ Observation of Hot Tearing In situ observation of hot tearing has been reported by a number of researchers. Pellini [46] used x-ray radiography to make the first observations of hot tear formation in aluminum-copper alloys. Fredriksson [61] performed hot tensile tests inside a scanning electron microscope (SEM), showing that hot cracks occur if the alloy contains a eutectic liquid with good ability to wet the solid grain boundaries. Davidson et al. [62] recorded the formation of hot tears in an aluminum- copper alloy during solidification using a CCTV camera, and determined that hot tearing begins to occur with very small applied loads at fraction solids between 0.93 and 0.96. Mitchell et al. [63] used the same apparatus as Davidson to examine strain accumulation during solidification in a number of aluminum alloys. They determined that strain accumulation is not homogenous at the scale of the microstructure but is accompanied by a process of strain localization. Farup et al. [42] combined an optical microscope and hot stage to study hot tear formation in succinonitrile- acetone. Hot tearing observations by Davidson, Mitchell, and Farup were captured real-time, while both Pellini, and Fredriksson captured images every 30 to 60 s. Introduction    9 While these observations have provided new fundamental insights into the mechanisms of hot tearing formation, the works to date have been limited by a number of factors. Firstly, the opacity of metals limits the observation of hot tears to those appearing on the free surface. Thus, in the metallic studies, the initiation of the hot tear is probably missed. Secondly, the correlation between an organic analogue and metals is unknown, since differences exist between the two systems (e.g. the formation of last eutectic and oxides in aluminum alloys). Thirdly, it is still quite difficult to quantify precisely the stresses and strains occurring during the hot tearing event. In Fig. 1.3, two examples of in situ hot tearing observation are provided. In (a), Farup’s observations [42] of hot tearing formation in succinonitrile-acetone are shown, and in (b), Mitchell’s observations [63] of surface strain contours associated with the formation of a hot tear on the surface of an Al–0.5wt% Cu casting are shown. Although (a) provides good evidence that hot tears begin as elongated pores, the 2D geometry of the apparatus does not allow for much liquid feeding. Furthermore, it is not clear in (b) which microstructural features, such as the number of grain bridges and the local fraction liquid, enabled the hot tear to form in its particular location. Thus, new in situ techniques need to be developed that will enable better observation of the internal damage accumulation which leads to hot tearing. Introduction    10 (1) (2) (3)  (a)   (b) Figure 1.3: In situ observations of hot tearing in (a) succinonitrile acetone (after [42]), and (b) Al–0.5Cu (after [63]). In (a), ‘A’ is a pulling stick, ‘B’ is a grain boundary, and ‘C’ is a bridge between the two grains. In (b), the surface strain contours are shown.  Introduction    11 1.3 Semi-Solid Constitutive Behaviour 1.3.1 Deformation Mechanics The constitutive behaviour of the mushy zone is complex, since it depends on many factors including fraction solid, structure of the solid network, viscosity of the liquid, impurities, metallostatic pressure, and the solid-liquid interface. This behaviour has been broadly divided into four stages – mass feeding, interdendritic feeding, interdendritic separation, and interdendritic bridging – based on the permeability of the solid network [3, 40, 64]. Furthermore, these stages help to provide a broad understanding of semi-solid ductility: 1) In the mass feeding stage (high fraction liquid), both the liquid and solid are free to move, allowing for much ductility. 2) With increasing fraction solid, the configuration of the grains does not permit bulk movement yet the solid cannot bear much of the load because the liquid forms a mostly continuous film around each of the solid particles. However, there is still ductility, which is accommodated by liquid flow through the solid network. 3) As the permeability of the solid network decreases, the ductility also decreases since there is less liquid flow. However, there is still a continuous film of liquid surrounding each grain. When the permeability reaches zero, the ductility must also be close to zero since the solid cannot bear load due to the presence of the liquid films. 4) Once large amounts of dendrite interlocking occur, the material develops considerable strength and increased ductility. Magnin et al. [65] recently conducted a series of tensile experiments on an aluminum-copper alloy at various temperatures between the solidus and the coherency point. The ductility results from these tests appear to validate the above interpretation since the measured ductility increased with increasing fraction liquid, as shown in Fig. 1.4. The formation of hot tears in DC cast aluminum alloys is controlled in part by the feeding of liquid during solidification, and in part by the constitutive behaviour of the partially solidified alloy. The mushy zone mechanical properties of aluminum alloys have been investigated under compressive [37], shear [66], and tensile loading [65, 67-74]. The compressive properties of alloys have been studied extensively within the framework of thixoforming and rheocasting processes, in which the microstructure is semi-solid globular. However, the mechanical property Introduction    12 data generally does not relate well to DC cast alloys, which solidify with as-cast dendritic microstructure. While shear tests have been performed on DC cast alloys, they have typically occurred at values of fraction solid less than 0.6. Since hot tearing is thought to occur at high fraction solids, and under tensile loading, it is unlikely that the shear properties will have much effect on hot tearing. 1.3.2 Semi-solid Tensile Tests There are two basic approaches to conducting tensile tests in the semi-solid: (1) cooling from the liquid state prior to application of load (Solidification Tests) and (2) rapidly reheating standard tensile specimens machined from as-cast material to a temperature within the semi-solid phase field (Hot Tensile Tests). The first approach requires a container for the melt. This container must be carefully chosen to minimize both chemical reaction with the liquid metal, and melt–container friction forces. Furthermore, control of heat transfer is critical to ensure that the test is performed either isothermally or at a known cooling rate. In the second approach, the main challenges are to maintain the as-cast microstructure during reheating and knowledge of the Ductility Measurements C oh er en cy S ol id us Temperature (oC) D ef or m at io n (% ) 520 540 560 580 600 620 640 0.5 1 1.5 2  Figure 1.4: The ductility in the semi-solid state for an Al–4.5wt% Cu alloy (after [65]). Introduction    13 fraction liquid during testing. Furthermore, there is no liquid feeding occurring as in the case of the solidification tests, and also in the industrial process. In the text below, a number of semi- solid tensile test studies are reviewed, followed by a critical assessment of their results. Solidification Tests Of the two approaches, the solidification test method is more challenging experimentally because of the presence of significant liquid metal, and interpretation of the results. One of the first solidification test apparatus was developed by Ackerman et al. [74], who devised a semi- solid tensile tester made of two water-cooled copper cylinders which could be directly plunged into a melt. After a certain shell thickness has formed, the two cylinders are moved apart. Due to its unique design, the tensile load is applied in a direction perpendicular to the growth axis of the columnar grains, allowing measurement of the mushy zone strength. The force required to separate the two cylinders was used as the basis for determining properties for both pure aluminum and aluminum-magnesium alloys. The authors observed that the material transitioned from having appreciable strength above a fraction solid of ~ 0.95 to virtually zero strength below this value. Langlais and Gruzleski [73] measured the behaviour of semi-solid aluminum alloys under conditions that simulate the primary cooling conditions encountered during DC casting. This apparatus, named Direct Chill Surface Simulator (DCSS), consists of a solidification unit and copper chill plate attached via anchors near the melt surface to a conventional horizontal tensile testing machine. After the container is filled with liquid metal, it is rotated ninety degrees about the anchors to create the solidification sequence seen in vertical DC ingot casting. Tensile load is applied through the anchors to measure hot tearing susceptibility. Magnin et al. [65] measured the ductility of an Al–4.5wt%Cu alloy at low fraction solid by casting specimens into an insulated mould between two jaws, one of which could be moved to apply a strain once the test temperature was reached. To approximate the cooling conditions during DC casting, the chill plate could be removed, simulating the mould and air gap heat transfer zones. One feature common to these results is that only the force and displacement data is reported. The constitutive behaviour cannot be calculated directly because the relatively high thermal gradients and cooling rates make it difficult to ascertain the current fraction solid, and because the actual load carrying area is not known. These apparatus are thus good for inferring general Introduction    14 hot tearing susceptibility, but require further improvement for measuring the constitutive behaviour. Larouche et al. [75] tried to address this issue by adding strain gauges to the cast surface of the DCSS unit at specific temperatures, allowing for failure strains to be reported. Hot Tensile Tests A reliable testing machine for tensile tests of reheated aluminum alloys above the solidus was first reported Singer and Cottrell [67] in 1947. The setup consisted of a short sample oriented horizontally, completely surrounded by the tensile test grips in order to prevent shape distortion, inside a furnace. The results showed a smooth decrease in tensile strength with increasing temperature in the fully solid region. In the semi-solid region, the strength of the sample decreased to zero at temperatures between five to sixty degrees above the solidus. In the 1970’s, Weinberg [68] studied the high-temperature mechanical properties of steel alloys, and found results similar to those found by Singer and Cottrel [67], except that in many cases the fracture occurred below the equilibrium solidus predicted by the iron-carbon phase diagram. For steels containing 0.05 to 0.12 percent carbon, it was suggested that this result was due to incipient melting at the grain boundaries. Other researchers have explained the brittle failure of steels at high temperatures near the solidus by the difference in the micro-segregation of sulphur and phosphorus between the δ and γ phases [50]. Nakagawa et al. [71] also measured the high temperature tensile properties of carbon steels, by equipping a tensile test machine with an induction furnace. The results from Nakagawa’s tests seem to conflict with those found by Weinberg [68], since the steels tested retained strength until the fraction solid was approximately 0.67–0.7. No explanation for this discrepancy was provided, although one difference was that Weinberg performed his study in an argon environment while Nakagawa appears to have performed his experiments in air. More recently, a number of semi-solid tensile test apparatus have been developed for aluminum alloys. Twite et al. [69] used a Gleeble 1500 thermomechanical simulator connected to a low-force mechanical testing machine to examine the semi-solid tensile properties of three AA6061 alloys from different sources – laboratory DC cast, commercially DC cast, and cast via thixoforming – at different deformation rates. The strain was measured lengthwise based on a central gauge section of 20 mm length with the assumption of minimal thermal gradients in that section. The results showed that tensile strength and ductility of the laboratory DC cast and commercially DC cast specimens were similar. The measured strength of the thixoformed alloy Introduction    15 was lower than the other two, while its ductility was highly dependent on strain rate. In the DC cast alloys, the strength dropped to zero around 570°C, while in the thixoformed alloy, the strength dropped to zero around 560°C. From the subsequent SEM analysis of the fracture surface, it was suggested that the tensile property differences were dependent on the solid grain morphology, the distribution and continuity of the liquid phase, and the deformation rate. Colley et al. [39] also used a Gleeble 1500 thermomechanical simulator connected to a low- force mechanical testing machine to investigate the semi-solid constitutive behaviour of AA5182. In this work, a diametral strain measurement was used because it was difficult to eliminate the lengthwise thermal gradients within the specimen. The results were found to be in qualitative agreement to Twite et al. [69], with the temperature of zero ductility occurring at ~ 560°C, and the temperature of zero stress occurring at ~ 580°C. Van Haaften et al. [70] conducted tensile tests on semi-solid, as-cast AA3104 and AA5182 using a Gleeble 3500 thermomechanical simulator. Specimens were machined from a DC cast ingot ensuring that the direction of tensile loading was parallel to the casting direction. Strain rate sensitivity was observed at all temperatures up to 550ºC, above which the tensile strength became very small. Kron and Fredriksson [72] studied the tensile properties of AA6061. A radiant mirror furnace with three ellipsoidal mirrors was mounted on the tensile test machine, providing a limited heating zone and non-contact heating. Before testing was carried out, specimens were heated to 30°C below the liquidus, and then allowed to cool to the test temperature. This heating and cooling step promoted partial in situ solidification just before the application of load, in contrast to other hot tensile tests which have simply heated the specimen to the test temperature and then conducted the test. It was observed that using this apparatus and testing procedure, the transition to zero ductility occurred at temperatures twenty to thirty degrees lower than the transition temperatures measured by Twite et al. [69]. Kron proposed that other studies thus overestimate the maximum semi-solid strength of alloys. Unfortunately, the microstructure and fracture surface of these new specimens was not presented for comparison to other work. Fabrèque et al. [76] recently conducted a comparative hot tensile study on AA6056 in which some specimens underwent partial in situ solidification before testing, while others were simply reheated to the semi-solid test temperature. The partial solidification tests showed a significantly lower failure stress as compared to the reheating semi-solid tests, although the trends of strain Introduction    16 rate and fraction solid were similar. The authors proposed that this difference in constitutive behaviour was caused by the presence of continuous liquid films surrounding the grains in the partial solidification tests, and discrete pockets of liquid in the remelting tests. Micrographs of both types of samples were also presented, corroborating their hypothesis. In comparing the different hot tensile test apparatus, several observations can be made. Firstly, specimen orientation will affect the results, since horizontally loaded specimens will also be acted upon by gravity and semi-solid flow stresses are small. Specimens tested by Magnin [65], Singer [67], Van Haaften [70], and Fabrègue [76] were oriented horizontally, while tests by Colley [39], Twite [69], and Kron [72] were oriented vertically. Secondly, the choice of strain measurement is important. Colley [39], Magnin [65] and Van Haaften [70] determined a local value of strain by calculating the diametral strain at the location of minimum cross-section along the specimen gauge length. In contrast, Singer [67], Kron [72], Twite [69], and Fabrègue [76] calculated an average strain based on the change in gauge length. In normal tensile tests, where small changes in temperature have only a small effect on the flow stress, the lengthwise strain is appropriate. In semi-solid testing, the change in flow stress with temperature is significant and thus a diametral strain is most appropriate. Thirdly, void formation may be an issue. Although the definitions of stress and strain used in the above experiments assumes that large internal voids form only at high strain, preliminary work as part of this research has shown that much of the strain is accommodated by the formation of internal damage or voids. Thus, both the stress and strain results may be underestimated. One of the main challenges in determining semi-solid constitutive behaviour is reproducibility of the test results. Both Colley [39] and Van Haaften [70] have measured the tensile semi-solid behaviour of AA5182. The results of both experiments for strain rates of ~ 10-4 and 10-3 are shown in Fig. 1.5. The data is qualitatively similar since the flow stress decreases with increasing temperature and the temperature of zero stress occurs around 580°C. However, the stresses measured by Colley below the temperature of zero ductility are two to three times larger than those measured by Van Haaften. Similarly, both Kron [72] and Twite [69] have published different semi-solid property data for AA6061. Colley [77] found that there was good reproducibility for flow stress values, but poor reproducibility in the fracture strain measurements for tests conducted using the same test apparatus. Thus, it appears that semi-solid property data is highly dependent on both the experimental conditions and the testing apparatus. Introduction    17 van Haaften 10-4 Colleyε. 10-3 Temperature (oC) σ( M P a) 540 550 560 570 5800 5 10 15 20  Figure 1.5: Comparison of semi-solid failure stress for AA5182 measured by Colley [39] and Van Haaften [70].  As shown, a number of unique testing apparatus have been developed to measure semi-solid tensile deformation behaviour. Of the two approaches, the solidification tests are a better approximation to the DC casting process while the hot tensile tests are less similar to DC casting since there is no supply of liquid metal to counteract hot tearing growth and the temperature dependence on fraction solid may be significantly different between solidification and melting. One inherent advantage with the hot tensile test apparatus is that accurate measurement of the stresses and strains is much simpler. 1.3.3 Semi-solid Constitutive Models In order to understand semi-solid deformation mechanisms, and to use this experimental data for process models and hot tearing criteria, a number of equations have been developed to approximate the semi-solid constitutive behaviour. Some authors [70, 78, 79] have proposed to use a creep law, based on the assumption that load is supported by the solid skeleton. Drezet and Eggeler [78] modified the exponential creep law such that the load carrying area was directly proportional to the fraction solid. Van Haaften [70] proposed that since the low melting phases Introduction    18 are located at the grain boundaries, the load carrying area during semi-solid deformation should be proportional to the amount of grain boundary area covered by a liquid film  exp 1 n LGB QA f RT σε ⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟− ⎝ ⎠⎝ ⎠&  (1.1) where fLGB is the fraction of grain boundary area covered with liquid. The predictions of Eq. (1.1) match up quite well to the experimentally measured data for as-cast AA3104 and AA5182. Eq. (1.1) also produced satisfactory results when used by Fabrèque et al. [76] to approximate the constitutive behaviour in an Al–Mg–Si–Cu alloy. However, since this law assumes that the load is supported only by the solid, the predictions will be incorrect once a substantial portion of the grain boundaries are wetted with liquid. A number of authors [32, 80, 81] have proposed a semi-solid model based on viscoplastic solid mechanics and consisting of a porous solid skeleton saturated with liquid. Early models of this type were able to characterize the compressibility and hardening / softening behaviour of the semi-solid material. Suery et al. [82] described the strong anisotropic behaviour by shifting the stress space to account for the lack of semi-solid tensile ductility. However, this model added four independent functions, and it is unclear how these functions would be determined experimentally. Martin et al. [83] have modelled this behaviour by introducing an internal function that represents the partial cohesion of the mush into a constitutive viscoplastic equation for porous solid skeleton plus liquid. This internal function relates the macroscopic strain rate to the averaged plastic strain rate of the solid phase and accounts for the increase of dendrite interlocking with increased fraction solid. At fraction solids below the coherency point, C = 0 since all the strain is accommodated in the liquid. In this study, the authors validated the cohesive function model with semi-solid compression, shear, and tensile experiments. While the two-phase, sponge-like model seems to account for many of the semi-solid deformation phenomena, there are a number of limitations. Firstly, while strain seems to play a major role in semi-solid material failure, the model does not include strain effects. Secondly, failure criteria are still required for hot tearing prediction since the model does not predict damage evolution. Thirdly, the model is not user-friendly since it requires considerable programming skill in order to implement it into industrial process models. Lahaie and Bouchard [84] developed a model specifically covering the tensile constitutive behaviour of regular hexagonal grains separated by a continuous liquid film. This model predicts Introduction    19 the viscosity-induced average stress at different levels of global strain based on deformation caused solely by liquid flow:  ( ) ( )3 31 2 129 m mvisc s sf fμε εσ ε− −⎡ ⎤= − − + − +⎢ ⎥⎣ ⎦&  (1.2) where μ is the viscosity of the liquid phase, ε is the global strain, ε&  is the strain rate, and m is a microstructural parameter with a value of 0.5 for columnar grains and 0.333 for equiaxed grains. No strain is accumulated in the solid, and the model is applicable only until the hexagonal grains contact each another. While the authors tested their model over the parameter-space of grain size, fraction solid, and strain rate, no validation to experimental data was presented. The major limitations of the Lahaie and Bouchard model were corrected by Larouche et al. [75]. Firstly, the regular hexagonal grain structure was replaced with a log-normal probability distribution function to capture the effect of different channel thicknesses. Secondly, for very thin liquid channels, the solid grains were allowed to deform if the viscous forces exceed the creep forces. Thirdly, a rule of mixtures was used to account for plastic deformation in grains where solid bridging exists. The new semi-solid constitutive equation is shown below:  ( ) ln ln 0 1 trans trans h yield creep creep h dh dhσ κ σ κ σ σ ∞⎡ ⎤= ⋅ + − ⋅ Ψ + Ψ⎢ ⎥⎢ ⎥⎣ ⎦∫ ∫  (1.3) where σ  is the average stress, κ is the proportion of grains that have bridged, htrans is the critical channel thickness for creep forces, and lnΨ  is the log-normal probability distribution. The authors also provided experimental data acquired using the DCSS apparatus previously described. The model proposed by Larouche et al. [75] is a good starting point for further work. However, in its current form, it provides a poor comparison to the DCSS experimental data, since the model predictions show an exponential relationship while the experimental reaches a steady-state maximum. Furthermore, it is not clear how one would measure the mean value and standard deviation of the liquid channel thickness required for the log-normal distribution. Assessment of the previously-developed models has demonstrated that the challenge to modeling semi-solid constitutive behaviour lies in its three-part behaviour of liquid flow through irregular-shaped channels, increasing grain cohesion as a function of both strain and fraction solid, and solid plastic deformation. Model validation is also an issue since good apparatus do not yet exist to measure tensile properties in the range of fraction solids below ~ 0.95. Introduction    20 1.4 X-ray Micro-Tomography Observing and characterizing the evolution of both microstructure and defects during solidification is particularly challenging for experimentalists. The reason for this has been the difficulties posed by the high temperatures involved, containment of reactive molten metals and problems with radiation from the metal surfaces. The classic technique has been the test-and- quench approach, with the final microstructure being observed afterwards using electron microscopy equipment. Recent advancements in techniques for the evaluation of microstructural evolution during solidification allow for in situ observation. These techniques include confocal scanning laser microscopy, in situ x-ray transmission radiography, and x-ray micro-tomography [85]. This last technique has been used as part of this research project and is outlined below. X-ray micro-tomography (XMT) is a non-destructive, three-dimensional characterization method that has been applied to a number of fields within materials science (e.g. [86-91]). The technique allows for the imaging of internal microstructural features by measuring variations in intensity of a transmitted x-ray beam through a rotating specimen. The rotational motion consists of a discrete number of rotations of constant angle, typically between 0.25° and 1°. At each step, a two-dimensional transmission radiograph is acquired, and the resultant series of images are used to reconstruct a three-dimensional representation of the material microstructure.  The most common reconstruction method is the inverse Radon transformation and is generally referred to as a filtered back-projection method [92]. Scans of objects that appear externally to be axi- symmetric are preferred because shadowing from any protruding portions of the specimen during rotation will decrease the quality of the tomograph. Furthermore, minimizing the rotation angle ensures quality results but increases both the scan time and the time required for reconstruction. There are two distinct flavours of XMT systems: laboratory scale units, such as the Phoenix| X-ray system at Imperial College London; and dedicated units setup at synchrotrons, such as beam lines ID15 and ID19 at the European Synchrotron Radiation Facility (ESRF) in Grenoble, FR. Laboratory scale XMT units are simply extensions to transmission radiography systems used in a variety of industries worldwide with a micro focal x-ray source for high-resolution. While these units are powerful tools, the long times required to make one full scan of the specimen (2+ hrs) limits their use for in situ tomography of solidification events. Dedicated synchrotron units benefit greatly from the high x-ray flux available at such facilities. The high flux, along Introduction    21 with the development of high-speed x-ray detectors, and associated computer hardware, has allowed the scan time to be reduced to ~10 s for tomographs with a resolution of ~2 μm. Although the tomography technique has not been used in conjunction with semi-solid tensile deformation, a number of other recent solidification studies have been performed using XMT. Lee et al. characterized the intermetallics and porosity in DC cast aluminum-magnesium alloys using XMT [85]. Bernard et al. [43] assessed the interdendritic permeability in binary aluminum- copper alloys using a solid skeletal geometry obtained from XMT. This was done by quenching samples from different temperatures within the mushy zone, scanning the samples to create a tomograph, and then separating the aluminum dendrite skeleton from the liquid phase using 3D image analysis. Flow simulations were then performed using the aluminum dendritic skeleton geometry. While the above two studies used low-speed tomography to make post-solidification observations, Ludwig et al. [7] recently observed in situ and in 3D the solidification of an aluminum-copper alloy at ESRF using a hot stage and a transparent mould. A 10 s scan time was used for the tomographs, while the melt was continuously cooled from 700°C to 500°C at a cooling rate of 0.1°C/s. Using the tomography data, Ludwig et al. [7] were able to calculate directly the fraction solid, shrinkage, and also the composition of the solid and liquid. A fair to good comparison was found between these results and prior knowledge. 1.5 Scope and Objective of Thesis The hot tearing defect is of major concern for the aluminum industry, since it affects both the quality and productivity of the casting process. While this defect has been extensively investigated, many of the findings have been empirically determined via industrial castings and macroscale experiments, and the relationship between semi-solid microstructure and hot tearing has been ill-defined. It is challenging to observe and characterize hot tearing mechanisms because of the high temperatures and opacity associated with liquid metals. Furthermore, the computational models of hot tearing are lacking, in part due to difficulties in accurately measuring semi-solid constitutive behaviour. With these issues in mind, the objectives of this thesis are as follows: (a) to characterize the early stage development of hot tears; (b) to examine the relationship between microstructure and hot tearing; (c) to develop a robust technique for measuring semi-solid constitutive behaviour; Introduction    22 (d) to develop an improved semi-solid constitutive equation. In Chapter 2, a preliminary study is presented in which hot tearing is characterized by x-ray radiography and micro-tomography of semi-solid AA5182 specimens deformed in tension. The formation and interconnectivity of the deformation induced damage is quantified, as is the initial as-cast porosity. However, a full quantitative characterization could not be performed because the sample geometry did not accommodate repeated testing on the same specimen. In Chapter 3, semi-solid tensile testing combined with x-ray micro-tomography is performed to characterize the growth of hot tears as a function of total macroscopic strain. Furthermore, the limitations of Chapter 2 are addressed. The basic aim of this chapter is to study the initial stages of void nucleation, growth and coalescence in AA5182. Novel techniques are presented to quantify the size, morphology and orientation of the internal damage in specimens reheated to ~ 0.98 fraction solid at increasing levels of strain. This quantification is then combined with finite element analysis of these microstructures to determine the mechanisms controlling the nucleation and early stage growth of damage, as well as the final stages of damage localization and failure. In Chapter 4, an experimental methodology for conducting semi-solid temperature tensile deformation tests is described. The emphasis is put on accurate measurement of diametral change via a laser dilatometer, and accurate control of the cross-head displacement. The effect of the liquid phase and as-cast porosity on strain accommodation is examined by comparing results from as-cast AA5182 and hot isostatic pressing AA5182. In Chapter 5, a three phase microstructural model for prediction of semi-solid constitutive behaviour via finite element simulation is presented. The effect of fraction solid and grain size is incorporated using a Voronoi tessellation-type algorithm for geometry generation. The Voronoi diagram also allows for random placement of grain nuclei for a given average grain size. The effect of fraction porosity is included by removing liquid material at the grain triple-junctions. The model predictions are compared to the experimental measurements from Chapter 4. In Chapter 6, the results of the microstructural model simulations are compiled and used to formulate a semi-solid empirical equation which describes the effect of the following microstructure variables – fraction solid, fraction porosity and grain size – on the semi-solid constitutive behaviour. The equation incorporates geometric strain hardening, a rule of mixtures to combine the solid and liquid behaviour, and a term describing porosity. Introduction    23 1.6 References 1. Rappaz, M., Int. Mater. Rev. , 34, 1989, (93). 2. Lee, P. D., Chirazi, A. and See, D., J. Light Metals, 1, 2000, (15). 3. Eskin, D. G., Suyitno and Katgerman, L., Prog. Mater. Sci., 49, 2004, (629). 4. Sengupta, J., Cockcroft, S. L., Maijer, D. M., Wells, M. A. and Larouche, A., Metall. Mater. Trans. B, 35B, 2004, (523). 5. Sengupta, J., Cockcroft, S. L., Maijer, D. and Larouche, A., Mater. Sci. Eng. A., 397, 2005, (157). 6. Phillion, A. B., Maijer, D. and Cockcroft, S. 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SIAM, New York, 2001.     26 Chapter 2: X-ray Micro-Tomographic Observations of Hot Tear Damage in an Al–Mg Commercial Alloyα 2.1 Introduction and Background The Direct Chill (DC) casting process [1] is commonly used to solidify non-ferrous alloys into primary ingots (rectangular cross-section) and billets (circular cross-section). Although this process has been used successfully by industry for many years, certain defects remain technically challenging, such as cold cracks, ingot distortion, and hot tears. Hot tears are commonly encountered during the start-up phase of the casting process and are most prevalent in long freezing range alloys [2]. It appears that these defects both initiate and propagate in regions of the casting that are at temperatures just above the solidus [3] and that are subjected to thermally or mechanically induced stresses acting on material with limited ductility [3, 4]. A number of criteria have been developed to predict the occurrence of hot tears (see review by Eskin et al. [5]). These approaches can be divided into two classifications: those relating to the mechanical aspects of the problem, such as total strain [6]; and those relating to solidification aspects, such as freezing time [7]. Recently, Suyitno et al. [8] evaluated a number of these criteria by implementing them into a finite element model of the DC billet casting process, and concluded that their predictive ability was qualitative at best. This lack of quantitative correlation suggests that our understanding of the mechanisms of hot tearing is insufficient, and that there is a need for more fundamental experimental studies. While most prior research has focused on quantifying the macroscopic aspects of hot tearing (e.g. load as a function of fraction solid [9, 10]), only a few studies have examined hot tearing in situ. For example, one study focused on an Al–0.5wt% Cu alloy in which surface cracking was observed in a partially solidified alloy subject to tensile loading [11]. A second study, using a transparent organic analogue [12], identified three different hot tearing initiation mechanisms – directly as elongated tears, on pores caused by solidification shrinkage, and as a restarted hot tear over a previously healed hot tear – illustrating the benefits of in situ observation. This study examines the applicability of x-ray micro-tomography (XMT) for the 3D quantification of damage formed during hot tearing.  α A version of this chapter has been previously published. Phillion, A.B., Cockcroft, S.L., and Lee, P.D., (2006), Scripta Materialia, 55, pp. 489-492. X-Ray Micro-tomographic Observations of Hot Tear Damage in an Al–Mg Commercial Alloy    27 2.2 Experimental Methodology In the current study, four tensile specimens of commercially DC cast AA5182 were heated to 528°C, corresponding to a fraction solid (fs) of ~ 0.98 [13]. The specimen geometry is shown in Fig.  2.1. Subsequent to reaching the test temperature, each specimen was loaded in tension to different values of strain as shown in Table  2A, using a Mushy Zone Tensile Tester (MZTT). The crosshead displacement rate was 0.085 mm/s. The MZTT is essentially a modified Instron1 mechanical testing frame connected to a Gleeble 35002 thermomechanical simulator. Full details of the apparatus are provided in [14]. The key feature of the apparatus is that a parabolic temperature gradient prevails along the specimen, promoting strain localization near the center of the gauge length. XMT scans using a commercial laboratory scale unit3 were performed on an undeformed specimen, and four specimens deformed while semi-solid to strains ranging from 0.02 to 0.2 (see Table  2A). Reconstructed images consisting of ~ 800 x 800 x 400 voxels (at a resolution of ~9 μm), were collected from the deformed region of each specimen (section “A” in Fig.  2.1). A planar sample along the centerline, 1.0 mm in thickness and 40 mm in length was then machined out of each specimen (section “B” in Fig.  2.1) and a transmission radiograph taken. 2D and 3D image analysis was then performed on the tomographs using the software VG Studio Max4.  Table 2A: Summary of analysis of XMT data. Test No. D initial (mm) D final (mm) ε xmt ε diametral 1 7.98 7.91 0.02 0.018 2 7.99 7.77 0.06 0.054 3 7.99 7.53 0.16 0.118 4 8.00 ~7.35 ~0.20 0.169 5* 8.00 8.00 %P = 0.32% *Undeformed specimen  1 Instron Inc, Norwood, MA, USA 2 Dynamic Systems Inc, Poestenkill, NY, USA 3 Phoenix|x-ray Systems and Services GmbH, Germany 4 Volume Graphics GmbH, Germany X-Ray Micro-tomographic Observations of Hot Tear Damage in an Al–Mg Commercial Alloy    28  Figure 2.1: Diagram of specimen geometry, including location of tomography and radiography scans. X-Ray Micro-tomographic Observations of Hot Tear Damage in an Al–Mg Commercial Alloy    29 2.3 Results and Discussion 2.3.1 Observations in 2D and in 3D of a Highly Damaged Specimen Fig.  2.2 shows a scanning electron microscopy (SEM) image of the fracture surface of specimen 4 (ε ~ 0.20). The surface has a smooth, glassy-like morphology, which is indicative of some liquid being present during deformation and failure and confirms that the specimen was semi-solid during testing. Transmission radiographs of specimens 3 and 4 (Fig.  2.3) show that the damaged region of each specimen contains a significant void fraction, which was not present in the undeformed specimen. The strain is accumulated both by diametral reduction and also internal void growth. Previous studies have used the diametral strain to characterize semi-solid properties [14]. However, the assumption of a fully dense structure during mushy zone testing ignores the internal accumulation of strain. Strain can be more accurately estimated by calculating the logarithm of the ratio of the initial cross sectional area to the number of voxels representing metal in each slice of the reconstructed volumes:  ( )ln inixmt vox A voxels A ε ⎛ ⎞= ⎜ ⎟⎜ ⎟⋅⎝ ⎠∑  (2.1) Therefore, in this study both the internal and diametral strains in each of the four tensile specimens were measured (Table  2A). At low strains, the diametral and XMT measurements are similar, with an increasing divergence at high strains due to increased internal damage. The initial void content prior to tensile deformation was ~ 0.32 % ± 0.05, as quantified from the XMT data of the undeformed specimen. This corroborates earlier findings, where DC as-cast Al– Mg alloys were found to contain 0.4–0.7 % voids [15] arising from a combination of gas, shrinkage and / or thermo-mechanical loading during cooling. Since it is not possible to differentiate in a single tomography scan between the as-cast void population, and voids formed during subsequent application of strain, the term porosity will be used to refer to the voids that formed during DC casting, and the term damage will refer to voids visible after tensile testing. Further examining the transmission radiograph of specimen 3 (ε ~ 0.16) in Fig.  2.3(a), the central region is highly necked and heavily internally damaged, containing both many small hot tears and a few large ones. These hot tears are oriented normal to the loading direction, as would be expected in a tensile test. One of the hot tears extends through a large portion of the cross- X-Ray Micro-tomographic Observations of Hot Tear Damage in an Al–Mg Commercial Alloy    30 section of the specimen and it is probable that only a small increment in the strain would have caused this specimen to fracture. It should also be noted that the damage is fully constrained to the center (~ 5 mm) portion of the radiograph. The damage is localized to this region because of the temperature profile that exists along the length of the specimen during testing – i.e. during testing the damaged region is above the solidus temperature (Tsolidus), while the remainder of the sample is below Tsolidus. The tomographs of all five specimens are shown in Fig.  2.4(a-e). Fig.  2.4(d) shows the 3D reconstruction of the damaged area of the sample strained to 0.16. The same area was previously presented in cross-section in the low magnification radiograph shown in Fig.  2.3(a). In this image, two cross-sectional slices, one parallel to the loading direction, and a second, perpendicular to the loading direction, are displayed behind the isosurfaces. The voids intersecting the surfaces of the planes have been coloured black, while in front of these planes the solid AA5182 has been removed. Thus, only the isosurface of the voids appear in the foreground. The loading direction is vertically oriented. This tomograph shows a very large void / hot tear on the right side, having a complex 3D structure with many interconnections. On the left side, a few other large voids can also be seen with varying degrees of interconnectivity. There are also many smaller isolated voids distributed throughout the sample. It appears from Figs.  2.3(a) and  2.4(c) that the strain in this specimen has been accommodated both by bulk deformation of the material (necking) and also by the formation and growth of a significant internal damage (voids), some of which have coalesced and become interconnected to form hot tears. X-Ray Micro-tomographic Observations of Hot Tear Damage in an Al–Mg Commercial Alloy    31 A  Figure 2.2: SEM image of specimen 4, showing the presence of liquid at the crack  interface. The marker ‘A’ highlights one area where liquid was present.  (a) (b)  Figure 2.3: Radiograph images showing the axial extent of damage in specimens 3 and 4, strained to values of: (a) 0.16 and (b) 0.20. X-Ray Micro-tomographic Observations of Hot Tear Damage in an Al–Mg Commercial Alloy    32  Figure 2.4: 3D micro-tomographic reconstruction of the damage and porosity in the specimens strained to values of: (a) as-cast, (b) 0.02, (c) 0.06, (d) 0.16, and (e) 0.20. The marker ‘A’ highlights the large hot tear from the radiograph in Fig.  2.3(a). X-Ray Micro-tomographic Observations of Hot Tear Damage in an Al–Mg Commercial Alloy    33 2.3.2 Qualitative Comparisons The radiographs shown in Fig.  2.3, in conjunction with the tomographs shown in Fig.  2.4 can be used to gain insight into the development of hot tears. Fig.  2.3 illustrates that the damage becomes highly interconnected at a strain close to fracture. Any of the medium-sized hot tears visible in both  2.3(a) and  2.3(b) could have led to a fracture event. From the tomograph of the unstrained specimen shown in Fig.  2.4(a), it is clear that the initial as-cast porosity is well distributed throughout the specimen and is of roughly uniform size. A comparison of Figs.  2.4(a) (unstrained) and  2.4(b) (ε ~ 0.02) indicates that the internal damage at low strain manifests itself as an increase in the size of the voids, because it appears that the number of large voids at a strain of 0.02 and the total number of voids in the unstrained specimen are similar. There also appears to be nucleation of new voids in Fig  2.4(b), as shown by the presence of small voids around the large ones. The image appearing in Fig.  2.4(c), suggests that at a strain of ε ~ 0.06 there has been new void formation and significant void growth and or coalescence, leading to several large voids. At a relatively high strain of 0.16, Fig.  2.4(d), the damage has coalesced extensively in one portion of the specimen. Internal damage at the point of final failure is shown in Fig.  2.4(e) (note: the fracture surface is at the top of the vertical plane). Qualitatively, it appears that both the number and size of the damage sites increases with increasing strain. Moreover, the amount of interconnectivity between the damage sites also increases with strain, leading to void texture and localization. These observations support the notion of nucleation, growth, and coalescence of damage sites in these materials leading to final fracture by hot tearing. 2.3.3 Quantitative Comparisons The evolution of hot tears is given quantitatively in Fig.  2.5 by plotting the distribution of internal hot tear voids at each strain level for voids with a radius greater than ~ 12.5 µm (≥ 8 voxels), normalized by the volume of material captured in each tomograph, as a function of equivalent void radius. Voids with radius smaller than ~ 12.5 µm were not included in the analysis since they approach the resolution limit of the tomographic scan.  Bearing in mind the missing small voids, the distributions appear to have the general form of the right-hand-side of a log normal distribution. It would appear from this plot that during the application of strain, there is a continuous increase in internal damage resulting from both an increase in the number density and also the size of the voids. Most notably, there is a substantial increase in the number of small X-Ray Micro-tomographic Observations of Hot Tear Damage in an Al–Mg Commercial Alloy    34 voids, and an increase in the size of the largest void with increasing strain. It should be pointed out that in the case of the failed specimen (ε ~ 0.20), the largest voids most probably became part of the fracture surface and hence would no longer be counted as void. This explains the observed decrease in maximum pore radius as compared with the specimen strained to ε ~ 0.16. The comments below on nucleation are hypotheses rather than certainty due to spatial resolution limit of ~ 12.5 µm, which precludes characterization of small voids. From Fig.  2.5, there appears to be a consistent increase in the number of voids at the limit in resolution with increasing strain, which is a combination of the growth of pre-existing small as-cast porosity below the XMT resolution limit, as well as the nucleation of new damage-based voids. There are two possible mechanisms for void growth: 1) void growth proper and 2) coalescence of one or more voids. The increase in the number and size of the voids visible in Fig.  2.5 illustrates that individual void growth is one of the mechanisms by which damage is accumulated, reinforcing the qualitative observations from Fig  2.4. Similarly, there is strong evidence of coalescence being a key mechanism for void growth at high strains. Perhaps the most convincing evidence of coalescence is the texturing of the pores into long connected voids at a strain of 0.06, observed in Eq. Radius (μm) N v (m m -3 ) 101 102 103 0 10 20 30 Total Strain as cast 0.02 0.06 0.16 0.20  Figure 2.5: Void density distribution. X-Ray Micro-tomographic Observations of Hot Tear Damage in an Al–Mg Commercial Alloy    35 Fig.  2.4(c), and localization at strains greater than this (e.g. Fig.  2.4(d)). This type of coalescence is frequently observed during fatigue when the crack tip will branch across regions of stress concentration between neighbouring voids. It is clear from the analysis of the data that the total amount of damage increases dramatically with strain. Quantification indicates that there is continuous nucleation of small voids coincident with growth of the larger voids. With increasing strain, there is a shift towards increased coalescence and thus a few large voids begin to accommodate the majority of the applied strain. 2.3.4 Hot Tearing Insights To correlate these observations to hot tearing, it should be noted that the tests were performed at a very high fraction solid (~ 0.98). The majority of the molten material present at this fraction solid is eutectic and is physically located along grain boundaries and within the interdendritic regions as isolated pockets of liquid. Moreover, the same region will also contain most of the as- cast hydrogen based porosity [15]. On application of load, both the pores and liquid may act as small cracks and concentrate stress. Given this structure, strain could be accommodated by both growth of the as-cast porosity into the surrounding eutectic liquid and nucleation of new damage- based voids in areas containing large pockets of liquid. From the data presented in Fig  2.5, it is clear that the number of small voids is continuously increasing at different levels of strain. Thus, if the increase in small voids is due in part to nucleation processes, it would appear that the tensile strain at even small deformations is large enough to cause void nucleation within the structure. It follows that material with fewer tendencies to form shrinkage-based porosity (e.g. short freezing range alloys) and material with a finer grain size (i.e. smaller and more evenly distributed pockets of residual liquid at high fractions solid) would be less prone to hot tearing. It may be postulated that the critical strain to avoid hot tearing is the strain at which significant coalescence of large voids begins. Based on the data presented in Figs.  2.4 and  2.5, this would lie between strains of 0.02 and 0.06 under the conditions examined in this study. X-Ray Micro-tomographic Observations of Hot Tear Damage in an Al–Mg Commercial Alloy    36 2.4 Conclusions X-ray micro-tomography was used to observe the development of damage and strain during mushy zone tensile loading in aluminum alloy AA5182 specimens at 528 °C. The growth of hot tearing damage was qualitatively assessed in 2D and 3D and quantitatively characterized in 3D. The results show that in the early stages of loading, strain is accommodated mainly by specimen diameter reduction with only limited internal damage accumulation. At intermediate strains it is accommodated by a combination of as-cast porosity growth and damage-based void nucleation in both the residual liquid present in the microstructure and near the intermetallics. At high strains, void coalescence and continued growth appears to be the main mechanism of damage accumulation and strain accommodation. X-Ray Micro-tomographic Observations of Hot Tear Damage in an Al–Mg Commercial Alloy    37 2.5 References 1. Sengupta, J., Cockcroft, S. L., Maijer, D. M., Wells, M. A. and Larouche, A., Metall. Mater. Trans. B, 35B, 2004, (523). 2. Viano, D., St. John, D., Grandfield, J. and Caceres, C., Light Metals 2005, H. Kvande, Ed., San Fransisco, CA, TMS, 2005, (1069). 3. Campbell, J., Castings. Butterworth Heinemann, 2 ed., 1991. 4. Ludwig, O., Drezet, J. M., Martin, C. L. and Suery, M., Metall. Mater. Trans. A, 36A, 2005, (1525). 5. Eskin, D. G., Suyitno and Katgerman, L., Prog. Mater. Sci., 49, 2004, (629). 6. Pellini, W. S., Foundry, 1952, (125). 7. Clyne, T. W. and Davies, G. J., Brit Found., 74, 1981, (65). 8. Suyitno, Kool, W. H. and Katgerman, L., Metall. Mater. Trans. A, 36A, 2005, (1537). 9. Instone, S., St.John, D. H. and Grandfield, J., Int. J. Cast Metals Research, 12(6), 2000, (441). 10. Guven, Y. F. and Hunt, J. D., Cast Metals, 1, 1988, (104). 11. Davidson, D., Viano, D., Liming, L. and St.John, D., Shape Casting: The John Campbell Symposium, M. Tiryakioglu, P. N. Crepeau, Eds., San Francisco, CA, TMS, 2005, (175). 12. Farup, I. and Mo, A., Metall. Mater. Trans. A, 31A, 2000, (1461). 13. Thompson, S., Cockcroft, S. L. and Wells, M. A., Mater. Sci. Techn., 20(4), 2004, (497). 14. Colley, L. J., Wells, M. A. and Maijer, D. M., Mater. Sci. Eng. A, 386(1-2), 2004, (140). 15. Nagaumi, H., Komatsu, K., Uematsu, M., Hasisawa, N. and Nishikawa, Y., J Japan Inst Light Metals, 6, 1999, (13).     38 Chapter 3: Quantitative Assessment of Deformation- Induced Damage in a Semi-Solid Aluminum Alloy via X-ray Micro-Tomographyα 3.1 Introduction and Background The control of defects during the processing of metals is critical to the production of high quality products. In the context of strain induced defects, damage initiates as vacancies, with coalescence and subsequent growth of these defects leading to macroscopic flaws. One example of this process is the formation of internal voids when semi-solid material is strained. For example, even though the metal is not externally constrained during solidification of aluminum alloys in direct chill casting, the thermal stress is sufficient to induce localized damage leading to product rejection. This phenomenon, often termed hot tearing or hot cracking, is an important defect in a range of processes from shape casting [1] to welding [2], but is poorly understood. Over the years, a number of researchers have experimentally investigated hot tearing. Pellini [3] was the first author to demonstrate that hot-tears form in the semi-solid. Feurer [4] examined the influence of alloy composition and solidification conditions on hot tearing, and proposed that this defect was a result of the inability of liquid to feed solidification shrinkage. Warrington and McCartney [5] examined the effect of grain refining on hot tearing, and found that hot tears formed easily in columnar and equiaxed-globular grain structures, but not in equiaxed-dendritic structures. Using experimental data from ring castings, Guven and Hunt [6] showed that hot tears initiate in a thin film of liquid between two grains. However, only a few studies have tried to determine the underlying mechanisms controlling hot tear formation. One recent study designed to identify hot tearing mechanisms was performed by Farup et al. [7]. In this work, the researchers used transparent organic analogues to directly observe the formation of hot tears under applied load. They concluded that there are three mechanisms of tear nucleation in these analogues: “1. directly as elongated pores or tears, 2. on pores caused by solidification shrinkage, or 3. as round pores nucleated in the liquid constituting a healed hot tear.”  α A version of this chapter will be submitted for publication. Phillion, A.B., Lee, P.D., and Cockcroft, S.L. Quantitative Assessment of Deformation-Induced Damage in a Semi-Solid Al Alloy via X-ray Tomography    39 A few authors have designed experiments to observe crack formation on the surface of solidifying metals. Pellini [3] used x-ray radiography to make the first observations of hot tear formation in aluminum-copper alloys. Fredriksson and Lehtinen [8] performed hot tensile tests inside a scanning electron microscope, showing that hot cracks occur if the alloy contains a eutectic liquid with good ability to wet the solid grain boundaries. Davidson et al. [9] recorded the formation of hot tears in an aluminum-copper alloy during solidification using a video camera and determined that hot tearing begins to occur with very small applied loads at fraction solids between 0.93 and 0.96. While these in-situ tests have been revealing, they have been limited by a number of factors. Firstly, in each experiment the observation of hot tears is limited to those appearing on the free surface. Thus, in the metallic studies, the initiation of the hot tear is probably missed. Secondly, the correlation between an organic analogue and a metal has not been demonstrated, since differences exist between the two systems (e.g. the formation of last eutectic and oxides in aluminum alloys). Several authors have used computational modeling to predict the onset of hot tearing [10-17] However, most of these have involved the development of semi-empirically based criteria to predict the onset of hot tearing as a function of the local stress state or accumulated strain at a macroscopic level (e.g. [14]). One of the un-resolved issues related to the prediction of hot tears is the relationship between semi-solid microstructures and crack development. Recently, the modeling of solidification microstructures has made major advances [18]; however, very few of these models incorporate the influence of applied stress. One study which does try to relate microstructure to semi-solid behaviour is Vernède et al. [19]. In this work, the researchers developed a microstructural model which illustrated that localization of intergranular liquid may occur, and suggested that this localization may be linked to hot tearing. In this study, x-ray micro-tomography (XMT) is performed on interrupted semi-solid tensile tests to quantify the evolution of internal damage in 3D and thus to develop insight into the early stages of hot tearing. This methodology overcomes the problems of earlier 2D surface observations, and allows for direct 3D meshing of some of the microstructural features relevant to hot tear formation. The meshed structure was used to simulate the strain localization driving void coalescence. Quantitative Assessment of Deformation-Induced Damage in a Semi-Solid Al Alloy via X-ray Tomography    40 3.2 Experimental Methodology The experimental approach employed for the investigation involved the following three steps: (1) partially remelting a previously cast aluminum alloy specimen, (2) deforming the specimen in tension, and (3) performing x-ray micro-tomography on the portion of the gauge length where damage localization occurred. The tensile tests can be described as ‘interrupted’ tests, since each test was stopped during the deformation process to conduct an off-line tomography scan before reheating and then continued application of load. The semi-solid deformation tests were conducted at The University of British Columbia, Vancouver, Canada, while the tomographic data was collected at Imperial College, London, United Kingdom. 3.2.1 Materials and Geometry A commercially Direct Chill (DC) cast aluminum AA5182 rectangular ingot, of composition Al–4.63%Mg–0.49%Mn–0.17%Fe–0.04%Cu was chosen as the source for the as-cast samples. The semi-solid constitutive behaviour of this alloy has been previously characterized, by Colley et al. [20] and by Van Haaften et al.[21]. Colley’s work showed that the alloy exhibits tensile strength up to a temperature of ~ 575°C, corresponding to a fraction solid of ~ 0.95, and exhibits some tensile ductility up to a temperature of ~ 565°C. Note that since the material is in the as- cast state, it is not damage free; it has been subjected to a strain history associated with the DC casting process. Tensile specimens of ~ 100 mm in gauge length and ~ 4.0 mm in radius, shown in Fig. 3.1, were machined out of the ingot, with their long axis orientated normal to the casting direction, and parallel to broad face of the ingot. They were extracted from material approximately ~ 6–10 cm below the surface of the ingot. This orientation was chosen such that deformation in the specimens takes place in the same direction and in the same region as the occurrence of hot tearing in ingot DC casting. The central 10 mm of each specimen was further reduced in radius to 3.5 mm to ensure that the thermal hot-spot and corresponding strain occurred at a known location. A total of three specimens were prepared for testing. Quantitative Assessment of Deformation-Induced Damage in a Semi-Solid Al Alloy via X-ray Tomography    41 3.2.2 Semi-solid Deformation The semi-solid deformation tests were performed using a previously developed Mushy Zone Tensile Tester (MZTT). This apparatus consists of a modified Instron 1  mechanical testing machine connected to a Gleeble 35002 thermomechanical simulator to rapidly heat the specimen via I2R resistive heating. Further apparatus details are provided in ref. [20].  The temperature chosen for semi-solid deformation was 528°C. This temperature corresponds to a fraction solid of approximately ~ 0.98 [22], and was chosen based on prior experience with AA5182 [23]. The experimental campaign is shown in Table 3A, which provides the strain at which the tensile test was interrupted for tomographic characterization. Prior to hot deformation, each specimen was subjected to a tomographic scan to characterize the initial as-cast void distribution (subscript '0' in Table 3A). Specimens were then heated at a heating rate of 1.5°C s-1 using the MZTT until the test temperature was reached. Semi-solid deformation was subsequently applied, at a displacement rate of 0.085 mm s-1 to create internal damage. After ~ 12 s loading time, the test was interrupted, the specimen was air-cooled and then removed from the MZTT to perform a tomographic scan on the first displacement level (subscript ‘1’ in Table 3A). This procedure was repeated to create further internal damage in the reduced gauge region (subscript ‘2’ in Table 3A). Note that specimen C did not undergo the second level of displacement application since the sample appeared to be heavily damaged following the first displacement level C1. One feature of the apparatus is that a parabolic temperature gradient prevails along the specimen, promoting strain localization near the center of the gauge length (this arises due to conduction of heat to the water-cooled copper grips). The addition of the reduced gauge region containing the control thermocouple ensured that the hot spot, and thus strain localization, occurred at the same location for both displacement levels (subscripts ‘1’ and ‘2’). 3.2.3 Tomographic Imaging X-ray micro-tomography on the reduced gauge region of each specimen was performed using a commercial laboratory-scale XMT unit3. To fully capture the entire reduced gauge region, three successive sub-scans were performed at a voxel resolution of ~ 9 μm. The resolution was  1 Instron Inc, Norwood, MA, USA 2 Dynamic Systems Inc, Poestenkill, NY, USA 3 Phoenix|X-ray Systems and Services GmbH. Quantitative Assessment of Deformation-Induced Damage in a Semi-Solid Al Alloy via X-ray Tomography    42 constrained by the diameter of the tensile specimens. In total, 2160 radiographs, scanning 360 degrees, were taken along the loading axis at 0.5-degree increments and at three different z- positions. Image slices were reconstructed from the series of projections based on the filtered back-projection method [24], to create digital volumes consisting of 850 x 850 x 1100 voxels at a 32-bit floating point greyscale range. In addition, two high-resolution tomographic scans, at a voxel resolution of ~ 2.5 μm, were performed using the same XMT unit on both the initial as-cast material and a deformed specimen. These digital images were obtained from small cylinders, 2.0 mm in diameter, machined out of both the as-cast ingot and also specimen C1. 100 mm 10 mm φg = 8 mmφr = 7 mm  Figure 3.1: Interrupted tensile test specimen geometry, showing the reduced area region.   Table 3A: Specimen text matrix and results of x-ray tomographic analysis. Specimen Test Level εtot εd %P Nv (mm-3) Max Void Size (μm) A A0 A1 A2 0 0.09 0.39 0 0.06 0.20 0.52 3.22 16.49 26 38 60 113 391 1108 B B0 B1 B2 0 0.11 0.39 0 0.09 0.25 0.26 1.92 12.55 8 76 49 99 173 1012 C C0 C1 0 0.32 0 0.18 0.74 13.07 16 94 188 1012  Quantitative Assessment of Deformation-Induced Damage in a Semi-Solid Al Alloy via X-ray Tomography    43 3.2.4 Analysis of the Tomographic Data To evaluate the microstructural effects quantitatively, the eight reconstructed tomographic datasets of the entire reduced gauge region and the two high-resolution datasets were subjected to image analysis using the software packages ImageJ [25] and Amira4. A series of filters was applied to each dataset to enhance the contrast between void, metal, and exterior. Firstly, the full dataset was transformed from 32-bit to 8-bit grayscale to reduce the size of the digital file. Secondly, a circle was inscribed on each of the image slices, approximating the exterior surface of the specimen. Voxels outside this circle were given a grayscale value of 255, while voxels inside this circle remained at their original value (between 0 and 254). This reclassified all external voids, i.e. voids that interfaced both metal and the exterior, as internal voids.  Thirdly, an edge-preserving-smoothing filter was applied to the full dataset to remove noise in both the metal and void regions while preserving the metal-void boundaries. Further details on this filter can be found in [26]. Finally, a threshold was imposed on the dataset and labelled to explicitly identify each voxel as void, metal, or exterior. Unfortunately, it was not possible to follow individual voids during deformation because the rotational alignment of the specimen with respect to the x-ray detector in the XMT unit was not consistent between the tomographic scans. It was observed in the datasets for specimens A and B, taken after the second level of semi- solid deformation, that most of the damage was contained within a small portion (~ 2 mm in length) of the reduced gauge region. A child volume of this region was cropped out of the full dataset, and child volumes of the identical region were cropped out of the datasets taken from as- cast and first level of semi-solid deformation for all four specimens. The child dataset was then relabelled with each void assigned a different greyscale value, allowing for the volume of each void to be determined. Furthermore, the average external diameter was calculated based on the average area of the circle approximating the exterior specimen surface, while the percentage porosity was calculated based on the total volume of voids found in the child dataset.  4 Mercury Computer Systems, Chelmsford, MA, USA Quantitative Assessment of Deformation-Induced Damage in a Semi-Solid Al Alloy via X-ray Tomography    44 3.3 Results and Discussion Analysis of the tomographic datasets allowed for both qualitative and quantitative comparisons of the damage formed during semi-solid tensile deformation as a function of strain. In this work, two definitions of strain have been used:  02 lnd i d d ε ⎛ ⎞= ⋅ ⎜ ⎟⎝ ⎠  (3.1)  ( ) 2 0 2 lntot voxi d voxels A π ε ⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟= ⋅⎜ ⎟⎜ ⎟⎝ ⎠ ∑  (3.2) In the first definition, εd, strain is estimated based on the average external diameter, di at each level of deformation (i = 0, 1, 2). In the second definition, εtot, strain is estimated based on the number of voxels representing metal in each slice of the reconstructed datasets, and thus takes into account the internal damage forming during semi-solid deformation. 3.3.1 Qualitative Assessment The evolution of damage as a function of strain during semi-solid tensile deformation of as- cast AA5182 is shown qualitatively in Figs. 3.2 and 3.3. The images shown are from specimen A, and are typical of the results seen in all three specimens. In Fig. 3.2, a 2D cross-sectional image of the 3D tomographic data at each strain level is shown, with the loading direction normal to image. The dark areas represent voids, while the small bright white areas represent solute phases. In Fig. 3.3, the 3D morphology of a quarter-section of the internal damage / void network at each strain level is presented via a threshold and segmentation process. This morphology is thought to play an important role in the final formation of macroscopic hot tears. Two symmetry planes and a cross-sectional slice are also displayed in the background of Fig. 3.3 to provide an indication of the specimen external dimensions. Note that in both Figs. 3.2 and 3.3, the same location within the specimen is shown but at three different levels of deformation. Fig. 3.2(a) shows the initial as-cast state of specimen A, which contains 0.52 % porosity, evenly distributed throughout the cross-section. After the material has been deformed in the semi-solid state to a total strain, εtot, of 0.09, Fig. 3.2(b), the amount of internal damage has Quantitative Assessment of Deformation-Induced Damage in a Semi-Solid Al Alloy via X-ray Tomography    45 increased substantially to 3.22 %. With a further increase in semi-solid deformation to a total strain, εtot,  of 0.39, Fig. 3.2(c), the internal damage has become quite extensive and localized. Fig. 3.3(a) shows the initial distribution of voids in specimen A in 3D. As can be seen from the figure, the voids are quite tortuous and their maximum length is much longer than is apparent from the 2D cross-section, Fig. 3.2(a).  In the early stages of semi-solid deformation, Fig. 3.3(b), it appears that both discrete growth of the pre-existing voids and formation of new voids has occurred. It is unclear whether the formation of new voids is a consequence of void nucleation, or of the rather large voxel size of 9 μm. With the large voxel size, small voids which were below the resolution limit in Fig. 3.3(a) may have simply grown to the point where they are now large enough to appear in the tomography scan. With a further increase in strain, Fig. 3.3(c), the void morphology has now become a highly complex, localized, and interconnected network of internal damage. Figs. 3.2 and 3.3 provide new insight into the extent of void formation, growth and coalescence occurring during semi-solid deformation processes. The as-cast porosity seems to play an important role, acting as pre-existing nuclei for void growth. As strain is applied, the growing voids appear to be preferentially orientating towards each other facilitating their eventual coalescence. Unfortunately, the 9 μm voxel size of the images shown in Figs. 3.2 and 3.3 is too large to indirectly observe the role of the liquid in void formation, growth and coalesence. To investigate liquid – void interaction, tomographic datasets of the initial as-cast material and a small portion of specimen C1 were obtained at a voxel size of 2.5 μm. 2D cross-section images of these tomography scans are shown in Fig. 3.4. The higher resolution allows for observation of some salient features. Firstly, small bright white areas are clearly visible in both Fig. 3.4(a) and 3.4(b), and represent the Mn/Fe/Cu enriched eutectic and intermetallic phases. Although the grain boundaries are not visible, these secondary phases mark the triple points associated with the grain boundaries. Secondly, the as-cast porosity appears to be clustered. This clustering is an artefact of the 2D sectioning of highly tortuous voids which weave in and out of the cut plane. The grain size was measured using optical metallography of anodized samples, with d  ~ 225– 250 μm, and correlates well to the distance between the white phases in Fig 3.4(a). Finally, the voids formed during deformation appear to have formed near the triple points, Fig. 3.4(b).  Voids (1), (2), and (3) are linked by relatively small channels which may have been grain boundaries or Quantitative Assessment of Deformation-Induced Damage in a Semi-Solid Al Alloy via X-ray Tomography    46 interdendritic eutectic and thus liquid during semi-solid deformation. Note that the ‘tail’ of material in both images is due to the process of wire electric discharge machining, which was used to make the cylinders. These high resolution scans clearly show that the damage forms predominately at the grain boundaries, correlating well with the perceived mechanism by which hot tears form - i.e. tearing along the last to solidify intergranular liquid. While a second interpretation could be that the liquid phase is trapped between the interdendritic arms, the length of the void channels joining voids (1), (2), (3) is similar to the grain size, and thus provides evidence of tearing along the intergranular liquid. These scans also allow for a semi-solid failure mechanism to be proposed that is dependent on the presence or absence of as-cast voids. When an as-cast void is present, it acts as a stress riser and allows strain to be accommodated by growth of this pre-existing damage through the liquid along the grain boundary – i.e. no nucleation step is required.  However, in the absence of a pre-existing void, then nucleation of voids will have to occur. Once the stress concentration becomes appreciable, these new voids act identically to as-cast porosity. As can be deduced from Fig. 3.4, it is the presence of the solid-liquid interface along the grain boundary, and intermetallics, which greatly enables both void nucleation and growth and leads to early material failure. Quantitative Assessment of Deformation-Induced Damage in a Semi-Solid Al Alloy via X-ray Tomography    47     (a)               (b)  (c) Figure 3.2: Transverse sections from the tomographic reconstruction of specimen A showing the initial porosity in the specimen, and then the development of further damage with application of strain: (a) ε = 0, %P = 0.52 (A0), (b) ε = 0.09, %P = 3.22 (A1), and (c) ε = 0.39, %P = 16.49 (A2). Quantitative Assessment of Deformation-Induced Damage in a Semi-Solid Al Alloy via X-ray Tomography    48  (a)  (b) Figure 3.3: 3D morphology of the internal damage in specimen A observed by tomography at various levels of strain in a quarter-section of the deformed region at strain levels of: (a) ε = 0, %P = 0.52 (A0), (b) ε = 0.09, %P = 3.22 (A1), and (c) ε = 0.39, %P = 16.49 (A2) (next page). Quantitative Assessment of Deformation-Induced Damage in a Semi-Solid Al Alloy via X-ray Tomography    49  (c)   (a)       (b) Figure 3.4: Transverse sections from the high resolution x-ray micro- tomography scans of (a) as-cast AA5182, and (b) specimen C1, ε = 0.32.  Quantitative Assessment of Deformation-Induced Damage in a Semi-Solid Al Alloy via X-ray Tomography    50 3.3.2 Quantitative Assessment The 3D image-analysis software allows for calculation of the volume of individual voids in the deformed region of each specimen. The bulk porosity, void number density, and the maximum void size for specimens A, B, and C, at different levels of strain, is shown in Table 3A. The variation in the distribution of void radii as a function of strain for specimens A, B, and C is shown in Figs. 3.5-3.8. These figures are plotted such that the data displayed has been divided into 15 different bins on a log scale Beginning with the as-cast material, it can be seen from Table 3A that there is a wide range in the initial porosity level (0.26 < %P < 0.74), the maximum initial void radius (99 μm < r < 188 μm), and also the initial void number density (8 < Nv < 26) in the three specimens. These initial voids are most probably a combination of shrinkage and hydrogen- based porosity [27, 28] and may also have been caused or augmented by some strain accumulated during the casting process. While the local solidification conditions in all three specimens were similar, small variations in alloying elements and inclusions can cause large voids to form in a particular location, explaining the observed variation in as-cast porosity. This variation in porosity will have a large effect on the nucleation and localization of damage, and hence on the semi-solid deformation behaviour. As shown in Table 3A, the porosity, maximum void radius and void number density increase significantly with tensile deformation, as would be expected, although the behaviour of the three specimens is not exactly the same. One interesting observation is the interaction between dimetral reduction and internal damage. As would be expected due to conservation of volume, the specimen diameter is decreasing with increasing deformation. During solid ductile yielding, small voids only develop after significant yielding, leading to strain localization. In contrast, these experiments show that in semi-solid deformation, a different sequence occurs in which significant internal damage develops to accommodate the deformation. For example, the dimetral strain, εd, for specimen A2 was 0.25, while the total strain, εt, was 0.39. Thus, the internal damage accounts for approximately one-half of the total strain obtained during deformation. In order to properly characterize semi-solid constitutive behaviour, this internal damage must also be included. To determine the spatial resolution of the tomographic datasets, a comparison of the void radii distribution observed in the high-resolution dataset for as-cast AA5182 and the lower resolution Quantitative Assessment of Deformation-Induced Damage in a Semi-Solid Al Alloy via X-ray Tomography    51 datasets for specimens A-C in the undeformed state was performed. The results from this comparison are shown in Fig. 3.5. As can be seen, voids sized between ~ 3 μm and ~ 100 μm equivalent radius were found in the high-resolution dataset, while voids sized between ~ 12 μm and ~ 220 μm equivalent radius were found in specimens A–C. It is clear from the figure that the computed number density of voids with a radius greater than ~ 25 μm was similar at both scan resolutions. However, far more small voids exist in the material than can be accurately observed at the lower resolution. Thus, results shown in Figs. 3.6 and 3.7 only include voids greater than 64 (43) voxels, which corresponds to a radius of ~ 25 μm. The variation in the number density of voids as a function of equivalent void radius for the as- cast and semi-solid deformed states is presented in Figs. 3.6 and 3.7 for specimen A and B. The distributions appear consistent with the right-hand-side of a log normal-type distribution. Note that in curves A1 and A2, Fig. 3.6, the first bin has a lower value for void number density than the second bin. This apparent peak value in the curve is due to the spatial resolution limits of the tomographic scan, and is not indicative of an actual peak value in the number of voids per unit volume with a certain radius. From the analysis of specimen B, Fig. 3.7, the processes of void growth and coalescence can be observed.  The results also are consistent with nucleation of new voids however, this cannot be established conclusively owing to limits in the resolution of the scans.  After the first semi- solid deformation (B1), the maximum number density in bin one, containing the smallest size range, has risen from 2 mm-3 to over 14 mm-3, and the total number density of voids has increased by ~ 850 %. This large increase in the number density of voids supports the view that at least some portion of the new voids arise as a result of nucleation processes and not exclusively growth of small as-cast voids. After the second semi-solid deformation (B2), some very large voids were found in the deformed region, and the total number density of voids had decreased by ~ 45 % as compared to B1. These two observations clearly indicate that void coalescence was the major damage mechanism active during the transition from B1 to B2. The results for specimen A, shown in Fig. 3.6, are quite different than for B. In A, there has only been moderate void formation and growth after the first semi-solid deformation (A1), since the maximum number density in bin one has only increased from ~ 4 mm-3 to ~ 6.5 mm-3. The transition from A1 to A2, therefore resulted in significant void growth and coalescence with relatively less void formation. Quantitative Assessment of Deformation-Induced Damage in a Semi-Solid Al Alloy via X-ray Tomography    52 In Fig. 3.8, the variation in the number density of voids as a function of equivalent void radius is shown for specimens A, B and C. The data shown represents the second round of deformation for specimens A and B and first round of deformation for specimen C. The total strain was similar in all three specimens. As can be seen from Fig. 3.8, the structure that results from applying these levels of deformation to a semi-solid material contains many small and medium sized internal voids. Only a few voids greater than 225 μm have formed (< 5 in each bin), although they contribute to creating most of the internal damage. Each specimen had one large void, which was the result of coalescence of many smaller voids. The similarity in the shape of the curves for all three specimens provides a reliable indicator that all three were deformed at similar fraction solid. However, since specimen C required only ~ 10 s of deformation time, while specimens A and B were deformed for ~ 20 s, the effect of the initial as-cast porosity on the deformation behaviour must have been to increase the void growth rate in sample C, and thus to increase the hot tearing susceptibility. The qualitative and quantitative analysis of strain-assisted void development have shown that the semi-solid deformation process is controlled by discrete growth and coalescence of voids, and support the notion of void formation by nucleation. In the deformed region of each of the specimens, significant amounts of strain have been accommodated by internal damage formation and growth in addition to dimetral reduction. At some critical strain, localized coalescence of voids occurs dominating damage accumulation. This was shown in specimen B, Fig. 3.7, with the decrease in void number from the first to the second round of semi-solid deformation. To further understand the mechanisms of semi-solid deformation, two additional features of the internal damage were examined: the spatial orientation of each void relative to the loading direction, and the concentration of stress around voids in the deformed region. Quantitative Assessment of Deformation-Induced Damage in a Semi-Solid Al Alloy via X-ray Tomography    53 Eq. Radius (μm) N v (m m -3 ) 100 200 10-2 10-1 100 101 102 φ = 2 mm φ = 7 mm  Figure.3.5: The influence of tomographic spatial resolution on the void number density as a function of equivalent radius. The resolution for diameters 2 and 7 mm is 2.5 and 9 μm. Eq. Radius (μm) N v (m m -3 ) 101 102 103 0 2 4 6 8 10 12 14 16 18 A0 (εmax = 0) A1 (εmax = 0.09) A2 (εmax = 0.39)  Figure 3.6: The effect of total strain on the void number density distribution in specimen A. Quantitative Assessment of Deformation-Induced Damage in a Semi-Solid Al Alloy via X-ray Tomography    54 Eq. Radius (μm) N v (m m -3 ) 101 102 103 0 2 4 6 8 10 12 14 16 18 B0 (εmax = 0) B1 (εmax = 0.11) B2 (εmax = 0.39)  Figure 3.7: The effect of total strain on the void number density distribution in specimen B. Eq. Radius (μm) N v (m m -3 ) 101 102 103 0 2 4 6 8 10 12 14 16 18 A2 (εmax = 0.39) B2 (εmax = 0.39) C2 (εmax = 0.32)  Figure 3.8: Comparison of the void number density distribution for specimens A2, B2, and C2. Quantitative Assessment of Deformation-Induced Damage in a Semi-Solid Al Alloy via X-ray Tomography    55 3.3.3 Internal Damage Spatial Orientation The spatial orientation of the internal damage refers to the direction cosine of each void’s major axis, termed the morphological texture. The results for specimen A at strain levels A0, A1, and A2 are presented in a series of equal-area pole figures similar to a texture plot for grain orientation, in Fig. 3.9. The loading direction is perpendicular to the pole figure, with points at (0,0) corresponding to voids having a major axis parallel to the loading direction, and points on the circle’s perimeter corresponding to a major axis perpendicular to the loading direction. The major axis of each void was calculated from the eigenvectors of a covariance matrix based on the x, y, and z coordinates of all the voxels in that void. The eigenvectors were  determined using the principal component analysis technique, ITK libraries [29], and in-house coding. Note that to reduce noise, only the results for voids containing more than 1000 voxels (103) are reported. The analysis of as-cast porosity, Fig. 3.9(a),  shows that on average their major axis is located in one plane. This bias is thought to arise due to the direction of solidification during DC casting.  With increasing strain, the morphology evolves as the voids grow preferentially normal to the loading direction (i.e. the majority of voids now have their major axis closer to the circumference of the pole plot, Fig. 3.9(b)). The number of voids exceeding the size threshold also increases significantly due to their growth. At higher strain, Fig. 3.9(c), it is clear that almost all the voids are becoming preferentially oriented perpendicular to the loading direction. This is an indication that at this level of strain, internal damage is increasing via void coalescence and growth across the liquid channels in a direction normal to the loading direction. The degree of void orientation can be quantified by calculating the variation in the major axis angle relative to the loading direction at different levels of strain. Fig. 3.10 presents the results from this calculation for specimen A. As can be seen in the figure, voids oriented from 0° to 90° are found in the as-cast material, with a mean orientation of ~ 62° and a standard deviation of ± 19. With a moderate amount of strain (A1), the proportion of voids oriented near 90° increases dramatically, with the mean orientation increasing to 70° ± 17. Coalescence is clearly occurring at higher strains (A2), since the mean orientation has increased to 75° while the standard deviation has decreased to ~ 12°. As is shown in Fig. 3.10, there are very few voids at A2 with a major axis oriented less than 45° to the loading direction. The orientation assessment for both specimen B (55° ± 20 for B0, 68° ± 18 for B1, and 76° ± 11 for B2) and specimen C (58° ± 22 for C0, and 73° ± 15 for C1) were similar to the results for specimen A. Quantitative Assessment of Deformation-Induced Damage in a Semi-Solid Al Alloy via X-ray Tomography    56  Angle w.r.t. Load Dir Fr ac tio n of V oi ds 15 30 45 60 75 90 0 0.05 0.1 0.15 0.2 B0, ε = 0 B1, ε = 0.09 B2, ε = 0.39 Angle w.r.t. Load Dir Fr ac tio n of V oi ds 15 30 45 60 75 90 0 0.05 0.1 0.15 0.2 B0, ε = 0 B1, ε = 0.09 B2, ε = 0.39  (a)      (b) Angle w.r.t. Load Dir Fr ac tio n of V oi ds 15 30 45 60 75 90 0 0.05 0.1 0.15 0.2 B0, ε = 0 B1, ε = 0.09 B2, ε = 0.39  (c)  Figure 3.9: Pole figures showing the evolution of the morphological texture of the voids in specimen A as a function of strain: (a) ε = 0 (A0), (b) ε = 0.09 (A1), and (c) ε = 0.39 (A2). Quantitative Assessment of Deformation-Induced Damage in a Semi-Solid Al Alloy via X-ray Tomography    57 Angle w.r.t. Load Direction Fr ac tio n of V oi ds 15 30 45 60 75 90 0 0.05 0.1 0.15 0.2 A0 (ε = 0) A1 (ε = 0.09) A2 (ε = 0.39)  Figure. 3.10: The effect of strain on the orientation of the voids relative to the loading direction for specimen A.  The void spatial orientation plots provide insight into void development in the semi-solid. In this material, isolated void growth occurs along directions in which the most strain energy is available for new surface creation hence the preferred orientation perpendicular to the loading direction. The role of the semi-solid material, located preferentially at grain boundaries and at triple points, is probably two-fold: 1) as less energy is required to create a surface within the liquid it will act to provide a preferential path for void nucleation and crack propagation and 2) in certain areas it may also act as a stress riser helping to localize strain. Thus, the residual liquid is able to influence the crack growth and will add some randomness to the crack propagation path. The results shown in Fig. 3.10 provide a clear indication of this, since the mean orientation angle is increased from A0 to A1, without a corresponding decrease in standard deviation. The tendency of cracks to propagate along paths associated with residual liquid may also contribute to coalescence, and ultimately leads to the low ductility observed at certain critical fractions solid. Quantitative Assessment of Deformation-Induced Damage in a Semi-Solid Al Alloy via X-ray Tomography    58 3.3.4 Stress Development around Individual Voids To support the coalescence hypothesis, a finite element model of the stress behaviour of the tortuous deformation damage was implemented. A 3D ABAQUS5 model of a subset (80 x 60 x 80 voxels) of specimen A1 was developed. A one percent displacement was applied in the vertical or z direction to simulate deformation, and symmetry boundary conditions were applied to the x and y directions. The subset was carefully chosen so that its properties in terms of void number and percentage porosity were representative of the bulk properties. The mesh, which consisted of 1.5 million tetragonal elements, approximated the semi-solid structure as either solid or void; the influence of the 2 % liquid between the grains was not included. All solid elements had the same elastic-perfectly plastic constitutive behaviour, with E = 55 GPa and σyield = 10 MPa [30]. The tortuosity of the voids and their location within the child volume is shown in Fig. 3.11. The evolution of the stress fields around two voids, V1 and V2, is discussed below. In Fig. 3.12, the von Mises stress contours at three different relative depths within the child volume are presented. As shown in the legend, the darkest areas experience a stress of 10 MPa, and correspond to elements that have plastically yielded. Through analysis of the stress contours, it is possible to infer the deformation behaviour leading to the coalescence of the two voids identified in Fig. 3.11. As shown in Fig. 3.12(a), the material along the major axis of the void V2 has yielded at a 45° angle with respect to the loading direction. The stress contours are quite complex, and the material immediately to the left of the void will also soon yield. Furthermore, the peak stresses around this void are ~ 650 times larger than the bulk stress (as determined from the load applied to the entire domain), driving void growth. In Fig. 3.12(b), 40 μm away from (a) in the y direction, it can be seen that the complex stress contour in (a) was shielding void V1. The material between the voids V1 and V2 has yielded much earlier than the rest of this child volume, indicating that these two voids will coalesce. Further evidence of stress shielding and ligament failure are also evident between voids found at the bottom of this Fig. 3.12(b). In Fig. 3.12(c), 100 μm away from (a) in the y direction, voids V1 and V2 are slightly further apart, and the material between these two voids has not yet fully yielded. Since both the voids and the liquid metal will be present along the grain boundaries, it is probable that the interligament region will  5 SIMULIA, Providence, RI, USA Quantitative Assessment of Deformation-Induced Damage in a Semi-Solid Al Alloy via X-ray Tomography    59 be liquid, and will not be able to withstand the applied load. Thus, these two voids will soon coalesce. The finite element model of the deformed region of specimen A has shown that voids act as stress risers and that interaction of the stress fields from adjacent voids in the interligament region further enhances stress concentration. The former will lead to individual void growth in an orientation perpendicular to the loading direction and the latter will lead to void coalescence.    V1 V2  Figure 3.11: 3D morphology of the internal damage in the specimen A1 which was used for the FE analysis. Two separate voids are marked – V1 and V2. Quantitative Assessment of Deformation-Induced Damage in a Semi-Solid Al Alloy via X-ray Tomography    60 (a) (b) (c) X Z von Mises Stress 5 MPa 10 MPa 250 μm V1 V2 V1 V2 V2  Figure 3.12: Contour plots of the von Mises stress at different relative depths inside specimen A1 at the onset of plastic yielding: (a) 0 μm, (b) +40 μm in y-dir, and (c) +100 μm in y-dir. The locations of voids V1 and V2 from Fig. 3.11 are also marked in (b). Quantitative Assessment of Deformation-Induced Damage in a Semi-Solid Al Alloy via X-ray Tomography    61 3.4 Conclusions The first 3D observations of the development of internal damage with strain in a semi-solid commercial Al–Mg alloy have been presented. The combination of interrupted semi-solid tensile tests with x-ray micro-tomography has enabled new insights into the processes involved in semi- solid material deformation. Using this combination of techniques allowed for the relative importance of damage growth and coalescence to be qualitatively and quantitatively assessed. Moreover, the approach also revealed significant void formation, which is believed to be in part due to nucleation processes. A mechanism was hypothesized to explain the influence of pre-existing voids on the failure of semi-solid material.  The as-cast voids act as stress risers and allow strain to be accommodated by growth of this pre-existing damage through the liquid along grain boundaries. In the absence of pre-existing voids, nucleation of voids would occur to create damage. This ‘new’ damage behaves identically to the as-cast porosity. The qualitative and quantitative analysis performed in this work all support this hypothesis. Firstly, the high resolution tomography scans showed that voids grow and coalesce along the thin channels demarking grain boundaries. Secondly, the quantitative assessment of void distributions showed that semi-solid deformation is initially controlled by growth of as-cast porosity and nucleation of damaged-based voids. These voids then grow preferentially in a direction normal to the applied load via void coalescence and unzipping along the liquid at the grain boundaries. This preferential growth was quantified by the void spatial orientation plots while finite element modeling supported the mechanism of void coalescence. At some critical strain, localized coalescence of the voids occurs leading to a decrease in the void number density and final failure. In summary,  the combination of x-ray micro-tomography and semi-solid deformation is an effective tool for assessing damage evolution in semi-solid aluminum alloys. For example, different alloy systems and compositions can be characterized to determine the interplay between internal damage development and final failure. Further, the stress-strain behaviour of the material can be measured directly, including both cross-sectional area and internal void growth. The main limitations of the current technique are two-fold: one, the use of interrupted tensile tests on partially remelted material in combination with tomography, rather than continuous in-situ observation of the material as it solidifies; and two the inability to conclusively distinguish between void formation by growth of as-cast porosity and nucleation of new voids. Quantitative Assessment of Deformation-Induced Damage in a Semi-Solid Al Alloy via X-ray Tomography    62 3.5 References 1. Campbell, J., Castings. Butterworth Heinemann, 2 ed., 1991. 2. David, S. and Debroy, T., Science, 257, 1992, (pp. 497). 3. Pellini, W. S., Foundry, 1952, (125). 4. Feurer, U., Giesserei-Forschung, 1976, (75). 5. Warrington, D. and McCartney, D. G., Cast Metals, 3, 1991, (202). 6. Guven, Y. F. and Hunt, J. D., Cast Metals, 1, 1988, (104). 7. Farup, I., Drezet, J. M. and Rappaz, M., Acta Mat., 49, 2001, (1261). 8. Fredriksson, H. and Lehtinen, B., International Conference on Solidification and Casting of Metals, Sheffield, UK, Metals Society, 1977, (260). 9. Davidson, D., Viano, D., Liming, L. and St.John, D., Shape Casting: The John Campbell Symposium, M. Tiryakioglu, P. N. Crepeau, Eds., San Francisco, CA, TMS, 2005, (175). 10. Clyne, T. W. and Davies, G. J., Brit Found., 74, 1981, (65). 11. Farup, I. and Mo, A., Metall. Mater. Trans. A, 31A, 2000, (1461). 12. Eskin, D. G., Suyitno and Katgerman, L., Prog. Mater. Sci., 49, 2004, (629). 13. Suyitno, Kool, W. H. and Katgerman, L., Metall. Mater. Trans. A, 36A, 2005, (1537). 14. Rappaz, M., Drezet, J. M. and Gremaud, M., Metall. Mater. Trans. A, 30A, 1999, (499). 15. Rindler, W., Kozeschnik, E., Enzinger, N. and Buchmayr, B., Mathematical Modeling of Weld Phenomena 6, H. Cerjak, H. K. D. H. Bhadeshia, Eds., Vienna, AUS, Maney Publishing, 2002, (819). 16. Katgerman, L., J. Metals, 34, 1982, (46). 17. Braccini, M., Martin, C. L. and Suery, M., Model. Cast. Weld. Adv. Solid., 2000, (18). 18. Provatas, N., Greenwood, M., Badrinarayan, A., Goldenfeld, N. and Dantzig, J., Intl. J. Mod. Phys. B, 19, 2005, (pp. 4525). 19. Vernede, S., Jarry, P. and Rappaz, M., Acta Mat., 54, 2006, (4023). 20. Colley, L. J., Wells, M. A. and Maijer, D. M., Mater. Sci. Eng. A, 386(1-2), 2004, (140). 21. Van Haaften, W. M., Kool, W. H. and Katgerman, L., Mater. Sci. Eng. A, 336, 2002, (1). 22. Thompson, S., Cockcroft, S. L. and Wells, M. A., Mater. Sci. Techn., 20(4), 2004, (497). 23. Phillion, A. B., Cockcroft, S. L. and Lee, P. D., Scripta Materialia, 55, 2005, (489). 24. Toda, H., et al., Metall. Mater. Trans. A, 37A, 2006, (1211). 25. Abramoff, M. D., Magelhaes, P. F. and S.J., R., Biophotonics International, 11(7), 2004, (36). 26. Perona, P. and Malik, J., IEEE- PAMI, 12, 1990, (629). 27. Lee, P. D., Atwood, R. C., Dashwood, R. J. and Nagaumi, H., Mater. Sci. Eng. A, 328, 2002, (213). 28. Lee, P. D. and Hunt, J. D., Acta Mat., 45, 1997, (4155). 29. Yoo, T. S., Ackerman, M. J., Lorensen, W. E., Schroeder, W., Chalana, V., Aylward, S., Metaxes, D. and Whitaker, R., Proc Medicine Meets Virtual Reality, J. Westwood, Ed., Press Amsterdam, 2002, (586). 30. Sengupta, J., Cockcroft, S. L., Maijer, D. and Larouche, A., Mater. Sci. Eng. A., 397, 2005, (157).       63 Chapter 4: A New Methodology for Measurement of Semi-Solid Constitutive Behaviour and its Application to Examination of As-Cast Porosity and Hot Tearing in Aluminum Alloysα 4.1 Introduction and Background During casting of aluminum alloys, the semi-solid material is frequently exposed to a tensile stress state due to temperature gradients and mechanical constraints. The constitutive response of the semi-solid phase to these tensile stresses is critical in controlling many solidification defects such as hot tearing [1] and porosity [2]. These defects in turn affect productivity and the mechanical properties of the cast material [3]. Both hot tearing and porosity are related to semi-solid constitutive behaviour. Hot tearing is a stress-related defect [4], while porosity is related to stress via the coefficient of thermal expansion and solidification shrinkage. Although a few authors have shown experimentally that hot tears nucleate at specific intergranular voids (e.g. [5]), the role of as-cast porosity on hot tearing is currently unknown. One study which examined both hot tearing and porosity was performed by Suyitno et al. [6], who developed a model to differentiate between void formation leading to microporosity and void formation leading to hot tearing. While this model presented a sophisticated approach to hot tearing prediction, it assumed no interaction between hot tearing and porosity, nor was any experimental evidence presented. In order to study the interaction between porosity and hot tearing, a semi-solid deformation apparatus is required that is able to apply small amounts of deformation to semi-solid materials. Many different techniques have been developed to measure the shear (e.g. [7]), compressive (e.g. [8]), and tensile (e.g. [9]) behaviour of the semi-solid material, and examine hot tearing susceptibility. In terms of the hot tearing defect, it is the tensile behaviour which is most relevant since this behaviour relates to openings in the semi-solid microstructure leading to material failure. Results from these tensile experiments have shown that semi-solid strength is strongly dependent on the fraction solid, microstructure size, morphology, and especially alloy chemistry [9-16]. Furthermore, at high fraction solid, the results show two characteristic transition points.  α A version of this chapter will be submitted for publication. Phillion, A.B., Cockcroft, S.L., and Lee, P.D. A New Methodology for Measurement of Semi-Solid Constitutive Behaviour    64 The first characterizes the ability to sustain tensile stress, and is known as the mechanical de- coherency point. The second characterizes the ability to accumulate tensile ductility and is known as the point of zero-ductility. Many researchers have reported that tensile mechanical de- coherency occurs at a fraction solid of ~ 0.90–0.95 [9, 16], and that appreciable ductility is possible only at fraction solids close to unity [14, 17]. However, one issue with previous experimental observations is repeatability since the reported tensile strength results for the same alloy often differ considerably, depending on the technique used. For example, Colley’s [9] semi-solid measurements of AA5182 are two to three times larger than those measured by Van Haaften [12]. Kron [14] and Twite’s [11] results for AA6061 also differ by similar amounts. There are three inherent challenges during measurement of semi-solid constitutive behaviour: 1. Thermal Control – Thermal control is critical during semi-solid testing because small changes in temperature and thus fraction solid greatly affect constitutive behaviour. Historically, temperature has been measured using a thermocouple to provide the ability for direct monitoring of the deformation region. Surface-welded thermocouples are generally used since embedded thermocouples can induce defects and affect the measured properties [9]. However, the challenge is that the weld between the thermocouple and the specimen will often fail due to localized surface melting, which has tended to place an upper bound on the testing temperature. Furthermore, welded thermocouples may induce internal defects. 2. Ductility Measurement – Measurement of the semi-solid ductility is complex. A lengthwise measurement cannot be used because the deformation localizes even at small values of strain. Similarly, a dilatometer cannot be used because the force that acts to close the dilatometer jaws could be greater than the compressive strength of the material. Another aspect of ductility measurement, as shown in Chapters 2 ([18]) and 3, is the possibility of internal damage accumulation during semi-solid tensile testing. This damage is not accounted for when macroscopically measuring diametral or length changes. 3. Grip Design – Fixed grips cannot be used since this would apply a compressive load during heating due to thermal expansion. Additionally, if the specimen is in a vertical orientation, the design should consider the mass of the grip since this mass could enable accumulation of creep strain. The development of a successful technique for measuring semi-solid constitutive behaviour should address the above three challenges. Previous researchers have adopted two basic A New Methodology for Measurement of Semi-Solid Constitutive Behaviour    65 approaches: partial remelting of the solid sample (e.g. [19]), and in situ solidification samples (e.g. [20]). Both of these approaches have significant advantages and serious drawbacks which are listed in Table  4A. One major issue is the measured properties, since a lower flow stress and ductility is reported for the solidifying melt specimens [21]. This difference is attributed to the existence of thicker liquid films in the in situ specimens for similar temperatures. It is clear from Table  4A that the partial remelting tests are best suited for measuring semi-solid constitutive behaviour, while the in situ solidification tests excel in addressing hot tearing susceptibility. In this chapter, a new methodology for conducting semi-solid tensile tests using the partial remelting approach is presented. This methodology incorporates novel techniques for temperature and displacement control within a Gleeble 35001 thermomechanical simulator. The methodology is then applied to investigate the constitutive behaviour of a semi-solid commercial Al–Mg alloy both with and without as-cast porosity, elucidating the role of as-cast porosity on hot tearing.  Table 4A: Advantages and disadvantages of the two testing approaches. Constrained Solidification Tests Partial Remelting Tests Advantages Disadvantages Advantages Disadvantages As-cast structure and defects Liquid flow from low to high fs Industrial solidification conditions Loading area is not known Container–sample surface friction effects fs is changing with time and may be unknown at crack Loading area is known No melt friction effects Good geometrical and load control Testing is possible only at high fs The fs – temperature relationship on heating is different than on cooling   1 Dynamic Systems Inc., Poestenkill, NY, USA A New Methodology for Measurement of Semi-Solid Constitutive Behaviour    66 4.2 Experimental Methodology 4.2.1 Materials and Analysis The alloy AA5182 was chosen for testing because of its long freezing range and susceptibility to hot tearing. To examine the effect of as-cast porosity on the semi-solid behaviour of this alloy, tests were performed on both as-cast material and hot isostatic pressing (HIP) material. The HIP process consists of a high temperature, high pressure cycle designed to remove as-cast porosity. For this work, the proprietary DENSALL II (Bodycote, Inc) HIP process was used. Cylindrical tensile specimens of as-cast material were machined from a Direct Chill (DC) cast ingot, normal to the casting direction, and parallel to and ~ 6-10 cm below the surface of the ingot. The HIP condition specimens were machined from two blocks of as-cast material that had undergone the HIP cycle. These blocks were cut from same cast ingot as the as-cast specimens such that the as-cast and HIP specimen orientation and positioning relative to the surface of the ingot were similar. The test specimen geometry, shown in Fig.  4.1, contains a central gauge region 15 mm in length and 5 mm in diameter. The remainder of the specimen has a diameter of 10 mm. A series of experiments were conducted between 500 and 580°C at a strain rate of ~ 0.001 s-1 on the as-cast and the HIP materials to measure the semi-solid constitutive behaviour. The experimental summary is provided in Table  4B. The steady-state flow stress is defined as the maximum measured stress value over the range of strain 0 < ε < 0.01, while strain is calculated based on the diametral reduction in the gauge as measured by a laser dilatometer. To compare the microstructure of the two materials, x-ray micro-tomography (XMT) and scanning electron microscope – energy dispersive spectroscopy (SEM-EDS) techniques were used to characterize the porosity and homogenization of the initial undeformed material. 3D tomographic images of the internal damage caused by the semi-solid deformation process were also acquired for as-cast specimens tested at 500 and 550°C, and a HIP specimen tested at 560°C. To obtain quantitative results, the tomographic datasets were subjected to image analysis using the software packages ImageJ [22] and Amira2, while the composition data from SEM-EDS was subjected to a rank sort analysis using a technique previously developed by Ganesan et al. [23].  2 Mercury Computer Systems, Chelmsford, MA, USA A New Methodology for Measurement of Semi-Solid Constitutive Behaviour    67 120 mm Φ = 10 mm Φg = 5 mm TC #2 TC #1 TC #3 15 mm  Figure 4.1: Specimen geometry showing dimensions and locations of control thermocouples.   Table 4B: Experimental conditions and results. (The flow stress is given in MPa) As Cast Material HIP Material Temp. (°C) Flow Stress Diametral Strain Temp. (°C) Flow Stress Diametral Strain 570 1.75 0 580 1.2 0 560 7.2; 1.4; 1 0; 0; 0 575 1.2 0 557 6.4 0 572 1.2 0 548 11.5 0 570 2.71; 4.12; 9.5 0; 0; 0 545 12.4 0.01* 565 9.3; 10.0; 11.7 0; 0.01; 0.021* 540 13.1 0.01* 560 12.5; 12.2; 12.3; 12.9; 12.0 0.008; 0.022; 0.085; 0.09; 0.1* 535 14.5 0.015* 550 13.0; 13.3; 13.5; 14.0; 13.1; 13.3 0.008; 0.009; 0.028; 0.045 0.124; 0.142* 527 14.8 0.035 535 15.6 0.011* 520 16.2 0.057* 515 17.6 0.192* 500 20.4 0.075* 500 21.4 0.184* *Denotes semi-solid ductility limit A New Methodology for Measurement of Semi-Solid Constitutive Behaviour    68 4.2.2 A New Semi-Solid Deformation Methodology The focus of the new methodology is to improve the thermal control. The Gleeble 3500 uses DC electric current to heat tensile specimens via resistive heating, and a servo-hydraulic system for deformation. Operation of the Gleeble requires a continuous feedback loop between the specimen temperature and software controlling the electric current input. This feedback is obtained via a control thermocouple welded near the area of interest on the specimen. If the thermocouple fails, the control software will abort the experiment for safety.  Prior experience has shown that localized surface melting can cause the thermocouple wires to detach from the specimen, resulting in a loss of control feedback and thus termination of the experiment. The new methodology also addresses difficulties associated with precision in regards to specimen deformation. The challenge with control of the deformation is that the position of the free end of the specimen will move during heat-up due to thermal expansion. The exact amount of this thermal expansion will vary from one test to the next due to variations in the contact pressure between the grips and specimen, symmetry with respect to specimen placement, and the location of the control thermocouple. One feature of the Gleeble apparatus is that during specimen heat-up, an asymmetric parabolic temperature profile develops along the specimen length. This parabolic shape results from both electrical-resistive heating along the specimen length and localized heat loss at the ends of the specimen via the water-chilled copper grips. Furthermore, the shape can vary from test to test. To address this issue, a reduced area region was added to the specimen geometry. This region of reduced cross-section will have the highest current density and thus will be hottest. By taking advantage of the parabolic temperature distribution and the near isothermal conditions within the reduced area region, it has been possible to develop a solution to the problem of control thermocouple detachment (failure). The new semi-solid deformation methodology incorporates a two thermocouple control strategy. Thermal control is provided by one thermocouple (TC#1, in the centre of the gauge region, Fig. 4.1) during heating up to a set temperature near the solidus, and by a different thermocouple (TC#2, just outside the gauge region, Fig. 4.1) to complete heating to the test temperature and during deformation. Fig. 4.2 outlines the methodology used to conduct the experiments. This methodology was written as a program in the Gleeble Scripting Language, software which is part of the Gleeble digital control A New Methodology for Measurement of Semi-Solid Constitutive Behaviour    69 system. In the first stage, the specimen is quickly heated until the reading at TC#1 is 400°C. In the second stage, the temperature of the specimen is controlled by TC#2 and is increased at a slow rate. Since the temperature needed at TC#2 to achieve the test temperature in the reduced area region near TC#1 is not known a priori, the difference in temperature between the two thermocouples, ΔT, is continually recorded during the second stage of the control program. A linear correlation for ΔT as a function of the temperature at TC#2 is developed and is continually refined until TC#1 stops functioning. This methodology allows for an accurate estimate of the set-point temperature at TC#1 based on the temperature at TC#2, and thus the test can be completed even if TC#1 fails. Due to the parabolic temperature distribution and the difference in cross-sectional area, TC#2 remains at a significantly lower temperature during testing and thus the weld fixing the thermocouple is maintained. Also shown in Fig. 4.2 is the new methodology for applying specific amounts of strain to the specimen, by firstly moving the cross-head precisely to offset the thermal expansion.  After a 30 s hold at the test temperature for thermal stability, the cross-head is displaced at a rate of 50 μm s-1, in increments of 25 μm. After each increment in displacement, the force is measured. If the load cell records a force of 15 N or greater, it is assumed that the cross-head is aligned with the end of the specimen. At this point, the cross-head is displaced for a set distance in a set time in order to create the desired deformation conditions within the semi-solid specimen. One of the potential drawbacks of this Two Thermocouple Technique (2TC) is that ΔT is related to strain since the ratio of cross-sectional area at TC#2 to TC#1 will increase as the specimen is deformed in tension. As a result, the reduced area region could be hotter than calculated based on the original correlation, and thus be a source of error in the temperature measurement. This error in the temperature measurement can be calculated based on the change in cross-sectional area at TC#1 due to tensile deformation. For example, for a set-point temperature of 550°C, a strain of 0.15 will lead to a temperature increase due to deformation of 6°C. However, strains of this magnitude were rarely achieved in the current experiments. The Gleeble setup for measuring semi-solid constitutive behaviour included the following options: a low-force jaw set for application of small loads, a 1000 lb load cell for accurate force A New Methodology for Measurement of Semi-Solid Constitutive Behaviour    70 measurement, and a Beta LaserMike3 162-100 laser dilatometer for remote measurement of the diametral change as a function of load. The typical evolution of temperature, diametral change in the gauge region, and force with time obtained from the 2TC technique is shown in Figs. 4.3 and 4.4. The results shown are for a test temperature of 550°C and a total strain of 0.02. In most of the experiments, TC#1 failed completely before the start of deformation. However, in this particular case, the weld at TC#1 did not fail and thus the thermocouple continued functioning during the entire deformation process. Furthermore, this specimen was instrumented with a third thermocouple, TC #3, on the shoulder opposite to TC#2. Referring firstly to the evolution of temperature with time, it can be seen from Fig. 4.3 that the specimen heats up to the test temperature in a controlled manner. The changes in heating rate at 75 and 120 s are part of the normal control heat-up algorithm. The large thermal spike in all three thermocouples at ~ 100 s was abnormal and was interpreted by the control algorithm as a failure of TC#1, and thus ΔT was fixed at 72°C and no longer updated. The sample was then heated to the calculated set-point temperature based on TC#2 alone and held there for 30 s. At the start of deformation, the actual temperature at TC#1, which was still functioning, was 550°C. At the end of deformation (55 s later and ε = 0.028), the temperature in the reduced area region had changed by 1°, to 549°C.  This value, while indicating a trend, is within the measurement error of the Gleeble apparatus. Since the desired temperature at the location of TC#1 was 550°C, it is clear that control based on a set-point temperature at TC#2 can successfully be used to control the temperature with the gauge length to a reasonable degree of accuracy. Referring next to the evolution of diametral change and force, three different regions can be seen in Fig. 4.4. In the first part of the curve, the specimen diameter increases at a constant rate, corresponding to the thermal expansion. In the second part of the curve, the specimen diameter continues to increase but at an irregular rate, due to localized melting within the gauge length. In both these regions, the measured force is close to zero, and the observed oscillations are caused either by experimental noise or friction related to apparatus–specimen interaction during thermal expansion. In the third part of the curve, the specimen yields and deforms in tension. The diameter decreases and the load cell measures the force required for this deformation to occur.  3 Beta LaserMike, Dayton, OH, USA A New Methodology for Measurement of Semi-Solid Constitutive Behaviour    71    Two TC Approach Software Flowchart Inputs: Test Temp, Δ , Δ Heat TC#1 to 400°C @ 5°C/s Heat by 1°C @ 1°C/s Record ΔT = TTC1 – TTC2 Is TTC2 +ΔT = Test Temp? Hold at Temp for 30 s Move Cross Head 50 mm @ 50 μm / s Is Force > 15 N ? Move Cross Head Δ in time Δ Record F and Δf using 1000 N load cell and laser l t l t   Figure 4.2: Flowchart outlining the Two Thermocouple Control Technique for conducting semi-solid deformation tests in the Gleeble 3500 thermomechanical simulator. A New Methodology for Measurement of Semi-Solid Constitutive Behaviour    72 Time (s) Te m pe ra tu re (o C ) 100 150 200 250400 450 500 550 600 TC#1 TC#2 TC#3 A  Figure 4.3: An example of the temperature-time profile obtained using the 2TC technique on HIP material. The arrow at ‘A’ marks the start of deformation. Time (s) Fo rc e (N ) Δφ (μm ) 50 100 150 200 250 -50 0 50 100 150 200 250 300 350 -40 -20 0 20 40ΔφForce A  Figure 4.4: An example of the force and dimetral change during semi-solid deformation on HIP material. The arrow at ‘A’ marks the start of deformation. A New Methodology for Measurement of Semi-Solid Constitutive Behaviour    73  4.3 Results and Analysis The experimental results are presented below. Firstly, the as-cast and HIP microstructure in the initial undeformed state is characterized. Secondly, a characterization of the semi-solid deformation process is presented via (a) analysis of fracture surfaces obtained using an SEM, (b) analysis of the semi-solid tensile constitutive behaviour, and (c) analysis of the tomographic images of tensile specimens deformed in the semi-solid region. 4.3.1 Initial Conditions of the As-Cast and HIP Materials The as-cast porosity and micro-segregation of the solute elements were characterized in the initial undeformed state of both the as-cast and HIP materials to determine the effect of HIP processing on these solidification phenomena. Fig.  4.5 shows 2D cross-sectional slices from 3D tomographs of (a) undeformed as-cast material and (b) undeformed HIP material at a resolution of ~ 2.5 μm per voxel side. The black areas of Fig.  4.5 denote voids, while the light or white areas denote regions enriched in the solute phases and may represent intermetallics or grain boundary triple points. Fig.  4.6 shows the content of the major solute element, Mg, at 100 different locations within the microstructure. This figure provides an estimate of Mg segregation during solidification by assuming that the location with the lowest Mg content is the dendritic core and thus fs = 0, and that the location with the highest Mg content is the last liquid to solidify in the interdendritic region and thus fs = 1. Furthermore, the analysis was performed using the Weighted Interval Rank Sort (WIRS) method [23], which is an averaging technique for determining the degree of chemical in-homogeneity within an alloy. It is clear from Fig.  4.5(a) that industrially DC cast aluminum alloys contain significant microporosity. This porosity, which forms due to a combination of gas, shrinkage and thermomechanical loading during the casting process, appears well distributed with morphologies ranging from circular to elongated and tortuous. In contrast, the HIP material, Fig.  4.5(b), does not appear to contain any porosity. It would appear that the HIP processing method has ‘closed’ the original as-cast porosity, an effect which has been previously documented (e.g. [24]). Although there may still be voids remaining in the HIP material, these voids are below the resolution level of the tomographic scan. A New Methodology for Measurement of Semi-Solid Constitutive Behaviour    74 Image analysis of the 3D tomographs was used to quantify the initial state of porosity in as- cast AA5182. The material contains an initial porosity (%P) of 0.3–0.5%, a void number density (nv) of ~430 mm-3, and an average void radius (ravg) of 7.5 μm. Due to a resolution limit of 2.5 μm per voxel side, the small voids with radius less than 3.1 μm were not included in the analysis. Thus, there is clearly much microporosity distributed throughout the as-cast material, and some of the voids can be quite large, on the order of 50–74 μm. Maire et al. [25] have also quantified the distribution of voids in as-cast AA5182, but using synchrotron XMT. In their work, the following values were measured: %P ~ 0.1%, nv ~ 2025 mm-3, and ravg ~ 4 μm. The results from the two studies are comparable apart from the void number density. This difference can be explained by comparing the resolution in the two studies. In Maire’s study, the tomography resolution, at 0.7 μm per voxel, was much higher than in the current work. Consequently, the myriad of small voids occurring around second phase particles, observed by Maire, were too small to be included in the void number density calculation performed in the current analysis. The effect of the HIP processing on micro-segregation is shown in Fig.  4.6. As can be seen in the figure, the composition of Mg in the as-cast material varies between ~ 3.0 at% at the dendrite core to 8.9 at% in the interdendritic liquid. In contrast, the compositional variation in the HIP material was between 4.9 at% and 5.7 at%. From this plot, it is clear that the HIP processing method is significantly homogenizing the microstructure. Fig.  4.5 also provides visual evidence of this homogenization, since Fig.  4.5(b) clearly contains fewer of the light or white solute-rich regions as compared to Fig.  4.5(a), and these areas appear more rounded in shape. 4.3.2 Fracture Surface Observations SEM images of fracture surfaces taken from as-cast AA5182 specimens tested via the 2TC methodology are shown in Fig.  4.7. Four images are shown, corresponding to test temperatures between 500 and 560°C. In (a), 500°C, the fracture surface contains complex topography, indicative of ductile fracture. In (b), 520°C, the fracture surface is similar to that at 500°C, although it appears that some of the isolated regions of material appear smooth and glassy-like. In (c), 540°C, much of the fracture surface is covered in small protuberances, representative of interdendritic melting. In (d), 560°C, the fracture surface is dominated by a smooth and glassy- like appearance, providing evidence of much liquid during deformation. A New Methodology for Measurement of Semi-Solid Constitutive Behaviour    75  (a)       (b) Figure 4.5: 2D cross-sectional slices from 3D tomography scans of the undeformed material: (a) as-cast AA5182, and (b) HIP AA5182.  fs So lu te C on te nt (a t% ) 0 0.2 0.4 0.6 0.8 10 2 4 6 8 10 HIP Material As-Cast Material  Figure 4.6: Solute profiles of Mg content in AA5182 in both the as-cast and HIP conditions. A New Methodology for Measurement of Semi-Solid Constitutive Behaviour    76  (a)      (b)  (c)      (d) Figure 4.7: SEM Images of the fracture surface of as-cast AA5182 specimens tested using the 2TC methodology: (a) 500°C, (b) 520°C, (c) 540°C, (d) 560°C. A New Methodology for Measurement of Semi-Solid Constitutive Behaviour    77 4.3.3 Semi-Solid Constitutive Behaviour The stress-strain behaviour of four as-cast specimens, tested at 520, 527, 535, and 545°C, is shown in Fig.  4.8. The results show typical behaviour: fully solid with ductility (520°C), semi- solid with ductility (527 and 535°C), and semi-solid with little ductility (545°C). It appears that the material is softening with increasing strain. This softening is due to the accumulation of internal damage during deformation, as previously discussed in Chapters 2 ([18]) and 3. Since stress is determined from a cross-sectional area estimate based on the external specimen diameter, its true value is underestimated because of the internal damage. The steady-state flow stress and ductility results for both the as-cast and HIP AA5182 semi- solid experiments are shown in Fig.  4.9 as a function of temperature. The steady-state flow stress was defined as the maximum true stress observed in the measured stress-strain curves. The ductility was also determined from the stress-strain curves, and was defined as the strain prior to specimen fracture. Referring firstly to the stress data, it can be seen from Fig.  4.9 that the measured flow stresses for the two materials are quite similar in the range 500 to ~ 530°C. In this temperature range, the alloy is (almost) fully solid so both materials should have similar failure stresses. Above 525°C, melting begins to occur. In the as-cast material, the strength continues to slowly decrease with increasing temperature and fraction liquid until 560°C where the mechanical de-coherency point is reached and the material loses all its strength. In the HIP material, the mechanical de-coherency point occurs at a higher temperature, 570°C. Referring next to the ductility data, it can be seen from Fig.  4.9 that the HIP material has significantly more ductility at all temperatures tested as compared to the as-cast material. The temperature of zero- ductility was measured to be 548°C in the as-cast material and 565°C in the HIP material. A series of experiments were also carried out to assess the repeatability of the semi-solid strength measurements of the 2TC technique. A total of eleven tests were carried out at 550 and 560°C on the HIP material. The measured values of flow stress were found to be quite consistent, at 13.3 ± 0.40 MPa for 550°C and 12.4 ± 0.35 MPa for 560°C. Thus, the 2TC technique provides both accurate and repeatable data. A New Methodology for Measurement of Semi-Solid Constitutive Behaviour    78 ε measured σ( M P a) 0 0.005 0.01 0.015 0.02 0.0250 5 10 15 Temperature 520oC 527oC 535oC 545oC  Figure 4.8: Stress-strain curve for as-cast AA5182 at 520, 527, 535 and 545°C. Temperature (oC) σ( M P a) ε m ax 500 520 540 560 580 0 5 10 15 20 25 0 0.05 0.1 0.15 0.2 As-cast σε HIP σε  Figure 4.9: Variation in (a) flow stress and (b) true strain with temperature for as-cast AA5182 and HIP AA5182. A New Methodology for Measurement of Semi-Solid Constitutive Behaviour    79 4.3.4 X-ray Micro-Tomography To examine the impact of as-cast porosity and the liquid phase on deformation, XMT was used to compare the extent of internal deformation in the as-cast and HIP specimens. Fig.  4.10 shows 2D cross-sectional slices from 3D tomographs of (a) as-cast material deformed at 550°C (ε = 0.0008), (b) HIP material deformed at 560°C (ε = 0.09), and (c) as-cast material deformed at 500°C (ε = 0.08) at a resolution of ~ 12 μm per voxel side. Note that the slices shown in Fig.  4.10 are from a region of material 1 to 2 mm away from the fracture surface. The effect of as-cast porosity on semi-solid tensile deformation can be seen clearly in the 2D cross-sectional slices of deformed material, shown in Fig.  4.10. In comparing these images, it appears that the as-cast semi-solid specimen, Fig.  4.10(a), contains voids which are larger and much more numerous as compared to the HIP semi-solid specimen, Fig.  4.10(b). In fact there was only one large crack in this HIP specimen, and it was located near the weld point for the control thermocouple. This weld point is believed to have acted as a stress riser during semisolid deformation, causing the crack to form. The effect of the liquid phase on semi-solid tensile deformation can be seen by comparing the two images  4.10(a) and  4.10(c). Three salient observations can be made. Firstly, the amount of internal damage in the fully-solid specimen, Fig.  4.10(c), appears to be much less as compared to the semi-solid specimen, Fig.  4.10(a). Secondly, the voids in Fig.  4.10(c) appear much more rounded as compared to Fig.  4.10(a). Thirdly, some of the voids in Fig.  4.10(a) appear to have formed along common lines, possibly the grain boundaries that would have contained liquid during deformation, whereas in Fig.  4.10(c), the voids remained isolated. Thus, it would appear that the role of the liquid is to aid void nucleation, growth and coalescence. A New Methodology for Measurement of Semi-Solid Constitutive Behaviour    80    (a)       (b)  (c) Figure 4.10: 2D cross-sectional slices from 3D tomography scans of deformed specimens: (a) as-cast AA5182 (T = 560°C, ε = 0.0008), (b) HIP AA5182 (T = 560°C, ε = 0.09), and (c) as-cast AA5182 (T = 500°C, ε = 0.08). A New Methodology for Measurement of Semi-Solid Constitutive Behaviour    81 4.4 Discussion 4.4.1 Evolution of Fraction Solid with Temperature One of the difficulties with the semi-solid deformation methodology and subsequent analysis relates to the quantification of fraction solid. In general, the evolution of fraction solid with temperature is experimentally determined via cooling from the liquid to solid phase. In the current approach, the experiments were conducted via heating and partial remelting of the solid specimen. Since the relationship between fraction solid and temperature is going to be different during solidification and melting – especially at high fraction solid under non-equilibrium conditions due to the kinetic effect – the actual fraction solid using the reheating methodology is difficult to ascertain. As preliminary work for this study, a number of different techniques including two-pan and single-pan [26] differential scanning calorimetry were used to measure the evolution of fraction solid with temperature of both as-cast and HIP AA5182 during melting processes. Unfortunately, none of the results were consistent. In the range of interest relevant to hot tearing, 0.9 < fs < 0.98, the latent heat associated with the phase change can be small relative to the heat capacity of the solid material. Thus, this portion of the fraction solid – temperature relationship is difficult to measure and interpret. Due to the difficulties associated with measuring fraction solid, the fracture surface images shown in Fig.  4.7 were used to qualitatively validate the presence of the liquid phase during semi-solid tensile testing of AA5182. These images clearly show that the solidus temperature is between 520 and 540°C, and that by 560°C, much of the grains boundaries are covered in liquid. These observations are similar to the relationship between fraction solid and temperature published by Thompson et al. [27], who determined that the solidus temperature for as-cast AA5182 was 525°C. 4.4.2 Semi-Solid Constitutive Behaviour The experimental data presented in Fig.  4.9 has provided a good description of the semi-solid constitutive behaviour of as-cast and HIP AA5182 at temperatures below the mechanical de- coherency point. The results have shown that both the flow stress and ductility of semi-solid AA5182 is significantly decreased as compared to the fully-solid state. Similar trends have A New Methodology for Measurement of Semi-Solid Constitutive Behaviour    82 previously been reported for AA5182 [9, 28], although their magnitude was found to be different due to compositional variation. The effect of the liquid phase on ductility is readily apparent when comparing the results from the SEM images provided in Fig.  4.7 to Fig.  4.9, and when analyzing the stress-strain curves in Fig.  4.8. The fracture surfaces of specimens deformed at 500 and 520°C, Fig.  4.7(a) and (b), contain little evidence of liquid. At these temperatures, the material exhibits considerable ductility. The specimen tested at 527°C, Fig.  4.8, also exhibits considerable ductility. While this last test was performed above the solidus temperature, it contained only a small amount of liquid. The microstructure in the specimen consisted of small isolated liquid pockets, small regions of liquid film and thus significant bridging between the grains. As the temperature is further increased, the measurements show decreased ductility, while fracture surfaces provide increasing evidence of the presence of liquid at the time of failure. At 545°C, Fig.  4.8, failure occurred at low ductility, indicating that the liquid phase existed as a continuous film with limited solid bridging between the grains. At 560°C, Fig.  4.7(d), the fracture surface appears to have been largely covered by liquid during deformation, leading to a complete loss in ductility. The effect of liquid on semi-solid flow stress is more subtle in nature as compared to ductility since the experimental results, showed in Fig.  4.9, indicate a gradual loss of flow stress as the temperature is increased towards the mechanical de-coherency point. In this case, the presence of liquid reduces the effective load-bearing area of the tensile specimen, resulting in increased local flow stresses and thus increased strain and strain rate. Material failure occurs when the strain at the local scale exceeds the (limited) semi-solid ductility. These experiments have also revealed that processing via HIP significantly increases both ductility of AA5182 in the fully-solid and semi-solid states, and the temperature for mechanical de-coherency. The increase in ductility in the fully solid state is in good agreement with previously reported results [24]. The increase in mechanical de-coherency occurs because of the homogenization undergone by the material during HIP processing, as shown in Figs.  4.5 and  4.6. It has previously been found by a number of researchers that de-coherency occurs at a fraction solid of ~ 0.95 [9, 16]. The results from the as-cast material experiments corroborate those earlier findings since the de-coherency temperature in the as-cast material, 560°C, corresponds to a fraction solid of ~ 0.95 using the fraction solid – temperature relationship measured by Thompson [27]. Due to the HIP process, the low melting temperature phases are not present and A New Methodology for Measurement of Semi-Solid Constitutive Behaviour    83 thus these specimens must be heated to a higher temperature to reach a fraction solid of 0.95 and exceed the de-coherency limit. In this case, the HIP process appears to have shifted the relationship between fraction solid and temperature by ~ 10°C. There are two possible mechanisms which would produce the observed gains in semi-solid ductility obtained from the HIP process. Firstly, as shown in Fig.  4.6, the shift in the relationship between fraction solid and temperature due to homogenization will increase the ductility in the HIP material as compared to the as-cast material for a given temperature. Secondly, as shown in Fig.  4.5, the HIP process is removing the as-cast porosity. Assuming that the grains are completely surrounded by liquid films, the pre-existing voids found in the as-cast material will act as damage nucleation sites and initiate failure along the grain boundaries. Even with significant solid bridging between grains, the presence of voids will still provide stress concentration, and will promote the loss of ductility at small values of strain. 4.4.3 Hot Tearing and Porosity The observed effects of as-cast porosity on tensile deformation, shown in Figs.  4.8– 4.10, indicate that its presence significantly reduces semi-solid ductility. While the ductility behaviour reported in Table  4B provides a macroscopic overview of the as-cast and HIP material behaviour, analysis at the micro-scale level provides further insight into the relationship between porosity and hot tearing. A quantitative comparison of the voids found in the three specimens shown in Fig.  4.10 was performed to provide a description of the evolution in internal damage caused by deformation both with and without porosity. This quantification, shown in Table  4C, characterizes the effect of fraction solid and material condition on the fraction porosity (%P), maximum void radius (rmax), and the void to strain ratio (VSR) in the three deformed specimens from Fig.  4.10. The VSR is defined as 1 d t ε ε− , where εd is the diametral strain assuming a fully dense specimen and εt is the strain including both diameter reduction and growth of internal voids, and provides a measure of the percentage of strain being accumulated internally. A description of the quantitative analysis methodology was previously provided in Chapter 3. The results shown in Table  4C indicate that the voids are more prevalent in the semi-solid as- cast material, specimen ‘A’, as compared to the other two specimens. For example, the fraction porosity in A was ~ 2.5% although the measured diametral strain was quite small. A New Methodology for Measurement of Semi-Solid Constitutive Behaviour    84  This increase in porosity, from ~ 0.3% in the as-cast undeformed material, Fig.  4.5(a), to 2.5% in the as-cast semi-solid deformed material, Fig.  4.10(a), demonstrates the effectiveness of liquid in enabling void growth and internal damage. Although specimens B and C were subjected to much more tensile deformation than A, this deformation was not accompanied by a significant increase in fraction porosity. Thus, it is clear from both Table  4C and Fig.  4.10 that it is not just the presence of liquid or as-cast porosity but the combination of the two that enables extensive internal damage to form in semi-solid materials leading to a hot tear. As shown qualitatively in Figs.  4.10(b) and (c), and quantitatively in Table  4C, if neither liquid nor voids are present then there is limited distribution of internal damage. Away from the fracture surface, for example at the locations of Figs.  4.10(b) and (c), internal deformation-based damage does not occur. The effect of as-cast porosity on hot tearing is clearly related to the relative locations of the voids, the liquid, and grain boundaries. At high fraction solid, the majority of the remaining molten material is eutectic liquid located as films along grain boundaries and as isolated pockets within the interdendritic regions. These same regions will also contain most of the as-cast porosity [2, 29]. On application of load, it may be postulated that both the isolated liquid pockets and the pores will act as small cracks and concentrate stress, while the continuous films will concentrate strain. As proposed in Chapter 3, the presence of free surface in the liquid (as-cast porosity) acts as a stress riser and allows strain to be accommodated by growth of this pre- existing void through the liquid along the grain boundary. When this free surface is missing, as is the case in the HIP material, voids must nucleate within the liquid, or at the solid-liquid interface to allow for development of internal damage. It can be hypothesized that when liquid is not present, as in the specimen shown in Fig.  4.10(c), void nucleation does not occur until the intermetallic particles separate from the matrix phase, or fracture [30], and void growth is limited because there is no liquid to concentrate strain, and allow for grain boundary tearing. Table 4C: Comparison of damage-related measurements via XMT in select as-cast and HIP specimens. Specimen Material T (°C) εd %P Rmax VSR A As-cast 550 0.0008 2.52 % 274 1 B HIP 560 0.09 0.47 % 223 0.022 C As-cast 500 0.08 0.31 % 140 0.03 A New Methodology for Measurement of Semi-Solid Constitutive Behaviour    85 In semi-solid material with pre-existing porosity, stresses will concentrate between the voids, and coalesce to form a hot tear. Without pre-existing porosity, more strain must be applied to incur void nucleation in a two-phase material composed of liquid and a highly compliant solid. It follows that material with fewer tendencies to form shrinkage-based porosity (e.g. short freezing range alloys) and material with a finer grain size (i.e. smaller and more evenly distributed pockets of residual liquid at high fractions solid) would be less prone to hot tearing. The interplay between porosity and hot tearing can also be discussed based on aluminum industrial processing. In the case of DC cast cylindrical billets, hot tearing occurs in the central part of the casting, when high casting rates are used. High casting rates lengthen the mushy zone and increase the solidification time, thus creating a coarse grain structure, insufficient feeding to compensate the solidification shrinkage [3, 31, 32] and the likelihood of increased porosity. The introduction of porosity acts as nucleation sites for hot tearing. In the case of DC cast rectangular ingots, hot tearing occurs under conditions which can be described as a ‘hot casting’ [33] (i.e. reduced bottom block filling time and reduced cooling water flow rate). In this geometry, semi- solid stresses and strains are concentrated on the rolling face [4] where solidification rates are high and there is little difficulty in liquid feeding. Hot tears form on the cast surface because of inhomogeneous strain distribution at the scale of the semi-solid surface microstructure, which leads to strain localization and discrete strain-concentrated areas [34]. While the casting surface is not normally prone to porosity, the hot casting conditions increase the surface solidification time and thus porosity. The presence of porosity would seem to exacerbate the inhomogeneous strain distribution, and greatly increase hot tearing susceptibility.  Therefore, in both cases, greater control over the as-cast porosity would lead to a reduction in hot tearing by reducing nucleation sites and thus increasing the amount of deformation required for hot tear propagation. A New Methodology for Measurement of Semi-Solid Constitutive Behaviour    86 4.5 Conclusions A new experimental methodology has been developed to measure the tensile constitutive behaviour of semi-solid aluminum alloys. Specimens machined from the aluminum alloy AA5182 in the as-cast and as-cast plus hot isostatic pressing states were tested at temperatures between 500°C and 580°C at a strain rate of ~ 0.001. X-ray micro-tomography was performed to quantify both the initial porosity distribution in the materials tested, and the effects of porosity and liquid on the development of internal damage during deformation. This damage is a pre- cursor to hot tearing. The following conclusions can be drawn: 1. The two thermocouple technique provides reasonable thermal control during semi-solid tensile deformation. This technique eliminates earlier issues associated with the thermocouple weld detachment and thermal expansion. 2. The two thermocouple technique also improves cross-head displacement control, allowing for application of specific amounts of strain while eliminating the uncertainty associated with lengthwise thermal expansion. 3. The semi-solid ductility was improved by the HIP process by as much as a factor of 7. This ductility increase is due to the removal of the as-cast porosity and homogenization of the microstructure. 4. The HIP process increased the mechanical coherency temperature by ~ 10° to 570°C. It is thought that this increase is due to a shift in the relationship between fraction solid and temperature, caused by chemical homogenization. 5. The application of the HIP process to the as-cast material has greatly reduced the amount of internal damage accumulated during semi-solid deformation, indicating that pre-existing as-cast porosity plays an important role in the formation of hot tears. Reducing or eliminating this porosity would make aluminum alloys more resistant to hot tearing. A New Methodology for Measurement of Semi-Solid Constitutive Behaviour    87 4.6 References 1. Campbell, J., Castings. Butterworth Heinemann, 2 ed., 1991. 2. Lee, P. D. and Hunt, J. D., Acta Mat., 45, 1997, (4155). 3. Suyitno, Eskin, D. G., Savran, V. I. and Katgerman, L., Metall. Mater. Trans. A, 35A, 2004, (3551). 4. Sengupta, J., Cockcroft, S. L., Maijer, D. and Larouche, A., Mater. Sci. Eng. A., 397, 2005, (157). 5. Farup, I., Drezet, J. M. and Rappaz, M., Acta Mat., 49, 2001, (1261). 6. Suyitno, Kool, W. H. and Katgerman, L., 8th International Conference on Aluminum Alloys, P. J. Gregson, S. Harris, Eds., Cambridge, UK, Trans Tech Publications Ltd, CH, 2002, (179). 7. Dahle, A. K., Instone, S. and Sumitomo, T., Metall. Mater. Trans. A, 34A, 2003, (105). 8. Drezet, J.-M., Ludwig, O., M'hamdi, M., Fjaer, H. G. and Martin, C. L., Light Metals 2004, A. T. Taberaux, Ed., Charlotte, NC, TMS, 2004, (655). 9. Colley, L. J., Wells, M. A. and Maijer, D. M., Mater. Sci. Eng. A, 386(1-2), 2004, (140). 10. Singer, A. R. E. and Cottrel, S. A., J. Inst. Metals, 73, 1946, (33). 11. Twite, M. R., Spittle, J. A. and Brown, S. G. R., Intl J Forming Processes, 2004, (233). 12. Van Haaften, W. M., Kool, W. H. and Katgerman, L., Mater. Sci. Eng. A, 336, 2002, (1). 13. Nakagawa, T., Suivanchai, P., Okane, T. and Umeda, T., Mater. Sci. Forum, 215-216, 1996, (377). 14. Kron, J. and Fredriksson, H., International Symposium of Liquid Metals and Casting, P. D. Lee, A. Mitchell, A. Jardy, J. P. Bellot, Eds., Nancy, Fr, SF2M-Paris, 2003, (393). 15. Langlais, J. and Gruzleski, J. E., Mater. Sci. Forum, 167, 2000, (331). 16. Ackermann, P., Kurz, W. and Heinemann, W., Mater. Sci. Eng. , 75, 1985, (79). 17. Magnin, B., Maenner, L., Katgerman, L. and Engler, S., Mater. Sci. Forum, 217-222, 1996, (1209). 18. Phillion, A. B., Cockcroft, S. L. and Lee, P. D., Scripta Mat. , 55, 2006, (489). 19. Spittle, J. A., Brown, S. G. R., James, J. D. and Evans, R. W., Proceedings of the 7th International Symposium on Physical Simulation of Casting, Hot Rolling and Welding, H. G. Suzuki, T. Sakai, F. Matsuda, Eds., Tsukuba, JP, National Research Institute for Metals, 1997, (81). 20. Instone, S., St.John, D. H. and Grandfield, J., Int. J. Cast Metals Research, 12(6), 2000, (441). 21. Fabregue, D., Deschamps, A., Suery, M. and Poole, W. J., Metall. Mater. Trans. A, 37A, 2006, (1459). 22. Abramoff, M. D., Magelhaes, P. F. and S.J., R., Biophotonics International, 11(7), 2004, (36). 23. Ganesan, M., Dye, D. and Lee, P. D., Metallurgical and Materials Transactions A, 36A, 2005, (2191). 24. Staley, J. T. J., Tiryakoglu, M. and Campbell, J., Mater. Sci. Eng. A, 460-461, 2007, (324). 25. Maire, E., Grenier, J. C., Daniel, D., Baldacci, A., Klocker, H. and Bigot, A., Scripta Mater. , 55, 2006, (123). 26. Dong, H. B., Shin, M. R. M., Kurum, E. G., Cama, H. and Hunt, J. D., Metallurgical and Materials Transactions A (USA), 34A, 2003, (441). 27. Thompson, S., Cockcroft, S. L. and Wells, M. A., Mater. Sci. Techn., 20(4), 2004, (497). 28. Van Haaften, W. M., Kool, W. H. and Katgerman, L., J. Mater. Eng. Perf., 11(5), 2002, (537). A New Methodology for Measurement of Semi-Solid Constitutive Behaviour    88 29. Nagaumi, H., Komatsu, K., Uematsu, M., Hasisawa, N. and Nishikawa, Y., J Japan Inst Light Metals, 6, 1999, (13). 30. Huber, G., Brechet, Y. and Pardoen, T., Acta Mat., 53, 2005, (2739). 31. Clyne, T. W. and Davies, G. J., Brit Found., 74, 1981, (65). 32. Rappaz, M., Drezet, J. M. and Gremaud, M., Metall. Mater. Trans. A, 30A, 1999, (499). 33. Sengupta, J., Cockcroft, S. L., Maijer, D. M., Wells, M. A. and Larouche, A., Metall. Mater. Trans. B, 35B, 2004, (523). 34. Mitchell, J. B., Cockcroft, S. L., Viano, D., Davidson, C. J. and St John, D., Metall. Mater. Trans. A, 38, 2007, (in press).   89 Chapter 5: Three-Phase Microstructural Model for Prediction of Semi-Solid Constitutive Behaviour in Aluminum Alloysα 5.1 Introduction Hot tearing is a casting-related defect in which cracks form in the semi-solid temperature region. This defect is industrially detrimental since both the productivity of the process and the quality of the final product are affected. Over the past 30 years, both industry and academia have endeavoured to develop a hot tearing criterion, with only limited success [1]. There are several requirements that must be met in order to predict the formation of a hot tear. Firstly, one must have a quantitative understanding of semi-solid constitutive behaviour [2]. Secondly, one must have a means of quantifying the development of stress or strain in a given alloy system subject to a prevailing set of casting process conditions [3]. Thirdly, one must be able to link the strain to the accumulation of damage and understand the limit of damage that a given alloy can tolerate prior to failure. Semi-solid constitutive behaviour is typically challenging to measure due to a combination of factors including the high temperatures, the presence of a liquid phase, and the uncertainties in the relationship between fraction solid, temperature, and microstructure. Two classes of experiments have been developed to measure semi-solid constitutive behaviour: (a) partial remelting tests in which a specimen is reheated up to the semi-solid region from room temperature (e.g. Fig. 2, [4]), and (b) constrained casting tests in which liquid metal is poured into a specialized mold and cooled (e.g. Fig. 3, [5]). Unfortunately, both techniques have their limitations. In the case of partial remelting tests, the results may differ from those actually arising during solidification because there is no liquid feeding to counter-act deformation – the liquid feeding allows for healing of hot tears and may enable higher flow stress and ductility [6]. In the case of constrained casting tests, our inability to completely negate the mechanical interaction of the casting with the mould, and to link the measured load to fraction solid and displacement makes interpretation of the load data challenging. One methodology which has not previously been explored in terms of semi-solid constitutive behaviour is the combination of experimentally derived measurements and a numerical  α A version of this chapter will be submitted for publication. Phillion, A.B., Cockcroft, S.L., and Lee, P.D. Three-Phase Microstructural Model for Prediction of Semi-Solid Constitutive Behaviour in Al Alloys    90 microstructural model. On their own, both techniques suffer from many limitations. However, the combination of the two becomes powerful since the use of a validated model allows for the completion of ‘virtual’ experiments, significantly extending the range of conditions that may be examined. The technique hinges on the development of realistic microstructures and on inclusion of a sufficient number of grains, perhaps one hundred or more. Recently, a technique for modeling the solidification of a large number of globular grains was applied to hot tearing by Mathier et al. [7], and extended by Vernède et al. [8, 9]. In this technique, grains are approximated by polyhedrons based on a Voronoi diagram of the nuclei centres. This diagram describes the area of points closest to each nuclei, with borders halfway between neighbouring nuclei. Solidification is carried out by advancing the grain edges towards the border along a linear segment connecting the nuclei with a Voronoi vertex. This style of model has previously been used to describe semi-solid ductility [10, 11] and liquid feeding [12], but these previous works assumed a regular arrangement of grains. In contrast, a model based on the Voronoi diagram is able to create a random grain geometry for examination. Furthermore, model results for an Al–1wt%Cu alloy have been shown to closely match predictions made using the phase- field method but with substantially less computational cost [9]. In this work, a two-dimensional model is described that numerically predicts semi-solid constitutive behaviour using the finite element (FE) method. The model includes the following microstructural phenomena: grain boundaries comprised of liquid; liquid encapsulated at grain triple-junctions due to curvature effects; variable grain size with irregular geometry; and variable fraction porosity and void size. To achieve improved hot tearing susceptibility predictions, the semi-solid constitutive behaviour in the range 0.75 < fs < 0.95 must be known. The commercial software package ABAQUS1 was chosen to perform the FE calculations, because of its ability to generate multiple model domains and geometries based on an input script file. Following a description of the model formulation, the model predictions at high fraction solid are discussed and then verified against semi-solid constitutive behaviour measurements made on the commercial aluminum–magnesium alloy AA5182 in the as-cast state.  1 SIMULIA, Providence, RI, USA Three-Phase Microstructural Model for Prediction of Semi-Solid Constitutive Behaviour in Al Alloys    91 5.2 Semi-Solid Equiaxed-Globular Grain Geometry The geometry for predicting semi-solid constitutive behaviour was constructed using a Voronoi tessellation approach based on the two-dimensional solidification model proposed by Mathier et al. [7]. However, in that work the Voronoi tessellation was used to model the solidification process from fs = 0 to fs = 1, whereas in this work the Voronoi tessellation is used to create semi-solid geometry at a specific fraction solid. A Voronoi tessellation is a special kind of geometric entity determined by distances to a specified set of objects, e.g., a discrete set of points. Given a set of N points in a plane, a Voronoi tessellation divides the domain into a set of polygonal regions, the boundaries of which are the perpendicular bisectors of the lines joining the points. The Voronoi regions can be thought of as the equiaxed-granular grains within an alloy, and the discrete points can be thought of as the grain nuclei. In order to match the microstructural model geometry to semi-solid microstructure obtained during casting processes, two additional features have been included: rounded grain corners, and as-cast porosity. The first feature is needed because the geometry generated directly from the Voronoi tessellation leads to polyhedral grains with sharp corners. Sharp corners are unrealistic since during solidification they would not exist due to surface tension effects. As a result there is proportionally a higher fraction of liquid that develops at the triple junctions. The second feature, as-cast porosity, is needed because it is an important microstructural phenomenon and was shown in Chapters 2 ([13]), 3, and 4 to be highly relevant to the problem of hot tearing. Creation of the equiaxed-globular geometry consists of the following steps: (a) producing a two-dimensional Voronoi diagram; (b) reducing the grains in size based on the fraction solid; (c) rounding the grain corners via the Gibbs-Thomson effect; and (d) adding porosity at the triple-junctions. 5.2.1 Creation of the Voronoi Diagram The Voronoi diagram is assembled using the qhull [14] program. This program requires as input the number of discrete points, or grain nuclei, and the bounding box which encloses these grain nuclei. The number of discrete points is calculated based on the required average grain size assuming circular grains. The output is a single text file containing the region numbers and the coordinates of the vertices making up the perimeter of each region. Since the vertices can be Three-Phase Microstructural Model for Prediction of Semi-Solid Constitutive Behaviour in Al Alloys    92 anywhere in space, including outside the bounding box, their coordinates have to be further modified to ensure that they remain inside the model domain. A Voronoi diagram for nine nuclei randomly placed in a 0.15 mm2 domain is shown in Fig.  5.1(a).   (a)       (b)  (c)       (d) Figure 5.1: Schematic showing the stages of the semi-solid microstructural model; (a) Voronoi diagram, (b) fs = 0.90 with sharp corners, (c) round corners and (d) porosity. Three-Phase Microstructural Model for Prediction of Semi-Solid Constitutive Behaviour in Al Alloys    93 5.2.2 Accounting for the Fraction Solid The geometry shown in Fig.  5.1(a) represents a fully solidified microstructure. To model semi- solid microstructure at a specific fraction solid, the grains are reduced in size and the remaining area is assumed to be liquid. This new geometry, for a fraction solid of 0.90, is shown in Fig.  5.1(b); the liquid region is represented by the dark shade. Each grain is reduced in size independently of the others by moving the coordinates of each vertex towards the nucleation point until the area of this new grain relative to the area of the Voronoi region is equal to the fraction solid of interest. Within a grain, each vertex is moved proportionally the same amount:  1 2 1 2 CM CM CO CO = uuuur uuuuur uuuur uuuur  (5.1) where C corresponds to the coordinates of the grain nuclei, O1 and O2 correspond to the original coordinates of two vertices, and M1 and M2 correspond to the moved coordinates of two vertices. The fraction solid calculation is checked in vertex movement intervals of 0.25 % by comparing the area of the grain with vertices Mi to the area of the grain with vertices Oi. The area of the each grain is calculated using the .getSize command within the ABAQUS scripting language. Different grains will have their vertices moved proportionally different distances, depending on the grain geometry. The resulting geometry consists of a random configuration of grains fully surrounded by a continuous film of liquid. The liquid channels between the grains vary in thickness depending on the size of the grains, and the distance between the two grain nuclei in any particular channel. The relationship between fraction solid and temperature used in this work to model semi-solid AA5182 is taken from Thompson [15], and provided in Table  5A.  Table 5A: The relationship between fraction solid and temperature for AA5182 after Thompson [15]. Temp. (°C) fs 561 0.95 578 0.90 588 0.85 596 0.80 602 0.75  Three-Phase Microstructural Model for Prediction of Semi-Solid Constitutive Behaviour in Al Alloys    94 5.2.3 Rounding the Grain Corners The geometry shown in Fig.  5.1(b) does not yet represent a semi-solid microstructure because the solid grains are unrealistically shaped as polygons. Vernède et al. [9] have developed a solute flux model for rounding the grain corners based on a balance between solute flux induced by the Gibbs-Thomson effect and the geometrical advantage of a rounded corner for diffusion. A brief outline of this model is presented below. Since spherical grains do not generally occur, the grain corner is modelled by a curved boundary with radius R, while elsewhere the boundary is flat and parallel to the Voronoi edge segment. The classical Gibbs-Thomson effect provides a means of assessing the reduction in solidification temperature necessary due to a curved surface, e.g. a grain corner as:  slrT R ΓΔ =  (5.2) where ΔTr is the difference in solidification temperature between the curved and the flat interfaces, and slΓ  is the Gibbs-Thomson coefficient. Alternatively, under isothermal conditions, the Gibbs-Thomson effect can be used to assess the compositional variation in the liquid phase near the curved interface as compared to the flat interface:  R sll lC C Rm ∞ Γ= +  (5.3) where RlC  and lC ∞  are the solute concentrations in the liquid regions near the curved and flat interfaces, and m is the slope of the liquidus line. This compositional variation will induce a solute flux between the two regions:  ( )R l sll l l~ 0DD C C Rm∞ ΓΦ − = − >  (5.4) where Φ  is the solute flux between the two zones and Dl  is the liquid diffusion constant. In addition to the solute flux due to Eq. (5.4), the curved interface also represents an geometric advantage for diffusion between the solid and the liquid because it contains additional liquid as compared to a sharp corner. The solute flux based on this advantage can also be estimated as:  ( )2 tan 2 T R m α αΦ = ⋅ −&  (5.5) Three-Phase Microstructural Model for Prediction of Semi-Solid Constitutive Behaviour in Al Alloys    95 where T&  is the cooling rate, and α is the angle shown in Fig.  5.2. Combining Eq. ( 5.4) and Eq. ( 5.5), and including some geometrical constraints, the radius of curvature for the equiaxed- globular grain corners derived from a Voronoi tessellation is given by:  ( )3 l sl 2 tan DR A Tα α Γ= − − &  (5.6) where A is a dimensionless constant and assumed to be one. The semi-solid geometry created by rounding the grain corners is shown in Fig.  5.1(c). It is obvious that the triple-junctions shown in Fig.  5.1(c) are much larger than those shown in Fig.  5.1(b). For example, the top-left triple junction in the sharp-edged model can inscribe a circle of radius 4.9 μm whereas in the rounded corners model, a circle of radius 6.5 μm can be drawn. Thus, the grain corner rounding increases the amount of liquid at the triple-junctions for a given fraction solid. Since the rounded interface increases the overall volume of liquid at the triple- junctions, the grain vertices must be moved slightly forward to achieve the desired fraction solid. A comparison of a sharp-edged grain and a rounded corner grain is shown in Fig.  5.2. Notice that the sharp-edged grain vertices lie outside the rounded corners but the flat interfaces of the sharp-edged grain lie inside that of the rounded corners grain. It is also now possible for the flat interfaces of two adjoining grains to impinge upon each other at fraction solids less than one due to the movement of solid material from the grain corners to the flat interfaces. This phenomenon has been included in the model and represents the experimentally observed grain bridging effect. In ABAQUS, the grain corners are rounded using the fillet command within the CAE module. The radius of curvature is calculated from Eq. ( 5.6), with lD  = 10 -9 m2 s-1, slΓ  = 5 x 10-7 K m, and T&  = 1 K s-1 [9]. Since the calculated radius of curvature will differ depending on whether the vertex is approached from the left or the right, an average value is used to fillet the vertex. Once the corners are rounded, the area of the grain is compared against the corresponding Voronoi region, and the vertices then moved outward in an iterative process until the required fraction solid is achieved. A Voronoi region is a complex shape, and in 2D usually consisting of five to seven vertices. If the radius of curvature calculated by Eq. ( 5.6) exceeds the value that is obtainable geometrically, i.e. the length of the flat interface becomes negative, then the radius is incrementally decreased until a value is found that is geometrically possible. The odd-shaped corner identified by Marker ‘A’ in Fig.  5.2 is one corner where the radius of curvature calculated by Eq. ( 5.6) exceeded the geometrically obtainable value. Three-Phase Microstructural Model for Prediction of Semi-Solid Constitutive Behaviour in Al Alloys    96 A α  Figure 5.2: A comparison of the grain geometry with sharp edges vs. round corners. The grain nuclei, rays to the vertices, variable α and marker ‘A’ are also shown.  5.2.4 Adding Porosity The last step in constructing the semi-solid microstructure is to add porosity. There are four variables that must be included: size distribution, shape, area fraction, and location. The first three variables were quantified in Chapter 3 in as-cast aluminum alloys using x-ray micro- tomography. This previous work showed that as-cast porosity is highly tortuous in shape, with a size distribution ranging from ~ 2 to 175 μm in equivalent radius, a volume fraction ranging from 0.003 to 0.006, an average radius of 7.5 μm, and generally located at the triple-junctions. Porosity is added to the model geometry by removing material from the liquid, as shown in Fig.  5.1(d). Thus, the fraction liquid decreases to accommodate the porosity, while the fraction solid stays the same assuming that fs + fl + fp = 1. Voids of radius 7.5 μm are added at both triple- junctions, and the midpoint of a liquid channel between two Voronoi vertices; circular voids of constant radius are used for simplification. Void fraction is used as a variable in the simulations. Another aspect of the geometry is that material is removed only from the liquid portion of the geometry, not the solid grains. This means that although circular porosity is cut from the geometry, the resulting void may not be circular since the triple-junction may not accommodate a void of 7.5 μm in radius. Three-Phase Microstructural Model for Prediction of Semi-Solid Constitutive Behaviour in Al Alloys    97 5.3 Finite Element Model The methodology outlined in Section 5.2 provides the two-dimensional model geometry necessary for conducting virtual semi-solid deformation experiments using the FE method. A python script was written to create this geometry within the ABAQUS framework based on three user-input variables: fraction solid, fraction porosity, and grain size. Once the geometry exists, the FE model can be constructed by generating a mesh, determining appropriate solid and liquid material properties, and applying loads and constraints. 5.3.1 Analysis Formulation The virtual experiments are conducted at constant fraction solid, thus the semi-solid microstructural model examines only one temperature at a time. For this analysis, ABAQUS solves the equations of virtual work for the entire geometry by simultaneously relating all of the nodal displacements and forces through the stiffness matrix. The stress-strain state can then be calculated within any element using the interpolation functions and the relationship between strain and displacement. If the material behaviour is in the elastic region, then the stiffness matrix incorporates the linear Hooke’s law relationship between total stress and total strain:  ( ) ( )e eDσ ε⎡ ⎤= ⎣ ⎦  (5.7) where [ ]D  is the stiffness matrix, the superscript ‘e’ refers to the elastic behaviour, ( )σ  is the stress tensor and ( )eε  is the total elastic strain tensor. If the material’s deformation occurs in the plastic region, then the stiffness matrix [ ]D  incorporates the relationship between the current increment of stress, ( )dσ , and the current increment of plastic strain ( )pdε :  ( ) ( )p pd D dσ ε⎡ ⎤= ⎣ ⎦  (5.8) where the superscript ‘p’ refers to the plastic behaviour. 5.3.2 Material Properties The input constitutive behaviour of the solid and liquid phases is a critical aspect of the semi- solid microstructural model since the purpose of the model is to combine the solid and liquid mechanical properties, using the appropriate geometry, to approximate semi-solid constitutive behaviour. The mechanical properties required for the analysis include Poisson’s Ratio, the Three-Phase Microstructural Model for Prediction of Semi-Solid Constitutive Behaviour in Al Alloys    98 elastic modulus and stress-strain behaviour for both the solid and the liquid. Furthermore, the model requires both solid and liquid flow stress data at temperatures where the material is semi- solid. In this model, the constitutive behaviour of the solid grains is described via an elastic rate- dependent plastic model; while the liquid is described via an elastic perfectly plastic model. Elastic Modulus The choice of elastic modulus for the liquid constituent is critical to the model predictions because the liquid network is fully interconnected, and is much weaker than the solid. A liquid modulus of 0.7 GPa was chosen to offer little resistance to stress. The modulus of the grains was assumed to be E = 70 GPa [16], that of a low temperature solid. Poisson’s Ratio One of the features of liquid displacement is that volume should be conserved in response to an applied load because the liquid is supposed to flow instead of deform. To conserve volume within the liquid, Poisson’s ratio, υ, should be equal to 0.5. However, this value cannot be used in the FE analysis because the resulting stiffness matrix will contain a zero in the denominator. Consequently, the model will not work. By trial and error, υ = 0.45, provides a compromise between conserving volume and minimizing convergence issues.  In the solid grains, υ =  0.30. Flow Stress Behaviour of the Solid Grains To provide the required solid flow stress behaviour, an empirical constitutive equation for as- cast AA5182 was used. The equation is based on the work by Chaudhary [17] but has been modified using the experimental measurements from Chapter 4. Chaudhary developed an empirical model based on the extended Ludwik equation:  n(T) m(T)( , , ) ( , , )T K Tσ ε ε ε ε ε ε= ⋅ ⋅& & &  (5.9) where σ is the stress, ε is the total plastic strain experienced by the material, ε&  is the strain rate, K is a flow stress coefficient, n is the strain hardening parameter and m is strain rate sensitivity of the material. These parameters all vary as a function of temperature, and thus the equation is able to predict the constitutive behaviour from room temperature up to the mechanical coherency temperature over a large range of strain and strain-rate. The correlations developed by Chaudhary for m, n, and K are shown in Table  5B. Note that above 350°C, n was found to have a value of zero. Thus, in the semi-solid regime, n = 0, and the stresses predicted by the Ludwik equation are a function of strain rate only, i.e. a material which is elastic perfectly plastic. Three-Phase Microstructural Model for Prediction of Semi-Solid Constitutive Behaviour in Al Alloys    99 A comparison of the flow stresses predicted by Chaudhary’s empirical equation to the hot isostatic pressing (HIP) experimental measurements from Chapter 4 is provided in Fig.  5.3. The HIP results were chosen for development of the semi-solid flow stress predictions because this material had a higher mechanical coherency point than the as-cast material, and thus may provide a better approximation of the flow stress of the solid material at temperatures above the solidus. Also shown in Fig.  5.3 and Table  5B are the predictions made using a modified version of Chaudhary’s equation. The modification consisted of adjusting the y-intercept of the K parameter correlation by ~ 5 %. As can be seen from Fig.  5.3, the Ludwik equation yields a much better correlation to the experimental data using the modified K as compared to the base- line equation. This result is not surprising, since Chaudhary’s empirical equation was developed for the temperature range 25–470°C. Thus, in the semi-solid microstructural model, the Ludwik equation with n = 0, Chaudhary’s correlation m, and the improved correlation for K, is used to determine the flow stress behaviour of the solid at temperatures above the solidus. Furthermore, a lower bound of 1.5 MPa is placed on the solid flow stress to ensure sufficient differentiation between the behaviour of the solid and the liquid.  Table 5B: AA5182 Ludwik parameter correlations for K, n, and m [17]. K (MPa) N M ( ) } } 200 448 0.63 200 500 458.2 0.77 483.5 0.77 oT C K T T K T Original K T Enhanced < = − < < = − = −   ( ) 150 0.25 0.0005 150 350 0.29 0.0008 350 500 0 oT C n T T n T T n < = − < < = − < < =  ( ) 150 0 150 400 0.1497 0.001 400 500 0.205 0.00006 oT C m T m T T m T < = < < = − < < = +   Three-Phase Microstructural Model for Prediction of Semi-Solid Constitutive Behaviour in Al Alloys    100 ε true σ( M P a) 500 520 540 560 580 5 10 15 20 25 Exp. Data Base-Line Enhanced  Figure 5.3: Variation in flow stress with temperature for the experimental data from Chapter 4, and the base-line [17] and enhanced Ludwik equations; ε&  = 0.0015 s-1.  Flow Stress Behaviour of the Liquid In the semi-solid microstructural model, the liquid exists as semi-continuous liquid films. The liquid flow stress will depend on the both the film thickness and the interfacial energy [18]. One approach for estimating the liquid flow stress is to use the Young and Laplace equation [10]:  sl2 h γσ =l  (5.10) where σl is the liquid flow stress, γsl is the solid-liquid interfacial energy with γsl = 1 J m-2, and h is the thickness of the liquid film. Unfortunately, h does not have a unique value due to the Voronoi-related randomness of the grain structure. Since it would be difficult to calculate h for every liquid channel, a flow stress of 0.5 MPa was used in the liquid. This value is an averaged value, calculated from Eq. ( 5.10) based on a range of fraction solids, grain size, and the corresponding liquid channel thickness [11]. The Young and Laplace equation is generally used to calculate the stress necessary to separate two plates bonded by a thin liquid film, and thus provides an upper bound to the flow stress of the liquid. The implications of this equation as a yield stress predictor will be examined using a sensitivity analysis presented in Section 5.4.2. Three-Phase Microstructural Model for Prediction of Semi-Solid Constitutive Behaviour in Al Alloys    101 5.3.3 Constraints and Loading The model domain is a 2D square geometry, an example of which is shown in Fig.  5.4. Symmetry boundary conditions are imposed on the left and bottom sides of the model domain, while the right side is free to move in all directions. The top side of the model is linked to a reference node for displacement in the y-direction. Thus, the model is constrained as though it were a quarter-section of a plate of material. Note that only boundary nodes that are linked to elements with solid constitutive behaviour are constrained. The liquid nodes are not constrained, to facilitate liquid flow both in and out of the liquid channels bordering the domain boundary. To deform the model, a fixed displacement is applied to the reference node, corresponding to a bulk strain of 0.0025 and a bulk strain rate of 0.0015 s-1. Furthermore, individual grains and elements will deform at different strain rates because the semi-solid microstructural model contains three distinct phases. To obtain the bulk semi-solid constitutive behaviour, the force required to apply the fixed displacement to the reference node is recorded. There are also a number of internal model constraints. Since each grain is meshed separately from the other grains, and from the liquid, the different geometric entities must be stitched together to facilitate the transfer of load. Thus, liquid and solid nodes on the solid-liquid interface are mathematically tied together, i.e. displacement for the liquid nodes at the interface are constrained to match displacement for the corresponding nodes in the solid grains. 5.3.4 Finite Element Mesh The FE mesh, shown in Fig.  5.4, was constructed from the geometry described in Section 5.2. The solid grains were meshed with a combination of triangular and quadrilateral plane strain elements, while the liquid films were meshed with triangular plane strain elements. The average liquid element length was 2 μm. From the results of a sensitivity analysis, to be presented in Section 5.4.2, a value of 5 μm was chosen as the solid element edge length. The number of grains (Ng) modelled in the simulation also had a large effect on the predicted constitutive behaviour. This is due to the random nature of the Voronoi tessellation, which for a specified Ng will alter the geometric configuration of the grains based on the randomly generated grain nuclei. At low values of Ng, the number of liquid channels is too low, and the results will be biased to that of the solid rather than of a semi-solid mixture. Based on a sensitivity analysis, presented in Section 5.4.2, the size of the model domain was 56 grains. Three-Phase Microstructural Model for Prediction of Semi-Solid Constitutive Behaviour in Al Alloys    102   Figure 5.4: An example model domain, constructed using the geometric methodology outlined in Section 5.3; fs = 0.90, d = 150 μm, and Ng = 56 (Note that the mesh has been removed from the liquid for clarity). Three-Phase Microstructural Model for Prediction of Semi-Solid Constitutive Behaviour in Al Alloys    103 5.4 Results and Discussion The semi-solid microstructural model was run under a variety of scenarios in order to predict the constitutive behaviour of as-cast AA5182. Firstly, results showing the effect of fraction solid on constitutive behaviour are presented. Secondly, a sensitivity analysis of some of the critical model assumptions and the model limitations are discussed. Thirdly, the model is validated via a comparison to the experimental data from Chapter 4 and to ductility measurements [19]. Finally, unique results are presented whereby the model is used to explore the effects of porosity and grain size on semi-solid constitutive behaviour. The predictions are presented in terms of the force–displacement curves provided by the finite element software for the reference node. 5.4.1 Model Predictions The effect of fraction solid on the stress-strain predictions in the range 0.75 < fs < 0.95 is shown in Fig.  5.5 in the absence of porosity and for d  = 225 μm. As can be seen in the figure, the semi-solid flow stress decreases with decreasing fraction solid. This stress decrease is related to changes occurring in both the solid grains and the liquid film. In the solid grains, there is a decrease in flow stress due to the effect of increasing temperature. In the liquid, decreasing the fraction solid corresponds to thicker liquid channels which allow for more strain accumulation in the liquid before the solid grains begin to interlock. The predictions made by the semi-solid microstructural model and shown in Fig.  5.5 can be described as elastic–plastic with no well-defined yield point, and considerable hardening. This behaviour is not the classical work hardening effect, but rather geometric hardening due to an increasing number of grains interacting with each other instead of freely moving about the liquid. Similar mechanisms would occur during deformation of actual solidification microstructures, although the strain accumulating in the liquid results instead in intergranular liquid flow. The microstructural model is able to simulate the constitutive behaviour in semi-solid as-cast material under conditions that would normally be a challenge to measure experimentally. The data shown in Fig.  5.5 provides new knowledge regarding semi-solid deformation such as geometric hardening, flow stress at fs < 0.95, and the effect of the solid on semi-solid behaviour. For example, although there is much liquid present at fs = 0.85, the bulk stress-strain curve shown in Fig.  5.5 exhibits a flow stress significantly higher than that of the liquid alone. Three-Phase Microstructural Model for Prediction of Semi-Solid Constitutive Behaviour in Al Alloys    104 ε true σ( M P a) 0.005 0.01 0.015 0.020 2 4 6 8 10 0.75 0.80 0.85 0.90 0.95 Fraction Solid  Figure 5.5: Effect of fraction solid on the predicted semi-solid constitutive behaviour in the range 0.75 < fs < 0.95 ( d  = 225 μm, fp = 0).  5.4.2 Sensitivity Analysis and Model Limitations During the development of the semi-solid microstructural model, a number of critical assumptions regarding construction of the model geometry were required. The effects of these assumptions on the model predictions, as well as model limitations, are discussed below. A comparison of the effect of liquid flow stress on the bulk stress-strain curve predicted by the semi-solid microstructural model is shown in Fig.  5.6, for a fraction solid of 0.95 and a temperature of 570°C. As can be seen in Fig.  5.6, the bulk properties are highly influenced by the choice of liquid flow stress. The predicted bulk flow stress at ε = 0.004 is predicted to increase from 5.83 to 6.32 MPa in conjunction with a liquid flow stress increase from 0.5 to 1.0 MPa. Flow stress values lower than 0.5 MPa were also attempted (e.g. 0.1 MPa). However, these low values resulted in numerous convergence issues. The effect of element size on the predicted semi-solid stress-strain curve is shown in Fig.  5.7 for a geometry with Ng = 56 grains, fs = 0.90, d = 150 μm, and T = 580°C. Six deformation simulations were performed, with the size of the solid elements varying from 2 μm to 32 μm in Three-Phase Microstructural Model for Prediction of Semi-Solid Constitutive Behaviour in Al Alloys    105 length. As can be seen in Fig.  5.7, the solid element size has a large effect on the results since larger mesh sizes predict higher semi-solid stresses. Below 2 μm, the results showed decreased sensitivity to mesh size. The effect of number of grains within the model domain on the predicted constitutive behaviour was investigated through analysis of the relationship between geometric configuration and model predictions. In Fig.  5.8, stress-strain predictions are presented from 25 different simulations of 56 grains, fs = 0.90 and d  = 150 μm – each with a different geometric configuration. As can be seen in the figure, the different geometric configurations provide stochastic variability to the constitutive behaviour. For example, at ε = 0.01, the flow stresses vary between ~ 4 and 5 MPa, which corresponds to a maximum flow stress variation, Δσ , of 0.98 MPa. This analysis was repeated for Ng = 14, 28, 56, and 100, to examine the dependence of Δσ on Ng. The results, provided in Table  5C, clearly show that Δσ is significantly larger with 14 or 28 grains in the model domain as compared to 56 grains, and slightly less for 100 grains. The choice for element size, 5 μm, and Ng, 56 grains, were made as a compromise between speed of the simulation run time, and accuracy of the result. As shown in Figs.  5.7, and  5.8, and in Table  5C, these choices appear reasonable considering many of the inherent assumptions within the model. For example, in the case of simulated number of grains, increasing the model domain to 100 grains would decrease the variation in stress-strain response due to geometric variability by approximately fifteen percent. However, this improvement would be offset by a twofold increase in the number of elements required to model the larger domain, and a four-fold increase in computation time. The choice of liquid flow stress, σl = 0.5, seems somewhat more arbitrary and certainly has a significant effect on the stress-strain predictions. This value was chosen based on calculations from Eq.  5.10, and the need to have a value that is much lower than the solid flow stress while avoiding convergence issues. There are a number of model limitations. Those relating to the equiaxed globular geometry can be found in [9]. One major limitation of the model is the 2D geometry. In reality, solidification microstructure has 3D geometry. The 2D model geometry will predict grain interlocking earlier than a 3D structure and thus the current model predictions will be stiffer. A second limitation, relating to the constitutive behaviour, is that the model predictions are valid only at low values of strain. In the model, the accumulation of too much strain within the liquid enables this phase to behave as a weak solid instead of a flowing-liquid. This is because extensive strain accumulation Three-Phase Microstructural Model for Prediction of Semi-Solid Constitutive Behaviour in Al Alloys    106 deforms the elements to such an extent that the element distortion becomes an issue. In addition, in the real situation, new voids will form and the existing voids will grow and coalesce. This effect has not been incorporated into the current finite element model and yet is known to have a large impact on strain localization and material failure by hot tearing. In the liquid, to achieve the large strains seen later in Fig.  5.10(b), the elements must be significantly deformed and are approaching or exceeding their distortion limit.  Table 5C: The variation in Δσ as a function of Ng, calculated from 25 different geometric configurations (fs = 0.90, d  = 150 μm). Ng Δσ @ ε = 0.01 14 2 28 1.5 56 0.98 100 0.82   ε true σ( M P a) 0 0.002 0.0041 2 3 4 5 6 7 1.5 1.0 0.5 0.1 σl (MPa)  Figure 5.6: The effect of σl on the bulk stress-strain curve predicted by the semi- solid microstructural model; fs = 0.95, T = 570°C, and d  = 150 μm. Three-Phase Microstructural Model for Prediction of Semi-Solid Constitutive Behaviour in Al Alloys    107 ε true σ( M P a) 0 0.01 0.021 2 3 4 5 6 32 um 16 um 8 um 5 um 4 um 2 um ELlength  Figure 5.7: The effect of solid element size on the bulk stress-strain curve predicted by the model; fs = 0.90, T = 580°C, d  = 150 μm, and Ng = 56. ε true σ( M P a) 0 0.01 0.023 4 5 6 Δσ = 0.98 MPa  Figure 5.8: Variation in predicted tensile response of the semi-solid material as a function of geometric configuration; fs = 0.90, d  = 150 μm, and Ng = 56. Three-Phase Microstructural Model for Prediction of Semi-Solid Constitutive Behaviour in Al Alloys    108 5.4.3 Model Validation In order to validate the semi-solid microstructural model, the model predictions in the absence of porosity have been compared to experimentally-obtained semi-solid stress-strain data determined from HIP AA5182. Three sets of experimental data were taken from Chapter 4. This data was reported to have been obtained at a set-point temperature of 570°C, yet the data showed a rather large variability in the flow stress – i.e. values of 2.7, 4.1, and 9.5 MPa were measured. Due to the testing methodology used in Chapter 4, it is not unreasonable to assume that the actual test temperature may have been few degrees higher or lower than the set-point value of 570°C. Since the temperature 570°C was found in Chapter 4 to correspond to the mechanical coherency point for HIP AA5182, small changes in temperature will greatly alter the semi-solid structure, including reducing the number of grain bridges and increasing the number of grains completely wetted by liquid films and thus significantly modifying the constitutive behaviour. To examine this data, three different model simulations were conducted corresponding to fs = 0.95, 0.98, and 1.0. Other model parameters were: fp = 0, and d  = 225 μm, and were chosen based on Chapter 4. A comparison of the experimental measurements and the model predictions is shown in Fig.  5.9. Note that the experimental data for fs = 0.95 and 0.98 is available only over a limited range since the material contained little ductility. As can be seen in the figure, the model results bound the experimental data. The stress-strain predictions for fs = 0.98, and 0.95 match the experimental measurements up to the point of fracture, while the predictions for fs = 1 match only at relatively higher values of strain (ε > 0.001). The sharp yield point predicted by the model at fs = 1 occurs because the constitutive behaviour is given as elastic perfectly plastic. However, the experimental data appears to exhibit some strain hardening and no well-defined yield point. The fraction solid in this experiment is thought to have been very high, and thus the specimen would have consisted of small and isolated pockets of liquid. Unfortunately, the semi- solid microstructural model does not support the range 0.98 < fs < 1 due to geometry and convergence issues. Therefore, despite relatively large assumptions in its formulation, the semi- solid microstructural model seems to satisfactorily predict the constitutive behaviour of AA5182. Another area that can be examined with the model is the distribution of strain and limits in ductility. Using an optical strain technique, Mitchell [19] measured surface strain during solidification in a constrained casting for a number of aluminum alloys. The strain profiles were measured along a line bisecting a hot tear. These prior results showed that the bulk material is Three-Phase Microstructural Model for Prediction of Semi-Solid Constitutive Behaviour in Al Alloys    109 able to accommodate strains in the range of 0.002 to 0.004 before strain localization, and that hot tears began to form when the local strain reaches a value of 0.016 to 0.019. An example of the strain localization with time for an aluminum–magnesium–silicon alloy, AA6111, measured by Mitchell [19] is shown in Fig.  5.10(a). As can be seen in the figure, the strain development is initially randomly distributed. With increasing time, the strain becomes localized in a few areas until at 4.4 s all the strain is accommodated in the region where a hot tear forms. A similar technique can be used to investigate the development of strain in the semi-solid microstructural model. To perform this investigation, a simulation was run in which fs = 0.90, fp = 0, and d  = 150 μm. In Fig.  5.10(b), the local strain is shown as a function of distance along a line perpendicular to the loading direction, with the line drawn at mid-height along the geometry. The strain profiles are plotted for similar times as Fig.  5.10(a). As can be seen in the figure, there are four regions of strain concentration, corresponding to the liquid channels and a number of regions of low strain, corresponding to solid grains. The two highest peaks represent triple-junction locations. In comparing the experimental data and the simulation results, there are a few salient observations that may be made. Firstly, the results presented in Figs.  5.10(a) and (b) are qualitatively similar, with both figures exhibiting regions of strain localization that evolve with time. Secondly, the magnitude of strain predicted by the simulation is an order of magnitude larger than the strain accumulated in the experimental data prior to failure. Thirdly, the simulation has small regions of localized strain, ~ 25–100 μm, while experimental data shows that strain is localized on a length scale of ~ 400–800 μm.  Finally, the local strain in some of the simulated triple-junctions (ε ~ 0.3–0.4), far exceeds the measured hot tearing ductility [19]. Much of the difference between the experimental data and the semi-solid microstructural model with respect to strain distribution may be attributed to the fact that the model is predicting results for an isothermal or iso-fraction solid test in which the level of damage does not evolve with time, whereas the experimental results show the evolution in strain for non-isothermal material undergoing solidification and accumulation of internal damage. In reference to the localized region size, the model is able to support strain localization in most of the liquid channels. In contrast, it is hypothesized in Chapter 3 that the experimental results reflect localization of internal damage within a few of the liquid channels. It is also possible that the experimental results are not resolving strain localization to within individual liquid channels, but Three-Phase Microstructural Model for Prediction of Semi-Solid Constitutive Behaviour in Al Alloys    110 rather to within grain clusters, i.e. the groupings of grains proposed by Vernède that are fully surrounded by liquid films [8]. These grain clusters may have the capacity to both localize strain to within the cluster, and to partition strain between solid and liquid. In reference to the strain accumulation prior to failure, it is clear from the experimental data in Chapter 4 and Mitchell et al. [19] that the model need only to simulate the semi-solid deformation conditions at small strains. Therefore, it provides a good qualitative description of the local strain accumulation in semi-solid materials.   ε true σ( M P a) 0 0.0005 0.001 0.0015 0.0020 2 4 6 8 10 12 Exp. Data Model Data fs = 1.0 fs = 0.98 fs = 0.95  Figure 5.9: Comparison of the predicted and experimentally measured semi- solid constitutive behaviour of AA5182; T = 570°C, fp = 0, and d  = 225 μm. Three-Phase Microstructural Model for Prediction of Semi-Solid Constitutive Behaviour in Al Alloys    111 Distance (mm) ε lo ca l 0 1 2 3 4 5 0 0.005 0.01 0.015 0.02 Time (s) 0.13 1.01 2.01 4.40  (a) X X X X X XX X X X X X X X X X X X X X Distance (μm) ε lo ca l 200 400 600 800 1000 0 0.05 0.1 0.15 0.2 Time (s) 0.13 0.92 2.01 4.47  (b) Figure 5.10: Local Strain vs. Distance (a) on the surface of a solidifying AA6111 alloy (after [19]) and (b) as calculated from the microstructure model; fs = 0.90, fp = 0, and d  = 150 μm. Three-Phase Microstructural Model for Prediction of Semi-Solid Constitutive Behaviour in Al Alloys    112 5.4.4 The Effect of Grain Size and Porosity on Model Predictions In Fig.  5.11, the results of a ‘virtual’ experiment, showing the effects of grain size and porosity on semi-solid constitutive behaviour, are presented. These results are unique since the effects of grain size  and porosity cannot easily be quantified using experimental techniques. For the grain size assessment, Fig.  5.11(a), the simulation was run with the following parameters: T = 580°C, fs = 0.90, d  = 75, 150, 225, and 300 μm, and fp = 0. As shown in the figure, the semi-solid flow stress decreases with increasing grain size. This decrease occurs because for a given fraction solid, larger gains must necessarily have thicker liquid films surrounding the grain boundary. These thicker films allow for more deformation in the liquid before the grains interact with each other. Larouche [11] has proposed that the average liquid channel thickness hi is directly proportional to grain size:  ( )0.331 sh d f= −  (5.11) The simulations presented in Fig.  5.11(a) have average channel thicknesses h ranging from 2.5 μm to 10 μm. Thus, in the simulations with d  = 75, the grains will begin to impinge on each other almost immediately while simulations with larger grains accommodate more liquid flow. For the fraction porosity assessment, presented in Fig.  5.11(b), the simulation was run with the following parameters: T = 560°C, fs = 0.95, d  = 75, 150 μm, and fp = 0, 0.002, 0.004, and 0.006. The results indicate that as-cast porosity will decrease the flow stress of the semi-solid material, though the effect is relatively small in comparison to the effect of grain size. This is because porosity at the triple-junctions creates internal surface area within the model, which allows the liquid to flow more easily in response to the applied deformation. The porosity has the secondary effect of strain localization. This effect is evident in Fig.  5.12, which shows strain contours from the model run with fp = 0.004. As can be seen in the figure, strain localization has occurred in the liquid area between voids A and B. It is clear that this localization would cause the two voids to preferentially grow in the interligament region and coalesce to form a single larger void. The consequence of ignoring the process of coalescence significantly affects the effect of porosity on semi-solid constitutive behaviour. However, this feature cannot be added into the model geometry as currently formulated. Fig.  5.12 also highlights two other semi-solid deformation phenomena captured by the model: shear banding on some of the grains at higher strains, and the accumulation of strain along the liquid channels at approximately 45° to the loading direction. Three-Phase Microstructural Model for Prediction of Semi-Solid Constitutive Behaviour in Al Alloys    113 ε true σ( M P a) 0 0.005 0.01 0.015 0.022 3 4 5 6 75 150 225 300 Grain Size (μm)  (a) ε true σ( M P a) 0 0.002 0.004 0.006 0.008 0.014 5 6 7 8 0 0.002 0.004 0.006 Fraction Void  (b) Figure 5.11: Effect of (a) grain size and (b) void area fraction on the predicted semi-solid constitutive behaviour. Three-Phase Microstructural Model for Prediction of Semi-Solid Constitutive Behaviour in Al Alloys    114 A B   Figure 5.12: Finite element simulation showing strain localization between voids A and B; fs = 0.95, fp = 0.004, and d  = 150 μm. Three-Phase Microstructural Model for Prediction of Semi-Solid Constitutive Behaviour in Al Alloys    115 5.5 Conclusions A three phase finite element microstructural model has been presented which describes the deformation behaviour of semi-solid materials. The model geometry is based on a Voronoi diagram with round corners to approximate equiaxed-globular grain structure, and circular voids at the triple-junctions representative of as-cast porosity. The model has been validated by comparing the model predictions to experimentally measured semi-solid stress-strain curves from HIP AA5182. Sensitivity analyses with respect to the liquid flow stress, element size, and number of grains were also performed to further validate the model’s predictive capability. This new model has allowed for exploration of the effect of fraction solid, fraction porosity, and grain size on semi-solid constitutive behaviour. The results show that increasing grain size and fraction porosity lead to reductions in flow stress for a given fraction solid because it becomes easier for strain to accumulate within the liquid. The semi-solid constitutive behaviour is predicted to be highly dependent on both fraction solid and grain size, while only moderately affected by porosity. Thus, the model appears capable of describing the effects of two of the three phenomena that appear important in semi-solid constitutive behaviour and hot tearing. The third phenomenon, porosity, has been largely suppressed since the voids cannot evolve via coalescence processes. Furthermore, these simulations reveal that it is local strain accumulation that is important for hot tearing, since strain localizes in the liquid very early on in the process. As-cast porosity tends to exacerbate the problem of strain localization since it encourages strain concentration leading to internal damage growth and coalescence. Three-Phase Microstructural Model for Prediction of Semi-Solid Constitutive Behaviour in Al Alloys    116 5.6 References 1. Eskin, D. G., Suyitno and Katgerman, L., Prog. Mater. Sci., 49, 2004, (629). 2. Ludwig, O., Drezet, J.-M., Martin, C. L. and Suery, M., Metall. Mater. Trans. A, 36A, 2005, (1525). 3. Phillion, A. B., Cockcroft, S. L., Sengupta, J. and Maijer, D. M., Light Metals 2005, H. Kvande, Ed., San Francisco, CA, TMS, 2005, (1063). 4. Spittle, J. A., Brown, S. G. R., James, J. D. and Evans, R. W., Proceedings of the 7th International Symposium on Physical Simulation of Casting, Hot Rolling and Welding, H. G. Suzuki, T. Sakai, F. Matsuda, Eds., Tsukuba, JP, National Research Institute for Metals, 1997, (81). 5. Instone, S., St.John, D. H. and Grandfield, J., Int. J. Cast Metals Research, 12(6), 2000, (441). 6. Fabregue, D., Deschamps, A., Suery, M. and Poole, W. J., Metall. Mater. Trans. A, 37A, 2006, (1459). 7. Mathier, V., Jacott, A. and Rappaz, M., Modelling Simul. Mater. Sci. Eng., 12(3), 2004, (479). 8. Vernede, S., Jarry, P. and Rappaz, M., Acta Mat., 54, 2006, (4023). 9. Vernede, S. and Rappaz, M., Acta Mat., 55, 2007, (1703). 10. Lahaie, D. J. and Bouchard, M., Metall. Mater. Trans. B, 32, 2001, (697). 11. Larouche, D., Langlais, J., Wu, W. and Bouchard, M., Metall. Mater. Trans. B, 37B, 2006, (431). 12. Dijkstra, W. O., Vuik, C., Dammers, A. J. and Katgerman, L., Solidification Processes and Microstructures, M. Rappaz, C. Beckermann, R. Trivedi, Eds., Charlotte, NC, TMS, 2004, (151). 13. Phillion, A. B., Cockcroft, S. L. and Lee, P. D., Scripta Mat. , 55, 2006, (489). 14. Barber, C. B., Dobkin, D. P. and Huhdanpaa, H. T., ACM Trans. Math. Soft., 22, 1996, (469). 15. Thompson, S., Cockcroft, S. L. and Wells, M. A., Mater. Sci. Techn., 20(4), 2004, (497). 16. Sengupta, J., Cockcroft, S. L., Maijer, D. and Larouche, A., Mater. Sci. Eng. A., 397, 2005, (157). 17. Chaudhary, A. and Wells, M. A., Proceedings of the International Conference on Advances in Materials and Meterials Processing, U. K. Chatterjee, B. K. Dhindaw, Eds., IIT Kharagpur, India, 2006, (763). 18. Grandfield, J., Davidson, C. J. and Taylor, J. A., Light Metals 2004, J. L. Anjier, Ed., New Orleans, LO, TMS, 2001, (895). 19. Mitchell, J. B., Cockcroft, S. L., Viano, D., Davidson, C. J. and St John, D., Metall. Mater. Trans. A, 38, 2007, (in press).         117 Chapter 6: A New Semi-Solid Constitutive Equation for AA5182α 6.1 Introduction Hot tearing is a casting-related defect in which cracks form in the semi-solid temperature region. This defect is of industrial importance since it usually results in rejection of the entire casting. Furthermore, it is academically challenging due to the interplay between many semi- solid phenomena. Recently, hot tearing in Direct Chill (DC) casting of aluminum alloys has received much interest due to the development of complex alloys and the need to produce them economically. With regards to hot tearing susceptibility, there are two issues which must be addressed. Firstly, one must have a quantitative understanding of the semi-solid constitutive behaviour. Secondly, one must have a means for assessing the development of stress and strain during the casting process. A number of researchers have performed semi-solid tensile tests on aluminum alloys. In some of these tests, specimens were partially remelted from solid samples [1-4], while other tests used in situ solidification [5-8]. In many instances, the results have been difficult to interpret quantitatively and show poor reproducibility. For example, the flow stresses measured by Colley [1] on semi-solid AA5182 were 2 to 3 times larger than those measured by Van Haaften [2]. Furthermore, there are challenges in measuring semi-solid deformation behaviour at fractions solid below ~ 0.95, as shown in Chapter 4. The results of these tests have shown that semi-solid material transitions from having virtually no strength at fraction solid less than ~ 0.95 to having significant strength at higher values of fraction solid. In terms of ductility, the semi-solid material is known to have a minimum in the range of fraction solid between ~ 0.90 and 0.98. At fraction solid below this range, deformation occurs due to mass feeding of the solid-liquid mixture. At fraction solid above this range, there is sufficient bridging to accommodate strain between the solid grains. Based on the semi-solid experimental measurements, a number of constitutive equations have been developed. Some researchers [2, 9-11] have proposed models based on a modified creep law where the load carrying area is related to the fraction solid or to the proportion of grain  α A version of this chapter will be submitted for publication. Phillion, A.B., Cockcroft, S.L., and Lee, P.D.  A New Semi-solid Constitutive Equation for AA5182    118 boundaries covered by liquid films. Other researchers [12-16] have proposed two-phase constitutive models consisting of a porous solid skeleton saturated with liquid. In these models, a number of sophisticated internal functions, such as the partial cohesion parameter devised by Martin et al. [16], have been introduced to deal with anisotropy in the tensile–compressive behaviour, and the increased dendrite interlocking with increasing fraction solid. Unfortunately, model validation continues to be an issue since good apparatus do not yet exist to measure tensile properties at fraction solids less than ~ 0.95. In terms of modeling techniques for characterization of stress and strain during DC casting, there has been significant progress. Sengupta et al. [17] used a low elastic modulus in combination with a large yield stress to minimize semi-solid plastic strain accumulation. M’Hamdi et al. [14] implemented a two-phase model allowing for both dilatation and densification of the solid skeleton. Ludwig [18] extended M’Hamdi’s formulation to include a coalescence parameter which links the effects of stress triaxiality on semi-solid flow stress. While the results of the above studies are important contributions towards our understanding the phenomena of hot tearing in DC casting of aluminum alloys, it is clear that more research is required. For example, in [19], M’Hamdi et al. explored the effect of casting velocity on hot tearing in DC cast AA6060 alloy extrusion billets via a finite element thermal-stress model. A comparison with experimentally derived results, also reported in [19], indicates that the model predictions appear to correlate well with the crack severity data. There were, however, a few noteworthy inconsistencies. Firstly, the largest semi-solid stresses predicted by the model along the centerline occur at around 0.01 m from the base, whereas hot tears are typically found to initiate in proximity to the base. Secondly, the peak semi-solid tensile stresses are predicted to occur at approximately the mid radius and not at the centre of the ingot, which is the region prone to hot tearing. Thus, while the work by M’Hamdi shows a high level of correlation with the industrial measurements, there is room for improvement. One of the major deficiencies with the above research is the assumption that semi-solid stress- strain behaviour is dependent only on temperature and strain-rate, but is independent of the microstructure geometry and conditions. In Chapter 5, a microstructural model was presented which quantifies semi-solid constitutive behaviour at fraction solids where experiments are not viable and includes local microstructural phenomena. This model uses an equiaxed-granular model for generation of the semi-solid liquid network geometry, which is meshed and solved  A New Semi-solid Constitutive Equation for AA5182    119 using the finite element method. In the current work, the model is applied to predict the semi- solid constitutive behaviour of AA5182 over a range of grain sizes, fraction solid, and percentage porosity. These virtual experiments have enabled the development of a new semi-solid constitutive relationship which takes into account grain size and solidification defects. 6.2 Three-Phase Semi-Solid Microstructural Model 6.2.1 Model Description The microstructural model developed in Chapter 5 predicts semi-solid constitutive behaviour for AA5182 via a finite element simulation using a square domain consisting of solid grains surrounded by intergranular liquid; critical model features are shown in Fig.  6.1. As can be seen in the figure, the model contains equiaxed-globular grains of variable size with rounded corners, surrounded by liquid. The as-cast porosity defect has been included and one is able to vary the size of individual voids and also the percentage porosity. Model Geometry The model geometry is based on previous work by Mathier et al. [20], and Vernède et al. [21], and is derived from a Voronoi tessellation since the Voronoi regions have been found to share many similarities with equiaxed-granular grain structure [22]. The domain is divided into a subset of polygonal regions, the boundaries of which are the perpendicular bisectors of lines joining a set of N randomly placed discrete points. Each Voronoi region simulates a grain, with the discrete points representing grain nuclei. In [20] and [21], the Voronoi tessellation methodology was used to explore grain coalescence by cooling the geometry to simulate solidification. The methodology was extended in Chapter 5 to stress-strain prediction by constraining the domain to a specific fraction solid, and then applying a bulk deformation to the model domain. The model geometry consists of 56 Voronoi regions. To create a semi-solid structure, each Voronoi region within the Voronoi tessellation is reduced in size via a simple scaling law; the resulting extra material surrounding each region is assumed to be liquid. The Voronoi regions contain sharp vertices which have been rounded based on a solute flux balance between the Gibbs-Thomson effect and enhanced solute diffusion at the corners. Removing solid material from the corners has a secondary effect whereby liquid concentrates at the corners creating  A New Semi-solid Constitutive Equation for AA5182    120 triple-junctions. Individual voids of radius 7.5 μm are then randomly added at the triple-junctions to approximate the effect of as-cast porosity on semi-solid behaviour. Finite Element Simulation The mesh generation and finite element simulation are performed using the software package ABAQUS1. The flow stress of the solid grains is both strain rate and temperature dependent, and is based on the Ludwik equation developed by Chaudhary for AA5182 [23] and modified in Chapter 5 for application to higher temperature. The flow stress in the liquid is assumed to be 0.5 MPa. The model predictions (i.e. the bulk semi-solid flow stress curves as a function of strain) have been calculated based on the force required to displace the top side of the square, with symmetry boundary conditions on the left and bottom side, a distance corresponding to a tensile strain of 0.0225 at a rate of 0.0015 s-1.    Figure 6.1: Critical features of the three phase semi-solid microstructural model (Note that the mesh has been removed from the liquid for clarity).  1 SIMULIA, Providence, RI, USA  A New Semi-solid Constitutive Equation for AA5182    121 6.2.2 Model Application The three-phase microstructural model has been used to predict the constitutive behaviour of semi-solid AA5182 over a large range of fraction solid (fs), fraction porosity(fp), and grain size ( d ). Firstly, a number of preliminary model runs were conducted to determine the effect of the above variables on constitutive behaviour. Secondly, a comprehensive series of model runs was performed to simulate tensile deformation in the range 0.75 < fs < 0.97, 0 < fp < 0.006, and 75 μm < d  < 300 μm; where fs is the volume fraction solid, fp is the volume fraction void and d  is the average grain diameter or grain size. A summary of the values chosen for fs, fp, and d  in the comprehensive series is provided in Table  6A. In total, 50 different stress-strain predictions from model runs, corresponding to 50 different microstructure geometries, were acquired. Note that the effect of porosity was investigated only in conjunction with grain sizes of 150 and 225 μm. The stress-strain predictions obtained by the comprehensive series of model runs are shown in Figs.  6.2– 6.4. It is clear from the figures that both fraction solid and grain size have a large effect on constitutive behaviour while the effect of porosity is comparatively small. It can also be seen in Figs.  6.2– 6.4 that the semi-solid flow stress increases with increasing strain. This hardening behaviour is termed geometric hardening and occurs due to interaction effects between the solid grains. Similar mechanisms would occur during deformation of actual solidification microstructure, although the strain accumulating in the liquid would be replaced by intergranular liquid flow. Fraction Solid The effect of fraction solid (fs = 0.75, 0.80, 0.85, 0.90, and 0.95) on the stress-strain predictions is shown in Fig.  6.2 for the case with d  = 150 μm. As can be seen from the figure, the semi-solid flow stress decreases with decreasing fraction solid. This flow stress decrease is related to changes occurring in both the solid grains and the liquid film. Firstly, there is a decrease in the flow stress of the solid grains due to an increase in temperature. The relationship between fraction solid and temperature [24] is provided in Table  6B, and the constitutive behaviour of the solid is as described in Chapter 5. Secondly, to accommodate more liquid, the thickness of the liquid channels must increase which allows for more strain accumulation in the (softer) liquid before the solid grains , and thus decrease the overall bulk flow stress.   A New Semi-solid Constitutive Equation for AA5182    122 Grain Size The effect of grain size ( d  = 75, 150, 225, and 300 μm) is shown in Fig.  6.3 for the case with fs = 0.90. As can be seen from the figure, the semi-solid flow stress decreases with increasing grain size. Larger grain sizes weaken the semi-solid since they result in thicker liquid channels for a given fraction solid, allowing for increased strain accumulation in the liquid before the solid grains deform. Although not shown in Fig.  6.3, the effect of grain size decreases with decreasing fraction solid. This is because at high fraction solid, the relative difference in liquid channel thickness between the large and small grain sizes is significant, while at smaller fraction solid, the liquid films are already quite thick and changing the grain size provides a relatively smaller change in film thickness. Fraction Porosity The effect of fraction porosity on the stress-strain predictions is provided in Fig.  6.4 for the cases with fs = 0.80 and 0.95, and d  = 150 μm. At fs = 0.95, the behaviour corresponding to fp = 0, 0.002, and 0.006 is shown, while at fs = 0.80, the curves for to fp = 0 and 0.006 are given. As can be seen in the figure, the addition of porosity to the model geometry results in a modest decrease in the semi-solid bulk flow stress, with more of an effect at fs = 0.95 as compared to fs = 0.80. This flow stress decrease is related to changes within the liquid phase of the model domain. Firstly, adding voids at the triple-junctions creates individual liquid films which are isolated from the main liquid network. While not all voids create individual films (see Chapter 5 for more details), these individual films lead to increased strain localization and deformation resulting in decreased flow stress. Secondly, while the fraction porosity is quite small, the voids are placed at critical locations within the geometry and thus the effective cross-sectional area of the model domain is reduced. Thirdly, the effect of porosity is more pronounced at high fraction solid because the liquid films are thinner. It is important to point out, that as shown in Chapter 4, porosity and its evolution with strain will have a large impact on ductility although the model does not currently account for this effect. One of the interesting aspects of the microstructural model is that for specific values of fs, fp, and d , multiple semi-solid geometries can be created because the set of N points which define the grain nuclei are randomly placed within the model domain. Due to interaction between the solid and liquid, these individual geometries will predict different stress-strain behaviour. This stochastic variability was investigated by conducting 25 model runs at fs = 0.90, fp = 0, for each  A New Semi-solid Constitutive Equation for AA5182    123 of the four grain sizes (75, 150, 225, and 300 μm). The location of the grain nuclei was varied between runs providing each run with a different geometric grain configuration. Table  6C shows the results of this analysis, by providing both the median flow stress (σ ) at a strain of 0.01, and the percentage variability resulting from the 25 different model runs, for all four grain sizes.  As can be seen in the table, geometric configuration can have a significant effect on semi-solid constitutive behaviour, especially at the larger grain sizes with a variability of 25 % for a grain size of 300 μm. The larger grain sizes are more vulnerable to geometric configuration variation because the liquid channels are thicker which provides for more variability in the interaction between the solid and liquid components.  Table 6A: Comprehensive series of model runs conducted to predict the semi- solid constitutive behaviour of AA5182. Grain Size Fraction Solid fp 75 0.75, 0.80, 0.85, 0.90, 0.95, 0.97 0 150 0.75, 0.80, 0.85, 0.90, 0.95, 0.97* 0, 0.002, 0.004, 0.006 225 0.75, 0.80, 0.85, 0.90, 0.95, 0.97* 0, 0.002, 0.004, 0.006 300 0.75, 0.80, 0.85, 0.90, 0.95, 0.97 0 *Note that this value of fs was only used when fp = 0  Table 6B: Variation in fs(T) for AA5182 after Thompson [24]. Fraction Solid Temperature 1.0 525 0.97 550 0.95 561 0.90 578 0.85 588 0.80 596 0.75 602   A New Semi-solid Constitutive Equation for AA5182    124 Table 6C: Results showing the effect of grain geometric configuration at ε = 0.01 on model predictions. Grain Size (μm) σ  (MPa) % Variability 75 5.52 15 % 150 4.71 20 % 225 4.16 24 % 300 3.87 25 %    ε true σ( M P a) 0.005 0.01 0.015 0.02 0.0250 1 2 3 4 5 6 7 8 fs = 0.75 fs = 0.95  Figure 6.2: Effect of fraction solid on the predicted semi-solid constitutive behaviour in the range 0.75 < fs < 0.95 ( d  = 150 μm, fp = 0).  A New Semi-solid Constitutive Equation for AA5182    125 ε true σ( M P a) 0.005 0.010 1 2 3 4 5 6 7 8 75 150 225 300 Grain Size (μm)  Figure 6.3: Effect of grain size on the predicted semi-solid constitutive behaviour in the range 75 < d  < 300 μm (fs = 0.90, fp = 0). ε true σ( M P a) 0.005 0.010 1 2 3 4 5 6 7 8 0 0.002 0.006 0.001 0.0061 Fraction Porosity fs = 0.95 fs = 0.80  Figure 6.4: Effect of fraction porosity on the predicted semi-solid constitutive behaviour at fs = 0.80 and fs = 0.95 ( d  = 150 μm).  A New Semi-solid Constitutive Equation for AA5182    126 6.3 Semi-Solid Empirical Constitutive Relationship The results from the three-phase semi-solid microstructural model provide a clear picture of the effects of microstructure and process defects on the semi-solid constitutive behaviour of AA5182. However, this model is not suitable for implementation in a macro-scale model as it requires a fine mesh resolution and thus, the model domain can only be on the order of a few millimeters. In contrast, some industrial process models are a few meters in dimensions. One approach to circumvent this problem is to develop an empirical-type constitutive relationship, using the results predicted by the microstructural model in lieu of experimentally derived data, The empirical-type model could be formulated to account for the variables investigated with the microstructural model including fraction solid, fraction porosity, and grain size. Within the macro-scale model, variations in these parameters can be calculated at each finite element integration point. For example, the evolution of fraction solid could be obtained from the temperature, the grain size from the cooling rate and thermal gradient (e.g. Ares et al. [25]), and the fraction porosity from a two-stage porosity model incorporating the effects of hydrogen solubility and encapsulated liquid (e.g. Zhu et al. [26]). 6.3.1 Form of the Constitutive Relationship The model results have identified four critical characteristics of semi-solid constitutive behaviour: (a) the flow stress of the solid is an important element of semi-solid constitutive behaviour; (b) the effect of the liquid on the semi-solid is governed by the thickness of the liquid films; (c) the semi-solid material experiences strain hardening; (d) porosity plays a minor but potentially important role in semi-solid constitutive behaviour. Based on these characteristics, it is clear that a new empirical constitutive relationship should include a strain hardening term of the form shown in Eq. ( 6.1) to account for the geometric hardening behaviour, a rule of mixtures term of the form shown in Eq. ( 6.2) to combine the solid and liquid material properties, and a term of the form shown in Eq. ( 6.3) characterizing the effect of fraction porosity on flow stress such that this effect decreases with decreasing fraction solid.  nσ ε∝  (6.1)  ( )1s s s lf fσ σ σ∝ + −  (6.2)  A New Semi-solid Constitutive Equation for AA5182    127  1 p s f fσ ∝ −  (6.3) The flow stress for solid grains can be calculated from the same Ludwik equation used to model solid AA5182 in the microstructural model, where σ has units of MPa, ε&  is the bulk strain rate, and T is the temperature:  ( ) 0.205 0.00006483.5 0.77 Ts Tσ ε += − ⋅ &  (6.4) One approach for estimating the liquid flow stress is the Young and Laplace equation, which is used to calculate the stress necessary to separate two plates bonded by a thin liquid film [27]:  2 sll h γσ =  (6.5) where γsl = 1 J m-2 is the solid-liquid interfacial energy and h is the liquid film thickness. This equation provides an upper bound to the liquid stress since it predicts the yield stress of the liquid surface instead of the shearing stress required for liquid flow. Furthermore, since h does not have a unique value, Larouche [28] proposed that an average h for an undeformed semi-solid body, could be related to the grain size, λ1 in units of μm, and fraction solid:  ( )1/31 1 sh fλ= −  (6.6) Eq. (6.6) can also be used to determine the degree to which geometric strain hardening occurs with respect to both fraction solid and grain size. Thus, the hardening parameter, n, in Eq. (6.1) was described via:  2n C h D h= ⋅ + ⋅  (6.7) where C and D are fitting parameters. Given the requirements of Eq. ( 6.1)–( 6.3), the results from the microstructural model simulations were compiled and used to formulate a new empirical constitutive relationship for the aluminum alloy AA5182 in the semi-solid state, 0.75 < fs < 0.95:  ( )( ) ( ) ( ) 2 1 1 1 Ch Dh p s s s l s s ff f Af B fσ σ σ ε + ⎛ ⎞= + − ⋅ ⋅ − −⎜ ⎟−⎝ ⎠  (6.8) where σs is the solid flow stress calculated by Eq. ( 6.4), σl is the liquid flow stress calculated by Eq. (6.1), h is the average liquid channel thickness calculated by Eq. ( 6.6), and A–D are fitting parameters. The values of the fitting parameters, determined using a least-squares approximation, is provided in Table  6D.  A New Semi-solid Constitutive Equation for AA5182    128 Table 6D: Fitting parameters for the empirically-based constitutive relationship. Fitting Parameter Value A 0.15, fp >0 0, fp = 0 B 1 C -635 D 20.24  6.3.2 Comparison with Microstructural Model Predictions To validate the new relationship, a comparison of the semi-solid constitutive behaviour predicted by the empirical relationship and the microstructural model is shown in Figs.  6.5, and  6.6. Fig.  6.5 provides this comparison for two grain sizes: (a) d  = 300 μm, and (b) d  = 75 μm, in the range 0.75 < fs < 0.95 (note: the domains used in this comparison did not contain any as- cast porosity). The symbols represent the microstructural model results, and the lines represent the flow stresses as calculated by the empirical relationship. Furthermore, these two grain sizes were chosen since they represent extremes within the model parameter space. Fig.  6.6 provides this comparison whereby the effects of porosity are included, for (a) fs = 0.85 and (b) fs = 0.95. As can be seen from Fig.  6.5, the stresses predicted by the empirical relationship match quite well the microstructural model predictions. In Fig.  6.5(a) ( d  = 300 μm), the empirical relationship provides an excellent fit to the model predictions. In Fig.  6.5(b) ( d  = 75 μm), the fit is good for all but fs = 0.95. For the case of fs = 0.95, the empirical relationship flow stresses are significantly higher than the microstructural model predictions. This difference occurs because the empirical constitutive relationship considers only the bulk strain rate within the model domain, while the solid grains in the microstructural model contain local strain rate sensitivity. At fs = 0.95, the combination of small grain size and high fraction solid create very thin liquid films. Thin liquid films allow for significant strain accumulation in the solid grains, leading to a high degree of strain rate sensitivity. At lower fraction solid, or larger grain sizes, this effect of strain rate on constitutive behaviour is less of an issue because the liquid films are thicker. As can be seen in Fig.  6.6, the results for the case with as-cast porosity included in the empirical constitutive behaviour are similar to the results from Fig.  6.5. In this case, the empirical relationship is able to match fairly well the microstructural model predictions at fs = 0.85, but does not match well the results at fs = 0.95. In contrast to the results presented in  A New Semi-solid Constitutive Equation for AA5182    129 Fig.  6.5, the empirical relationship underestimates the flow stress at fs = 0.95 as compared to the model predictions including porosity. The porosity term in Eq. ( 6.8) was the most difficult aspect of the empirical constitutive relationship to quantify because the actual locations of individual voids have a significant effect on stress-strain response. Similar to the analysis presented in Table  6C for grain geometric configuration, an analysis on 25 model runs with different void placement was conducted. The microstructural model predictions varied by ~ 0.40 MPa at fs = 0.90. From an application standpoint, the void geometric configuration with the lowest flow stresses and therefore the one that would predict the highest strains, was chosen as input data for the constitutive equation. In this manner, the model would be conservative in terms of its ability to predict hot tearing. 6.3.3 Comparison with Experimental Data The new empirical constitutive relationship has been successfully validated against the microstructural model predictions. As a final check on its accuracy, a comparison between the flow stresses predicted by the empirical relationship and experimental measurements from Chapter 4 is presented in Fig.  6.7. The measurements were made on as-cast AA5182 at various fraction solid in the range 0.95 < fs < 0.98, while the empirical relationship calculations were performed for the case with d  = 225 μm, ε&  = 0.0015, and ε = 0.005. The results for two values of fraction porosity are provided: 0.003 and 0.006. These values represent the range of porosity reported in Chapter 2 ([29]) and 3 for as-cast AA5182. As can be seen in the figure, the empirical relationship underpredicts the flow stresses for fs > 0.95 as compared to experimental data. At fs = 0.95, the empirical relationship seems to match the experimental measurements. At fractions solid less that 0.95, the empirical relationship underestimates the reduction in flow stress. The comparison shown in Fig.  6.7 highlights some of the limitations in both the semi-solid microstructural model and the empirical model. At fraction solids less than 0.95, the experimental measurements show a significant loss in flow stress due to the rapid accumulation of internal damage. In contrast, the liquid films in the microstructural model do not simulate this evolution in damage and contain intergranular liquid flow. At fraction solids above fs = 0.95, the empirical relationship underpredicts the semi-solid flow stresses because the microstructural model fails to account for the effective grain bridging, proposed by Rappaz, that occurs when the thickness of a liquid channel falls below a certain threshold [30].  A New Semi-solid Constitutive Equation for AA5182    130 X X X X X X X X X X X X ε true (d = 300 μm) σ( M P a) 0 0.005 0.010 1 2 3 4 5 6 7 8 9 10 fs = 0.75 fs = 0.95  (a) X X X X X X X X X X X ε true (d = 75 μm) σ( M P a) 0 0.005 0.010 1 2 3 4 5 6 7 8 9 10 fs = 0.75 fs = 0.95  (b) Figure 6.5: Validation of the empirical constitutive relationship via the model predictions for the case without porosity: (a) d  = 300 mm, (b) d  = 75 mm. (Note that the lines represent the empirical equation relationship)  A New Semi-solid Constitutive Equation for AA5182    131 ε true σ( M P a) 0.005 0.010 2 4 6 8 10 12 0.002 0.006fs=0.85 0.002 0.006fs=0.95 Fraction Porosity  Figure 6.6: Validation of the empirical constitutive relationship for the case where porosity is included at fs = 0.85 and fs = 0.95 ( d  = 150 μm). (Note that the lines represent the empirical equation relationship) Fraction Solid σ flo w (M P a) 0.88 0.9 0.92 0.94 0.96 0.98 2 4 6 8 10 12 14 Experimental fp = 0.003 fp = 0.006  Figure 6.7: Validation of the semi-solid AA5182 empirical constitutive via the experimental measurements from Chapter 4.  A New Semi-solid Constitutive Equation for AA5182    132 6.4 Implementation within Industrial Process Models and Application to Other Alloys In order to improve the industrial predictions of semi-solid deformation and thus enable reduction of hot tearing, the new constitutive relationship must be implemented within an industrial process model. As discussed earlier, implementation requires techniques for predicting grain size, fraction porosity, and fraction solid at the scale of the mesh employed in the numerical solution. This in turn requires knowledge of the temperature, cooling rate, and thermal gradient – variables which are generally available in finite element software. However, while implementation of the new constitutive relationship is conceptually trivial, it accounts only for deformation behaviour at fraction solids greater than 0.75. Therefore in order to fully solve the problem one would have to grapple with the potential for strain to accumulate at low fraction solid, which does not physically make sense as mass feeding is still active. Presumably, one would have to develop an algorithm to remove any strain accumulated at temperatures above the limit in mass feeding. Another shortcoming of the present model is that it does not account for the strain-induced accumulation of damage (porosity) on the constitutive response of the material. The new empirical constitutive relationship has provided a mechanism for incorporating semi- solid microstructure configuration and as-cast defects into the prediction of semi-solid deformation for the aluminum alloy AA5182. There are many other alloys that are also susceptible to the hot tearing defect and would benefit from a similar empirical constitutive relationship. In calculating the values of the fitting parameters A–D from microstructural model results from these other alloys, three key characteristics are required: a temperature and strain rate dependent constitutive relationship for fully solid material; experimental measurements on semi-solid material to allow for extrapolation of the fully solid constitutive relationship into the semi-solid temperature range; and the relationship between fraction solid and temperature.  A New Semi-solid Constitutive Equation for AA5182    133 6.5 Conclusions A new empirical constitutive relationship for the semi-solid aluminum alloy AA5182 has been formulated. This relationship is valid for high fraction solid, where the solid phase plays a major role in the deformation properties. Furthermore, this relationship includes the relevant effects of fraction porosity and grain size on the stress-strain predictions. These features have not previously been incorporated in a semi-solid constitutive relationship. This new relationship was developed based on virtual experiments conducted using the finite element-based semi-solid microstructural model developed in Chapter 5 in the range 0.75 < fs < 0.95, for various grain sizes and various fraction porosity. Since it is relatively easy in a process model to calculate a local value for fraction porosity and grain size, the new constitutive relationship allows for the input semi-solid constitutive behaviour to vary as a function of the solidification conditions. It is anticipated that implementation of this model will enable improved hot tearing predictions.  A New Semi-solid Constitutive Equation for AA5182    134 6.6 References 1. Colley, L. J., Wells, M. A. and Maijer, D. M., Mater. Sci. Eng. A, 386(1-2), 2004, (140). 2. Van Haaften, W. M., Kool, W. H. and Katgerman, L., Mater. Sci. Eng. A, 336, 2002, (1). 3. Kron, J. and Fredriksson, H., International Symposium of Liquid Metals and Casting, P. D. Lee, A. Mitchell, A. Jardy, J. P. Bellot, Eds., Nancy, Fr, SF2M-Paris, 2003, (393). 4. Twite, M. R., Spittle, J. A. and Brown, S. G. R., Intl J Forming Processes, 2004, (233). 5. Ackermann, P., Kurz, W. and Heinemann, W., Mater. Sci. Eng. , 75, 1985, (79). 6. Langlais, J. and Gruzleski, J. E., Mater. Sci. Forum, 167, 2000, (331). 7. Magnin, B., Maenner, L., Katgerman, L. and Engler, S., Mater. Sci. Forum, 217-222, 1996, (1209). 8. Mitchell, J. B., Cockcroft, S. L., Viano, D., Davidson, C. J. and St John, D., Metall. Mater. Trans. A, 38, 2007, (in press). 9. Drezet, J.-M. and Eggeler, G., Scr. Metall. Mater., 31, 1994, (757). 10. Braccini, M., Martin, C. L. and Suery, M., Model. Cast. Weld. Adv. Solid., 2000, (18). 11. Fabregue, D., Deschamps, A., Suery, M. and Poole, W. J., Metall. Mater. Trans. A, 37A, 2006, (1459). 12. Martin, C. L., Favier, D. and Suery, M., Intl. J. Plas, 13, 1997, (237). 13. Martin, C. L., Favier, D. and Suery, M., Intl. J. Plas, 13, 1997, (215). 14. M'Hamdi, M., Mo, A. and Martin, C. L., Metall. Mater. Trans. A, 33A, 2002, (2081). 15. Suery, M., Martin, C. L., Braccini, M. and Brechet, Y., Adv. Eng. Mater., 3, 2001, (589). 16. Martin, C. L., Braccini, M. and Suery, M., Mat. Sci. and Eng. A, A325, 2002, (292). 17. Sengupta, J., Cockcroft, S. L., Maijer, D. and Larouche, A., Mater. Sci. Eng. A., 397, 2005, (157). 18. Ludwig, O., Drezet, J.-M., Martin, C. L. and Suery, M., Metall. Mater. Trans. A, 36A, 2005, (1525). 19. M'Hamdi, M., Benum, S., Mortensen, D., Fjaer, H. G. and Drezet, J.-M., Metall. Mater. Trans. A, 34A, 2003, (no. 9). 20. Mathier, V., Jacott, A. and Rappaz, M., Modelling Simul. Mater. Sci. Eng., 12(3), 2004, (479). 21. Vernede, S., Jarry, P. and Rappaz, M., Acta Mat., 54, 2006, (4023). 22. Vernede, S. and Rappaz, M., Acta Mat., 55, 2007, (1703). 23. Chaudhary, A. and Wells, M. A., Proceedings of the International Conference on Advances in Materials and Meterials Processing, U. K. Chatterjee, B. K. Dhindaw, Eds., IIT Kharagpur, India, 2006, (763). 24. Thompson, S., Cockcroft, S. L. and Wells, M. A., Mater. Sci. Techn., 20(4), 2004, (497). 25. Ares, A. E., Caram, R. and Schvezov, C. E., TMS Light Metals 2003, San Diego, CA, 2003, (1055). 26. Zhu, J. D., Cockcroft, S. L. and Maijer, D., Metall. Mater. Trans. A, 37A, 2006, (1075). 27. Lahaie, D. J. and Bouchard, M., Metall. Mater. Trans. B, 32, 2001, (697). 28. Larouche, D., Langlais, J., Wu, W. and Bouchard, M., Metall. Mater. Trans. B, 37B, 2006, (431). 29. Phillion, A. B., Cockcroft, S. L. and Lee, P. D., Scripta Mat. , 55, 2006, (489). 30. Rappaz, M., Jacot, A. and Boettinger, W. J., Metall. Mater. Trans. A, 34A, 2003, (467).     135 Chapter 7: Summary and Conclusions 7.1 Summary and Conclusions Hot tears form in the last stages of solidification, in a region where the grains are surrounded by a continuous film of liquid. Application of load causes strain to be concentrated within this liquid film, thus limiting ductility and creating a hot tear. In this study, the mechanisms of semi- solid tensile deformation and hot tearing were investigated in the aluminum–magnesium alloy AA5182. To examine the effects of strain accumulation, the evolution of void size distribution and void number density were quantified using x-ray micro-tomography on material in both the as-received condition and at various stages of semi-solid deformation. To analyze the role of porosity, as-cast material in both the as-received condition and following a hot isostatic pressing (HIP) treatment (which removed the porosity) were rapidly heated and then deformed at semi- solid temperatures. To acquire accurate load-displacement data at temperatures just below and within the semi-solid regime, a new two thermocouple tensile test methodology was developed. The above experimental work was complimented and extended to a wider range of fraction solid and microstructural conditions through creation of a three-phase finite element-based semi- solid microstructural model. This model explored the effects of fraction solid, grain size and fraction porosity on semi-solid constitutive behaviour. The results obtained via x-ray micro- tomography were used as input to the three-phase microstructural model, while the semi-solid stress-strain behaviour obtained from the two thermocouple tensile test methodology was used for model validation. Combined, these unique techniques have enabled new insight into the processes involved in semi-solid deformation, and have allowed the relative importance of as- cast porosity, discrete growth of damage, and damage growth via a process of coalescence to be qualitatively and quantitatively assessed. The findings and achievements of the current thesis can be summarized as follows: • The first 3D observations of the development of internal damage with strain in a semi-solid material have been presented. From these images, it is clear that semi-solid deformation is initially controlled by discrete growth of as-cast porosity and an increase in the number of small voids. These voids grow preferentially in a direction normal to the applied load via void coalescence and unzipping of the liquid at the grain boundaries. Void growth was quantified by void spatial orientation plots, while finite element modeling on a small volume Summary and Conclusions    136 of deformed material supported the mechanism of void coalescence. At some critical strain, localized void coalescence leads to a decrease in the void number density and final failure. • A new experimental methodology has been developed to measure the tensile constitutive behaviour of semi-solid aluminum alloys. This two thermocouple technique provides accurate thermal control during semi-solid tensile deformation, accurate control over cross- head displacement and eliminates earlier issues associated with thermocouple weld detachment and thermal expansion. • The semi-solid constitutive behaviour of AA5182 has been quantified in both the as-cast and HIP states. The results from this analysis showed little difference in the flow stress associated with porosity other than a shift in the relationship between flow stress and temperature to higher temperatures by ~ 10°C, including the mechanical coherency temperature. The shift is consistent with the chemical homogenization that occurred and the associated increase in the solidus temperature. In contrast, and as a result of the removal of the as-cast porosity, the HIP material exhibited a significantly higher ductility than the as-cast material. • The application of the HIP process to the as-cast material has greatly reduced the amount of internal damage accumulated during semi-solid deformation, indicating that pre-existing as- cast porosity plays an important role in the formation of hot tears. The evidence points to as- cast voids acting as stress risers, allowing strain to be accommodated by growth of this pre- existing damage through the liquid along grain boundaries. In the absence of pre-existing voids, nucleation of voids would need to occur to create damage increasing material ductility. • A three-phase microstructural model geometry based on a Voronoi diagram has been developed. Additional geometric features include rounded corners to approximate equiaxed- globular grain structure, and circular voids at the triple-junctions representative of as-cast porosity. The flow stress of the solid grains is both strain rate and temperature dependent, and is based on a Ludwik equation developed by Chaudhary for AA5182 which has been extended for application to higher temperatures. The flow stress in the liquid is assumed to be a small value. Virtual experiments were conducted to predict the bulk constitutive behaviour in the range 0.75 < fs < 0.95, 0 < fp < 0.006, and grain sizes between 75 and 300 μm. • To facilitate application of the microstructural model results within macro-scale thermo- mechanical process models, an empirical relationship was proposed which accounts for the effect of grain size and fraction porosity on constitutive behaviour. Local variation in these Summary and Conclusions    137 quantities can be estimated within an industrial process model from the cooling rate and thermal gradient. This new relationship was developed based on the virtual experiments conducted using the three-phase semi-solid microstructural model. In conclusion, it appears that hot tearing is directly related to both the accumulation of strain within the liquid and the evolution of damage within the liquid. The former has been shown to be dependent on the grain size through its influence on flow stress and the latter is strongly influenced by the number of initiation sites in the semi-solid material, such as hydrogen-based and shrinkage-based microporosity. In industrial aluminum castings, a reduction in hot tearing sensitivity could thus be accomplished by several strategies including decreasing the hydrogen content in the liquid, increasing the cooling rate / gradient during solidification, and decreasing the grain size via increased grain refining.  7.2 Future Work The conclusions drawn from the experimental portion of this research have been based on the use of interrupted tensile tests on partially remelted material in combination with tomography, rather than continuous in situ observation of the semi-solid material during deformation. To provide a more complete picture of growth and coalescence phenomena, research should be initiated to develop an apparatus capable of partially melting, and then deforming in tension, an aluminum specimen while inside an x-ray micro-tomography unit. Furthermore, an Al–Cu alloy should be used, to allow for better x-ray contrast between the solid and liquid phases. These experiments would enable fundamental insight into damage formation during semi-solid processing through quantification of the kinetics of damage growth. The three-phase microstructural model was run at constant temperature, to facilitate meshing of the semi-solid microstructure. However, this scenario limits the model’s capabilities to stress- strain prediction at a specific fraction solid. To further explore semi-solid deformation mechanisms, the model should be expanded to include the effects of solidification. This expansion would require use of new meshing techniques such as adaptive re-meshing, and development of interpolation schemes to account for plastic strain since elements would transition from liquid to solid properties. Finally, the new empirical constitutive relationship should be implemented into an industrial process model to verify that improvements in the hot tearing predictions are obtained.

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