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I 3.52... . 9 a??? Lfihfllwfstll‘nflal!\ . 31!?“ . \ ‘ 2.33.. .3095... {Egg .1? . Ii! .. afgfifiznx,..rus3. aagfkg 9 Hanan; . . .hfifiihufio .2. (an I. 11%.}... ‘ p9. ., fig 211%. ‘33»;ng . |(.Ill.b|l+v \i- » gum, 123;,1? . V 7 ‘ .ro. {It 4:4??33401. giving , . L Jaufiwifwllfxwuflwfififle . ‘ x h 3.3.... . t1?! . 284.1%... ‘ lllflEl'l Hull‘l 11111111111 "'Es's E 12923 01074 3346 LIBRARY ' Michigan Stan: University r”. This is to certify that the thesis entitled Residual Stress Factors in Torsional Failure Modes of Induction-Hardened Steel Axial Shafts presented by Stephen A. Zayac has been accepted towards fulfillment of the requirements for Ph.D. Materials Science degree in (a 1m ,4 “5237 Major professor Date 8’ NWWM(Q77 O-7639 1‘1, "fl-‘ll\\\\'.‘ b ' )5 Juicy,” 3:4 ..,I .‘\l/I. My new SI'ATE WWII!!!" '..‘ ‘- .‘.p~v “(A DEC, 0315998 0.1mm: 25¢ per day per item RETURNING LIBRARY MATERIALS: Place in book netum to ream charge from circulation recov RESIDUAL STRESS FACTORS IN TORSIONAL FAILURE MODES OF INDUCTION-HARDENED STEEL AXLE SHAFTS By Stephen Adam Zayac A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Metallurgy, Mechanics and Material Science 1979 ABSTRACT RESIDUAL STRESS FACTORS IN TORSIONAL FAILURE MODES INDUCTION-HARDENEgFSTEEL AXLE SHAFTS BY Stephen Adam Zayac A quantitative relation between surface residual stress and ultimate torsional strain of induction-hardened SAE 1038 steel axle shafts has been established experimentally. Shafts which met the same heat treatment specifications and had comparable ultimate torsional strengths exhibited a wide variation in shear strain at failure. Two distinct failure modes were observed: a brittle mode, controlled by a maximum shear strain criterion; and a ductile mode, controlled by void coalescence. No correlation was found between the bulk properties and the ultimate torsional strain. Residual stress measurements, obtained using a dual diffractometer technique prior to torsional testing, revealed that large local grad- ients exist in the surface residual stress distribution. Nevertheless, a mean residual stress level could be associated with each.axle shaft. This mean residual stress level, determined by an average of eight equispaced measurements on a transverse cross section, was constant along the shaft except at the flange, where heat treatment conditions vary, and at the spline, where the material was severely cold worked. Comparison of these mean residual stress measurements revealed that the angle-of—twist at failure increased as the compressive residual stress level increased, and that the failure mode was a function of this residual stress level. Analysis demonstrated that this mean residual stress level, and consequently, the angle-of—twist at failure, can be controlled by process selection and quench conditions. To my Father and Mother whose love and sacrifices have been a source of strength throughout my life. ii ACKNOWLEDGMENTS Dr. Robert Summitt, my academic advisor, whose persistent pursuit of excellence weathered this investigation to completion and whose studies of surface properties suggested the direction of this research; The late Dr. Denton McGrady, my previous academic advisor, for invaluable assistance on the practical aspects of industrial metallurgy, and for unmeasurable support, patience and a positive attitude whenever needed; Mr. Carl Langenberg of Oldsmobile Division, General Motors Corporation, for suggesting initial research areas, and for providing materials and research facilities; Mr. Jack LaBelle of Detroit Diesel Allison Division, General Motors Corporation, for technical advice and experimental assistance; Dr. Terry Triffet, for guidance in developing an understanding of continuum models and dislocation theory; Dr. K. N. Subramanian, for guidance in structural aspects of fracture and in fractography; Dr. Nicholas Altiero, for his assistance in fracture mechanics; Dr. Chuan-Tseng Wei, for developing fundamental approaches to material behavior; Dr. Donald Montgomery, whose technology assessment group provided the techniques for evaluating real-time production problems; Mr. Leo Tankersley, for sharing his practical expertise and for assistance with the scanning electron microscopic studies; Mr. Edison Searles, for his assistance in photography and final preparation, Mrs. Mary Wood, for the care and patience that she displayed in preparing this manuscript; Mrs. Thelma Liszewski, without whose experience and last—minute assistance publication would still be pending; Finally, to my friends and family, without whose love, encouragement and assistance, this work could not have been completed, and, especially to my wife, Marilyn, who sustained us through the simultaneous genesis of this thesis and births of our daughters, Nicole and Elise. TABLE OF CONTENTS Page LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . vi LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . viii NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . xi CHAPTER I INTRODUCTION . . . . . . . . . . . . . . 1 CHAPTER II BACKGROUND . . . . . . . . . . . . . . . 4 2.1 Induction Heat Treatment . . . . . . . . . . . 4 2.1.1 Heat Treatment Effects . . . . . . . . 4 2.1.2 Process Parameters . . . . . . . . . . 6 2.1.3 Process Description . . . . . . . . . . 10 2.2 Residual Stress . . . . . . . . . . . . . . . 14 2.2.1 Origins of Residual Stress . . . . . . 14 2.2.2 Residual Stress Distribution . . . . . 17 2.2.3 Residual Stress Effects . . . . . . . . 21 2.3 Fracture . . . . . . . . . . . . . . . . . . . 25 2.3.1 General Concepts . . . . . . . . . . . 25 2.3.2 Static Torsion . . . . . . . . . . . . 27 2.3.3 Structural Aspects . . . . . . . . . . 29 CHAPTER III EXPERIMENTAL . . . . . . . . . . . . . . . 32 3.0 Sample Preparation . . . . . . . . . . . . . . 32 3.1 Physical Testing . . . . . . . . . . . . . . . 35 3.1.1 Experimental Procedure . . . . . . . . 35 3.1.2 Statistical Analysis . . . . . . . . . 37 iv 3. 3. 3.2 Re 3. CHAPTER IV 4.1 Re 4.2 Re 4.3 Re CHAPTER V APPENDIX I APPENDIX II APPENDIX III APPENDIX IV 1.3 Data and Observations . 1.4 Test Results sidual Stress 2.1 Sample Preparation . . . .2.2 Measurement Theory .2.3 Experimental Apparatus .2.4 Data and Observations . .2.5 Test Results .2.6 Fractography ANALYSIS sidual Stress—-Distribution sidual Stress--First Order Effects sidual Stress--Second Order Effects CONCLUSIONS . PHYSICAL DATA . "ERRROR" PROGRAM "COMPARE" PROGRAM . STATISTICAL RESULTS . BIBLIOGRAPHY . Page 39 43 46 46 46 50 54 64 71 84 84 87 94 95 98 101 108 111 132 Number IV.1 IV.2 IV.3 IV.4 IV.5 IV.6 IV.7 IV.8 IV.9 IV.10 IV.11 IV.12 IV.13 IV.14 IV.15 IV.16 IV.17 LIST OF TABLES Angle-of-Twist-at-Failure Correlations . Summary of Experimental Results Physical Data . . . . . . . . . . . Angle-of—Twist-at-Failure Distribution . Torsional Ultimate Strength Distribution . Torsional Yield Strength Distribution Torsional Yield Point Distribution . Surface Hardness Distribution . . . Case Hardness Distribution Core Hardness Distribution Depth to Rc 45 Distribution Hardenability Distribution Step Approximation Distribution . Ramp Approximation Distribution Angle-of—Twist—at-Failure Correlation With Torsional Ultimate Strength . Angle-of—Twist—at-Failure Correlation With Torsional Yield Strength Angle-of-Twist-at-Failure Correlation With Torsional Yield Point . . . . Angle-of—Twist-at-Failure Correlation With Surface Hardness . . . . Angle-of-Twist—at-Failure Correlation With Case Hardness . Angle—of—Twist-at-Failure Correlation With Core Hardness . vi Page 44 69 98 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 Number IV.18 IV.19 IV.20 IV.21 Angle-of—Twist-at-Failure Correlation with Case Depth . . . . . . . . . . . Angle-of-Twist-at—Failure Correlation with Hardenability . . . . . . . . . Angle-of—Twist-at-Failure Correlation With Step Approximation . . . . . . Angle-of—Twist-at-Failure Correlation With Ramp Approximation . . . vii Page 128 129 130 131 LIST OF FIGURES 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. Page Torsional Stress vs Yield Strength . . . . . , , , , 5 Temperature after Induction Heat Treatment, , , , 7 Austenitic Region of the "Iron-Carbon" Phase Diagram. 9 Progressive Hardening Process . . . . . . . . . ll Single-Shot Process . . . . . . . . . . . . . . . . . 13 Specific Volumes of Steel 03 Carbon Content , , , , , 15 Dilatometric Measurements During Heat Treatment , . 16 Residual Stress Fatigue Effects . . . 18 Factor Affecting Residual Stress . , . . 20 Residual Stress Fatigue Effects , , 23 Mohr's Envelope and Combined Stress . , , 30 Rear Axle Shaft . . . . . . . . . . . . . 33 Residual Stress Analysis by the Normal-Oblique Technique . . . . . . . . . . . . . . . . . 48 "Fastress” Automated Diffractometer . , 51 "Fastress” Test Setup . . . . . . . 53 "Fastress" Output . . . . . . 55 Consistency of Mean Residual Stress for Scan-Hardened Shaft . . . . , . . 57 Consistency of Mean Residual Stress for Single-Shot Hardened Shaft.. . . . . . . . . . . . . 58 Local Residual Stress Variation for Scan-Hardened Shaft . . . . . . . . . . . . . . . . 50 Data Establishing Mean Residual Stress for Scan-Hardened Shafts , , 61 viii 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. Local Residual Stress Variation for Single-Shot Hardened Shaft . . . . . . . . . . . . . . . Data Establishing Mean Residual Stress for Single-Shot Hardened Shafts Torque-Twist Plots for Furnace Tempered Shafts Torque-Twist Plots for Induction Tempered Shafts Spline Hardness Traverse for Furnace Tempered Shafts O O O O O O O O O I O O O O O O O O Spline Hardness Traverse for Induction Tempered Shafts , . , _ , , , , , , , , Ultimate Strain Variation with Mean Residual Stress Level . . . . . . . . . . . Ductile Mode Fractograph——Radial View of Spline Fracture Surface , Ductile Mode Fractograph--Axial View of Spline Fracture Surface , , , , , , . . , Scanning Electron Micrograph of Ductile Mode Fracture Showing Shear Dimples Scanning Electron Micrograph of Ductile Mode Fracture Showing Microvoid Coalescence Brittle Mode Fractograph--Radia1 View of Spline Fracture Surface , , , , , , , . . . . . Brittle Mode Fractograph--Radia1 View of Shaft Fracture Surface , Scanning Electron Micrograph of Brittle Mode Fracture . . . . . . . . . . . . . . . Mixed Mode Fractograph--Radial View of Spline Fracture Surface Mixed Mode Fractograph--Axial View of Spline Fracture Surface 0 o o o g o o 0 ix Page 62 63 65 66 67 68 7O 72 73 74 75 77 78 79 81 82 37. 38. 39. 40. Quench Patterns Unflawed Yield Strength and Fracture Toughness and Fracture Ductile Mode Fractograph Page 85 89 9O 92 A ',A ",A '1' C. KIIIc K! C CORE 50’R150 WWW W NOMENCLATURE upper critical temperature for non-equilibrium heating AC for heating rates of 100, 500 and 3 1000 °F/sec lower critical temperature for equilibrium magnetic transition (Curie) temperature upper critical temperature for equilibrium magnetic field vector ideal critical diameter Young's modulus slope of linear region of torque-twist curve ~ shear modulus electrical current material toughness critical value of material toughness in torsion stress constant electrical resistance Rockwell "C" hardness R measured on flange R: measured 0.050 and 0.150 inch below surface torsional ultimate strength torsional yield point torsional yield strength coefficient of variation ductile-brittle residual stress level ductile 1imit--10% of transition brittle limit-#9OZ of transition flaw size inherent flaw size xi Q=ICC>’CD O B QQQQ "U "d ‘6) N Q ‘4 Q N 'Gr-IQI F3 U) E lattice spacing equilibrium lattice spacing lattice spacing for grains oriented perpendicular to the z, wx and $2 axes frequency radius time reference depth nominal strain components of strain in x, y, z,w x’ W2 directions diffraction angle diffraction angle for equilibrium spacing diffraction angle for grains oriented perpendicular to z, wx’ my directions angle—of—twist-at—failure wavelength magnetic permeability Poisson's ratio a constant-~3.l416 . . . nominal stress, electrical conductivity stress required for stable crack growth maximum interatomic stress perfect plastic fracture stress components of stress in x, y, 2, directions unflawed yield strength mean compressive residual stress level nominal shear stress angular displacement of oblique detector from normal surface energy plastic component of w elastic component of w xii I. INTRODUCTION "In our description of nature the purpose is not to disclose the real essence of the phenomena but only to track down, so far as it is possible, relations between the manifold aspects of our experience." Niels Bohr "Atomic Theory and Description of Nature" (1934) Relations between the conditions under which a component fails and its properties are a major aspect of materials science. A good engineering design must specify not only a material which will with- stand the required design loads, but also must consider the material response if stressed to failure. A major cause of failure in components is the presence of unan- ticipated residual stress. Such stresses may result from design flaw, inappropriate processing, or poor material selection. More insidious sources are variations inherent within the manufacturing process and material specification. The effect of these process variables on the residual stress state must be examined critically. Unfortunately, the effect of the various mechanical, chemical and thermal manufacturing processes on the residual stress state is not understood quantitatively for most materials and manufacturing processes. Residual stresses produced during the manufacturing process may be either tensile or compressive. Newton's Law of equilibrium for a free body under no external load, however, requires that the sum of these internal stresses be zero. This fundamental principle requires that the residual stresses generated cannot be uniform, but does not stipu- late what distribution will result. This inhomogeneity necessitates a quantitative analysis of the residual stress distribution if its importance in failure is to be established, and suggests that a single measurement may be insufficient. From the Greco-Roman period1 to contemporary times 2, cold working on forgings, bronze swords to alloy pinions, has improved component durability and ability to withstand impact. Myriad examples of fatigue life improvement exist with perhaps the world war II National Defense Research Committee Report (NA-115) being the most extensive early investigation. This report attributes fatigue life improvements of up to 700,000% to compressive residual stresses on the surface. Unfortu- nately, this report contains no residual stress measurements. Further— more, no quantitative analysis is currently available which successfully correlates residual stress variations and fracture resistance. Industrial applications and theoretical analyses, conducted by Almen at the General Motors Research Laboratories 3 show that the maximum beneficial effect is obtained with components whose surfaces bear the maximum stress. In torsion, for example, the stress, zero at the neutral axis, increases linearly to a maximum at the surface for circular shafts elastically loaded. Horger and Maulbetsh 4 report that cold-rolled railroad axles exhibit doubled fatigue life and attribute the increase to the presence of compressive residual stresses on the surface. Similarly, Osborn 5 and Shklyarov 6 credit compressive residual surface stresses induced by induction heat treatment with doubling the fatigue life of the case-hardened rear axle shafts over that of alloy shafts which possessed equivalent maximum yield strengths. Unfortunately, a quantitative evaluation of the role residual stress plays in torsional failure is not available. In this study we analyzed the proposition that for a given set of similarly processed, induction-hardened axle shafts, the residual stress variation is significant and is critical in defining the con- ditions of torsional overload failure. Measurements were made on commercially available automotive rear axle shafts. Residual stress measurements, prior to testing, were compared with the results of the standard torsion test, metallurgical and failure analysis. This choice of subject exploits the inherent symmetry and geometric simplicity of the component, the surface hardened layer, and the applied loading to reduce the complexity of the stress analysis. Furthermore, since the vase majority of rear axle shafts manufactured since World war II have been surface hardened by induction heat treatment 5, any potential cost savings or reliability improvement is important commercially. In the 1978 automotive model year, production of induction-hardened rear axle shafts, sold in the United States market, totals thirty million units 7 with an estimated value of approximately one half billion dollars II. BACKGROUND 2.1 Induction Heat Treatment 2.1.1 Heat Treatment Effects The primary effect of induction heat treatment, which must be isolated if the residual stress effects are to be understood, is the production of a case-hardened surface layer. Properly processed, this exterior case consists almost entirely (> 99%) of martensite. The extent of the case is indicated by the distance from the surface to the reference depth, 6 , where the structure is 50% martensite. For quantitative analysis, this 50% martensitic structure is measured by a Rockwell "C" hardness reading of 45. The importance of this hardened case, other than providing wear resistant bearing surfaces, is a dramatic increase in the yield strength of the case material. The effect of this increased yield strength is to provide load- bearing capabilities comparable with more expensive materials. Under torsional load, the surface experiences the maximum stress. The results of stress analysis for a circular shaft 8, illustrated in Figure 1, reveal that the stress is not constant on a cross—section but decreases linearly to zero on the central axis. Consequently, choosing a material whose yield strength matches the maximum elastic design stress is not necessary. It is sufficient that the yield strength of the material chosen exceed the stress level at every point on the cross-section. Induction hardening provides the most flexibility in selecting a yield strength distribution. Regulation of process parameters deter- mines whether yielding occurs initially at the surface, Figure 1a, or in the transition zone at the case-core interface, Figure lb. The Figure 1. a. A Yield Strength Distribution Stress Distribution A! Yield Strength Distribution E! cl Stress Distribution DI Surface 25% 50% 75% Center Depth Torsional Stress vs Yield Strength Curve ABC represents shear yield strength distribution for a deep hardened shaft. Curve AD represents the maximum stress distribution for which the shaft responds elastically. Yielding initiates at the surface (A). Curve A'B'C' represents shear yield strength distribution for a shallow hardened shaft. Curve E'D' represents the maximum stress distribution for which the shaft responds elastically. Yielding initiates at the core (3'). shaftHsresponse beyond initial yielding has been analyzed by Olszak9 for 10’11 to determine various yield strength distributions and by Klosowicz the optimum yield strength distribution. Their analyses, however, terminate with the shaft fully plastic and consider only stress and strain-rate relations. Behavior at failure is not considered; although, the important variables of the analyses, yield strength, case depth, and plastic rigidity, must be considered if the importance of residual stress at failure is to be determined. 2.1.2 Process Parameters Induction heat treatment of hardenable steels provides flexibility in obtaining optimum hardness and yield strength distributions. Heating is direct and selective throughout a cross-section rather than dependent on thermal transfer and diffusion mechanisms at the surface. With induction, heating results from the power dissipation of electrical currents induced to flow in the workpiece (IZR losses). These currents, limited by the magnetic diffusion equation, 1 2 _ §§_ no V E _ dt ’ (l) diminish exponentially with distance below the surface. The exponential constant or reference depth, 6 , where 6 = 3160(uof)'1/2, (2) defines the region within which 85% of the heating occurs. Extensive research has been done to characterize the temperature 12’13’14. Figure 2 displays the distribution for various geometries temperature distribution which should result if the Optimum heating rates as suggested by the ASM Metals Handbook 15, are employed in Temperature (°F) Figure 2. (a) (b) r3 ..4 g 'H 4b 35 CD .3 ..6 m H .17 U .83, '< 1600 "9 800 Frequency - 3kHz Power Density - 15 kw/sq in Heat Time - 3 sec 0 0.2 0.4 0.6 Surface Depth (inch) Center Temperature after Induction Heat Treat Temperature distribution for axle shaft with circular cross section as calculated using Kasper's equation (15), a 1700°F surface temperature and setup data recommended by ASM(14). The martensitic grain size is determined by the maximum temperature reached during this process. The Correspon- dence between temperature and grain size was measured by Wuerfel (21). Kasper's calculations 14 for a circular shaft with a 1.2 inch 0.D. and which reaches a maximum.surface temperature of 1700°F. This prediction compares favorably with the experimental measurements of lshii et a1 16. This controlled temperature profile enhances the steel's hardenability and increases the maximum hardness obtainable 17 because minimal heat is retained within the core to temper the surface throughout the quench cycle. Induction heating, normally at rates between 100 and 1000°F per second, supresses the diffusion controlled transformations Ae3 and A81. This results in higher solution temperatures than are common with furnace heat treatments. Figure 3 illustrates the increase which Feurstein and Smith 18 measured in the Ae upper critical temperature 19 3 for heating rates of 100, 500, and 1000°F per second. The Ae transi- 3 tion temperature increases to AC ', AC " and AC "' respectively for 3 3 3 these normalized 0.38% carbon steel forgings. Material preparations that alter a steels electrical conductivity or magnetic susceptibility also yield different solution temperatures 18. These higher solution temperatures, however, do not result in grain coarsening. The high energy input during induction heating, as opposed to the interface regulated thermal diffusion of isothermal heat treatments, requires only 0.5 to 1.0 seconds at solution temperatures to uniformly disperse the dissolved carbon 20 and thus produceszistructure with fine grain size 21. The best steels for induction hardening contain between 0.35 and 0.40% carbon and enough manganese to harden to the required depth 22. Lower carbon content produces a less saturated martensite which restricts the maximum surface hardness 23, and consequently, the maximum yield 2000 1800 1600 1400 1200 A 1 q»- 0 F 0.4 0.6 1:2 1.6 . 2.0 Weight—Percent Carbon Figure 3. Austenitic Region of the ”Iron—Carbon” Phase Diagram Austenitic region of the iron-carbon phase diagram 19 modified to illustrate the effect of rapid heating on the upper critical temperature. The Ae3 equilibrium transition temperature increases to AC3' (A) for a heating rate of 100°F/sec, to AC3" (B) for 500°F/sec, and to Ac3"' (C) for 1000°F/sec, for the 0.38% carbon steel indicated 10 strength obtainable 24. Higher carbon content results in lower strength because the resulting structures contain retained austenite 22. Higher carbon may be required for large cross—section parts, however, if additional hardenability is required. The optimum tempering cycle, a trade-off between retaining high strength and reducing quench embrittle- ment, which typifies the extreme heating and cooling rates of most induction heating processes, requires one hour at temperatures between 300 and 350°F 22. 2.1.3 Process Description Two separate induction processes produce similar results--pro- gressive and single-shot hardening. With progressive hardening equip- ment, the axle rotates constantly and moves through a circular, water- cooled copper coil, called an inductor, which establishes the heating magnetic field. The hot zone, once established, moves along the length of the shaft. A spray quench follows and progressively hardens the workpiece. This system, alternately referred to as scan hardening, provides considerable flexibility in hardening shafts of various lengths without complicated tooling changes. A schematic representation of this process is shown in Figure 4. With single-shot hardening equipment, the axle rotates constantly, heats and quenches in position. Two hot zones, induced by a focused magnetic field of a laminated, water cooled c0pper tube (the inductor), positioned along the length of the shaft, pass around the axle as it is rotated. Once the shaft reaches theaustenitizhugtemperature, heating ceases and quenching, accomplished by pressurized spray along the entire length of the shaft, hardens the entire shaft simultaneously. Single-shot ll Direction of Travel 392°F 752°F .\\\\\\\s l472°F +—-Workpiece Quench ----—-€:\\\\\ ////// Impingement Angle 30° Pressure 30 psi Temperature 70-95°F L Inductor +— Workpiece Figure 4. Progressive Hardening Process Isotherms after Ishii, Iwamato, Shiriawa and Sakamoto l6 12 hardening reduces induction heating times by up to 80% and allows rapid subsequent induction tempering. Tooling costs and change-over time, however, limit application to product lines which require minimal flexibility. A schematic representation of the single-shot process is provided in Figure 5. l3 : Ferromagnetic Inductor-———————9 Laminations Water Cooled Copper Tube Workpiece \\\\‘__flf,////1 Rotation Quench: Water Pressure 30 psi Temperature 70-95°F p" ‘r ‘ ~ ~ / Figure 5. Single—Shot Process Density of lines on upper drawing is proportional to the heating rate. 14 2.2 Residual Stress 2.2.1 Origins of Residual Stress A primary source of failure of components is the presence of unanticipated residual stress. Residual stress develops in induction hardened axle shafts as the result of rapid metallurgical changes in the exterior structure of the shafts. Localized geometric misfit, one source of residual stress, arises from variation in the specific 6)25 and from in— volumes among the various microconstituents (Figure sufficient relaxation time at grain boundaries 26. Large thermal gradiants, inherent in the process, provide another source. Dilatometric measurements (Figure 7) by Bther and Scheil 28 indi— cate that the relative coefficients of thermal expansion and the various transformation temperatures determine whether any strain results from heat treatment. Their data reveal the importance of material selections- the addition of 16% Ni to their plain carbon steel switched the strain from tension to compression. This strain in a transformed exterior case acts upon the non—transformed interior core structure to produce residual stresses within an axle shaft. The maximum residual stress that can be generated by heat treatment is a function of the material's high temperature yield strength. Plastic flow acts to relieve stresses that are induced beyond yielding. The maximum value that residual stress can reach is a function of the thermal diffusivity of the material, the characteristic temperature distribution of the process, and the time at transformation temperature. With induction heat treatment, the temperature distribution can be con— trolled to allow minimal time-at—temperature and rapid heat extraction. 15 .. Low Carbon Martensite Martensite Epsilon 130 “ Carbide Ferrite & Epsilon Carbide 128 ._ Q; Ferrite & Cementite i" 8 126 E o E a 1-1 0 >» 0 £3 124 8 Austenite c. U) 122 120 O l 2 Weight Percent Carbon Figure 6. Specific Volumes of Steel Versus Carbon Content (25) 16 17% Ni Steel 20 .. AC1 Ac3 10 I BCC .5 FCC 91 O A O O. V BCC f: O ‘3 m -10 - a 3-4 0 Lu Q) Q -20 1 .L 1 0 400 800 1200 Temperature (°F) Figure 7. Dilatometric Measurements During Heat Treatment (28) 17 This process choice inhibits plastic flow and tempering of the case to produce maximum values of residual stress. 2.2.2 Residual Stress Distribution The residual stress distributions which result from induction hardening--both scan and single-shot processes--have been measured by Vatev 29. Vatev analyzed 0.45% C, plain carbon steel shafts. His measurements, illustrated in Figure 8, indicate that the residual stress varies smoothly. Tangential and longitudinal components are comparable throughout the hardened zone. Both of these components range from a high compressive stress on the surface to zero stress at the boundary of the heat-affected zone. Tensile stresses are present in the core region. The hardness transverse also indicates that the depth to 50% martensite, as indicated by RC45, corresponds to a 50% reduction in the maximum compressive residual stress. The radial com- ponent, zero on the surface and tensile in the interior, does not exhibit a strong dependence on the cross-section structure. These measurements indicate that increases in the case depth yield increases in the value of the resultant residual stresses. It should be noted, however, that this residual stress-case depth relation was achieved by varying heating times and solution temperatures, not by varying penetration depth. Vatev's experiments also isolate residual stress variations between scan and single-shot hardening 29 and between furnace and single-shot tempering 31. Between the hardening processes, the major difference observed is in the radial component. The axles which were single-shot hardened possess significantly higher radial stresses Residual Stress Components (ksi) Figure 8. 18 80 o .0 INK. Hardness(Rc) A l l 50 t ' ' r . 50 o 100 150 200 Tangential 50 -50 -100 -150 -200 Longitudinal 100 50 ‘50 1» -100 “ Radial .15 .30 .45 .60 .75 Depth Below Surface (in) Residual Stress on Axle Cross—section A 1.6 in 0.D. plain carbon steel of 0.45C, axle single-shot hardened with 69 kHz source to depth of 0.10 in. Electroetch and x-ray diffraction techniques of Christenson 30 were used to measure residual stress. 19 throughout the cross section. The tangential and longitudinal compo- nents also differ. The single-shot hardened axles obtain higher levels of compressive stress; however, the decrease throughout the heat affected zone is more abrupt. The single-shot hardened axles also possess a less pronounced maximum in the tensile stress in the core. Between the axles tempered using the two different methods, no signi- ficant differences were observed for comparable processing. The net effect of the tempering process is a general decrease in both compres- sive and tensile residual stress values. Vatev's experiments, however, underline the complexity in achieving this equivalence. The effects that variations in shaft diameter, quench conditions and surface structure cause in the residual stress distribution have been investigated by Liss, Massieon and McKloskey 32 at Caterpillar Tractor Research. Their experiments were conducted on as-quenched shafts of circular cross-section that had been austenitized for one hour under a protective atmosphere. Residual stress was determined by x-ray diffraction techniques 30 and only the longitudinal component of the surface stress was analyzed. Figure 9 summarizes their results. For equivalent quench conditions, the compressive stress on the surface increased with increases in the shaft diameter. The increases are attributed to increased thermal plastic strains that result as the ratio of core—to-case cross-sectional area increases. This hypothesis is based on their calculation that stresses produced by the specific volume differences between martensiteamd ferrite-carbide aggregates in a 0.50% C steel should not exceed 100,000 psi. For equivalent steels, the compressive stresses on the surface increased with in- creases in quench severity. Nonuniform quench resulted in variations (a) -50 r .100.. -150 a —200 a f: 1.0 2.0 m (‘5 Bar Diameter (in) m m a O .. (b) H m H g —50 w 'u °H 3 94 “100 .1. I SAE 1345 o o m u}: -150 0 a SAE 1045 -2001E 20 30 40 50 Figure 9. Quench Pressure (psi) Factors Affecting Residual Stress Experimental measurements by Liss (32) indicating that increasing (a) bar diameter and (b) quench rate increases the maximum residual stress on a shaft's surface. 21 in surface hardness and a corresponding variation in residual stress. For similar grade steels, surface decarburization drastically dimin- ished the compressive stress obtainable on the surface. A thin layer of free ferrite on the surface reduced the surface residual stress to approximately one-third of that measured on a fully martensitic surface. 2.2.3 Residual Stress Effects The major improvement associated with the use of induction— hardened axle shafts is the fatigue life increase which case-hardened plain carbon steels exhibit over alloy steels with comparable yield strength. Industrially reported improved service life, as cited by Osborn 5, has been verified by Shklyarov 6 in a series of controlled torsion experiments, Shklyarov's experiments compare the static and fatigue properties of induction-hardened 0.40—0.45%C plain carbon steels with alloy steels of similar carbon content. Fatigue results indicate that the plain carbon, induction‘hardened axles can withstand a 100% increase in loading without decreasing fatigue life. Static torsion tests results, however, indicate no apparent residual stress dependence. Failure loads are comparable to Olszak's predictions and are correspondingly less than the alloy axles. Strain variation is not considered. Fatigue life improvement also can be measured by the endurance limit - the maximum stress at which continuous cycling produces no failure. Liss at al.32 analyzed the comparative importance of carbon content and surface residual stress on the endurance limit of steel bars. The longitudinal component of the residual stress, determined 22 by x-ray diffraction, was utilized in the comparison. Constant moment, reverse bending fatigue tests were conducted on notched circular bars 1.750 inch in diameter. As illustrated in Figure 10a, their tests indicated that surface residual stress, and not carbon content, is critical in controlling the endurance limit. Their analysis, however, does not exclude dependence on hardenability or yield strength distri- bution. Their experiments, as illustrated in Figure 10b, do analyze what effects tempering causes in fatigue. As-quenched SAE 1045 steel shafts exhibited a higher endurance limit, whereas tempered SAE 1045 steel shafts demonstrated a capacity to withstand more severe loads. Although no relation between residual stress and failure in static torsion has been observed for induction-hardened axle shafts, experimental evidence exists which relates static fracture and residual stress. Littleton's experiments 33 demonstrated a four-fold increase in the bending strength of glass when properly quenched. Photoelastic comparison of annealed and quenched glass indicated the surface of the quenched glass is in a state of compression, approximately 25 ksi, higher than the annealed glass. Littleton attributes this change in fracture resistance to the residual stress difference. Experiments by Kaplan and Rowell 34, investigating the shear con- straint and macrosc0pic fracture crietrion for ductile metals, reveal anomalous behavior in the angle-of—twist-at-failure. Their torsion measurements, conducted with 2024-T3 aluminum tubing, produced failure strains which varied from 0.17 to 0.30 on the outside diameter for equivalently machined tubes. Further testing, conducted with similarly machined tubes, which were chemically treated to diminish the effect of surface finish, produced strains which varied from 0.31 to 0.33 on 23 (a) 160 " 1045 untempered :2: 120 -> 014335 E 1045 o o 1036 + Boron o I“ 1036 0 o G U) g L”, 8360 00 -o 4 u ,5 0 o 8645 00 Quench: 8660 0Q 0Q - 011 , . . . . Others - Water —200 -120 -8'0 ' 0' 1 Residual Stress (ksi) (b) 160“ 150-. T m d k//e pere U) m " As Quenched § ,5 140 ‘// "3 ”:3 130.. .4 .z m v F, 120» 2: 110“ 104 105 106 107 Figure 10. Number of Cycles Residual Stress Fatigue Effects Experimental measurements by Liss (32) indicating; (21) Increased compressive residual stress on the surface increases endurance limit (carbon content not related to endurance limit); and, (b) Tempering increased fatigue life at high loads. 24 the outside diameter. Kaplan and Rowell purported that residual stresses introduced by machining were the source of the variation. 25 2.3 Fracture 2.3.1 General Concepts Ultimate strength is determined by the strength of interatomic bonds. Theoretically, the maximum resistance to fracture can be repre- sented as the stress required to separate adjacent planes of atoms. A model, developed by Orowan 35, equates the energy required to overcome the lattice binding energy or the potential barrier with the surface energy required to form the two new surfaces. The fracture energy is calculated by integrating the stress-displacement curve from the equilib- rium position,cr),to infinity. Orowan approximates the actual distri- bution with a triangular barrier. Then, the separation energy is equal to omAd where am is the maximum stress and Ad is the displacement at this peak stress. A strain, 8 = Ad/do,can be introduced so that the separation energy becomes deom. For materials whose deformation can be modelled as elastic, the separation energy becomes szdéE where E is Young's modulus. At rupture, the energy input to separate the material is equated to the surface energy, 2wa, on the two new surfaces. Using this energy balance, OmzdéE = 2mg, (3) the maximum fracture stress is given as O = 212000‘ (4) m dO ' The predictions of this model correspond closely to the fracture strength measured by Brenner 36 for single crystal "whiskers." Predictions, based on this model, however, grossly overestimate both yield and ultimate strengths for commercially prepared materials. This . . . 7 . . discrepancy, recognized by Griffith 3 , results from imperfect atomic 26 order. Using continuum arguments to model the fracture resistance of glass, Griffith's analysis employs a similar energy balance argument. The application of uniform stress does not produce a homogeneous stress distribution. Pre-existing f1aws--dislocations, voids, grain boundaries, inclusions, microcracks--act as stress concentrators. The strain energy released as cracks grow is equated to the surface energy of the crack. Griffith's calculations assume an elliptical flaw and produce a result similar to the atomistic approximation, O=K-—a- , (5) o where K is a geometric constant, a0 is the inherent flaw size, and m is the surface energy term. Griffith's model considers only fracture following elastic behavior and sets m = mg. Irwin 38 and Orowan extended Griffith's model to consider fracture following plastic behav- ior. Their results are similar except that m = ma + mp where mp is the energy irreversibly consumed as plastic flow per unit area. Usually, mp >> ma , so letting m = mp introduces minimal error. Predictions, based on this Griffith-Orowan-Irwin model, conform closely to observed fracture behavior 39 and form the basis of fracture mechanics. The effect of residual stress on the Griffith—Orowan—Irwin criterion has been studied analytically by Jahsman and Field 40 and experimentally by Ebert, Krotine and Troiano 41. Jahsman at al. proposed that the re- sidual stress directly affects the surface energy term, W. Their calcu- lations for the effects of residual stress in tempered glass under tension indicated a general decrease in critical stress level. Ebert et al. proposed that residual stress produces triaxial stress below the surface in tension. Their experiments with through-hardened tensile bars indicated temperature embrittlement similar to tests conducted 27 with notched tensile bars. Both analyses indicated the presence of residual stresses, whether tensile or compressive on the surface, decrease the critical fracture stress. These analyses, however, did not consider the effect of residual stress in either bending or torsion. . . . 42 . For tor81ona1 loading of a Circular shaft , fracture mechanics analysis establishes a failure criterion which is based on the size of the pre-existing flaws (a0), the material's toughness (K ), and the IIIC unflawed yield strength (0 ). Fracture occurs by unstable crack ys growth of pre-existing flaws if the local stress, 0, equals the critical stress 0C, where (6) Comparison with Orowan's and Irwin's equation suggest KIIIc is dependent on the surface energy. If the flaw size is sufficiently small, the crack stability criterion breaks down and the material fractures if its unflawed yield strength is exceeded. 2.3.2 Static Torsion Modelling of the fracture behavior of induction-hardened axle shafts under severe torsional loads must consider the possible effects of the inherent residual stress distribution. An analytical model of the com- bined stress state which would include both pure torsional loading and the actual residual stress distribution is beyond the scope of this investigation. Relationships between measurable material characteristics and observable fracture conditions are sought. The actual stress state will be modelled as a simple state of combined compression (or tension) 28 and torsion. This approximation allows utilization of well-developed analyses. Makky 44 derives a torsional failure criterion based on the potential for slip instability along the planes of principle stress--1ongitudina1, transverse, and inclined at an angle of 45° to the shaft generatrix. She dismisses fracture along the shaft's axis as not realizable; and although this surface does not seem important for static torsion, the ASM Metals Handbook 4 presents examples that demonstrate its importance in torsional fatigue. Makky finds that for rigid-plastic materials, the surface of slip instability coincides with the principle planes which intersect the shaft's axis at 45° . This rigid-plastic assumption requires minimal strain before failure and corresponds to the behavior of brittle materials 45. Makky, however, advises 46 that if the small strain con- dition breaks down, triaxial stresses, generated if expansion along the longitudinal axis is restrained, cause the surface of latent instability to coincide with the principle planes perpendicular to the shaft's axis. This loosening of constraints to allow large strains simulates the behavior of ductile materials 43. Nadai 47 cites the experimental work of B5ker, who explored the influence of hydrostatic and axial compression on the torsional failure of solid marble cylinders. Under torsion alone, the cylinders fractured along a surface which intersects the surface of the cylinder in a helix inclined at an angle of 45° with respect to the longitudinal axis. This fracture occurred by cleavage on a principle plane of maximum tensile stress. Under the application of compression during torsional tests, plastic deformation was recorded and fracture occurred on two surfaces which intersected at 45°. These fractures, one by cleavage and one by shear, corresponded to the directions of calculated principle stresses. 29 The influence of compressive stress on torsional failure, and possibly the effect of residual stresses on the torsional response of induction-hardened axle shafts, can be analyzed by strength of material arguments 48’39’49. Consider the Mohr's enve10pe, which represents a combination of maximum shear stress and maximum tensile stress failure criteria, for a brittle material. For pure torsion, and for combined tension and torsion, failure, indicated as the point tangency of the maximum principle Mohr's circle and Mohr's envelOpe in Figure 11a, occurs by cleavage as the maximum tensile strength of the material is exceeded. For combined compression and torsion, if the compressive stress is sufficient, failure, as indicated in Figure 11a, occurs if the maximum shear strength is exceeded. Next, consider the Mohr's envelOpe for a ductile material. For combined compression and torsion, and for pure torsion, failure, as indicated in Figure llb, occurs if the maximum shear strength is exceeded. For combined tension and torsion, if the tensile stress is sufficient, failure, as indicated in Figure 11b, occurs by cleavage if the maximum tensile strength of the material is exceeded. 2.3.3 Structural Aspects Both fracture mechanics and continuum arguments allow that a variation in residual stress could change the fracture behavior of induction-hardened axle shafts. Both analyses predict the presence of dual failure modes—~yie1d and crack controlled propagation. The occurrence of two independent fracture modes would have to be linked to the microstructural existence of competing mechanisms. For the torsional failure of steel shafts, experiments by Yokobori and Otsuka 50 (a) (b) Figure 11. 3O T Mohr's Envelope Mohr's Envelope and Combined Stress Mohr's envelOpe indicates maximum shear stress criteria for failure. Failure represented as point of tangency between principle Mohr's circle and envelope. (a) (b) For brittle material. Combined tenSion and torsion (i) failures by maximum tensile stress. Pure torsion (ii) failure by maximum tensile stress. Combined compression and torsion (iii) failure by maximum shear stress. For ductile materials. Combined tension and torsion (i) failure by maximum tensile stress. Pure torsion (ii) failure by maximum shear stress. Combined compression and torsion (iii) failure by maximum shear stress. 31 demonstrate the existence of dual failure modes and the dependence of failure mode on ambient temperature. Low temperature torsional failures appeared bright, crystalline and granular. The fractures occurred on a helical surface that was inclined at an angle of 45° to the shaft's generatrix. Etching the fracture surface with a 25% nitric acid solution produced square etch pits which indicate cleavage along the {100} plane. Scanning electron micrographic studies by Yokobori et a1. 51 revealed a plateau and ledge morphology typical of brittle fracture on the plane of maximum tensile stress. McClintock 52 proposed that this brittle fracture occurs as slip is blocked by pinning sites. Local stress increases until sessile dis- locations break loose. Cleavage relieves the local stress and crack growth is arrested. As the applied torque increases, the local stress again increases until the shear strength is exceeded at the next barrier. The presence of these shear lips was not investigated by Yokobori. High temperature torsional failures appeared grey, silky and fibrous. The fracture occurred on a transverse plane. Etching the fracture surface with a 25% nitric acid solution produced rectangular etch pits which indicate shear along the {110} plane. Scanning electron microscopic studies cited by Hertzberg 51 indicated ductile fracture on the plane of maximum shear stress proceeds as mobile dislocations coalesce into voids. Final fracture occurs when the effective cross- sectional area can no longer support the applied load. III. EXPERIMENTAL ,3.0 Sample Preparation The experimental work was performed on commercial grade S.A.E. 1038 steel axle shafts, forged, machined and heat—treated by Oldsmobile Division of General Motors Corporation. These shafts, illustrated in Figure 12, measured 30.5 inches in length. At the transverse section of minimal area, the spline, the outside diameter measured 1.22 inches. The spline contained 28 teeth which measured 0.050 inch deep with a minimal radius of curvature at the Spline root circle, which measured 0.015 inch. These axle shafts were hardened to Rockwell "C" 50—58 on the surface and to Rockwell ”C" 45 between 0.100 and 0.150 inch below the surface from the flange radius to the spline end of the shaft. All shafts were processed similarly, except that either of two different induction hardening processes, progressive or single-shot, were utilized. The hardening cycles chosen represent the optimal heat treatment conditions as specified in the ASM Metals Handbook 15. Both hardening cycles used three kHz power sources. For the progressive hardening process, the axle shafts were heated with an effective surface power density of fifteen kilowatts per square inch for three seconds. The hot zone was held at 1700°F for one second. Quenching was accomplished with a 30 psi, room temperature water spray incident at 30 degrees. Overall heating time was 52 seconds. For the single—shot hardening process, the axle shafts were heated with an effective surface power density of four kilowatts per square inch for twelve seconds. The hot zone was held at solution temperature for one second. Quenching was accomplished using a room temperature, 30 psi, water spray. Following 32 dues are name NH SHE 34 the hardening cycle the shafts were tempered, either one hour at 300°F ambient or equivalent induction cycle. 35 3.1 Physical Testing 3.1.1 Experimental Procedure Test samples were selected randomly from commercially available axle shafts over a three-year period which commenced in September 1972. All shafts were induction hardened and subsequently tempered. A Tinius—Olson torsion testing machine was used to evaluate the torsional performance of each shaft. The shafts were bolted and twisted at the flange end. Torque was applied by holding the spline end rigid with a mating side gear. A strain rate of 0.2 degrees per second was used. This choice, according to the analysis of Kardos 54, should not intro— duce any dynamic effects and thus allows application of static analyses. Each shaft was twisted to failure. The test instrumentation provided a plot of applied torque versus angle—of—twist and a direct display of the maximum torque and strain. Since no change in cross-sectional area accompanied the deformation, these torque-twist curves were used to determine the yield point and the yield strength. The yield strength was identified as that point on the torque-twist curve at which a line parallel to the linear region of the curve but offset two degrees intersects the experimental trace. Subsequent to failure, the extent of the hardened zone adjacent to the fracture zone was measured by a series of Rockwell "C" hardness measurements on each shaft. These measurements extended in 0.010 inch increments from a depth 0.010 inch to 0.150 inch below the surface. Case hardness was measured with a Brinell 3000 kg tester at the flange. These hardness data were used to assess the effect of the yield strength variation on failure. 36 For plain carbon steels, the relation between hardness and yield strength is well established 24. The hardness 0.050 inch below the surface was used as a measure of the maximum yield strength. The hardness 0.150 inch below the surface and the depth to Rockwell "C" 45 were chosen as measures of the case depth. The core hardness was used as the measure of the core strength. In addition, the maximum torque, which each shaft should sustain if circular and perfectly plastic, was estimated using the hardness data. A step and a ramp approximation which measured torque in RC—in2 were used to evaluate the effect of the transverse yield strength variation. The step approximation modelled the cross-sectional hardness variation as constant from the surface to the penetration depth with a value equal to that measured at a depth of 0.500 inch and as constant from the penetration depth to the shaft's midpoint with a value equal to that measured for the core. The penetration depth was taken as the depth to Rockwell ”C" 45. The ramp approximation modelled the cross- sectional hardness variation as: first, constant from the surface to a depth of 0.050 inch; second, varying linearly between the depths of 0.050 inch and 0.150 inch with the endpoint hardness as measured; and, third, varying linearly between a depth of 0.150 and the central axis with the end point hardness as measured. Then, using the cross section of minimal area where the radius equals 0.610 inch, the maximum torque, 10 n G = O'(r) rd0dr, (7) pp 0 can be calculated. For the step approximation, 37 _ . —6 2 2 Opp — n 10 {R5O(12206—6 ) + Rcore (610-5) } , (8) and for the ramp approximation, + 70,533R } (9) _ -6 o — w 10 {111,167R 150 core pp 50 + 190,400R where R are the hardness measurements at depths of 50’ R150’ and Rcore 0.050 inch, 0.150 inch and the core, respectively, and 5 is the depth to Rockwell "C" 45. Chemical analyses were performed on a random subset of the test axles. A Leco Carbon—Sulfur Analyzer was used to measure carbon and sulfur content. Spectrographic analysis was used to determine residuals. Based on the experimental work of Liss at al. 32, no direct link between chemical composition and failure should be expected, and was not evaluated. Since their analysis did not exclude failure dependence on hardenability, these chemical analyses and the algorithm of Jatczak 55 H were used to determine the ideal critical diameter" and this standard measure of hardenability was compared with the failure conditions. 3.1.2 Statistical Analyses In—process variables were monitored and regulated to yield axle shafts with equivalent behavior under load. Since some variation within the heat treatment was unavoidable, the failure loads, the maximum deformations and the variables thought to affect failure were analyzed to determine whether their distributions were random. The mean, the standard deviation, the skewness, the kurtosis and the Kolmogorov— Smirnov "d" 56 were calculated for each distribution (Appendix I) by 38 standard techniques (Appendix II). The "Errror" program compared the distribution with a normal distribution whose mean and standard distri- bution were similar, compared the actual range of the variable with the six-sigma limits, calculated the maximum confidence at which the Kolmogorov—Smirnov test indicated normality and plotted the actual distribution. This Kolmogorov-Smirnov test for normality compares the actual distribution with the associated normal distribution. The Kolmogorov- Smirnov statistic represents the maximum difference between the normal- ized actual and Gaussian distributions. Based on Lilliefor's calcu— lations 56, this statistic provides a more powerful test of normality than that provided by the standard chi-squared test. The "Errror" program tests at the .20, .15, .10, .05 and .01 levels of significance. The lowest level (.01) is tested first, and if accepted at this level, is tested at progressively higher levels to determine the maximum acceptance level. 9 Previous analyses have established the ultimate torsional strength's dependence on the yield strength variation that induction hardening induces. Since no relation has been established for the ultimate torsional strain, linear regression analysis between the angle-of—twist-at-failure and those variables indicated by previous analysis as critical were performed (Appendix III) to isolate their effects. The "Compare" program calculated the correlation coefficient, standard deviation in predicted failure angle, and the coefficients for the least squares linear approximation. _Scatter diagrams were plotted to insure the correlation calculations were not biased and to detect any trends not revealed by the calculations, Furthermore, the 39 coefficient of variation (V), determined by the ratio of the standard deviation to the mean, was used to compare the relative dispersion of the various distributions. 3.1.3 Data and Observations* The deformation and load at failure for the test group exhibited dissimilar statistical behavior. The angle-of—twist-at-failure distri- bution (Table 1) did not meet the minimum Kolmogorov-Smirnov "d" cri- terion for normality. The individual data showed much scatter as indi- cated by the high coefficient of variation, V = 0.422, for the distri— bution. The torsional ultimate strength distribution (Table 2) met the most stringent Kolmogorov-Smirnov "d" criterion for normality. The individual data were grouped closely as indicated by the low coefficient of variation, V = 0.070, for the distribution. Correlation analysis (Table 12) between the angle-of—twist—at-failure and the torsional ultimate strength yielded a correlation coefficient of —0.076. Linear regression analysis yielded a standard deviation in the failure angle estimate, which was 19.5% of the actual distribution range, and a slope which indicated that increasing the ultimate strength decreases the failure angle. Analysis of the scatter diagram confirmed the apparent independence of these physical measures of failure. The torsional yield strength distribution (Table 3) met the K01— mogorov-Smirnov "d" criterion for normality at the .01 significance level. The individual data were grouped closely as indicated by a low coefficient of variation, V = 0.093, for the distribution. Correlation * References to tables in this section refer to tables in Appendix IV. 40 analysis (Table 13) between the angle-of—twist-at-failure and the torsional yield strength yielded a correlation coefficient of 0.010. Linear regression analysis yielded a standard deviation in the failure angle estimate that was 19.6% of the actual distribution range and a slope which indicated that increasing the yield strength decreases the failure angle. Analysis of the scatter diagram revealed no observable trend in the data. The torsional yield point distribution (Table 4) met the Kolmo- gorov—Smirnov "d" criterion for normality at the .01 significance level. The individual data were grouped closely as indicated by a low coefficient of variation, V = 0.169, for the distribution. Correlation analysis (Table 14) between the angle~of~twist~at~failure and the torsional yield point yielded a correlation coefficient of 0.284. Linear regression analysis yielded a standard deviation in the failure angle estimate which was 18.8% of the actual distribution range and a slope which indicated that increasing the yield point increases the failure angle. Analysis of the scatter diagram revealed no observable trend in the data. The surface hardness distribution (Table 5) did not meet the mini- mum Kolmogorov-Smirnov criterion for normality. The individual data, however, were grouped closely as indicated by a low coefficient of vari- ation, V = 0.041, for the distribution. Correlation analysis (Table 15) between the angle-of—twist—at—failure and the surface hardness yielded a correlation coefficient of —0.085. Linear regression analysis yielded a standard deviation in the failure angle estimate which was 19.5% of the actual distribution range and a slope which indicated increasing the surface hardness decreases the failure angle. Analysis of the 41 scatter diagram revealed no observable trend in the data. The case hardness distribution (Table 6) met the Kolmogorov- Smirnov "d" criterion for normality at the .01 significance level. The individual data were grouped closely as indicated by a low coefficient of variation, V = 0.194, for the distribution. Correlation analysis (Table 16) between the angle-of—twist-at-failure and the case hardness yielded a correlation coefficient of -0.242. Linear regression analysis yielded a standard deviation in the failure angle estimate which was 18.4% of the actual distribution range and a slope which indicated that increasing case hardness decreases the failure angle. Analysis of the scatter diagram revealed no observable trend in the data. The core hardness distribution (Table 7) did not meet the minimum Kolmogorov—Smirnov "d" criterion for normality. The individual data were not scattered as indicated by the coefficient of variation, V = 0.346, for the distribution. The distribution appears single-sided. Correlation analysis (Table 17) between the angle—of—twist-at-failure and the core hardness yielded a correlation coefficient of -0.265. Linear regression analysis yielded a standard deviation in the failure angle estimate which was 19.9% of the actual distribution range and a slope which indicated that increasing case hardness decreases the fail- ure angle. Analysis of the scatter diagram revealed no observable trend in the data. The case depth distribution (Table 8) did not meet the minimum Kolomogorov—Smirnov "d" criterion for normality. The individual data were not scattered as indicated by the coefficient of variation, V = 0.150, for the distribution. Correlation analysis (Table 18) between the angle-of—twist-at-failure and the case depth yielded a 42 standard deviation in the failure angle estimate which was 18.6% of the actual distribution range and a slope which indicated that increasing the case depth decreases the failure angle. Analysis of the scatter diagram revealed no observable trend in the data. The hardenability distribution (Table 9) met the Kolmogorov— Smirnov "d" criterion for normality at the ,05 significance level. The individual data were grouped closely as indicated by a low coefficient of variation, V = 0.103, for the distribution. Correlation analysis (Table 19) between the angle-of—twist-at-failure and the hardenability yielded a correlation coefficient of -0.345. Linear regression analysis yielded a standard deviation in the failure angle estimate which was 18.5% of the actual distribution range and a Slope which indicated that increasing the hardenability decreases the failure angle. Analysis of the scatter diagram revealed no observable trend in the data. The step approximation distribution (Table 10) met the Kolmogorov- Smirnov "d" criterion at the .01 significance level. The individual data were grOUped closely as indicated by a low coefficient of variation V = 0.130, for the distribution. Correlation analysis (Table 20) between the angle-of—twist-at-failure and the step approximation yielded a correlation coefficient of -0.137. Linear regression analysis yielded a standard deviation in the failure angle estimate which was 19.2% of the actual distribution range and a slope which indicated that increases in the step approximation should result in decreases in the failure angle. Analysis of the scatter diagram revealed no observable trend in the data. The ramp approximation distribution (Table 11) did not meet the minimum Kolmogorov-Smirnov "d” criterion for normality. The individual 43 data, however, were grouped closely as indicated by a low coefficient of variation, V = 0.093, for the distribution. Correlation analysis (Table 21) between the angle-of—twist-at-failure and the ramp approxi- mation yielded a correlation coefficient of -0.290. Linear regression analysis yielded a standard deviation in the failure angle estimate which was 18.2% of the actual distribution range and a slope which indi- cated that increases in the ramp approximation should result in decreases in the failure angle. Analysis of the scatter diagram revealed no observable trend in the data. 3.1.4 Test Results No obvious correlation between the angle—of—twist-at-failure and the investigated parameters was apparent. Summarizing the test results (Table 1), however, revealed some interesting trends. Those variables that were affected by the heat treatment (i.e., all except the core hardness as measured at the flange) exhibited very little dispersion compared with the failure angle. This difference in data scatter suggested that perhaps some other variable had been overlooked. The variables, except those that specifically measured the elastic limit (i.e., both the torsional yield point and the torsional yield strength), possessed negative correlation coefficients. This difference indicated that increases in material hardness tends to reduce the ultimate twist that can be sustained, whereas raising the elastic limit tends to in- crease the twist that can be sustained. Since elastic limit increases usually accompany hardness increases, the difference in correlation trend was probably structure dependent. The slopes of the linear re- gression analyses corroborate these trends except for the torsional 44 .mms.o u > .muemsca> so samsosmsaoo aossssscumse ossssmslsauumsauumoumswea .m .mompwov am pmusmmma ousammmlumlumwsuIMOIonsm .H U mm.m1 mmm s.mm omm.1 mmo. sss mes- m .xaaa same 0 mm.s1 mms s.sm mms.- oms. mm mas- m .xaas amum mm.o- mom m.mm mam.1 mos. ms moos\es so sm.o1 0mm m.mm mmm.- oms. om ooos\ss spawn ammo 0 No.41 ems m.oo mem.1 mam. sms msee m mmmcesmm . muoo U ms.m- mmm m.mm mam.1 ems. sss oms.® a mmmeesmm ommo U mm.m1 omm m.ms mmo.- sso. sms omo.® m mamaesme oomMHSm sm.s ms s.om 4mm. mos. sms ass .a.m.e om.o1 mms m.mo oso. moo. sms ass .m.m.e os.s1 osm m.mo smo.1 Ono. sms ass .m.:.e Amuse amass maoam .uale .>mo .mooo .wooo oumm mums: manmmum> .pum .ppoo .um> oHQEmm maomumaouuoo Housammmlumlummselmoumaws< .H manme 45 yield strength; however, the correlation coefficient and the slope of the least-squares linear approximation that are associated with this variable and the angle-of—twist-at-failure were so close to zero that the change is insignificant and no trend could be ascertained. The predicted errors in the failure angle estimate, varying from :18.2%. to jfl9.6%, were remarkably consistant and quite large. This estimation error also suggested that other variables were necessary to explain the angle-of—twist-at-failure behavior. 46 3.2 Residual Stress 3.2.1 Sample Preparation Fourteen rear axle shafts were removed from production for residual stress analysis and for physical testing. The shafts comprised three test troups. The first group, consisting of samples A and B, were from the same heat and were not tempered. Sample A was induction hardened using a vertical scanning process, and sample B was hardened using the single—shot technique. The second group, consisting of samples C, D, E, F, G, and H, were from the same heat as the first group but were furnace tempered. Samples C, D, and E were hardened using a vertical scanning process, and samples F, G, and H were hardened using the single—shot technique. The third group, consisting of samples I, J, K, L, M, and N, were from a heat different than the previous groups. These samples were induction tempered. Samples I, J, and K were induction hardened using a vertical scanning process, and samples L, M, and N were induction hardened using the single-shot process. Processing within each group was sequential to duplicate the heat treatment conditions. 3.2.2 Measurement Theory The residual stress on the surface of the various induction hardened axle shafts, samples A through N, was evaluated by the normal— oblique x-ray diffractometer technique which was developed by Glocker 57. This experimental procedure, applicable to polycrystalline materials, measured the position of the diffracted x-rays for two slightly different angular exposures. These two measured angles were 47 coupled with Hooke's law and the appropriate elastic constants to calculate the strain on grains with different crystallographic orien— tations with respect to the surface. These strains and the relative invariance of the diffracted planes determined the local residual stress. In the schematic diagram (Figure 13) the residual stress is to be evaluated at point 0. A rectangular coordinate system, chosen so that the x and y axes determine the tangent plane at 0 and coincide with the directions of the principle residual stresses, simplifies the calculation. Since the z direction is perpendicular to the surface at 0 and since stresses cannot act across a free surface, the 2 component of the residual stress is equal to zero at 0. Further, since the x and y axes are the principle directions of residual stress and since this is a plane stress problem, Hooke's law yields the following relations between stress and strain for the usual conditions of homogeneity, isotropy and linear elasticity: l+v . 2 \) = o __ o _ _ + ewx 0X E Sln w E (ox 0y), (10) - 0 .____1+\) 0 ' 2 _ _\) EWY — 0y E Sln w E (OX + 0y), and (11) = _ 31 82 E (oX + 0y). (12) Subtracting Equation 12 from Equation 10 in order to eliminate the stress and strain components in the y direction yields: 5 _ e = o - 1:3- - sinzw. (13) 0x z x E Solving for the x component of the shear stress gives: 0 = —— - sinzw -(e - ez) . , (14) wx Source 48 Z \ T ID...- \ I \ / Detector Principle Directions x ? Tangent Plane Figure 13. Residual Stress Analysis by the Normal-Oblique Technique 49 This means that measurement of the strain in the normal or z direction combined with measurement of the strain in an oblique direction deter- mines the stress state uniquely. In the neighborhood about point 0, grains, oriented at the appropriate angle with respect to the surface, diffract the incident x-rays at an angle 02. Since the surface is stressed the lattice spacing differs from the standard lattice parameter. The residual strain in the normal direction, expressed in terms of the lattice parameter is: e = —EL———13 . (15) In the oblique direction, grains at a different orientation diffract the x-rays incident at angle 6W . Since the stress that acts on these x grains is the same, the lattice spacing, reflecting the change in orientation of the crystallographic planes, is different. Relating this change in spacing to the residual strain in this oblique direction yields: 1.. ”‘21— (16> Combining Equations 15 and 16, the difference in strain between the normal and oblique directions is: d - d . .=—L—— z (17> wx z do . . 58 . . . Prev1ous x-ray experiments that measured lattice parameter variation under stress indicate that the change in lattice parameter is less than 1%. Using the Bragg law, A ZSin0 d = , (18) 50 to represent the lattice spacing in terms of the diffracted angle, the variation in lattice spacing may be represented by: . (19) These relationships between the Bragg angle and the lattice spacing allow the residual stress to be written: 6 — e = cote (0 z o WK - 02) . (20) 0x substituting Equation 20 into Equation 14, the residual stress at 0 may be expressed: cote (0 - 02) . (21) wx sin w For a specific material and experimental setup, E, v, 00, and w are are constant and the residual stress' dependence on the diffraction angles, zewx and 202, 18 Simply: _ 1 __ Ox — K (20 202) , (22) 0x where K', known as the stress constant, is a property of the material examined and the experimental setup only. Thus, evaluation of the residual stress on the surface requires measurements of two angles 20 and 20 to determine 0 and two angles 20 and 20 to determine 0x 2 x my 2 O' . V 3.2.3 Experimental Apparatus An American Analytical Fastress machine was utilized to measure the , 59 . . . surface reSidual stress . This deVice (Figure 14) used two chromium- source x-ray diffraction tubes with an effective penetration depth of 0.5 x 10-3 inch. The incident x-ray beams covered an area Figure 14 "Fastress" Automated Diffractometer. 52 which measured 0.090 inch in diameter. Appropriate shimming of the v-block fixtures insured that the plane which is determined by the x-ray beams normally intersects the plane which is tangent to the shaft. An indicator gauge, fixed with respect to the x-ray sources and graduated in 0.1 x 10.3 inch intervals, was used to position the shaft's surface in the focal plane of the x-ray beams. Linear and angular position along the shaft are referenced from the machined end surface that is adjacent to the spline. The Fastress automates the normal-oblique diffractometer measure— ments. Two separate x-ray sources and detectors, illustrated in Figure 15, are positioned to measure the diffracted intensity from those planes which are oriented parallel to and 45° to the surface, and that are oriented normally to that plane which contains the incident x—rays. Each detector contains two x—ray sensors, co-planar with the incident x—ray beams and separated by an angular distance which corresponds to the width of the diffraction peak at half maximum. Each sensor gener— ates a voltage proportional to the incident diffracted intensity. Within each detector unit, an error signal is generated if the dif— fracted beam is off center and the signal strength is prOportional to the difference in diffracted intensities registered at the sensors. This error signal directs a servomotor angular drive which moves the de- tector toward the center of the diffraction peak. The control system is essentially critically damped so that the balance point is reached rapidly and without hunting. Fastress is electronically calibrated to provide a graphical read- out which plots residual stress in pounds per square inch. The residual stress is measured according to the prescription provided by 1 "Fastress" Test Set—Up} Figure 15 54 Equation 22. The diffraction angles are determined from voltages that are proportional to the angular displacement of the detectors. The difference between these voltages is proportional to the residual stress. The output is calibrated experimentally through the use of two references. The zero stress level is established by blocking the x-ray sources and adjusting the output display to indicate zero. Calibration is established by adjusting the output to display the residual stress which corresponds to the calibrated standard supplied by the Timken Roller Bearing Company. This calibrated voltage, used to plot the residual stress, is measured over intervals of at least five minutes. The residual stress at the point of measurement is taken to be the graphic average over this period. This measurement technique is illustrated in Figure 16. Previous experiments 60 indicate that this value is accurate to wihin :_2000 to 3000 pounds per square inch. 3.2.4 Data and Observations Initial experiments characterized the variation in the residual stress along the surface of the axle shafts. For all measurements, zero reference was established as the plane determined by the spline end of the axle shafts. No measurements were attempted at the spline be- cause the surface irregularity in this region is large compared with the x-ray spot size and no measurements were made adjacent to the flange because the heat treatment differs in this region. The residual stress was measured in the longitudinal direction only. Although the fixturing of the Fastress prevented measurement of the corresponding tangential component, the longitudinal component itself proved significant. 55 usmuso :mmouummm: .0H mwswmm poo: Hoe comm q wmoam uumru Ho>me mo comuoopmn .v om ow pomoso mwouuszm a ammo muouusnm < oom mcomummom umaswc< poomammsvm m A maaemm 56 Further, experiments by Vatev 29 imply that, for induction-hardened shafts of circular cross section, the longitudinal and tangential com— ponents of the residual stress at the surface are comparable. The first group of samples was used to characterize the variation of residual stress that may be associated with the different induction hardening processes. Four measurements, spaced 90° apart, were made at each of the ten cross sections which were chosen. As shown in Figures 17 and 18, the residual stress was found to vary as much as 40 ksi from point-to-point. However, if the average value of the four residual stress measurements at each cross section are compared and if the measurements immediately adjacent to the spline are neglected, a mean value of residual stress may be associated with each shaft. For sample A, a comparison of the cross-sectional averages yielded a mean value of 92.3 ksi compressive for the residual stress with a standard deviation of 2.4 ksi for the nine cross—sectional averages. For sample B, a comparison of the cross—sectional averages yielded a mean value of 51.2 ksi compressive for the residual stress with a standard deviation of 5.4 ksi for the nine cross-sectional averages. The cross- sectional average adjacent to the spline was significantly higher in both cases. The local variation of the surface residual stress for both samples was evaluated as close as possible to the most probable fracture surface. In the preliminary experiments all fractures occurred within the spline, so in these residual stress experiments the machined surface that borders the spline was examined. First, the longitudinal variation of the surface residual stress was determined by a series of twenty measurements that were spaces 0.10 inch apart. Second, the circum- ferential variation of the surface residual stress was determined by a Compressive Residual Stress (ksi) 57 Spline Flange 80 a Sample A Equispaced Angularly A o n O Calculated Mean 0 40 - 0 10 20 30 Longitudinal Position (inches) Figure 17. Consistency of Mean Residual Stress for Scan Hardened Shaft Compressive Residual Stress (ksi) 58 Spline Flange -‘ Sample B I Equispaced Angularly A 0 0 0 Calculated Mean 0 80 “ ‘1‘ 40 D 4 0 .‘h - 10 20 30 Figure 18. Consistency of Mean Residual Stress for Single—Shot Hardened Shafts 59 series of twenty-eight equispaced angular measurements. Observations from this data for sample A (Figure 19) indicated for the progressively hardened axle shaft that the residual stress varies sinusoidally. Since this pattern also repeats circumferentially (Figure 20A), the cross-sectional average should be valid. Observations from this data for sample B (Figure 21) indicated that for the single-shot hardened axle shaft, the residual stress varies with an irregular period along the length of the axle shaft but that the variation circumferentially (Figure 22B) provides a representative cross—sectional average. The data in Figures 19 and 21 revealed an increase in compressive residual stress level near the spline. Subsequent experiments examined whether the maximum strain which can be sustained before failure can be related to the residual stress distribution. For each of the test samples C through N, eight residual stress measurements, spaced every 45° at a distance that is two inches from the spline end of the axle shaft, were used to establish a mean residual stress level for each axle shaft. For the initial test samples A and B, twenty eight equispaced measurements at the same longitudinal position were taken. The results, presented in Figures 20 and 22 showed: first, that the average residual stress level varied little for axle shafts subjected to the same induction hardening conditions; second, that, within each test group, induction hardening by the vertical scanning process produced less variation in residual stress than by the single—shot process; and third, that, within the first group, the standard deviation of those samples which were not tempered was significantly larger than the standard deviation of those samples in the second group which were similarly processed but tempered. 60 umwzm pocopumm :mom pom comummum> mmouum Hmspmmom Hwoos .mm owuwmm Ammfiosmv somummom Hmsmwsumwcog mm mm mm . . . v o t om m maaamm O o OOH (rsx) sseiis {enprsag anrssaidmog 61 muwwsm monopwmm scum Mom mmowum Hospmmmm cum: wcmsmmmsmumm comm .om opnwmm tam mammam Sosa mmsosm N muomummom smaswa< poommmmscm w w w w w m mm mama mo wonasz m.m m.m m.o m.m c.q w.m m.w Ammxv .>oa .pum o.mm ¢.©m o.m ¢.wm H.© H.HOH H.Hm Ammxv emu: -. Vs m s m a o < uses-Wm O O 0 O. o o o o o o o oo o .. o . o o 0 o Orr 00 00 o o o o O o O . O O O OO 0 0 6 so 0 oo o o o o . O O O O O o ON as oo ow OOH (rsx) 838133 {enprsag anrsseidmog 62 H Amoeocfiv coaummom Hmampsumwcoq .H. . ><\l\t/ .1 i m onEMm cs 00 Ge co oq on cs 00 388118 Isnprsag snrssaidmog (1831) 63 mummsm monopumm uonmzaomcmm How mmmuum Hmsvmmmm emu: wamfimmanmumm mumm mam ocmaam Bonn mesons N maomummom Hmaswsd poommmmswm .ss D.D.s. w ¢.¢H w.¢o 2 w H.0H w.mm m some mo monasz Asmxv .>0m .eum Ammxv cmoz maaamm . 00 o o o 10 om cs 00 om OOH sseiig Isnpisag snrsssidmog (Isa) 64 Following the residual stress determination, the test samples were torsion tested to failure. Torque-twist plots are presented in Figure 23 for the furnace-tempered group and in Figure 24 for the induction-tempered group. Next, the extent of the metallurgical trans- formation was measured on a transverse section through the test shafts' splines. This data, Rockwell "C" hardness traverses, are presented in Figure 25 for the furnace-tempered shafts and in Figure 26 for the induction tempered shafts. Finally, the chemical composition of each shaft was determined and used to evaluate its hardenability through the ideal critical diameter calculations. 3.2.5 Test Results These test results, summarized in Table 2, indicate a correspon— dence between the angle-of—twist-at—failure and the mean residual stress level. A comparison of the mean residual stress level and its variation with angle-of—twist-at-failure is presented in Figure 27. The indicated level represents the mean value and the error bar represents the two sigma deviation for each shaft. This correspondence suggests that for induction-hardened axle shafts which meet the same heat treatment specifications, the mean residual stress level determines the maximum strain which can be sustained. Furthermore, these data (Figure 27) indicate that a ductile- brittle transition may exist. For comparison purposes, three limits were identified. A lower limit, chosen at the 10% point in the tran- sition region, was identified as Z and should represent a brittle 10 fracture limit for lower levels of compressive residual stress. An upper limit, chosen at the 90% point in the transition region, was 65 mummnm mouomaoa oomsuum How muomm ummBHIoswuoe Amooswmpv umm3H we meQ< co oe om D d1 I4! 11- pococumm cmomlamumuwo> Illllllll. saucepan nosmnmsmcsm 11111 IIIIIIIIIIIIII ‘II‘II‘IIIU ‘ R u 1.11.. 1111 .s m c .mm msnmse ON oq (Spunod—qoul puesnoql) anbiol 66 4+ mummzm pmpoaeoe compospaH pom muosm Amoouwomv ummae we oaws< 00 ca D I I d m I. monopwmm cmomlamomuuo> summons: soemamswesm 11111 umHBHIosvuoe ON .em muswsm J- ON . oq (Spunod—qoul puesnoql) BHDIOL, mummsm wouoafioauoomfipsm sow omum>mu9 mmospumm ocmamm .mN owswmm Asoesv enema Houaoo Os ON oumwpsm bl b D I 67 possess: smomlmmomuuo> possess: uOSmIoncmm 1 1 1 1 1 0 ON Os (3 IIamxoog) ssaupieH 68 muwwcm wouomfime1eomuo=ch pow omuo>muw mmocwumm ocmamm .om ouuwmm Asuesv enema Hmucoo «.0 N.o ovumunm . p . — b p a in ON H 1r .1: cc monopum: cmomlamomuuo> IIIIIIIII. //V/ .r vacuums: uozmloawcmm 1 1 1 1 1 ILWWV (3 IIamxoog) ssaupieH 69 m.ss m.ss oms c.6s mm o.mm m.mm mm mm.s smmsm z m.ss m.ss mms m.6s em m.sm o.ms ms mm.s smmmm z m.m 0.4m mas o.ms mm o.mm o.ms ms mm.s smmmm s m.m o.mm oss m.ms mm m.mm o.mm as mm.s sm>6m e m.m m.sm oss m.6s ms o.mm o.mm as mm.s smsmm m m.s o.mm mos m.ss cm o.om o.mm as mm.s sm>sm s m.m m.mm ems m.ss mm s.mm 0.00 ms me.s ammsa a m.m muss .ams m.6s mm 0.6m s.os ms me.s emmma o s.os m.mm mms s.ss sm m.mm m.mm as ss.s ammmm a m.m s.mm mmm o.es am m.mm m.em sm ms.s em>sa m 6.. s.ea mmm m.ms as o.mm s.mm as ms.s amemm a m.m s.sos ssm o.ms mm o.mm e.mm cs cm.s em>so o s.ss s.es Ne s.ms om o.mm s.mm as cm.s zmmmz m m.m s.sm sss o.ss om o.sm s.sm as mm.s Zm>sz a Am>smmmsaaoo sass soc A.\asxv Asses Aassv Aassv some sass >mn new zamz so u see mme may mmao sa mace .oz mmmmem sasosmma mmsemmmomm saosmmmm mmmUOMm emme muasmom amusoamuoaxm wo >umEEDm N omnme | 70 1 q- 1- I uh . «1- 4h .ummsm comm wow nOHumHHm> meHm oBu ecu ucomowmou mums uossm Ho>oH mmouum Hmstmom duo: Sums :OHOMHpm> chsum oumEHuHO .mN osswmm Amoopwmmv opsHHmMIumlumHBHIOOIQch< OON omH OOH (D l 1 1 uh uh owsuompm oHuqum .. OHN at: ounuompm mHHuosm oq OO Om OOH (rsx) 888133 {enprseu eArsseidmog 71 identified as 290 and should represent a ductile fracture limit for higher levels of compressive residual stres. Finally, a central limit, chosen at the 50% point in the transition region, was identified as Z, the transition stress level, in order to facilitate further discussion. For those axle shafts represented by the data (Figure 27), interpo- lation yielded: Zlo = 60 ksi compressive (23) Z = 70 ksi compressive, and (24) 290 = 85 ksi compressive. (25) The existence of the suspected ductile-brittle transition was evaluated by fractography. 3.2.6 Fractography Analysis of the fracture surfaces on the test samples corroborated the suspected ductile-brittle transition. Evidence for two distinct fracture mechanisms was found. Test samples with low levels of com- pressive residual stress (5 < 210) failed by a brittle cleavage RES mechanism whereas samples with high levels of compressive residual stress (ERES > 290) failed by a ductile void coalescence process. Those test samples with a mean compressive residual stress level in the trans- ition zone (Zlo < 5 < Z90) ultimately failed by cleavage but RES exhibited evidence of some void coalescence. Fractographs were made of sample A to illustrate the morphology of the fracture surface for mean compressive residual stress levels which exceed the ductile limit (5 > Z Examination of Figure 28 RES 90)° indicated that: first, fracture occurred on a surface that is oriented H C w 5 O Ductile Mode Fractograph——Radial View of Spline Fracture Surface. Figure 28. Note transverse fracture, spline distortion and spline root ruptures. H- 025 INCH Figure 29 Ductile Mode Fractograph——Axia1 View of Spline Fracture Surface. Note radial crack propagation from spline root. .memEHQ Mums-Hm wcHSOsLm musuomum owoz oHHuoua mo smwuwouofiz souuoon waHaamum OM mustm E simN .ooawomonou pHo>oson maHBonm ousuomum opoz mHHuossQ mo :amuwowuds nowuoon wuHeswom Hm ouumHm 76 90° to the axle shaft's axis; second, distortion of the spline in the direction of the applied torque was considerable; and third, ruptures in the material's surface occurred at the spline root. Subsequent examination of Figure 29 revealed cracks which extend from the spline root, through a glossy region which was smeared during fracture, and into a region characterized by shear dimples as shown in Figure 30. These cracks terminate in the central fibrous core. Scanning electron microsc0pic analysis of the shear dimple region (Figure 31) suggests that the ductile fractures are controlled by a void coalescence mechanism 53. Apparently, if the mean compressive residual stress level exceeds the ductile limit (5 RES > 290), surface crack growth is stable or suppressed as the shaft is twisted. This inhibition allows dislocations to pile-up and microvoids to coalesce. This process continues and reduces the effective cross—sectional area until the applied load can no longer be supported. Fracture then occurs. Etch pit studies 50 on torsionally induced fractures in low carbon steels indicate that dislocation motion proceeds along the {100} slip planes. Fractographs were made of sample B to illustrate the morphology of the fracture surface which results if the mean compressive residual stress level falls below the brittle limit (5R < Examination ES 210)' of Figure 32 indicated that: first, fracture occurred on a surface that is oriented 45° to the axis of the axle shaft; second, distortion of the spline was minimal; third, ruptures in the material's surface occurred at the spline root and extended across the spline teeth; fourth, chevron lines on the outer spiral surface indicated that the fracture originated at the spline root; and fifth, a crack extended to 45° across 77 Figure 32 Brittle Mode Fractograph——Radial View of Spline Fracture Surface. Note helical fracture, minimal spline distortion, spline root ruptures, crack propagation at 45°, and chevrons on helical fracture surface. ‘—————+-— 025 INCH Figure 33 Brittle Mode Fractograph——Radial View of Shaft Fracture Surface, Note shear lips. .mwOHonmuoa owva use smwOMHa wuoz .muuuomwm owoz oHuuHum mo aemuwouon souuoon mcHanmom cm ouswmm 510m Tull! 80 the spline from this initiation site. Subsequent examination of the mating fracture surface, which is illustrated in Figure 33, suggested that the chevron lines which were referenced in Figure 32 are shear lips which were generated as the crack front propagated. Scanning electron microscopic analysis of these shear lips (Figure 34) reveals a plateau and ledge morphology that typifies brittle fractures which are controlled by cleavage mechanisms 52. Apparently, if the mean compressive residual stress level falls below the brittle limit (5 surface crack growth is suppressed as RES > 210)’ the shaft is twisted until the maximum shear strain that the material can sustain is reached in the plastic zone at the root of the spline. If twisting continues, these sessile dislocations break loose from the pinning sites. This incipient crack extends a microscopic amount and relieves the local shear strain. At this point, the crack would arrest, except that the material is subject to continually increasing strain. Thus, at the crack tip, the local strain increases until the critical shear strain is reached and, once again, the crack front advances. This process continues until fracture is complete. Etch pit studies 50, indicating dislocation motion along the {100} cleavage planes, allow that a Cotrell mechanism 61 may act as the pinning site. Fractographs were made of sample H to illustrate one of the possible morphologies that the fracture surface may possess if the mean level of the compressive residual stress lies within the transition zone (Z < < z 10 ORES Examination of Figure 35 indicated that: 10)° first, the fracture occurred on a surface oriented perpendicular to the shaft's axis near the surface and on a surface oriented at 45° to the Figure 35 H 0.25 INCH Mixed Mode Fractograph-—Radial View of Spline Fracture Surface. Note compound fracture, minimal spline distortion, and spline root ruptures. H 0.25 INCH Figure 36 Mixed Mode Fractograph—-Axial View of Spline Fracture Surface. Note radial crack propagation from spline root to central helix. 83 axle shaft's axis near its central axis; second, distortion of the spline was minimal; and third, ruptures in the material's surface occurred at the spline root. Subsequent examination of Figure 36 revealed that cracks extended radially from the spline root of many teeth and that one, the source of final fracture, extends from the spline root to the edge of the internal spiral. Apparently, if the mean compressive residual stress level is in this transition region (Z the variation in residual < 8 < z 10 RES 90)’ stress is sufficient to allow localized dislocation pile—ups as the overall cross section is subject to microvoid coalescence. Once the critical shear strain is exceeded at a pile-up, unstable crack growth and fracture occurs. IV. ANALYSIS 4.1 Residual Stress-—Distribution Unanticipated residual stresses are recognized as a primary source of failure, but some models used to evaluate the effects of beneficial residual stresses, introduced by chemical, thermal or mechanical pro- cessing, neglect the inhomogeneities introduced by the process, material and geometric constraints 62. Such is the case with induction-hardened axle shafts. Shklyarov 6, and others cited by Almen 63, assume that on equivalent cross sections, the residual stress is a function of depth alone. Further credence to this angular consistency proposition stems from the experimental works of Vatev 26 and Ishii et al.16. The radial variation which they measured, however, should not be assumed to assure a constant value of residual stresses at a particular depth. Indeed, the experimental evidence indicates that on equivalent cross sections, the residual stress distributions on the surface do differ and that the variation significantly affects failure. Our residual stress measurements reveal large, local gradients on the surface of induction-hardened axle shafts. These point-to-point residual stress variations reflect local variations in quench condi- tions. Inspection of the observable quench patterns (Figure 39) indi- cates that for single-shot hardened shafts (Figure 39a) the quench pattern, as well as the residual stress distribution (Figure 21) appears random, and that for progressively hardened shafts, the quench pattern (Figure 39b,39c) as well as the residual stress distribution (Figure 19) appears periodic. Furthermore, progressively hardened shafts with significant residual stress variation (Figure 211) exhibited high contrast quench patterns (Figure 39c) whereas those with minimal 84 85 seesaw showmesaos sums vmawpumnlumom AoO mauouuum accuse mm ousmwm soamdv EHOMHn: nuHB podoOHMSIaMUm Any wmumwumn uonmlonamm Amy 86 residual stress variation (Figure 20E) exhibited low contrast quench patterns (Figure 39b). Quench composition, pressure, temperature, flow and impingement angle are but a few of the variables which affect uniformity. Quench irregularities locally reduce cooling rates and introduce minima into the anticipated uniform residual stress distribution. The minima reduce the mean compressive residual stress level on the surface but, because thermal conduction minimizes cooling rate variations, introduce minimal distortion beneath the surface. Whether increased time, labor and material costs are warranted to increase uniformity should be evaluated by potential reliability enhancement. In the analysis of Shklyarov,6 and those cited by Almen 63, the residual stress distribution is treated as uniform at a particular depth. Our physical testing and statistical analyses demonstrate that this assumption is consistent with elastic behavior and failure loads; however, this consistency results not from uniformity of the residual stress distribution on the surface, but from the relative insensitivity of these bulk properties to local surface variations. In order to understand the torsional strain behavior of induction-hardened axle shafts, however, these surface residual stress variations, suspected by Kaplan and Rowell 34, must be considered. 87 4.2 Residual Stress--First Order Effects The residual stress variation within this set of similarly pro- cessed induction—hardened axle shafts is critical in determining the mode and strain at fracture. Extant models--Littleton for tempered glass33--use superposition arguments to explain increased load bearing capability realized with an increase of compressive residual stress on the surface. The success of this model for brittle materials within the linear elastic range does not transfer to the plastic behavior of ductile materials subjected to similar compressive residual stress on the surface. Neither the tests of Shkylarov 6 nor Liss at al. 32 successfully relate increased compressive residual stress with improved static fracture resistance. Both, however, show increased fatigue life performance with increased compressive residual stress on the surface. Our experiments suggest that fracture mechanics arguments can resolve these inconsistencies. Fracture mechanics analyses the growth of pre-existing flaws and establishes failure criteria which depend only on the material tough- ness, the size and distribution of the flaws present, and the nominal stress. The inherent flaw size and distribution in the test shafts are assumed to depend on those forging and manufacturing Operations which precede heat treatment. The insensitivity of the stress sustained before final fracture to prior processing, as indicated in Table 2, is assumed to be evidence that flaw size is constant. Thus, if the vari— ation of strain sustained before failure is combined with this consis— tency of flaw size, then either the material's unflawed yield strength or toughness controls failure. 88 Variation in the unflawed yield strength for constant flaw size is illustrated in Figure 38. If the unflawed yield strength is high (Figure 38a), then the failure mechanism is determined by the crack stability criterion (Equation 6). This condition precipitates brittle failure as load increases. If the unflawed yield strength is low (Figure 38b), then the shaft fails as the unflawed yield strength is reached. This criterion determines ductile fracture. These results demand that the mean level of compressive residual stress decrease with increasing yield strength. This conclusion contradicts Littleton's data and, thus, this relation can be rejected. Illustrated in Figure 39 are two different values of material toughness chosen to test the proposed correspondence between KIIIc and ORES . If ORES is less than the brittle limit (210), by assumption, KIIIc is low, and, as indicated in Figure 39a, if the applied stress is increased until failure, unstable crack proPagation governs and brittle fracture results. If aRES is greater than the ductile limit (290), by assumption, K is large, and as indicated in Figure 49b, if the IIIc applied stress is increased until fracture, the unflawed yield strength governs and ductile fracture results. These arguments are consistent with the experimental residual stress results and reduce to the stress superposition model for brittle materials with sufficiently large inherent flaw size. This residual stress dependent ductile—brittle transition is quite analagous to the temperature induced embrittlement that Yokobori observed 50’51. Low levels of mean compressive residual stress on the surface yield the least modification of the case martensitic properties. Martensite, a rigid structure with minimal fracture toughness, should Nominal Stress Nominal Stress Figure 38. 89 El 0 B = KIIIcB (b) IIIcA = KIIIcB ao Flaw Size Unflawed Yield Strength and Fracture (a) For high unflawed yield strength, the flaw line intersects the line repre- senting failure controlled by unstable crack growth. (b) For low unflawed yield strength, the flaw line intersects the line repre— senting failure controlled by the maximum yield strength. Nominal Stress Nominal Stress 90 ‘ 1 E K 4 I O = IIIcA (a) oysA \‘ ' GA ME.— \I K \Ig‘g__oB= IIIcB 0 B \\ C V 2fla (b) 378 I \ = O B ys I IIIcA < KIIIcB aO Flaw Size Figure 39. Toughness and Fracture. a. For low toughness, the flaw line intersects the line representing failure controlled by unstable crack growth. b. For high toughness, the flaw line intersects the line representing failure controlled by the maximum yield strength. 91 precipitate a brittle fracture, controlled by cleavage along a helical surface, under extreme torsional loads. This behavior is confirmed by the observed plateau and ledge morphology (Figure 32) on the spiral fracture surface (Figure 34). High levels of mean compressive residual stress on the surface should inhibit crack opening and allow extended crack propagation before fracture. With sufficient stable crack growth, the crack tip will penetrate the core and final fracture should occur as the tough core material ultimately yields to the shear forces on the transverse surface. This crack growth is confirmed by the observed shear dimples (Figure 30) and microvoid coalescence.(Figure 31) on the transverse fracture surface.(Figure 40) The effect of compressive residual surface stress on the fracture strain of induction-hardened axle shafts also can be analyzed by stress superposition arguments, provided the inherent non-linearities are included in the analysis and provided fracture is initiated at the surface. Baker's findings47 that, under sufficient hydrostatic com- pressive stress, torsional fracture switched from cleavage to shear for brittle materials parallels the conclusions drawn from Figure 27. Both transitions can be understood using the Mohr's envelope analysis which Nadai developed 48 and is reproduced in Figure lla. This envelope, unlike the fracture toughness criterion, could be established experi- mentally by residual stress and fracture stress measurements or established by Altiero's calculations 43’46 also allow for Similarly, Makky's slip—instability arguments dual fracture paths. The only modification required is the realization that processing, in addition to the boundary condition constraints she analyzed, can introduce the tri-axial stresses necessary to switch from 05 INCH ' Figure 40 Ductile Mode Fractograph. 93 the helical to the transverse principle direction. These triaxial stresses disrupt the symmetry of the calculations and allow slip in the z direction. The presence of slip in the z direction favors trans- verse fracture. Clearly, since similar residual stress distributions, varying primarily in magnitude, exist in both cases, continuum argu- ments break down and microstructural effects must be considered to evaluate the residual stress influence on 2 direction slip. 94 4.3 Residual Stress—-Second Order Effects Analysis of the origins of residual stress and comparison of the surface residual stress with the surface quench pattern indicated that fast quench rates, characterized by a quenchant with high heat capacity, low quenchant temperature and adequate pressure, yield high compressive residual stresses on the surface. For all test induction-hardened axle shafts, cracks initiated at the spline root at the cross-section of minimum area and high levels of compressive residual stress on the surface correspond to extended ultimate strains. These results are consistent with the fracture mechanics, strength of materials, and slip instability analyses. The free body equilibrium condition, however, demands an inhomogeneous residual stress distribution. Both Vatev and Ishii et al.16 measure significant internal tensile residual stresses which these models neglect. Internal tensile stresses, as noted by Jahsmams et al. 40 and Ebert at al. 41, produce diminished tensile properties and also increase a shaft's susceptibility to Hertz stress failure 64. Although no direct comparison between this susceptibility and 5 is available, a RES comparison of quench rates and probability of cracking in a 0.38% carbon steel by Kobasko 65 demonstrated that quench rate, and by analogy, compressive residual surface stress, cannot be chosen arbitrarily. Thus, if internal residual stresses are sufficient, crack initiation and growth occurs internally, and the models proposed to eXplain the observed compressive residual stress dependence do not apply. V. CONCLUSIONS "Real materials are enormously complex in their response to stress even under isothermal conditions. ... The key to successful analysis or design is to choose the simplest permissible idealization of the behavior of the material not to obtain the best description over the widest range of environmental conditions." Daniel C. Drucker "Edgar Marburg Lecture" (1966) The primary effects of induction hardening on the fracture of steel axle shafts could be attributed to the increased yield strength in the case. For these similarly processed axle shafts, the maximum loads sustained before failure were distributed normally and exhibited little dispersion. The results of these static torsion tests confirmed the 21’22 and thoeries of Olzak 9. The deformation experiments of Wuerful at failure, however, could not be anticipated by sole reliance on their analysis. The residual stress, generated during the heat treatment, significantly affected the maximum deformation sustained before failure. Detailed investigation of the surface residual stress distribution for induction—hardened rear axle shafts reveals significant point-to- point variation in residual stress. These potentially extreme and random local variations eliminate the practical application of any theory that requires complete specification of the stress state. How- ever, if the symmetry inherent in the process and the applied loads allow, a judicious set of measurements can define a mean compressive residual stress level which may prove useful in defining experimentally verifiable fracture criteria. Any model based on this "scalar" ERES must be an energy model. Mohr's envelope, slip instability and fracture mechanics can be 95 96 applied successfully. The analyses place similar restrictions on crack initiation and growth in static and fatigue torsion. At the crack tip, the compressive residual stress affects the energy release rate which governs whether cleavage or shear fracture occurs. Theoretical evalu- ation of the critical residual stress level, Z, requires either modifi- cation of the Griffith-Orowan-Irwin theory or a statistical mechanics comparison of dislocation motion on competing slip planes. For those induction hardening processes investigated, the following conclusions are experimentally significant: 1. A mean level of compressive residual stress can be associated with each shaft. 2. Increasing ERES increases the torsional strain sustained before fracture. 3. A critical value of ERES exists such that: a. If ERES < Z, then fracture is governed by a maximum shear strain mechanism and appears brittle; and, b. If ERES > Z, then fracture is governed by a void coalescence mechanism and appears ductile. 4. The value of ERES can be controlled through the quench process variables. This experimental relationship between the mean level of com- pressive residual stress on the surface and the angle-of—twist-at- failure implies that if residual stress measurements are made prior to torsional testing, the mode of fracture and the ultimate strain can be predicted. For this to be valid, the axle shafts tested must meet Specification. Eddy current inspection to evaluate case depth and 97 ultrasonic inspection to verify material integrity, are non-destructive methods which can be employed prior to torsional testing to determine whether specification is met. This work demonstrates that residual stress measurements are necessary to characterize fracture behavior of inductionahardened axle shafts. Extension to different heat treatment specifications, forging conditions, materials and geometries requires the develOpment of an exten31ve GRES data base and a complementary theory relating KIIIC w1th ORES' APPENDIX I PHYSICAL DATA PHYSICAL ANG FAIL DEG 116 88 193 107 50 90 94 123 195 146 95 115 300 135 168 195 175 205 134 U L T TOR NIP 61.0 68.1 57.7 63.9 33400 54.1 59.3 60. 50. 12' -..J 2 6 041.: $0. 1a.; -V' o v? or 32.1714”th a; L‘! i. :._\ .. r- J};- :..x -- ..H I). .- -p uxm J} 51. CasuassdoJOcnxdu 57 58. 100 U5. 148 55. 190 52. 759 51.0 160 52.7 180 57 . .7340 170 180 7370 195 180 146 177 t 198 194 973 233 270 147 182 115 12’. 8 7 146 183 59.6 ‘3' 7 E.- -...‘ \.. 9 ..J 58.5 59.1 It.“ - II." \J .‘r a... ":1 .. I.» 6 56.? DATA TOR NIP 37.0 34.0 34.3 39.0 36.0 24.0 37.0 33.5 33.3 32.5 40.0 31.5 32.0 37.0 393.0 40 .. 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ZAYAC 1010 PRINT 'INSTRUCTIONS (1=YES)'3 1020 INPUT G4 1030 IF G4€§1 GOTO 1190 1040 PRINT 1050 PRINT 'THIS PROGRAM CALCULATES MEAN! STANDARD DEUIATION! SKEUNESS' 1060 PRINT 'AND KURTOSIS (IF REQUESTED)! SAMPLE LOU! SAMPLE HIGH! SAMPLE' 1070 PRINT 'RANGE! LOUER 3-SIGMA LIMIT! UPPER 3-SIGMA LIMIT! AND 6-SIGMA' 1080 PRINT 'RANGE. IF REQUESTED! COMPARES DISTRIBUTION UITH SPECIFICATION-' 1090 PRINT 'COUNT! ACTUAL PERCENTAGE! AND PROBABLE PERCENTAGE BELOU AND' 1100 PRINT ‘ABOUE SPECIFICATION ARE INDICATED. IF REQUESTED! COMPARES' 1110 PRINT 'DISTRIBUTION UITH NORMAL DISTRIBUTION FUNCTION USING THE' 1120 PRINT 'CORRECTED KOLMOGOROU-SMIRNOU D STATISTIC.’ 1130 PRINT 1140 PRINT 'INPUT DATA (USE LINES 1 THROUGH 999)‘ 1150 PRINT ' FOR RAU DATA ENTER: A(1)!A(2)!...!A(N)!B(1)!B(2)!...B(N)!...' 1160 PRINT ' ENTER: 909090 FOR MISSING DATA UITHIN SET' 1170 PRINT ' FOR FREQUENCIED DATA ENTER: F(1)!X(1)!F(2)!X(2)!...!F(N)!X(N)' 1180 GOTO 4980 1190 DATA 999999 1200 DIM U(20)!F(20)!U(20!5)!B(5)!C(999)!T(999)!E2(20) 1210 DIM 0(999) 1220 S=S7=88=R=B6=F5=F6=LO=H1=Iq= 1230 Y=I1=1 1240 PRINT 'DATA: 0=RAU OR 1=UEIGHTED‘5 1250 INPUT E4 1260 PRINT 'OUTPUT (1=HISTOGRAM 8 STATISTICS or 2=STATISTICS ONLY)‘: 1270 INPUT E5 1280 O9=2XE4+E5 1290 PRINT 'UNITS OF MEASURE (10 CHARACTER FIELD)'3 1300 INPUT U43 1310 PRINT 'CALCULATE SKEUNESS? KURTOSIS (1=YES)'; 1320 INPUT GI!G2 1330 PRINT 'TEST NORMALITY OF SAMPLE (1=YES)'3 1340 INPUT G3 1350 IF 0932 GOTO 1380 1360 PRINT 'NUMBER ASSOCIATED DATA POINTS? VARIABLE ANALYZED': 1370 INPUT E1!E3 1380 PRINT 'SPECIFICATION LIMITS (LOU!HIGH) (ENTER 0 IF NONE)'§ 1390 INPUT B9!88 1400 IF 09%3 GOTO 1450 1410 PRINT 'DATA ADDITIUE CONSTANT'? 1420 INPUT B6 1430 PRINT ‘DATA MULTIPLICATIUE CONSTANT'? 1440 INPUT Y 1450 IF E5€§1 GOTO 1500 1460 PRINT 'HISTOGRAM LOU LIMIT-USE 1 MORE DECIMAL DIGIT THEN REST OF DATA'? 1470 INPUT F5 1480 PRINT 'HISTOGRAM INTERUAL*USE SAME NUMBER OF DECIMAL DIGITS AS DATA'3 1490 INPUT F6 1500 PRINT 'DATE'? 1510 INPUT U1$ 1520 PRINT 'TITLE'; 1530 INPUT U2$ 1540 PRINT 'ENTER ANY CHARCATER. POSITION PAPER. PRESS RETURN. '3 1550 INPUT U35 1560 PRINT 1570 IF O9<3 GOTO 1710 1580 READ U 1590 IF w=999999 GOTO 1840 1600 1610 1620 1630 1640 1650 1660 1670 1680 1690 1700 1710 1720 1730 1740 1750 1760 1770 1780 1790 1800 1810 1820 1830 1840 1850 1860 1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 2020 2030 2040 2050 2060 2070 2080 2090 2100 2110 2120 2130 2140 2150 2160 2170 2180 2190 102 ERRROR READ 2 IF Z=>B9 GOTO 1630 LO=L0+U IF Z<=BS GOTO 1650 H1=H1+U FOR I=Il TO 11+U-1 LET C(I)=Z LET I2=I NEXT I LET I1=I1+U GOTO 1580 FOR E0=1 TO E1 READ E2(E0) IF E2(1)=999999 GOTO 1840 IF E2(E0)=999999 GOTO 4960 NEXT E0 IF E2(E3)=909090 GOT01710 C(I1)=E2(E3) IF C(I1)=>B9 GOTO 1800 LO=L0+1 IF C(I1)fi=88 GOTO 1820 H1=H1+1 Il=Il+1 GOTO 1710 I2=Il-1 N1=I2 IF O9P2 GOTO 1950 FOR J=1 T0 N1 FOR I=1 TO N1-1 IF C(I){C(I+1) GOTO 1930 E6=C(I) C(I)=C(I+1) C(I+1)=E6 NEXT I NEXT J E9=C(1) E8=C(N1) FOR I=1 T0 N1 LET C(I)=C(I)*Y+86 LET S=S+C(I)“2 LET R=R+C(I) LET T(I)=C(I) O(I)=C(I) NEXT I LET A=R/N1 LET C=S/N1 LET D=A92 LET G=U(L)+.000001 THEN 2310 LET F9=F9+1 LET C(I)=100000. LET I= I+1 IF I f= N1 THEN 2280 LET F(L)=F9 LET F8=F8+F9 IF F(L)}81 THEN 2850 IF U9=4 THEN 2400 IF F(L)}54 THEN 2830 IF U9=3 THEN 2400 IF F(L)>27 THEN 2810 NEXT L G9=0 G9=G9+1 IF G9=1 GOTO 2660 PRINT TAB(13)!'CELL MAX PRCT GUAN FOR L=1 TO 20 IF L=20 THEN 2770 IF ABS(U(L))<..000001 THEN 2890 PRINT USING 2490!U(L)5100*F(L)/N1!F(L)! ####¢.#¢*O ### ### IF F(L)= 0 THEN 2600 FOR M= 1 TO INT((F(L)+1)/U9+.5) 0 O 0 O O O 0 000000000.0.00000000000000000000000 IF M=INT((F(L)+1)/V9+.5) THEN 2590 IF Mk1 THEN 2560 PRINT -:-T 60 TO 2570 PRINT -x'; IF M=27 THEN 2870 NEXT M IF H>1 THEN 2620 PRINT 'z- GO TO 2630 PRINT -*- IF ABS(U(L)){.000001 THEN 2920 NEXT L PRINT TAB(13)!'CELL MAX PRCT GUAN IF U9=4 THEN 2750 IF U9=3 THEN 2730 IF U9=2 THEN 2710 PRINT TAD<32)!'0 5 10 1 ON G9 GOTO 2420!4580 PRINT TAB(32)!'0 10 20 30 ON G9 GOTO 2420!4580 PRINT TAB<32)!‘0 15 30 4 ON G9 GOTO 2420!4580 PRINT TAB(32)!'0 20 4O 60 ON 69 GOTO 2420!4580 O 0 O O O 0 0 000000900000000.0000000000000000... 4O 60 80 25 30' 50 60' 75 90' 100 120' PRINT USING 2780!100*(N1-F8+F(L))/N1§N1-F8+F(L)i ABOVE ### ### LET F(L)- N1- F8+F(L) GOTO LET GOTO LET GOTO LET GOTO PRIN GOTO PRINT GOTO PRINT 0 104 ERRROR 2500 U9=2 2400 09:3 2400 u9=4 2400 T '2- 2630 USING 2900!100*F(L)/N13F(L)3 0.0000 000 #00 2510 USING 2930.v/N1;F(L+1); 00000.0000 000 0:0 LET L=L+1 GOTO PRIN FOR LET LET LET LET NEXT LET LET LET LET LET FOR FOR READ NEXT NEXT DATA DATA DATA DATA DATA DATA DATA DATA DATA DATA LET LET LET LET LET B LET LET LET IF (B IF 2 GOSU LET LET IF (B IF 2 GOSU LET 2500 T I=1 T0 N1 C(I)=T(I) S9=C(I)-A 88=88+S923 S7=S7+S924 I 86=K122 SB=88/(N1-1) 88:98/(S6XK1) S7=S7/(N1-1) S7=S7/(8622) M=1 TO 20 M1=1 TO 5 U(M!H1) M1 M .3ro319!o352!.391!.417ro285!.299!.315!.337!o405 .265!.277!.294!.319!.364!.247!.258!.276!.3!.348 0233’ 02447 0261’ 0285' 0331! 0223’ 0233’ 0249’ 0271! 0313 .215!0224!.239!.258!.294!o206!.217!o23!o249!.284 01997 02 2,022370242702757 019! 0202! 02147 0234! 0268 0.183! 0194' 0207' 0227' 0261! 01777 0187’ 0201’ 022! 0257 0173’ 0182' 0195! 0213! 025! 0169' 0177! 0189! 0206’ 0245 0166! 0173! 0184! 02! 0239' 0163! 01699 0179! 0195’ 0235 016! 0166' 0174' 0197 0231! 0149! 0153’ 01657 018! 0203 01317 0136, 0144' 01617 0187! 0736' 0768' 0805' 0886I10031 B(l)=.25483 B(2)=-.284497 8(3)=1.42141 B(4)=-1.45315 (5)=1.0614 P=.327591 K1=K1*1.41421 Z=(A-B9)/K1 8+B9)=0 GOTO 3330 i= 0 THEN 4930 B 4660 A3=50—50352 Z=(BB-A)/K1 8+B9)=0 GOTO 3380 q: 0 THEN 4930 B 4660 A2=50*50*E2 3400 3410 3420 3430 3440 3450 3460 3470 3480 3490 3500 3510 3520 3530 3540 3550 3560 3570 3580 3590 3600 3610 3620 3630 3640 3650 3660 3670 3680 3690 3700 3710 3720 3730 3740 3750 3760 3770 3780 3790 3800 3810 3820 3830 3840 3850 3860 3870 3880 3890 3900 3910 3920 3930 3940 3950 3960 3970 3980 3990 105 ERRROR FUR I3=1 T0 N1 LET C(I3)=T(1) LET K0=1 FOR I4=2 T0 N1 IF C(I3) fi= T(I4) THEN 3470 LET K0=I4 LET C(I3)=T(I4) NEXT I4 LET T(K0)=1.E+25 NEXT I3 LET Z=(A-C(1))/N1 GOSUB 4660 LET US: 05_E2*OS LET 16=1 FOR I=I6 T0 N1 IF C(I)}C(I6) THEN 3580 IF I=N1 THEN 3790 NEXT I LET Z=(ABS(C(I-1)*A))/K1 GOSUB 4660 IF C(I-1){A THEN 3630 LET D6=0J+E2*05 GOTO 3640 LET D6=0d-E2*05 LET O2=(I-1)/N1 LET D6=ABS(02-06) IF D52D6 THEN 3680 LET D5=D6 LET Z=(ABS(C(I)*A))/K1 GOSUB 4660 IF C(I){A THEN 3730 LET 06=0J+E2*05 GOTO 3740 LET D6=.5~E2*.5 LET D6=ABS(02-D6) IF D5306 THEN 3770 LET D5=D6 LET I6=I GOTO 3540 LET Z=(C(N1)-A)/K1 GOSUB 4660 LET D6=.3+.5*E2 LET D6=1-D6 IF D5>D6 GOTO 3850 LET D5=D6 15=15+1 IF 15=6 GOTO 4050 I3=18 IF N1 {= 20 THEN 3920 IF N1=30 THEN 3940 IF N1330 THEN 3960 GOTO 3970 LET I3=N1~3 GOTO 3970 LET I3=19 GOTO 3970 LET I3=20 LET D6=U(I3!I5) IF I3420 GOTO 4000 LET D6=D6/SQR(N1) 4000 4010 4020 4030 4040 4050 4060 4070 4080 4090 4100 4110 4120 4130 4140 4150 4160 4170 4180 4190 4200 210 4220 4230 4240 4250 4260 4270 4280 4290 4300 4310 4320 4330 4340 4350 4360 4370 4380 4390 4400 4410 4420 4430 4440 4450 4460 4470 4480 4490 4500 .4510 4520 4530 4540 4550 4560 4570 580 4590 106 ERRROR IF D6<=D5 GOTO 3850 C2=020—005*(IS‘ 1) IF C2fi C2=.01 PO GOTO 4060 GOTO4060 C2=0 PRINT PRINT PRINT PRINT PRINT PRINT PRINT PRINT PRINT PRINT PRINT PRINT PRINT IF G1fiP1 GOTO 4270 PRINT IF PRINT F'RINT TAB(17)!U1$ TAB(17)!U2$ USING 4120 TAB(17)!'SAMPLE USING 4160!N1 DESCRIPTION' QUANTITY MEASURED.............. USING 4180!U4$ UNITS OF MEASURE............... TAB(17)!'STATISTICAL MEASURES' USING 4210!A THE MEANOOOOOOO900.00.000.00... USING 4230!G8 STANDARD DEVIATION............. USING 4260!88 SKEUNESS (-1 TO +1 NORMAL)..... G2(}1 GOTO 4300 USING 4290987 KURTOSIS (+2 TO +4 NORMAL)..... TAB(17)!'DISTRIBUTION LOU HIGH IF (BB+B9)= 0 GOTO 4340 PRINT PRINT PRINT USING 4330989!88!BS- B9 SPECIFIED. USING 4350!E9!E8!E8- E9 ACTUAL.... USING 4370!L3!U3!6*G8 6- SIGMA... #%#f.#¢## #*#¢.#$## #*##.##¢1 ####.#### i¢¢#.##*# ##*#.#¢¢# IF (B8+B9) =0 GOTO 4460 PRINT PRINT PRINT PRINT IF 63%? IF C22 PRINT PRINT GOTO 45 PRINT PRINT PRINT PRINT IF E5= FOR I= FHINT TAB(17)!'SPECIFICATION COMPARISON USING 4410!L0!H1 ACTUAL COUNT.......... USING 4430!L0*100/N1!H1*100/N1 ACTUAL PERCENTAGE..... USING 4450!A3!A2 PROBABLE PERCENTAGE... 1 GOTO 4550 }0 GOTO 4520 TAB<17)!'NORMALITY TEST-REJECT 8 USING 4500!D5 KOLHOGOROU_SMIRNOU 000000000000 BELOW ##4## 40*.# 400.0 50 USING 45307C2 NORMALITY TEST- ~ACCEF‘T G USING 4500! D5 .## USING 4120 1 GOTO 2170 1T0 15 ‘3 ######*### ’CCCCCCCCCC #####.##¢# #*#¢#.#¢## ####¢.#¢¢# 44444.4444 RANGE' 4444.4444 4444.4444 4444.4444 ABOUE' 44444 444.4 $00.! .01 CONFIDENCE LEVEL' ##4##.#### CONFIDENCE LEVEL 4600 4610 4620 4630 4640 4650 4660 4670 4680 4690 4700 4710 4720 4730 4740 4750 4760 4770 4780 4790 4800 4810 4820 4830 4840 4850 4860 4870 4880 4890 4900 4910 4920 4930 4940 4950 4960 4970 107 ERRROR NEXT I IF 38<-1.1 THEN 4750 IF 8831.1 THEN 4750 IF S7fi2 THEN 4750 IF S724 THEN 4750 GO TO 4980 LET T3=1/(1+P*Z) LET P2=0 FOR J2=1 TO 5 LET J3=6~J2 LET P2=P2¥T3+B(J3) NEXT J2 LET P2=P2*T3 LET E2=1~P2¥EXP(*Z22) RETURN PRINT 'PRINT MEDIAN RANKS (1=YES)'3 INPUT E7 IF E7<31 GOTO 4980 PRINT PRINT U15 PRINT PRINT U23 PRINT PRINT 'THE SKEUNESS-KURTOSIS TEST FOR NORMALITY HAS' PRINT 'IMPLIED NON—NORMALITY. MEDIAN RANKS WILL BE GIUEN.‘ PRINT PRINT ' NUMBER OBSERVATION MEDIAN RANK' PRINT FOR Q7=1 T0 N1 LET Q3=909090 GOTO 1&00 E=E+1 GOTO 1620 PT.NO.'!' TO GROUP SIZE.‘ 109 COMPARE 1600 PRINT H9J9X4J1yY(J) 1610 u=u+1 1620 NEXT J 1630 IF 2(112090999 GOTO 1650 1640 GOTO 2570 1650 PRINT 1660 RESTORE 1670 PRINT 'UARIARLE NAMF?1 X UNITSTv TABLE NUMBER'? 1680 INPUT Y$9U59T$ 1690 PRINT IUARIABLE ANALYZEDTy FIRST CELL?! CELL UIDTH'= 1700 INPUT nyopu 1710 SO=N 1720 91=R 1730 82184 1740 4:41 1750 R=AO 1760 PRINT 'ENTEP ANY CHARACTER1 POSITION PAPER. PRESS RETURN.'; 1770 INPUT 9% 1780 PRINT TAB<1711T$ 1790 PRINT 1900 PPTNT TAB<17iv'ANGLE-OF-TMISTuéT—FAILURE' 1810 PRINT TAR<1711'CORPELATION MITH' 1370 PRINT TOBt171-¥$ 1830 PRINT 1940 PRINT USING 1850 1950 2 ———————————————————————————————————————————————————————— 1060 PRINT 1870 PRINT USING 1880.80 1880 . SAMPLE SIZE-................ ##* 1890 PRINT USING 1900181 1900 2 CORRELATION COEFFICIENT..... ##.### 1910 PRINT USING 1990.11?» 1920 t UNITS TO MEASURE X.......... ’RRRPRRRRRR 1970 PRINT TAB(17)1‘HNITS TO MEASURE Y.......... DEGREES' 1940 PRINT USING 1950vS2 1950 ’ STANDARD DEVIATION (YX)..... ##.# DEGREES 1950 PRINT 1°70 PRINT USING 1950 1990 PRINT 1900 PRINT USING 2000,x0,10+101w9x0+201wvx0+3oxw 7000 : ¢##.# 111.: 001.1 101.1 ?010 PRINT TABI19>,-o z....:....:....:....:....:....:- 2020 FOR N=l TO 15 7030 O=291N coao PRINT USING zofioon: 2050 : 111 0040 PRINT -:-: “070 FOR M=1 TO 29 7090 F=0 2000 FOR 1:; TO 131 2100 READ T111~Z(TIw?/31o2’41v7f51o2(5112(7)v2(81vZf9) 2110 Y=T.14159xr7f7)¥(1220*?(91—If91“71+Z(8)*<610-2(91)“21/1000000 2120 IF Y 100 GOTO 2180 2130 IF ZISTIOSIIN-I) OOTO 2180 2140 IF 21q12=251u OOTO 7180 2100 IF 7/715X0+m*rM—11 OOTO 2190 7150 IF Z(<)¥=¥0+M*M OOTO 2190 7170 c=c+1 7190 NEXT L 2100 RFSTORF 2200 2210 2220 2230 5240 2250 2240 2270 2280 2290 2300 2310 2320 2330 2340 2350 2360 2370 2390 2300 2400 2410 2420 2430 2440 2450 2440 2470 2480 2490 2500 2510 2520 2530 2540 2550 2560 2570 ON 8+1 GOTO PRINT ' ‘4 60TH 2440 PRINT 'A'; GOTO 2440 PRINT '8': GOTO 2440 PRINT '0'; GOTO 2440 PRINT 'D'; GOTQ 221110 PRINT 'F'; GOTO 1440 PRINT 'F'; GOTOQAAO PRINT '8'; GOTO 2440 PRINT ‘ '3 SOTO Q440 PRINT '1': OOTO 2440 PPINT '1'; GOTO 2440 PRINT 'P'i NEXT M PRINT '1' NEXT N 110 COMPARE 22109223002250-2270v229012310v233072350v2370v23909241092430 PRINT TAR’173-'400 3....2....1....14...I....3...43' PRINT USING PRINT 2000.x0,x0+101w940+20*w.¥0+30*w PRINT TAB(17)9'LEAST—SOUARES LINEAR APPROXIMATION' PRINT USING 9 FOR O=1 TO TO PRINT ' NEXT 0 UATA 000990 ENH ZCQOvAyB Y = (11.0%) X + (##1) APPENDIX IV STATISTICAL RESULTS 111 TABLE 1 ANGLE~OF-TUIST~AT-FAILURE DISTRIBUTION —-“—cut—u—n-nu-u-a—I-u-uo—ou-u—u-n—n—u-n-n-“a—“u~n—m——-——_-~~uuu‘~a-nu-.—~o.~noun~—I-—-_ SAMPLE DESCRIPTION QUANTITY MEASURED.............. 131 UNITS OF MEASURE............... DEGREES STATISTICAL MEASURES THE MEANOOOO00090000004000.0000 14801603 STANDARD DEUIATION............. 62.4557 SKEUNESS (*1 TO +1 NORMAL)..... .7537 KURTOSIS (+2 TO +4 NORMAL)..... 3.1724 DISTRIBUTION LOU HIGH RANGE ACTUAL.... 50.0000 370.0000 320.0000 6-SIGMA... “39.2069 335.5375 374.7344 NORMALITY TEST-REJECT O .01 CONFIDENEE LEVEL KOLMOGOROU-SMIRNOU D........... .1272 0 10 20 30 40 50 60 CEL'... ”AX F'RCT GLJAN :6 O 6 O .2 O 0 0 9 z 0 0 O O 2 O O O O 2 6 0 O 9 1: O 9 4' 0 g 0 O O 0 25.0000 0 0 2 50.0000 1 1 75.0000 9 12 t****** 100.0000 15 20 3*********# 125.0000 23 30 2*************** 150.0000 11 14 £******* 175.0000 8 10 t#**** 200.0000 17 22 :*******#R#* 225.0000 3 4 2** 250.0000 6 8 2**** 275.0000 5 7 2*** 300.0000 2 2 3* 325.0000 0 O 2 350.0000 0 0 2 375.0000 1 1 2 400.0000 0 0 2 425.0000 0 0 2 450.0000 0 0 3 475.0000 0 0 1 ABOVE 0 0 2 CELL MAX PRCT OUAN 2....2....2....1....2....2....2.... 0 10 20 30 40 50 60 112 TABLE 2 TORSIONAL ULTIMATE STRENGTH DISTRIBUTION “m--‘u-’-*‘n---m-_“-*~——_~——-——0—--~—————-_—*~_~-C~“~mfl_--_ SAMPLE DESCRIPTION QUANTITY MEASURED.............. 131 UNITS OF MEASURE............... 10001N~LBS STATISTICAL MEASURES THE MEANooooooooooooooo09.6009¢ 5709298 STANDARD DEVIATION............. 4.0685 SKEUNESS {—1 TO +1 NORMAL)..... .6470 KURTOSIS (+2 TO +4 NORMAL)..... 3.9769 DISTRIBUTION LOU HIGH RANGE ACTUAL.... 50.0000 72.0000 22.0000 NORMALITY TEST~ACCEPT G .20 CONFIDENCE LEVEL KOLMOGOROV-SMIRNOV D........... .0580 _-‘_-——~_I-—-‘__“-D-_-_‘_———-_—-~——*“--~‘—~_~*~“--*—~“*-*- 15 30 45 60 75 90 CELL MAX PRCT OUAN ....3....:....t....:....t....2.... 5.0000 10.0000 15.0000 20.0000 25.0000 30.0000 35.0000 40.0000 45.0000 50.0000 55.0000 60.0000 65.0000 70.0000 75.0000 80.0000 85.0000 90.0000 95.0000 ABOVE CELL MAX PRCT OUA o. oo o» 99 o. o. 90 99 v9 ... o. o 91 bl—‘OOOOOOOOO 3*********** :*************#****** 3******X*** f-J b 1'.) 0000 O?~J:’-JJ>\JO~5-'-OOOOOOOOO '3‘???“ 20000 OF-Jf-JI 6 9 9 f 9 00096906000000090.000.000.000 15 30 45 60 75 90 000 O 90 Q0 9. 9.0. o. 99 9° 113 TABLE 3 TORSIONAL YIELD STRENGTH DISTRIBUTION SAMPLE DESCRIPTION QUANTITY MEASURED.............. 131 UNITS OF MEASURE............... 1000IN-LBS STATISTICAL MEASURES THE HEANO00096000999900.9900... 3502863 STANDARD DEVIATION............. 3.2809 SKEUNESS (-1 TO +1 NORMAL)..... .2515 KURTOSIS (+2 TO +4 NORMAL)..... 6.3572 DISTRIBUTION LON HIGH RANGE ACTUAL.... 24.0000 50.4000 26.4000 6-SIGMA... 25.4436 45.1289 19.6852 NORMALITY TEST-ACCEPT @ .01 CONFIDENCE LEVEL KOLMOGOROU"SHIRNOK’ I]. o o o + a» o o o o o .0880 —-—‘——-u—u-I--—-——*~—~_—o—n—n-—_“_-_—an—--—.mu—-_o~~u——u-—m~u—m~mnnu-uu—_—“~ 15 30 45 60 75 90 CELL MAX PRCT QUAN ....:....2....2....2....:....2.... 5.0000 10.0000 15.0000 20.0000 25.0000 30.0000 35.0000 40.0000 45.0000 50.0000 55.0000 60.0000 65.0000 70.0000 75.0000 80.0000 85.0000 90.0000 95.0000 ABOVE CELL MAX PRCT QUA 0 oo 90 to 99 99 99 o. C) 9(- 9(- 57 :****************** 61 3******************** b5 OOOOOOOOOHONMbLflI—‘OOOO 0 0 0 0 0 O 0 N O 0 9 O 9 #OOOOOOOOOOOOOOOOOOOOQOO60600 15 30 45 60 75 90 9 9 O C) o. to o. o. o. 90 99 90 99 o. o. 00 o. 114 TABLE 4 TORSIONAL YIELD POINT DISTRIBUTION SAMPLE DESCRIPTION QUANTITY MEASURED.............. 131 UNITS 0F MEASUREOOOOO+OOOOOOOOO 10001N“LBS STATISTICAL MEASURES THE MEANooooooooooooooooo00.99. 2496981 STANDARD DEVIATION............. 4.1861 SKEUNESS (-1 TO +1 NORMAL)..... .3456 KURTOSIS (+2 TO +4 NORMAL)..... 3.0192 DISTRIBUTION LON HIGH RANGE ACTUAL.... 15.2000 37.1000 21.9000 6-SIGMA... 12.1397 37.2565 25.1168 NORMALITY TEST-ACCEPT G .01 CONFIDENCE LEVEL KOLMOGOROV—SMIRNOU Doo+ooooooo§ 90853 15 30 45 60 75 90 0 O 0 0 O 09000600000900690009000900.0006... 9 CELL MAX PRCT QUAN to to +9 90 0 5.0000 0 0 10.0000 0 0 15.0000 0 0 20.0000 16 21 2**#*** 25.0000 43 56 :****************** 30.0000 33 43 :*******X****** 35.0000 10 2*** 40.0000 1 45.0000 0 50.0000 0 55.0000 60.0000 65.0000 70.0000 75.0000 80.0000 85.0000 90.0000 95.0000 ABOVE CELL MAX PRCT QUA OOOOOOOOOOOOHED 0 0 O 0 O 0 0 0 0 0 N 9 O 9 9 0 9 00060000669000OOOOQOOOOOOOOOOOOOOO 15 30 45 60 75 90 O 99 0.9 90 90 9. ¢¢ 9+ +9 9. 0.; 9'. 9o 90 §¢ 115 TABLE 5 SURFACE HARDNESS DISTRIBUTION SAMPLE DESCRIPTION QUANTITY MEASURED.............. 131 UNITS 0F MEASUREooooooooooooooo RCQOOOSOIN STATISTICAL MEASURES THE MEANOOOOOOOOOOOOOOOOOOOOO09 5404198 STANI’ARI} ['EUIATIONO O O 0 O O 0 O O O O O 0 202359 SKEUNESS (-1 TO +1 NORMAL)..... ~.5611 KURTOSIS (+2 TO +4 NORMAL)..... 3.3815 DISTRIBUTION LOU HIGH RANGE ACTUAL.... 48.0000 60.0000 12.0000 6-SIGMA... 47.7121 61.1276 13.4155 NORMALITY TEST-REJECT G .01 CONFIDENCE LEVEL KOLMOGOROV‘SMIRNOV D........... .1812 15 30 45 60 75 90 CELL MAX F.RCT QUAN .900:00.0:000000000:090.:000030909 5.0000 10.0000 15.0000 20.0000 25.0000 30.0000 35.0000 40.0000 45.0000 50.0000 55.0000 60.0000 65.0000 70.0000 75.0000 80.0000 85.0000 90.0000 95.0000 ABOVE CELL MAX PRCT QUA 9. 9o 9¢ +0 9. o. 66 to 9+ 99 9+ <§ it- *- i- 3************************ 3************** (AU! oooooooo»vmooooooooo .b\l'r-'- zoooooooommwooooooooo O 0 O 0 0 00000000000000.00000000009000 15 3O 45 60 75 90 000 C) o. o. o. 00 o. to o. 99 co 116 TABLE 6 CASE HARDNESS DISTRIBUTION -—~‘_—“_‘““—"‘*--fi--_~“fl.--..-—-——_——-‘—_-——-_-~-—_-*-~-—u. SAMPLE DESCRIPTION QUANTITY MEASURED.............. 111 UNITS OF MEASURE............... RCGO.15OIN STATISTICAL MEASURES THE HEANooooooooooooooooo000.90 4307207 STANDARD DEVIATION............. 6.7233 SKEUNESS (*1 TO +1 NORMAL)..... -.4184 KURTOSIS (+2 TO +4 NORMAL)..... 2.5460 DISTRIBUTION LOU HIGH RANGE ACTUAL.... 27.0000 56.0000 29.0000 6-SIGMA... 23.5507 63.8907 40.3399 NORMALITY TEST-ACCEPT G .01 CONFIDENCE LEVEL KOLMOGOROU“SMIRNOU D. c- o o o o o o o o o .0841. 10 20 30 40 50 60 CELL MAX PRCT QUAN ....t....:....3....2....:....i.... 5.0000 10.0000 15.0000 20.0000 25.0000 30.0000 35.0000 40.0000 45.0000 50.0000 55.0000 60.0000 65.0000 70.0000 75.0000 80.0000 85.0000 90.0000 95.0000 ABOVE CELL MAX PRCT QUA Q 9. O. o. 90 99 o¢ <3 .** :XXX* 23 2******#**** 27 1***#********* 31 3*************** 3******** {13500000 grunJm o<3<>o<>c>o<>Hmbaabbéahbc>o<>c>o 0H2} 0 0 0 0 O 0 0 N 0 0 0 9 O 0 OOOOOOOOOOOOOOOO960000000000900000 10 20 30 40 50 60 C) co 9. oo 90 09 90 90 00 99 90 117 TABLE 7 CORE HARDNESS DISTRIBUTION SAMPLE DESCRIPTION QUANTITY MEAS‘JREI‘OOOOOOOO’OOGOO 13... UNITS OF MEASURE............... RCCOREGFLG STATISTICAL MEASURES THE HEANOOOOOOQOOO0OOOOOOOO0000 1109160 STANDARD DEVIATION............. 4.1265 SKEUNESS (~1 TO +1 NORMAL)..... .8349 KURTOSIS (+2 TO +4 NORMAL)..... 2.7347 DISTRIBUTION LOU HIGH RANGE ACTUAL.... 5.4000 21.7000 16.3000 6-SIGMA... ~.4636 24.2956 24.7592 NORMALITY TEST-REJECT G .01 CONFIDENCE LEVEL KOLMOGOROV-SMIRNOV D........... .1902 15 30 45 60 75 90 0 O 9 O 9 6 00000000090000.000000609600064-940 CELL MAX PRCT QUAN 5.0000 0 0 10.0000 51 67 15.0000 24 31 20.0000 16 21 25.0000 12 30.0000 0 35.0000 0 40.0000 0 45.0000 0 50.0000 55.0000 60.0000 65.0000 70.0000 75.0000 80.0000 85.0000 90.0000 95.0000 ABOVE CELL MAX PRCT QUA 0' ****************X***** ********** ****** XXX OOOOOOOOOOOOOOO‘O 0 0 0 0 0 O 0 0 0 0 O N 0 O Q 0 9 O00.0900000000009009000906006 15 30 45 60 75 90 O 5 . 4) O 90 oo- +4» 9o «00 9. ve- +o 96 9° 99 co 0° 99 90 v9 90 90 +0 0+ 9¢ +0 0 118 TABLE 8 DEPTH TO RC 45 DISTRIBUTION SAMPLE DESCRIPTION QUANTITY MEASURED.............. 86 UNITS OF MEASURE............... IN/1000 STATISTICAL MEASURES THEHEANOOOOOOQOOOOOOOO906900691330.19.77 STANDARD DEVIATION............. 20.0451 SKEUNESS (-1 TO +1 NORMAL)..... ~.1988 KURTOSIS (+2 TO +4 NORMAL)..... 4.6899 DISTRIBUTION LON HIGH RANGE ACTUAL.... 70.0000 200.0000 130.0000 6-SIGHAO o 9 73906.23 1.9303331. 1.3092708 NORMALITY TEST*REJECT Q .01 CONFIDENCE LEVEL KOLMOGOROV*SMIRNOU D........... .1196 0 5 10 15 20 25 30 CELL MAX PRCT OUAN 3....3....3....3....X....3....3.... 10.0000 0 0 3 20.0000 0 0 3 30.0000 0 0 3 50.0000 0 0 3 60.0000 0 0 1 70.0000 1 1 2* 80.0000 0 0 3 90.0000 2 2 3** 100.0000 6 5 3***** 110.0000 2 2 IXX 120.0000 8 7 i******* 130.0000 22 19 3******X************ 140.0000 29 25 1***********R********Bki** 150.0000 21 18 3***************$** 160.0000 2 2 3** 170.0000 2 2 2#* 180.0000 2 2 3** 190.0000 0 0 3 ABOVE 1 1 3* CEI...L.MAX F‘RCT QIJQN :2.006.244-060940630094200903660020600 0 5 10 15 20 25 30 119 TABLE 9 HARDENABILITY DISTRIBUTION —_-“—-~~_--"_—-o__—.—_——--—.-_————-.“~__--—--*-_“*““~"~~~_~M— SAMPLE DESCRIPTION QUANTITY MEASURED.............. 43 UNITS OF MEASURE............... Di~IN/1000 STATISTICAL MEASURES THE MEANOOOOOOOQOOOOQObfi0000090 6540:2558 STANDARD DEUIATION............. 67.4357 SKENNESS (~1 TO +1 NORMAL)..... —.4493 KURTOSIS (+2 TO +4 NORMAL)..... 2.5229 DISTRIBUTION LOU HIGH RANGE ACTUAL.... 492.0000 764.0000 272.0000 6~SIGMA... 451.9487 856.5629 404.6143 NORMALITY TEST-ACCEPT G .05 CONFIDENCE LEUEL KOLMOGOROU~SMIRNOU D........... .1261 0 5 10 15 20 25 30 CELL ”fix FIRCT QUAN :6090:9000:OOOQ:OOOO:OOOO:OOO§:OOOO 400.0000 0 0 2 425.0000 0 0 2 450.0000 0 O 3 475.0000 0 0 3 500.0000 2 1 2* 525.0000 0 O 2 550.0000 12 5 :***X* 575.0000 2 1 2* 600.0000 5 2 3** 625.0000 5 2 :** 650.0000 19 8 1******** 675.0000 21 9 3********* 700.0000 0 O 3 725.0000 23 10 :**X******* 750.0000 5 2 2** 775.0000 7 3 2*** 800.0000 0 O 2 825.0000 0 0 : 850.0000 0 0 2 ABOVE O O 3 CELL MAX PRCT QUAN 2....2....2....2....:....2....3.... 0 5 10 15 20 25 30 120 TABLE 10 BTEP APPRQXIMATION DISTRIBUTION ”“~‘Q~~*”*"-n~I-OQQ~-u*“.l-ln—m.’C--to“...I.-00-.“--—-“.n—--_—-Q_‘~O__I-~--~—-u—-~_~ SQHPLE DESCRIPTION QUANTITY MEASUREDoooooo+ooee+oe 86 UNITS OF MEASUREO+¢¢¢¢O¢9099¢¢¢ RC”INMQ STATISTICQL HEASURES THE HEAN¢¢+ 32.7950 4.2760 *.4305 3.1564 RANGE STfiNflfiRD UEUIATIONF+9++¢0...0.. SNEUNESS (”1 TO +1 NURMAL)..++. KURTUSIS (+2 TU +fi NURMRL)...+. DISTRIBUTIDN LON HIG} ACTUAL,4.. 19°7207 41.8649 22.1435 6*SIGMQ.++ 1999é69 45.5230 25.6561 NORMALITY TEQTmACCEPT 9 .01 CBNFIUENCE LEUEL HOLMOGURUUmSNIRNUU 91101 [‘6$+‘é§6+*06 1~--.~u~-QIO .—...c~.~an»...“cave-cum...-mono-co.—.~~~—wu——-‘~—...—nnn-o—uuumoo—“u-uu~uwu~o~no——~Mo—~o_uu~_~m“_ 5 10 15 20 25 30 0 0 O 9 O 9 #699690¢¢6§+9+69.660006990060666. -.~¢ o '> CELL MAX PRCT QUAN 1H.7500 0 0 BOAHOOO 1 1 21.2500 0 O 22,5000 0 2367500 0 25.0000 3 726 .~1-35tf)0 5 27.5000 6 28.'500 1 \I v:- 0- 99 #9 9. ‘6'. 't. 99 #W* 9**** z**m** t* 59401} 330 a 00 00 31. 4 33355300 3245000 33.17500 135.(N300 3&«2500 37.W000 38.7500 ~40 o 0000 43 .1. . 95300 ABOVE CELL MAX PRCT é‘fl\463b‘flh39?l fl +L$ UNA? :W* :******* §#************* tkfifiX$$$$$$R$$$¥¥$$ :******* $******* 2**** 2**#*W t****** :* 2* 0 5 10 15 20 25 3O 121 TABLE 11 RAMP AF'F'ROXI NATION DISTRIBUTION “uuunwmnnmnwnnwuwmnun-“~I-unuomn-nhn-0—mno-uuboon-monn—‘wo-u-“w*fl‘mwwnu--fi~c~~nmo~~_m*-*m SAMPLE DESCRIPTION QUANTI TY MEASUREIII. . . . . UNITS 0F MEASURE. . . . . . STATISTICAL MEASURES THE HEANobooooooeooooe STQNDAFTD IIEUIF‘ITION. . . . SKELJI-JESS (*1 If U F: T 08 I S ( +1? II I S TR‘ 1' EU T I UN MZTUAL. . . _. . «.fi-SIGf-hi‘v . . . LON 3.75.6 . 33.334 55-4 . 4741 TO +1 NORMAL) . . . . . 1”0 +4 NORMAL.) O 9 9 O O 111 Far-IN"? 47.8454 4.4571 “.4521 2.7606 RANGE 20.4393 26.7427 §§OO¢O90§ OOQQ‘OQOQ HIGH 536 . .7717 (51. .2168 —..-.--o '"n 1., “N L, N l") R M ICI L I T Y ..-. ...- ... ...-a .-.. oo— ...— ... an.- —~ .4 .... no. ELL MAX 36.3500 3'7 . 5000 338 . 7500 40 . 0000 41.2500 42.5000 47:? . 7500 43?} . 0000 4d,) . £5300 4'7 .13000 49.7500 50.0000 51.2500 55.3.2? . 373000 73175 . 735300 5355.3 . 0000 56.9500 5"? . 3.55000 58 . 73300 QBOUE ELL PRCT 0 T E S T --- F: 5.115517?!" I? If I] L i"? 0001-? DU W 3 M I F: N C] U [-1 U A N m¢~o¢0 h.) OJ L»! 0* 1;} .3} 0V OO-‘ifléflibifil M A X F' R 1.“. T Q U A N l... E UE L. . 1 0 3 9 . 0 .1. C: If] N F I I! I? N C If: lJé-196560-500’.‘ -—.u—o—~—-.«a—noo-“ooununooo—tovnu-oqun—non-u—uuoo—c—h-Ic—o—o-‘oom“—~~o*—n~—-“——u_ '- m n” . 0 5 10 lb 20 20 30 9 0 0 0 0 9 000099990999990"}9099909999909 9t- * 0 $6 4 fifi + "r§ '66 Q. 90' 3* :*#** :##***#** :xxm ZXXXXXX :*mm**#******m :*%***#** :m##m*******m :#**#** :***#****#***#**$**$$***$* :*****# :******* :**** :xm* :* 6 O O O 0 O "099006h0¢9990¢9+666090§00§660.000 5‘5 1 0 .1. “.3 C? 0 ’53 1’3 '5 0 O 90 99 «>0 122 T #5 .174 L E 1 .13.? (A N F} L F -~ D F - T M I 33 T :9: T -~' F A I l... U F! E IT? If] F: R‘ F? I. (4" T I U N M I T H T D F! ‘73 I E} N i3) I... U L T I M :55; T E 53 T R E N l”?- TH .uouuw..~ww-..u.-ngnu—o-“nuns-nom-gated-000......-Ila-v...*uuuuonumnnnoanun-ounrcnucm-“un-“ocnto‘IuI-uq**~*~I*o~w*o~mo*~wuu SAMPLE SIRE..........+...~.. 131 DflfififiLmTIUN CUEFFICIENT..... ~.07é UNITS T0 MEHSHPE x.......... IoooxNuLBR UNI a To MEAQURF Y....,...., DEGREES QTANHARH DEUIATIHN (YN)..... a2.5 DEGREES .... .... .... .n— .... .... .40. ...-u .... o... .... ~00. oohn a... un— .... ...-o O... m. .... o..- a.” .--. n... on.- .... a... non. I... .... D... can: "an no. .... .o— 30.. ~~ no” u..- non. .... p... ..- no- --.- u—o ~00 ac— uu 0». you: ~- In- I.- la— R 4d _u'l "\ \J ' P. s.‘ .1 z' E I "N “a ::-. p A ..r "4 '1 0 f) > w) '2' I ‘ 0 1. )..A AA a [.5 9+0 l.\ >¢.\a.\x\ ""IIZT " '.'.-..| 7 .“fi (i) 13' TI R '8 1.00 .-’. 0.11:1". I"? (1‘: f: H r5. 1:"- :31 I} T'. 5‘1 1 A A A If! B if? ‘0 l"? 7.3114} F' {3) (J) 1. 3‘3 0 f. Bimini-3073.133 RR (3 .1. 7' 9'7} 3. {:3 :1"; (31000511 A A (43 fiéfimfififfififié A A f-‘s Fx‘ {-3 s’iz P: A P. 1?: (ix 1'?! A A A {413 (1‘. 7;? 0 0 3 (To T7." f7? '53 a "z '3; .1) .-, .. '.. \ f' ->+ any 7* 4-9 9.9 >9 1:6 #9 -.>o> f:: «X 3v r _ I" .--! I“. ‘J \- O ’9 3‘v vx -) . A 4001...-.'.’.r..~.l'..~....3....§.-~..3...~. 0:” 1.1 A ' . 0 {3.335.} .. 0 61'7": . 0 7113' . 0 -?9 3"- '>9 ‘3'. 9* P6 ’9 3'9 l...F§.'s%9T‘.'|"-!l..|n’-'\F!E:".53 L I N Eff-H3} flwF-‘F' F? If} 3': I MAT I {I N Y 1:: ( --1 . 1141 ) X + (' (331(6) 123 TABLE 13 ANGLfiwflFwTUI8TnAT~FAILURE CORRELATION MITH TORSIONAL YIELD STRENGTH .... ... —c - o~ u... .... page .0.- 0— u..- a..- --o. .... .... ...— 00-. '0'- -0- an. IQ. no. I.“ a... c... 0... am ...- ...n u..- no. —‘- -a. .... can. a..- .... oun- I... —-. n—aa no. no. .... .... ... .... un- n... ..o. .... no.- «.0 a... I..- .... SAMF'I-F f312':66¢64‘¢909§900690 1“{1 CORRELATION COFFFIOIRNT..... “.010 UNITS TO MEASURE X.......... 1000INWLBS UNITS TO MEASURE Y.......... UKGREES STANOARD DEUIATION (YX)..... 62.7 URGREES ~“~*~~-m“~mq-wu—~uw...-a—mo~~t~~~nm“nos-muoo—vm—ooanoounnoun-...no.-~uo-u-uounnoun-mo...no—oo-cu—c-u-ooo‘mpnoooa-~-u-.o~~uom~* J' J 0 C) ... . , _.. 1'3 :3 9 (3 '3 A? 0 0 4 33.? (t () a 9 9 ') 3‘ 9 z. 65"’~\'-¢-‘6(-4"+*¢6500066990000090 “~12? \',Iu.' C." -v.. . "09’: o.) .99 90 ;’-_‘ 'CT .I 100 135 150 I75 900 225 250 375 3 0 0 3 I? 5 350 375 4 0 0 BB BEA A A ARAAECARAAA B BBAAUCFH EB A ARR CC C A A BARAA A AAF HBURA AA AA A A AADAA AA 300 A A 0 O O (L g. é A O " 6 O A 0 o 6 45 "I" D0 ).5 -)¢ '7”? '3'9 .,¢ 90 39 1'9 9* "’0' >6 “P9 '5. 66 ’r. I» o o- ~9~a~e~ao~¢4+t~~¢+o+¢+a~¢aoo $32? . f) 131? . (I 5313 . (J 2) I. s 7. > ‘9 139 V -3 >9 1'. 99 ~>o «>0 éq- 9 l. E A 33' T -- S ['3 U A R F. 8 L I N E: A R A R F' R I") X I N A T I ON Y x ( “.90) X + (155) TABI...E 124 .‘I. -<'I A N G I... 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( ' 3': \ ..l i? t”) ., a~¢¢¢ no. cv-I coco -..”..o-anooua-ou-n-nolon-flmmouua '1. 1’5 71. ."'. r" .. “'.ifir}.l I“ HEBREES DFfiHFFR 0 >4 .3. .g. >6 ’4':- 4’.‘ 'I-c- 49 0+ ~>~'- 94' 44? +9 96 9'6 04- "Is’r 94- ? r) (- O 126 TABLE 16 HNGLE"UE~TMISTmfiTwFQILURE CDRRELfiTIUN HITH CASE HERDNEES (RC @ O+150 in) -— .... .... .... .... ..- .... ..- .... .... ...- ..— .... .-.. .... ...- ...- .... ...- ..u I... --oa u.- an u-n .... u- a... .... .... .... ou- .... .... ...— u... ... . a... .... .... .... a... .... tun ...- ~0- .... I... u- uon no. - u‘. noon n.- SAMPLE SIZE¢¢o+o¢+o+éfi¢o+¢+¢ 111 CURRELQTIUN COEFFICIENT.¢§¢o *9242 UNITS TU MEASURE X ¢a¢++¢¢§¢ ROCKNELL C UNITS Tn MEASURE Yo.¢¢.+¢.»+ DEGREES STANDARD DEUIATIUN (YX)+«¢.. 58+? DEGREES .... pun ...- ‘4. .... .000 u... ”a. .0- -~~ a... o..- no“ 9..- “u an our: .0.- oo” coo. .... I... not. ecu. "on not: 0.. none 0... a... .... «on no“ u- —00 o... a... o... .... ...- a... nu u... 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"r-a- flu;- ~:-<- '2'9 ‘i'b “I” "I (- s‘ x“ 0 BIBLIOGRAPHY 132 BIBLIOGRAPHY 1. Forbes, R. J., Metallurgyin Antiquity, Brill, Leiden, 1950. 2. Shot Peening, Wheelabrator-Frye, Mishawaka, Indiana, 1977. 3. Almen, J. 0., "Improving Fatigue Strength of Machine Parts," Machine Design, Vol. 14, No. 6, pp. 124-129, 1943. 4. Horger, O. J. and Maulbetsch, J. L., "Improving the Fatigue Strength of Press-Fitted Axle Assemblies by Surface Rolling," A.S.M.E. Transactions, Vol. 48, pp. 91-98, 1936. 5. Osborn, H. B., Jr., "Surface Hardening by Induction," Metals Engineering Quarterly, August 1971. 6. Shklyarov, I. N., "Surface Quenching in the Case of Bulk Induction Heating ZIL-130 Axles," Metal Science and Heat Treatment, 1967, (UDC 621.735.545.4). 7. Smith, D. C., "Rating the 79's," wards Auto World, Vol. 14, No. 10, p. 39, 1978. 8. Crandall, S. H., Dahl, N. L. and Lardner, T. J., An Introduction to the Mechanics of Solids, McGraw-Hill, 1972, p. 366. 9. 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R., "A Behavioral Model for the Fracture of Surface Hardened Components," Trans- actions of the A.S.M.E. - Journal of Basic Engineering, Dec. 1965, pp. 871—874. Rolfe, S. T. and Barsom, J. M., Fracture and Fatigue Control in Structures, Prentice-Hall, 1977, p. 32. Makky, S. M., "Plastic Flow and Fracture in Round Bard in Pure Torsion," Journal of Mathematics and Mechanics, Vol. 10, No. 2, 1961, p. 199ff. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 135 Fractographs 4918, 491 , 5032, A.S.M. Metals Handbook, Vol. 9, A.S.M., 1974. "Interpretation of Light—Microscope Fractographs," A.S.M. Metals Handbook, Vol. 9, pp. 36-48, A.S.M., 1974. Makky, S. M., "Application of the Theory of Fracture on the Surface of Instability," Angew. Math. und Mech., Vol. 46, No. 7, 1966, p. 432ff. Nadai, A., Theory_of Flow and Fracture of Solids, Vol. 1, Chap. 17, p. 243ff, McGraw-Hill, N.Y., 1950. Nadai, A., op. cit. Chap. 15.13, p214ff. Altiero, N. J., "On Edge — Fracture Problem of Rock Mechanics," Mechanics Research Committee, Vol. 3, pp. 345-352, Pergamon Press, 1976. Yokobori, T. and Otsuka, T., Proceedings of the First International Congress on Experimental Mechanics, Pergamon Press, 1963, p. 353. Yokobori, T., Takahaski, T. and Kishimoto, H., Journal of the Australian Institute of Metals, Vol. 8, p. 184, 1963. McClintock, F. A., "Plasticity Aspects of Fracture," Fracture, An. Advanced Treatise, Vol. III, Academic Press, 1971, p.111ff. Hertzberg, R. W., Deformation and Fracture Mechanics of Engineering Materials, J. Wiley and Sons, 1976, p. 248. Kardos, G., "Design Criterion for Generalized High Strain," Transactions of the ASME-Journa1 of the Engineering for Industry, Vol. 90, August 1968, pp. 485-490. Jatczak, C. F., "Determining Hardenability from Composition," Metals Progress, September 1971. Lilliefors, H. W., "0n the Kolmogorov-Smirnov Test for Normality with Mean and Variance Unknown," American Statistical Association Journal, pp. 399-402, June 1967. Glocker, R., Material Prufung mit Rontgenstrahlen, Springer, 1958, cited in J. T. Norton, "X-ray Determination of Residual Stresses," Materials Evaluation, February 1973, p. 21Aff. Cullity, B. D., Elements of X-ray Diffraction, Addison-Wesley, 1959, p. 431ff. Weinman, E. W., J. E. Hunter and D. D. McCormack, "Determining Residual Stresses Rapidly," Metal Progress, July 1969, p. 88ff. 60. 61. 62. 63. 64. 65. 136 Anderson, K. G., "Practical Accuracy Considerations in Use of the Fastress Residual Stress Analyzer," Chrysler Fastress Seminar, April 4, 1978. Patch, N. J., "Metallurgical Aspects of Fracture," Fracture: An_ Advance Treatise, Vol. 1, Academic Press, 1968, p. 381ff. Cellitti, R. A., "A Study of the Effect of the Induction Hardening Variables on the Residual Stresses and Bending Fatigue Strength of Final Drive Gears," SAE Transactions, Vol. 76, 1968, p. 124ff, (SAE Paper No. 670504). Almen, J. D. and Black, F. H., .Residual Stresses and Fatigue in ,Mgtala, McGraw—Hill, 1963. Landau, L. D. and E. M. Lifshitz, Theory of Elasticity, Pergamon Press, 1959, p. 30. Kobasko, N. 1., "The Development of Quench Cracks in Steel," Metal Science and Heat Treatment, 1971, p. 30. MICHIGAN STATE UNIV. LIBRnRIES I11111111|111111111111HIM/111| 31293010743346