UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Multiaxial deformation of AZ80 magnesium alloy Tomlinson, Philip S. 2013

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.

Item Metadata


24-ubc_2013_fall_tomlinson_philip.pdf [ 20.85MB ]
JSON: 24-1.0165637.json
JSON-LD: 24-1.0165637-ld.json
RDF/XML (Pretty): 24-1.0165637-rdf.xml
RDF/JSON: 24-1.0165637-rdf.json
Turtle: 24-1.0165637-turtle.txt
N-Triples: 24-1.0165637-rdf-ntriples.txt
Original Record: 24-1.0165637-source.json
Full Text

Full Text

Multiaxial Deformation of AZ80 Magnesium AlloybyPhilip S. TomlinsonB. Mechanical Engineering, University of Western Ontario, 2005M. Mechanical and Materials Engineering, University of Western Ontario, 2007A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Materials Engineering)The University Of British Columbia(Vancouver)October 2013c? Philip S. Tomlinson, 2013AbstractThe multiaxial deformation of magnesium alloys is important for developing reli-able, robust models for both the forming of components and also analysis of in-service performance of structures, for example, in the case of crash worthiness.This work presents a combination of unique biaxial experimental tests and biaxialcrystal plasticity simulations using a visco-plastic self-consistent (VPSC) formu-lation conducted on AZ80 magnesium alloy in two different conditions - extrudedand a more weakly textured as cast condition. The experiments were conductedon tubular samples which are loaded in axial tension or compression along thetube and with internal pressure to generate hoop stresses orthogonal to the axialdirection. The results were analyzed in stress and strain space and also in terms ofthe evolution of crystallographic texture. In general, it was found that the VPSCsimulations matched well with the experiments, particularly for the more weaklytextured cast material. However, some differences were observed for cases wherebasal < a > slip and {101?2} extension twinning were in close competition suchas in the biaxial tension quadrant of the plastic potential. The evolution of texturemeasured experimentally and predicted from the VPSC simulations was qualita-tively in good agreement. Finally, experiments and VPSC simulations were con-ducted in which samples of the extruded AZ80 material were subjected to a smalluniaxial strain prior to biaxial loading in order to further explore the competitionbetween basal slip and extension twinning.iiPrefaceCited figures appearing in Chapter 2 are used with permission from applicablesources. Portions of the abstract text are used with permission from Tomlinsonet al. (2013) of which I am an author. Portions of the Abstract, Chapters 4 and6 have previously published as P. Tomlinson, H. Azizi-Alizamini, W. J. Poole, C.W. Sinclair and M. A. Gharghouri, ?Biaxial Deformation of the Magnesium Al-loy AZ80? MetTransA, 2013. Experiments and simulations conducted herein wereconceived by the research group consisting of myself, Dr. Warren Poole, Dr. ChadSinclair, Dr. Hamid Azizi-Alizamini and Dr. Michael Gharghouri. The experi-mental rig was assembled by myself from purchased components and componentsfabricated by the Materials Engineering Machine Shop. I wrote all control anddata acquisition software. I conducted all biaxial experimentation including thatdone at the Canadian Neutron Beam Centre, which was conducted jointly with Dr.Gharghouri. The VPSC code of Lebensohn and Tome? (1993) was modified byDr. Sinclair and Dr. Azizi-Alizamini and simulations were run by myself and Dr.Azizi-Alizamini. I prepared the neutron diffraction samples, the neutron diffrac-tion experiments themselves were conducted by Dr. Gharghouri at the CanadianNeutron Beam Centre and finally plotting and interpretation of their results wasconducted by myself.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Deformation of Magnesium Alloys . . . . . . . . . . . . . . . . . 52.2.1 Crystallography . . . . . . . . . . . . . . . . . . . . . . . 52.2.2 Slip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.3 Twinning . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Crystallographic Texture . . . . . . . . . . . . . . . . . . . . . . 122.3.1 Texture Measurement Techniques . . . . . . . . . . . . . 132.3.2 Polycrystal Deformation . . . . . . . . . . . . . . . . . . 162.3.3 Multiaxial Deformation . . . . . . . . . . . . . . . . . . 202.4 Iso-Work Surfaces for Anisotropic Materials . . . . . . . . . . . . 232.5 Biaxial Testing Methods . . . . . . . . . . . . . . . . . . . . . . 282.5.1 Cruciform Samples . . . . . . . . . . . . . . . . . . . . . 29iv2.5.2 Combined Tension Torsion . . . . . . . . . . . . . . . . . 302.5.3 Tension/Compression with Internal Pressure . . . . . . . . 312.5.4 Polycrystal Deformation Modelling . . . . . . . . . . . . 322.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 Scope and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Biaxial Test Rig . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.3 Strain Gauge Calibration . . . . . . . . . . . . . . . . . . . . . . 404.4 LaVision Digital Image Correlation . . . . . . . . . . . . . . . . 444.5 Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.6 LabVIEW Control Program . . . . . . . . . . . . . . . . . . . . . 484.6.1 Iso-Work Surfaces . . . . . . . . . . . . . . . . . . . . . 504.7 Neutron Diffraction Studies . . . . . . . . . . . . . . . . . . . . . 514.8 Test Material and Thermomechanical Preprocessing . . . . . . . . 524.9 VPSC Simulations . . . . . . . . . . . . . . . . . . . . . . . . . 545 System Validation and Examination of Al 6061 Alloy . . . . . . . . . 565.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.2 Experimental Scope . . . . . . . . . . . . . . . . . . . . . . . . . 565.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . 575.4 von Mises Equivalent Stress Surfaces . . . . . . . . . . . . . . . 585.5 Neutron Diffraction Studies . . . . . . . . . . . . . . . . . . . . . 605.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626 Mechanical Testing of AZ80 Alloy Under Constant Biaxial Ratios . 636.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.2 Cast and Heat Treated AZ80 . . . . . . . . . . . . . . . . . . . . 666.2.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.2.2 Deformation Texture of Cast AZ80 . . . . . . . . . . . . 72v6.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 836.3 Extruded AZ80 . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.3.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.3.2 Deformation Texture of Extruded AZ80 . . . . . . . . . . 956.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 1046.3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 1156.4 Effect of Pre-Strain on Extruded AZ80 . . . . . . . . . . . . . . . 1166.4.1 Results and Discussion . . . . . . . . . . . . . . . . . . . 1166.4.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 1237 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 1257.1 Summary of Observations . . . . . . . . . . . . . . . . . . . . . 1257.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129viList of TablesTable 2.1 Slip systems and associated CRSS . . . . . . . . . . . . . . . 8Table 2.2 Characterisitc parameters of commonly observed twin types inmagnesium alloys . . . . . . . . . . . . . . . . . . . . . . . . 11Table 4.1 VPSC input parameters used for both cast and extruded AZ80 . 55viiList of FiguresFigure 2.1 HCP Unit Cell . . . . . . . . . . . . . . . . . . . . . . . . . 6Figure 2.2 Important HCP directions and planes . . . . . . . . . . . . . 7Figure 2.3 Twinning involves a reorientation of the crystal lattice. . . . . 10Figure 2.4 Twin types and planes . . . . . . . . . . . . . . . . . . . . . 12Figure 2.5 Example pole figures for extruded and rolled materials. . . . . 14Figure 2.6 Sample of an EBSD orientation map. . . . . . . . . . . . . . 16Figure 2.7 Stress-strain curves for extruded AZ61 samples pulled in ten-sion at different angles ? to the extrusion direction. . . . . . . 18Figure 2.8 Correlation between the amount of strain accommodated byextension twinning and the tilt angle between extrusion andloading directions in AZ61 magnesium alloy. . . . . . . . . . 19Figure 2.9 Comparison of Tresca and von Mises yield criteria . . . . . . 25Figure 2.10 Twinning can have a large effect on the yield surface of a ma-terial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Figure 2.11 Effect of variation of the variable k on the CPB06 yield criterion. 27Figure 2.12 Relationship between yield surface and test types. . . . . . . . 29Figure 2.13 Sample configuration for testing via tension with internal pres-sure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Figure 4.1 Sample with speckle pattern applied for DIC. . . . . . . . . . 38Figure 4.2 Loading of sample achieved with axial load and pressure. . . . 38Figure 4.3 Combination of axial load and internal pressure results in axialand hoop stresses. . . . . . . . . . . . . . . . . . . . . . . . . 39Figure 4.4 Data and control pathways within the test rig . . . . . . . . . 41Figure 4.5 Quarter bridge strain gauge configuration . . . . . . . . . . . 42viiiFigure 4.6 Two axially mounted strain gauges mounted one quarter of theway around the sample surface will reveal any misalignmentof the grips by recording differing strain levels. This figureillustrates good alignment. . . . . . . . . . . . . . . . . . . . 43Figure 4.7 Camera configuration for the DIC setup. . . . . . . . . . . . . 45Figure 4.8 DIC validation using extensometer . . . . . . . . . . . . . . . 46Figure 4.9 Numerical integration of biaxial stress-strain data. . . . . . . . 51Figure 4.10 Optical micrograph of cast AZ80 alloy . . . . . . . . . . . . . 53Figure 4.11 Optical micrograph of extruded AZ80 alloy . . . . . . . . . . 53Figure 5.1 Comparison of axial component of aluminum biaxial tests con-ducted at UBC and at CNBC. . . . . . . . . . . . . . . . . . 58Figure 5.2 Comparison of Al6061 experimental data with von Mises yieldcriterion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Figure 5.3 Pole figures for initial Al6061 texture. . . . . . . . . . . . . . 60Figure 5.4 Pole figures for Al6061 sample deformed axially in tension tofailure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Figure 5.5 Pole figures for Al6061 sample subjected to a biaxiality ratioof 2 until failure. . . . . . . . . . . . . . . . . . . . . . . . . 61Figure 6.1 Initial texture for cast and extruded samples. . . . . . . . . . . 64Figure 6.2 Discretized texture for cast and extruded samples used for VPSCsimulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Figure 6.3 Stress-strain results for cast AZ80. . . . . . . . . . . . . . . . 67Figure 6.4 Plastic work surfaces for cast AZ80 alloy . . . . . . . . . . . 68Figure 6.5 Strain-Strain plot for cast AZ80 alloy . . . . . . . . . . . . . 70Figure 6.6 Evolution of strain ratio with respect to applied plastic work incast AZ80 alloy. . . . . . . . . . . . . . . . . . . . . . . . . 71Figure 6.7 Experimental and VPSC textures for cast AZ80 after testingvia a number of uniaxial and biaxial loading paths. . . . . . . 77Figure 6.8 Slip system and twinning activity, as well as twinned fractionpredicted by VPSC simulation for deformation of cast AZ80subjected to various loading conditions. . . . . . . . . . . . . 82ixFigure 6.9 < 0001> Pole figures for axial tension samples pulled to strainsof 0.00, 0.04 and to failure respectively. . . . . . . . . . . . . 83Figure 6.10 < 0001 > Pole figures for 0.9 biaxiality samples pulled to axialstrains of 0.00, 0.025 and to failure respectively. . . . . . . . . 83Figure 6.11 Stress-strain results for extruded AZ80. . . . . . . . . . . . . 93Figure 6.12 Experimental plastic work surface for extruded AZ80 alloy . . 94Figure 6.13 Plastic work surface for extruded AZ80 alloy . . . . . . . . . 95Figure 6.14 Normalized plastic work surface for extruded AZ80 alloy . . . 96Figure 6.15 Strain-Strain responses for extruded AZ80 alloy . . . . . . . . 97Figure 6.16 Experimental and VPSC textures for extruded AZ80 after test-ing via a number of uniaxial and biaxial loading paths. . . . . 103Figure 6.17 Slip system and twinning activity, as well as twinned fractionpredicted by VPSC simulation for deformation of extrudedAZ80 subjected to various loading conditions. . . . . . . . . . 105Figure 6.18 Comparison of strain-strain data for cast and extruded AZ80alloy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110Figure 6.19 Comparison of cast and extruded material strain-strain data for0.4 and 0.9 biaxiality tests. . . . . . . . . . . . . . . . . . . . 111Figure 6.20 Comparison of basal slip and extension twinning activity forcast and extruded materials. . . . . . . . . . . . . . . . . . . 112Figure 6.21 VPSC predicted texture for axially pre-strained samples . . . 117Figure 6.22 Axial and hoop stress strain data for pre strained extruded AZ80.118Figure 6.23 Strain ratio evolution for pre strained extruded AZ80. . . . . . 119Figure 6.24 Strain ratio evolution for pre strained extruded AZ80. . . . . . 121Figure 6.25 VSPC predicted extension twinning activity and volume frac-tion for extruded AZ80 with the measured initial texture aswell as the texture predicted for after an axial tensile strain. . . 122Figure 6.26 Extruded AZ80 iso-work data showing effect of pre-strains. . 123xAcknowledgmentsThis research has been made possible thanks to the generous financial support ofThe MagNET Strategic Research Network in turn funded by the Natural Scienceand Research Council.I wish also to thank my supervisors Dr. Warren Poole and Dr. Chad Sinclairfor their time and patience over these past years. I would further like to express mygratitude to Dr. Michael Gharghouri for his continued patience with frantic pleasto get me just one more set of pole figures before a deadline. Finally, I would liketo thank the Mr. Ross Mcleod, Mr. Carl Ng and Mr. Dave Torok of the machineshop for fabricating my samples, Dr. Hamid Azizi-Alizamini for his unflaggingwillingness to pass on his wealth of experience and administrators Mrs. DebbieBurgess and Ms. Michelle Tierney who helped me navigate the byzantine maze ofbureaucracy that is UBC.xiChapter 1IntroductionHistorically, the production of magnesium peaked during the first and second worldwars [1] and then fell off sharply during peace time as magnesium alloys struggledto remain price competitive with aluminum alloys. As such, research in magnesiumlargely languished, particularly through the 1970s to 1990s and it is only the rapidlydeveloping concern over the production of green house gases (GHGs) over the last10-15 years that has revived interest in magnesium alloys for applications in thetransportation sector.One of the motivations for the renewed interest in the behaviour of magnesiumalloys principally revolves around fuel efficiency in automobiles and light trucks.The increasing level of pressure from government and public bodies to increasetransportation efficiencies through directives such as the United States? 2012 Cor-porate Average Fuel Economy (CAFE) Standard which will require an averagefuel economy of 54.5 MPG (4.35 L/100km) by the year 2025 [2] and similar com-mitments from the Canadian Government [3] has spurred significant research intomagnesium alloys, which represent an opportunity for significant weight savings1without compromising performance. Magnesium, with a density of approximately1.8g/cm3 is 33% less dense than aluminum and 77% less dense than steel, presentsa particularly interesting possibility in the lightweighting of both non-structural andstiffness-in-bending constrained components where the modest elastic modulus ofmagnesium (45 MPa) is more than offset by its low density.Currently, the principal barriers to the adoption of magnesium alloys rest withtheir strong anisotropy, generally poor room temperature ductility, corrosion be-haviour, liquid reactivity and cost.The detailed deformation behaviour of magnesium is generally poorly under-stood compared with steel and aluminum. This lack of knowledge frustrates at-tempts to manage or improve the anisotropy and room temperature properties ofmagnesium alloys. Magnesium possesses a hexagonally close packed (HCP) crys-tal structure. The crystallographic structure of an HCP metal is comprised of al-ternating layers of hexagonally closed packed planes. The two primary crystallo-graphic orientations are the a direction which lies within the closed packed planesand the c direction which is perpendicular to it. Magnesium possesses a c/a ratioof 1.624 and three primary slip modes; basal slip in the < a > direction, prismatic,also in the < a > direction and < c+ a > 2nd order pyramidal slip. It is gener-ally accepted that for general arbitrary slip based deformation, five independentdeformation modes are required which is not the case for magnesium alloys whichfeature easy slip only in the < a > direction. Accommodation of deformationalong the < c > axis is dependent on either deformation twinning or the difficultto activate 2nd order pyramidal slip system.Twinning represents an alternative to slip and plays a significant role in mag-nesium alloys. There are two principal twins which have been observed; {101?2}2extension twins and {101?1} contraction twins. Basal slip is understood to providethe easiest deformation mechanism within magnesium and is closely followed byextension twinning. Prism slip, pyramidal slip and contraction twins are all muchharder to activate but as the material work hardens and the ability of twins to formis exhausted, stresses can begin to activate these more difficult deformation modes.Ultimately, the deformation of magnesium alloys is governed by the complex inter-play and competition between the various deformation modes. This dearth of slipsystems and reliance on complex twinning behaviour makes the understanding andprediction of the deformation behaviour of magnesium alloys extremely difficultand on the other hand it is absolutely crucial to understand this for the productionand in service behaviour of these alloys.The cost of manufacturing dies in the auto-motive sector is expensive so thereis a significant incentive to conduct simulations to guide die development. Further,full scale vehicle collision tests are expensive and extensive crash simulations aredone in advance of testing to minimize potential problems. If magnesium is tomove forward as a viable candidate for applications in the transportation sector,reliable material models are required which can accurately predict both formingand in-service behaviour. These materials models are highly dependent on accurateexperimental data. Due to the complex nature of deformation in magnesium alloys,in which anisotropy and twinning feature prominently, multi-axial testing methodsare required for successful forming and understanding of in service behaviour. Theobjective of this work is to develop, validate and utilize a biaxial test rig whichfacilitates the measurement of this much needed data. Subsequently, the ability ofa current polycrystal model will be compared to this experimental data in orderto ascertain its efficacy in predicting the deformation behaviour of AZ80 which is3a common commercial magnesium alloy that is used in the as cast and extrudedconditions.4Chapter 2Literature Review2.1 IntroductionThis literature review focuses on the structure and deformation behaviour of hexag-onally close packed materials with an emphasis on magnesium alloys. This is fol-lowed by a review of experimental techniques relevant to the current work. It isthen concluded with a survey of the current state of the art regarding modelling ofyield and deformation behaviour for magnesium alloys.2.2 Deformation of Magnesium Alloys2.2.1 CrystallographyFundamentally, the deformation of low symmetry materials such as magnesium isgoverned by the hexagonal crystal structure. The hexagonal closed packed (HCP)unit cell consists of alternating A,B layers in a hexagonally closed packed configu-ration [4]. The HCP unit cell is made up of three primative unit cells as illustrated5Figure 2.1: The unit cell for a hexagonally close packed structure consists ofthree primitive cells (example shown in bold lines). The atomic struc-ture consists of alternating A and B planes.in Figure 2.1 [5]. The ratio of the c axis to the length of the a direction is 1.632 inan ideal HCP material [6]. This c/a ratio is near the ideal value in magnesium at1.623 [6]. The system of Miller indices used for denoting crystalographic planesand directions, is slightly different in HCP materials as it is usual to use a fourindex (h,k,i,l) system where h + k + i = 0 [7].Important crystallographic directions and planes can be seen in Figure 2.2. The[112?0] and [011?0] directions are in the basal plane while the [0001] direction liesalong the c axis [4]. Figure 2.2b illustrates basal {0001}, prismatic {101?0} andpyramidal {101?1} planes, all of which facilitate deformation with a type Burgersvectors [4] i.e. the Burgers vector lies entirely within the basal plane. In Figure2.2c, 2nd order pyramidal {112?2} and extension {101?2} twinning planes are pre-sented which facilitate c axis deformation.6(a) (b) (c)Figure 2.2: Important directions and planes in HCP crystals. a) Example di-rections in the HCP crystal structure < 112?0 > and < 011?0 > are closepacked directions. b) Planes with Burger?s vectors in the basal plane. c)Planes with burgers vectors in the c direction.2.2.2 SlipThe HCP crystal structure of magnesium alloys presents a problem for generaldeformation. This can be understood by a consideration of the number of indepen-dent slip systems available to accommodate a general shape change. An indepen-dent slip system is one which cannot be reproduced by any combination of otherslip systems. Whereas von Mises [8] argued that five independent slip systemsare required for arbitrary homogeneous slip, magnesium has only two easily acti-vated independent slip systems, with Burgers vectors of the type {0001}?112?0?[9]. This type of slip is known as basal slip and involves slip on the close packedplane, in closed packed directions. There are additional slip systems provided byprismatic slip - {101?0}?112?0?in which slip occurs on planes parallel to the c axis,but in the close packed a direction and 1st order pyramidal slip - {101?1}?112?0?, inwhich slip occurs on the inclined {101?1} plane in the close packed a direction, and2nd order pyramidal slip - {112?2}?112?2?[9] in which slip takes place on the in-7clined {112?2} plane, but with a burgers vector in the < c+a> direction. However,these slip systems are not as easily activated, with critical resolved shear stressesmeasured on single crystals that are many times greater than basal slip, as seen inTable 2.1 [4, 10?13]. The critical resolved shear stress (CRSS) for a slip system isdefined by resolving the macroscopic stress on to the slip plane in the direction ofthe Burgers vector. In polycrystalline materials, local geometric effects can reducethe ratio of the CRSS values for basal and non-basal slip to something on the orderof 1.5 to 5.5, with basal slip remaining the easier mode [10, 13, 14].Table 2.1: Slip systems and associated CRSS [13, 15?20]SlipplaneSlipdirectionBurger?svector typeTotal slipsystemsNo. of inde-pendent sys-temsCRSS(MPa){0001}?112?0??a? 3 2 0.49{101?0}?112?0??a? 3 2 44{101?1}?112?0??a? 6 4 -{112?2}?112?3??c+a? 12 5 2.3-40The slip direction for basal slip and prismatic slip lies in the?112?0?direction.The Burger?s vector corresponding to slip on 1st order pyramidal planes is also oftype?112?0?and provides a total of four independent slip systems, none of whichhave a component in the ?c? direction. While slip on 1st order pyramidal planes canprovide four independent slip systems on its own, these are equivalent to combina-tions provided by basal slip and prismatic slip [10, 14, 21] and is therefore oftenignored in polycrystal plasticity codes.In addition, slip on 2nd order pyramidal slip systems i.e. {112?2}?112?3?is pos-sible. This slip mode alone can provide the requisite five independent slip systems[21]. However, 2nd order pyramidal slip ( {112?2}?112?3?type) features a Burger?s8vector with a magnitude almost twice that of?112?0?type slip systems due to theaddition of the < c > component. This results in this slip type being very difficultto activate [10, 22].2.2.3 TwinningSlip is not the only deformation mechanism available to magnesium. Mechanicaltwinning provides a mode by which deformation along the ?c? direction can beaccommodated [4, 22, 23]. Twinning provides a deformation mode which maybe more easily activated than 2nd order pyramidal to accommodate deformationparallel to the c-axis [23]. Whereas slip allows deformation by the movement ofdislocations, twinning accommodates deformation through a reorientation of thecrystal lattice [6]. A twin is a region in which this crystallographic reorientationhas taken place in order to accommodate either extension or contraction of the caxis of a grain. The growth of a twin may occur via dislocation motion as will bebriefly described later in this section. In some case, a twin can grow until the entiregrain has been consumed by the twin.The crystallographic structure remains thesame, but the orientation of the twinned portion of the grain is altered. The crys-tallographic structure remains the same, but the orientation of the twinned portionof the grain is altered. This concept is illustrated in Figure 2.3.There are three important planes to consider in a twin. The first is the twinningplane or first undistorted plane as it does not change in length or shape duringtwinning; it is referred to with the symbol K1. Secondly, the plane that intersectsK1 and lies in the shear direction is known as the shear plane. The next is the secondundistorted plane K2, which intersects and makes equal angles with K1 before andafter shear from twinning. In addition to these three planes, there are two important9Figure 2.3: Twinning involves a reorientation of the crystal lattice.directions to consider, the first is the direction of shear which corresponds to thedirection in which the lattice shears during twinning. The other important directionis given by the intersection of the shear plane and K2. These two directions arereferred to as ?1 and ?2 respectively [24]. These characteristic parameters arepresented for twin systems observed in magnesium alloys in table 2.2.The amount of macroscopic deformation accommodated by twinning is depen-dent on the volume of grains that have twinned [23]. Twinning is a directionalprocess - it can only occur if the applied load is in the correct direction relative tothe crystal orientation. There are so called extension twins and contraction twinswhich are formed depending on the applied load. Extension twins can accommo-date extension along the < c > axis, but not compression - thus their directionalnature. Extension twins result in extension along the < c > axis of the grain, con-traction twins result in contraction along the < c> axis. In the event of the loading10Table 2.2: Characterisitc parameters of commonly observed twin types inmagnesium [25].Twin type Twinningplane, K1Twinningshear di-rection,?12nd un-deformedplane, K2Direction ofintersectionof plane ofshear withK2, ?2Avg.CRSS(MPa)?c? axisextensiontwin{101?2}?101?1?{1?012}?101?1?2.4?c? axiscontractiontwin{101?1}?101?2?{101?3}?303?2?114being reversed, it is possible for detwinning to occur, in which formed twins shrinkor entirely collapse [26]. Extension twins are twins formed by tension along the< c > axis and twinning takes place on the {101?2} plane [27]. Contraction twinsmeanwhile (resulting from compression along the ?c? axis) occur on the {101?1}plane [27]. Both extension and contraction twins are schematically represented inFigure 2.4. Contraction twins are significantly harder to activate than extensiontwins and are therefore far more rarely seen [28, 29]. The extent of the crystalreorientation resulting from twinning is 86o [30] and 56o [31] for extension andcontraction twins respectively.While the nucleation of twins is not fully understood [33], for modelling pur-poses the use of an effective CRSS has been proposed [34] and values for this arepresented in table 2.2. While the accuracy of describing the triggering of twinnucleation with a CRSS is under debate, it has been argued as a necessary simpli-fication within current modelling regimes [35].Twin growth occurs via motion of the fault at the border of the twin within the11Figure 2.4: Twin types and planes: (a) Extension twinning and (b) Contrac-tion twinning [32]encompassing grain [36]. This motion has been proposed to be resultant of slipdislocations? decomposition at the twin border [36, 37] or alternatively related tonon-slip behaviour involving the large scale coordinated movement of planes ofatoms moving from matrix positions to twinned positions [38, 39].2.3 Crystallographic TextureIn polycrystal materials, particularly those with strong deformation anisotropy suchas magnesium alloys, an understanding of the distribution of the orientations of thegrains within a polycrystalline sample is paramount. Pole figures present a tech-nique by which the orientation of an arbitrary number of grains can be presentedin a manner that can be interpreted either qualitatively or quantitatively. A polefigure is generated by first determining the orientation of each grain with respectto the sample being investigated, and the pole figure itself is a projection (stereo-graphic or equal angle) of the orientations of a particular crystallographic direction(< 0001 > for example) with respect to the sample. For ease of interpretation thisis then generally converted to an intensity plot such as that presented in Figure 2.512which provides < 0001 > and < 101?0 > pole figures for various alloys and thuspresents the orientations of the < c > axes for all measured grains in the sample inthe first and < a > directions in the second. As the number of grains measured isarbitrary, it is standard practice to present the data as contours denoting intensityrelative to a random distribution. For extruded materials, the direction parallel tothe extrusion direction is known as ?ED while the direction perpendicular to theextrusion direction is defined as the transverse direction, ?TD. Ultimately, it can beseen in these pole figures that the peak intensity observed in the < 0001 > polefigure for extruded AZ80 is 6 times a random distribution and the minimum is 0and that there is a preferential orientation perpendicular to the extrusion direction.Maximum and minimum intensities for the < 101?0 > pole figure are 5.5 and 0.53respectively. The more strongly a sample?s deviation from a fully random texturedistribution, the stronger its texture is said to be. Figure 2.5a provides examplepole figures for an extruded AZ80 alloy. Figure 2.5b shows the texture developedby a similar alloy when cold rolled. Instead of a banded structure as in Figure2.5a, there is a strong, central peak indicating what is known as a basal texture(all basal planes are roughly aligned perpendicular to the sheet plane). Rare-earthalloying additions have been shown to result in some splitting of the central peak[40] along the TD direction as seen in Figure 2.5c which provides pole figures forrolled ZEK100 alloy.2.3.1 Texture Measurement TechniquesThree commonly used techniques for establishing the texture of a metal sample areelectron back scattered diffraction (EBSD), x-ray diffraction and neutron diffrac-tion.13(a) Extruded AZ80(b) Cold rolled AZ80(c) ZEK100Figure 2.5: Example < 0001 > and < 101?0 > pole figures for recrystallizedtextures of a) extruded AZ80, b) cold rolled AZ80 and c) industriallyprocessed ZEK100 sheet.EBSD is a mature technique that has been successfully used for characterizingthe texture of magnesium [41?46]. EBSD orientation maps of a sample surface,such as Figure 2.6, provide information on both the grain size, and orientation ofgrains. The technique works by progressively scanning the electron beam across14the sample surface and measuring the orientation from the back scattered diffrac-tion pattern which is collected on a detector. While this technique has the advantageof providing grain size information, it is only capable of measuring grains on thesurface of the sample, providing a single plane of information. Furthermore, thesampling population is relatively low (tens to hundreds of grains, whereas thou-sands are required to properly characterize a material?s texture) and thus good sta-tistical significance can require stitching together multiple maps, increasing thetime required for measurements, in which case bulk sampling techniques such asneutron diffraction can prove preferable.A significant advantage provided by EBSD is that as individual grains are di-rectly identified, it is possible to measure the missorientation angles between neigh-bouring grains. When magnesium undergoes twinning, it causes a significant ro-tation of the crystal lattice. This rotation can be identified by EBSD, allowing forthe size and sites of twins to be identified.Neutrons, unlike electrons can penetrate magnesium readily. This allows diffrac-tion measurements to investigate the texture of a three dimensional sample, unlikethe two dimensional data provided by EBSD. This three dimensional data doesnot, however, provide information on individual grains. Grain size and placementcannot be determined by this technique. Neutron diffraction does, however, allowfor the sampling of a much larger number of grains than EBSD (millions versustens or hundreds of grains), providing statistically significant results quickly [47].Neutron diffraction techniques have been well documented as being successfullyused for establishing the texture of bulk samples of magnesium alloys [47?52].X-ray diffraction is in many ways similar to neutron diffraction in that a largenumber of grains can be sampled simultaneously (hundreds to thousands of grains).15Figure 2.6: Sample of an EBSD orientation map. In this case the sample wasextruded AZ31 alloy pulled in tension to failure [42].However, x-rays cannot significantly penetrate metals, having a maximum penetra-tion depth of roughly 15 ?m in magnesium, and the result is that x-ray diffractionpatterns, like those from EBSD, give information only about grains at the surfaceof the sample. The advantage of x-ray diffraction over neutron diffraction though,is the readily available nature of x-rays. Whereas neutron diffraction requires anuclear source, x-ray diffraction can be done in a normal lab setting.2.3.2 Polycrystal DeformationThe resolved shear stress ???, on a single crystal, resultant from an applied stress?? ? is given by the Schmid law, i.e. Equation (2.1) [53]16? = ? cos? cos? (2.1)Where ? and ? are the angles between the applied stress and the slip plane nor-mal and direction of slip respectively. It is immediately obvious that the directionof the applied load relative to crystallographic orientation will have a significanteffect, particularly in materials like magnesium alloys which feature significantdeformation anisotropy.In polycrystal materials, which represent the vast majority of real world appli-cations, the crystallographic orientation relative to an applied load is not constant.On one extreme, a truly random distribution of orientations within the polycrystalwould see every possible orientation with respect to the applied load and as suchthe material response, in any direction, would be composed of an aggregate of theresponses of each orientation. The effect of these varying orientations is joinedby the concept of geometric compatibility. The resolved shear stress predicted by(2.1) would, in the absence of other grains, induce a given amount of deformationin a grain. However, the presence of other grains with responses dependent ontheir own orientations will impose geometric constraints on each other [54]. Thus,the texture of a polycrystal sample can have a significant effect on the deformationbehaviour of that sample [28].Extruded materials provide an excellent vehicle for an investigation into theorientation dependence of deformation in polycrystalline magnesium. Extrudedmaterials tend to have the basal planes oriented parallel to the extrusion direction[55?57] (see Figure 2.5a for a typical example). When samples of extruded AZ61alloy were pulled in tension at different angles to the extrusion direction, the re-17Figure 2.7: Stress-strain curves for extruded AZ61 samples pulled in tensionat different angles ? to the extrusion direction [56].ported stress strain behaviour varied significantly. Yield stress ranged from 100 to200 MPa, and work hardening rates saw similarly significant variations as seen inFigure 2.7. The orientations that exhibited the easiest yield were 45 and 90 de-grees, these samples were favourably oriented to take advantage of basal slip andextension twinning respectively [56].The reason for the decreased yield stress is that the basal plane is oriented inthe direction of maximum shear which facilitates easy basal slip. This is supportedby Equation (2.1), which, for the case in which the applied stress is oriented at45 degrees to the slip plane (basal) and the slip direction (basal) yields a result of? = 0.5? .In addition to the effect of the loading direction relative to the alignment of the18Figure 2.8: Correlation between the amount of strain accommodated by ex-tension twinning and the tilt angle between extrusion and loading direc-tions in AZ61 magnesium alloy, indicating that for tension, a tilt angleof 90 yields the most strain by twinning, while a tilt angle of 0 yields tomost in compression [56]basal plane on the activation of basal slip, one must also consider the activationof twinning. As twinning is dependent on the tensile or compressive nature of thetwins being formed, as well as the orientation of the load relative to the < c > axisof the grain, the activation of twinning in samples with strong texture would beexpected to be very anisotropic. Figure 2.8 clearly demonstrates this as the straindue to twinning during tension and compression is plotted as a function of the tiltangle between the extrusion and loading directions.In tension, with the majority of the basal planes aligned with the extrusiondirection, the loading is not oriented in such a manner that there is a tensile loadalong the < c > axis of the grains resulting in the absence of easy {101?2} twins[56].19Tension - Compression AsymmetryMagnesium and its alloys have been shown to exhibit asymmetry in their tensionand compression responses [26, 40, 58?62], particularly in single crystal [63] andand wrought conditions [26, 40, 60, 63, 64]. The reason for this is the directionalnature of twinning. As shown in Table 2.2, extension twins are much easier toactivate than contraction twins. In materials with completely random texture, com-pression tends to activate more twins than tension [65, 66] In wrought materialspossessing typical extrusion textures, compression tends to activate extension twins(extension occurring at 90o to the applied load). As such in most cases one tendsto see lower yield stresses in compression than in tension [58, 59].2.3.3 Multiaxial DeformationThe effect of multiaxial loadings (either biaxial or triaxial) must be understoodin order to predict the behaviour of magnesium alloys during forming and for inservice behaviour. In sheet materials, there has been a body of work conductedwhich attempts to investigate biaxial response of magnesium alloys by employingforming limit diagrams [67]. These tests suffer in that consistent biaxial stressratios cannot be maintained, rather, it is the biaxial strain ratio which is controlledthrough the selection of test geometry. Nevertheless useful information can beobtained from these tests and in the case of Kim et al. [68], bulge testing wassuccessfully used to investigate the deformation properties of 50mm thick rolledplate by creating reduced thickness samples. The relatively thick plate allowedbulge test samples to be fabricated both parallel and perpendicular to the rollingdirection. This permitted testing in the RD-TD plane as well as the TD-ND planeresulting in 1mm thick samples with a diameter of approximately 50mm. The20material exhibited a classic rolled texture with a strong < 0001 > intensity in theND direction. The results indicated lower yield stress from the samples cut fromthe TD-ND planes. This is likely due to the availability of suitably oriented grainswith basal poles aligned with the ND direction which would be therefore ideallyoriented for extension twinning during tension [68].Weiler [69] employed uniaxial testing, shear testing and in-plane compressionin an attempt to characterize the yield surface of as cast AM60B magnesium al-loy. Due to the largely random texture of the material, there was success in fittingthe data to a von Mises yield surface which is an isotropic yield criterion (fur-ther discussed in Section 2.4). Similar, but far more robust work was completedby Safi-Naqvi et al. [70] on three different extruded magnesium alloys. Tests in-cluded tension and compression as well as plane strain tension and compression.The strong anisotropy inherent in extruded alloys resulted in data which could notreadily be plotted on a von Mises equivalent surface. These techniques are highlylimited in the range of biaxiality ratios (biaxiality ratio being defined as the ratioof the stresses in the two loading directions) which can be investigated and furtherappear to make direct comparison of data difficult. Regardless there are some keyfindings in this work. Namely, it is observed that in ZM61 and Mg-3Nd alloys,increased precipitation hardening affects twinning more than basal slip. The resultis that there is an observable decrease in the material?s yield anisotropy.Steglich et al. conducted biaxial studies of AZ31 wherein biaxial loadings wereinduced either utilizing cruciform samples, or bulge testing [71]. Their resultswere then compared with visco-plastic self consistent (VPSC) modelling results(VPSC is introduced in section 2.5.4). Interestingly, they saw very little texturalevolution through the investigated small strain regime (< 0.03) which ran contrary21to VPSC predictions. It has been proposed that the adjustment of simulation inputparameters could reduce this perceived discrepancy [71].Chun et al. [72], employed uniaxial tension and compression tests to modelthe yield surface of AZ31 plate. Employing the idea that uniaxial compressionalong ND produces a deviatoric stress state analogous to that of balanced biax-ial tension in the RD-TD directions (and equivalently, that ND tension producesa deviatoric stress state analogous to that of RD-TD compression), as previouslyused by [70]. These experimental results were then fit to a von Mises yield surfacewhich employed a non-zero back stress so as to accommodate the asymmetric na-ture of twinning. While this technique appears somewhat capable at small strains(< 0.01), it breaks down as the disparate work hardening rates associated with dif-ferent deformation modes (i.e. different slip modes or twinning types) manifestthemselves.In order to capture the large strain behaviour of AM30 alloy tubes, Jiang etal. [73] employed uniaxial tension and compression tests that were completedalong the ED direction of extruded samples. The specific aim of this work wasto capture and characterize the twinning behaviour of the material. The resultsindicated significant twinning in the compressive samples and minimal twinningin tensile samples. These results were then extrapolated by Levesque et al. [74]to a simulated ring hoop tension test (RHTT) which employs mechanical grips toachieve dilation of the tube, however, due to a lack of experimental data, the resultsof the simulations could not be compared with experimental data [74].Overall, the status of the understanding of multiaxial deformation of magne-sium alloys is that there has been limited work completed on a limited set of alloys.From this some general observations can be made. First of all, the presence of twin-22ning distorts the yield surface such that they do not readily agree with the von Misescriterion. Second, it is clear that twinning can strongly affect texture and given thesheet and plate textures normally observed, this is particularly prevalent in com-pression, but much less so in tension. The vast majority of currently publishedwork is constrained to the tension-tension quadrant of stress space [68, 69, 71].Alternatively, attempts have been made to equate the results of various differenttest methods based on equating similar deformations (such as uniaxial tension be-ing equivalent to an orthogonal biaxial compression) [70, 72, 74]. Finally, there iseffectively no data at all capturing the tension-compression regions of stress spaceleaving this region very poorly understood.2.4 Iso-Work Surfaces for Anisotropic MaterialsNumerous models have been developed to attempt to describe both when and howa material will plastically deform for complex loading states. These models all, inone form or another, attempt to describe a yield criterion which can be used to con-struct a theoretical yield surface for a material against which experimental resultscan be compared [75]. The yield surface of a material exists within three dimen-sional stress space, whose axes are aligned with the principal stresses of the system.Any point on that surface describes a loading condition resulting in the onset ofplastic deformation [76]. Traditionally, yield has been defined experimentally byarbitrary criteria, for example the proportionality limit [77], the elastic limit [78]or using some level of plastic offset strain, such as the traditional 0.002mm/mm[79]. None of these can be easily adapted to multiaxial deformation wherein thereare multiple stress-strain curves for a single test. A biaxial test for example, hastwo stress-strain curves, one for either principal loading axis. While equivalent23stresses and strains can be estimated for anisotropic materials or even anisotropicmaterials with well characterized responses, this is clearly not feasible for inves-tigative work where the nature of the material response is simply unknown [80].For this reason, iso-work surfaces are generally used when dealing with multiaxialmaterial response [71, 81?87]. Iso-work surfaces provide contours of equal plasticwork. If the yield stress is constant, regardless of orientation or stress state, thena 0.002mm/mm offset strain yield surface will be identical to an iso-work surfacegenerated for a level of plastic work corresponding to that experienced by a sampleplastically deformed to 0.002mm/mm plastic strain. Once an iso-work surface hasbeen experimentally determined, it can then easily be compared to various yieldmodels as the generated surface is analogous to a yield surface.The simplest model of a yield surface is the Maximum Shear-Stress Condition,more commonly known as the Tresca Yield Criterion which is shown in Figure2.9. This model is based on the premise that whenever the maximum shear stressin a sample (?max) reaches a critical value (?y), yielding of the sample will occur(Equation (2.2)) [88]. The outcome of this theory is a locus of points which, inthree dimensional stress space, create a hexagonal prism whose long axis followsa line described by ?1 = ?2 = ?3 i.e. hydrostatic stress does not affect yieldingwithin this theory [89].?max =?1??32= ?y (2.2)A more popular model for the onset of yield is the von Mises Yield Criterion,also presented in Figure 2.9 ? Equation (2.3) [91]. The Tresca Yield Criterion failsto incorporate the effect of the intermediate principle stress (which would be ?224Figure 2.9: Comparison of Tresca and von Mises yield criteria for a materialwith yield stress ?Y ? [90].in Equation (2.2)). The result is the magnitude of the equivalent stress predictedfor some triaxial loading conditions can be overestimated. The von Mises modelincreases the predicted yield stress when ?1 = ??2 by roughly 15% [92], whilepredicting the same yield stress under uniaxial loading and balanced biaxial stress.2?y2 = (?1??2)2 +(?2??3)2 +(?3??1)2 (2.3)The yield surface described by Equation (2.3) assumes that the yield strengthof the material is isotropic [89]. The effect of anisotropy on yielding is to changethe shape of the yield surface [93].Once a material has begun to yield, the required stress to continue deforma-tion increases; which is known as work hardening. Isotropic hardening involvesan expansion of the yield surface whereas kinematic hardening causes a lateraldisplacement of the yield surface [75], generally in the direction of the imposedloading. An example of this is the Bauschinger effect [75], where a sample, ini-25Figure 2.10: Twinning can have a large effect on the yield surface of a mate-rial [94].tially loaded in tension and then subjected to a compressive load, yields prior tothe normally expected yield stress. Twinning on the other hand can cause quitesignificant alterations in the shape of the yield surface as shown in Figure 2.10.This is due to the loading specific nature of twinning [65, 66] which was discussedin section 2.5.3.In order to account for the tension-compression asymmetry resultant from twin-ning, phenomenological approaches, such as those taken by Cazacu, Plunkett andothers [93, 95?97] have been developed. In this case, the yield surface has beenmodelled for materials which exhibit tension-compression asymmetry; and havethus far shown promising agreement with experimental results. The core of thiswork is what is known as the CPB06 yield criterion which takes the form of (2.4).F = (|S1|? kS1)a+(|S2|? kS2)a+(|S3|? kS3)a (2.4)26Figure 2.11: Variation of the variable k modifies the yield surface allowingit to describe the effect of tension-compression asymmetry resultingfrom twinning. In the case where k = 0, the resultant yield surface isthat described by the von Mises criterion. [95].Where F is a function of the size of the yield surface, and S1 ? S3 are theprincipal stress deviators as given in equation (2.5).Si = ?i?13(?i+?2 +?3) (2.5)Where i = 1, ...,3. Furthermore, k is a material constant corresponding to thematerial?s tension-compression asymmetry and a is a positive integer. In the specialcase that a = 2 and k = 0, the criterion simplifies to the von Mises criterion.It can be shown that [95]:27k =1?h(?T?C)1+h(?T?C) (2.6)where:h(?T?C)=??2a?2(?T?C)a2(?T?C)a?2??1a(2.7)which demonstrates the physical significance of k (graphically illustrated inFigure 2.11) as (for a given a) it is purely a function of the tension-compressionasymmetry. While purely a phenomenological model, the capacity to fit some ofthe effects of twinning is of significant value in the description of magnesium de-formation [98]. The model as presented, fails to accommodate anisotropic tensionor compression responses in materials which experience orientation dependence.This is of vital import in magnesium alloys possessing non-random textures whichcan result in twinning activation being highly dependent on the direction of theapplied load with respect to the starting texture [70].2.5 Biaxial Testing MethodsWhen determining the yield surface of a material, different tests produce strainpaths through different quadrants. As illustrated in Figure 2.12, biaxial testingtechniques are required in order to explore regions of the yield surface which liebetween the results of uniaxial tests.Four established methods for conducting biaxial testing are bulge testing, com-bined tension torsion, cruciforms and tension with internal pressure. Bulge testing,in which sheet samples are deformed by either pressure [99?102] or a punch into28a bulge shape which induces biaxial deformation can provide valuable informationof biaxial behaviour, however, the strain path be cannot effectively controlled dur-ing the test, as the path is dependent on the geometry of the sample. As such, thesetests will not be considered further here.Figure 2.12: Relationship between yield surface and test types.2.5.1 Cruciform SamplesA cruciform sample, subjected to biaxial tensile loading has principal axes alignedwith the applied load and the alignment is maintained throughout the test, and has,in fact been successfully used in the investigation of the biaxial response of AZ31magnesium alloy [71]. The disadvantage is that the load frame necessary for thistype of testing can be quite complex. First of all, the equivalent of two standardlinear load frames is required as the sample must be pulled in orthogonal directionssimultaneously [103]. Secondly, whereas a normal tensile test can be conducted29with one end of the sample held by an unmoving grip while the other grip appliesload via displacement, for cruciform samples must remain centered during plasticdeformation so that bending loads are not induced in the arms of the cruciform.This is generally achieved by having four linear actuators applying the load whilemaintaining the sample?s position [104], resulting in the need for simultaneousfour way control. Alternatively if two linear actuators are used, one must be ableto move independently of the other so that that the sample can remain centered[86, 105] which poses significant technical difficulties in its own right. A finalsolution which involves only two linear actuators is a pantograph style setup whichovercomes many of the limitations of the other setups [106, 107], but requireslinkages which result in increased compliance in the test setup. A further issuewith cruciform samples is that depending on exact sample geometry, the stressstate within the gauge section of the sample is not necessarily homogeneous whichcan complicate analysis of test results[108?110]. The stress state and deformationin cruciform samples is complex enough that the use of finite element models isgenerally required in order to assist in interpreting the results [103].2.5.2 Combined Tension TorsionCombined tension torsion has the advantage of simplicity, samples are generallyshort, hollow cylinders and the sample only undergoes a linear displacement alonga single axis [111] which eliminates the biaxial centering difficulties experienced incruciform testing. The significant disadvantage posed by this method of perform-ing biaxial testing is that the principal axes of the applied load rotate during theapplication of torsion [112]. When conducting a uniaxial tensile or compressivetest, the principal stress ?1 is aligned with the loading orientation. If a torsional30load is then applied, the result is that the principal axis originally aligned with theloading direction, rotates away [112]. This is of particular importance to highlyanisotropic materials such as magnesium.2.5.3 Tension/Compression with Internal PressureThe combination of tension or compression with internal pressurization of hollowcylindrical samples addresses the difficulties posed by both combined tension tor-sion and cruciform or bulge based biaxial testing. While, as with combined tensiontorsion, testing of sheet samples is not possible, sample centering does not pose anissue, nor does rotation of the principal axes [113]. This method of testing operatesby pressurizing hollow cylindrical samples while applying a simultaneous tensileor compressive load along the sample?s axis [84]. If wall thickness and radius canbe controlled such that rt < 0.1, then thin walled assumptions and calculations canbe used [114, 115]. These assumptions assume that the stresses remain constantthroughout the sample wall. The pressure causes hoop stress, which, if using thinwalled assumptions, is given by Equation (2.8), and is analogous to a tensile loadtangential to the sample wall. In addition, if the axial length of the sample is notconstrained, an axial stress, given by Equation (2.9), and associated strain, will alsodevelop. This axial stress is independent of any stresses applied by the load frame.??? =Prt(2.8)?zz =Pr2t(2.9)Figure 2.13 illustrates the relative simplicity of this loading arrangement.31Figure 2.13: Sample configuration for testing via tension with internal pres-sure [113]The principal disadvantage of this testing method is that only the first and sec-ond quadrants of the yield surface can be investigated due to the system?s inabilityto generate compressive loads in the hoop direction.2.5.4 Polycrystal Deformation ModellingPolycrystal deformation modelling can facilitate in the understanding of experi-mental data. Information such as texture can be measured at discrete intervals,however understanding how one gets from one point to the next can be difficult andreliant on conjecture. Polycrystal models can provide a physically based approachby which yield and iso-work surfaces and their evolution can be examined. Twopolycrystal deformation simulation techniques are crystal plasticity finite elementmethods (CPFEM) and viscoplastic self consistent models (VPSC).CPFEM techniques have been employed in order to model polycrystal systems[113, 116?119]. The principal advantage of these models is that they account forthe interaction between neighbouring grains as opposed to VPSC models whichdo not. Furthermore, CPFEM simulations can employ more realistic geometries32[120] than other modelling techniques. This comes with computational cost how-ever. Recent work by Quey et al. [120] on a 3000 grain system required 100 hoursto complete on a 128 core computation cluster. In comparison, biaxial VPSC sim-ulations of a 16416 grain system on a single core computation platform requiredonly 30 minutes.The VPSC model originally developed by Lebensohn and Tome [34] consid-ers each individual grain as an ellipsoidal inclusion within a homogenous effectivemedium (HEM) possessing the average properties of all the grains. This modelconsiders the orientation of each grain which is particularly relevant for an ex-amination of materials which possess distinct hard and soft orientations such asmagnesium. The model is self consistent in that global stresses must balance andstrains must be globally compatible. The self consistent nature of the model tiesthe response of the HEM to that of individual grains through Eshelby?s inclusionformalism [121]. In this manner the stress and strain response of the individualgrains is summed at each step in the simulation and used to predict the macro-scopic response. The basis of the model is a rate sensitive constitutive law which isused to relate the plastic strain rate to the applied stresses on the individual grains,and is known as the viscoplastic equation given in Equation (2.10).??i j = ??0?smsi j(mskl?kl?s0)n(2.10)In Equation (2.10), ?i j is the plastic strain rate in response to the applied stress?kl on the grain and ?s0 is the CRSS for the given deformation system. The super-script s denotes the deformation system. Finally, n is a strain rate exponent, ??0 is areference strain rate (1s?1) and msi j is a function of the normal nsi and the Burger?s33vector bsi such that msi j =12(nsibsj +nsjbsi).Prior work has been conducted in which VPSC models have been used withsome success in the prediction and interpretation of the response of magnesiumalloys, in particular AZ31 [23, 27, 50, 65, 122?125]. The results have demonstratedoverall good prediction of deformation induced textural evolution.2.6 ConclusionThe deformation behaviour of magnesium alloys is complex owing to the stronganisotropy of the material. This anisotropy stems from the limited number of easyslip systems and the influence of mechanical twinning. Crystallographic textureis directly related to the anisotropic mechanical response observed and thereforeaccurate texture measurements are crucial. Neutron diffraction studies allow thepolling of large numbers of grains and are not constrained to the measurement ofgrains at the surface of the sample. The use of tension with internal pressure asa testing mechanism allows independent control of axial and hoop stresses andprovides easier to interpret results than either cruciform samples or tension-torsiontests. The use of polycrystal modelling can then facilitate the interpretation ofresults by providing insight into the texture evolution and the evolution of iso-work surfaces. The development of an understanding of how texture relates todeformation and how that texture evolves as deformation progresses is of vitalimportance for successful forming and in service behaviours of these alloys. Thesetechniques provide the tools with which testing and analysis were completed.34Chapter 3Scope and Objectives3.1 ScopeThis work covers the uniaxial and biaxial deformation of an as cast AZ80 magne-sium alloy and a more strongly textured extruded AZ80 magnesium alloy. Samplesare to be tested in axial tension, hoop tension, axial compression, and over a rangeof positive and negative biaxialities. Additional testing of a 6061 aluminum al-loy provides a nearly isotropic counterpoint to the anisotropic magnesium alloyresponse and the extruded AZ80 provides a material with stronger texture than thecast material. Testing is to be conducted at room temperature. The AZ80 in as-cast condition has been the focus of prior research with regards to its deformationbehaviour [27, 30, 58, 126] and thus literature exists both characterizing its uni-axial behaviour and the results of modelling efforts which employed VPSC codes.Conversely, the extruded material has not seen the same level of study; modellingparameters have not been developed and there is limited experimental data.353.2 ObjectiveThe primary objective of this work is to characterize the deformation behaviourand textural evolution of AZ80 magnesium alloy under uniaxial and biaxial loadingconditions.Understanding the deformation and yield behaviour of this material necessi-tated the development of a biaxial test rig. After the consideration of various op-tions, a tension/compression with internal pressure type scheme was selected. Ex-isting systems at other research institutions have thus far featured independent con-trol of the pressure source and axial load frame components, resulting in changingbiaxiality ratios throughout the test. The system developed for this work eliminatesthis issue.Ultimately in order to achieve the primary objective, this work aims to achievethe following:1. Develop a test rig capable of conducting biaxial tests on bulk samples with aconstant or variable biaxiality ratio.2. Conduct biaxial testing on as-cast and extruded samples in order to developexperimental data sets against which numerical models can be compared toevaluate their predictive capabilities.3. Employ neutron diffraction testing, in conjunction with VPSC models ofbiaxial tests to measure and predict the textural evolution these materialsundergo. This will assist in developing a quantitative knowledge of the de-formation mechanisms in play and how this effects the yield and early plasticbehaviour of these alloys.36Chapter 4Methodology4.1 IntroductionThe biaxial test rig utilized during the course of testing was modelled off of anexisting rig in use at the Canadian Neutron Beam Centre [113]. The rig at UBCconsists of an Instron 8874 servo-hydraulic load frame, a SITEC motor drivenpiston and controller, a National Instruments data acquisition unit and a controland acquisition computer.4.2 Biaxial Test RigBiaxial stresses are induced in hollow cylindrical samples (Figure 4.1) by employ-ing a combination of axial load and internal pressure to produce biaxial stresseswithin the sample wall as illustrated in Figures 4.2 and 4.3. The samples featureda gauge length of 50mm, outer diameter of 18mm and a wall thickness of 0.9mmin accordance with thin walled pressure vessel calculations which call for a ratioof radius to wall thickness of 10:1. Wall thickness was measured via direct mea-37surement of the external diameter and then the calculation of the internal diameterby applying a small axial tensile load prior to the commencement of the biaxialportion of the test. Using the elastic modulus of AZ80 magnesium alloy (45 GPa)and the stress-strain data from this axial tensile loading, the internal radius is easilycalculated.Figure 4.1: Test sample with speckle pattern applied for digital image corre-lation.Figure 4.2: Loading of the sample is achieved via a axial load F and a pres-sure P provided by a high pressure source.The biaxiality ratio of a given test is defined as the ratio of its hoop stress toits axial stress (????zz ). A 0.4 biaxiality ratio test therefore follows a loading path inwhich ????zz = 0.4 throughout the test.The biaxial test rig consists of 4 major components. An Instron 8874 servo-hydraulic load frame and controller, a SITEC high pressure, electric motor drivenpiston and controller, a National Instruments Data Acquisition (NI-DAQ) unit and38Figure 4.3: The combination of axial load and internal pressure results in ax-ial and hoop stresses. There are also radial stresses (not shown) actingbelow the sample surface.a computer running a LabVIEW program. A Bridgeman seal (not shown in fig-ures) is used to seal the ends of the sample within the grips while still permittinghydraulic fluid to enter the sample. This seal consists of a Viton ring and a stainlesssteel washer. The Viton ring fits over the flanges at the end of the sample, visiblein Figure 4.1 and illustrated in Figure 4.2 and is of slightly greater length than theflange. The washer then fits between the end of the sample and the bottom of thegrip. As pressure within the system increases, the washer pushes against the Vitonring, creating a seal.The control process for the rig is as follows:1. The test conditions are input into the computer and all controllers are initial-39ized.2. The load frame, operating in displacement control, begins to apply an axialdisplacement to one end of the sample. This step continues until the test isterminated.3. Simultaneous with step 2, the applied load measured by the load cell isrecorded at a rate of 1Hz by the NI-DAQ system. The LabVIEW programthen calculates the applied axial stress and then determines the pressure nec-essary to maintain the desired biaxiality ratio. This value is passed to theSITEC Controller which attempts to seek the desired pressure using its ownindependent PID based control system.System instrumentation consists of: 2 pressure transducers, 4 strain gauges(two CEA-13-062UT-350 gauge packages), a 25kN Instron Dynacell load cell, aMTS 25? 5mm extensometer and a 3-dimensional LaVision digital image corre-lation system.4.3 Strain Gauge CalibrationStrain gauges were installed on the sample surface in accordance with InstructionBulletin B-127-14 by Vishay Micro-Measurements [127]. The model of straingauge used is the Vishay Micro-Measurements CEA-13-062UT-350. Each gaugepackage is a tee rosette configuration and consists of two independent gaugesoriented at ninety degrees to each other. Despite being on the package, thesegauges were treated independently. The individual gauges possess a gauge lengthof 1.57mm and width of 2.03mm. Nominal resistance is 350 ? and the gauge factoris 2.125. These tee rosettes minimize alignment issues by ensuring that the gauges40Figure 4.4: Data and control pathways within the biaxial test rig. A Lab-VIEW program acts as the central controller and collects data from sys-tem sensors and outputs from the load frame.posses an orthogonal layout. The strain gauges possess a known nominal resis-tance (Ro) and deviation from this value must be compensated for prior to testing.The compensation process is conducted automatically by LabVIEW and consistsof applying an excitation voltage to the gauge and measuring its resistance. Thenominal resistance of the gauges is 350 ?; deviation from this value is due to leadresistance. This resistance is subtracted from the measured resistivity for the pur-poses of strain calculation. The strain gauges are each mounted in a quarter bridgeconfiguration, illustrated in Figure 4.5.Vo =[R3R3 +(Ro+?R)?R2R1 +R2]Vex (4.1)If R1 = R2 = R3 = Ro and ?R = 0, then the bridge is balanced and thereforeVo = 0. However, as the gauge deforms, the resistance of the strain gauge changes.41Figure 4.5: A quarter bridge strain gauge configuration consists of a sup-plied excitation voltage Vex, three resistors (R1 = R2 = R3) and a straingauge with resistance Ro+?R;Vo is then measured and used to calculatestrain.As a result ?R either increases or decreases depending on the nature of the inducedstrain.?R = RoG f ? (4.2)Substituting (4.2) into (4.1) and rearranging provides an expression by whichthe strain within the sample can be measured as a function of Vo.? =?4 VoVex(1VoVex2G f +G f)(4.3)The measurement of sample strain is contingent on measured resistance. Withthe configuration used, lead resistance can distort strain measurements, as can anydamage done to the gauge during application. The resistance is therefore measuredafter the sample is loaded into the test rig and any deviation from the nominal gaugeresistivity is removed via a calibration offset in order to zero the output.The purpose of the strain gauges was to not only ensure proper functioning42of the DIC system by providing a secondary measurement (albeit only capable ofmeasuring small strains), but also to ensure that the sample grips were properlyaligned during the test. In the event of a misalignment of the grips, this wouldinduce bending of the sample which would manifest as uneven strains measuredby strain gauges mounted at one quarter circumference intervals from each otherabout the circumference of the sample surface. The results of such a measurementare shown in Figure 4.6 in which the results of a successful grip alignment areshown in the balanced deformation of the sample. This process was not requiredfor each sample, but only if the grip assembly had been removed from the loadframe and then reinstalled, i.e. to check the initial alignment.Figure 4.6: Two axially mounted strain gauges mounted one quarter of theway around the sample surface will reveal any misalignment of the gripsby recording differing strain levels. This figure illustrates good align-ment.434.4 LaVision Digital Image CorrelationA LaVision digital image correlation (DIC) system was employed to measure strainsbeyond those that can be recorded using the strain gauges. The DIC system em-ploys two digital cameras mounted on a vertical post shown in Figure 4.7. Thespread angle between the cameras was maintained between 60 and 80 degrees,comfortably exceeding the LaVision recommendation of not less than 30 degress.This angle is necessary to develop a proper stereoscopic image of the sample per-mitting sample curvature and radial dilation of the sample to be measured.Sample preparation for DIC consisted of applying a matte white layer of enamelspray paint to the sample and then applying a speckle pattern using black enamelspray paint. This provided a randomized speckle pattern for image analysis.LaVision?s DaVis software package was used for both capturing images as wellas processing. After the cameras had been manually focused on the sample, a cal-ibration card featuring a regular array of crosses was placed in front of the sampleand five pairs of images captured by the camera. The calibration card was tiltedand rotated slightly between each capture in accordance with the DaVis instruc-tions. The calibration card itself consisted of 1.38mm crosses at 4.29mm spacing.Given the field of view, itself a function of the camera focal length and po-sitioning, as well as the camera resolution and processing parameters employed,the claimed strain calculation accuracy for the system is on the order of 0.0001mm/mm.The DIC results can be validated on a test by test basis by comparing the DICsystem?s calculated axial strain to that measured using the extensometer, as pre-sented in Figure 4.8. In all cases throughout testing, the strain data was extracted44Figure 4.7: Camera configuration for the LaVision DIC setup. The spreadangle between the two cameras is maintained between 60 and 80 degreesin order to capture radial dilation and curvature of the sampleby calculating the maximum and minimum normal strains on the surface of thesample ensuring their alignment with the applied loads.The region over which the software calculated a 3D surface, and resultantstrains, was roughly 22mm by 9mm. During processing, each image was correlatedagainst the first image. This technique, as opposed to the summation of incremen-tal strains between subsequent images, minimizes the accumulation of error, butis computationally more expensive as the area searched for each correlation point45Figure 4.8: Example comparison of axial DIC results to those measured us-ing an extensometer for an axial tension test on extruded AZ80 demon-strates strong agreement.must be much larger due to the much larger displacement of these points between,for example, the first and last images compared with subsequent images. Once thestrains were calculated, they were averaged over an area of approximately 18mmby 5mm. An outer region of roughly 2mm on all sides was discarded as data nearthe edge of the calculated surface is less reliable due to reduced correlation pointsand the increased angle of the surface relative to the image plane of the cameracaused by the curvature of the sample.4.5 Data ProcessingThe axial stress to which the sample is subjected is composed of two parts (4.4),that induced by the load frame ? lzz, and that induced by the internal pressure ?pzz.?zz = ? lzz+? pzz (4.4)46The applied axial load from the load frame creates an axial stress within thesample ? lzz.? lzz =Fpi(r2o? r2i) (4.5)Simultaneously, pressure within the sample induces axial and hoop stresseswhich, assuming thin-walled pressure vessel assumptions are made, take the formof (4.6),(4.7) which are functions of the induced pressure P, the average sampleradius r and the sample?s wall thickness t. The load frame induced stress and thepressure induced stress combine yielding a total axial stress of (4.8).? pzz =Pr2t(4.6)??? =Prt(4.7)Combining the axial load from the load frame, (4.5) with the axial load due tothe induced pressure, (4.6) gives the total axial stress.?zz =Pr2t+Fpi(r2o? r2i) (4.8)The above calculations all assume that thin walled pressure vessel equationsare used. Samples were designed such that these simplified equations could beused. The resulting error is equal to an approximate underestimation of 5% at theinner wall and an overestimation of approximately 10% at the outer wall. This errorscales nearly linearly between these two points and thus they represent the very ex-tremes of the calculation errors. These errors represent the difference between thin47and thick walled calculation methods. Thick walled calculation methods cannot beused after deformation has commenced as the internal radius cannot be accuratelydetermined during the test.Measured values for stress and strain were converted to true stress and truestrain according to equations (4.9) and (4.10). The elastic component of the strainwas then removed according to equation (4.11).?t = ? (1+ ?) (4.9)?t = ln(1+ ?) (4.10)?1p = ?1t ??1t ??(?2t +?3t)E(4.11)In Equation 4.11, superscript numbers represent principal directions.Strain-strain data was differentiated by fitting a polynomial to the data (rangingfrom second to eighth order) and then differentiating the polynomial. This wasuseful for investigating the agreement between experimental and VPSC results.4.6 LabVIEW Control ProgramA control program was written in LabVIEW which monitors inputs, generates set-points and records data. Biaxiality values are achieved by monitoring the loadcell data and then calculating the necessary pressure to achieve the given biaxialityratio.Given equation (4.8), for a given biaxiality ratio B, ??? must also take the form48of (4.12).??? = B(Pr2t+Fpi(r2o? r2i))(4.12)PRt= B(Pr2t+Fpi(r2o? r2i))(4.13)Solving for the pressure P gives equation (4.14), which provides a target pres-sure as a function of the load cell output F in order to maintain a given biaxialityratio B. It should be noted that this equation does not account for either radialexpansion, nor wall thickness changes and therefore the actual biaxiality ratio willdeviate to a certain degree throughout the test. It has been found however that thiseffect is minor.P =B Fpi(r2o?r2i )(1? B2) ?tR; (4.14)The LabVIEW program then passes this pressure P to the controller for theSITEC pressure source as a target set point. The strain rate at which these testsare run is approximately ?? = 10?4. There is some variability before the ultimatetensile strength is reached due to the nature of the loading in these tests - namelythat the axial load frame operates in displacement control (constant strain rate) butthe pressure source operates in load control and thus there is unavoidable variationin the strain rate. This variability was on the order of a factor of 2. Upon theonset of necking, the strain rate is much more difficult to characterize, howeverthis stress-strain data is not considered in this work.494.6.1 Iso-Work SurfacesAs previously addressed in Section 2.4, it is difficult to define a yield surface for amaterial which is anisotropic. Traditional techniques such as calculated equivalentstresses and strains using von Mises expressions are obviously not applicable tomaterials known to have strong anisotropy. The necessity to be able to plot equiv-alent points on a surface led to the adoption of plastic work surfaces. While alu-minum itself displays good mechanical isotropy, magnesium does not and thereforetechniques for analyzing data which do not require reliance on material isotropy orprior knowledge of material properties were required. These iso-work surfaceswere constructed by calculating the plastic work done in a given sample, in boththe axial and hoop directions, by integrating their true stress - true plastic straincurves. The plastic work done to a sample tested in axial tension, to a given plasticstrain was measured and the equivalent points for all other tests were determined tobe the point at which the same level of plastic work had been done to the sample.The level of plastic work was calculated according to equation (4.15) where theincremental plastic work is calculated for subsequent images in a DIC data set ofsize n. This represents a piecewise linearization of the true stress-true plastic strainresponse with a strain increment on the order of 5x10?5. This numerical integra-tion scheme is illustrated in Figure 4.9 which presents a coarsened and simplifiedrepresentation of the integration. Regardless, this provides a means by which sam-ples undergoing differing deformation paths can be compared at similar levels ofdeformation.W =n?i=2{[(?i? ?i?1)??i]zz+[(?i? ?i?1)??i]??} (4.15)50Figure 4.9: Biaxial stress-strain data was numerically integrated as schemat-ically shown here. It should be noted however that this is for illustrativepurposes only and the actual integration was far finer in nature as datawas numerically integrated at every data point.4.7 Neutron Diffraction StudiesSamples were prepared for neutron diffraction studies by removing a 1 cm an-nular section from the centre of the gauge section of the sample. This was thenradially sliced into eight segments. These segments were stacked and secured us-ing an aluminum tape. This process was necessary in order to approximate the10mmx10mmx10mm volume required for neutron diffraction studies at CNBC.Neutron diffraction data was then returned to UBC and plotted using the MTexOpen Source MatLab Toolbox for Quantitative Texture Analysis [128]. Plottedpole figures were equal area spherical projections.514.8 Test Material and Thermomechanical PreprocessingAZ80 magnesium alloy was the material principally investigated in this course ofstudy. Two batches of material were obtained from different sources; one was inan as cast condition whereas the other had been extruded into 34mm bars. In allcases, the material underwent a heat soak at 415oC with the intention of improvinghomogeneity by eliminating the ? Mg17Al12 phase. Machining was accomplishedby boring out the inner radius and then turning the sample on a lathe. Once ma-chined, the samples were subjected to a further heat treatment of 385oC in orderto remove surface deformation imbued during the manufacturing process. The castmaterial, shown in Figure 4.10 was found to have a grain size of 32?m in accor-dance with the ASTM E112 standard while the extruded material had a grain sizeof 23 ?m and also exhibited stringer particles believed to consist of undissolved ?phase which can be seen in Figure 4.11.Neutron diffraction samples were prepared by cutting an 10mm long annularsection from tube samples and then subdividing the resultant ring into 8 segmentswhich were then stacked and fastened with aluminum tape. This procedure wasnecessary in order to roughly achieve the required 10x10x10mm sample volume re-quired for testing. Neutron diffraction testing indicated peak pole figure intensitiesfor the cast and extruded material of 1.8 x random and 2.3 x random respectively(Pole figures are presented in Chapter 6).Optical microscopy specimens were prepared by polishing using silica papersfrom 600-1200 grit followed by 6?m and 1?m diamond slurries. Samples werethen etched to reveal grain boundaries using a 4% Nitol solution for 20 seconds.52Figure 4.10: Optical micrograph of the cast AZ80 alloy reveals a grain sizeof 32?m. Sample has been homogenized, machined and heat treated.Figure 4.11: Optical micrograph of the extruded AZ80 alloy reveals a grainsize of 23?m. Sample has been homogenized, machined and heattreated and exibits534.9 VPSC SimulationsVPSC input parameters used for this material were those proposed by Jain et al.[27] for the same material and are presented in Table 4.1. In addition to thosevalues presented in Table 4.1, no latent hardening parameter was used for slip-slipinteractions but a value of htt = 4 [27] was used for twinning related interactions.Finally, an inverse rate sensitivity parameter n = 20 and a grain-matrix interac-tion parameter ne f f = 10 [27] was used. These parameters were not modified andwere used as presented by Jain et al. The purpose of this simulation work was toinvestigate the ability of existing codes to predict the behaviour of AZ80 in castand extruded conditions and to facilitate the interpretation of the experimental re-sults. The code was modified to permit biaxial loading conditions to be simulated.VPSC simulations were used to calculate the predicted response of this materialacross the majority of possible biaxiality ratios, however some ratios (between 2and -2) failed to converge with the present code base. This is resultant from themanner in which biaxial stress ratios are maintained by the code. For each step ofthe simulation, a given level of strain is applied in the axial direction, the resultantstress necessary to induce this level of strain is then calculated. This value of axialstress is then used to determine the appropriate hoop stress for the given biaxialityratio at which point the hoop strain resulting from this stress is calculated. Theissue preventing convergence through some regions of the yield surface is the ap-plication of an axial strain as for some biaxiality ratios this the level of axial straincan be either zero (at the hoop maxima of the yield surface) or negative. The im-position of the small positive strain therefore results in failure to converge in caseswhere this type of deformation is not a possible solution. As such, results have54been interpolated through these regions.Table 4.1: VPSC input parameters used for both cast and extruded AZ80 [27]Slip/Twin System ?0 (MPa) ?1 (MPa) ?0/? ?1Basal 33 86 1/133 0Second-order pyramidal 224 23 1/133 0Prismatic 145 195 1/100 0Extension Twinning 43 0 0 0Iso-work surfaces were calculated using the results of VPSC simulations byfitting a curve to calculated data points.55Chapter 5System Validation andExamination of Al 6061 Alloy5.1 IntroductionAn aluminum alloy provides an ideal material with which to validate the biaxialtest rig and experimental techniques due to its low mechanical anisotropy and wellcharacterized properties. A series of samples were tested both at UBC using thedeveloped biaxial rig, and also at the Canadian Neutron Beam Centre (CNBC) ina series of off line experiments. This permitted the behaviour of biaxial rig to bebenchmarked against that of a similar system and it permitted the establishment ofa robust experimental process.5.2 Experimental Scope6061-T6 aluminum alloy was selected for the benchmarking of the system. Sam-ples were prepared in accordance with the dimensioning provided in section 4.256and possessed an average grain size of approximately 30?m.Two sets of samples were tested, one set was tested on an existing rig at theCanadian Neutron Beam Centre. The purpose of testing these samples was toverify the proper functioning of the rig at UBC where a second set of samples wererun. The test rig at CNBC was incapable of simultaneous control of the pressureand load frame systems and therefore samples were pressurized (producing bothaxial and hoop stresses) before being pulled axially. This resulted in a biaxialloading ratio of 2 which then decreased as the axial displacement (via load framein displacement control) was applied. Samples were at pressures of 0, 6.89 and17.2 MPa. A test employing the same independent control as the CNBC systemwas conducted at UBC and further tests employed proportional control permittinga constant biaxiality ratio to be maintained.5.3 Experimental ResultsThe CNBC biaxial rig was incapable of simultaneous control of internal pressureand the load frame whereas the rig developed herein does have that capability. Assuch, tests were conducted at UBC employing independent control, in the samemanner as the CNBC rig, in order to verify the proper functioning of the rig, usingthe CNBC rig as a bench mark. Subsequent tests employed proportional controlin which pressure was controlled in accordance with the axial load. A furtherlimitation of the CNBC rig was its inability to monitor hoop strains, which onceagain was possible using the UBC rig.In Figure 5.1, it can be seen that the two rigs showed good agreement betweeneach other with measured stress and strain values agreeing within 1%. This verifiedthe proper functioning of the rig developed at UBC with respect to its instrumenta-57Figure 5.1: Comparison of the axial component of aluminum biaxial testsconducted at UBC and at CNBC. This test consisted of internally pres-surizing samples to 17.2 MPa before subsequently deforming them ax-ially to an approximate strain of 0.03 before unloading. Data showsagreement within 1% indicating good agreement between the rig devel-oped at UBC and an existing rig at CNBC.tion.5.4 von Mises Equivalent Stress SurfacesThe results of tests completed at UBC and CNBC are presented in figure 5.2. Thedata agrees well with the von Mises yield criterion for a material with a yield stressof 264 MPa, the value determined for the 6061 aluminum alloy tested via uniaxialtensile test. While there are other, more complex models which better describe58Figure 5.2: Comparison of data gathered at on the UBC experimental rig,shown in blue, and the CNBC experimental rig, shown in red. A corre-sponding von Mises surface is shown in grey.the behaviour of aluminum alloys, the von Mises surface provides a good pointof comparison for these purposes. Iso-work values could not be calculated for theCNBC data due to the lack of any means by which to measure hoop strains. Assuch, a traditional 0.2% offset technique was used to determine the yield point. Ashown in Figure 5.1, the work hardening rate for this material is quite low and thuserrors associated with employing this technique are lower than would be observedin materials with high work hardening rates.59Figure 5.3: Neutron diffraction pole figures for the initial Al6061 aluminumalloy.5.5 Neutron Diffraction StudiesNeutron diffraction was used to measure the texture of the samples before andafter testing. The initial texture for this material is provided in Figure 5.3. InFigures 5.4 and 5.5, pole figures are presented for samples deformed via axialtension and a biaxiality ratio of 2. The results indicate that the texture is largelythe same for both samples, with a slight weakening of the texture in the biaxialytested sample, however the peak intensities do not change appreciably. It can beseen from this that the 6061 aluminum alloy samples do not experience significanttextural evolution associated with deformation. There is a decrease in intensityfrom the intial pole figures, however the overall character is consistent across alltests. This behaviour is consistent with expectations as aluminum alloys do notgenerally display twinning behaviour [129].60Figure 5.4: Neutron diffraction pole figures for an Al6061 aluminum alloysample deformed via axial tension until failure.Figure 5.5: Neutron diffraction pole figures for an Al6061 aluminum alloysample subjected to a biaxiality ratio of 2 until failure.615.6 ConclusionResults presented in figure 5.2 indicate that the biaxial test rig developed at UBCprovides accurate experimental data and agrees with that obtained from the rigat CNBC. Testing of this well characterized material indicates good agreementbetween the experimentally determined iso-work data and a corresponding vonMises surface indicating that as expected, this material exhibits nearly isotropicbehaviour when subjected to biaxial loading. Texture analysis conducted usingneutron diffraction techniques indicates minimal textural evolution is associatedwith different loading paths.62Chapter 6Mechanical Testing of AZ80 AlloyUnder Constant Biaxial Ratios6.1 IntroductionThis chapter presents the results of testing the two AZ80 alloys presented in Section4.8 under a range of constant biaxiality ratios. Neutron diffraction experimentswere then used to determine the starting and as deformed textures for samplestested at each biaxiality ratio.The starting texture for the cast material is presented in Figure 6.1a and for theextruded material in Figure 6.1b. It can be seen that in both cases, there exists aband in the < 0001 > pole figure which is perpendicular to the axial direction ofthe sample which is parallel to CD in cast specimens and parallel to ED in extrudedspecimens. It can be seen that despite one being cast and the other extruded, theirtexture is actually quite similar between the two. The major difference that shouldbe observed is that the texture in the extruded material is somewhat stronger than63that of the cast material with a peak intensities in the < 0001 > pole figures of 2.3and 1.8 respectively.(a) Cast Texture(b) Extruded TextureFigure 6.1: Initial texture for cast and extruded samples.The textures measured via neutron diffraction for the cast and extruded mate-rials, presented in Figure 6.1, needed to be discretized prior to use in VPSC simu-lations. This was accomplished by calculating the orientation distribution functionfor the measured texture and then discretizing it into 16416 5x5x5 degree blocksover which the texture for each block was averaged. These block sizes correspondto the angular increments used during neutron diffraction measurements. Each ori-64entation bin was considered a discrete grain with a weight corresponding to theintensity associated with that block, resulting in a discrete grain orientation fileconsisting of 16416 grains, presented in Figure 6.2. It can be seen that while thediscretization into 5x5x5 blocks does result in some smoothing of the texture, theoverall character is preserved.(a) Discretized cast texture(b) Discretized extruded textureFigure 6.2: Discretized texture for cast and extruded samples used for VPSCsimulations.Biaxial samples were produced from the cast and extruded AZ80 materials andwere tested in accordance with the techniques outlined in Section 4.8.656.2 Cast and Heat Treated AZ806.2.1 ResultsFigure 6.3 presents the true stress - true plastic strain response of cast AZ80 sam-ples subjected to both uniaxial and biaxial loadings. Tests were conducted untilfailure of the sample which was defined as either strain localization and failureof the sample or buckling of the sample in the case of compression and negativebiaxiality tests. In Figure 6.3a, the results of uniaxial testing consisting of axialtension, hoop tension and axial compression are presented. With the current ex-perimental setup it is not possible to test samples in hoop compression. Figures6.3b through 6.3d present the results of biaxial tests. In each of these cases, twostress-strain curves are presented, one for the axial response and one for the hoopresponse. The end points of each plot indicate the onset of strain localizations intension dominated tests which was rapidly followed by failure. It should be notedthat it is not possible with this sort of plot to directly compare axial and hoop re-sponse for given levels of strain other than that at failure, in which case the endpoints of each data set can be directly compared.It is immediately obvious from Figure 6.3a that the axial tensile orientation hasa yield strength ? 60 MPa higher than hoop tension or axial compression. In allcases, serrations can be observed in the stress-strain behaviour. This is consistentwith observations by other groups [130?133] across a variety of magnesium alloys.The displayed behaviour, commonly referred to as the Portevin-Le Chatelier effect,has been observed in similar alloys including AZ91 [130] and is normally attributedto dynamic strain aging [134?136].Plastic work surfaces were created using the stress-strain results from the uni-66(a) Uniaxial (b) 0.4 Biaxiality(c) 0.9 Biaxiality (d) -0.5 BiaxialityFigure 6.3: True stress - true plastic strain results for cast AZ80 samplestested in a) uniaxial loadings, b) 0.4 biaxiality, c) 0.9 biaxiality andd) negative 0.5 biaxiality. Black lines indicate the axial deformationcomponent and red lines indicate the hoop deformation component.axial and biaxial tests. Levels of equal plastic work were calculated using equation(4.15) equal to that of axial tensile strains of 0.005 and 0.03 with resultant values of? 0.5MJ/m3 and 4.6MJ/m3 respectively. The principle of normality [137] permitsthe slope of the experimentally determined plastic work surface to be calculatedat the points investigated. The slope for each experimental data point is indicatedwith a short red line segment.These results are presented in Figure 6.4 in which lines represent VPSC data67and symbols represent experimental data. The solid black symbol in this figurepresents a prior result by Jain et al. [27] measured by compression on this materialand is a data point which cannot be achieved using the current experimental rig.Figure 6.4: Plastic work surfaces for cast AZ80 alloy corresponding to axialstrains of 0.005 and 0.03. Symbols represent experimental data whilelines represent corresponding VPSC predictions. Red line segmentspassing through symbols represent the experimentally determined slopeof the plastic work surface at that point. The solid black symbol is fromJain et al. [27] who used the same material.Examining the experimental data presents a number of observations. First ofall, there is significant anisotropy. Tension/compression anisotropy observed inthese experiments is consistent with prior work on this material [27] starting at ap-68proximately 1.4 at a plastic work level of 0.5 MJ/m3 and decreasing to 1.3 for aplastic work of 4.6 MJ/m3. This is consistent with the availability of deformationmodes as the axial tension orientation has significantly fewer grains favourablyoriented for twinning and therefore is predominantly dependent on basal slip untilprismatic or pyramidal slip systems can be activated. Conversely, the hoop andaxial compression tests do have significant numbers of grains favourably orientedfor twinning and therefore, possessing easier deformation mechanisms, yield atlower stresses. Relying on VPSC data for compression in the hoop direction indi-cates tension/compression anisotropy ratios of 0.9 decreasing to 0.75 in the hoopdirection at the lower and higher levels of plastic work, respectively. It is nec-essary to rely on VPSC data in these cases as they can not be readily measuredexperimentally using the current experimental setup. Overall, the agreement be-tween experimental and VPSC results is good, with values falling within 10% ofeach other. Further, a qualitative comparison of the VPSC surface and that of theexperimentally determined tangency vectors is good.It is useful to examine the strain-strain response exhibited by different biax-iality ratios and thus, Figure 6.5 presents both experimental and VPSC results.Strain-strain plots are directly related to the evolution of the shape of the plasticwork surface, and as such, comparing the VPSC and experimental results facilitatesan evaluation of the success with which the VPSC results agree with experimentalresults at discrete points as with plastic work surfaces, and also throughout the test.It can be seen that by and large the agreement between experimental and VPSCresults is good in most cases with some divergence of the results as strains getlarger. The only test in which the VPSC model does not capture the experimentalresponse is in the case of a biaxiality ratio of 0.4. Referring to Figure 6.4, it can be69seen that this sample corresponds to a nearly vertical tangency - a region in whichvery little hoop deformation would be expected. The VPSC simulation predictsnegligible hoop deformation at small strains, but then an increasing amount after acritical point corresponding to an axial strain of approximately 0.005.Figure 6.5: Experimental and VPSC strain-strain responses for cast AZ80 al-loy show generally good agreement. The notable exception is the caseof a biaxiality ratio of 0.4.In order to examine this behaviour in greater detail, a polynomial was fit tothe strain-strain data and then numerically differentiated. The result is presentedin Figure 6.6 where the derivative? p??? pzzis plotted as a function of the applied plasticwork. This permits qualitative analysis of the evolution of the ratio of the hoop and70axial strains as the sample undergoes deformation as well as a clearer picture of thebehaviour at small plastic strains.Figure 6.6: Evolution of experimental and VPSC strain ratios with respect toapplied plastic work in cast AZ80 alloy.From this figure, it can be seen that the strain ratio for the axial compressioncase maintains a near constant value of -0.5 throughout the test and is in keepingwith expectations for this orientation as there should be little difference in the hoopand radial response of the material as neutron diffraction texture measurementspresented in Figure 6.1a have indicated little variation of the < 0001 > pole figurethrough the TD plane and thus it would be expected that axial strain would beevenly accommodated by hoop and radial strains.A similar response was observed for axial tension, though this is not presentedin Figure 6.6 as it lies entirely on top of the compression data. In the case of the71biaxiality ratio of 0.4, the experimental strain ratio starts at 0.3 and subsequentlydecreases to a strain ratio of 0.1. The VPSC prediction for this case on the otherhand, starts at a value of 0.1 and then steadily increases to a value of 0.5. Thereasons for this discrepancy - mirrored in Figure 6.5 are likely associated withthis biaxiality ratios proximity to a region in which the slope of the plastic worksurfaces presented in Figure 6.4 rapidly change. Comparison of the tangents forexperimental points corresponding to biaxiality ratios of 0.4 and 0.9 indicates that arapid change in the tangency of the plastic work surface must occur between thesetwo points.The 0.9 biaxiality ratio test however, does see good agreement in strain ratioresponse between experimental and VPSC results. Interestingly, the initial agree-ment between experimental and VPSC results for the hoop test sees the value of thederivatives differing by 11.5% , however they converge somewhat eventually show-ing slightly better agreement with a difference of approximately 10%. Converselyto the case of 0.4 biaxiality this suggests an overestimation of the initial hoop re-sponse, however unlike the case of 0.4 biaxiality, the issue appears to resolve itselfsomewhat.6.2.2 Deformation Texture of Cast AZ80In order to understand the response of materials such as magnesium alloys whichare highly anisotropic and thus highly dependent on crystallographic orientation,an understanding of the evolution of texture of the material is useful. Neutrondiffraction studies were conducted on samples in the as tested condition as wellas on samples which had been machined and heat treated in accordance with theprocess described in Section 4.872Figure 6.1a provides the initial texture against which all experimental andVPSC pole figures presented in Figure 6.7 can be compared. In Figure 6.1a, aweak band across the TD plane with a peak intensity of 1.8 times random. Afteruniaxial tension, the texture, presented in Figure 6.7a remains similar in character,but the initial, weak banded texture has increased in strength and narrowed relativeto the TD-ND plane, with a peak intensity of 2.7 times random. The associatedVPSC prediction shows generally good agreement, with a peak intensity of 2.6and with overall a stronger banded structure - with decreased intensity at the northand south poles of the figure, which are parallel to the axial direction.Two additional plots are presented in addition to the experimental texture andthat predicted by VPSC. The third figure in each set presents the most active de-formation system (based on VPSC calculations). The final image indicates themagnitude of the strain increment undergone in the first deformation step of theVPSC simulation. Examining the plot of dominant deformation mechanism andstrain increment reveals much about the nature of the deformation. For example,for uniaxial tension, the north and south poles of the pole figure, which are parallelto the sample?s axial direction are predicted to exhibit deformation predominantlyvia deformation twinning, and the relative magnitude of this deformation is high.This explains the decreased intensity in these areas and the increased intensity ofthe central band - grains are twinning away from the north and south poles as theyundergo extension twinning, which features a reorientation angle of 86o, and endup largely randomly distributed through the TD-ND plane. Prismatic slip, shownin red, is predicted to be the dominant deformation mechanism along the centralband, however it can be seen from the activity plot, that there is negligible defor-mation activity in this region and thus prismatic slip plays a minor role, at least for73low strains. Basal slip (blue) is the dominant deformation mechanism throughoutthe remainder of orientations.When a hoop component is added, as in the 0.4 biaxiality ratio presented inFigure 6.7b, the applied hoop stress causes not only a strengthening in the TD-NDplane, but also causes grains oriented with their c-axes parallel to the hoop direc-tion (East and West poles of the figure) to rotate towards the centre of the polefigure which manifests itself as a significant reduction in intensity along the rimof the pole figure. This weakening along the rim of the figure is accompanied bya strengthening of the intensity at the centre of the pole figure with peak intensityrising to 3.2. As in the case of uniaxial tension, there is good agreement betweenthe VPSC and experimental results, though the VPSC results once again predict astronger peak intensity of 4 and the overall texture is sharper, the central peak issteeper than that seen in the experimental results. Extension twinning remains thedominant deformation mechanism at the North and South poles, but it?s influencespreads in the TD direction and East-West poles which are parallel to the hoopdirection in the sample now also feature extension twinning as the dominant defor-mation mechanism. It can be seen however, that activity in this region is relativelylow compared with the activity at the North-South poles, which explains the over-all oval profile of the central peak in the pole figure. Basal slip clearly plays animportant role in this biaxial loading condition as seen by the strong peaks in thestrain increment plot which coincide with regions of predominantly basal slip.74(a) Axial Tension(b) 0.4 Biaxiality75(c) 0.9 Biaxiality(d) Hoop Tension76(e) -0.5 Biaxiality(f) Axial CompressionFigure 6.7: Experimental and VPSC < 0001> pole figures for cast AZ80 after testing via uniaxial and biaxial loadingpaths. Left to right: experimentally determined texture, VPSC predicted texture, VPSC predictions for the ori-entation dependent slip system with the highest activity (blue = basal slip, red = prism slip, yellow = 2nd orderpyramidal slip and green = extension twins) and the VPSC orientation dependence of the magnitude of strainincrement at an imposed far field strain of 0.01. These final two plots can be useful for understanding the textureevolution observed.77In the case of the 0.9 biaxiality test (Figure 6.7c), the hoop stress is now nearlyequal to the axial stress. The resulting experimental texture has a nearly circularpeak with an experimental peak of 4.2 and a peak in VPSC predictions of 5.2. TheVPSC prediction indicates stronger texture in the centre of the figure and weakeralong the rim. Extension twinning is the dominant deformation mechanism alongthe entire perimeter of the pole figure, and can be seen to be nearly uniform inmagnitude on the strain increment plot which explains the strong, circular peakin the observed and predicted textures. Basal slip accounts for the bulk of thedeformation and is the dominant slip mechanism throughout an annular region ofthe pole figure with relatively uniform distribution other than slight peaks alignedwith the hoop direction.Figure 6.7d presents the case of hoop tension. Consistent with the 0.4 and 0.9biaxiality cases, in the case of hoop tension it can be seen that there is a significantdecrease in intensity of grains with their c axis oriented parallel with the hoop di-rection (East/West in the pole figure). However, unlike the biaxial cases which alsosaw decreased intensity parallel to the axial direction, in the case of hoop tension,in which there is no axial tensile load, the intensity of grains oriented parallel tothe axial direction of the sample is increased. The peak intensity is 2.8, similarto that of the axial tension case, though with a significantly different distribution- effectively rotated by 90o. The VPSC simulation presents qualitatively similarresults, however the simulation predicts a more even vertical banded texture distri-bution and a peak intensity of 3.4. The distribution of dominant slip mechanismsand strain increments is the same as for axial tension, other than a 90o rotation,which is consistent with the effective 90o rotation of the loading direction.When the axial load switches from tensile in nature to compressive as in the78case of a biaxiality ratio of -0.5, presented in Figure 6.7e, there is a very strongchange in the nature of the orientation distribution. The increase in intensity par-allel to the sample?s axial direction observed in the hoop tension case is muchstronger in the -0.5 case, with the peak intensity of 3.5 in the experimental resultsand 5.6 predicted by VPSC. In both cases the peak is oriented parallel to the axialdirection as opposed to the centre of the pole figure as in previous cases, thoughthere are still a number of grains oriented with c axes parallel to the radial directionat the centre of the pole figure. Extension twinning is the dominant deformationmechanism through the TD-ND band which is also the region of highest initialtexture intensity, this explains the starkly decreased intensity through this band,however the strain increment plot indicates that there is a central region which seesmuch lower levels of deformation which explains the continued presence of grainsof this orientation. Basal slip remains a very significant deformation mechanism,though deformation is largely restricted to lobes along the perimeter of the polefigure.The behaviour observed in the -0.5 case is even stronger in the axial com-pression case (Figure 6.7f) with a peak intensity of 5 for experimental and 6.2 forVPSC, in both cases again parallel to the axial CD direction and weaker texturein the centre of the pole figure. Despite a distribution of dominant deformationmechanisms that is very similar to that of the -0.5 case, it can be seen that the ac-tual strain increments are quite different. The low levels of deformation right at thecentre of the figure in the -0.5 case are now gone, with extension twinning takingplace right through the TD-ND plane resulting in minimal grains remaining withthis orientation. While grains oriented parallel to the axial direction see minimaldeformation, there is significant basal slip taking place between these poles and79the band of grain orientations which see predominantly extension twinning. In allcases it can be seen that basal slip and extension twinning are responsible for thebulk of the deformation with negligible contributions from prism slip or 2nd orderpyramidal slip.VPSC predictions have been shown to reasonably agree with experimental val-ues for plastic potentials and the strain response for the range of biaxiality ratiosexamined. Figure 6.8 presents activity charts for the various deformation modes,including < a > basal slip, < a > prismatic slip, < c+ a > 2nd order pyramidalslip and extension twinning. From Figure 6.8a, it can be seen that initially the levelof basal slip decreases with the level of axial tension applied to the sample - axialtension sees the greatest level of basal slip and axial compression the least. Theexception to this observation is hoop tension, which sees greater levels of basalslip than the 0.9 biaxiality ratio case.The contribution of prism slip (Figure 6.8b) is initially negligible in all cases,however this system becomes more significant in key loading conditions - axialtension, hoop tension and -0.5 biaxiality. This is due to the effect of work hardeningon slip systems as well as the consumption of grains suitably oriented for extensiontwinning. Pyramidal slip activity, presented in Figure 6.8c starts and remains lowin all cases due to the very high CRSS associated with this deformation mode.The initial activity of extension twinning (Figure 6.8d) appears to be almostthe reverse of basal slip. Cases such as axial tension, which saw the highest basalslip activity, correspondingly has the lowest extension twinning activity. As de-formation progresses, all loading conditions display decreasing levels of twinningactivity commensurate with the exhaustion of suitably oriented grains. Unsurpris-ingly, examining Figure 6.8e, which presents the fraction of twinned grains, shows80that the loading conditions which displayed the highest levels of twinning activityaccumulate the greatest fraction of twinned grains. In all cases it can be seen thatthe twinned fraction appears to be in the process of saturating as the number ofun-twinned, favourably oriented grains decreases.Intermediate StrainsIn order to investigate the speed with which texture develops in the deformed sam-ples, cast AZ80 specimens were tested to levels of plastic equal to roughly halftheir failure strain.The first such sample was an axial tension sample which was deformed axiallyto a strain of 0.04. Qualitatively, it can be seen in Figure 6.9b that the < 0001 >band has increased in intensity as compared with the starting texture in Figure 6.9a,though it has not fully developed to the point of the sample pulled to failure, shownin Figure 6.9c; there are still significant numbers of grains whose < c > axes areoriented in the region approximately half way between CD and TD - a region withless intensity in the sample deformed to failure.A second sample was deformed at a biaxiality ratio of 0.9 until an axial strainof 0.025 was reached. This represents half the axial strain at which prior samplestested with this biaxiality ratio failed. The level of axial strain was used as thecontrol in the case as hoop strains cannot be monitored in real time. In this case, thetexture presented in Figure 6.10b appears to be nearly fully developed as comparedwith that in Figure 6.10c. There is the same central peak as seen in the sampledeformed to failure and the initial banded structure seen in Figure 6.10a is nearlygone, with only minor spreading of the peak in the hoop direction.81(a) < a > Basal Slip (b) < a > Prism Slip(c) < c+a > Pyramidal Slip (d) Extension twinning(e) Twinned FractionFigure 6.8: Slip system and twinning activity, as well as twinned fraction pre-dicted by VPSC simulation for deformation of cast AZ80 subjected tovarious loading conditions.82(a) Initial Texture (b) Intermediate Texture (c) Final TextureFigure 6.9: < 0001 > Pole figures for axial tension samples pulled to strainsof 0.00, 0.04 and to failure respectively.(a) Initial Texture (b) Intermediate Texture (c) Final TextureFigure 6.10: < 0001 > Pole figures for 0.9 biaxiality samples pulled to axialstrains of 0.00, 0.025 and to failure respectively.6.2.3 DiscussionDespite possessing relatively weak texture, the stress - strain behaviour observedin this cast AZ80 material exhibits strong anisotropy in both tension - compressionand tension - tension. The ratio of axial tension and axial compression stresses at aplastic work level corresponding to an axial tension strain of 0.005 is 1.4. The ratioof axial tension to hoop tension stresses is 1.5. This section will attempt to explainthis behaviour as a product of the material?s texture as it relates to the complexinterplay and competition between deformation modes.83The deformation mechanisms observed in magnesium alloys such as the onesbeing investigated here include basal < a> type slip, prismatic < a> type slip, 2ndorder < c+ a > slip and extension twinning [10, 23, 48, 138]. Contraction twinshave also been observed in magnesium [43], however they are not commonly ob-served in most magnesium alloys [139]. A detailed study by Jain et al. on thismaterial showed very little evidence of contraction twins. When present contrac-tion twins have been found to have minimal effect on the material?s texture due totheir low volume fraction (due to both scarcity of the twins themselves as well astheir very thin profiles when present [140, 141]) and thus it is believed that theircontribution can be safely ignored.As discussed in Chapter 2 in Sections 2.2.2 and 2.2.3, basal slip and extensiontwinning have the lowest CRSS of the available deformation modes and as such itis unsurprising to note in Figure 6.8 that they are predicted by VPSC simulationsto be the most active deformation mechanisms, regardless of applied loading. Ithas been previously shown that for this material, non-basal slip occurs only ingrains with close to ideal Schmid factors for these systems [126]. VPSC simulationdeformation system activity data, presented in Figure 6.8 is generally consistentwith these observations, apart from the activity of the < c+a> 2nd order pyramidalslip system, which was observed experimentally in this material through slip traceanalysis [126] as well as a similar magnesium alloy [48].Stress-Strain ResponseThe starting texture observed in the cast AZ80 material (Figure 6.1a) is weakerin intensity, but of similar character to that of the extruded material (Figure 6.1b).The majority of grains are oriented with their c axis perpendicular the the sample?s84axial direction which is aligned with the casting direction.A key point is that the texture is not random. Extension twinning is the secondmost easily activated deformation mechanism after basal slip, but can only con-tribute to deformation while there are grains suitably oriented due to their polarnature. The results presented in Figure 6.7a indicate that when subjected to axialtension, the grains oriented with their basal poles parallel to the axial sample di-rection are best oriented for extension twinning, yet this is the orientation regionwith the weakest intensity in Figure 6.1a and correspondingly the extension twin-ning activity for axial tension (Figure 6.8d) is the lowest of any of the consideredloading conditions. However, it can further be seen in the strain increment plot inFigure 6.7a, that extension twinning in suitably oriented grains is predicted to berapid and in fact represents the region of the pole figure with the greatest strainincrement.Basal slip accounts for the majority of the deformation and while the strainincrement through basal dominated regions is not as great as that of extensiontwinning, it occurs over a much larger area of the pole figure accounting for itsoverall greater activity.A number of inferences can be made from this. First of all, when subjectedto axial tension, extension twinning presents a favourable deformation mechanism,but given the lower likelihood of grains existing with this orientation it does not oc-cur with as great an activity as other loading conditions. Secondly, basal slip is thedominant deformation mechanism. Finally, despite there being a well populatedorientation band perpendicular to the casting direction which is favourably ori-ented for prismatic slip, it initially makes a negligible contribution to deformationas imposed strains are accommodated through other easier to activate deformation85mechanisms due to the much higher CRSS associated with this deformation mode.As the sample work hardens, prismatic slip is predicted by VPSC to become astronger contributor to deformation.The results of measuring the texture for a sample loaded in axial tension to astrain of 0.04 appear to further substantiate these findings as the banded charac-ter observed at failure in Figure 6.9c has already manifested itself in Figure 6.9bindicating that extensive twinning has already taken place. An axial tensile strainof 0.04 corresponds to a plastic work of 6 MJ/m3; at this stage VPSC predictionsestimate extension twinning activity to have dropped below a contribution of 0.1and total twinned fraction to also be approximately 0.1. This is qualitatively inkeeping with observed experimental results which see some intensification of thebanded structure, but not a complete reorientation of the grain structure as seen insome cases.In summary, when undergoing tension parallel to the casting direction, it cantherefore be seen that basal slip provides the greatest contribution to deformation.The grains which are suitably oriented for twinning appear to begin to do so earlyon. As the material work hardens prismatic slip becomes a contributor to deforma-tion with a predicted activity of 0.2 at a plastic strain of 0.04 while twinning con-tinues to provide a modest contribution to deformation. 2nd order pyramidal slipis not predicted to contribute significantly with a predicted activity of 0 through-out the test suggesting that all < c > axis deformation is accommodated throughextension twinning.As hoop tension is added into the loading scenario, a marked change in be-haviour is observed. In the 0.4 biaxiality case, prismatic slip is no longer the pre-ferred deformation mechanism for any grains, as shown in Figure 6.7b and there86is a very small region at the centre of the pole figure in which 2nd order pyramidalslip is predicted to be favourable, however this region is so small, and the CRSS forthis deformation mode so high, that it is not expected to contribute in a meaningfulway, as seen in Figure 6.8c in which 2nd order pyramidal slip activity rises to only0.001 at plastic work of 6 MJ/m3. The region of the pole figure in which extensiontwinning represents the most favourable deformation mechanism has grown andnow includes regions aligned with the hoop direction. The strain increment plot inFigure 6.7b indicates very low values in this region however, and thus basal slipremains the dominant deformation mechanism with an initial activity of 0.8 whichremains relatively constant up to a plastic work of 10 MJ/m3. As in the case ofaxial tension, prismatic slip is not predicted to contribute to deformation initially,however as the material work hardens prismatic slip becomes more active, reachingan activity of 0.1 at a plastic work of 10 MJ/m3.When the proportional level of hoop tension is further increased to a biaxialityratio of 0.9, the entire perimeter of the < 0001 > pole figure becomes favourablyoriented for extension twinning and the remaining orientations are predicted to ini-tially deform via basal slip. The strain increment plot suggests significant twinningrelated deformation at the periphery, particularly in line with hoop tension. Thiscoincides with a large number of favourably oriented grains and thus substanti-ates the VPSC prediction that twinning initially accounts for 30% of deformationwith basal slip accounting for the remaining initial deformation. The fraction oftwinned grains is predicted to reach 30% at a plastic work of 10 MJ/m3 - a levelwhich roughly coincides with the test stopped at an axial strain of approximately0.025. As opposed to the 0.4 biaxiality case, prism slip is now completely sup-pressed. When viewed in tandem with the 0.4 biaxiality data, this change deforma-87tion mechanisms illustrates the competition between basal slip, prism slip and ex-tension twinning. The close competition between these deformation modes couldaccount for the relatively poor VPSC prediction of strain - strain behaviour for the0.4 biaxiality test as shown in Figure 6.5. As with other tests, 2nd order pyramidalslip is not predicted to contribute to deformation in any meaningful manner and infact the activity of this slip system is not predicted to reach more than 0.008 at aplastic work of 10 MJ/m3.In the hoop tension case, the axial load is removed and the result is signifi-cant. As seen in Figure 6.7d, the layout of predicted slip systems is rotated 90oas compared to that of axial tension, the result of which is that twinning is nowthe predicted deformation mechanism in an area of the pole figure in which theprobability of grains existing is quite high. The result of which is that the predictedtwinning activity is expected to start at a relatively high level of 0.28 of all activitywith the remaining deformation being accommodated by basal slip. As deforma-tion continues, extension twinning remains nearly as active in the hoop tension caseas in the 0.9 biaxiality case. Interestingly, it can be seen in Figure 6.8a that at aplastic work of 3.5 MJ/m3, the level of basal slip activity which had until this pointbeen increasing to a level of 0.79, is predicted to begin to drop and is accompaniedby an equivalent increase in prismatic slip activity.The addition of axial compression further modifies the material response. Fig-ure 6.7e indicates that for a biaxial ratio of -0.5 nearly the entire intensity band inthe starting texture is favourably oriented for twinning in this case, however the plotof the initial strain increment suggests a very complex material response. Exten-sion twinning is an extremely important deformation mechanism in this case andinitially accounts 43% of deformation, second only to strain axial compression. As88deformation progresses, the twinning activity remains high and by a plastic work of10 MJ/m3 the twinned fraction reaches 0.4. When the hoop stress is then removedin the axial compression case, the deformation mode map remains the same, andin fact extension twinning remains roughly 43% of deformation, but the strain in-crement plot is quite different than in the -0.5 biaxiality case. The strain incrementplot for axial compression shares some similarities with that of axial tension, how-ever, grains with basal poles oriented parallel to the axial direction are now inactiveand the previously inactive grains located in the intensity band are now active andexpected to primarily deform via extension twinning.Texture EvolutionExtension twinning plays a crucial role in the textural evolution of magnesiumalloys and the texture results in Figure 6.7 as they relate to the starting texture inFigure 6.1a can be largely explained through an understanding of the activationof twinning as a deformation mechanism and the accompanying 86.3o orientationrotation.When undergoing axial tension, there are a limited number of grains which aresuitably oriented for extension twinning and thus most deformation is accomodatedthrough basal slip. However, as shown in the pole figures in Figure 6.7a, the TD-ND band observed in the as cast material intensifies as some of the CD orientedgrains twin and reorient themselves into the band. The increase in intensity from1.8 to 2.7 is accordingly modest in comparison to other loading cases.In the 0.4 biaxiality case, the presence of grains with c axes oriented in the hoopdirection facilitates an increase in twinning. The result of which is a transformationof the TD-ND intensity band into an oval peak. The axial tension component of the89test also induces twinning of CD oriented grains and the result of grains twinningfrom both the hoop and axial direction is a stronger intensity increase of 1.8 to3.2. As the hoop component increases as in the 0.9 biaxiality case, this behaviourintensifies as all grains located along the periphery of the pole figure are now suit-ably oriented for twinning. Hoop deformation induces extension twinning of highprobability orientations aligned with the hoop direction and axial tension inducestwinning in the low probability orientations aligned with the axial/cast direction.Intensity increases from 1.8 to 4.2 in accordance with increased twinning activityaround perimeter of the pole figure.Hoop tension does not experience the effect of axial tension; whereas in the 0.9biaxiality case, the nearly balanced biaxial loads drives any grains with orientationsaround the rim of the pole figure towards the center, in hoop tension, it is only thoseoriented towards the East and West poles which are expected to twin. Without theinput of axial tension, twinning towards the casting direction - aligned with theaxial direction, becomes equally favourable to twinning into the centre. The resultof the twinning related reorientation is now that grains can twin into any positionoriented along a vertical band in the pole figure. Because of the lower constraintson final orientations, peak intensity only increases to 2.8 and a faint vertical bandwith a central peak is observed in Figure 6.7d.The addition of a compressive load oriented along the axial direction producesanother dramatic change in the texture evolution associated with deformation inthis material. The entire band of higher intensities observed in Figure 6.1a is nowfavourably oriented for twinning (Figure 6.7e), although the continued presenceof hoop tension appears to retard twinning activity at the center of the pole figureand thus some grains remain in this orientation. Peak intensity is no longer located90at the centre of the pole figure and instead is now aligned with the casting/axialdirection with an intensity of 3.5. When the hoop component is removed as in theaxial compression case, the intensity jumps to 5, the highest measured intensity forany orientation, the starting intensity band has almost completely emptied due totwinning activity. As shown in Figure 6.8 the axial compression loading conditionexperiences the lowest level of basal slip of any loading condition, and accordinglyexperiences the greatest degree of twinning activity.6.3 Extruded AZ80Extruded AZ80 samples were tested in the same manner as cast samples. Axialtension, axial compression and hoop tension were tested using biaxial tube samplesas well as biaxiality ratios of 0.3, 0.4, 0.9, 2 and -0.9. Figure 6.11 presents thestress-strain results of a subset of these tests - specifically the uniaxial tests andbiaxiality ratios of 0.4, 0.9 and - ResultsIn Figure 6.11a, the stress-strain results for the uniaxial tests are presented. It canbe seen that a similar pattern to the cast material is observed - the hoop tensionand axial compression samples exhibit similar yield stresses (with hoop tensionbeing slightly stronger than axial compression) while axial tension yields at a stressapproximately 60 MPa higher. However, each orientation in the extruded materialhas a yield strength approximately 40 MPa higher than its cast equivalent. Thegrain size of the extruded material was observed to be smaller than that of the castmaterial (23 ?m versus 32 ?m) and thus the smaller grain size will contribute to theincreased overall strength. Both materials exhibit basal texture with a < c > axis91band along the TD-RD plane and as such, the limited availability of grains suitablyoriented for twinning (as compared with the hoop tension and axial compressioncases) is likely the cause for the higher strength in the axial tension case.Figures 6.11b - 6.11d present the stress-strain behaviour for a selection of bi-axial tests. As with cast samples, all tests display serrations resultant from thePortevin - Le Chatelier effect. Experimental iso-work values are shown in Figure6.12 for plastic work levels equal to 0.005, 0.03 and 0.06. Note that not all samplesreached a work level equal to 0.06 and thus experimental data does not exist for allbiaxiality ratios at this level.Employing the discretized texture shown in Figure 6.2b, VPSC simulations wasused to create a theoretical plastic work surface corresponding to an axial tensileplastic work level of 0.005 against which the experimental data was plotted. Theresults of this exercise are presented in Figure 6.13. A similar surface was createdfor an axial tension plastic work of 0.03, but is omitted here for clarity due tooverlap. The level of agreement between VPSC and experimental data is weakerthan that seen in the cast material. The iso-work surface predicted by VPSC fallsentirely inside the hull created by the experimental data.When the data is normalized according to the level of axial stress at a plasticwork of 0.005, as presented in Figure 6.14, then the situation changes. Experi-mental data was normalized to the experimental stress at an axial strain of 0.005and the VPSC predictions similarly normalized to the VPSC predicted stress at anaxial plastic strain of 0.005. The input parameters being used had not been alteredin any manner to those developed by Jain et al. [65] and thus there had been noaccommodation for the grain size. An analogous technique would have been toincrease the CRSS value for each deformation mode by a given percentage. While92(a) Uniaxial (b) 0.4 Biaxiality(c) 0.9 Biaxiality (d) -0.9 BiaxialityFigure 6.11: True stress - true plastic strain results for extruded AZ80 sam-ples tested in a) uniaxial loadings, b) 0.4 biaxiality, c) 0.9 biaxiality andd) negative 0.9 biaxiality. Black lines indicate the axial deformationcomponent and red lines indicate the hoop deformation component.the VPSC simulation appears to fail to capture the full extent of the corner ob-served in the tension-tension quadrant - a region in which strain normality vector israpidly changing - throughout the rest of the surface, agreement is much stronger.Examining the strain-strain data in Figure 6.15 reveals a lower level of agree-ment between experimental data and VPSC predictions as compared with the castmaterial. This indicates that ratio of axial and hoop strains are not as well predictedas in the cast material. Axial tension and compression as well as hoop tension show93Figure 6.12: Experimental plastic work surfaces for extruded AZ80 alloycorresponding to an axial strains of 0.005, 0.03 and 0.06. Line seg-ments passing through symbols represent the experimentally deter-mined slope of the plastic work surface at that point.good agreement, other loading conditions show a failure by the VPSC simulationsto capture experimental data. Interestingly, the 0.4 biaxiality tests in both cast andextruded materials demonstrate similar disagreements between experimental andVPSC data. Experimental data indicates a higher ratio of axial to hoop strainsearly on in the deformation process, which then tapers, while VPSC predictionsbeing largely the inverse - early deformation is predicted to be hoop dominatedwith increasing axial activity. The 0.9 and -0.9 biaxiality ratio tests both appear to94Figure 6.13: Plastic work surface for extruded AZ80 alloy correspondingto an axial strains of 0.005. Symbols represent experimental datawhile lines represent corresponding VPSC predictions. Red line seg-ments passing through symbols represent the experimentally deter-mined slope of the plastic work surface at that point.exhibit more hoop deformation than is predicted by the VPSC simulations.6.3.2 Deformation Texture of Extruded AZ80Figure 6.1b provides the initial texture for this material after machining and heattreating the samples and thus a base line against which the deformed and predictedtextures can be compared. As with the cast material, there is an intensity band in theTD plane, though it is stronger in this material with a peak intensity of 2.3. Axial95Figure 6.14: Normalized plastic work surfaces for extruded AZ80 alloy cor-responding to axial strains of 0.005and 0.03. Experimental data wasnormalized to experimental stress value for axial strain at 0.005 andVPSC to the VPSC predicted stress at an equal strain. Symbols repre-sent experimental data while lines represent corresponding VPSC pre-dictions. Red line segments passing through symbols represent the ex-perimentally determined slope of the plastic work surface at that point.tensile deformation causes an intensification of the TD intensity band as seen inFigure 6.16a with peak intensity rising to 3.9 and developing a oblong central peak.This is similar to the texture predicted by VPSC, though the VPSC predictions donot show the oblong peak and the predicted texture is somewhat weaker at 3.2 timesrandom. The dominant deformation mode plot and strain increment plots (3rd and4th plots respectively) predict that the twinning is responsible for the strengthening96Figure 6.15: Experimental and VPSC strain-strain responses for extrudedAZ80 alloy.of this intensity band. This can been seen as twinning is dominant at the Northand South poles of the deformation mode plot, regions which are coincident withthe highest levels of activity in the strain increment plot. As in the cast material,this explains the decreased intensity at the North and South poles of the pole figure(parallel to the axial direction) and the increased intensity of the TD band whichcorresponds with the 86o reorientation angle associated with extension twinning.Prismatic slip is predicted to occur through a narrow band aligned with the TD band(parallel to the plane connecting the hoop and radial directions). The remainder oforientations are predicted to deform by basal slip.97The addition of a hoop component induces extension twinning in grains withc-axes parallel to the hoop direction (East and West poles of Figure 6.16b) with theresult that they rotate towards the centre of the pole figure which further displayssome splitting. The result is a decrease in intensity in the hoop direction and anincrease in the intensity at the center of the pole figure whose intensity reaches 4.9.This corresponds well with the textural evolution predicted by the VPSC modelwhose peak intensity reaches 4.6. The VPSC texture is somewhat sharper thanthat predicted experimentally, with the minimum intensity reaching zero comparedwith a minimum of 2.6 in the experimental data. Twinning is now the dominantdeformation mechanism for grains with c-axis aligned with either the axial or hoopdirections. Other than a very small region at the centre of the pole figure which ispredicted to undergo deformation via second order pyramidal slip (and according tothe strain increment plot sees negligible activity), the remainder of the orientationsundergo deformation via basal slip.Increasing the hoop component until it is nearly equal to the axial stress, as inthe case of the 0.9 biaxiality test results in an experimental texture which in Figure6.16c, features a strong central peak with an intensity of 4. This peak is quite broad.The peak predicted by VPSC is much sharper in character and has a much higherpeak intensity of 5.9. All grains with orientations along the rim of the pole figureare now predicted to deform via extension twinning, with the remainder of grainsdeforming via basal slip, apart from a negligible region at the centre predictedto undergo deformation via second order pyramidal slip. Looking at the strainincrement plot, it appears that the bulk of the deformation occurs via basal slip,accompanied by significant twinning activity, particularly by grains oriented in thehoop direction.98The hoop tension case is very similar to that of axial tension, but rotated by90o. Presented in Figure 6.16d, grains oriented in the hoop direction are expectedto undergo significant deformation via extension twinning. Some prismatic slipis expected from grains oriented in a band connecting the North and South poles,though examination of the strain increment plot indicates that this deformationmechanism is not expected to significantly contribute to overall deformation. Theremainder of deformation is expected to take place via basal slip. Peak intensity isfound experimentally to be 3.8, and which compares with a VPSC predicted valueof 4.2. As in other cases, the overall character of the predicted texture is sharperthan that observed experimentally.Applying a compressive axial component as in the -0.9 biaxiality case (Figure6.16e) yields a somewhat vertically banded structure with peaks occurring at theNorth and South poles, aligned with the axial direction and sub peaks occurringmid way between these pole sand the centre of the pole figure. This behaviour isalso observed in the VPSC simulated textures. Peak intensity is 3.9 in experimen-tal data and 6.5 in the VPSC simulated texture. The bulk of the discrepancy isin the intensity of grains aligned with the axial direction. These are grains whichare predicted to have rotated out of the central band through extension twinning.Sub peaks are expected to deform via basal slip, however examination of the strainincrement plot suggests that these sub peaks likely do not account for a significantcomponent of the deformation. Twinning is expected to be the favourable deforma-tion mechanism for most grains in the initial intensity band scene in Figure 6.1b.Axial compression of the extruded AZ80 yields similar results to that of thecast material. Per Figure 6.16f, twinning is the favourable deformation mechanismthroughout the initial central band while basal slip is favourable for the remainder99of crystallographic orientations, bar very small regions at the top and bottom ofthe pole figure which are expected to provide negligible contributions accordingto the strain increment plot. The 86o crystallographic rotation of the grains suit-ably oriented for extension twinning results in grains twinning away from the TDplane towards the North and South poles of the pole figure, aligned with the axialdirection of the sample.100(a) Axial Tension(b) 0.4 Biaxiality101(c) 0.9 Biaxiality(d) Hoop Tension102(e) -0.9 Biaxiality(f) Axial CompressionFigure 6.16: Experimental and VPSC < 0001 > pole figures for extruded AZ80 after testing via uniaxial and biaxialloading paths. Left to right: experimentally determined texture, VPSC predicted texture, VPSC predictions forthe orientation dependent slip system with the highest activity (blue = basal slip, red = prism slip, yellow = 2ndorder pyramidal slip and green = extension twins) and the VPSC orientation dependence of the magnitude ofstrain increment at an imposed far field strain of 0.01. These final two plots are useful for understanding thetexture evolution observed.103Examining the predicted deformation system activity garnered from VPSCsimulations in figure 6.17 reveals a number of interesting observations. First ofall in the axial tension case, the basal slip contribution (figure 6.17a) to deforma-tion is predicted to drop off more quickly in the extruded material than in the castmaterial. This is largely matched by a more rapid increase in prismatic slip typedeformation (figure 6.17b). Prismatic slip is predicted to contribute to deformationin both axial tension and hoop tension. When these two are combined, as in thecase of the 0.4 biaxiality simulation, prismatic slip is still predicted, but in the caseof the 0.9 biaxiality test, no prismatic slip is predicted. 2nd order pyramidal slip isnot predicted to contribute to the deformation in any case, as seen in figure 6.17c.Axial compression and -0.9 biaxiality are both predicted to exhibit a great deal oftwinning with twinning predicted to initially account for nearly 50% of deforma-tion activity. Hoop tension and 0.9 biaxiality follow at 30% of initial deformationactivity followed by 0.4 biaxiality at 20% of deformation accommodated throughtwinning at yield.6.3.3 DiscussionThe extruded AZ80 provides a material with a similar texture to that of the castmaterial, but with stronger texture - the band along the transverse plane of the ma-terial is both sharper and has a peak intensity nearly 30% higher than that of thecast material. The material displays a stronger tension-compression asymmetry,with a ratio between tension and compression stresses in the axial direction being1.7 at a strain of 0.005 but only 1.2 in the hoop direction, though this is reliant onVPSC data for the hoop compression value which can not be garnered experimen-tally with the current experimental setup. The tension-tension asymmetry is also104(a) < a > Basal Slip (b) < a > Prism Slip(c) < c+a > Pyramidal Slip (d) Extension twinning(e) Twinned FractionFigure 6.17: Slip system and twinning activity, as well as twinned fractionpredicted by VPSC simulation for deformation of extruded AZ80 sub-jected to various loading conditions.105quite strong, with the ratio of stresses at a strain of 0.005 being 1.7. The tension-compression asymmetry in the axial direction, which starts at 1.7 at a strain of0.005 drops to 1.4 at a strain of 0.06. The tension-tension asymmetry likewisedrops from 1.7 to 1.3. This section will attempt to explain how the increased in-tensity of the texture results in this asymmetry through the availability of grainsfavourably oriented for different deformation mechanisms.Stress-Strain ResponseThe sharper, banded texture of the extruded material possesses fewer grains suit-ably oriented for extension twinning during axial tension. The result is that duringthe axial tension test, VPSC simulations predict twinning activity is decreased byapproximately 11% at the beginning of the test as compared with predicted be-haviour for the cast material. This is accompanied by a nearly equal increase inthe prism slip activity. The axial stress at 0.005 plastic strain is equal is 67 MPahigher for the strain material. There is decreased availability of grains suitably ori-ented for twinning and basal slip due to the sharper texture, this results in a higherstresses due to these less favourable orientations. The higher stresses in turn allowfor prism slip, with its higher CRSS, to be activated, though its role is still limited.As before, the largest strain increment is associated with twinning, but basal slipoccurs over a much larger swath of the pole figure and thus has the highest overallactivity, as shown in Figure 6.17.Extension twinning remains a favourable deformation mode, however there arefewer grains suitably oriented for this type of deformation and thus its activity isdecreased for the axial tension case. Basal slip remains the dominant deforma-tion mode for this loading condition, as well as all of the others. Prism slip is the106favourable deformation mode for a large region of the central band. In the castmaterial, the observed stresses where initially not high enough to activate this de-formation mode, however, in the extruded material, the higher observed stressesare sufficient to activate this deformation mode at the onset of plastic deformation.At an iso-work level equal to an axial tensile strain of 0.03, the predicted activityof prismatic slip is increased from 0.233 in the cast material to 0.304. This is con-sistent with the overall higher stress levels observed in the extruded material whichpermit increased activity of this deformation mode.Hoop tension has a significant effect on the stress-strain behaviour of the ex-truded AZ80. In the 0.4 biaxiality case, the increased stresses observed in the ex-truded material facilitates more grains undergoing extension twinning - the rangeof orientations which are favourably oriented for extension twinning is increasedin the extruded material as compared with the cast material (referring to Figures6.7b and6.16b). Despite increased stresses as compared with the cast material,the stresses are not high enough to initially activate prismatic slip, though as thesample work hardens, this deformation mode is predicted to become active. Asin all cases in which second order pyramidal slip is the favourable deformationmode, the CRSS associated with this deformation mode prevents its activity. Fur-ther increasing the biaxiality ratio to 0.9, the favourable slip system plot in Figure6.16c is effectively the same as that of its cast counterpart in Figure 6.7c and thecorresponding initial activity of extension twinning is nearly the same for the castand extruded material at 0.293 and 0.289 respectively. Prism slip is completelysuppressed in this case, consistent with the cast material.The removal of the axial load in the hoop tension case reintroduces prismaticslip as a favourable deformation mode. The cast AZ80 VPSC simulations predicted107an activity for prismatic slip of 0.033 at the onset of plastic deformation. In theextruded case, the prismatic slip activity is predicted to be 0. The reason for this isthe availability of grains suitably oriented for extension twinning. Referring to thedeformation mode plot in Figure 6.16d indicates that the East and West poles of thepole figure, aligned with the hoop direction, are favourable for extension twinning.In the extruded material, these regions are more densely populated and thus thereare more grains favourably oriented for twinning. Whereas there was a 67 MPadifference in stress between the cast and extruded materials at a strain of 0.005,in the hoop orientation, this difference is reduced to only 28 MPa. Figure 6.17indicates that particularly in the hoop tension case, there is intense competitionbetween basal slip, prismatic slip and extension twinning. At low levels of work,the competition exists between basal slip and extension twinning, however at aplastic work of about 2.5-3 MJ/m3 prismatic slip also enters the competition andwould appear to be accompanied by decreasing activity of both slip and extensiontwinning.The -0.9 biaxiality test is predicted to induce twinning in grains whose orien-tation matches that of the highly populated intensity band in the extruded material(Figure 6.16e). Correspondingly, twinning is predicted to be very active for thisbiaxiality ratio, predicted to initially account for 44% of deformation. The strainincrement plot for this loading condition is very similar in nature to that of the castmaterial denoting a very complex deformation behaviour. When the hoop stress isremoved as in the axial compression case, the the strain increment plot indicatesa simplification of the initial deformation behaviour. According to Figure 6.16f,the region of grains suitably oriented for extension twinning has expanded to in-clude more of the intensity band present in the extruded material. The result is108that twinning is predicted to account for 52.5% of the deformation at the onset ofplasticity, the only case for either material in which basal slip is not predicted to bethe dominant deformation mode.Ultimately however, the accuracy of the VPSC predictions is not as good as forthe cast material. As previously seen in Figure 6.13, there is significant disagree-ment between the VPSC predicted stress levels and the experimentally determinedvalues. Normalization of the iso-work contours, as in Figure 6.14 suggests thatwhile the stresses may not be in perfect agreement at least some of the behaviouris being captured. Conversely, it can be seen that there is intense competition be-tween the various deformation mechanisms which is largely regulated by the CRSSof the mechanisms, the texture of the material (i.e. the presence of suitably orientedgrains) and the stress state. Imperfect prediction of the stress levels therefore couldresult in inaccurate prediction of deformation mechanism activity. The axial ten-sion and 0.4 biaxiality ratio tests for example are expected to induce prismatic slip,despite significantly underestimating the stress levels, it is quite possible thereforethat prismatic slip plays a larger role than anticipated, as could second order pyra-midal slip for that matter which presently is not predicted to contribute.Comparing the strain-strain data for cast and extruded materials in Figure 6.18shows that the two materials behave similarly in axial tension and axial compres-sion, but that for hoop tension and the presented biaxiality ratios, the extrudedmaterial shows increased hoop deformation. This is believed to be resultant fromthe increased intensity of the East and West poles in Figure 6.1b which as previ-ously discussed, indicates increased availability of grains favourably oriented forextension twinning or basal slip in the presence of loading in the hoop direction.Figure 6.19 presents both experimental and VPSC data for the cast and ex-109Figure 6.18: Comparison of strain-strain data for cast and extruded AZ80 al-loy.truded materials for the 0.4 and 0.9 biaxiality tests. Here it can be seen that forthese tests in which there is significant competition between basal slip and exten-sion twinning, the VPSC simulations under predict the level of hoop deformationfor both the cast and extruded materials. For the 0.4 biaxiality case, the VPSCsimulations initially under predict the level of hoop deformation (as with the 0.9biaxiality case), but that this scenario begins to reverse itself as strain levels in-crease. This would suggest that extension twinning is easier to activate than iscurrently being predicted. Easier to activate extension twinning would increasethe level of hoop deformation at low strains and would then decrease in effect as110twinned volume fraction saturates.Figure 6.19: Comparison of cast and extruded material strain-strain data for0.4 and 0.9 biaxiality tests.In Figure 6.20 the VPSC predicted activity of the two most active deformationmodes, basal slip and extension twinning, is presented. It indicates that the activityof the two deformation modes is nearly identical for the two different materials.This is intriguing as it does not account for the greater hoop deformation compo-nent observed in the extruded material in Figure 6.19, which at an axial strain of0.05 sees a hoop strain 13% higher in the extruded material than in the cast materialfor VPSC simulations of both the 0.4 biaxiality test and the 0.9 biaxiality test. It islikely that the discrepancy is due to the fewer grains favourably oriented for axial111deformation.Figure 6.20: Comparison of basal slip and extension twinning activity forcast and extruded materials during 0.4 and 0.9 biaxiality VPSC sim-ulations.Texture EvolutionAs with the cast material, extension twinning has a strong effect on texture evo-lution. The majority of the different final textures presented in Figure 6.16 andhow they evolved from the starting texture in Figure 6.1b is driven by the mannerin which grains undergo extension twinning due to the reorientation of the latticeassociated with this deformation mode.112Axial tension provides a limited number of grains suitably oriented for twin-ning, the result of which is that this loading profile has the least effect on the sam-ple texture. There is a strengthening of the TD band as what grains exist thatare favourably oriented for twinning, do so. In Figure 6.17e it can be seen that thetwinned fraction appears to saturate quickly. The twinned fraction at a plastic workof 5 MJ/m3 is 0.053, but only increases by 0.018 at a plastic work of 10 MJ/m3.Experimentally measured peak intensity rises from 2.3 to 3.9, though develops acentral peak. The reason for the development of this peak is not completely under-stood as it would normally be expected to require a hoop component in order todevelop.The 0.4 biaxiality ratio case does provide a hoop component and the result isa marked increase in the oval peak character of the experimental pole figure inFigure 6.16b. Peak intensity rises to a value of 4.9, higher than the peak intensityof 3.2 observed in the cast material for the same loading condition. The hooptension component causes grains with their c axes aligned in the hoop directionto twin towards the centre, so the peaked nature of the pole figure is consistentwith expectations. The increased intensity as compared with the cast material isresultant from the increased twinning activity which in turn is due to the strongertexture which sees more grains favourably oriented for twinning, and also due tothe higher overall stresses observed which serves to increase the region over whichstresses are adequate to activate twinning. Increasing the hoop component to abiaxiality ratio of 0.9 results in a broad but defined central peak with a maximumintensity of 4 (shown in Figure 6.16c), and less oval in character than that of the0.4 biaxiality case. The entire rim of the pole figure is expected to deform viaextension twinning which explains the nearly circular peak. Peak intensity for the113extruded material is actually somewhat weaker than that of the cast material whichhas a peak intensity of 4.2, but examination of the two figures (Figures 6.16c and6.7c) shows that the peak extruded material has a much larger, high intensity areaaccompanied by a lower minimum intensity value (2.5 versus 3.2 for the extrudedand cast materials respectively) indicating that more twinning has occurred in theextruded material.The hoop tension case is quite interesting as the hoop tension drives grains withc axes parallel to the hoop direction to twin, but the loss of the axial componentmeans that they are no longer driven to only twin towards the centre of the polefigure and thus can twin to any position on the pole figure that is 86.3o away. Theresult is that grains reorient to any position along the vertical axis of the pole figure.As intensity is already strong in the centre of the figure, the final character of thefigure is that of a central peak accompanied by increased intensity along the entireaxis. Intensity at the North and South poles of the figure have increased from astarting value of approximately 0.26 to a value of nearly 1. A similar increasein intensity is seen throughout the ED plane (the vertical axis of the pole figure)indicating that twinning does not appear to result in reorientation to any favouredpositions along the band.Axial compression combined with hoop tension, as in the -0.9 biaxiality caseyields an even more pronounced vertically banded structure in Figure 6.16e. Thisis due to the manner in which twinning occurs in this sample. There is a band overwhich twinning is favourable that coincides with the intensity band present in thematerial. This band flares near the East and West poles of the pole figure whichis parallel to the hoop direction. The hoop tension component of the applied loadcauses grains to twin as in the previous hoop tension case - they can reorient to114any point along the vertical axis of the pole figure, however the axial compressioncomponent drives some further emptying of the central band. The result is thatgrains do not reorient from the hoop direction to the centre of the pole figure, butare free to twin to any other point along the vertical axis of the pole figure. Grainswhich already occupied the centre of the pole figure are driven by the axial com-pression component to twin and reorient to the North and South poles of the polefigure, regions which were originally quite weak. The net effect is the generationof the observed vertically banded structure which features particular intensity at theNorth and South poles. Eliminating the hoop component has the effect of drivingnearly all grains within the starting intensity band to reorient such that their c axesare aligned with the sample?s axial direction. This agrees with the VPSC predic-tion for twinning activity in this sample, which predicts a twin fraction of 50% at aplastic work of 10 MJ/m3.6.3.4 ConclusionThe preceding sections have illustrated the strong tension-compression and tension-tension anisotropy present in AZ80 magnesium alloy. This behaviour has beenexplained as being resultant of combination of the limited deformation modesavailable to magnesium alloys as well as the material?s texture wherein the morestrongly textured extruded material also exhibited more anisotropy than the moreweakly textured cast material. Samples of both materials were subjected to a rangeof uniaxial and biaxial loading tests and neutron diffraction used to measure theirfinal textures. VPSC simulation have helped explain how the starting texture trans-forms into the final observed texture. It has been shown that these textural changesare largely due to extension twinning and that this deformation mode is highly de-115pendent on the availability of suitably oriented grains. Tests in which extensiontwinning can be activated tend to have lower initial flow stresses, but as twinningis exhausted as a deformation mode, work hardening serves to decrease the ob-served anisotropy. The VPSC model used proves itself a valuable tool in termsof facilitating the understanding of the materials behaviour. Agreement with thecast material, for which optimal input parameters were determined by Jain et al.[65] is generally good; however the agreement with the extruded material is muchweaker. It can be seen that currently, VPSC simulations remain a valuable tool forinterpretation, but cannot yet be used in a truly predictive fashion.6.4 Effect of Pre-Strain on Extruded AZ80It was shown previously that the more strongly banded texture typical of the ex-truded material results in a deformation behaviour that sees a relative increase inhoop deformation due to the decreased grains available for extension twinning re-lated to axial deformation and a corresponding increase in those grains availablefor extension twinning associated with hoop deformation. Applying an axial pre-strain to the extruded material can further this trend by forcing what grains areavailable for axial extension twinning to do so prior to the application of a biaxialload. The situation is of course not completely clear cut as it comes with the naturalwork hardening associated with the induced plastic work, but nevertheless, usefulinferences can be made based upon the observed behaviour.6.4.1 Results and DiscussionSamples were initially subjected to axial displacements resulting in strains of 0.02and 0.04. Samples were then unloaded and then reloaded at a biaxiality ratio of 0.9116until failure. The texture predicted by VPSC simulations for the 0.02 and 0.04 arepresented in Figure 6.21. It can be seen that after a pre-strain of 0.02 the texturein Figure 6.21a has somewhat intensified with a peak intensity of 2.7, but has notfully developed the higher intensity band or overall intensity seen in Figure 6.16awhereas the predicted texture for the sample pre-strained to 0.04 features the fullintensity band seen in Figure 6.16a and has a slightly higher peak intensity of 2.8as compared with the predicted intensity at failure of 3.2. This indicates that themajority of twinning due to axial deformation is complete and this hypothesis issupported by the extension twinning activity plot presented in Figure 6.17.(a) 0.02 axial strain (b) 0.04 axial strainFigure 6.21: VPSC predicted textures for extruded AZ80 alloy subjected toaxial strains of 0.02 and 0.04. Increasing strain corresponds with anincrease in the intensity of the banded texture.The stress strain response presented in Figure 6.22 shows that with increasingpre-strains, the point at which the material yields axially increases. With no pre-strain, the 0.9 biaxiality sample, at a plastic work equal to 0.005 axial strain, theaxial stress is at 117 MPa, in the 0.02 prestrain sample this value rises to 135 MPaand then rises further to 152 MPa when the sample is prestrained to 0.04. The117corresponding hoop stresses are 110 MPa, 125 MPa and 135 MPa respectively.Figure 6.22: Axial and hoop stress and strain data for pre strained extrudedAZ80. Axial data is presented with solid lines, hoop data with dashed.The strain ratio progression data presented in Figure 6.23 indicates that thisis only part of the story however, as increasing the level of pre-strain appears tohave a particularly strong effect on the early deformation behaviour and results indeformation which is increasingly dominated by hoop deformation. A 0.005 axialstrain in tension for this material corresponds to a plastic work of 0.84 MJ/m3 andat this level, the strain ratios for the three conditions are 4.1, 4.6 and 7.8 in order ofincreasing pre-strain. This indicates that the effects of axially pre-straining samplesis far higher on the axial direction than on the hoop direction. For example, in orderto achieve an axial plastic strain of 0.005, the axial stress must be 149 MPa, for the0 pre-strain sample and 173 MPa and 199 MPa for the 0.02 and 0.04 pre-strain118samples respectively, increasing by 24 MPa and 26 MPa between each test. In thehoop direction, the same level of plastic strain is achieved at hoop stresses of 94MPa, 116 MPa, and 121 MPa for pre-strains of 0, 0.02 and 0.04 respectively, anincrease of 22 MPa between the 0 and 0.02 pre-strained tests and 5 MPa betweenthe 0.02 and 0.04 tests.Figure 6.23: Strain ratio evolution for pre strained extruded AZ80.It was seen in Figures 6.21a and 6.21b that for the 0.02 pre-strain condition,it is not predicted that the ultimate banded structure is fully developed and thatit is believed that there are still grains suitably oriented for extension twinningrelated to axial tension. In the 0.04 pre-strain case, the texture is far more ma-ture, with the final banded structure nearly fully formed. This indicates that thereshould be fewer grains available for extension twinning related to the axial tension119component of the subsequently applied 0.9 biaxiality loading due to the increas-ing levels of pre-strain. Simultaneously, the pre-straining of samples is actuallyincreasing the number of grains suitably oriented for extension twinning relatedto hoop tension (refer to Figure 6.11c for information on favourable deformationsystems related to orientation for this loading condition). Ultimately, Figures 6.22and 6.23 indicate that the effect of the increasingly strong texture resultant frompre-straining, results in fewer grains suitably oriented for the accommodation ofaxial deformation through extension twinning or basal slip while simultaneouslyincreasing the number of grains suitably oriented for these deformation modes toaccommodate hoop strain.In order to attempt to deconvolute the various mechanisms at play, VPSC simu-lations were run using a 0.9 biaxiality ratio using both the starting texture in Figure6.1a as well as the VPSC predicted texture after an axial strain of 0.04 shown inFigure 6.21b. The VPSC simulation does not account for any of the work harden-ing that has happened, but only looks at any differences in deformation behaviourresultant from the modified texture alone. In Figure 6.24, strain-strain data is pre-sented for the two simulations and it is immediately obvious that the texture re-sultant from pre-straining is expected to exhibit more hoop deformation than thestarting texture.An examination of the predicted twinning activity, shown in Figure 6.25 how-ever shows that the twinning activity and volume fraction are actually expectedto be somewhat lower for the pre-strained texture. The reason for the predictionof greater hoop deformation is unrelated to hoop oriented extension twinning andinstead related to the fact that more grains exist in the regions expected to deformvia basal slip in this particular orientation. However, it must be remembered that120Figure 6.24: Strain ratio evolution (as predicted by VPSC) for extruded AZ80with the measured initial texture as well as the texture predicted for thismaterial after experiencing an axial tensile strain of 0.04.the simulation uses only the predicted texture arising from an axial pre-strain of0.04 and does not account for any of the work hardening associated with that de-formation and therefore a useful avenue of future work would be to attempt toincorporate the effects of work hardening on the basal slip system and investigatethe manner in which that effects the competition between basal slip and extensiontwinning in the hoop direction. It is postulated that the effect would be an increasein the proportion of deformation accommodated via extension twinning during theearly stages of deformation before twinning saturates in that region.121Figure 6.25: VSPC predicted extension twinning activity and volume frac-tion for extruded AZ80 with the measured initial texture as well as thetexture predicted for this material after experiencing an axial tensilestrain of 0.04.Iso-work data is presented in Figure 6.26 for iso-work levels equal to axialtension to 0.005 and 0.03. The plastic work done to the sample during the pre-straining portion of the test is not included in the calculation of these values. Thisfigure indicates that the effect of pre-straining is greater at small strains than itis at higher strains, an observation that would be consistent with the theory thatthe principal effect of pre-straining is an initial texture modification and the corre-sponding effect on both extension twinning and basal slip. As subsequent loadingof the sample takes place, with its own associated texture modifications, the effectof starting texture diminishes.122Figure 6.26: Extruded AZ80 iso-work data showing effect of axial pre-strainsof 0.02 and ConclusionPre-straining samples via axial tensile deformation serves to strengthen the alreadybanded structure of the extruded AZ80 material. The effect of this pre-strainingand texture strengthening is a reduction in the number of grains suitably orientedto accommodate axial tensile deformation through basal slip and particularly ex-tension twinning. The reduction of available easy slip systems is an increase in theobserved stresses in this direction which is greater than that seen in the hoop direc-tion. The reason for this is believed to be that the initial axial deformation increases123the number of grains suitably oriented to accommodate hoop deformation throughbasal slip and extension twinning in this direction. Ultimately, while in the presentwork the effects of axial cold work on hoop deformation cannot be extracted fromthe data, it presents an avenue of study well worthy of future consideration.124Chapter 7Concluding Remarks7.1 Summary of ObservationsIn this work a biaxial test rig has been developed, validated and subsequently usedto explore the behaviour of AZ80 magnesium alloy possessing two different start-ing conditions - a cast material and an extruded material. Neutron diffraction tech-niques and VPSC simulations have been used to facilitate the interpretation of ex-perimental data. This work represents the first time that a rigorous course of ex-periments has been conducted in order to investigate the behaviour of magnesiumalloys subjected to constant biaxial loadings.The key findings of this work are as follows:? This work has confirmed that the combination of tension or compressionwith the internal pressurization of a hollow cylinder provides an attractivetechnique for investigating material response over a wide range of loadingconditions particularly when combined with the proportional control systemdeveloped as part of this work. The combination of independently controlled125axial displacement and internal pressurization offers a level of control notprovided by other biaxial testing techniques and facilitates the careful explo-ration of deformation behaviour. Further, digital image correlation has beenshown to be an excellent tool for investigating the resulting deformation be-haviour.? From this data, experimental iso-work surfaces have been generated for ma-terials with both cast and extruded textures. Despite possessing relativelyweak texture, significant anisotropy was observed in these materials. Theshape of the iso-work surfaces appears to be strongly influenced by thecompetition between basal slip and extension twinning. This is particularlystrong in the more strongly textured extruded material.? Crystallographic texture data was captured for a wide range of tests usingneutron diffraction techniques and it was observed both that there is signifi-cant evolution of the texture during deformation and that it is highly depen-dent on the stress state.? The VPSC model developed by Tome? et al. [34] was employed in conjunc-tion with material parameters developed for the cast AZ80 material by Jainet al. [27] in order to investigate the ability of this model to predict both thestress-strain response and crystallographic texture evolution of these materi-als. VPSC simulations showed generally good results for cast materials, butVPSC parameter adjustment could potentially lead to more accurate captureof the experimental behaviour of the extruded material. In general, the weak-est agreement for both materials was observed in biaxial tension, a region ofstress space in which there is significant competition between basal slip and126extension twinning for these materials. While no changes were made to theparameters of Jain et al. [27], it is likely that parameter adjustment couldimprove the quality of the fit between experimental and VPSC simulationsthrough this region.7.2 Future WorkA number of regions exist in which future work is warranted.? Biaxial testing of magnesium alloys possessing different or stronger textures.The extruded AZ80 examined in this work had relatively weak texture for amagnesium extrusion. The extruded AZ80 shown in Figure 2.5a for examplehas a far stronger texture and would make a fascinating material to study asthe grains available for axial deformation via basal slip or extension twinningare minimal. Additionally, rare-earth alloys such as ZEK100 have displayedinteresting behaviour during sheet tests, in particular, a splitting of their cen-tral peak as seen in Figure 2.5c. The biaxial testing could potentially helpunderstand this phenomenon.? The work presented in Section 6.4 examined the effects of axial pre-strainson a single biaxiality ratio. It would be of interest to test generate a full ex-perimental iso-work surface for a material pre-strained in this manner. Hooptension after an axial pre-strain in particular would be of significant interestas it eliminates most axial extension twinning and therefore would facilitatethe deconvolution of the competition between basal slip and extension twin-ning in this case as the biaxiality ratio studied includes extension twinningnot only in the hoop direction, but also an unknown component in the axial127direction.? With the experimental data collected, the VPSC input parameters could nowbe refined for both materials. The extruded material in particular would ben-efit from this exercise.128References[1] I. Polmear, ?Magnesium alloys and applications,? Materials Science andTechnology, vol. 10, no. 1, pp. 1?16, 1994.[2] NHTSA, ?Obama administration finalizes historic 54.5 mpg fuel efficiencystandards,? 2012.[3] S. McCarthy and G. Keenan, ?Car prices soar as ottawa takes aim atemissions,? The Globe and Mail, 2012.[4] P. G. Partridge, ?The crystallography and deformation modes of hexagonalclose-packed metals,? International Materials Reviews, vol. 12, no. 1,pp. 169?194, 1967.[5] R. Messler, The Essence of Materials for Engineers. Jones & BartlettPubl., 2010.[6] R. Hertzberg, R. Vinci, and J. Hertzberg, Deformation and FractureMechanics of Engineering Materials. Wiley, 2012.[7] D. Askeland, P. Fulay, and W. Wright, The Science and Engineering ofMaterials: Si Edition. Delmar Cengage Learning, 2011.[8] R. von Mises Zeitschrift fr Angewandte Mathematik und Mechanik, vol. 6,p. 85, 1928.[9] T. Mayamal, T. Ohashi, K. Higashidaz, and Y. Kawamura, CrystalPlasticity Analysis on Compressive Loading of Magnesium withSuppression of Twinning. John Wiley & Sons, Inc., 2011.[10] J. Koike, T. Kobayashi, T. Mukai, H. Watanabe, M. Suzuki, K. Maruyama,and K. Higashi, ?The activity of non-basal slip systems and dynamicrecovery at room temperature in fine-grained AZ31B magnesium alloys,?Acta Materialia, vol. 51, no. 7, pp. 2055 ? 2065, 2003.129[11] K. Ma?this, K. Nyilas, A. Axt, I. Dragomir-Cernatescu, T. Unga?r, andP. Luka?c, ?The evolution of non-basal dislocations as a function ofdeformation temperature in pure magnesium determined by x-raydiffraction,? Acta Materialia, vol. 52, no. 10, pp. 2889 ? 2894, 2004.[12] H. Friedrich and B. Mordike, Magnesium Technology: Metallurgy, DesignData, Applications. Springer London, Limited, 2006.[13] W. Hutchinson and M. Barnett, ?Effective values of critical resolved shearstress for slip in polycrystalline magnesium and other hcp metals,? ScriptaMaterialia, vol. 63, no. 7, pp. 737 ? 740, 2010.[14] S. R. Agnew and A?. Duygulu, ?Plastic anisotropy and the role of non-basalslip in magnesium alloy AZ31B,? International Journal of Plasticity,vol. 21, no. 6, pp. 1161 ? 1193, 2005.[15] W. Sheely and R. Nash, ?Mechanical properties of magnesiummonocrystals,? Trans. Metall. Soc. AIME, vol. 218, pp. 416?423, 1960.[16] A. Akhtar and E. Teghtsoonian, ?Solid solution strengthening ofmagnesium single crystalsi alloying behaviour in basal slip,? ActaMetallurgica, vol. 17, no. 11, pp. 1339?1349, 1969.[17] A. Akhtar and E. Teghtsoonian, ?Solid solution strengthening ofmagnesium single crystalsii the effect of solute on the ease of prismaticslip,? Acta Metallurgica, vol. 17, no. 11, pp. 1351?1356, 1969.[18] T. Obara, H. Yoshinga, and S. Morozumi, ?{112?2}< 1?1?23 > slip system inmagnesium,? Acta Metallurgica, vol. 21, no. 7, pp. 845?853, 1973.[19] S. Ando, K. Nakamura, K. Takashima, and H. Tonda, ?{112?2}< 1?1?23 >slip in magnesium single crystal,? Journal of Japan Institute of LightMetals(Japan), vol. 42, no. 12, pp. 765?771, 1992.[20] B. Bhattacharya, Plastic deformation behaviour of pure Mg in thetemperature range 4.2K-300K. PhD thesis, McMaster University, 2006.[21] M. Yoo, ?Slip, twinning and fracture in hexagonal close-packed metals,?Metallurgical and Materials Transactions A, vol. 12, no. 3, pp. 409 ? 418,1981.[22] M. Gharghouri, G. Weatherly, J. Embury, and J. Root, ?Study of themechanical properties of mg-7.7at.% al by in-situ neutron diffraction,?Philosophical Magazine A, vol. 79, pp. 1671 ? 1695, 1999.130[23] D. Brown, S. Agnew, M. Bourke, T. Holden, S. Vogel, and C. Tome?,?Internal strain and texture evolution during deformation twinning inmagnesium,? Materials Science and Engineering: A, vol. 399, no. 1-2,pp. 1 ? 12, 2005.[24] R. Abbaschian, L. Abbaschian, and R. Hill, Physical MetallurgyPrinciples. Cengage Learning, 2008.[25] W. Hosford, Mechanical Behavior of Materials. Cambridge UniversityPress, 2010.[26] L. Wu, A. Jain, D. Brown, G. Stoica, S. Agnew, B. Clausen, D. Fielden,and P. Liaw, ?Twinning-detwinning behavior during the strain-controlledlow-cycle fatigue testing of a wrought magnesium alloy, ZK60A,? ActaMaterialia, vol. 56, no. 4, pp. 688 ? 695, 2008.[27] J. Jain, W. Poole, and C. Sinclair, ?The deformation behaviour of themagnesium alloy AZ80 at 77 and 293k,? Materials Science andEngineering: A, vol. 547, no. 0, pp. 128 ? 137, 2012.[28] R. Gehrmann, M. M. Frommert, and G. Gottstein, ?Texture effects onplastic deformation of magnesium,? Materials Science and Engineering A,vol. 395, no. 1-2, pp. 338 ? 349, 2005.[29] Y. Wang and J. Huang, ?The role of twinning and untwinning in yieldingbehavior in hot-extruded mg-al-zn alloy,? Acta Materialia, vol. 55, no. 3,pp. 897 ? 905, 2007.[30] J. Jain, J. Zou, C. Sinclair, and W. Poole, ?Double tensile twinning in aMg8Al0.5Zn alloy,? Journal of Microscopy, vol. 242, no. 1, pp. 26?36,2011.[31] L. Jiang, J. J. Jonas, A. A. Luo, A. K. Sachdev, and S. Godet,?Twinning-induced softening in polycrystalline am30 mg alloy at moderatetemperatures,? Scripta Materialia, vol. 54, no. 5, pp. 771 ? 775, 2006.[32] J. Jain, ?Ph.d. proposal.?.[33] J. Wang, J. Hirth, and C. Tome, ?Twinning nucleation mechanisms inhexagonal-close-packed crystals,? Acta Materialia, vol. 57, no. 18,pp. 5521 ? 5530, 2009.[34] R. Lebensohn and C. Tome?, ?A self-consistent anisotropic approach for thesimulation of plastic deformation and texture development of polycrystals:131Application to zirconium alloys,? Acta Metallurgica et Materialia, vol. 41,no. 9, pp. 2611 ? 2624, 1993.[35] R. Lebensohn and C. Tome?, ?A self-consistent viscoplastic model:prediction of rolling textures of anisotropic polycrystals,? MaterialsScience and Engineering: A, vol. 175, no. 12, pp. 71 ? 82, 1994.[36] L. Capolungo, I. Beyerlein, and C. Tome?, ?Slip-assisted twin growth inhexagonal close-packed metals,? Scripta Materialia, vol. 60, no. 1, pp. 32 ?35, 2009.[37] A. Serra and D. J. Bacon, ?A new model for 1012 twin growth in hcpmetals,? Philosophical Magazine A, vol. 73, no. 2, pp. 333?343, 1996.[38] S. Song and G. G. III, ?Structural interpretation of the nucleation andgrowth of deformation twins in zr and tii. application of the coincidencesite lattice (csl) theory to twinning problems in h.c.p. structures,? ActaMetallurgica et Materialia, vol. 43, no. 6, pp. 2325 ? 2337, 1995.[39] J. Wang, I. Beyerlein, N. Mara, A. Misra, and C. Tome, Deformationtwinning mechanisms in FCC and HCP metals, pp. 88?91. DestechPublications Incorporated, 2011.[40] J. Bohlen, M. R. Nu?rnberg, J. W. Senn, D. Letzig, and S. R. Agnew, ?Thetexture and anisotropy of magnesium?zinc?rare earth alloy sheets,? ActaMaterialia, vol. 55, no. 6, pp. 2101?2112, 2007.[41] T. Al-Samman and G. Gottstein, ?Room temperature formability of amagnesium AZ31 alloy: Examining the role of texture on the deformationmechanisms,? Materials Science and Engineering: A, vol. 488, no. 1-2,pp. 406 ? 414, 2008.[42] S. Yi, S. Zaefferer, and H.-G. Brokmeier, ?Mechanical behaviour andmicrostructural evolution of magnesium alloy AZ31 in tension at differenttemperatures,? Materials Science and Engineering: A, vol. 424, no. 1-2,pp. 275 ? 281, 2006.[43] M. Barnett, ?Twinning and the ductility of magnesium alloys: Part ii.?contraction? twins,? Materials Science and Engineering: A, vol. 464,no. 1-2, pp. 8 ? 16, 2007.[44] M. Barnett, M. Nave, and C. Bettles, ?Deformation microstructures andtextures of some cold rolled mg alloys,? Materials Science andEngineering A, vol. 386, no. 1-2, pp. 205 ? 211, 2004.132[45] M. Barnett, Z. Keshavarz, A. Beer, and D. Atwell, ?Influence of grain sizeon the compressive deformation of wrought mg-3al-1zn,? Acta Materialia,vol. 52, no. 17, pp. 5093 ? 5103, 2004.[46] B. Beausir, S. Suwas, L. S. Tth, K. W. Neale, and J.-J. Fundenberger,?Analysis of texture evolution in magnesium during equal channel angularextrusion,? Acta Materialia, vol. 56, no. 2, pp. 200 ? 214, 2008.[47] S. R. Agnew, J. A. Horton, T. M. Lillo, and D. W. Brown, ?Enhancedductility in strongly textured magnesium produced by equal channelangular processing,? Scripta Materialia, vol. 50, no. 3, pp. 377 ? 381, 2004.[48] S. R. Agnew, C. N. Tome?, D. W. Brown, T. M. Holden, and S. C. Vogel,?Study of slip mechanisms in a magnesium alloy by neutron diffraction andmodeling,? Scripta Materialia, vol. 48, no. 8, pp. 1003 ? 1008, 2003.[49] S. Agnew, D. Brown, and C. Tome?, ?Validating a polycrystal model for theelastoplastic response of magnesium alloy AZ31 using in situ neutrondiffraction,? Acta Materialia, vol. 54, no. 18, pp. 4841 ? 4852, 2006.[50] S.-B. Yi, C. Davies, H.-G. Brokmeier, R. Bolmaro, K. Kainer, andJ. Homeyer, ?Deformation and texture evolution in AZ31 magnesium alloyduring uniaxial loading,? Acta Materialia, vol. 54, no. 2, pp. 549 ? 562,2006.[51] J. Bohlen, S. Yi, J. Swiostek, D. Letzig, H. Brokmeier, and K. Kainer,?Microstructure and texture development during hydrostatic extrusion ofmagnesium alloy AZ31,? Scripta Materialia, vol. 53, no. 2, pp. 259 ? 264,2005.[52] T. Liu, Y. Wang, S. Wu, R. L. Peng, C. Huang, C. Jiang, and S. Li,?Textures and mechanical behavior of mg-3.3%li alloy after ecap,? ScriptaMaterialia, vol. 51, no. 11, pp. 1057 ? 1061, 2004.[53] R. Abbaschian, L. Abbaschian, and R. Hill, Physical MetallurgyPrinciples. Cengage Learning, 2008.[54] L. Priester, Grain Boundaries and Crystalline Plasticity. Wiley, 2013.[55] T. Mukai, M. Yamanoi, H. Watanabe, and K. Higashi, ?Ductilityenhancement in AZ31 magnesium alloy by controlling its grain structure,?Scripta Materialia, vol. 45, no. 1, pp. 89 ? 94, 2001.133[56] S. Kleiner and P. Uggowitzer, ?Mechanical anisotropy of extruded Mg-6Znalloy,? Materials Science and Engineering: A, vol. 379, no. 1-2, pp. 258 ?263, 2004.[57] M. Barnett, ?Twinning and the ductility of magnesium alloys: Part i:Tension twins,? Materials Science and Engineering: A, vol. 464, no. 1-2,pp. 1 ? 7, 2007.[58] J. Jain, W. Poole, C. Sinclair, and M. Gharghouri, ?Reducing thetension-compression yield asymmetry in a mg-8al-0.5zn alloy viaprecipitation,? Scripta Materialia, vol. 62, no. 5, pp. 301 ? 304, 2010.[59] J. Bohlen, P. Dobron?, J. Swiostek, D. Letzig, F. Chmel??k, P. Luka?c?, andK. U. Kainer, ?On the influence of the grain size and solute content on theae response of magnesium alloys tested in tension and compression,?Materials Science and Engineering: A, vol. 462, no. 1, pp. 302?306, 2007.[60] B. Han and D. Dunand, ?Microstructure and mechanical properties ofmagnesium containing high volume fractions of yttria dispersoids,?Materials Science and Engineering: A, vol. 277, no. 12, pp. 297 ? 304,2000.[61] D. L. Yin, J. T. Wang, J. Q. Liu, and X. Zhao, ?On tension?compressionyield asymmetry in an extruded mg?3al?1zn alloy,? Journal of Alloys andCompounds, vol. 478, no. 1, pp. 789?795, 2009.[62] M.-G. Lee, R. Wagoner, J. Lee, K. Chung, and H. Kim, ?Constitutivemodeling for anisotropic/asymmetric hardening behavior of magnesiumalloy sheets,? International Journal of Plasticity, vol. 24, no. 4,pp. 545?582, 2008.[63] S. Graff, W. Brocks, and D. Steglich, ?Yielding of magnesium: From singlecrystal to polycrystalline aggregates,? International Journal of Plasticity,vol. 23, no. 12, pp. 1957?1978, 2007.[64] E. Ball and P. Prangnell, ?Tensile-compressive yield asymmetries in highstrength wrought magnesium alloys,? Scripta Metallurgica et Materialia,vol. 31, no. 2, 1994.[65] A. Jain and S. Agnew, ?Modeling the temperature dependent effect oftwinning on the behavior of magnesium alloy AZ31B sheet,? MaterialsScience and Engineering: A, vol. 462, no. 12, pp. 29 ? 36, 2007.134[66] S. Agnew, M. Yoo, and C. Tome?, ?Application of texture simulation tounderstanding mechanical behavior of mg and solid solution alloyscontaining li or y,? Acta Materialia, vol. 49, no. 20, pp. 4277 ? 4289, 2001.[67] W. Hosford and R. Caddell, Metal Forming: Mechanics and Metallurgy.Metal Forming: Mechanics and Metallurgy, Cambridge University Press,2011.[68] H. L. Kim, W. K. Bang, and Y. W. Chang, ?Effect of initial texture ondeformation behavior of AZ31 magnesium alloy sheets under biaxialloading,? Materials Science and Engineering: A, vol. 552, pp. 245 ? 251,2012.[69] J. Weiler, The development of comprehensive material models of thestructure-property relationship for die-cast magnesium alloy AM60B. PhDthesis, University of Western Ontario, 2009.[70] S. H. Safi-Naqvi, W. B. Hutchinson, and M. R. Barnett, ?Texture andmechanical anisotropy in three extruded magnesium alloys,? MaterialsScience and Technology, vol. 24, no. 10, pp. 1283?1292, 2008.[71] D. Steglich, Y. Jeong, M. Andar, and T. Kuwabara, ?Biaxial deformationbehaviour of AZ31 magnesium alloy: Crystal-plasticity-based predictionand experimental validation,? International Journal of Solids andStructures, 2012.[72] Y. Chun and C. Davies, ?Twinning-induced anomaly in the yield surface ofhighly textured mg-3al-1zn plate,? Scripta Materialia, vol. 64, no. 10,pp. 958 ? 961, 2011.[73] L. Jiang, J. Jonas, R. Mishra, A. Luo, A. Sachdev, and S. Godet, ?Twinningand texture development in two mg alloys subjected to loading along threedifferent strain paths,? Acta Materialia, vol. 55, no. 11, pp. 3899 ? 3910,2007.[74] J. Le?vesque, K. Inal, K. Neale, and R. Mishra, ?Numerical modeling offormability of extruded magnesium alloy tubes,? International Journal ofPlasticity, vol. 26, no. 1, pp. 65?83, 2010.[75] R. Jones, Deformation Theory of Plasticity. Bull Ridge Publishing, 2009.[76] A. Singh, Mechanics Of Solids. Prentice-Hall Of India Pvt. Limited, 2007.[77] J. Davis, Tensile Testing. ASM International, 2004.135[78] B. Punmia and A. Jain, Mechanics of Materials. Laxmi Publications PvtLimited, 2002.[79] R. Tilley, Understanding Solids: The Science of Materials. Wiley, 2004.[80] H. Vegter and A. Van den Boogaard, ?A plane stress yield function foranisotropic sheet material by interpolation of biaxial stress states,?International journal of plasticity, vol. 22, no. 3, pp. 557?580, 2006.[81] R. Hill and J. Hutchinson, ?Differential hardening in sheet metal underbiaxial loading: a theoretical framework,? J. Appl. Mech, vol. 59,pp. S1?S9, 1992.[82] R. Hill, S. Hecker, and M. Stout, ?An investigation of plastic flow anddifferential work hardening in orthotropic brass tubes under fluid pressureand axial load,? International journal of solids and structures, vol. 31,no. 21, pp. 2999?3021, 1994.[83] A. S. Khan, R. Kazmi, A. Pandey, and T. Stoughton, ?Evolution ofsubsequent yield surfaces and elastic constants with finite plasticdeformation. Part-I: A very low work hardening aluminum alloy(Al6061-T6511),? International Journal of Plasticity, vol. 25, no. 9,pp. 1611 ? 1625, 2009.[84] A. S. Khan, A. Pandey, and T. Stoughton, ?Evolution of subsequent yieldsurfaces and elastic constants with finite plastic deformation. Part II: Avery high work hardening aluminum alloy (annealed 1100 Al),?International Journal of Plasticity, vol. 26, no. 10, pp. 1421 ? 1431, 2010.[85] A. S. Khan, A. Pandey, and T. Stoughton, ?Evolution of subsequent yieldsurfaces and elastic constants with finite plastic deformation. Part III: Yieldsurface in tension-tension stress space (Al 6061-T6511 and annealed 1100Al),? International Journal of Plasticity, vol. 26, no. 10, pp. 1432 ? 1441,2010.[86] M. Teaca, I. Charpentier, M. Martiny, and G. Ferron, ?Identification ofsheet metal plastic anisotropy using heterogeneous biaxial tensile tests,?International Journal of Mechanical Sciences, vol. 52, no. 4, pp. 572 ? 580,2010.[87] M. O. Andar, T. Kuwabara, and D. Steglich, ?Material modeling of az31mg sheet considering variation of? i? r?/i?-values and asymmetry of theyield locus,? Materials Science and Engineering: A, vol. 549, pp. 82?92,2012.136[88] H. Tresca, ?On the yield of solids at high pressure,? Comptes RendusAcademie Science, vol. 59, p. 754, 1864.[89] D. Franc?ois, A. Pineau, and A. Zaoui, Mechanical Behaviour of Materials:Volume I: Elasticity and Plasticity. Solid Mechanics and Its ApplicationsSeries, Kluwer Academic Pub, 1998.[90] R. Smallman and A. Ngan, Physical Metallurgy and Advanced Materials.Elsevier Science, 2011.[91] R. von Mises, ?Mechanik der festen ko?rper im plastisch deformablenzustand,? Go?ttin. Nachr. Math. Phys, vol. 1, pp. 582?592, 1913.[92] T. Chung, Continuum Mechanics. General Continuum Mechanics,Cambridge University Press, 2007.[93] O. Cazacu and F. Barlat, ?A criterion for description of anisotropy andyield differential effects in pressure-insensitive metals,? InternationalJournal of Plasticity, vol. 20, no. 11, pp. 2027 ? 2045, 2004. Daniel C.Drucker Memorial Issue.[94] W. Backofen, Deformation Processing. United States of America:Addison-Wesley Publishing Company Inc., 1972.[95] O. Cazacu, B. Plunkett, and F. Barlat, ?Orthotropic yield criterion forhexagonal closed packed metals,? International Journal of Plasticity,vol. 22, no. 7, pp. 1171 ? 1194, 2006.[96] B. Plunkett, R. Lebensohn, O. Cazacu, and F. Barlat, ?Anisotropic yieldfunction of hexagonal materials taking into account texture developmentand anisotropic hardening,? Acta Materialia, vol. 54, no. 16, pp. 4159 ?4169, 2006.[97] B. Plunkett, O. Cazacu, R. Lebensohn, and F. Barlat, ?Elastic-viscoplasticanisotropic modeling of textured metals and validation using the taylorcylinder impact test,? International Journal of Plasticity, vol. 23, no. 6,pp. 1001 ? 1021, 2007.[98] B. Plunkett, O. Cazacu, and F. Barlat, ?Orthotropic yield criteria fordescription of the anisotropy in tension and compression of sheet metals,?International Journal of Plasticity, vol. 24, no. 5, pp. 847?866, 2008.[99] S. Mahabunphachai and M. Koc?, ?Investigation of size effects on materialbehavior of thin sheet metals using hydraulic bulge testing at137micro/meso-scales,? International Journal of Machine Tools andManufacture, vol. 48, no. 9, pp. 1014?1029, 2008.[100] T. Sokolowski, K. Gerke, M. Ahmetoglu, and T. Altan, ?Evaluation of tubeformability and material characteristics: hydraulic bulge testing of tubes,?Journal of Materials Processing Technology, vol. 98, no. 1, pp. 34?40,2000.[101] G. Gutscher, H.-C. Wu, G. Ngaile, and T. Altan, ?Determination of flowstress for sheet metal forming using the viscous pressure bulge (vpb) test,?Journal of Materials Processing Technology, vol. 146, no. 1, pp. 1?7, 2004.[102] A. Ranta-Eskola, ?Use of the hydraulic bulge test in biaxial tensile testing,?International Journal of Mechanical Sciences, vol. 21, no. 8, pp. 457?465,1979.[103] D. Green, K. Neale, S. MacEwen, A. Makinde, and R. Perrin,?Experimental investigation of the biaxial behaviour of an aluminumsheet,? International Journal of Plasticity, vol. 20, no. 8, pp. 1677?1706,2004.[104] A. Makinde, L. Thibodeau, and K. Neale, ?Development of an apparatusfor biaxial testing using cruciform specimens,? Journal of ExperimentalMechanics, vol. 32, no. 2, pp. 138?144, 1992.[105] D. N. Fang, W. Lu, W. Y. Yan, T. Inoue, and K. C. Hwang, ?Stress-strainrelation of CuAlNi SMA single crystal under biaxial loading?constitutivemodel and experiments,? Acta Materialia, vol. 47, no. 1, pp. 269 ? 280,1998.[106] E. Shiratori and K. Ikegami, ?Experimental study of the subsequent yieldsurface by using cross-shaped specimens,? Journal of the Mechanics andPhysics of Solids, vol. 16, no. 6, pp. 373?394, 1968.[107] A. Hannon and P. Tiernan, ?A review of planar biaxial tensile test systemsfor sheet metal,? Journal of Materials Processing Technology, vol. 198,no. 1-3, pp. 1 ? 13, 2008.[108] A. Smits, D. Van Hemelrijck, T. Philippidis, and A. Cardon, ?Design of acruciform specimen for biaxial testing of fibre reinforced compositelaminates,? Composites science and technology, vol. 66, no. 7,pp. 964?975, 2006.138[109] D. Lecompte, A. Smits, H. Sol, J. Vantomme, and D. Van Hemelrijck,?Mixed numerical?experimental technique for orthotropic parameteridentification using biaxial tensile tests on cruciform specimens,?International Journal of Solids and Structures, vol. 44, no. 5,pp. 1643?1656, 2007.[110] E. Hoferlin, A. Van Bael, P. Van Houtte, G. Steyaert, and C. De Mare?, ?Thedesign of a biaxial tensile test and its use for the validation ofcrystallographic yield loci,? Modelling and simulation in materials scienceand engineering, vol. 8, no. 4, p. 423, 2000.[111] I. Barsoum and J. Faleskog, ?Rupture mechanisms in combined tension andshearexperiments,? International Journal of Solids and Structures, vol. 44,no. 6, pp. 1768?1786, 2007.[112] H. Cezayirlioglu, E. Bahniuk, D. Davy, and K. Heiple, ?Anisotropic yieldbehavior of bone under combined axial force and torque,? Journal ofbiomechanics, vol. 18, no. 1, pp. 61?69, 1985.[113] T. Marin, P. Dawson, M. Gharghouri, and R. Rogge, ?Diffractionmeasurements of elastic strains in stainless steel subjected to in situ biaxialloading,? Acta Materialia, vol. 56, no. 16, pp. 4183?4199, 2008.[114] J. Den Hartog, Strength of Materials. Dover books on engineering andengineering physics, Dover Publ., 1961.[115] A. Samuel and J. Weir, Introduction to Engineering Design. ElsevierScience, 1999.[116] A. Beaudoin, P. Dawson, K. Mathur, and U. Kocks, ?A hybrid finiteelement formulation for polycrystal plasticity with consideration ofmacrostructural and microstructural linking,? International Journal ofPlasticity, vol. 11, no. 5, pp. 501?521, 1995.[117] K. K. Mathur and P. R. Dawson, ?On modeling the development ofcrystallographic texture in bulk forming processes,? International Journalof Plasticity, vol. 5, no. 1, pp. 67?94, 1989.[118] D. P. Mika and P. R. Dawson, ?Effects of grain interaction on deformationin polycrystals,? Materials Science and Engineering: A, vol. 257, no. 1,pp. 62?76, 1998.139[119] S.-Y. Huang, S.-R. Zhang, D.-Y. Li, and Y.-H. Peng, ?Simulation of textureevolution during plastic deformation of fcc, bcc and hcp structured crystalswith crystal plasticity based finite element method,? Transactions ofNonferrous Metals Society of China, vol. 21, no. 8, pp. 1817?1825, 2011.[120] R. Quey, P. Dawson, and F. Barbe, ?Large-scale 3d random polycrystals forthe finite element method: Generation, meshing and remeshing,? ComputerMethods in Applied Mechanics and Engineering, vol. 200, no. 17,pp. 1729?1745, 2011.[121] J. D. Eshelby, ?The determination of the elastic field of an ellipsoidalinclusion, and related problems,? Proceedings of the Royal Society ofLondon. Series A. Mathematical and Physical Sciences, vol. 241, no. 1226,pp. 376?396, 1957.[122] G. Proust, C. N. Tom, A. Jain, and S. R. Agnew, ?Modeling the effect oftwinning and detwinning during strain-path changes of magnesium alloyAZ31,? International Journal of Plasticity, vol. 25, no. 5, pp. 861 ? 880,2009.[123] B. Hutchinson, J. Jain, and M. Barnett, ?A minimum parameter approach tocrystal plasticity modelling,? Acta Materialia, vol. 60, no. 15, pp. 5391 ?5398, 2012.[124] S.-H. Choi, E. Shin, and B. Seong, ?Simulation of deformation twins anddeformation texture in an AZ31 Mg alloy under uniaxial compression,?Acta Materialia, vol. 55, no. 12, pp. 4181 ? 4192, 2007.[125] S. Choi, D. Kim, and B. Seong, ?Simulation of strain-softening behaviorsin an AZ31 Mg alloy showing distinct twin-induced reorientation before apeak stress,? Metals and Materials International, vol. 15, pp. 239?248,2009.[126] J. Jain, W. Poole, and C. Sinclair, ?Study of deformation modes in AZ80magnesium alloy,? Magnesium Technology 2008, pp. 9?13, 2008.[127] V. Micro-Measurements, Instruction Bulletin B-127-14: Strain GaugeInstallation with M-Bond 200 Adhesive, 2005.[128] F. Bachmann, R. Hielscher, and H. Schaeben, ?Texture analysis withmtex?free and open source software toolbox,? Solid State Phenomena,vol. 160, pp. 63?68, 2010.140[129] S. Hai and E. Tadmor, ?Deformation twinning at aluminum crack tips,?Acta Materialia, vol. 51, no. 1, pp. 117 ? 131, 2003.[130] C. Corby, C. Ca?ceres, and P. Luka?c?, ?Serrated flow in magnesium alloyAZ91,? Materials Science and Engineering: A, vol. 387, pp. 22?24, 2004.[131] D. Wu, R. Chen, and E. Han, ?Serrated flow and tensile properties of amggdzn alloy,? Materials Science and Engineering: A, vol. 532, no. 0,pp. 267 ? 274, 2012.[132] C. Wang, Y. Xu, and E. Han, ?Serrated flow and abnormal strain ratesensitivity of a magnesium-lithium alloy,? Materials Letters, vol. 60,no. 24, pp. 2941 ? 2944, 2006.[133] Z. Trojanova and C. Caceres, ?On the strain to the onset of serrated flow ina magnesium alloy,? Scripta Materialia, vol. 56, no. 9, pp. 793 ? 796, 2007.[134] P. McCormick, ?Theory of flow localisation due to dynamic strain ageing,?Acta Metallurgica, vol. 36, no. 12, pp. 3061 ? 3067, 1988.[135] P. McCormick and Y. Estrin, ?Transient flow behaviour associated withdynamic strain ageing,? Scripta Metallurgica, vol. 23, no. 7, pp. 1231 ?1234, 1989.[136] K. Prasad and S. Kamat, ?Transient flow behaviour in a near alpha titaniumalloy timetal 834 in the dynamic strain aging regime,? Materials Scienceand Engineering: A, vol. 490, no. 12, pp. 477 ? 480, 2008.[137] A. Phillips and R. L. Sierakowski, ?On the concept of the yield surface,?Acta Mechanica, vol. 1, pp. 29?35, 1965.[138] S. Agnew, ?Plastic anisotropy of magnesium alloy AZ31B sheet,?Magnesium Technology, vol. 169, p. 174, 2002.[139] A. Levinson, R. K. Mishra, R. D. Doherty, and S. R. Kalidindi,?Microstructure evolution during roller hemming of AZ31B magnesiumsheet,? Metallurgical and Materials Transactions A, pp. 1?10, 2012.[140] J. J. Jonas, S. Mu, T. Al-Samman, G. Gottstein, L. Jiang, and E?. Martin,?The role of strain accommodation during the variant selection of primarytwins in magnesium,? Acta Materialia, vol. 59, no. 5, pp. 2046?2056, 2011.[141] M. Knezevic, A. Levinson, R. Harris, R. K. Mishra, R. D. Doherty, andS. R. Kalidindi, ?Deformation twinning in AZ31: Influence on strain141hardening and texture evolution,? Acta Materialia, vol. 58, no. 19,pp. 6230?6242, 2010.142


Citation Scheme:


Citations by CSL (citeproc-js)

Usage Statistics



Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            async >
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:


Related Items