Engineering Research Institute University of Michigan Ann Arbor EIGHTH PROGRESS REPORT TO MATERIALS LABORATOR Y WRIGHT AIR DEVELOPMENT CENTER ON NOTCH SENSITIVITY OF HEAT-RESISTANT ALLOYS AT ELEVATED TEMPERATURES by H. R. Voorhees J. W. Freeman Project 2024 Air Force Contract No. AF 18(600)-62 Expenditure Order No. R-614-15, SR-7j September 30, 1954

..SUMMARY Work.previously reported for -this investigation. under Contract AF 18(600).-62 showed a qualitative relationship between notched.bar rupture properties and relaxation characteristics. The question remained unanswered as to.how the.data obtairned might be affected by residual-stresses developed during preparation of notches. Moreover, the relative dependence of notch behavior upon relaxation strength, ductility, or.other material property was not established. Tests are now in progress to study effects of notch preparation methods.on notched-bar rupture data. No general conclusions have yet been d.rawn, Smooth and notched b.ars from three different heats of Waspaloy are being tested at 1350'F in an attempt to learn whether variations in other properties can be related to reported differences in notch behavior between heats. Tests on smooth bars.only, show material from the one vacuum melted heat being used to have higher.creep strength and rupture strength than do the two heats melted in air. A tentative.quantitative.analysis of notch behavior has been d~evel. oped, based on initial stress distribution around a notch and on changes.-of this stress pattern during testing. Incomplete calculations indicate satisfactory agreement with observed behavior for conditions of notch strengthening ex. amined to date.

2 INTR ODU C TION This report covers progress made during the third quarter of 1954 under Contract AF 18(600).-62 in a study of factors affecting notch sensitivity of heat-resistant alloys. Past work -has.shown a qualitative relationship between notchedbar rupture properties and relaxation characteristics, but did not establish relative dependence of notch behavior upon relaxation strength, ductility, or other material properties, Further studies are to investigate the relative significance on notch behavior of variations in such properties. For this purpose two additional heats of. Waspaloy have.now been obtained. Notch strength.at 1350'F of one is reputed to be greater than for the.heat tested previously, while the notch -strength.of. the other may be lower. Differences between heats in short-time tensile properties,: creep-rupture properties, ductility in the rupture test,.and relaxation. rates ~are to. be. compared with corresponding differences in notch behavior. Specimens are to be compared both for conventional heat treatment and for this treatmeant without the 4-hour age at 1550'F. It has been reported that some heats of this alloy without the 1550"F aging treatment are notch weakened at test conditions where notch strengthening may be expected for conventional heat treatment.. One.question left unanswered by past experiments.was the magnitude of effects on notch rupture strength from residual stresses developed during machining of the notches. The present report gives results obtained to, date in tests with bars after various notch.-preparaUt on methods. It also introduces a tentative method for quantitative analysis of notch behavior based on initial stress distribution and changes of stress pattern during testing of notched bars..

3 CURRENT STATUS OF THE INVESTIGATION Study of Methods,of Notch Preparation It was reasoned that any residual stresses at the notch root would be largely eliminated for high stresses where extensive yielding occurs at the base of the notch during loading. Also for alloys with low relaxation strength stress. concentrations should be reduced quickly and notch strength should not be much affected even though residual stress-es augmented the initial level at the notch root, In accordance with these considerations, tests have been limite.d to a condition of high yield strength and low relaxation rate (In.conel X-550 at 1350~F). A single notch root radius (0.005 in.) and a single.stress (40, 000 psi) are being studied first. This condition appeared to give about the greatest degree of notch embrittlement in tests using ground notches. Tests are still in progress at this first condition. A few experiments with other notch-root radii and/or other nominal stress levels will probably be necessary before definite conclusions may be drawn as to the magnitude of effects on test results caused by variations in notch preparation practices. Comparison of Notch Behavior Differences and.Smooth-Bar Property Differences for Different Heats of the Same Alloy In order to reduce effects of prior history, material from three heats of Waspaloy under study were given the same hot rolling into bars of a common size from which specimen blanks are to be cut, Limited creep-rupture tests have been completed for all three alloys with the two different heat treatments planned. Notched-bar rupture tests are being delayed until effects of various notchpreparation methods

4 have been evaluated. The only observation of note on tests so far is that the one vacuummelted heat has higher creep strength and.rupture strength than do the two heats melted in air. Development of Correlations Between Notch Behavior and Smooth-Bar Properties A tentative analysis explaining notched-bar behavior in terms of initial stress distribution and.rate of change of this stress with time has been worked out in detail and is covered in a later section of thisreport. Incomplete calculations now in progress indicate that notch strengthening can be satisfactorily explained by this analysis for the conditions examined to date. Further calculations are to extend.to cases of notch weakening. The step-wise.analysis.developed is too detailed and lengthy to serve as a quality control method. An empirical expression relating notch strengthening or weakening to easily-obtained smooth-bar properties is being sought. A/s yet but limited effort has been placed on this subject and no general criteria of notch weakening or strengthening have been found. EXPERIMENTAL RESULTS New Rupture Data at 1500'F for S.-816 with.Special Heat Treatment Previous reports indicate notch weakening under some cmalditions for S-816 with the following heat treatment: 2325'F, 1 hour, WQ + 13. 5% Reduction at 1200'F + 1400'F, 12 hours, AC. To better establish smoothbar rupture properties at 1500-F for comparison with notch characteristics

5 at that temperature, the following additional data have now been obtained: Stress Level (psi) Rupture Life (hours) 65, 000 0. 2 50, 000 2.5 25,000 1053.2 Notched-Bar Rupture Life at a Single Nominal- Stress, but with Different Methods of Notch.Preparation Three techniques were employed to finish the notch root after prior turning to within 0. 020-inch of the final root diameter: (1.) Turning with a formed lathe tool, (2.) Rough grinding to within about 0. 005 inch, followed by a quick light finish grind.to dimension, with flood cooling at all times. (3.) Turning to within about 0. 005 inch on diameter with a specially-formed lathe tool, then lapping with a wire 0. 009-0,. 010 inch in diameter, using a fine lapping compound, as illustrated in Figure 1. The most highly.stressed portion of the worked metal left at the notch root by the turning operation can be quickly and easily removed in a few minutes by this form-lapping technique. Specimens tested had the notches prepared at any of three stages in the.heat treatment: (a) after.conventional solution and aging treatments, (b) after solution, but- before aging steps,... (c) before either solution or aging. (d) one pair of bars notched after a complete conventional heat treatment was placed in a 1600F furnace 20 minutes and then air cooled prior to testing at 1350~F, Test results are shown in Table 1.

6 TABLE I NOTCHED-BAR RUPTURE TESTS FOR SPECIMENS:WITH DIFFERENT METHODS OF NOTCH PREPARATION (All specimens tested.at 40,000 psi stress and 1350'F. Shank Diameter 0.600", Nctch Diameter 0. 424", Notch.Root Radius 0. 005", Notch.Angle 60') Order and Method of Notch Preparation Rupture Life, Hours ~..,..,..... ".'',.,...!':.........'........... -:''_.'....., Conventional H. T. + Turned Notch 176.4 Conventional H, T. + Ground Notch 110.7 Conventional H. T. + Lapped Notch 74. 5 Conventional H. T. +-Turned Notch + 20 min. in 1600'F Furnace 168. 6 Conventional H. T. + Ground Notch.+ 20 min. in 1600~F Furnace 87. 5 (2150'F, 1 hr,- AC + Turned Notch + Convy. Ages) 76.4 (2150'F, 1 hr, AC + Ground Notch + Conv. Ages) 30. 7 a(As Received + Turned Notch + Conv. H. T. in Atmosphere) 50. 2 a(As Received + Ground Notch + Conv. H, T. in Atmosphere) 65. 1 a. Test to be repeated. Furnace coil burned after 3 (+0. 5) hours of the 1600~F age, Cooled to room temperature. and given usual aging;at. 1350~F before testing. The data indicate a moderate increase in notch strength for turned notches over ground notches when specimens are machined after all heat treatments are completed. The lapped notch gave a somewhat shorter rupture life than even the ground notch. A short "stress relief" at 1600~F after notching appeared to have little effect on relative strengths of turned and gr~ound notches. Longer stress reliefs arenotplanned at this time in view of probable alterationn of material properties before appreciable relaxation of residual stresses. Comments.on other results appear premature, but solution and aging after notch preparation seemn to give lives of the same magnitude as

7 for bars carefully machined after heat treatment. It might be noted that the notch ground after conventional treatment ruptured at almost the identical time (109.4 hours) reported by Carlson, MacDonald, and-Simmons (1) for the.same stress and notch geometry. A number of the tests are being repeated where the aging time at 1600'F was uncertain due to furnace burnout. ANALYSIS OF CHANGES IN STRESS DISTRIBUTION FOR A NOTCHED BAR DURING LOADING AND DURING CREEP TO RUPTURE This investigation has been based on the belief that variable response of materials to notches at elevated temperatures must be closely related to initial stress concentrations and their change with time, as controlled by creep and relaxation,.At the outset it was reas.oned that a notch introduces nothing inherently new into properties of an alloy, but only changes the stress-strain history of fibers in the notched bar. If one were to reproduce in a smooth bar the history experienced by a fiber of a notched bar, the life of each should be the same. In the following:development attention is.focused on a thin circular section containing the plane of a circumferential notch. It is proposed to follow changes with time of the stress levels in representative fibers located in this disk at various radii from the axis.of the notched bar, Consideration is being limited to deep notches with a 6.0 included angle and with notch root radii between 0. 005 and 0. 100 inches. The diameter at the notch has been chosen to give a notch. cros.s section half that f the shank, The.widely adopted analysis of Neuber (See Ref. 2) for a deep

8 hyperbolic notch should give a very close solution for stresses in the elastic region. When this analysis is applied to a notch with 0, 005 inch radius of curvature at the root of a notch in a specimen with minimum diameter of 0.424 inches, the fiber nearest.the notch root is.found to have an axial stress slightly more than six -times the nominal axial stress, For the same fiber the hoop stress (in the circumferential direction) is about twice the nominal axial stress. General Behavior During Plastic Flow One may inquire as to how high the tensile load may be raised before this most highly stresses.fiber will yield plastically, and how high an initial localized stress concentration may be achieved in a bar with a deep sharp notch, The general plastic behavior of metals at room tempera.ture is treated.by Hill. (See Ref. 3).. There is no reason immediately apparent why his mathematical development should not remain valid at elevated temperatures for changes occurring so rapidly that creep effects are relatively small, provided the necessary physical constants are evaluated at the temperature under consideration. The following observations are taken from..Hill's treatment: (1,o) By expe.riment, the extent of yielding is but little affected by a moderate superimposed hydrostatic pre.ssure.. From this observation it may be reasoned that the component of plastic strain in a given direction depends not on the magnitude of total stress, but rather on the "deviator" stress in the given direction. This deviator stress (SI'j) is defined as the component of total stress less the arithmetical average of the three princiP pal stresses.. (2. ) The start of yielding appears to depend only on differences between individual principal stresses (SI>S2>S3) For ductile materials the

best correlations to date seem to indicate the following combination of individual stresses as giving the.proper measure of the effective stress (S) at onset of yielding: 2 (S)2 = (Slsz)2 + (S23)2 + (S3-S1)2. (1) In pure tension the value of S is simply the axial stress S1.. A similar yield criterion in terms of strains can be written: (9/2)(.,e) = (el-e)2 + (e 1-e3)2 + (e2-e3), (2) where e is the axial strain for pure tension and el, e2, e3 are the three principal strains, (Some.authors prefer to use the octahedral shear stress and strain, given respectively by ~ 7 and e (3. ) When an increment of plastic strain occurs, the directions of the principal components of this strain coincide with the axes of the total principal stresses existing at that moment, independent of the direction of the added increment of stress. Moreover, the magnitudes of plastic strain increments depend on the existing total stresses and not on the stress incre:ment. As a consequence, the plastic-strain history must be followed in a step-wise manner. After each.smpll plastic deformation the new stress pattern is determine.d, Then the strain pattern may be evaluated for the next stress addition. The incremental plastic strain (deij) in any direction is related to the incremental effective strain (d ) by the following relationship develN oped on page 39 of reference 3; deij=:eP (3/2)(Sij/~) (3) (4.) If an element "unloads", i. e., if the effective stress in a fiber decreases, all changes follow the laws of elasticity until such time as the effective stress is again raised to the value from which unloading began0

10 Stress Patterns During Loading of a Notched Cylindrical Bar The above general behavior patterns apply to all classes of plastic flow, However, specific expressions for the distribution of stresses and strains have been published for only a limited number of examples. Lacking ia rigorous analytical solution for notched tensile bars with axial symmetry, one must resort to approximate methods, Fried and Sachs (Ref. 4) took hardness traverses on sections from large notched bars of carbon steel pulled to fracture at room temperature Contour lines of constant hardness in the.fractured bars were.observed to correspond closely to the photo-elastic pattern showing lines of constant shear stress in flat notched specimens pulled under tension within the elastic range. From this.coincidence it was.concluded that "...in a notched body (under load) the lines of constant maximum shear stress are almost identical in the plastic and in the elastic state." At conditions studied in the present program, changes in notch geometry during loading have been shown previously to be too small to affect elastic stress concentration factors significantly. Under these circumstances, the distribution of deviator stress components corresponding to the elastic state will be assumed to continue during the small plastic strains of the loading period. When a notched specimen is stressed below the elastic limit for. all fibers, nearly all the energy associated with straining is recoverable. During plastic.deformaticons1 energy is used in changing shape of the body, with unknown energy losses in the process. Neglecting any energy-loss during plastic deformation, the attainable plastic strain in any fiber would appear to bethat value at which the area under the actual stress-strain curve just: equals the -elastic strain energy which would have been expended had elastic conditions

persisted throughout loading. This approximate method should satisfy the needs of the present analysis, Application of Plasticity Correlations to Creep and Rupture The usual methods of handling plastic flow at room temperature seem to be adequate to correlate creep rates under triaxial loading provided an experimental relation is available between stress and creep rate for the state of strain present. Johnson.(Ref. 5) compared creep rates of magnesium at 20~C for flat plates pulled in two directions and for combined tension. torsion runs in thin tubes. The two systems were designed to give like values of effective stress, but with the principal stress components in the two cases differing by a constant value of hydrostatic stress. Creep rates were observed to be the same for these two quite different stress patterns. For tests on thin cylinders of four alloys at two or three temperatures each, a plot of octahedral stress versus octahedral strain gave common curves.for pure shear, pure tension and variable ratios of shear and tension. Similar good agreement between theory and experiment was found by Sode~rberg (Ref. 6) for thin steel cylinders under internal pressure when creep components were compare.d with corresponding deviator stresses, The.situation is less clear with respect to rupture under combined stresses at creep conditions. From short-time fracture tests on an aluminum alloy in combined tension and torsion, Johnson and Frost (Ref. 7) concluded-,,,..the criterion appears to be between the octahedral stress and the maximum shear stress, and is certainly not a direct function of the maximum principal stress." But incomplete tests on 0. 5% Mo steel and on copper indicated the criterion of fracture might be the maximum principal (tensile) stress of the system imposed.

12 Proposed Step-wise Treatment of the Creep.Relaxation Process When a notched bar is held under constant load at elevated temperature,a complex changes in stress and strain throughout the bar may be.expected, with gradual leveling of initial stress gradients. In the actual bar the stress levels will vary smoothly from point to point without discontinuities, but to facilitate.calculations.the cross section will be divided.into a sufficient number of concentric rings such that conditions at the centroid of any given ring are.quite.representative of that entire rings Further,.the actual continuous change in stress pattern will be replaced by an equivalent series of time intervals over each of which the creep rate and stress in a given fiber may be considered nearly constant. Immediately on loading, a fiber in the notched bar has a..unique creep rate determined by the initial effective.stress and effective strain. The corresponding creep rates in the three principal directions may then be calculated from a modification of Equation (3): e_ = eP (3/2)(Si IS) (4) (The dot over a symbol represents "rate," a prime indicates a deviator component, and a bar over a symbol refers to effective stress or strain. ) The component of plastic strain in any direction may result in elongation (creep) of the body,, but it. could also replace initial elastic strain, with resultant drop in the stress level of the fiber (relaxation). How the total plastic deformation splits, between creep and relaxation depends on the extent of stress gradients in the structure. In a convenational tensile.creep bar, where all fibers are subjected to the.same stresses until necking occurs, the body can creep as a unit with no reaction of one.fiber on another. Such is not the case, however, in

13 a notched bar where the stresses vary continuously from one fiber to the next, Consider three parallel bands with axial stresses.-S.>> S3 at their respective centroids. Corresponding axial creep rates,. if each band were separate from its neighbor s.,, would be C1>C2>C3. For continuity to be maintained between filaments, the same total deformation must exist on the two sides of'the "common interface. This does.not say that the deformations at the two edges of a particular band will be the same. The creep rate at different points across any such band in a notched bar will deviate slightly from the.rate at its centroid, but this latter value should be quite representative if the band chosen is not too wide. When band (1), has a total creep in excess of band (2), the difference in plastic strain must be made up by elastic strains in the two bands so long as the fibers of (2) do not become stressed above their yield point. This elastic interaction gives a stress reduction (or relaxation) in band (1) and a stress rise in (2). The absolute values of these two elastic stress..changes.will be distributed inversely as the areas of the two bands concerned. Sinultaneously,. the elastic stress in band (2) is relaxing and that in (3) increasing due to a similar interaction atthe 2-3 interface. The net stress change for band (2) can be taken as the difference between the gain from (1) and the loss to (3). Using this analysis, the physical requirement of constant total axial load in a notched-bar test is automatically met since.each.time one band drops a certain portion of the axial load by relaxation, a.neighbor picks up exactly the same amount of load. The procedure can be applied in turn to strain rates and stresses in all three principal directions. On occasion the loweristressed of two adjacent fibers will be above

14 the elastic limit. In this.event the stress interaction between the two bands will be reduced considerably below that for the pure elastic.case. The load gain of one band will still be set equal to the load loss of the other, but the stress-strain relationships for the ring with rising stress level must be determined from the short-time tensile curve at the existing stress level. Applying this approach to concentric rings in the plane of a circumferential notch, analysis will start with the two outermost rings and proceed inward toward the axis, The drop in load for each by relaxation and the gain in load by shift from its neighbor is found.for each ring for the same short time inte rval. -When all changes are known, the new stress levels in each.ring are calculated and the process repeated. During the first time interval considered, each ring has been subjected to some average.stress level for the given length of time, so that a fraction of the total rupture life.of each.ring has been used up. Initial correlation attempts will assume that the portion of rupture life consumed during any interval equals the fraction: actual time.at a given effective stress rupture time at this e ffective.stress. When the entire life has expired for any one fiber, failure of the notch bar will be considered to have initiated and rupture should be imminent. Part of the.data of Johnson and Frost referred to above indicates that rupture life.depends.on the largest single stress; i. e,, on the axial component in.the present case. If the analysis indicated here is correct, the axial component cannot fall below the nominal stress, This, plus the fact that other data refute the role of the largest principal stress as..con trolling rupture, was the reason for using the effective stress in.the above fraction,

15 SAMPLE CALCULATIONS ILLUSTRATING PROPOSED CORRELATION METHOD For the notches tested in this program the region of stress concentration occupies approximately the oute.r one tenth of the notched cross.section, The notched section was divided into a central core covering one half of the total area, surrounded by four rings each with one tenth of the area, plus four outer rings each with one fortieth of the total area, These rings are designated consec.utively by numbers from one to nine, When a Waspaloy specimen with 0. 040-inch root radius was loaded to a nominal stress of 35,000 psi at 1500-F, the stress distribution at the centroids of the several rings was estimated to be as follows: Ring No. Axial Stress, S Tangential Stress, St Radial (psi) (psi) Stress, S. (psi) 9 (outermost) 73, 300.22,900 2 180 8 69,400 23,150 5,760 7 65,900 23,400 9, 100 6 61, 200 22,700 10,640 5 51,150 20,900 13, 330 4 40,800 18,400.14,900 3 35,200 17,400 15,160 2 31,000 16,200 14,950 23, 650 13,870 13, 650 These values were.converted to deviator components and then combined to give.effective stress levels using relationships previously cited: S ='a - (1/3)(Sa+St+Sr), etc. = (1/2){ (S'a-Slt)2t(S'a-S' )2+(SltSr )2} Resualts of the above calcalations are included in Table Il, below0

16 TABLE II INITIAL CONDITIONS AT THE CENTROIDS OF THE SEVERAL RINGS CONSIDERED IN THE ANALYSIS OF A NOTCHED BAR OF WASPALOY LOADED TO 35,000 psi AT 1500'F. (NOTCH ROOT RADIUS 0. 040 IN.) Slope of Stress Components (psi) Creep Rate (in/in./hr) Tensile Curve Ring No. Sa -St -Sr i 6P H' (psi/in/in.) 9 40,510 9,890 3:"9410 63,500 0. 072. 6 8 36, 630 9,620 27,:010 57, 000 0. 029 13* 6'x 10 7 33,.100 9,400 23,.700 51,100 0.012 17.6 6x 10 6 27,690 10,810 22,870 45,650 0. 0054 21 x 10 5 22,690 7,560 15,130 34,600 0. 00097 21 x 106 4 16, 100 6,300 9,800 24,300 0.00019 21 x106 3 12,630 5,170 7,410 19,000 0.000 074 21 x 106 2 10,280 4,5.20 5,770 15,100 0.000 034 21 x 10 1 6,590 3, 190 3,410 9,900 0.000 009 7 21 x 10 During the experimental program on this alloy complete smoothbar creep curves were obtained for a range of stress levels. These were cross plotted to show stress versus creep rate for different percentages of total life expired. Thus, one.curve gave initial creep rates; a s.econd showed minimum creep rates (approximately 5 to 15% of total life expired.) Other curves showed creep rates for times equal to 0. 4, 0, 6, and 0. 8 of the rupture lives at the several creep stresses employed. From such plots the effective creep rate ep can be found corresponding to the effective stress S for each ring. The initial creep rate for each ring has been added to Table II. Values from that tabulation will now be used to illustrate the steps in a typical calculation cycle. Step 1. For any given ring-the deviator components of creep are

17 distributed in proportion to the deviator stresses: bj = (3/2)( P)(S / Thus for Ring 9: e" a = (3/2)(0. 072 in/in/hr)(40,510/63,500) = 0.0689 in/in/hr.:t = (3/2)(0. 072)( -9,890/63,500) = -0.0168 and for Ring 8:'a = (3/2)(0. 029)(36,630/57,000) = 0. 0279 e'-t (3/2)(0.029)( -9,620/57,000) = -0. 00734 Step 2. Perform subtractions for each component and.each pair of rings: Ring No. eta -et 9 0. 0689 0. 01690 8 0. 0279 0. 00724 9-8 0, 0410 0. 00946 Step 3. The differential creep-rate components may be.converted.to stress changes for a time interval chosen small enough that the assumption of constant conditions over the interval is not too far in error: Stress component change = (Component creep rate difference) (psi) (in/in/hr) (Time interval)(An area-modulus factor) (hr) (psi/in/in) The generalized relationship between components of stress and strain in the elastic region may be expressed: S.. e.. 6.. + 2G e.., where X is a proportionality constant 1ij 1i - ij and G the shear modulus. By the assumption of plastic incompressibility inherent to the criterion of yielding already adopted in this analysis, eii can be set equal to zero, It is further noted that the plastic strains in relaxation are

18 equal to but opposite in direction to the elastic strains replaced. Therefore, changes of principal stresses and principal strains for relaxation are re.lated by ASi = -2GAei Since differences in deviator components are the same as differences in total components, this same.expression relates deviator changes. The factor 2G is related to the elastic modulus (E) and Poisson's ratio (n) by: 2G =( - E l+n Available.data indicate that n has a value of about 0. 32 for the temperatures and alloys of interest to this program. When the stress gain of the lower-stressed fiber puts it into the yield range, the resulting deformation for a given stress.change is larger than for the elastic case by the ratio E/H', where H' is the slope of the stress-strain curve at the stress concerned. In the following the symbol M will be defined for both the elastic and plastic cases as M = H' /(l+n), with H'=E for elastic stresses. Distribution of stress changes in adjacent fibers with different creep rates may be seen from consideration of Figure 2, which shows bands (1) and (2) with respective cross sections A1 and A2, In a given time interval, creep would change the lengths from ab to acl and ac2 if the bands could creep independently. The two rings will now be strained amounts 1 and 62 in opposite directions until the creep difference Ac is "made up" -6 = Ac2 (I) ~1 52

19 But the load picked up by (2) must equal that dropped by (1) to preserve boundary conditions. alA1M1 +62A M = 0 2 2 61 = 62 AM / AM1 (II) 22 11 Substituting (II) into (I)o 62 (1 + A2M /AM) = =62 -Ac( AM 2-, A.-AXM ) 11 2 2 61 -Ac( ) A.lM1 + A2M2 The stress changes corresponding to these strains are: AS = M = ( A2M1M2 )(Ac) 1 M1'1 AllM + A2M2 A1M1M2 As2= 6M2 - ( AlMlM2 )(Ac) A1M1 + A2M2 The area-modulus factors for rings (1) and (2) are thus. seen to be, respectively: + A2M1M2 and A1M1Mz A1M1 +A2M2 A2M21 AM2M2 For elastic strain distribution between rings of equal area, the area-modulus factor is simply M/2 = G. At 1500~F Waspaloy has an elastic modulus of 21 x 106, while H' = 13.6 x 106 at- = 57,000 psi, Therefore, 21 x 106 6 M9 = = 15.9 x 10 9 1,.32 13.6x106 6'1 0"*''

20 For.a time interval of 0, 002 hour, the axial strain change in ring 9 is found to be: (-15. 9)(10, 3 x 106) (0. 041 in/in/hr)(0.002 hr)(3 x 10) psi/in/in = -510 psi. 15.9 + 10.3 The axial stress in Ring 8 is raised by 510 psi from the same 9-8 interaction while the 8-7 interaction would lower the axial stress in Ring 8 by 240 psi, determined by a calculation similar to the one shown. The net change in Ring 8 is thus +510 -240 = +270 psi during the first 0. 002 hr. Step 4. Combine the new values of the deviator stress for each ring to get the.new effective stress. Step 5. The percent of life expended during the above time interval is found for each ringo (a) - At the initial effective stress of 63, 500 psi the rupture life of Ring 9 would be 1. 3 hr. (b) At the new effective stress (62,800 psi) after the first time interval, the rupture life is 1. 5 hr. (c) 00 002 hr 100%o = 0, 14%o of life expended in Ring 9 (1. 3 + lo.5)/2 during the first 0. 002 hr of the notched-bar test. Step 6. Knowing the new effective stresses and the cumulative.expenditure of life to date, the new effective creep rate for each ring may be read from the plot of stress versus creep rate and the process repeated for a second interval.

21 BIBLIOGRAPHY on Notch-Sensitivity of Heat-Resistant Alloys at Elevated Temperatures. (Rupture Strengtho Notchd Bar s at High Tempe ratNures. Pr elimina'ry Copy, WADC Technical Report 54 -3391, June 1954. 2., Neuber, H. Theory of Notch.Stre.sses. J. W. Edwards, Ann Arbor, Michigan, 1946b'''. 3. Hill, R. The Mathematical Theorof Plasticity Oxford University Press, London, 1950. 4. Fried, M. L. and Sachs, G. Notched Bar Tension Tests on- Annealed Carbon Steel Specimens of Various Sizes and Contours, A.S. TM. pec:ial Technical Publication.No.' 87, Symposium on Deformation of Metals as Related to Forming and Service, 1948, pp. 83-117. 5. Johnson, A. E. Creep Unde.r Complex Stress Systems.At Elevated Temperatures. Proceedings, of the Institution of -Mechaniccal Engineers, VotT.. 164T (1951), pp. 432-447. 6.. Soderberg, C. Ro. Interpretation of Creep Tests on Tubes. Transactions, Americ-an.Society of Mechanical Engineers, Vol' 63, (1941), pp. 737-748. 7. Johnson, A. E. and Frost, N. E. Rheology of Metals at Elevated Temperatures, Journal of the Mechani-cs and Physics of Sol-ids, Vol. 1, (1952) s ppo 37-52.

22 Shade:d area.srhows.naterial removed during lapping. Shkape.: of notch::after turning with special lathe tool. Figure 1. Sketch Illustrating Notch Preparation by Form Lapping. a b cl 6 -i Band 1 AC - i t,'- "-62 Band 2 a b c2 Figure 2. Schematic Representation of Stress-Strain Interactions at an Interface between Two Bands with Different Creep Rates,