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Mathematical modeling of the fatigue life following rim indentation test in aluminum alloy wheels Bhatnagar, Mohit 2010

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Mathematical modeling of the fatigue life following rim indentation test in aluminum alloy wheelsbyMohit BhatnagarA THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIES(Mechanical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)January 2010© Mohit Bhatnagar, 2010iiAbstractThe volume of components manufactured from cast aluminum alloys in both the aerospaceand automotive industries has seen a tremendous rise in the past 10 years primarily due totheir high strength to weight ratio. In the case of the automotive industry, cast wheels alsoprovide aesthetically appealing designs for the consumers, which is another factor for theirincreased demand. The cyclic nature of in-service loading in some applications makesfatigue performance a key design consideration. Manufacturers use standardized testingprocedures (SAE J328) to assess the fatigue life of wheels for regular driving or corneringconditions. Recently, a leading North American wheel manufacturing company has begantesting the radial fatigue life of wheels following rim indentation. The development of thistest has highlighted the need for a tool to support the wheel design process that is capableof predicting the fatigue life of cast components following permanent deformation.   A considerable amount of research work has already been conducted on establishing theeffects of microstructure and casting defects on the fatigue life of cast aluminum alloyA356.  However, there has been limited work published on the effects of  initial plasticdeformation on the fatigue life of cast alloy A356.  Thus, the current research project aimsto quantify the effects of initial plastic strain on the fatigue life of automotive wheels man-ufactured from A356. Specifically, the research aims to enhance the understanding of theimpact of rim indentation on the fatigue life of a wheel. A finite element model was developed to predict the deformation occurring during appli-cation of a static load during the rim indentation test. The residual stress distribution occur-ring after the T6 heat treatment process was used as an input to the rim indentation model.The model was then extended to calculate the stress state of the wheel under radial fatiguetest conditions. Lab-scale experiments were performed to characterize the fatigue behaviorof alloy A356 following different amounts of pre-strain (plastic strain). This data was usedto develop an empirical relationship to relate the fatigue life to initial plastic strain and thecyclic stress state. Industrial-scale rim indentation and radial fatigue tests were used to val-idate the overall model predictions.iiiTable of ContentsAbstract..........................................................................................................................iiTable of Contents..........................................................................................................iiiList of Tables.................................................................................................................viList of Figures..............................................................................................................vii  Acknowledgements........................................................................................................xDedication.....................................................................................................................xiCHAPTER 1. Introduction..................................................................................... 11.1. Automotive wheels ..................................................................................... 11.2. Wheel fatigue testing .................................................................................. 21.3. Design tools ................................................................................................ 2CHAPTER 2. Literature Review ........................................................................... 42.1. Fatigue life of cast aluminum alloys........................................................... 42.1.1. Micro-structural features .......................................................................... 62.1.1.1. Secondary dendrite arm spacing (SDAS) .............................................. 82.1.1.2. Silicon eutectic particles ........................................................................ 92.1.1.3. Fe-intermetallics .................................................................................... 92.1.2. Defects.................................................................................................... 112.1.2.1. Oxide films .......................................................................................... 112.1.2.2. Micro-porosity ..................................................................................... 132.2. Fatigue modeling ...................................................................................... 152.3. Fatigue life of pre-strained aluminum alloys............................................ 202.4. Wheel testing ............................................................................................ 212.4.1. Radial fatigue test................................................................................... 212.4.2. Cornering test ......................................................................................... 232.5. Radial fatigue modeling............................................................................ 24CHAPTER 3. Scope and Objectives .................................................................... 283.1. Scope of research work............................................................................. 283.2. Objectives ................................................................................................. 30ivCHAPTER 4. Experimental Work ...................................................................... 314.1. Industrial scale experiments ..................................................................... 314.1.1. Rim indentation test ............................................................................... 314.1.1.1. Test setup ............................................................................................. 324.1.1.2. Test results ........................................................................................... 334.1.2. Radial fatigue test................................................................................... 354.1.2.1. Test preparation ................................................................................... 354.1.2.2. Test procedure...................................................................................... 384.1.2.3. Test results ........................................................................................... 404.2. Laboratory-scale experiments................................................................... 444.2.1. Tensile test.............................................................................................. 444.2.1.1. Tensile test set up................................................................................. 444.2.1.2. Test procedure...................................................................................... 464.2.1.3. Results.................................................................................................. 464.2.2. Fatigue life experiments......................................................................... 484.2.2.1. Fatigue sample and test setup .............................................................. 484.2.2.2. Pre-strain levels and technique ............................................................ 524.2.2.3. S-N data ............................................................................................... 534.2.2.4. Modified fatigue life equation ............................................................. 554.3. Summary................................................................................................... 62CHAPTER 5. Model Development ...................................................................... 635.1. Rim indentation model ............................................................................. 635.1.1. Geometry................................................................................................ 635.1.2. Material properties ................................................................................. 655.1.3. Initial conditions..................................................................................... 665.1.4. Boundary conditions .............................................................................. 665.1.5. Loading description................................................................................ 675.1.6. Contact definition................................................................................... 685.2. Radial fatigue test model .......................................................................... 705.2.1. Geometry................................................................................................ 705.2.2. Material properties ................................................................................. 705.2.3. Initial conditions..................................................................................... 705.2.4. Boundary conditions .............................................................................. 715.2.5. Loading description................................................................................ 71CHAPTER 6. Results and Discussion.................................................................. 756.1. Verification of the rim indentation finite element model ......................... 756.1.1. Contact simulation.................................................................................. 796.1.2. Sensitivity analysis................................................................................. 826.1.2.1. Effect of flow stress of aluminum alloy A356..................................... 826.1.2.2. Effect of change in Young’s modulus ................................................. 836.1.2.3. Numerical sensitivity ........................................................................... 876.2. Verification of the radial fatigue test model............................................. 89v6.2.1. Sensitivity analysis................................................................................. 956.2.1.1. Effect of change in Young’s modulus ................................................. 956.2.2. Summary ................................................................................................ 966.3. Fatigue life prediction............................................................................... 976.3.1. Thermal model ....................................................................................... 986.3.2. Residual stress model........................................................................... 1006.3.3. Summary .............................................................................................. 1046.3.4. Rim indentation model including residual stress ................................. 1056.3.5. Radial fatigue model following rim indentation .................................. 1086.3.6. Numerical fatigue life........................................................................... 1126.3.7. Verification........................................................................................... 1126.3.8. Sensitivity............................................................................................. 1146.3.8.1. Variation in rim indentation load....................................................... 1146.3.8.2. Variation in radial fatigue load .......................................................... 1176.3.8.3. Variation in the empirical parameters................................................ 1206.3.8.4. Variation in material properties ......................................................... 123CHAPTER 7. Conclusions and Future Work................................................... 1247.1. Conclusions ............................................................................................ 1247.1.1. Rim indentation model......................................................................... 1247.1.2. Radial fatigue test model...................................................................... 1257.1.3. Empirical fatigue life relation .............................................................. 1257.1.4. Combination of models ........................................................................ 1267.2. Future work............................................................................................. 1277.2.1. New rim indentation test ...................................................................... 1277.2.2. Modified fatigue life equation.............................................................. 1287.2.3. Multi-axial fatigue loading................................................................... 129References..... ........................................................................................................... 130Appendix A................................ ............................................................................... 133Appendix B................................ ............................................................................... 139viList of TablesTable 4-1. Gauge specifications......................................................................... 36Table 5-1. Flow stress A356 aluminum alloy.................................................... 65Table 6-1. Comparison of deformation.............................................................. 76Table 6-2. Comparison of rim indentation model results for different flow stress......................................................................................... 83Table 6-3. Comparison of rim deformation sensitivity analysis for variation in Young’s modulus .............................................................................. 85Table 6-4. Comparison of displacement for the rim indentation test .............. 106Table 6-5. Change in total fatigue life with parameter C2 .............................. 120Table 6-6. Change in total fatigue life with parameter w ................................ 121viiList of FiguresFigure 2-1. Fatigue life curve [9]..........................................................................  5Figure 2-3. Modified microstructure of A356 [11]...............................................  7Figure 2-2. Unmodified microstructure of A356 [11] ..........................................  7Figure 2-4. Effect of SDAS on fatigue life [11] ...................................................  8Figure 2-5. Fatigue life A356 with varying Fe-content [14] ..............................  10Figure 2-7. New oxide film [18].........................................................................  12Figure 2-6. Old oxide film [18] ..........................................................................  12Figure 2-8. A typical cast A356 pore [8]............................................................  13Figure 2-9. Finite element generated stress field around a pore [8] ...................  14Figure 2-11. Fatigue life comparison for HIPed A356 [3] ...................................  19Figure 2-10. Fatigue life comparison for non-HIPed A356 [3]............................  19Figure 2-12. Radial fatigue test setup [27] ...........................................................  22Figure 2-13. Cornering fatigue test setup [28] .....................................................  23Figure 2-14. Tire under a vertical load [31]..........................................................  24Figure 2-15. Axisymmetric model of wheel [29] .................................................  25Figure 2-16. Radial load schematic at the bead area [29] ....................................  26Figure 3-1. Research methodology.....................................................................  29Figure 4-1. Sintech universal test machine.........................................................  32Figure 4-2. Rim indentation graphs....................................................................  34Figure 4-3. Strain gauge locations ......................................................................  36Figure 4-5. Strain gauges at location 5, inboard rim ..........................................  37Figure 4-4. Strain gauges at locations 1 to 4 ......................................................  37Figure 4-6. Strain gauges at location 6, spoke....................................................  38Figure 4-7. Radial fatigue test setup...................................................................  39Figure 4-8. Strain graphs for gauges 1 and 4......................................................  40Figure 4-9. Strain graph for gauges 7 and 10 .....................................................  41Figure 4-10. Strain graph for gauge 13.................................................................  42Figure 4-11. Strain graph for gauge 16.................................................................  43Figure 4-12. Tensile test specimen .......................................................................  45Figure 4-13. Instron 8874 servomechanical test platform....................................  45Figure 4-14. Flow curves alloy A356...................................................................  47Figure 4-16. Sonntag fatigue machine..................................................................  49Figure 4-15. Fatigue specimen .............................................................................  49Figure 4-17. Sonntag fatigue machine assembly..................................................  50Figure 4-18. Instrumentation on fatigue machine ................................................  51Figure 4-19. Axisymmetric model of fatigue sample...........................................  53viiiFigure 4-20. Fatigue life data ...............................................................................  54Figure 4-21. Trendlines- fatigue life.....................................................................  56Figure 4-22. Effect of pre-strain on fatigue life....................................................  57Figure 4-23. SEM picture of a pore in a fatigue sample.......................................  58Figure 4-24. Fatigue life for samples with 0, 5% and 10% pre-strain..................  60Figure 4-25. Variation of constants with pre-strain ..............................................  61Figure 5-1. Finished wheel with symmetry plane ..............................................  64Figure 5-2. Finite element assembly model........................................................  64Figure 5-3. Boundary conditions in rim indentation model ...............................  67Figure 5-4. Contact pair in rim indentation model .............................................  69Figure 5-5. Meshed model of the wheel with boundary conditions ...................  71Figure 5-6. Pressure load ....................................................................................  72Figure 5-8. Bead area..........................................................................................  73Figure 5-7. Radial loading as a cosine waveform ..............................................  73Figure 6-1. Comparison of modeling and experimental results .........................  76Figure 6-2. Predicted equivalent plastic strain following rim indentation testing...............................................................................................  78Figure 6-3. Contact pressure under platen predicted for rim indentation test ....  80Figure 6-4. Stress state of a node in the contact region......................................  81Figure 6-5. Comparison of sensitivity analysis results for model with different values of E .......................................................................................  84Figure 6-6. Sensitivity analysis with the rim indentation model for variation in E and flow strength .............................................................................  86Figure 6-7. Comparison of numerical models ....................................................  88Figure 6-8. Comparison of predicted and measured strains for gauge 1 during radial fatigue test..............................................................................  90Figure 6-9. Comparison of predicted and measured strains for gauge 7 during radial fatigue test..............................................................................  91Figure 6-10. Comparison of predicted and measured strains for gauge 10 during radial fatigue test..............................................................................  92Figure 6-11. Comparison of predicted and measured strains for gauge 13 during radial fatigue test..............................................................................  93Figure 6-12. Comparison of predicted and measured strains for gauge 16 during radial fatigue test..............................................................................  94Figure 6-13. Strain state of a node present at the inboard flange with variation in Young’s modulus..............................................................................  96Figure 6-14. Temperature variation during quench process .................................  99Figure 6-15. Stress state of the wheel after quench, air cool and machining processes ........................................................................................  101Figure 6-16. Node at inboard flange...................................................................  102Figure 6-17. Stress state of a node on the inboard flange...................................  103ixFigure 6-18. Comparison of modeling results with and without residual stress  105Figure 6-19. Stress state of a node in contact region..........................................  107Figure 6-20. Stress state of a node during radial fatigue test..............................  109Figure 6-21. Principal stress of the node with maximum strain.........................  110Figure 6-22. Strain state of the node during radial fatigue test ..........................  111Figure 6-23. Principal stress variation with angular position of radial load.......  115Figure 6-24. Variation of total fatigue life with rim indentation load ................  116Figure 6-25. Principal stress with different radial loads.....................................  118Figure 6-26. Effect of radial load on total fatigue life........................................  119Figure 6-27. Percentage change in fatigue life with change in parameter w......  122Figure A-1. Strain measurements for 18 gauges before and after air pressure .  134Figure A-2. Strain curves for gauges 2 and 5....................................................  135Figure A-3. Strain curves for gauges 3 and 6....................................................  135Figure A-4. Strain curves for gauges 9 and 12..................................................  136Figure A-6. Strain curves for gauges 14 and 15................................................  137Figure A-5. Strain curves for gauges 8 and 11 ..................................................  137Figure A-7. Strain curves for gauges 17 and 18................................................  138xAcknowledgementsI would like to extend my heartfelt gratitude towrads my supervisors Dr. Robert Hall andDr. Daan Maijer, who have patiently guided me in this research project. Their invaluableadvice and constructive feedback have helped me complete this project successfully. Iwould also like to thank Dr. Douglas Romilly and Dr. Edouard Asselin for agreeing to readthis thesis and provide their insightful comments. My sincere thanks to Dr. Yves Nadot forhis generosity in helping me in this project.Next, I would like to express my sincere thanks to the wheel manufacturing company fortheir co-operation in allowing me to use their facility and product as part of the researchproject. I would like to recognize Mr. Mathew Roy and Mr. Dave Marechad, Ph. D. students in theMaterials Engineering department at UBC, Vancouver, for their help in this researchproject. I would also like to acknowledge the Materials Engineering Machine shop person-nel who provided me with their technical assistance throughout this project. I want to thank all my friends and colleagues in the UBC Mechanical Engineering Depart-ment, especially the Applied Mechanics ROTA Group for their stimulating discussions andseminars. Your academic advice helped me complete this thesis, and your friendship hasmade my time here memorable. I want to thank my friends at Vancouver, especially Ms. Akila Kannan and my friends inIndia, especially Mr. Nakul Aggarwal and Mr. Tushar Mishra for being there for me, whenI needed them the most.Last but not least, I would like thank my parents and my sister for all their constant wordsof encouragement and having faith in me all the time.  xiTo my parents and sister11 Introduction1.1. Automotive wheelsThe use of cast aluminum alloys in wheel fabrication has resulted in a sharp increase in thedemand for aesthetic wheels with intricate designs. The casting process has provided wheeldesigners with the freedom to explore designs with complicated geometry that are impos-sible to fabricate by the forging process [1]. Wheels made out of cast aluminum alloys havea higher aesthetic appeal compared to traditional forged steel. Moreover, cast aluminumalloys have a higher strength to weight ratio compared to forged steel which can potentiallycontribute in production of lighter wheels with better fuel efficiency [2][3]. Forged andwelded steel wheels are cheaper than cast aluminum alloys wheels but lack the aestheticappeal. The type of wheel chosen depends on the consumer's needs and budget. Another trend in today's automotive industry is the increased demand for large diameterwheels. This demand is also driven by the aesthetic appeal of large diameter wheels withthin spokes and large rims. Owing to the high strength to weight ratio of cast aluminumalloys, these large diameter wheels have been made possible without sacrificing the struc-tural integrity of the wheels [1].Although, cast aluminum alloys are light and durable, their relatively low fatigue resis-tance hinders their application in structural components which undergo repetitive load-ing[2]. The cyclic nature of in-service loading in wheels makes fatigue life a key criterionfor design purposes. 21.2. Wheel fatigue testingAn aluminum alloy wheel design must pass three tests before it can be approved for pro-duction [4]. Two of these tests - the radial fatigue and bending (cornering) fatigue (SAEJ328) are the standard tests outlined by SAE for wheels used on passenger cars and lighttrucks to assess their in-service fatigue life characteristics. The third test is a drop test (SAEJ 175) which is performed to assess the damage to the tire-wheel assembly from a suddenimpact. Once in production, periodic quality assurance checks are also conducted onwheels to ensure that the standards continue to be met.Some manufacturing companies have designed additional tests to ensure wheel quality. Aspart of its continuous self improvement strategy, a leading North American wheel manu-facturing company has added rim indentation testing to its quality control program. In thistest, the rim is indented (permanently deformed) by applying a static load. The deformedwheel is then tested for radial fatigue life. Wheel designs are rejected following rim inden-tation if they deform more than the permissible limit of 3mm. This testing protocol wasdesigned to assess the ability of a wheel to withstand an event such as driving up a curb orthrough a deep pot hole which causes permanent rim deformation and may impact radialfatigue life.1.3. Design toolsOnce a new wheel design has been conceptualized, a prototype of the wheel is manufac-tured and tested through accelerated fatigue life tests prior to mass production. Based onthe results of the fatigue tests, modifications are made in the wheel design if required [5].This cycle of designing, manufacturing, testing and then modifying continues until adesign is reached which satisfies the fatigue life criteria. This iterative method of wheeldesign is cumbersome and cost inefficient. Moreover, there is an added pressure in theautomotive industry to minimize the time taken between the inception of wheel design andfull-scale production [6]. The recent advancements in the computational capabilities of computers as well as thedevelopment of analysis software has led many manufacturing companies to employ Com-puter Aided Engineering (CAE) techniques in conjunction with traditional testing to opti-3mize part designs [6]. It has also been reported that fatigue testing without concurrent stressanalysis does not yield optimum wheel designs [7]. The use of CAE analysis softwareenables a more thorough analysis of a system as compared to traditional methods. Thisresearch project aims to model the rim indentation test and radial fatigue test for automo-tive wheels using an available CAE tool. It should be understood that the current state ofthe art in fatigue modeling does not eliminate the traditional testing since fatigue life isdependent on several unpredictable factors [6]. This research work is focused on advancingthe efforts in the use of these advanced analytical tools for wheel design.42 Literature Review2.1. Fatigue life of cast aluminum alloysCast aluminum alloys are being used in the aerospace and automotive industries to replacecomponents made from forged steel and cast iron alloys. Cast aluminum alloys offer abetter strength-to-weight ratio and hence can potentially improve fuel efficiency. Howeverthe casting process produces a wide range of discontinuities such as porosity, intermetallicparticles, trapped oxide films as well as variations in microstructural features, all of whichare deleterious to mechanical properties-especially fatigue life [8]. These factors eitherindividually or in combination can result in the fatigue failure of cast aluminum alloys,hence the failure causing defect can be either one or combination of discontinuities.One of the common aluminum alloys used for mass production of automotive componentsis A356 which has the composition by wt %: 7.25Si, 0.32Mg, 0.06Fe, <0.01Cu, <0.01Mn,<0.01Cr, <0.01Ni, <0.02Zn, <0.01Ti and <0.01B with the rest being aluminum. Silicon ispresent in an aluminum-silicon eutectic bonded to the primary aluminum matrix. The pres-ence of magnesium (Mg) makes the alloy heat-treatable and can result in improvedmechanical properties of the alloy with proper heat treatment process. The T6 heat treat-ment process is employed for strengthening of the alloy A356. This process employed bythe wheel manufacturing company consists of 3 steps i) Solutionizing step - heating thealloy to a temperature of 540 °C and then holding it for 4 hours at that temperature ii) Waterquench iii) Artificial aging. Holding the alloy for an extended period of time results in dis-solution of Mg and Si in the α-Aluminum phase. The rapid water cooling inhibits the for-mation of unwanted phases and precipitation of Mg-Si particles, resulting in a non-equilibrium super saturated solid solution. Finally, in the last step of artificial aging, the5supersaturated solid solution is held at an intermediate temperature of 140 °C, for a speci-fied time till the desired material properties are achieved. One of the major concerns with the wide scale application of cast aluminum alloys is theirreduced fatigue life when compared to forged steel.Figure 2-1 shows a typical fatigue life curve for cast alloy A356, as published by Wang et.al. [9]. This type of graph provides important information about the fatigue properties ofA356. The fatigue life curve is plotted showing stress amplitude versus cycles to failure.Figure 2-1 shows that the fatigue life is inversely proportional to stress amplitude level. Ateach stress amplitude level, the fatigue life is not defined by a single point but rather a scat-ter of points that may exhibit a difference between the lower and upper limit of up to oneorder in magnitude. The scatter in the fatigue life can be attributed to the type, size, positionand shape of failure causing defects in the alloy [10]. A more detailed description of thesefactors is provided later in this chapter. Lastly, the arrow marks on the graph depict runout/no failure, i.e. fatigue samples which did not fail by 107 cycles. This does not mean thatthese samples will never fail as cast aluminum alloys do not show any fatigue endurancelimit but a limit of 107 cycles has been specified by ASTM standard E466 to measure thefatigue life of cast aluminum alloys, after which the test is stopped. Figure 2-1. Fatigue life curve [9]62.1.1. Micro-structural featuresThe main microstructural components of alloy A356 are a primary Al matrix, an Al-Sieutectic and Fe rich intermetallics. The volume fraction of the eutectic phase is dictated bythe chemistry of the alloy but the size and distribution of primary dendrites and eutecticstructures are controlled by the solidification time [11]. The fatigue failure of cast alumi-num alloys is dominated by discontinuities like pores and oxide films. In order to analyzethe effects of the micro-structural constituents on fatigue life, castings must be made withspecially clean and degassed melts. Many authors such as Wang et. al. [11], Yi et. al. [8]and Campbell et. al. [12] have used the Hot Isostatic Pressing (HIP) process to reduce thesize and distribution of porosity in the castings and thereby focus on the effects of micro-structure.Figure 2-2 shows the microstructure of an unmodified alloy A356 [11]. Fe-intermetallicsand silicon particles are present in the eutectic phase bonded to the aluminum matrix. Forslow solidification rates, the microstructure produced can be very coarse hence, strontiumis added in small quantities to modify the morphology of the eutectic from large flake-likecolonies as seen in Figure 2-2 to the fine fibres shown in Figure 2-3. The modification ofthe eutectic phase results in better fatigue and tensile properties for A356 [11].7Figure 2-2. Unmodified microstructure of A356 [11]Figure 2-3. Modified microstructure of A356 [11]82.1.1.1. Secondary dendrite arm spacing (SDAS)Secondary dendrite arm spacing (SDAS) is a characteristic feature of cast aluminum alloysthat is typically reported. It is the spacing between secondary arms of the primary alumi-num dendrite which defines a grain. The dendrite size and morphology depends upon thesolidification rate [9]. Cast aluminum alloys with a finer microstructure, i.e. lower SDAS,exhibit better mechanical properties like Yield strength, Ultimate tensile strength and elon-gation as compared to alloys with coarse dendrite structures [12]. Heat treatment, ageingand the HIPing process are known to have no effect on SDAS.In defect free castings, i.e. castings free of pores and oxides, fatigue cracks are known toinitiate in the dendritic microstructure of the alloy. Wang et. al [11] experimentallydeduced that fatigue life increases with a decrease in SDAS when SDAS is less than 60µm,whereas when the SDAS is greater than 60µm, the fatigue life increases with an increasein SDAS [11]. However, no explanation was provided to support the experimental results.Figure 2-4 shows the experimental results reported by Wang. et. al. All the samples testedwere HIPed to remove porosity before T6 heat treatment.Figure 2-4. Effect of SDAS on fatigue life [11]92.1.1.2. Silicon eutectic particlesThe Si- particles in the eutectic of alloy A356 are distributed around the α-Al dendrites toform a cell pattern. This repeated cell pattern across the metallographic surface is called amicro-cell [3] or a dendrite-cell. Generally, the shapes of these micro-cells are ellipticalwith the minor axis length equal to the secondary dendrite arm spacing. The major axis istwice the length of the minor axis and is hardly affected by solidification time or HIPing.Longer solidification time leads to larger and more acicular Si particles [3]. As noted earlier, strontium-modification is used to produce fibrous silicon particles. Themodified alloy A356 has better mechanical and fatigue properties [11]. For HIPed castingswith low Fe- content, the eutectic Si particles can act as initiation sites for fatigue failure.Gall et. al. [13] investigated the effects of bonded Si- particles in the Al- matrix using finiteelement analysis. Their work concluded that in the absence of voids, bonded particles likeeutectic Silicon can have sufficient plastic strain accumulated around them during cyclicloading to cause debonding. The debonded Si-particles result in high stress concentrationand can cause fatigue crack initiation at the site. 2.1.1.3. Fe-intermetallicsPlate like Fe-rich intermetallic particles are present in the castings and their size is propor-tional to Fe-content, i.e. higher Fe- content leads to large plate like intermetallics whereaslow Fe-content results in small needle like Fe- intermetallics in the eutectic region [14]. Feintermetallics are observed to debond from the Al-matrix in a crack initiating micro-cellbecause of accumulated plastic strain. The presence of Fe-intermetallic plates can causehigher localized plastic strain as compared to Si-eutectic particles in HIPed castingsbecause of their elongated shape. With an increase in Fe-content in the alloy, the fatiguelife decreases for coarse SDAS whereas for fine to intermediate SDAS, there is no apparenteffect of increased Fe- content [11]. Yi et. al. [14] observed that there exists a critical stress level below which high Fe-contentreduces the observed fatigue life, this critical stress level is reported to be 130-140 MPa(see Figure 2-5). However, if the applied stress is above the critical stress level, the highFe-content increases the fatigue life slightly [14]. The presence of high Fe-content at low10amplitude stress levels i.e. less than 130-140 MPa results in a large localized plastic strainconcentration at the crack initiation site. The localized plastic strain can result in debondingof the Fe- intermetallics from the Al-matrix and then act as a crack initiation site [13]. Atlow amplitude stress levels, in HIPed castings, fatigue life is dominated by crack initiationlife; a high Fe-content contributes to crack initiation thus reducing the number of cycles forfatigue crack initiation. On the other hand, when the applied stress levels are high, thefatigue life is dominated by the propagation phase. The presence of plate-like Fe-interme-tallics particles at an angle to, or perpendicular to the propagation crack tip can provideextra resistance to crack growth, hence resulting in higher fatigue life.Figure 2-5. Fatigue life A356 with varying Fe-content [14]112.1.2. DefectsCasting defects such as oxide films or pores are inevitably present in cast aluminum alloys,generally due to poor mold design, bad casting practices or turbulence [12]. Pores andoxide films are more harmful to fatigue life than any other microstructural feature. It is anongoing debate among various authors as to which defect is more detrimental to fatiguelife; pores or oxide films. On the one hand, authors Wang. et. al.[9], Couper et. al. [15],McDowell et. al. [16] believe that the pores have a more significant effect on fatigue lifeof cast aluminum alloys as compared to oxide films, whereas authors such as Campbell et.al. [17] and Nyahumwa et. al. [18] have the opposite opinion. It is still not clear which isthe most detrimental defect and more research work is required in this area. A brief descrip-tion of pores and oxide films is provided in the following sections.2.1.2.1. Oxide filmsOxide films exist on the surface of the melt and get entrained into the melt during fillingdue to turbulence or poorly designed runner systems [12]. Two types of oxide films havebeen found to exist in cast aluminum alloys, old films and new films [12]. Old films whichare formed during the melting process at the surface of the furnace or transfer ladle are gen-erally thicker with higher levels of magnesium and oxygen [18]. These old oxide films aremore likely to be flat and less convoluted than new films (see Figure 2-6). These films tendto partially bond with the Al matrix reducing the stress concentration at their edges [19].The new oxide films are formed due to folding-in of a thin oxide film formed on the liquidsurface. These are called bi-films [17]. When entrained in the melt, these films form anoxide-to-oxide (dry side-to-dry side) interface with varying amounts of air trappedbetween the two interfaces [18]. When viewed with a scanning electron microscope (SEM)(see Figure 2-7), they appear very thin and finely folded. These double films have no sig-nificant bond between the interfaces and can act as a crack initiation site for fatigue failure[12]. 12Figure 2-6. Old oxide film [18]Figure 2-7. New oxide film [18]  132.1.2.2. Micro-porosity Figure 2-8 shows a typical pore in an alloy A356 produced during the casting process. Thesize and distribution of pores is highly variable depending on casting process parameterssuch as local cooling rate and hydrogen content [3]. Pores associated with dissolved gasesare generally round in shape and occur as individual bubbles whereas shrinkage porosityoccurs in clusters of irregularly shaped pores [10]. The size of pores in the alloy A356varies from ten to hundreds of microns [2]. Various casting processes give rise to differentpore densities; abundant porosity has been seen in high pressure die castings whereassqueeze castings can be essentially pore free if the melt is handled with care. Semi-solidcastings can minimize defects but due to complex shapes of final products, ideal conditionsare not reached and defects tend to occur [10].The eutectic regions are the favored location for porosity [2]. The presence of eutectic sil-icon and intermetallic particles around pores help in the formation of cracks [9]. Yi. et. al.[8] used finite element modeling techniques to verify that a high stress concentration isdeveloped at the roots of pores in between the secondary dendrite arms which leads tofatigue crack formation. Figure 2-9 shows the normalized stress field around the concaveFigure 2-8. A typical cast A356 pore [8]14tips of the pore. Here, σ refers to the localized stress and σ∞ refers to the far-field amplitudestress applied to the sample. Figure 2-9 shows that the normalized stress ratio can be ashigh as 6. Thus, even if the far-field amplitude stress applied is below the yield strength,the localized stress at the pore tip may be much higher than the yield strength due to thestress concentration effect of the pore [8]. Therefore, it is predicted that a fatigue crack canreadily initiate in the early stages of loading and fatigue life when pores are present is dom-inated by crack propagation [9].Figure 2-9. Finite element generated stress field around a pore [8]152.2. Fatigue modeling One of the first attempts to quantify fatigue life was made by Paris in 1960. According tothe Paris law, the rate of crack growth is proportional to the stress intensity factor range atthe crack tip. Equation (2-1) shows the Paris law, (2-1)where da/dN represents crack growth rate, a is the crack size, N is the number of loadingcycles, C and m are material constants and ∆K is the stress intensity factor range at thecrack. In cast aluminum alloys, large pores present at or near the specimen surface are pre-dominantly the cause of fatigue failure [20]. The number of cycles required to initiate acrack from a defect is called fatigue initiation life. For large pores present at or near thesurface of a specimen, fatigue initiation life is almost negligible [19][21]. As a result, afracture mechanics or crack growth type approach, as in Equation (2-1) can be used for pre-dicting fatigue life of cast aluminum alloys. Many authors [15][22] have tried to model the fatigue life of cast aluminum alloys using amodified Paris law where ∆Keff is calculated considering the closure effect. The modifiedParis law is shown in Equation (2-2) (2-2)where ∆Keff is the closure corrected effective stress intensity range at the crack initiatingdefect. Couper et. al. [15] integrated Equation (2-2) from an initial crack size (ai) to a final cracksize (af) and estimated ∆Keff using the closure effect. They found that predictions forfatigue life were in good agreement with their experiments. The initial crack size (ai) wasmKCdNda )(∆=meffKCdNda )(∆=16determined from the pore responsible for the fatigue failure. Skallerud et. al [22] in a sim-ilar approach integrated Equation (2-2) and were able to predict fatigue life in good agree-ment with experiments. Fatigue life prediction by both the authors [15] and [22] were ingood agreement with the experimental data but both had a similar trend of over estimatingthe fatigue life. It was later realized that the Paris law is only applicable to long cracks [20] i.e. cracksgreater in size than 2mm. In cast aluminum alloys, an appreciable amount of time in fatiguecrack growth is spent in the small crack growth regime. A small crack is generally identi-fied as a crack less than 1 to 2mm in size. It has been shown that small cracks grow fasterand at smaller ∆K than long cracks [20]. Hence, small crack growth should be taken intoconsideration while modeling fatigue behavior. Thus, Linear Elastic Fracture Mechanics(LEFM) based on long crack growth theory is not suitable for accurately describing the fullfatigue life behavior of cast aluminum alloys.Other authors have also tried to model small crack growth using LEFM and introducingdifferent definitions of ∆K based on the crack closure effect, but in general the equationshave become more complicated and their use has been limited [20].To overcome these deficiencies, Nisitani et. al. [23] proposed the fatigue crack growth rela-tion (see equation (2-3)) independent of stress intensity factor range. In this equation, theywere able to successfully relate fatigue crack growth to crack size. (2-3)where σa and σy are applied amplitude stress and yield strength, respectively. C and n arematerial constants determined empirically. Caton et. al. [20] acknowledged that in equa-tion (2-3), the term within the parenthesis raised to power n is dimensionless and the driv-ing power for crack growth rate is in units of length but equation (2-3) neglected the effectsof local heterogeneities on crack growth. Caton et. al. [20] proposed equation (2-4)aCdNdanya =σσ17 (2-4)where εmax is the maximum total strain achieved during the loading cycle. Yi et. al. [8]expanded and integrated the equation (2-4) from an initial crack size (a0) to a final cracksize (af) determined by fracture toughness to give an expression for fatigue propagation life(Np): (2-5)Equation (2-5) was successfully used to predict fatigue propagation life when the initiatingdefects are pores in non-HIPed castings and dendrite-cell in HIPed castings [8] but totalfatigue life for cast aluminum alloys may include fatigue initiation life, as shown in equa-tion (2-6). Whenever, a large pore is present at or near the specimen surface, the fatiguecrack initiation life is almost negligible and total fatigue life can be considered equal tocrack propagation life [21]. In other fatigue crack initiation mechanisms i.e. fatigue cracksinitiating from oxide films, dendrites, eutectic particles, fatigue initiation life play animportant role in the total fatigue life of the specimen. For large pores present deep in thespecimen and low far field stress levels, initiation life can account for a large part of thetotal fatigue life. In this case, the total fatigue life may be defined as: (2-6)where Nf , Ni and Np represent total, initiation and propagation fatigue life, respectively.aCdNdanya =σσεmax( )110max1 +−+−−−= tftstyap aaCN σσεpif NNN +=18Fatigue crack initiation life is defined as the number of cycles required to develop a fatiguecrack from a pore or a micro-cell. Yi. et. al. [3] proposed a fatigue initiation life relationbased on the total accumulated equivalent plastic strain (εeqp) at the failure causing den-drite cell, see equation (2-7).  (2-7)where, λ2 refers to the secondary dendrite arm length, σa is the amplitude stress, C0, k0, αand β are material constants. Yi et. al. [8] combined equations (2-5) and (2-7) and gave a generalized fatigue life modelfor failure emanating from either eutectic particles (Si particles or Fe-intermetallics),SDAS or pores as shown in Equation (2-8). (2-8)Figure 2-10 shows a comparison between the fatigue life predicted by models (2-5), (2-7)and (2-8) with the experimental results for non-HIPed cast A356 samples [3]. It can beclearly seen that for non- HIPed castings, where pores act as the failure causing defects, thetotal fatigue life is dominated by the propagation life for σa > 80 MPa whereas initiationlife dominates at very low stress levels. It should also be noted that the fatigue life predic-tion models aptly match with the experimental results.Figure 2-11 shows a similar comparison between the empirical models (2-5), (2-7) and (2-8) with the experimental results for HIPed cast alloy A356 [3]. For the HIPed castings, itwas observed that Si-particles were the failure initiation sites [3]. It can be inferred fromFigure 2-11 that for HIPed castings, the fatigue life is dominated by the crack initiation lifefor σa < 140 MPa. βλασλ22020 1 += kCNai( )110max122020 1 +−+−−−+ +=+= tftstyaapif aaCkCNNNσσελασλβ19Figure 2-10. Fatigue life comparison for non-HIPed A356 [3]Figure 2-11. Fatigue life comparison for HIPed A356 [3]202.3. Fatigue life of pre-strained aluminum alloysAs explained in the Section 1.2, a leading wheel manufacturing company is interested indetermining the radial fatigue of plastically deformed cast A356 aluminum wheels.Section 2.1 introduced the prominent work done in the field of fatigue life of cast alumi-num alloys. This section will review the effects of plastic deformation on fatigue life.One of the earliest attempts to identify the effects of pre-straining of Al alloys on theirfatigue life was reported by A. F. Getman and Yu. K. Shtovba in 1982 [24]. They tried toinvestigate the effects of plastic deformation on forged Al alloys immediately after hard-ening. The specimens were aged artificially after deformation. After experimenting withdifferent degrees of pre-strains, they concluded that plastic deformation up to 2% does nothave any significant effect on the fatigue life of forged Al alloys. Plastic deformation ofmore than 2% leads to a decrease in fatigue life [24]. In a recent work, K. S. Al-Rubaie et. al. [25] and [26], showed the effects of pre-strain onfatigue life of forged aluminum alloys 7475- T7351 and 7050-T7451. They showed thatwith an increase in the level of pre-strain, the fatigue life decreased for the aluminumalloys. The fatigue life of pre-strained forged aluminum alloys is found to decrease with anincrease in the level of pre-strain. Significant work has been done on understanding thefatigue life characteristics of pre-strained forged aluminum alloys [24-26]. The effects ofpre-straining on the fatigue life of cast aluminum alloys is an area of research that hasreceived little attention and to the best of the author's knowledge there are no publicationson it. 212.4. Wheel testingWheels are considered a safety related component. Consequently, their fatigue perfor-mance under various loading conditions is an important issue. The current emphasis onmaking lighter wheels by either using cast aluminum alloys or making thinner gauge sec-tions has increased the amount of study in this area. Wheels made of cast aluminum alloystypically have improved aesthetic designs that are attractive to consumers. As the complex-ity of wheel designs increases, it is difficult to estimate the fatigue strength of the wheelsusing analytical methods [27]. Component-level fatigue tests are full-scale experimentsperformed by industry, either at a testing facility or at a wheel-manufacturing foundry, toassess the fatigue life or durability of a wheel. Two fatigue tests performed as part of typ-ical quality assurance testing are: 1) the radial fatigue test and 2) cornering test (SAEJ328a) [27]. Based on the results of these fatigue tests, the wheel designs can be modifiedto improve strength and fatigue life. A brief description to the tests will be provided in thefollowing section.2.4.1. Radial fatigue testA typical radial fatigue test setup according to standard (SAE J328a) is shown in theFigure 2-12. In this test, a test wheel and tire are mounted on a hub by lug nuts at a suitabletorque and placed in contact with a rotating drum. The drum axis is parallel to the axis ofthe test wheel. A hydraulic ram loads the test wheel and tire by pushing it normal to thesurface of drum and in-line radially with the center of the test wheel and the drum. Thedrum is rotated at a simulated road speed based on the size of the wheel. This test simulatestypical car operation where the weight of a car is balanced with a vertical reaction forcefrom the road through the tires. To accelerate fatigue testing and enhance the effectivenessof the radial fatigue test, the entire vehicle weight is applied to a single tire resulting inradial compressive loading. While the test is running, the radial load at each point on thewheel is cyclically varying with the rotation of the wheel leading to possible fatigue failure.22According to SAE J 328a standards, a wheel should maintain its structural integrity for aminimum 4 x 106 cycles under the radial load, given by equation (2-9). (2-9)where Q is the radial load applied on the wheel, S is the acceleration test factor accordingto SAE J328a standards (S=2.2) and F is the maximum possible load on the wheel (i.e. thefull weight of the vehicle).Figure 2-12. Radial fatigue test setup [27]FSQ ×=232.4.2. Cornering testThe cornering fatigue test (SAE J328a) is a durability test performed on wheels to assesstheir fatigue characteristics under cornering conditions. In this test, the dynamic forcesacting on the wheel during the cornering of an automobile on the road are simulated byapplying a varying or constant load to a load arm attached to a rotating wheel, as shown inFigure 2-13. In this test, the inboard rim flange is clamped securely to a rotating table. The hub of thewheel is attached to a rigid shaft running through the centre. The test load is applied to theshaft as shown in Figure 2-13. A cyclic bending load is generated by rotating the table.Figure 2-13. Cornering fatigue test setup [28]242.5. Radial fatigue modelingThe rim indentation test for wheels is followed by the radial fatigue test. In order to predictthe fatigue life of deformed wheels under radial load, the characterization of the stress stateof the wheel under radial load is required. Hence, the effect of radial load on the wheel isdiscussed in this section.The weight of a vehicle is supported on the four wheels of the automobile. The verticalreaction forces exerted by the road on the tires balance the total weight of the automobile.The reaction forces compress the tire, which in turn transmits the forces to the wheel. Whenthe tire is compressed, the contact between the road and the tire is a flat surface called thecontact patch. The contact patch is measured in terms of the angle swept by the contactpatch length at the centre of the wheel, shown as θ in Figure 2-14. The value of contactpatch (θ) depends on the tire pressure, vehicle weight and tire geometry. Figure 2-14. Tire under a vertical load [31]25The radial load on the wheel can be estimated as a pressure load applied to the bead area.Stearns et. al. [29] suggested the contact patch angle to be 80° and hence to quantify theeffect of the radial load on the wheel, a pressure load should be applied for the correspond-ing 80° degrees of the bead area on the wheel. Ramamurthy et. al. [27] demonstratedthrough experiments and a finite element model that the effect of radial load on the wheelis not limited to the corresponding angle on the wheel but should span the entire bottomhalf of the wheel i.e. pressure should be applied to 180° of the bead area. Stearns et. al. [29] investigated the effect of the radial load on the wheel. The contact regionbetween the tire and the wheel is the bead seats area as shown in Figure 2-15. Theyassumed the entire radial load was applied on the bead area of the wheel. They experimen-tally deduced that the radial load applied on the wheel by the tire follows a cosine curvei.e. the load is maximum at the centre of contact patch and decreases to zero following acosine curve, as shown in Figure 2-16. Figure 2-15. Axisymmetric model of wheel [29]26The variable pressure load Wr can be estimated as shown in equation (2-10). (2-10)where Wr is the pressure load on the bead area at a circumferential angle θ from the centreof the patch, W0 is the maximum pressure load at the centre of contact patch, θ is the cir-cumferential angle of the wheel ( -θ0 <θ < θ0) and θ0 is half of the angle swept by the con-tact patch at the centre. Distributed pressure load (Wr) can be integrated over the bead areato give total radial load (W), as shown in equation (2-11).Figure 2-16. Radial load schematic at the bead area [29] ××=00 2cos θθpiWWr27 (2-11)where rb is the bead radius and b is the bead width. Equation (2-10) can be substituted into equation (2-11) to calculate the maximum distributed pressure load (W0) at the centre ofthe contact patch, shown in equation (2-12). (2-12)Once, W0 has been determined, equation(2-10) can be used to calculate the pressure loadat any circumferential location. This same approach has been used in this research project.∫−×=00θθθdrWbW br00 4 θpi××××=brbWW283 Scope and Objectives3.1. Scope of research workThis research work aims to develop finite element models of the rim indentation and theradial fatigue tests for aluminum alloy automotive wheels. These models may be used asdesign tools for predicting the fatigue life of wheels early in the design stage in an attemptto reduce the wheel development time. In order to predict the fatigue life of wheels follow-ing rim indentation, a numerical model of the fatigue life for pre-strained cast aluminumalloys is required which relates the fatigue life to the level of pre-strain and stress state inthe wheel. Thus, a sub-section of this research project is to develop an empirical relationto quantify the effects of initial plastic strain on the fatigue life of A356-T6 alloy. Figure 3-1. shows a flowchart of the work completed during the course of this research.Industrial-scale rim indentation and radial fatigue tests were done on cast alloy A356wheels to obtain data required to model these industrial tests on a finite element platform. The rim indentation test was performed by deforming the inboard rim flange of the wheelwith a static load. The amount of plastic deformation sustained by the wheel is a measureof the quality of rim design. Rim indentation is a contact modeling problem between thewheel and the platen. This non-linear problem required an iterative solution technique. Thecommercial finite element software package ABAQUS™ has been employed for this workas it is well suited for this type of mechanical problem. The rim indentation test is followed by radial fatigue tests on the deformed wheels. Mod-eling radial fatigue required a solution technique which is capable of reproducing the load-ing conditions in the radial fatigue test to estimate the stress state of the wheel. The29complex geometry of the wheel makes this problem unsolvable with analytical solutiontechniques. Hence, ABAQUS was used to model the radial fatigue test.Laboratory-scale experiments were performed to determine the material characteristics ofA356-T6 alloy used in the wheels. Young's modulus, yield strength, flow strength andplastic strain for alloy A356 were determined by tensile tests. Laboratory-scale experiments were also performed to characterize the fatigue behavior ofA356 following different amounts of pre-strain (plastic strain). The data was used todevelop an empirical relationship between fatigue life and initial plastic strain. The sam-ples for the fatigue test were taken from the same type of wheels used in industrial tests.The amount of pre-strain applied to the fatigue samples was estimated from the rim inden-tation model for a plastically deformed wheel.Figure 3-1. Research methodology Rim Indentation Test Rim Indentation Radial Fatigue Test Industrial scale Tests FEA Modeling Radial Fatigue Lab-scale experiments  Flow stress v/s plastic strain Pre-strained Fatigue tests data data E, σ Nf 30Finally, models of the quench process [1], rim indentation and radial fatigue testing werecombined with the empirical fatigue life equation to predict the radial fatigue life of a rimindented wheel. 3.2. ObjectivesThe objectives of this research work are:1) To develop and validate finite element models of the rim indentation and radial fatiguetests for aluminum alloy automotive wheels.2) To develop an empirical relationship between fatigue life, level of pre-strain, andapplied stress amplitude for cast aluminum alloys.3) To combine the results of the rim indentation and radial fatigue models with the devel-oped fatigue life equation to predict the fatigue life of a wheel.314 Experimental WorkExperimental work was conducted during this project to provide data necessary to formu-late and validate the models. This test work included both laboratory testing and industrialscale component testing. The testing completed for this project and the results will be pre-sented in this chapter. Laboratory scale experiments were performed to characterize theconstitutive behavior of the A356 aluminum alloy of the wheels and to develop a fatiguelife equation for pre-strained cast aluminium alloys. Industrial tests were performed to pro-vide the data necessary to validate the finite element models of the rim indentation test andthe post-indentation radial fatigue test. 4.1. Industrial scale experimentsIndustrial scale experiments were performed in collaboration with a North American wheelmanufacturing company at their facility in British Columbia, Canada. The following indus-trial experiments were performed: i) Rim indentation testing on finished wheels, ii) Radialfatigue testing on a finished wheel 4.1.1. Rim indentation testThe rim indentation test is a new quality assurance test added by the wheel manufacturingcompany as part of their quality control program. In this test, a wheel is permanentlydeformed under a static load. The amount of load applied depends on the material andgeometry of the wheel. The permanent deformation sustained by the wheel is used as ameasure of the quality of wheel design. 324.1.1.1. Test setup A dual column Sintech 20/G universal testing machine was used to deform the wheel. Therim indentation test setup on this machine is shown in Figure 4-1. A custom designed fix-ture was specified by the company to locate the wheel under the cross-head. A cylindricalplaten was mounted in the cross-head. During a test, the wheel is aligned with the inboardrim flange placed directly under the platen. An optical sensor, mounted on the testmachine, measured the coordinates of the cross-head and a load cell (20,000 lbf or 88 kN)was used to measure the load applied by the platen. The position of the cross-head and theload applied by the platen was monitored by a computer. The data acquisition software alsorouted the signal to stop the test, if either the load or the displacement of the cross-headexceeded their pre-assigned limits. Figure 4-1. Sintech universal test machine33At the start of a test, the platen is placed just above the rim of the wheel. Testing is con-ducted by applying a pre-determined load with the platen moving at 0.1 mm/s. The wheelis unloaded and the permanent deformation is measured. If the deformation is less than 3mm, the wheel has passed the test.4.1.1.2. Test resultsTwo rim indentation tests were performed at Powertech, a commercial testing facility inSurrey, BC. As per the wheel manufacturing specifications, a maximum load of 20.5 kNwas applied in the tests.The load versus displacement curves measured during the rim indentation tests for the twowheels are shown in Figure 4-2. The load increases from zero at the start of the test whenthe platen contacts the inboard rim flange. Displacement increases with increasing loaduntil the peak load is reached and the wheel is unloaded. The amount of deformation in thewheel is equal to the displacement of the platen and the permanent deformation is deter-mined as the displacement offset at zero load after unloading. In the rim indentation test graphs, shown in Figure 4-2, both wheels were subjected to amaximum load of 20.5 kN. The peak displacement of wheel #1 under load was 8.87 mmwith 1.89 mm permanent displacement whereas wheel #2 deformed 8.32 mm with 1.4 mmpermanent deformation. The permanent displacement sustained by both the wheels waswithin the acceptable limit of 3 mm as outlined by the wheel manufacturing company.Although, care was taken to ensure the same test conditions for both the wheels but the dif-ference in the displacements of the wheels is very apparent. The discrepancies observedmay be attributed to variations in the experimental setup such as orientation of the wheeland / or variations in the wheel manufacturing process. Additional testing would be neces-sary to characterize the extent of the variability and isolate the specific cause. However,these results provided data suitable for validating a finite element model of the rim inden-tation test.34Figure 4-2. Rim indentation graphs354.1.2. Radial fatigue testThe radial fatigue test is an SAE standard test (SAE J328a) used to assess fatigue life ofwheels subjected to a radial load. It was mentioned in the literature review that radialfatigue is one of the durability tests each wheel design has to pass before being approvedfor full scale production. Wheel designs that do not satisfy the radial fatigue life criterion(see Section 2.4.1) are modified based on test results.The relevance of the radial fatigue test in this research project is that as part of the newquality assessment testing in the wheel manufacturing company, wheels must satisfy aradial fatigue criterion following rim indentation testing. Radial fatigue tests were per-formed on wheels to systematically measure strain at a variety of locations under differentloading conditions on the wheel to collect data for verifying the radial fatigue test model. 4.1.2.1. Test preparationThe strain experienced by a wheel during the radial fatigue test was characterized usingstrain gauges. The locations of the strain gauges were chosen using the following criteria:i) space available for mounting the gauges, ii) the surface was relatively flat and iii) thelocation was expected to experience significant strain. A preliminary version of the radialfatigue test model that will be described in the next chapter was used to evaluate locationsbased on the third criterion. Based on the previous criteria, the six locations shown inFigure 4-3 were selected. A standardized procedure was used according to the strain gaugemanufacturer's specifications to mount the strain gauges. The strain gauges were mountedby the quality control personnel from the wheel manufacturing company. Three straingauges were mounted at each location. The strain gauges used in the experiments weremanufactured by Kyowa Sensor System Solutions. The specifications of the strain gaugesare provided in Table 4-1. Strain gauges 1 to 12 were of type 1 whereas gauges 13 to 18were of type 2. As these strain gauges were readily available with the wheel manufacturingcompany, rosettes were not purchased. The orientation of the strain gauges was such thatall strain components on the plane could be calculated. Pictures of the installed gauges andtheir orientations are shown in refer Figure 4-4, 4-5 and 4-6. Figure 4-5 shows gauges 13,14 and 15, the angle between the longitudinal axes of gauges 13 and 14 as well as gauges3613 and 15 is same 135° such that gauges are in a 45° rosette formation. Same orientationwas used at other gauge locations. The gauges used had self compensation temperaturerange from 10 to 100 °C. Table 4-1. Gauge specificationsFigure 4-3. Strain gauge locationsType 1 Type 2Gage Length(mm) 3 1Resistance (ohms) 119.6+/- 0.4 119.6+/- 0.4Gage Factor 2.13 +/- 1% 2.17 +/- 1%Thermal expansion(/°C) 23.4 23.4Transverse sensitivity 0.5 2.337Figure 4-4. Strain gauges at locations 1 to 4Figure 4-5. Strain gauges at location 5, inboard rim384.1.2.2. Test procedureWith the strain gauges mounted on the wheel, strain measurements were taken under thefollowing four conditions:1) Base level: The first set of strain measurements were taken on a fully finished wheel without the tire.One might expect the strain readings to be zero as no load had been applied to the wheelbut generally they were not. The gauges tend to develop an initial strain due to a numberof reasons like installation, rough handing, temperature, etc. This is known as zero error orbase level strain. This base level strain was measured for all the strain gauges.Strain measurements for each of the 18 gauges at conditions 2, 3 and 4 were zero error cor-rected by subtracting the base level strain from them. 2) Tire mounted on the wheel but not inflated: Next, a tire was mounted on the wheel but not inflated. A second set of strain readings wererecorded for this condition. Figure 4-6. Strain gauges at location 6, spoke393) Tire inflated to test conditions i.e. 60 psi: The tire was inflated to the test pressure of 60psi and a third set of strain readings wererecorded. 4) Wheel rotated under radial fatigue test conditions:Finally, the wheel was mounted on the radial fatigue test setup, shown in Figure 4-7. Thewheel manufacturing company’s standard for their radial fatigue testing of the wheelmodel used for this investigation were:Tire pressure = 60 psiRadial Load = 1560 kgfA non-standard rotational speed of 3 km/hr was selected for this test. This speed wasselected because it was slow enough to provide enough strain data for almost 2 completerotations within the limitations of the DAQ system. A total of 4000 data points were col-lected for each strain gauge.The strain measurements from the Radial Fatigue Test were used to validate the radialfatigue model which was developed in ABAQUS.Figure 4-7. Radial fatigue test setup404.1.2.3. Test resultsStrain measurements for one strain gauge from each of the six locations, i.e. strain gauges1, 4, 7, 10, 13 and 16 at radial fatigue test conditions are presented here. These particulargauges are chosen because of their orientation; gauges 1 and 4 are mounted on the spokeperpendicular to radial direction, gauges 7, 10 and 13 provide an estimate of the hoop strainwhereas gauge 16 gives a measure of the radial strain in the spoke. A summary of the entirestrain measurement dataset can be found in Appendix A.Figure 4-8 shows the strain curves for strain gauges 1 and 4. As shown in Figure 4-4, thestrain gauges 1 and 4 are located on two different spokes but in the same orientation.Hence, the same trend and numerical values in the strain graphs are expected. The gapbetween the peaks of the two curves shows the time taken by the wheel to rotate the angulardistance between the two gauges.Figure 4-8. Strain graphs for gauges 1 and 4-6.0E-04-4.0E-04-2.0E-040.0E+002.0E-044.0E-046.0E-048.0E-040 1000 2000 3000 4000Data pointsStrainStrain Gauge 1Strain Gauge 441Figure 4-9 shows measurements for strain gauges 7 and 10, respectively. Strain gauges 7and 10 show similar trends but the strain levels of strain gauge 10 are much higher thanthat of strain gauge 7. The difference in strain levels can be attributed to their different loca-tions (see Figure 4-4). Strain gauge 7 is located near a stiff spoke region; hence the stressin that area gets well distributed and results in a lower strain level whereas strain gauge 10is located near the opening between two consecutive spokes, the stiffness of the areaaround strain gauge 10 is lower which is believed to have resulted in the higher strain val-ues.Figure 4-10 shows the measurements for strain gauge 13. Strain gauge 13 is of specialinterest in this study because of its position on the inboard bead area where rim deforma-tion takes place in the rim indentation model (refer to Figure 4-5). A large part of the straincurve lies in the tensile region. The maximum strain seen by strain gauge 13 is 1.03 x10-3.This is higher than any other tensile strain zone in the wheel.Figure 4-9. Strain graph for gauges 7 and 10-8.0E-04-6.0E-04-4.0E-04-2.0E-040.0E+002.0E-044.0E-046.0E-048.0E-041.0E-031.2E-030 1000 2000 3000 4000Data pointsStrainGauge 7Gauge 1042Figure 4-11 shows the measurements for the strain gauge 16. Strain gauge 16 was the onlystrain gauge in the experiment which did not show any tensile strain during the radialfatigue test. High compressive strains were seen in this spoke region. This can be attributedto the geometry of the wheel and high radial load applied on the wheel.These strain curves provide information on the strain characteristics of the wheel duringthe radial fatigue test. This data was used to validate the Radial Fatigue model.Figure 4-10. Strain graph for gauge 13-8.0E-04-6.0E-04-4.0E-04-2.0E-040.0E+002.0E-044.0E-046.0E-048.0E-041.0E-031.2E-030 1000 2000 3000 4000Data pointsStrain43Figure 4-11. Strain graph for gauge 16-1.6E-03-1.4E-03-1.2E-03-1.0E-03-8.0E-04-6.0E-04-4.0E-04-2.0E-040.0E+000 1000 2000 3000 4000Data pointsStrain444.2. Laboratory-scale experimentsTwo types of laboratory scale experiments were performed to: i) determine the mechanicalproperties of A356-T6 alloy and ii) develop an empirical relationship between fatigue lifeof pre-strained cast aluminum alloys and pre-strain level. 4.2.1. Tensile testLaboratory scale tensile experiments were completed to determine the mechanical proper-ties of A356 aluminium alloy used in the wheels. This was done because the mechanicalproperties like Young's modulus, and stress versus strain data for A356 available in the lit-erature showed a wide amount of variability. Also, these mechanical properties are depen-dent on the casting and heat treatment processes and hence, are specific to each foundry.In order to make sure that all the mechanical properties used in the modeling work pertainto the aluminum alloy used by the wheel manufacturing company, laboratory scale tensiletests were performed on test specimens machined from the inboard flange section of awheel manufactured at the foundry. 4.2.1.1. Tensile test set upTensile test specimens were machined by the machine shop in the Department of MaterialsEngineering at UBC. The dimensions of the tensile test specimen, shown in Figure 4-12,were selected according to the ASTM E8 standard. All dimensions are in mm. An Instron 8874 servomechanical test platform (refer Figure 4-13) was used to conductuniaxial tensile tests. The testing machine is equipped with a 25 kN load cell and inter-changeable, hydraulic wedge-shape grips for holding test specimens. Tests were conductedin displacement control mode based on crosshead position. The combination of the smallcross-sectional area of the gauge length and the alloy resulted in the expectation of lowpeak loads during these tests. Thus, limited machine compliance was expected. Howeverto accurately measure strain in the sample, extensometers were mounted on the samples.Wavemaker, the Instron machine control software, was used to control the cross-head dis-placement during the test.45Figure 4-12. Tensile test specimenFigure 4-13. Instron 8874 servomechanical test platform464.2.1.2. Test procedureA set of threaded adapters were fabricated to mount the tensile samples in the Instron 8874servomechanical test platform because the hydraulic grips would not accommodate thesample diameter. Locknuts were used with the adapters to ensure that the test samples wereheld rigid in the machine. The assembled sample and adapters were placed between thegrips of the Instron machine and the jaws of the machine were tightened. Two tensile testswere performed at a strain rate of 0.01s-1. The first sample was strained to 15% where asthe second was strained to failure.4.2.1.3. ResultsThe change in the gauge length of the extensometer and load applied to the tensile samplewere recorded. The change in length was converted to engineering strain and then conse-quently to true strain using equations (4-1) and (4-2). Similarly, engineering stress and truestress values were calculated using equations (4-3) and (4-4).  (4-1) (4-2)where e and ε refer to engineering and true strain, respectively. Lo refers to the initial length(initial extensometer gauge length) whereas Lf refers to the final length. ∆L and ∂L repre-sents total change in gauge length and infinitesimal change in length. (4-3)oLLe ∆=)1ln(ln eLLLLof +==∂=εAFeng =σ47 (4-4)where σeng and σtrue refer to engineering and true stress, respectively. F is the load applied,A is the initial area of cross-section whereas Ainst represents instantaneous area during thetest.The true stress versus true strain curves for the two tensile tests are shown in Figure 4-14.The similarity of the curves indicates an accurate and repeatable test. The Young's modu-lus, calculated as the slope of the engineering stress versus strain curve in the elastic rangewith low strain, were 67.53 and 66.25 GPa. A line was drawn with slope equal to theYoung's modulus offset by 0.2% engineering strain to intersect the engineering stress-strain curve to estimate the yield strength. The calculated yield strengths were 167 and 164MPa. The ultimate tensile strength were estimated from the true-stress versus strain curve(see Figure 4-14) to be 319 and 317 MPa with 16% strain to failure.Figure 4-14. Flow curves alloy A356)1( εσσ +== enginsttrue AF0501001502002503003500 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2True StrainTrue Stress (MPa)First testSecond test484.2.2. Fatigue life experiments The literature review revealed that there is an absence of research work available on fatiguelife of pre-strained cast aluminum alloys. It has also been noted that in order to predict thefatigue life of wheels following rim indentation testing, a numerical model of fatigue lifefor pre-strained cast aluminum alloys is required. In order to formulate such an empiricalrelationship, lab-scale fatigue experiments were performed with pre-strained fatigue sam-ples. 4.2.2.1. Fatigue sample and test setupSamples for fatigue testing were produced by the machine shop in the Department of Mate-rials Engineering at UBC. The samples, shown in Figure 4-15, with dimensions in mmwere machined according to ASTM standard E466 with dimensions are in mm. In the lit-erature review, it was noted that the fatigue life of cast aluminum alloys depends on thesize of its microstructure and defects. In the industrial radial fatigue tests on wheels follow-ing rim indentation testing performed by the wheel manufacturing company, fatigue fail-ures were observed on the inboard rim flange at the indented location. This was contraryto other radial fatigue tests performed by the wheel manufacturing company on the wheelsfree of any rim indentation, where fatigue failure occurred at the spoke region. It wasinferred from these tests that the microstructure, defects and the permanent deformation atthe inboard flange are together responsible for crack formation. To ensure that the labora-tory scale fatigue samples had the same defects and microstructure as the inboard flange ofthe wheels, fatigue samples were machined out of the inboard rim sections of a series ofwheels supplied by the manufacturer.49A Sonntag uniaxial fatigue testing machine (refer to Figure 4-16) was used for these exper-iments. In this machine, a cyclic force is generated by an eccentric mass spinning at 1800rpm. The offset of the eccentric mass from the center of rotation is adjusted to control theload amplitude. The specimen fixture is mounted on the oscillator housing. Only the ver-tical component of the dynamic force is transmitted to the specimen since the horizontalcomponent is absorbed by the flexplates (Manufacturer’s manual), refer Figure 4-17.Figure 4-15. Fatigue specimenFigure 4-16. Sonntag fatigue machine50The fatigue machine has been instrumented with an Omega LCHD- 2000 lb load cell anda Linear Variable Differential transformer (LVDT), to measure the load and oscillator tabledisplacement, respectively (refer to Figure 4-18). As the test is displacement controlled, adata acquisition system is used to monitor the LVDT during a test. The data acquisitionprogram written in LabView acquires a baseline amplitude of the displacement waveformin 3 cycles and compares that to the amplitude of displacement acquired during the test. Ifthe waveform changes by more than a prescribed amount (1-2%), the test is automaticallystopped. Using this system, tests can be stopped allowing the surface of the sample to bechecked for the presence of cracks using die penetrant inspection. If a crack is detected,testing is stopped, otherwise the test is restarted. Using this approach, testing can bestopped before catastrophic failure which destroys the fracture surface of the sample.Figure 4-17. Sonntag fatigue machine assembly51Before running the fatigue test, the load cell is used to check the pre-load applied to thefatigue sample, adjustments are made to the sample fixture to ensure zero pre-load. Thisstep is necessary as the fatigue tests are performed at fully reversed stress cycles, i.e. R=-1. During testing, the load amplitude waveform is checked to ensure that it is sinusoidal.Noise in the signal suggests loose assembly which can be rectified by checking all connec-tions.Figure 4-18. Instrumentation on fatigue machine524.2.2.2. Pre-strain levels and techniquePrior to fatigue testing, samples were pre-strained to impose inelastic deformation. In orderto make the laboratory scale fatigue experiments more representative of the strain condi-tions present in wheels following rim deformation tests, a maximum pre-strain level for thelaboratory scale experiments was selected using the rim indentation model that will bedescribed in Chapter 5. The peak plastic strain predicted by the model was 10%. Hence,the pre-strain levels selected for the laboratory fatigue experiments were 5% and 10%.The Instron 8800 servomechanical test platform (refer Figure 4-13) was used to pre-strainsamples prior to fatigue testing. The fatigue samples have an hourglass shape that resultsin a continuous variation of stress and strain along the gauge length. A 2D axisymmetricmodel of a fatigue sample was developed in ABAQUS to determine the required cross-head displacement to achieve the desired pre-strain. The model, shown in Figure 4-19,employed the flow stress data measured via the tensile tests described in Section 4.2.1.Boundary conditions were applied at the bottom edge and the vertical axis of the fatiguesample. The top edge of the sample was displaced. The effect of the displacement on thefatigue sample in terms of plastic strain was estimated. Figure 4-19 shows the equivalentplastic strain at the neck region of the fatigue sample. The middle of the sample sustainsmaximum plastic strain shown in red color and the majority of the sample does not undergoany plastic strain, shown in dark blue color. The peak load predicted at 10% strain was 5kN which agreed well with the measured load of 5.35 kN. Using these results the cross-head displacements for 5 and 10% pre-strain in the neck region of the fatigue samples weredetermined to be 0.5155 and 0.9787 mm, respectively.534.2.2.3. S-N data Figure 4-20 shows the fatigue life data for the 0 (unstrained), 5 and 10% pre-strainedfatigue samples. The tests were conducted at four different amplitude stress levels: 140,120, 100 and 80 MPa at a stress ratio (R) of -1. At each stress level, a minimum of 3 and amaximum of 6 samples were tested for each level of pre-strain. The data clearly shows adecrease in fatigue life with increase in pre-strain level. At a stress amplitude of 140 MPa,the shift in fatigue life with pre-strain is clearly delineated. At stress amplitudes of 120MPa and 100 MPa, the fatigue life of pre-strained samples is less than unstrained fatiguesamples but it is difficult to distinguish between 5 and 10% pre-strained samples as towhich have lower fatigue life. At a stress amplitude of 80 MPa, 3 out of 4 fatigue samplestested at 0% pre-strain did not display fatigue failure after testing for 10 million cycles.These points are showed as "runout" in Figure 4-20. As far as 5 and 10% pre-strained sam-Figure 4-19. Axisymmetric model of fatigue sample54ples are concerned at this stress level, most failed before 1 million cycles. The laboratoryscale experiments clearly show that unstrained A356 has longer fatigue life. Figure 4-20. Fatigue life data4060801001201401601.0E+04 1.0E+05 1.0E+06 1.0E+07 1.0E+08Fatigue Life (Nf)Stress Amplitude (MPa)0% pre-strain5% pre-strain10% pre-strainRunout554.2.2.4. Modified fatigue life equationThe pre-strained fatigue life data was analyzed to quantify the effect of pre-strain and todevelop an equation capable of predicting fatigue life based on stress amplitude and levelof pre-strain. Figure 4-21 shows the trendlines and their equations for the fatigue samples.The equations can be re-written as: (4-5) (4-6) (4-7)where, Nf0%, Nf5% and Nf10% represent fatigue life of 0, 5 and 10% pre-strained fatiguesamples, respectively. ( )021.1604.324%0−−=aeN fσ( )355.2076.364%5−−=aeN fσ( )907.2572.426%10−−=aeN fσ56To qualitatively assess the effect of pre-strain on the fatigue life, the difference betweenequations (4-6) and (4-7) and the 0% pre-strain equation were plotted and can be seen inFigure 4-22.Figure 4-22 shows a graph of the change in fatigue life versus the amplitude stress level.The graphs show that at high amplitude stress levels, the decrease in fatigue life due to pre-strain is almost zero, i.e. at high stress levels; pre-strain shows no prominent effect on thefatigue life for cast A356. From this observation, it can be implied that at high stress levels,the localized plastic stress and strain distributions around the defects dominate the fatiguecrack growth rate relative to the induced level of pre-strain concentration. On the otherhand, at low amplitude stress levels, a prominent effect of the pre-strain can be seen fromFigure 4-22. The fatigue life decreases with an increase in the level of pre-strain level asthe blue dashed line is above the solid pink line, which represent decrease in fatigue lifefor 10% and 5% pre-strained samples, respectively. It can also be inferred that at lowamplitude stress levels, localized stress and strain concentrations around the defects in castFigure 4-21. Trendlines- fatigue lifey = -16.021Ln(x) + 324.04y = -20.355Ln(x) + 364.76y = -25.907Ln(x) + 426.724060801001201401601.0E+04 1.0E+05 1.0E+06 1.0E+07 1.0E+08Fatigue Life (Nf)Stress Amplitude (MPa)0% pre-strain5% pre-strain10% pre-strain57A356 are not as high, as a result pre-strain plays a major role in crack initiation/propaga-tion. Section 2.1.2.2 discussed the effect of micro-porosity on the fatigue life. It was mentionedthat when pores are responsible for fatigue crack propagation, the fatigue initiation life isvery small and can be neglected. The fracture surfaces of the fatigue samples were ana-lyzed to identify the defect(s) responsible for fatigue crack initiation. Figure 4-23 shows aScanning Electron Microscope (SEM) picture of a fatigue sample fracture surface. Thepore shown in the figure was determined to be the fatigue initiating defect due to two rea-sons: i) the proximity of the pore to the surface was very close and ii) the area around thepore showed marks/ striations which validated the slow growth of fatigue crack. The initi-Figure 4-22. Effect of pre-strain on fatigue life58ation site in each fatigue sample was examined and it was found that microporosity presenteither near or at the free surface was the fatigue failure causing defect in nearly all cases. As microporosity was determined to be the cause of fatigue failure in the fatigue samples,the fatigue initiation life can be neglected for the cast A356 aluminum alloy used in thisresearch work and the total fatigue life for the material can be described by the empiricalfatigue propagation life relationship previously discussed (refer to equation (2-5)). A sim-ilar approach was used by Gao et. al. [2] where the fatigue propagation life (Np) calculatedusing equation (2-5) was compared to the experimental fatigue life. They also assumed thefinal crack length (af) to be constant since fatigue propagation life is insensitive to it. In thecurrent research project, all the fatigue samples were taken from the same location on thewheel, i.e. the inboard rim flange; therefore it has been assumed that initial crack lengthwhich is essentially the maximum pore size does not vary in the samples. As a result, initialFigure 4-23. SEM picture of a pore in a fatigue sample59crack length (a0) in equation (2-5) has been considered to be a constant in this researchproject. Hence, the fatigue life equation (2-5) can be modified to the empirical relation asshown in equation (4-8): (4-8)where Nf is the total fatigue life. σa is the amplitude stress, σy is the yield strength, εmax isthe maximum total strain during a fatigue cycle, C2 and w are parameters depending on thematerial and the level of pre-strain.In order to determine C2 and w for each level of pre-strain, equivalent stress (εmax σa/σy)must be calculated. The analytical method of strain calculations cannot be used to deter-mine the strain state of the fatigue sample because of the hour-glass shape of the fatiguesample. The axisymmetric model of the fatigue sample, shown in Figure 4-19 was used toestimate the equivalent stress at different stress levels for fatigue samples with varyinglevels of pre-strain. To simulate fatigue test conditions for samples with 5 and 10% pre-strain, the first step was to introduce pre-strain in the samples according to the techniquediscussed in Section 4.2.2.2. In the second step, a pressure load was applied on one end ofthe axi-symmetric model, keeping the other end fixed to produce the same stress conditionsas used in the fatigue experiments. Simulations were performed at 140, 120 and 100 MPaamplitude stresses. Similarly for the fatigue samples with no pre-strain, the first step wasskipped and directly, the second step was applied. Figure 4-24 shows the variation of total fatigue life (Nf) with the equivalent stress, as rep-resented by (εmax σa/σy) in the equation (4-8) for 0, 5 and 10% pre-strained samples. In theequation (εmax σa/σy), for a stress amplitude (σa) of 120 MPa, the ratio of amplitude andyield stresses (σa/σy) is same for all the fatigue samples, irrespective of the pre-strain. Thedifferentiating parameter in the equation (εmax σa/σy) is the total principal maximum strain(εmax ). For the un-strained (0% pre-strain) fatigue samples, the total maximum principalwyaf CN =σσεmax260strain (εmax ) is same as the elastic strain whereas for the 5 and 10% pre-strained samples,the total maximum principal strain contains both the elastic and plastic strains and it isdominated by the maximum principal plastic strain. Parameters C2 and w were determined by fitting a curve to the data points shown in Figure4-24. The resulting equation to calculate fatigue life Nf for samples with no pre-strain is: (4-9)The parameters C2 and w were determined to be 0.0007 and -3.0585 respectively. A similarprocedure was followed for fatigue samples with 5 and 10% pre-strain. The C2 and wderived for the 5% pre-strain level were 1.2336 and -3.6566 whereas for the 10% pre-strained samples, they were 2.5099 and -4.274, respectively.Figure 4-24. Fatigue life for samples with 0, 5% and 10% pre-strain0.0E+002.0E+054.0E+056.0E+058.0E+051.0E+061.2E+061.4E+061.6E+060.0001 0.001 0.01 0.1 1Maximum strain x Amplitude stress/ Yield stressFatigue life (Nt)No pre-strain5% pre-strain10% pre-strain 0585.3max0007.0−=yatN σσε61Figure 4-25 shows the variation of C2 and w with the level of pre-strain applied to thefatigue sample. It can be seen from the figure that both the parameters vary linearly withthe level of pre-strain. Equations were developed to quantify the variation of parameterswith the level of pre-strain which are shown as equations (4-10) and (4-11). (4-10) (4-11)where εp is the plastic strain in percentage.Figure 4-25. Variation of constants with pre-strain0065.02509.02 −×= pC ε0553.31216.0 −×−= pw ε-5-4-3-2-101230 2 4 6 8 10 12Level of pre-strain (%)ParametersC2wLinear (C2)Linear (w)62Hence, equation (4-8) can be used in conjunction with equations (4-10) and (4-11) to deter-mine the fatigue life of cast A356 aluminum alloy from the inboard rim flange locationupto 10% level of pre-strain.4.3. SummaryThe rim indentation test provided the load versus deformation curve for the wheel. Theradial fatigue test provided strain state at various locations on the wheel during the radialfatigue test.The material data such as Young's modulus, yield strength and flow strength data for theA356 aluminum alloy used in this research project was calculated from the tensile testexperiment. The empirical fatigue life relation required to estimate the fatigue life of deformed wheelswas developed by fatigue experiments on the pre-strained samples. 635 Model Development5.1. Rim indentation modelA model of the rim indentation test was developed to predict the permanent deformation,plastic strain and residual stress distributions in a wheel following this test. The procedurefor the industrial rim indentation test was explained in Section 4.1.1. Essentially, the rimis permanently deformed by the application of a predetermined static load in a mechanicaltesting machine. To solve this non-linear contact problem between the wheel and theplaten, the commercial finite element software ABAQUS was used. This model will serveas a design tool to help engineers predict the deformation behavior of the wheels during thedesign stage. 5.1.1. GeometryThe geometry of the finished wheel was obtained in electronic format from the wheel man-ufacturing company. As the wheel is symmetric about the central plane A-A, shown in theFigure 5-1, the wheel geometry in the model may be simplified to a half section. A three dimensional model of the platen was developed in ABAQUS. The dimensions ofthe platen geometry in the model were based on the specifications provided by the wheelmanufacturing company. During the model development stage, the half section of thewheel geometry was meshed with 234,646 nodes arranged into 145,153 10-node tetrahe-dral elements. Figure 5-2 shows the finite element assembly of the wheel and the platen.The rim indentation model simulates the contact behavior between two separate bodies, thestress state developed on the wheel is a result of the load transferred to the wheel from thecontact region between the platen and the wheel, hence a fine mesh was used in the contactregion of the wheel to precisely simulate the contact process.64Figure 5-1. Finished wheel with symmetry planeFigure 5-2. Finite element assembly model65The platen geometry was meshed with 18,689 rigid elements in ABAQUS CAE. Unlikethe wheel, the platen geometry was not simplified to provide a flat surface, free of sharpedges and corners along the contact interface. If the platen were cut along the wheel sym-metry plane, contact would occur along the cut edge. The simulation of contact betweentwo such edges is not recommended as it can lead to convergence errors. Hence, the fullgeometry of the platen was used. The centre point at the top surface of the platen waschosen as the reference point for the rigid platen5.1.2. Material propertiesThe material properties used for the A356 aluminum alloy of the wheel were determinedby the tensile tests described in Section 4.2.1. Table 5-1 shows the true stress data used foralloy A356 in the model. The Young’s modulus (E) of 66.25 GPa was also determinedfrom the tensile test data. A Poisson’s ratio used of 0.3 was employed based on publishedvalues [1].Table 5-1. Flow stress A356 aluminum alloyTrue stress (MPa) Plastic strain164 0196 0.01216 0.02231 0.03244 0.04255 0.05264 0.06272 0.07280 0.08286 0.09292 0.1297 0.11302 0.12306.7 0.13310.75 0.14314.7 0.15317.2 0.16665.1.3. Initial conditionsInitially, the platen was positioned 0.5mm above the inboard flange of the wheel.During the rim indentation model development stage, residual stresses developed duringthe quench stage of the T6 heat treatment and altered in the finish machining process werenot accounted for to reduce the complexity of the model. However in earlier work at UBC,Estey [1] developed a thermal-mechanical model to estimate the residual stresses and plas-tic strain in a wheel following heat treatment. This procedure was later improved by Li [32]to include the effects of material removal during machining. After the development of therim indentation model, the procedures developed by Estey and Li were followed to gener-ate residual stress and plastic strain data for use as initial conditions in the model. 5.1.4. Boundary conditionsFigure 5-3 shows the boundary conditions for the Rim Indentation model. The nodes onthe symmetry plane of the wheel were constrained in the X-direction. This constraint wasapplied to enforce symmetry on this plane during the simulation. In the industrial rim indentation experiment, the wheel is mounted on to a shaft and held inposition by bolts tightened on to the hub, refer Figure 4-1. In the Rim Indentation model,all nodes in the bolt region of the wheel were constrained in all degrees of freedom. The platen only moves along the vertical direction in the industrial rim indentation tests.Hence, in the model, motion of the platen in the X and Y directions was constrained. Theonly motion allowed for the platen during the simulation was in the Z direction.675.1.5. Loading descriptionThe application of load in the Rim Indentation model required 3 model steps: 1) Establish contact - In the first step, contact was established between the platen and thewheel. As mentioned in the Section 5.1.3, at the start of a rim indentation test, the platenwas 0.5 mm above the inboard flange of the wheel. In the first step, the platen was moveddown a distance of 0.6 mm. The extra 0.1 mm of translation ensured planar contact wasestablished between the platen and the wheel as opposed to a point contact. Figure 5-3. Boundary conditions in rim indentation model682) Apply load - In the second step, a concentrated force was applied at the reference pointof the platen. This load in turn deformed the wheel.3) Remove load - In the third and final step, the concentrated force at the reference pointof the platen was reduced to zero. 5.1.6. Contact definitionA contact condition was defined between the surface of the platen and the inboard rimflange of the wheel. A contact condition in ABAQUS consists of two interacting surfacescalled a contact pair. The two interacting surfaces are assigned either master or slave prop-erties. The general rule is to make the stiffer surface the master surface. Thus, in the rimindentation model, the bottom surface of the platen, defined with rigid elements wasdefined as the master surface. The top surface of the inboard rim flange was defined as theslave surface (refer to Figure 5-4). Although ABAQUS offers additional contact definitionoptions, this type of surface-to-surface contact was used because it eliminated large pene-tration errors that developed using other methods. The mesh of the contact surface on thewheel was made finer than the other regions to enable smooth contact between the platenand the wheel surfaces. Coarse mesh at the contact surfaces with surface-to-surface discret-ization can lead to very high computational time. Hence, finer mesh at the contact surfaceswas used to optimize computational time with accurate results.Finite sliding of the contact surfaces relative to each other was enabled in the rim indenta-tion model. The finite sliding tracking feature allows for arbitrary sliding, separation androtation at the contact surface. This approach was suitable for the modeling work becausein the rim indentation model some degree of sliding between the master surface of theplaten and the slave surface of the wheel was expected. 69Figure 5-4. Contact pair in rim indentation model705.2. Radial fatigue test modelThe radial fatigue model was developed to simulate the stress state of the wheel during test-ing. Using the predictions of this model, the fatigue life for rim indented wheels can be pre-dicted by the fatigue life equation described in Section 4.2.2.4. A description of theindustrial radial fatigue test for wheels was presented in the literature review.5.2.1. GeometryThe same type of finished wheel as used in the rim indentation model was used in the radialfatigue model. However, for this model, a mesh of the complete wheel was necessarybecause of the rotation that occurs during the test. The wheel was meshed with 72,485nodes arranged into 39,724 10-noded quadratic tetrahedral elements, refer to Figure 5-5.The number of nodes and elements used in the development of the radial fatigue model isless than the rim indentation model discussed in Section 5.1.1, the reason for the differencedepends on the complexity of the model. In the radial fatigue model, the load applied onthe wheel is precisely defined in terms of the amount and location on the wheel as dis-cussed in Section 5.2.5. The radial fatigue model is simpler in terms of its complexity,hence a fewer number of nodes were required in this model.5.2.2. Material propertiesThe material properties used in the radial fatigue model for the A356 aluminum alloy werethe same as described in Section 5.1.2. In this model, no plastic strain was expected todevelop because the loads are relatively low. 5.2.3. Initial conditionsDuring the development of the radial fatigue model, no initial conditions were consideredto simplify the model. However as discussed earlier, residual stresses and plastic strains aredeveloped in finished wheels due to the quench and machining processes. The distributionsof these quantities were calculated based on the procedures developed by Estey [1] and Li[32] and then used as initial conditions in the final radial fatigue model. 715.2.4. Boundary conditionsWhen the wheel is mounted on to the Radial Fatigue test machine, the hub of the wheel isheld on the shaft of the machine by bolts, hence in the radial fatigue model, all the nodesin the bolt region of the wheel were constrained in all degrees of freedom (refer to Figure 5-5)5.2.5. Loading descriptionThe application of load in the Radial Fatigue model occurred in two steps:1) Apply pressure load - A pressure load of 60 psi was applied on the wheel to simulate theloading caused by inflation of the tire. Figure 5-6 shows the area on the wheel where pres-sure load is applied. The pressure load also results in a transverse load on the inboard andoutboard flange which was taken in account during the early stages of model developmentbut its effect was found to be almost negligible on the stress state of the wheel.Figure 5-5. Meshed model of the wheel with boundary conditions722) Apply radial load - The radial load due to contact with the drum (refer to Figure 4-7)was applied on the wheel and the wheel was rotated. Based on the radial fatigue tests con-ducted, the radial load applied to the wheel was 1560 kgf and the wheel was rotated at 3km/hr. As discussed in Section 2.5, when the tire is pressed against the drum, load is applied to acontact patch or area on the tire. Stearns et. al. [29] defined the effect of radial reactionforce on the wheel as a pressure load using a cosine waveform spanning a total angle of80°. The pressure load is maximum at the centre and minimum at the two ends of contactarea (refer to Figure 5-7). The load applied to the tire is applied to the inboard and outboardbead area of the wheel, shown in Figure 5-8, where the tire is in contact with the wheel.Stearns et. al. [29] also proposed the load on the bead area should be applied normal to thesurface of the beads. Hence, the radial load in the radial fatigue model was applied normalto the bead surface.Figure 5-6. Pressure load73Figure 5-7. Radial loading as a cosine waveformFigure 5-8. Bead area74To simulate the rotational motion of the wheel, the wheel must be rotated about its centrepoint. Rotating the meshed wheel requires that the coordinates of all the nodes in the wheelcontinuously change. Initially, this strategy was implemented in the model but it was foundto be computationally intensive. To avoid excessive computational times, the wheel washeld stationary and the radial load was rotated around the circumference of the wheel. Auser subroutine was written in Fortran to vary the magnitude and location of the radial loadbased on step time. This method for simulating the test was much more efficient in termsof CPU time and accuracy. Appendix B contains the input files for the two models.756 Results and Discussion6.1. Verification of the rim indentation finite element modelThe rim indentation model was verified by comparing the modeling results with the rimindentation industrial tests performed at the wheel manufacturing company site. The rimindentation tests were explained in detail in Section 4.1.1 and the finite element modelingprocedure of the same test has been explained in Section 5.1.The load - displacement curves, shown in Figure 6-1, compare the model results with themeasurements from the industrial rim indentation tests. During the industrial tests, twowheels were tested by applying 20.5 kN. Figure 6-1 shows that the modeling results are ingeneral agreement with the industrial tests. Selected values from the load-displacementcurves are compared in Table 6-1. The numerical values of deformation as well as the trendof curves exhibit a good correlation between the experiments and modeling results. The load - displacement curves of the industrial tests coincide perfectly at the beginning(see dark blue and red curves in Figure 6-1), but at a displacement of 5 mm, the curves sep-arate leading to a maximum difference of 0.55 mm at the peak load. When the load isdecreased, the difference between the final permanent displacements of the two wheelswas 0.49 mm. Although both the wheels were of the same type, the difference in the dis-placement curves for the two wheels can be attributed to stochastic differences and generalvariability in the wheel manufacturing process in factors such as melt chemistry, melt han-dling, quench conditions, etc.The predicted load - displacement curve exhibits a similar slope to the industrial tests.However, differences between model results and industrial tests are apparent from the76beginning but the maximum displacement and final permanent displacement predictionsby the model match very well with the industrial tests. Table 6-1. Comparison of deformationFigure 6-1. Comparison of modeling and experimental resultsMaximum displacement (mm) Permanent displacement (mm)Industrial test wheel #1 8.87 1.89Industrial test wheel #2 8.32 1.4Modeling results 7.63 2.0277The results from the rim indentation model are a source of information not availablethrough traditional experimental work. To characterize the stress state of the wheel duringrim indentation tests through traditional methods involves mounting strain gauges on thewheel, setting up a data acquisition system and analyzing the recorded data. The developedmodel provides more detailed information and the ability to analyze different scenarioseasily. The plastic strain developed in the wheel is of particular importance for calculatingthe fatigue life of the wheel. The industrial tests only quantify the final permanent displace-ment sustained by the wheel and provide no quantification of the plastic strain distributionin the wheel. The rim indentation model was used to estimate the amount of plastic strain sustained bythe wheel. The maximum equivalent plastic strain generated in the wheel was 10% asshown in the Figure 6-2. As shown in Figure 6-2, the plastic strain is concentrated in asmall zone on the inboard rim flange consistent with where the platen contacted the wheel.The red color on the figure shows the nodes / area which sustained the maximum amountof plastic strain. The majority of the wheel does not experience any plastic deformation, asshown by the dark blue color in the figure. The rim indentation model was used to estimatethe range of pre-strain needed for the fatigue samples used in the laboratory scale experi-ments (refer Section 4.2.2.2). On the basis of the modeling results, the levels of pre-strainselected for the laboratory scale experiments were 5 and 10%.78Figure 6-2.  Predicted equivalent plastic strain following rim indentation testing796.1.1. Contact simulationThe contact area and contact pressure are valuable pieces of information predicted by therim indentation model. For wheel designs which do not pass the rim indentation test, thisinformation can be used to modify the geometry of the rim flange of the wheel. Figure 6-3 shows the progressive change of contact area on the wheel surface during a rim indenta-tion test. The platen has been removed from the figures to provide a better view of the con-tact area on the wheel surface. Figure 6-3 (a) shows the initial condition of the model wherethe platen is not in contact with the wheel. The blue surface in the figure shows the maxi-mum area on the wheel which can come in contact with the platen and was defined as theslave surface in contact formulation. The rest of the surface of the wheel which will nevercome in contact with the platen is free of any contact properties defined for it to minimizecomputational expense. At the end of step 1, contact is established between the platen andthe wheel as shown in Figure 6-3 (b) (small light blue surface). It should be noted that thecontact zone established is a surface and not a point contact. After step 1, the load appliedto the wheel is increased to the maximum test load. Figure 6-3 (c) shows the contact areaon the wheel surface, when half of the load has been applied (~ 5 kN). Figure 6-3 (d) showsthe contact area when the full load has been applied on the wheel. Figure 6-3 (e) shows thecontact area when the load has been removed from the wheel.The maximum contact pressure and area were observed at the maximum applied load of10.25 kN. The maximum contact area on the wheel inboard flange surface was calculatedto be 19.2 mm2. As the maximum contact area involved in the rim indentation test is small,the contact pressure in the area is very high, reaching a maximum of 1100 MPa at certainnodes. This contact pressure is responsible for the deformation in the wheel.   80Figure 6-3. Contact pressure under platen predicted for rim indentation test81Figure 6-4 shows the stress state of a node in the contact region of the wheel. The node ismarked in Figure 6-2. Prior to contact between the platen and the wheel, the stresses arezero. When contact is made and load is applied, a high compressive stress develops at thenode. These high compressive stresses are shown as S-XX, S-YY and S-ZZ in the Figure6-4 where XX, YY and ZZ represent circumferential, axial and radial directions, respec-tively. At the point of maximum load, all three of the stress components S-XX, S-YY andS-ZZ are above the yield strength and attain their maximum values of 567 MPa, 395 MPaand 666 MPa, respectively. The Von-Mises stress, shown as a dark blue curve in the graph,reaches a maximum of 328 MPa. This node sustained the highest amount of equivalentplastic strain of 10%. Following removal of the load, high tensile residual stresses in thecircumferential direction remain at this location. Figure 6-4. Stress state of a node in the contact region-8.0E+02-6.0E+02-4.0E+02-2.0E+020.0E+002.0E+024.0E+020.5 1 1.5 2 2.5 3Step TimeStress (MPa)S- MisesS-XXS-YYS-ZZ826.1.2. Sensitivity analysisThe sensitivity of the rim indentation model to the parameters involved in the modelingprocess was analyzed to establish their effects. Two different parameters were analyzed:6.1.2.1. Effect of flow stress of aluminum alloy A356The rim indentation model was used to assess the sensitivity of the result to variations inthe flow stress of the aluminum alloy. The results of the rim indentation tests performedfor the wheel manufacturing company (dark blue and dashed red colored curves in theFigure 6-1), show considerable variability. Hence a sensitivity analysis was carried out onthe rim indentation model to determine the effects of varied flow stress on the rim inden-tation test and explain the variability in curves in Figure 6-1. In this analysis, the flow stresswas increased and decreased by 10% while Young's modulus and Poisson's ratio remainedthe same. Table 6-2 shows the maximum displacement, permanent displacement and maximumequivalent plastic strain data for the two cases compared to the baseline result. As shownin the table, when the flow stress was increased by 10%, the amount of equivalent plasticstrain (PEEQ) decreased to 9.5% (a decrease of 5% relative to the baseline) whereas itincreased to 11.8% (an increase of 18% relative to the baseline) with the reduction in theflow stress. It can be concluded that a decrease in flow stress level has a more pronouncedeffect on the plastic strain sustained by the wheel.The deformation sustained by the wheel was also analyzed. With an increase of 10% in theflow stress, both the maximum and permanent displacements decreased. The total defor-mation decreased to 7.29mm and plastic deformation decreased to 1.72mm, a decrease of4.42% and 14.42% from the baseline displacement, respectively. On the other hand, witha 10% decrease in the flow stress, the maximum displacement increased to 8.02mm andpermanent displacement increased to 2.41mm which is an increase of 5.11% and 19.3%from the baseline, respectively. The displacement levels also support that a decrease inflow stress has a more pronounced effect than an increase in flow stress.83Table 6-2. Comparison of rim indentation model results for different flow stress6.1.2.2. Effect of change in Young’s modulusFigure 6-1 showed the comparison of model predictions with the experimental results forthe rim indentation test. Although, the peak and permanent displacements predicted by themodel were within the acceptable range of the experiments, the slope of the curve predictedby the model differs appreciably from the experiments. It can be inferred from the differ-ence in the curves that the stiffness of the aluminum wheel in the model is higher as com-pared to the experiments. Hence, a sensitivity analysis of the Young’s modulus used in therim indentation model was conducted. The Young’s modulus (E) as determined by the ten-sile test (see Section 4.2.1) was decreased by 5% to 62.1 GPa and 10% 59.6 GPa. The flowstrength data was unchanged. The rim indentation of the wheel was simulated with thechanged material properties for the aluminum wheel. Figure 6-5 shows the load versus displacement curves for the experiments and the rimindentation model’s sensitivity to the Young’s modulus. The blue colored solid line andthe red colored dashed line are the experimental results. The dark green colored dashed linewith circle markers is the displacement curve for the baseline rim indentation model withYoung’s modulus (E) = 66.25 GPa. The solid pink colored line with square markers is therim displacement predicted by the indentation model with a 10% reduction in E (59.6 GPa).It can be clearly seen from the graph that the pink curve matches better with the experimen-tal curves than the green curve. The slope of the pink curve is closer to the slope of theexperimental curves. Also, the peak displacement predicted by the rim indentation modelfor the 10% reduced value of E, is between the peak displacements measured during theexperiments. Finally, it should also be noted that the permanent displacement estimated forFlow stress Maximum displacement (mm) Permanent displacement (mm)PEEQ(%)10% decreased 8.02 2.41 11.8Baseline 7.63 2.02 1010% increased 7.29 1.72 9.584the 10% reduced value of E is higher than the permanent displacement values of the exper-iments and the baseline model.Table 6-3 shows the numerical values of the peak and permanent displacements as pre-dicted by the rim indentation model compared with the experiments. The baseline displace-ment predictions of peak and permanent displacements, i.e. 7.63 and 2.02 mm, respectivelycorrespond to a Young’s modulus of 66.25 GPa. When the Young’s modulus wasdecreased by 5% to 62.1 GPa, the peak and permanent displacements increased to 7.88 and2.06 mm, an increase of 3.27% and 1.9% respectively. Similarly, when the Young’s mod-ulus was decreased by 10% to 59.6 GPa, the peak and permanent displacements increasedto 8.36 and 2.14 mm, an increase of 9.56% and 5.9% respectively. Figure 6-5. Comparison of sensitivity analysis results for model with different values of E85With the 10% decreased Young’s modulus, the peak displacement of 8.36 mm was inbetween the peak displacements of 8.32 and 8.87 mm, as determined through experimentsbut the permanent displacement of 2.14 mm was more than the permanent displacements,1.89 and 1.4 mm, established through experiments. Table 6-3. Comparison of rim deformation sensitivity analysis for variation in Young’s modulusAfter further investigation, an improved fit between the model results and the experimentswas found when E was decreased by 15% to 56.3 GPa and the flow strength values wereincreased by 10%, keeping the plastic strain unchanged. Figure 6-6 shows the displace-ment prediction with the model in solid orange color line with circular markers and mea-sured during the experiments in solid blue and dashed red lines. The peak displacementpredicted with the rim indentation model was 8.38 mm and the permanent displacementwas 1.87 mm, both of which lie in between the experimental data: peak displacementbetween 8.32 and 8.87mm and permanent displacement between 1.4 and 1.89mm. Maximum displacement (mm) Permanent displacement (mm)Industrial test wheel #1 8.87 1.89Industrial test wheel #2 8.32 1.4Model E=66.25 GPa 7.63 2.02Model E=62.1 GPa 7.88 2.06Model E=59.6 GPa 8.36 2.1486Figure 6-6. Sensitivity analysis with the rim indentation model for variation in E and flow strength 876.1.2.3. Numerical sensitivityThe sensitivity of the rim indentation model to element type was assessed as a measure ofthe numerical sensitivity. The model was run with 4-node linear tetrahedral (C3D4) ele-ments by removing the mid-side nodes on each of the original 10-node quadratic tetrahe-dral elements. All the other conditions such as material properties and the contactdescription were the same for both the models. Figure 6-7 shows the comparison of numer-ical model results i.e. 4-node tetrahedral elements and 10-node tetrahedral elements withthe experimental results. The load - displacement predictions are very sensitive to the ele-ment type. The size of the elements in the contact region also effect the accuracy of results. Fine meshwith smaller element size as compared to a coarser mesh results in a better distribution ofcontact pressure over the contact area. Finer mesh is also recommended to avoid situationssuch as, a node from the master surface penetrating into the salve surface. In general, second order elements with finer mesh in the contact region should be used tomodel contact behavior between two bodies.88Figure 6-7. Comparison of numerical models896.2. Verification of the radial fatigue test modelThe radial fatigue test model was verified by comparing the simulation results with theindustrial tests done on a finished wheel. The industrial tests were performed to determinethe strain state of the wheel during the radial fatigue test. The test setup, instrumentationdetails and strain curves were discussed in Section 4.1.2. The loads, boundary conditionsand simulation steps required to model this test were discussed in detail in Section 5.2. Figure 6-8 shows a comparison of the strain measurements for strain gauge 1 to the pre-dicted elastic strain from a node in a corresponding location from the radial fatigue model.The location of strain gauge 1 on the wheel is on the spoke region (see Figure 4-4). InFigure 6-5, the solid blue line shows the experimental measurements whereas the dashedred line shows the simulation results for an applied radial load of 15.2 kN. The strain curvesshown for the model results are for two complete rotations of the wheel whereas the exper-imental strain data was recorded for just under two rotations of the wheel limited by thecapacity of the DAQ system. In Figure 6-8, the variation of strain between markers A andE represents the strain cycle at this location for one complete rotation of the wheel. Whenthe wheel rotates, the cycle from A to E repeats itself, making the strain repetitive in nature.At the markers A and E, strain gauge 1 is directly under the radial load i.e. centre of thecontact patch. Markers B, C and D are placed at points on the graph equivalent to 90°, 180°and 270° from point A. The maximum tensile strain measured at this location was 6.85 x10-4 during the experiments. The applied radial load of 15.2 kN was applied in the modelto duplicate the experimental loading conditions.It is evident from the strain curves of Figure 6-8 that the numerical values of the strain pre-dicted with the radial fatigue model for a radial load of 15.2 kN are low as compared to theexperimental strain measurements. Although, the strain values are different, the trends ofthe strain curves for both the experiments and modeling results are the same. The qualita-tive comparison of the strain curves confirms the correctness of the basic modeling meth-odology. In an effort to achieve a more quantitative fit, the radial load was increased. InFigure 6-8, the curves in dark green and orange color show the strain variation at the samenode with increased radial loads. As the radial load was increased, the strain amplitude,maximum and minimum strain values increased and came closer to the experiments.90Although, there is a scope for further increase in the radial load but the results with theradial load of 23 kN were accepted based on the fit of the curve for gauge 13 located at theinboard flange location which is critical to this research work.Similarly, strain measurements for gauges 7, 10, 13 and 16 (refer to Figure 4-4, Figure 4-5 and Figure 4-6) were compared to the strain estimated by the radial fatigue model.Figure 6-9 shows a comparison of the strain gauge 7 measurements with the strain pre-dicted by the radial fatigue model for a node in the same location on the wheel. Straincurves for radial loads of 15.2, 18.2 and 23 kN are plotted. As with strain gauge 1, the pre-Figure 6-8. Comparison of predicted and measured strains for gauge 1 during radial fatigue test -4.0E-04-2.0E-040.0E+002.0E-044.0E-046.0E-048.0E-040 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Step TimeStrainExperiments -- Gauge #1RF= 15.2 KNRF= 18.2 KNRF = 23 KNA EBCD91dicted strains show the same trends. At this location, the amplitude of the predicted strainvariation matches the amplitude of the measured strains for a radial load of 23 kN. Figure 6-10 shows a comparison of the experimental measurements for strain gauge 10with the strain estimated from the radial fatigue model. A similar trend can be seen in thisfigure as in Figure 6-8 and 6-7 where the radial load 15.2 kN resulted in very low strainsand as the radial load was increased, the predicted strains approached the experimentallymeasured strains.Figure 6-9. Comparison of predicted and measured strains for gauge 7 during radial fatigue test-8.0E-04-6.0E-04-4.0E-04-2.0E-040.0E+002.0E-044.0E-046.0E-040 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Step TimeStrainExperiments -- Gauge #7RF = 15.2 KNRF = 18.2 KNRF = 23 KN92Figure 6-11 shows the strain comparison for strain gauge 13 (refer to Figure 4-5) measure-ments with the radial fatigue model. This strain gauge was strategically located at theinboard rim flange where the rim is deformed in the rim indentation test. As mentioned inSection 4.1.1.1, fatigue failure for rim indented wheels was observed at this location, accu-rate prediction of the strains in this area is very critical. It can be clearly seen in Figure 6-11, strain predicted by the radial fatigue test model atgauge 13 for a radial load of 23 kN matched with the experiments better than other radialloads. As the focus of this research work is to predict the fatigue life of rim indentedwheels, the strain characteristics at the inboard rim flange location are of paramount impor-tance. Based on this fit of the curve, it can be inferred that the radial fatigue test model witha load of 23 kN gives: i) the exact match for the maximum strain between the model andFigure 6-10. Comparison of predicted and measured strains for gauge 10 during radial fatigue test-8.0E-04-6.0E-04-4.0E-04-2.0E-040.0E+002.0E-044.0E-046.0E-048.0E-041.0E-031.2E-030 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Step TimeStrainExperiments -- Gauge # 10RF= 15.2 KNRF = 18.2 KNRF = 23 KN93the experiments and ii) reliable estimations of the strain state at the inboard rim flange areaon the wheel can be made. In the fatigue model developed to estimate the fatigue life of wheels (see Section 4.2.2),the maximum strain is one of the parameters. Hence, with a radial load of 23 kN, the radialfatigue test model provides a good estimate of the maximum strain state at the inboardflange location of the wheel. Hence, the radial fatigue test model with a radial load of 23kN was accepted.Figure 6-12 shows the strain comparison for gauge 16. The numerical values predicted bythe radial fatigue model do not match with the experimental calculations but the trend ofthe curves match. This strain gauge is placed on the spoke region of the wheel (refer toFigure 4-6). The amplitude strain of the measurements matches closely with the radialfatigue model predictions with 23 kN load. Figure 6-11. Comparison of predicted and measured strains for gauge 13 during radial fatigue test -1.0E-03-8.0E-04-6.0E-04-4.0E-04-2.0E-040.0E+002.0E-044.0E-046.0E-048.0E-041.0E-031.2E-030 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Step timeStrainExperiments -- Gauge # 13RF= 15.2 KN RF = 18.2 KNRF = 23 KN94Figure 6-12. Comparison of predicted and measured strains for gauge 16 during radial fatigue test-2.0E-03-1.5E-03-1.0E-03-5.0E-040.0E+005.0E-041.0E-030 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Step TimeStrainExperimentsRF = 15.2 KNRF= 18.2 KNRF = 23 KN956.2.1. Sensitivity analysisA sensitivity analysis was performed on the radial fatigue test model for the wheel to estab-lish the effects changes in the material properties of the wheel, specifically Young’s mod-ulus on the strain state of the wheel.6.2.1.1. Effect of change in Young’s modulusAs discussed earlier, the Young’s modulus used in the radial fatigue test model was deter-mined through a tensile test to be 66.25 GPa, it was decreased by 5% to 62.1 GPa and 10%to 59.6 GPa, to estimate the change in strain state of the wheel during the radial fatigue testwith the change in Young’s modulus. Figure 6-13 shows the change in strain state of a node present at the inboard flange locationon the wheel with variation in Young’s modulus. The radial load applied on the wheel was23 kN. The solid orange colored line shows the strain state of the node with Young’s mod-ulus equal to 66.25 GPa, the red dashed curve shows the strain state of the node withYoung’s modulus decreased by 5% to 62.1 GPa and the solid blue line with square markersshows the strain state of the node for Young’s modulus decreased by 10% to 59.6GPa. AsYoung’s modulus decreases, the maximum and minimum strains increase. With a 5%decrease in the Young’s modulus, the maximum strain increased by 5.33% and with the10% decrease in the modulus, the maximum strain increased by 13.2%. As discussed in Section 6.1.2.2, when the Young’s modulus was decreased by 15% to 56.3GPa, the rim indentation model matched well with the experiments. Hence, for the radialfatigue test model, the Young’s modulus was reduced to 56.3 GPa, as a result, the radialload required to produce the same strains measured during the experiments decreased. Forthe strain gauge 13 located at the inboard flange, the load may be was reduced to 20.5 kN.The strain comparison has been done with the strain gauge 13 because it is placed at a crit-ical location on the wheel, for this research work. As during the rim indentation test, thewheel is deformed at the inboard flange location and has been observed to fail at the samelocation during the radial fatigue test following rim indentation, the correct strain estima-tion for gauge 13 were required. 96In Section 6.2, it was established that 23 kN radial load was required in the radial fatiguetest model to match with the experiments performed at 15.2 kN radial load, a difference of50.3%. After modifying the material properties, the radial load required to match the exper-imental data was decreased to 20.5 kN which matches better with the applied load of 15.2kN.6.2.2. SummaryThe radial fatigue test model was developed to predict the stress and strain characteristicsof the wheel during the radial fatigue test. The trend of strain curves developed at 15.2 kNradial load matched with the strain measurements but the numerical values did not match.The radial fatigue load was increased to 23 kN, in order to achieve the best match in termsof strain amplitude and numerical values for the strain gauge placed at the inboard rimflange location. As the inboard rim flange is the critical location in this research project,the radial fatigue model with 23 kN was accepted. With the sensitivity analysis it wasestablished that with the Young’s modulus decreased to 56.3 GPa, the radial load requiredin the radial fatigue test model decreased to 20.5 kN. Figure 6-13. Strain state of a node present at the inboard flange with variation in Young’s modulus-1.0E-03-5.0E-040.0E+005.0E-041.0E-031.5E-030.0 0.2 0.4 0.6 0.8 1.0Time (s)StrainE=59.6 GPaE=62.1 GPaE= 66.25 GPa976.3. Fatigue life predictionOne of the main objectives of this research is to predict the fatigue life of rim indentedwheels during the radial fatigue test. This objective may be accomplished by combiningthe validated models of rim indentation and radial fatigue testing with the empirical equa-tion for fatigue life of pre-strained cast aluminum alloys developed in Section 4.2.2.4. Inthis section, the integration of the models for rim indentation, radial fatigue, quenching andmachining with the fatigue life relationship will be described and used to predict the fatiguelife of rim indented wheels. Estey [1] developed a thermo-mechanical model to predict the stress state of a wheel afterthe quench process which is part of the T6 heat treatment to which all wheels are subjected.This model was used in conjunction with a technique developed by Li [32] to estimate theeffects of the material removal that occurs during machining on the stress state of a wheel.The quench process is performed on wheels that have been rough trimmed following cast-ing, but not machined to the final geometry. Thus, the trimmed wheel geometry wasrequired for this analysis.The trimmed wheel geometry obtained from the wheel manufacturing company wasmeshed with 149,605 nodes arranged in 669,070 4-node linear tetrahedral (C3D4) ele-ments. The C3D4 elements were used to reduce the computational time as compared to 10-node quadratic tetrahedral elements required to simulate the rim indentation and radialfatigue processes. For the work done by Estey [1] and Li [32], 4-node tetrahedral elementswere shown to perform adequately, especially because their work did not involve any con-tact simulation. To assist the reader in understanding the quenching and machining modeland their integration into the overall model, a brief description of them is provided next. Itshould be emphasized that the techniques developed by Estey [1] and Li [32] have beenused in this research work without any modification, hence these models were not dis-cussed in the model development section in Chapter 5.986.3.1. Thermal model The thermal modeling technique developed by Estey [1] was used to predict the tempera-ture response of the wheel during the quench process. The quench process is part of the T6heat treatment process which follows a solutionizing step where the wheel is held at a tem-perature of 540°C to dissolve Magnesium-based precipitates. During the quench process,the wheel is rapidly cooled in a water bath to obtain a supersaturated solid solution state.The quench process was modeled by applying the temperature dependent heat transfercoefficient boundary condition reported by Estey [1] to the exposed surface of the wheel.The quench process results in high thermal gradients at raised temperatures which inducesstress and plastic deformation in the wheel.Two heat transfer processes take place simultaneously, i.e. convective heat transferbetween the wheel and the water and heat conduction within the wheel. The surface of thewheel, which comes in contact with the water first, is most effectively cooled, whereas thecore region of the wheel takes more time to cool due to conduction. The predicted temper-ature variations with time for surface, core and flange nodes are shown in Figure 6-14. Thetemperature variation shown in solid blue line is for a surface node located on the spokeregion of the wheel. As shown in Figure 6-14, from 0 to 2s, the surface node has a fastercooling rate than the other two nodes. This can be attributed to the fact that only the surfacenode comes in direct contact with water. After 2s, the cooling rate slows down because oftwo reasons; firstly, as the surface temperature is reduced, the difference in temperaturesbetween the water and the wheel surface is reduced, resulting in a lower heat transfer rateand secondly, due to heat transfer from the core of the spoke to the surface. As a result, thesurface node on the spoke maintains a temperature which is determined by the balance ofconvection and conduction heat transfer. The green dash-dot curve represents the temper-ature state of a node present on the flange of the wheel. As this node is not in direct contactwith the water, the initial cooling rate is low but due to the small thickness of the flange,after 2s the cooling rate increases. The core node is surrounded by material on all sides andmust rely on the conduction to the surface of the wheel for the cooling. It has the slowestcooling rate among the three nodes shown in Figure 6-14.99Figure 6-14. Temperature variation during quench process1006.3.2. Residual stress modelThe mechanical model, also developed by Estey [1], was used to estimate the residualstress state in the wheel after the quench process. The results from the thermal model dis-cussed in Section 6.3.1 were used as an input in the residual stress model providing thetemperature variation of the wheel during the quench process. The flow stress as a functionof temperature, strain and strain rate were acquired from Estey's [1] work. The coefficientof expansion as a function of temperature was also obtained from Estey [1]. Boundary con-ditions were used to restrict rigid body motion of the wheel. These included constrainingthe nodes on the symmetry plane of the wheel to remain on this plane as well as constrain-ing a number of nodes on the inboard rim flange to move in the plane normal to the sym-metry plane of the wheel [1].Approximately 1 mm of material is removed during machining from the inner and outersurfaces of the rim flange and the back of the spokes which results in a re-distribution ofthe residual stresses in the wheel. During the quench process, the surface of the wheel isplaced in a state of compression whereas the core region of the wheel is in tension. Uponmachining, interior locations that were once in a state of tension may be exposed and theresulting redistribution of stress may place these locations in a state of compression. Acompressive stress state on the wheel surface can retard fatigue crack growth whereas thetensile stress in the core region can aid the propagation of fatigue cracks. The technique developed by Li [32] to estimate the effect of material removal duringmachining on the final residual stress state of a finished wheel was employed in thisresearch. This required a single modeling step where the elements representing the materialcut during machining were removed from the analysis. The boundary conditions were alsochanged to restrict the motion in all the directions of the nodes in the bolt holes [32]. Figure 6-15 shows the stress state of a finished wheel i.e. after the quench, air cool andmachining processes. The figure shows that compressive stresses are developed on the sur-face whereas tensile stresses develop at the interior locations. It may also be observed thatthicker regions in the spoke developed higher tensile stresses compared to the thinner sec-tions, where compressive stresses may develop. The complex geometry near the bolt holes101developed high tensile stresses. The rim surface connected to the spoke region of the wheeldeveloped tensile stress contrary to the compressive stress developed on the rest of the sur-face. Figure 6-15. Stress state of the wheel after quench, air cool and machining processes102As the focus of this research project is the response of the inboard flange area of the wheel,the stress state of a node (see Figure 6-16) on the inboard flange is shown in Figure 6-17.As shown in Figure 6-17, tensile stresses (S-XX, S-YY and S-ZZ) are developed at thislocation during the quench process (t ≤ 20s). Small changes in the stress level are observedduring the air cool down following quenching (t = 20 - 30s). During the machining step (t=30 - 35s), the node which was in the interior of the wheel is now on the surface of thefinished wheel. The re-distribution of stress takes place and the node develops a residualcompressive stress. It will be discussed later in the chapter but out of the three stresses, S-XX, S-YY and S-ZZ (directions shown in Figure 6-16), S-XX which is in the normal direc-tion of the plane shown in Figure 6-16 plays the most important role in fatigue life estima-tion. The stress values at the node are -28, -15 and -1 MPa in the directions X, Y and Zrespectively. Figure 6-16. Node at inboard flange103The stresses developed in the wheel model used in this research project were comparedwith the works of Li [32]. The residual stresses developed were in accordance with Li'sestimations for a different wheel geometry, but similar process conditions. According to Li[32], the highest tensile stresses developed in the thick spoke region connected to the rimof the wheel, were around 90 MPa. Similar values of 70 - 90 MPa were calculated for thewheel geometry used in this research project. Li [32] also predicted compressive stressesat the surface of the wheel with a maximum stress of -30 MPa, which is similar to the valueof -32 MPa calculated in this research project.The elements used in the model to predict the residual stress state of the wheel after thequench and machining processes were linear tetrahedral (C3D4). These elements providegood performance for the thermal-stress analysis, but were found to inaccurately describethe contact conditions occurring in the rim indentation test. Thus, prior to running the rimindentation model, the residual stress state of the wheel was mapped to the wheel meshFigure 6-17. Stress state of a node on the inboard flange -40-30-20-1001020304050600 5 10 15 20 25 30 35Time (s)Stress (MPa)S-XXS-YYS-ZZQuench process Air cooling Machining104formed with 10-node tetrahedral elements (C3D10) using a utility routine available inABAQUS.6.3.3. SummaryThe residual stress state developed in the wheel during the quench and machining pro-cesses affects the in-service stress state. As discussed, compressive stresses are known toretard the fatigue crack growth rate whereas the tensile stresses can increase the fatiguecrack propagation rate. Thus an estimation of the residual stress needs to be included whenstudying fatigue crack growth behavior. 1056.3.4. Rim indentation model including residual stressThe predicted residual stress state of the wheel after the quench and machining processeswas used as an initial condition for the rim indentation model. This was added as an initialstep of 1 second duration to the rim indentation model discussed in Section 6.3.2. Similarto the other steps in this model, the duration of this step is arbitrarily set at 1 s since theinterpolation process to import the residual stress results is independent of time. Figure 6-18 shows the predicted load versus deformation plot for the rim indentation model with andwithout the initial residual stress distribution compared with the industrial experiments.The deformation curve (pink solid line with square markers) for rim indentation with resid-ual stress state coincides with the deformation curve (dashed green with round markers)for rim indentation without the residual stress. Figure 6-18. Comparison of modeling results with and without residual stress106A comparisons of the numerical values of displacement from the rim indentation modelwith and without residual stress are presented in Table 6-4. The initial residual stress doesnot affect the displacement values for the modeling results, such a behavior may be attrib-uted to the low initial residual stress and strain values present in the inboard rim flangelocation as discussed in Section 6.3.2 in relation to the node shown in Figure 6-16. Table 6-4. Comparison of displacement for the rim indentation testFigure 6-19 shows the evolution of stress at a node on the surface in the contact regionwhere the highest plastic deformation occurs. During the initial step (0 - 1s), the residualstress state predicted with the model of the quench and machining processes is imported.In the second step (1 - 2s), the platen is moved into contact with the wheel. In the third step(2 - 3s), the load is applied to the wheel generating high compressive stresses as shown inthe curves for S-XX, S-YY and S-ZZ. When the platen is removed from contact with thewheel (3 - 4s), the compressive stresses decrease and eventually evolve into tensile residualstress. When compared with the results of the rim indentation model without initial residualstress (see Figure 6-4); the stresses presented in Figure 6-19 are very similar. Maximum displacement (mm) Permanent displacement (mm)Industrial test wheel #1 8.87 1.89Industrial test wheel #2 8.32 1.4Modeling results - NO initial residual stress 7.63 2.02Modeling results with residual stress 7.64 2.07107There was no significant change in the deformation values by including the residual stressdata. As discussed earlier, in the literature review, fatigue failure is most probable in thetensile zone of the wheel with highest principal stress and strain.Figure 6-19. Stress state of a node in contact region-8.0E+02-6.0E+02-4.0E+02-2.0E+020.0E+002.0E+024.0E+020 0.5 1 1.5 2 2.5 3 3.5 4Time (s)Stress (MPa)S-XXS-YYS-ZZ1086.3.5. Radial fatigue model following rim indentationAs the final step in this analysis, the radial fatigue test was simulated including the effectsof rim indentation and the initial residual stress distribution. Figure 6-20 shows the stressvariation for one complete rotation of the radial fatigue test at the same node discussed inFigure 6-18. This particular node sustained the maximum equivalent plastic strain duringthe rim indentation test. In Figure 6-20, the Y-axis represents stress in MPa and the X-axisrepresents time in seconds. For the simulation process it was arbitrarily chosen that onecomplete rotation of the wheel takes 0.5 second, as discussed earlier that step time is arbi-trary in this simulation. Figure 6-19 shows the first four steps of 1 sec time duration each.In the fourth step, the force on the platen was removed and the indented zone on the wheeldeveloped a tensile residual stress state as shown in Figure 6-19. In the fifth step (4 - 5s),the contact condition between the platen and the wheel was removed so that in the sixthstep (5 - 6s), the platen can be displaced away from the wheel. Although, these two stepsdo not change the stress distribution of the wheel and have not been shown on the graph,these steps are important in the model so that when radial fatigue test conditions are appliedto the wheel, the platen does not interfere with the deformation in the wheel. In the seventhstep (6 - 7 s), the pressure load consistent with the tire inflation is applied and in the eighthstep (7 - 8s), radial fatigue test conditions are applied on the wheel. The radial fatigue teststep (eighth step), has been setup to simulate two complete rotations of the wheel as dis-cussed in Section 6.2, Figure 6-20 shows the stress variations at the node for one of therotations (7.5 - 8s). During the radial fatigue test, only one stress component, i.e. S-XXexhibits significant variation with time. All other stress components such as S-YY, S-ZZare nearly constant i.e. amplitude stress is very small. Appendix B contains the input file for the full analysis.109To put this result in context, the variation of stresses at the indented location of the wheelwith respect to the position of the radial load may be plotted. Figure 6-21 shows the vari-ation of principal stresses at the same node discussed in Figure 6-18, 16 and 17 with theangular variation if the radial load. At θ = 0, the radial load is applied at the bead area dia-metrically opposite the indent location. At θ = 180, the radial load is applied at the beadarea closest to the node location. The principal stress estimations are shown in Figure 6-21. The maximum principal stress (solid blue curve) coincides with the stress oriented inthe hoop direction or S-XX at this location (dashed pink curve). The intermediate and min-imum principal stresses do not change significantly with time. Fatigue crack growth willbe most pronounced in the plane perpendicular to the maximum principal stress. Thus, itFigure 6-20. Stress state of a node during radial fatigue test-500501001502002503003507.4 7.5 7.6 7.7 7.8 7.9 8 8.1Time (s)Stress (MPa) S-XXS-YYS-ZZS-XYS-XZS-YZ110can be inferred that the stress state in the wheel at this location will lead to fatigue failureconsistent with uni-axial loading conditions.The laboratory fatigue experiments discussed in Section 4.2.2 and the empirical fatigue lifeequation (4-8) developed in the same section can be used to predict the fatigue life of thewheel. It should be noted that laboratory fatigue experiments were performed at load ratio(R) of -1. Based on Figure 6-21, R is estimated to be 0.53 in the radial fatigue tests whichis very different from the fatigue experiments. Gao et. al. [2] have conducted experimentsat R=0.1 and found the empirical relation (2-5) to predict fatigue life is valid. However, theeffect of higher load ratios on the fatigue life has not been investigated. In this researchproject, it has been assumed that the equation (4-8) can be used for higher load ratios. Figure 6-21. Principal stress of the node with maximum strain-500501001502002503003500 60 120 180 240 300 360Angular position of the radial load (°)Stress(MPa)S-Max principalS-XXS-Mid PrincipalS-Min Principal111The strain variation at a node on the inboard rim flange during the one rotation of the wheelduring a radial fatigue test is shown in Figure 6-22. The two components of maximum prin-cipal total strain (E-Max Principal), the maximum principal plastic strain (PE- Max Prin-cipal) are shown in Figure 6-22. The total strain at the node is dominated by the plasticstrain component. As seen from the figure, there is no significant change in the total strainduring the radial fatigue test, it can be attributed to the fact that during the radial fatiguetest, the amount f load applied on the wheel is not high enough to change the plastic strainstate of the wheel. Figure 6-22. Strain state of the node during radial fatigue test0.0E+002.0E-024.0E-026.0E-028.0E-021.0E-011.2E-017.4 7.5 7.6 7.7 7.8 7.9 8 8.1Time (s)StrainPE-Max PrincipalE-Max PrincipalPEEQ1126.3.6. Numerical fatigue lifeEquation (4-8) can be used to predict the numerical fatigue life for the rim indented wheels.Parameters C2 and w were determined based on the predicted plastic strain. Since fatiguecracks are most likely to grow in the plane normal to maximum principal stress, the max-imum principal plastic strain was used to determine the parameters C2 and w. For the node discussed in Section 6.3.5, the parameters C2 and w were calculated withequations (4-10) and (4-11) based on the maximum principal plastic strain shown inFigure 6-22 were 1.8145 and -3.937. The total maximum strain (εmax) extracted fromFigure 6-22 is 7.16 x 10-2. The stress amplitude (σa), 74.23 MPa was calculated based onthe variation of maximum principal stress shown in Figure 6-21. The numerical fatigue lifeestimated for the rim indented wheel during the radial fatigue test using equation (4-8) is1.32x106 cycles. Based on this fatigue life estimation, this particular wheel will pass thequality assurance check. 6.3.7. VerificationTo verify the fatigue life estimation for radial fatigue test on rim deformed wheels, exper-iments need to be performed on the wheels. First the wheel should be deformed with theindentation load of 20.5 kN as discussed in Section 4.1.1 and then radial fatigue testsshould be performed on the wheel as explained in Section 4.1.2. It is critical to the verification of the fatigue life model that the number of cycles requiredto fail by the wheel are determined as accurately as possible through experiments. Whenthe wheel is rotating on the radial fatigue test machine, the experiment needs to be pausedand wheel must be tested at regular intervals for any cracks using dye-penetration test. Thisis a cumbersome job because everytime the wheel is checked for cracks, it needs to betaken off from the radial fatigue setup, the tire should be deflated and then removed fromthe wheel and after which the wheel can be checked for any cracks. If a crack is detected,the number of cycles to failure is reported in terms of the two cycles between consecutivestops. Otherwise, if no crack is detected, the wheel is again loaded on the radial fatigue test113setup with test re-started. This is a labor intensive job requiring a lot of time and the avail-ability of fatigue setup.Currently, the experiments are in progress with the two wheels which have already beendiscussed in section 4.1.1. Till now, the wheels have been tested for 1.25x106 cycles. None, of the wheels have failedyet which agrees with the predictions made by the fatigue model discussed in section 6.3.6.Further testing of the wheels has been delayed by production constraints at the wheel man-ufacturing company but there are definite plans in future at the wheel manufacturing com-pany to complete the testing work for the wheels.1146.3.8. SensitivityThis research project contributed in three main areas: 1) Rim Indentation test 2) Radialfatigue test and 3) Empirical fatigue life equation for different levels of pre-strain. Each ofthese factors can have an effect on the final fatigue life estimation for the wheels. In orderto quantify the effects of these parameters, a sensitivity analysis was performed on thefatigue life estimation for wheels.6.3.8.1. Variation in rim indentation loadIn order to quantify the sensitivity of total fatigue life (Nf) with respect to rim indentationload, the load in the rim indentation model was increased by 10% to 22.55 kN and thendecreased by 25% to 15.375 kN with respect to the baseline load of 20.5 kN. The rimindentation model was followed by radial fatigue test model with a radial load of 23 kN.In one of the cases, the radial fatigue model with 23 kN radial load was also simulated foran undeformed wheel (i.e. rim indentation load = 0 kN).Figure 6-23 shows the principal stress variation of a node in the contact region of thewheel. For the curves with non-zero rim indentation load i.e. solid blue line with circularmarkers, dashed orange line with square markers and dashed dark green line with triangu-lar markers, rim indentation loads were 22.55, 20.5 and 15.375 kN respectively. This par-ticular node on all the three curves sustained the maximum strain. Whereas for the solidred line with zero rim indentation load and hence no plastic strain, this particular node ispresent at the same location as the node discussed for non-zero rim indentation loads. Also,the red curve represents maximum amplitude stress at that particular location (Maximumprincipal stress - Minimum principal stress). As shown in the Figure 6-23, the stress ampli-tude is the same for all the four cases because there is no change in the radial fatigue loadof 23 kN but as the maximum stress is varying in the four conditions, the maximum prin-cipal total strain and maximum principal plastic strain are different for the four conditions.For the 15.375 kN rim indentation load, maximum principal plastic strain was estimated tobe 5.71x10-2. Parameters C2 and w estimated based on the maximum principal plasticstrain using equations (4-10) and (4-11) were 1.427 and -3.75, respectively. The amplitude115stress extracted from the Figure 6-23 was 72.67 MPa and maximum principal total strain(εmax) was estimated to be 5.55x10-2. Based on these values, using equation (4-8), the totalfatigue life (Nf) for the wheel was estimated to be 1.45x106 cycles. The fatigue lifeincreased by 9.8% with a decrease of 25% in the rim indentation load. Similar procedurewas carried out for 22.55 kN rim indentation load and the total fatigue life (Nf) estimatedto be 1.25x106 cycles, a decrease of 5.3%. Figure 6-24 shows the variation of total fatiguelife with change in rim indentation load for radial fatigue test load of 23 kN, as the loadincreases, the total fatigue life (Nf) decreases.The fatigue life prediction for zero rim indentation load (i.e. undeformed wheel) failing atthe inboard flange location using parameters from equation (4-9) was estimated to be22x106 cycles, which is very high as compared to the fatigue lives shown in Figure 6-24and hence is not shown in the figure.  Figure 6-23. Principal stress variation with angular position of radial load-10001002003004005000 60 120 180 240 300 360Angular position of radial load (°)Stress (MPa)Rim Indentation load =22.55kNRim Indentation load = 20.5 kNRim Indentation load = 15.375 kNRim Indentation load = 0 kN116This fatigue life estimation is only applicable, if the fatigue failure occurs at the inboardflange. For an undeformed wheel, it is highly unlikely that the wheel will fail at the inboardflange, generally, the undeformed wheels have been observed to fail at the spoke region. Figure 6-24. Variation of total fatigue life with rim indentation load1.2E+061.3E+061.3E+061.4E+061.4E+061.5E+061.5E+0615 16 17 18 19 20 21 22 23Rim indentationload (kN)Fatiigue life (N f)1176.3.8.2. Variation in radial fatigue loadTo quantify the effect of radial fatigue load on the total fatigue life (Nf), the rim indentationload was maintained at 20.5 kN and the radial fatigue test load was changed to 15.2 kNfrom 23 kN. As the rim indentation load was unaltered, the residual stress state of the nodethat sustained the maximum plastic strain is unchanged.Figure 6-25 shows the principal stress variation of the same node that sustained the maxi-mum plastic strain for the two radial fatigue test loading conditions. As expected, bothcurves have the same trend. It should also be noted that both curves have the same numer-ical value for the maxima of principal stress but the minima for the curves is dependent onthe radial load applied. As both the curves have the same maximum principal stress, thetotal principal strain (εmax) is equal for both the conditions (7.16x10-2). The maximumprincipal plastic strain for the curve with radial load 15.2 kN was estimated at 7.25x10-2whereas the same quantity for 23 kN radial load was determined to be 7.27x 10-2. Thesevalues are almost equal and show that the plastic strain sustained by the wheel is unaffectedby the radial fatigue load upto 23kN. Parameters C2 and w calculated for radial load 15.2kN are 1.8195 and -3.94 using equations (4-10) and (4-11). The amplitude stress deter-mined from Figure 6-25 for a radial load 15.2 kN is 51.67 MPa which is significantly dif-ferent than 74.23 MPa amplitude stress determined for 23 kN radial load. The fatigue lifecalculated for wheels indented with a 20.5 kN deformation load and then subjected to 15.2kN radial fatigue tests was 5.58x106 cycles. 118Following the same procedure as discussed above, fatigue life was calculated for a 17.5 and20 kN radial load. Figure 6-26 shows the variation of total fatigue life with the radialfatigue test loading. The graph follows a non-linear path where the non-linearity is intro-duced by the different amplitude stress values due to variation in radial load. Figure 6-25. Principal stress with different radial loads0501001502002503003500 60 120 180 240 300 360Angular position of the radial load, θ (°) Principal stress (MPa)Radial load= 15.2kNRadial load = 23kN119Figure 6-26. Effect of radial load on total fatigue life0.0E+001.0E+062.0E+063.0E+064.0E+065.0E+066.0E+0614 16 18 20 22 24Radial load (kN)Total fatigue life (Nf)1206.3.8.3. Variation in the empirical parametersThe empirical relationship (see equation (4-8)) was used for estimating the fatigue life ofrim indented wheels under radial fatigue test as discussed in Section 6.3.6. To estimate theeffect of change in the empirical relation (4-8) on the total fatigue life (Nf), parameters C2and w were varied keeping the amplitude stress (σa), yield strength (σy) and maximum totalstrain (εmax) constant. The numerical values used for σa, σy and εmax were obtained for arim indentation test load of 20.5 kN and radial fatigue test load condition of 23kN, asexplained in Section 6.3.6. i) Parameter C2: The numerical value of parameter C2 used in determining the fatigue lifein the previous section was 1.8145. Keeping parameter w constant at -3.937, the value ofparameter C2 was increased by 10% to 1.9959 and then decreased by 10% to 1.633. Table6-5. shows the percentage change in total fatigue life with change in parameter C2 in com-parison to baseline fatigue life of 1.33x106. From the table, it can be seen that with a 10%decrease in parameter C2 fatigue life decreased by 10.02% and with the 10% increase inparameter C2, there was a 9.95% increase in fatigue life. A linear relationship between totalfatigue life (Nf) and parameter C2 can be inferred from the analysis. The analysis can beextended to conclude that a small error in determining the value of parameter C2 will intro-duce a linearly dependent small error in the numerical value of total fatigue life (Nf). Fur-ther, a safety of factor considered in reporting the allowed total fatigue should be able toeliminate the effect of parameter C2.Table 6-5. Change in total fatigue life with parameter C2Parameter C2 Nf % change 10% decrease 1.19E+06 -10.021.8145 1.33E+06 0.0010% increase 1.46E+06 9.95121ii) Parameter w: To estimate the effect of parameter w on the total fatigue life (Nf), param-eter C2 was kept constant at 1.8145 and parameter w was changed by +/-10%. When thenumerical value of parameter w was decreased by 10% to -3.5433, the total fatigue lifedecreased by 74.15%. On the other hand, an increase in the value of parameter w by 10%to -4.33, increased the total fatigue life by 284.78%. Table 6-6 summarizes the results ofthe analysis. It can be clearly seen that the dependence of total fatigue life (Nf) on parameterw is far from linear. To ascertain the effect of the parameter w on total fatigue life (Nf),Figure 6-27 was developed which shows the percentage change in total fatigue life (Nf)with change in parameter w. The figure shows that the change in fatigue life is more pro-nounced when there is an increase in parameter w. For example, in Figure 6-27, four datapoints have been marked at +/- 2% and 6% change in parameter w. At 2% increase inparameter w, the total fatigue life increased by 31% whereas for the 2% decrease in param-eter w, the total fatigue life decreased by 23%. A factor of safety can be defined which canaccommodate error in determining the value of parameter w upto +/- 2% but if the error indetermining the value of parameter is as high as +6%, the error in estimating total fatiguelife is very high to a value of 124%. Hence, maximum care should be taken in deciding theparameter w.Table 6-6. Change in total fatigue life with parameter wParameter w Nf % change 10% decrease 3.43E+05 -74.15-3.937 1.33E+06 0.0010% increase 5.11E+06 284.78122Figure 6-27. Percentage change in fatigue life with change in parameter w-100-50050100150200250300350-10 -8 -6 -4 -2 0 2 4 6 8 10Percentage change in parameter wPercentage change in total fatigue life (Nf)1236.3.8.4. Variation in material propertiesSections 6.1.2.1 and 6.1.2.2 describe the effect of changing material properties on the rimindentation model. It was established that with a 15% decrease in Young’s modulus to 56.3GPa and a 10% increase in the flow strength, the load versus displacement curve for thewheel matched well with the experiments.It was also established in Section 6.2.1.1 that radial fatigue model is sensitive to the mate-rial properties such that with a reduced Young’s modulus to 56.3 GPa, the radial loadrequired in the radial fatigue test model to match the experiments was estimated to be 20.5kN which is still larger than 15.2 kN radial load applied in the experiments but reducedfrom 23 kN, as required with a Young’s modulus of 66.25 GPa.For the node discussed in Section 6.3.5 and the material properties, E=56.3 GPa and 10%increased flow strength, the maximum principal plastic strain is estimated to be 6.342 x 10-2. The parameters C2 and w calculated with equations (4-10) and (4-11) are 1.526 and -3.824. The total maximum strain (εmax) extracted from the radial fatigue model is 6.035x10-2. The stress amplitude (σa) is estimated to be 70 MPa based on the variation of maxi-mum principal stress during the radial fatigue test. Finally, as the flow strength data wasincreased by 10%, the Yield strength used in equation 4-8 is 180.4 MPa. Based on theabove parameters, the fatigue life estimated for the rim indented wheel during the radialfatigue test using equation (4-8) is 1.92x106 cycles. In Section 6.3.6, the predicted the fatigue life was 1.32x106 cycles based upon the materialproperties determined through the tensile test (see Section 4.2.1) whereas with the changedmaterial properties (the Young’s modulus decreased by 15% and the flow strengthincreased by 10%), the fatigue life was estimated to be 1.92x106 cycles. The percentageincrease in the fatigue life with modified material properties as compared to original prop-erties is calculated to be 45.45%. Hence, it can be inferred that the fatigue life model is sen-sitive to the material properties of the aluminum alloy used for manufacturing. 1247 Conclusions and Future Work7.1. Conclusions7.1.1. Rim indentation modelA model of an industrial rim indentation test used during quality assurance testing ofwheels was developed to predict the permanent deformation of a wheel following a speci-fied load cycle. This finite element model was developed using ABAQUS, a commercialsoftware package. The modeling results were verified through comparison to data mea-sured during industrial tests. It was established that rim indentation produces high tensilestress (~300 MPa) and plastic strains (~10%) in a localized region defined by contact withthe platen. The indented location with high residual tensile stress was recognized as thepotential site for fatigue failure. A previously developed modeling technique was used topredict the residual stress distribution following the quench and machining processes foruse as an initial condition in the rim indentation model. Their inclusion of the residualstresses did not alter the results significantly at the indentation location but provided a morecomprehensive estimation of the deformation and stress state of the wheel during indenta-tion. It was established that due to the low stress levels present at the rim flange location (-30 MPa maximum as shown in Figure 6-16), residual stresses do not affect the results ofthe rim indentation. It was also established that second order 10-node tetrahedral elementswere better suited for contact simulation of the rim indentation test as compared to linear4-node tetrahedral elements. The rim indentation model was found to be sensitive to thematerial properties used for the wheel.1257.1.2. Radial fatigue test modelA model of the radial fatigue test for wheels was developed to estimate the characteristiccyclic stress and strain variation during this test. Although, the model does not predict thecorrect maximum and minimum strains at the applied test load, it was able to qualitativelypredict the strain curves. The radial load was increased to 23 kN to obtain the same strainamplitudes through model as in experiments. With a radial load of 23 kN, the strain ampli-tude, and numerical values for maximum and minimum strain matched well with the exper-iments for the strain gauge located at the rim flange which is critical to this project. It wasalso established that the radial fatigue test model is sensitive to the material properties ofthe aluminum alloy used in the model. With a reduced Young’s modulus to 56.3 GPa, theradial load required to match the model predictions with the experimental data was reducedto 20.5 kN.7.1.3. Empirical fatigue life relationUni-axial laboratory fatigue experiments were performed on samples cut from wheels andpre-strained with 0, 5 and 10% plastic strain. It was observed that with the increase in thepre-strain level of the fatigue samples, their fatigue life decreased. The defect(s) responsi-ble for the failure of the samples were identified using a Scanning Electron Microscope(SEM) and in all case were found to be micro-porosity in close proximity to the specimensurface. The fatigue data was used to modify an empirical relationship for total fatigue lifeof cast aluminum alloys to include levels of pre-strain in the samples.1267.1.4. Combination of modelsThe rim indentation model, radial fatigue model and empirical fatigue life equation werecombined to estimate the fatigue life of a rim indented wheel during radial fatigue testing.It was observed that the rim indentation load played a major role in deciding the plasticstrain state of the wheel which was crucial for determining the parameters in the empiricalfatigue life equation. The amplitude stress and total maximum strain during a radial fatiguecycle were the parameters obtained from the radial fatigue model. It was noticed that theamount of radial load during the radial fatigue test had a direct influence on the amplitudestress for the fatigue life estimation whereas the total maximum strain was found to beindependent of the amount of radial load. The numerical value of fatigue life predicted for20.5 kN indentation load and 23 kN radial load was 1.32x106 cycles. The sensitivity of the fatigue life was analyzed based on rim indentation load, radial fatigueload, the new empirical fatigue equation and material properties. With a reduced Young’smodulus to 56.3 GPa and 10% increased flow strength, the radail fatigue life for the rimindented wheel was estimated to be 1.92x106 cycles. 1277.2. Future work7.2.1. New rim indentation testDuring the course of this research project, the wheel manufacturing company has modifiedthe Rim Indentation test. The new rim indentation test has two changes over the earlier test:1) Instead of directly deforming the wheel using a platen as done in the earlier test, now, atire mounted wheel is deformed using a wedge shaped platen. The load applied to thewedge shaped platen deforms the tire first and as the load is gradually increased, the wheelis deformed.2) In the earlier test, only the inboard rim flange was deformed and the permanent defor-mation sustained at that section was measured whereas in the new rim indentation test, boththe inboard and outboard rim flanges are deformed. As discussed in the Section 1.2, the rim indentation test was developed to assess the effectsof rim deformation on the radial fatigue life of wheels. The deformation was intended tobe similar to that sustained by a wheel following events like hitting a curb or running intoa pothole etc. In all practical cases of rim deformation, a tire is mounted on the wheel.Hence, the changes made to the earlier rim indentation test are done with a point of viewof correlating the test more closely to the driving scenarios. In light of the new developments, the current rim indentation model can be modified toinclude the tire geometry and material behavior of rubber in the model. As both the inboardand outboard rim flanges are deformed in the new test, more fatigue experiments shouldbe done, in order to modify the empirical fatigue life relation to include the fatigue prop-erties of the outboard flange of the wheel. The radial fatigue test model developed in thisresearch project can be directly applied without change to predict the stress characteristicsof the wheel for the radial fatigue test after the new rim indentation test.1287.2.2. Modified fatigue life equationThe fatigue life relation (refer equation (4-8)) considering pre-strain that was developed inthis research project is specific to the casting parameters and defects seen in the inboardrim flange of the wheel.As discussed in the Section 4.2.2.2, all the fatigue samples for developing the mathemati-cal relation for fatigue life with pre-strain were taken from the inboard rim flange of thewheel. Within acceptable variations, the SDAS, pore density, size and distribution of theeutectic particles were same in all the fatigue samples tested. This was justified as thedeveloped fatigue life equation was used to calculate the fatigue life of wheels which weredeformed at the inboard rim flange. In order to develop, a generalized equation of fatigue life with pre-strain, varying SDASand pore sizes should be tested for fatigue behavior. The developed equation would showdistinctly the effect of pre-strain on fatigue life. In the current research work, fatiguebehavior of cast aluminum alloy was analyzed for only two levels of pre-strain (5 and10%), more pre-strain levels should be examined for a better understanding of the effect ofpre-strain on fatigue life. A more detailed analysis of the effects of pre-strain on the fatiguelife of cast aluminum alloys is an underexplored interesting topic of research.The current research work also exhibited the limited amount of work available on thefatigue life estimation for cast aluminum alloys with high load ratio. The fatigue life dataavailable in the literature ranges for a load ratio of -1 to 0.1. In the current research project,due to rim indentation test, high tensile stresses are developed which give rise to high loadratio (~ 0.5) during the radial fatigue test. Hence, the fatigue life empirical relation devel-oped in future should consider the effects of high load ratios on the fatigue life.1297.2.3. Multi-axial fatigue loadingIn the radial fatigue test model of the wheel, the radial fatigue test conditions were close toa uni-axial fatigue loading conditions with the maximum principal stress varying with timewhile the intermediate and minimum principal stresses were constant during the radialfatigue test. Although, the intermediate and minimum principal stress values did notchange with time and their effects on the fatigue life of the wheels were neglected for sim-plicity but in a general case, a constant load in a direction other than the uni-axial fatigueload direction will affect the fatigue life of the sample. Hence, in future, the fatigue liferelation should be modified to include the effect of intermediate and minimum principalstresses on the fatigue life of specimen. 130References[1] Christina M. Estey, “ Thermal mechanical analysis of wheel deformation induced fromquenching,” M.A.Sc. thesis, University of British Columbia, 2004[2] Y. X. Gao, J. Z. Yi, P. D. Lee and T. C. Lindley, “The effect of porosity on the fatiguelife of cast aluminum silicon alloys,” Fatigue Fract Engng Mater Struct 27, 559–57[3] J. Z. Yi, Y. X. Gao, P. D. Lee and T. C. Lindley, “Microstructure based Fatigue LifePrediction for Cast A356-T6 aluminum -Silicon Alloys,” METALLURGICAL ANDMATERIALS TRANSACTIONS B, VOLUME 37B, APRIL 2006-301[4] Yeh-Liang Hsu, Chia-Chieh Yu, Shang-Chieh Wu, “ Developing an automated designmodification system for alumiunm disc wheels,” Proceedings of the 10th InternationalConference on Computer Supported Cooperative Work in Design, IEEE 2006[5] P. Ramamurty Raju, B. Satyanarayana , K. Ramji , K. Suresh Babu, “ Evaluation offatigue life of aluminum alloy wheels under radial loads ,” Engineering Failure Analysis14 (2007) 791–800[6] M. Firat, U. Kocabicak, "Analytical durability modeling and evaluation -complementary techniques for physical testing of automotive components," EngineeringFailure Analysis 11 (2004) 655-674[7] V. Grubisic, G. Fischer, “Automotive wheels, methods and procedures for optimaldesign and testing,” SAE Technical Paper Series. 830135; 1984:1.508–1.525[8] J. Z. Yi, Y. X. Gao, P. D. Lee and T. C. Lindley, “Scatter in fatigue life due to effectsof Porosity in Cast A356-T6 aluminum silicon Alloys,” METALLURGICAL ANDMATERIALS TRANSACTIONS A, VOLUME 34A, SEPTEMBER 2003-1879[9] Q.G. Wang, D. Apelian, D.A. Lados, “Fatigue behavior of A356-T6 aluminum castalloys. Part I. Effect of casting defects,” Journal of Light Metals 1 - Copyright TheMinerals, Metals & Materials Society (TMS), (2001) 73-84131[10] Q.G. Wang, C. J. Davidson, J. R. Friffiths and P. N. Crepeau, “Oxide Films, Pores andthe Fatigue Lives of Aluminium Alloys,” METALLURGICAL AND MATERIALSTRANSACTIONS B, VOLUME 37B, APRIL 2006-887[11] Q.G. Wang , D. Apelian, D.A. Lados, “Fatigue behavior of A356/ 357 cast aluminumcast alloys. Part II. Effect of microstructural constituents,” Journal of Light Metals 1-Copyright The Minerals, Metals & Materials Society (TMS), (2001) 85-97[12] J. Campbell, “ Castings,”First edition, Butterworth_Heinemann, Oxford, U.K. 1991[13] Ken Gall, Mark F. Horstemeyer, Brett W. Degner, David L. McDowell and JinghongFan ,"On the driving force for fatigue crack formation from inclusions and voids in a castA356 aluminum alloy," International Journal of Fracture 108: 207-233, 2001[14] J. Z. Yi, Y. X. Gao, P. D. Lee and T. C. Lindley, “ Effect of Fe-content on fatigue crackinitiation and propagation in a cast aluminum–silicon alloy (A356–T6),” Materials Scienceand Engineering A 386 (2004) 396–407[15] M. J. Couper, A. E. Neeson and J. R. Griffiths, “Casting Defects and the Fatiguebehaviour of an aluminium casting alloy,” Fatigue Fracture Engineering MaterialStructures Vol. 13 No. 3 pp. 213-227, 1990[16] D.L. McDowell, K. Gall, M.F. Horstemeyer, J. Fan, "Microstructure-based fatiguemodeling of cast A356-T6 alloy," Engineering Fracture Mechanics 70 (2003) 49-80[17] J Campbell,” An overview of the feects of bifilms on the structure and properties ofcast alloys,” METALLURGICAL AND MATERIALS TRANSACTIONS B, VOLUME37B, DECEMBER 2006-857[18] C. Nyahumwa, N.R. Green and J. Campbell, “ Influence of Casting Technique andHot Isostatic Pressing on the Fatigue of an Al-7Si-Mg Alloy,” METALLURGICAL ANDMATERIALS TRANSACTIONS A, VOLUME 32A, FEBRUARY 2001—349[19] M. J. caton, J. W. Jones, J. M. Boileau and J. E. Allison, “ The Effect of Solidificationon rate on the growth of fatigue cracks in cast 319 type aluminium alloy,”METALLURGICAL AND MATERIALS TRANSACTIONS A, VOLUME 30A,SEPTEMBER 1999-3055[20] M. J. caton, J. W. Jones, J. M. Boileau and J. E. Allison, “Use of small fatigue crackgrowth analysis in predicting the S-N response of cast aluminium alloys,” ASTMInternational 2000[21] Y. X. Gao, J. Z. Yi, P. D. Lee and T. C. Lindley, “A micro-cell model of the effect ofmicrostructure and defects on fatigue resistance in cast aluminium alloys,” 2004 ActaMaterialia Inc. Published by Elsevier Ltd132[22] B. Skallerud, T. Iveland and G. Harkegard, “Fatigue Life Assessment of AluminumAlloys with Casting Defects,” Engng Fract Mech, Vol 44, pp 857-874, 1993[23] H. Nisitani, M. Goto and N. Kawagoishi, “ A small-crack growth law and its relatedphenomena,” Eng Fract Mech 41 (4) (1992), pp. 499–513[24] A. F. Getman and Yu. K. Shtovba, “ Influence of preliminary plastic deformation onthe fatigue properties of aluminium alloys,” Energy Scientific Production Association,Moscow, translated from Problemy Prochnosti, No. 2, pp70-73, 1982[25] Kassim S. Al-Rubaie, Emerson K.L. Barroso, Leonardo B. Godefroid, “ Fatigue crackgrowth analysis of pre strained 7475- T7351 aluminum alloy,” International Journal ofFatigue, V.28, p. 129-139, 2006[26] Kassim S. Al-Rubaie , Marcio A. Del Grande, Dilermando N. Travessa and Katia R.Cardoso, “ Effect of the pre strain on fatigue life of 7050-T7451,” Material Science andEngineering, V.464, p. 141-150, 2007[27] P. Ramamurty Raju, B. Satyanarayana, K. Ramji, K. Suresh Babu, “ Evaluation offatigue life of aluminum alloy wheels under radial loads,” Engineering Failure Analysis 14(2007) 791–800[28] Yeh-Liang Hsu, Shu-Gen Wang and Tzu-Chi Liu, “ Prediction of Fatigue Failures ofAluminum Disc Wheels Using the Failure Probability Contour Based on Historical TestData,” Journal of the Chinese Institute of Industrial Engineers, Vol. 21, No. 6, pp. 551-558(2004)[29] J. Stearns, T. S. Srivatsan, X. Gao, and P. C. Lam, “Understanding the Influence ofPressure and Radial Loads on Stress and Displacement Response of a Rotating Body: TheAutomobileWheel,” International Journal of Rotating Machinery, Volume 2006, ArticleID 60193, Pages 1–8[30] U. Kocabicak and M. Firat, “Numerical analysis of wheel cornering fatigue tests,”Engineering Failure Analysis 8 (2001) 339-354[31] M H R Ghoreishy, “ Steady state rolling analysis of a radial tyre: comparison withexperimental results,”Proceedings of IMechE Vol. 220 Part D: Journal of AutomobileEngineering, 2006[32] Peifeng Li, “ Through Process Modeling of Aluminum alloy Castings to predictfatigue performance.” PhD thesis, Imperial College, London, 2006133AppendicesAppendix ASection 4.1.2.3 provided strain measurements observed for gauges 1, 4, 7, 10, 13 and 16during a radial fatigue test. These strain gauges were chosen for discussion because of theirorientations where of primary interest for the model development. The rest of strain mea-surements are discussed in this appendix.As discussed in Section 4.2.1.2, strain measurements were taken for 18 gauges at four dif-ferent conditions:1) Base level (Zero error): Strain gauge measurements for all the 18 gauges at conditions2, 3 and 4 were zero error corrected by subtracting the base level strain from them 2) Tire mounted on the wheel but not inflated 3) Tire inflated to test conditions i.e. 60 psi 4) Wheel rotated under radial fatigue test conditionsFigure 1 shows the zero corrected strain measurements for all the 18 gauges before andafter the inflation pressure of 60 psi. The diamond shaped dark blue markers in the figurerepresent the strain measurements for all the gauges after the tire has been mounted on thewheel but not inflated. The numerical strain values are very low as no load has been appliedon the wheel yet. The square shaped dark red markers in the figure show the effect of infla-tion pressure of 60 psi on the strain levels of the wheel. It can be seen from Figure 1 thatgauges 1 to 6, which are located on the inner spoke region undergo compressive strainwhereas the strain gauges on the outer spoke region i.e. 16 to 18 sustain tensile strain.Gauges 11 to 13 did not have an appreciable change in strain level because of the air pres-sure. Gauges 7 and 8 sustained tensile strain whereas gauges 9, 10, 14 and 15 sustainedcompressive strain due to the inflation pressure. 134In Chapter 4 Figures 4-7 to 4-10 showed the strain sustained by gauges 1, 4, 7, 10, 13 and16 under radial fatigue test conditions i.e. radial load of 1560 kgf and a rotational speed of3 km/ hr. This appendix presents the strain curves for the gauges not discussed in chapter 4. Figure 2 shows the strain curves for gauges 2 and 5. Their similar location on two differentgauges (see Figure 4-4) results in similar trend as shown in the figure. The numericalvalues of strain for gauge 5 are lower than gauge 2, which may be due to  the angular ori-entation of the gauges not being exactly same. Figure 3 shows the strain curves for gauges3 and 6, as expected, they also have similar trend in their curves but it  is interesting to notethat the strain values for gauge 6 are higher than gauge 3. A simple explanation for thisbehavior can be summarized as the vector addition of strains in gauge 2 and gauge 3 shouldbe equal to similar addition for gauges 5 and 6. This should also be true as the numericalstrain values for gauges 1 and 4 were equal as shown in Figure 4-7.Figure A-1.Strain measurements for 18 gauges before and after air pressure135Figure A-2. Strain curves for gauges 2 and 5Figure A-3. Strain curves for gauges 3 and 6-8.0E-04-6.0E-04-4.0E-04-2.0E-040.0E+002.0E-044.0E-046.0E-048.0E-040 1000 2000 3000 4000Data pointsStrainGauge2 Gauge 5-1.0E-03-8.0E-04-6.0E-04-4.0E-04-2.0E-040.0E+002.0E-044.0E-046.0E-048.0E-040 1000 2000 3000 4000Data pointsStrainGauge 3Gauge 6136Gauges 9 and 12 are located on the rim area of the wheel as shown in Figure 4-4. Figure 4shows the strain levels and trends for these gauges. As explained for gauges 7 and 10 inSection 4.1.2.3 and Figure 4-8, the difference in the numerical values of the strain for thegauges can be attributed to high stiffness near the spoke region for gauge 9 and compara-tively lower stiffness for gauge 12 at the window location between two spokes..Figure 5 shows the strain curves for gauges 10 and 12. A similar reasoning can be appliedto explain their behavior and numerical values as applied to gauges 9 and 12. Figure 6 shows the strain data for gauges 14 and 15. These strain gauges are located oninboard rim flange as shown in the Figure 4-5. They show higher tensile strain and wererecognized as probable fatigue failure locationsFigure A-4. Strain curves for gauges 9 and 12-6.0E-04-4.0E-04-2.0E-040.0E+002.0E-044.0E-046.0E-048.0E-040 1000 2000 3000 4000Data pointsStrainGauge 9Gauge 12137Figure A-5. Strain curves for gauges 8 and 11Figure A-6. Strain curves for gauges 14 and 15-6.0E-04-5.0E-04-4.0E-04-3.0E-04-2.0E-04-1.0E-040.0E+001.0E-042.0E-043.0E-044.0E-045.0E-040 1000 2000 3000 4000Data pointsStrainGauge 8Gauge 11-6.0E-04-4.0E-04-2.0E-040.0E+002.0E-044.0E-046.0E-048.0E-040 1000 2000 3000 4000Data pointsStrainGauge 14Gauge 15138Figure 7 shows the strain curves for gauges 17 and 18. These gauges show high compres-sive strains which is consistent with strain gauge 16, see Figure 4- 10.Figure A-7. Strain curves for gauges 17 and 18-1.4E-03-1.2E-03-1.0E-03-8.0E-04-6.0E-04-4.0E-04-2.0E-040.0E+002.0E-044.0E-046.0E-040 1000 2000 3000 4000Data pointsStrainGauge 17Gauge 18139Appendix BInput file for the analysis******************************************************************************INITIAL PARAMETERS *****************************************************************************************************************Heading** Job name: Rim_indentation_radial_fatigue**Model name: Rim_indentation_radial_fatigue_model*Preprint, echo=NO, model=NO, history=NO, contact=NO******************************************************************************DEFINITION OF FIRST PART, FULL WHEEL************************************************************************************************** PARTS***Part, name=FULL_WHEEL_MESHED*Node****Definition of the nodes on the wheel***Element, type=C3D10****Definition of elements on the wheel and their connectivity to the nodes***NSET**140** Node sets defined for boundary conditions***ELSET**** Element sets for surface definition where pressure and bead loads are applied ** Set of elements which define the slave surface on the wheel***Surface, type=ELEMENT, name=Wheel_slave_surf**Defines the slave surface on the wheel***Surface, type=ELEMENT, name= FULL_WHEEL_MESHED-1.Pressure_area**Defines the surface on the wheel where pressure is applied***Surface, type=ELEMENT, name= FULL_WHEEL_MESHED-1.Bead_area**Defines the bead surface on the wheel where radial load is applied**** Section: Section-1-_WHEEL-1_PICKEDSET9*Solid Section, elset=_WHEEL-1_PICKEDSET9, material="A356 ALLOY"1.,**Selects the full volume of the wheel to assign the material properties***End Part**End of part FULL_WHEEL_MESHED******************************************************************************DEFINITION OF SECOND PART, PLATEN ***************************141************************************************************************Part, name=Platen*Node****Definition of the nodes on the platen***Element, type=R3D3****Definition of elements on the platen and their connectivity to the nodes***NSET**** Reference node selected for applying boundary conditions and concentrated load*****ELSET**** Set of elements which define the master surface on the platen***Surface, type=ELEMENT, name=Platen_mater_surface****Master surface on the platen***End Part**  142******************************************************************************DEFINITION OF ASSEMBLY OF TWO PARTS************************************************************************************************* ASSEMBLY***Assembly, name=Assembly**  *Instance, name=FULL_WHEEL_MESHED-1, part=FULL_WHEEL_MESHED*End Instance**  *Instance, name=Platen-1, part=Platen*End Instance**** Definition of a new cylindrical coordinate system *Orientation, name=FULL_WHEEL_MESHED-1-Cyl_sys, system=CYLINDRICAL          0.,           0.,           0.,           0.,          -1.,           0.1, 0.** Constraint: Constraint-1*End Assembly** ******************************************************************************MATERIAL AND INTERACTION PROPERTIES************************************************************************************************ MATERIALS** 143*Material, name="A356 ALLOY"*Elastic 66250., 0.3*Plastic   164.,   0.   196., 0.01   216., 0.02   231., 0.03   244., 0.04   255., 0.05   264., 0.06   272., 0.07   280., 0.08   286., 0.09   292.,  0.1   297., 0.11   302., 0.12  306.7, 0.13 310.75, 0.14  314.7, 0.15  317.2, 0.16** ** INTERACTION PROPERTIES** **Definition of type of contact between the wheel and the platen144***Surface Interaction, name=IntProp-11.,*Friction0.,*Surface Behavior, pressure-overclosure=HARD** ** Interaction: Int-1*Contact Pair, interaction=IntProp-1, type=SURFACE TO SURFACE, no thickness,adjust=0.0Wheel_slave_surf, Platen-1.Platen_mater_surface******************************************************************************BOUNDARY AND INITIAL CONDITIONS***************************************************************************************************** BOUNDARY CONDITIONS** ** Name: RP_fixed Type: Displacement/Rotation*Boundary****Fix the reference node on the platen in all directions except vertical** Name: Tie_node_fixed Type: Symmetry/Antisymmetry/Encastre***Boundary****Fix the bolt holes in all degree of freedom145** **Import the initial stress state of the wheel*MAP SOLUTION, STEP=1, INC=4 ** ******************************************************************************STEP1: ACHIEVE EQUILIBIRIUM************************************************************************************************************ ** STEP: Acheive_equilibirium** *Step, name=Acheive_equilibirium*Static0.1, 1., 1e-05, 0.5** ** INTERACTIONS** ** Interaction: Int-1*Model Change, type=CONTACT PAIR, removeWheel_slave_surf, Platen-1.Platen_mater_surface** ** OUTPUT REQUESTS** *Restart, write, frequency=0** ** FIELD OUTPUT: F-Output-1146** *Output, field, frequency=4*Node OutputRF, RT, U, UR, UT*Element Output, directions=YESE, EE, PE, PEEQ, S*Contact OutputCSTRESS, *Output, history, frequency=0*End Step******************************************************************************STEP2: ESTABLISH CONTACT BETWEEN PLATEN AND WHEEL ******************************************************************************** STEP: Establish contact** *Step, name="Establish contact"*Static0.025, 1., 1e-05, 0.025** ** BOUNDARY CONDITIONS** ** Name: Move_platen_establish_contact Type: Displacement/Rotation*Boundary****Apply boundary condition on the reference node of the platen147**** ** INTERACTIONS** ** Interaction: Int-1*Model Change, type=CONTACT PAIR, addWheel_slave_surf, Platen-1.Platen_mater_surface** ** OUTPUT REQUESTS** *Restart, write, frequency=0** ** FIELD OUTPUT: F-Output-1** *Output, field, frequency=4*Node OutputRF, RT, U, UR, UT*Element Output, directions=YESE, EE, PE, PEEQ, S*Contact OutputCSTRESS, *Output, history, frequency=0*End Step****************************************************************************STEP3: APPLY RIM INDENTATION LOAD***************************148************************************************************************* STEP: Apply_load** *Step, name=Apply_load*Static0.025, 1., 1e-05, 0.05** ** BOUNDARY CONDITIONS** **Remove the displacement boundary condition from the reference node of the platen** ** LOADS** ** Name: Load_applied   Type: Concentrated force*Cload****Apply concentrated load on the reference node of the platen** ** OUTPUT REQUESTS** *Restart, write, frequency=0** ** FIELD OUTPUT: F-Output-1** *Output, field, frequency=4149*Node OutputRF, RT, U, UR, UT*Element Output, directions=YESE, EE, PE, PEEQ, S*Contact OutputCSTRESS, *Output, history, frequency=0*End Step******************************************************************************STEP4: REMOVE RIM INDENTATION LOAD************************************************************************************************** STEP: Remove_load** *Step, name=Remove_load*Static0.1, 1., 1e-05, 0.1** ** LOADS** ** Name: Load_applied   Type: Concentrated force*Cload**Modify the concentrated force load to zero** ** OUTPUT REQUESTS** 150*Restart, write, frequency=0** ** FIELD OUTPUT: F-Output-1** *Output, field, frequency=4*Node OutputRF, RT, U, UR, UT*Element Output, directions=YESE, EE, PE, PEEQ, S*Contact OutputCSTRESS, *Output, history, frequency=0*End Step******************************************************************************STEP5: REMOVE CONTACT BETWEEN WHEEL AND PLATEN********************************************************************************** STEP: Remove_contact** *Step, name=Remove_contact*Static0.1, 1., 1e-05, 1.** ** ** Interaction: Int-1*Model Change, type=CONTACT PAIR, remove151Wheel_slave_surf, Platen-1.Platen_mater_surface** ** OUTPUT REQUESTS** *Restart, write, frequency=0** ** FIELD OUTPUT: F-Output-1** *Output, field, frequency=4*Node OutputRF, RT, U, UR, UT*Element Output, directions=YESE, EE, PE, PEEQ, S*Contact OutputCSTRESS, *Output, history, frequency=0*End Step******************************************************************************STEP6: MOVE PLATEN AWAY************************************************************************************************************** STEP: Move_platen_away** *Step, name=Move_platen_away*Static0.1, 1., 1e-05, 0.5152** ** BOUNDARY CONDITIONS** ** Name: Move_platen_away Type: Displacement/Rotation*Boundary****Apply displacement boundary condition on the reference node of the platen and move **it away from the wheel** ** OUTPUT REQUESTS** *Restart, write, frequency=0** ** FIELD OUTPUT: F-Output-1** *Output, field, frequency=4*Node OutputRF, RT, U, UR, UT*Element Output, directions=YESE, EE, PE, PEEQ, S*Contact OutputCSTRESS, *Output, history, frequency=0*End Step***********************************************************************153*******RIM INDENTATION MODEL'S STEPS************************************************************************************************************RADIAL FATIGUE TEST MODEL STARTS**********************************STEP7: APPLY PRESSURE ON THE WHEEL*************************************************************************************************** STEP: Apply_pressure** *Step, name=Apply_pressure*Static0.1, 1., 1e-05, 0.25** ** LOADS** ** Name: Pressure_load_wheel   Type: Pressure*DsloadFULL_WHEEL_MESHED-1.Pressure_area, P, 0.413685** ** OUTPUT REQUESTS** *Restart, write, frequency=0** ** FIELD OUTPUT: F-Output-1** *Output, field, frequency=4*Node Output154RF, RT, U, UR, UT*Element Output, directions=YESE, EE, PE, PEEQ, S*Contact OutputCSTRESS, *Output, history, frequency=0*End Step******************************************************************************STEP8: APPLY RADIAL LOAD ************************************************************************************************************** STEP: Apply_radial_Load** *Step, name=Apply_radial_Load*Static0.025, 1., 1e-05, 0.025** ** LOADS** ** Name: Radial_Load   Type: Surface traction*Dsload, follower=NO, orientation=FULL_WHEEL_MESHED-1-Cyl_sys, constantresultant=YESFULL_WHEEL_MESHED-1.Bead_surface, TRVECNU, 1., -1., 0., 0.** ** OUTPUT REQUESTS** 155*Restart, write, frequency=0** ** FIELD OUTPUT: F-Output-1** *Output, field*Node OutputRF, RT, U, UR, UT*Element Output, directions=YESE, EE, PE, PEEQ, S*Contact OutputCSTRESS, *Output, history, frequency=0*End Step

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