1、!ZI-2 radians.In reference 3 it was shown that the Neuber constant O for steelscould be taken to be a function of the tensile strength of the material,shown graphically as a curve of 0_V plotted against tensile strength.In the present investigation it was found that two curves are requiredfor wrough
2、t aluminum alloys (fig. 2), one for alloys in T condition(heat-treated) and one for alloys in 0 (annealed) or H (strain-hardened) condition. Like the curve for steels given in reference 3,the curves shown in figure 2 were obtained by a trial-and-error pro-cedure based on the analysis of notch-fatigu
3、e data. It may be seenthat for heat-treated aluminum alloys, the quantity o_ ranges fromabout 0.5 to 0.12 in.l/2; compared on the basis of equal strength-density ratios, the quantity _ for heat-treated aluminum alloysaverages about three times the corresponding value for low-alloy steelsas given in
4、reference 2.The factor KN constitutes the predicted value of the fatiguefactor for fully reversed loading at stresses near the fatigue limit.It also serves as the basis for computing static strength factors,which will be discussed later.L1743Provided by IHSNot for ResaleNo reproduction or networking
5、 permitted without license from IHS-,-,-llSharp Notches and CracksWhenthe notch radius p is small comparedwith the depth of thenotch and the width of the net section, all formulas for theoreticalfactors maybe written in the formL1743KT_l+Constantwhich is often useful when dealing with sharp notches.
6、For U-notches and slots, the flank angle _ is zero. For suchcases, substitution of the expression above into formula (1) yields inthe limit, as P approaches zero,limp_O KN- KTN = 1 +Constant (2)It will be noted that this expression is identical with the expressionfor KT, except that O has taken the
7、place of O- The value of pis finite except for perfectly brittle materials. Thus, if P isdecreased indefinitely for a notch while the other dimensions remainfixed, the value of KN given by formula (1) tends toward a finitelimit (KTN), as long as P is finite, while the theoretical factorKT tends towa
8、rd infinity. Because the expressions for KT and thelimiting value of KN are identical, the symbol KTN has been chosento denote the latter.For cracks, the tip radius is indefinite but extremely small;microscopic observations suggest that it is well under lO-4 inch. Cal-culations for aluminum alloys s
9、how that for p = l0-4 inch, the differ-ence between KTN and KN is only a few percent. Thus, the use ofKTN for cracks is justified. The use of KTN instead of KN mightalso be acceptable for a notch with a very small radius, but the com-putational advantage gained by using KTN instead of KN is usuallyn
10、egligible.Numerical examples for the computation of KTN are included inappendix B.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-12Remarkson Effect of Size VariationsIn order to demonstrate the large changes in factors which resultfrom varying the s
11、ize of a part, a set of calculated curves is showninfigure 3. The configuration assumedcorresponds to figure l(a) (sheetwith 60 V-notches), with _ = 0.15 and w either 1 inch or 36 inches.wThe material is assumedto have au = 68 ksi, which results in_ = 0.145 in.l/2 according to figure 2. Figure 3 sho
12、ws KTN, curvesof KN, and, so far as possible, KT. Twogeneral observations may bemadeon this figure.The first observation is that the relation between the Neuber factorand the theoretical factor is so tenuous that the theoretical factor KTcannot be regarded as a directly useful quantity when the notc
13、h radiuslies between the smallest value likely to be produced in a controlledmanner (say l0-4 inch) and the largest value shown (p = 0.1 inch). Thefactor KT approaches direct usefulness only when p becomeslargerthan 0.1 inch. Radii of this magnitude are encountered in actual partsof fair size; howev
14、er, actual parts for small machinery and notchedspecimensused for materials testing generally have notch radii wellbelow 0.1 inch.The second observation concerns geometric similarity. Points suchas A and A, or B and B, represent geometrically similar speci-mens. (The flank angle and the ratio d/w ar
15、e identical for bothcurves. Thus, to find a point A which represents a specimen geo-metrically similar to A, it is only necessary to find the point atwhich the radius D is 36 times larger than for point A.) It isobvious that there are great differences in the Neuber factors, inspite of geometric sim
16、ilarity; in fact, the range of factors for the1-inch-wide specimensdoes not even overlap the range of factors forthe 36-inch specimens. In the range of proportions where size effectis important, the law of mechanical similitude (geometrically similarstructures behave similarly when subjected to the
17、samestress) isinvalid, and it is futile to search for nondimensional plotting param-eters to simplify the relations as was sometimes done in the past byexperimenters.Finally, figure 3 indicates that fatigue factors can be very largeif the specimen is large; for instance, for a specimenwidth of 36 in
18、chesand a notch radius of 0.001 inch, the predicted fatigue factor is about 17.The frequently heard statement that large fatigue factors cannot be real-ized physically is based on improper generalizations of test resultsobtained on small specimens.L1743Provided by IHSNot for ResaleNo reproduction or
19、 networking permitted without license from IHS-,-,-13L1743Correction for Plasticity EffectIn the elastic range, the stress distribution in the vicinity of anotch shows characteristically a high peak at the bottom of the notch.As the load on a notched specimen is increased, the peak stress willeventu
20、ally reach the yield value and will no longer be proportional tothe load. With further increase in load, the stress distribution willbecome more and more uniform, and the factor of stress concentrationwill approach unity more and more closely as the plastic region extends.For the case of a circular
21、hole in an infinitely wide sheet, Stowellhas given a simple formula (ref. 7) which corrects the theoretical(elastic) factor for the effect of plasticity. For application to othernotch configurations, this formula was generalized in reference 8 to readKp:1+/Es,n(3)where Kp denotes the factor of stres
22、s concentration in the plasticrange, Es, p the secant modulus corresponding to the peak stress, andEs, n the secant modulus corresponding to the net-section stress. Forall practical applications, KN should be substituted for KT informula (3) in order to take care of size effect_ as written(with KTin
23、stead of KN) formula (3) is valid only for ideal material which isideally brittle (p = 0).In first approximation, a notched specimen may be assumed to frac-ture when the peak stress at the bottom of the notch becomes equal tothe tensile ultimate stress. If the subscript u is used to designatevalues
24、appropriate to this special case, the (size-corrected) equation (3)becomesKu:1+ u (4)Es, nwhere Es, u is the secant modulus corresponding to the stress at theultimate load of a simple tension specimen. In reference 4, values ofEs, u were obtained from stress-straln curves. In general, completestress
25、-straln curves (up to failure) are not available, but the perma-nent elongation measured after failure is a property which is univer-sally determined as a materials property. The modulus Es, u can beestimated from this elongation e by the formulaProvided by IHSNot for ResaleNo reproduction or networ
26、king permitted without license from IHS-,-,-14EEs_ u - keE (_)l+_uwhere E is the elastic (Youngs) modulus, while k is a factor some-what less than unity which corrects for the fact that the elongationmeasured after fracture includes some nonuniform elongation that takesplace after the maximum load (
27、“ultimate load“) is exceeded. For allcalculations shown in this paper, the value of 0.8 was used for k.The use of formula (5) eliminates the need for a stress-straincurve, provided the net-section stress is below the proportional limitso that Es, n = E in equation (4). This condition is fulfilled in
28、 thegreat majority of the tests discussed later in this paper; it may notbe fulfilled when the notch is a very short crack, or when a machinednotch is either very shallow or has a fairly large radius. In suchcases, with Es, n a variable, the direct use of equation (4) becomesawkward because the solu
29、tion must be effected by trial and error. Toavoid this difficulty, a curve of Ku against KN can be computed fora given stress-strain curve by solving equation (4) for KN and assuminga series of values for the net-section stress at failure. Some typicalcurves of this nature are shown in figure 4. Whe
30、n Ku is sufficientlylarge to keep the net-section stress below the limit of proportionality,the curves are straight lines. For 7075-T6 sheet, the curve deviatesfrom the straight line for KN 3 but only slightly. For the 2024 alloy,however, the straight-line portions of the curves begin near or beyond
31、 theright-hand border of figure 4.L1743Secondary CorrectionsThe corrections of the stress-concentration factor for materialsize effect and for plasticity effect are generally the most importantones. However, two other corrections may be sufficiently important torequire application in some cases. Bot
32、h are entirely empirical andbased on very meager data; they should therefore be regarded as stopgapformulas subject to improvement either by theory or by analysis of moreextensive sets of data when they become available. The two correctionsare a “flow-restraint“ correction and a buckling correction.
33、The theoretical factor of stress concentration is the ratio of peakstress to average stress over the net section; this factor is, by itsnature, always larger than unity. The experimental factor for ultimatestrength can be defined as the ratio of ultimate stress of the material,measured on specimens
34、without notches, to net-section stress at failureProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2B19L1745of the notched specimen. (It is presumed here that conventional engi-neering methods are used, that is, the load used for computing thestress is
35、 the maximum load developed in the test and the area used isthat measured before the test begins.)It has been known for decades that the experimental ultimate-strength factor for ductile materials can be substantially less thanunity. In the face of the fact that a stress-coneentratlon factor mustbe
36、larger than unity, this observation can be explained only by theassumption that material at a notched section may develop a higherstrength than a specimen devoid of notches. The increase in strengthis attributed to restraint against flow of the metal in the notch region,exerted by the material in th
37、e adjacent regions which are under a lowerstress.Although a large amount of research has been devoted to the prob-lems of fracture, particularly under conditions of restrained flow, thereis little or no information for practical design use. In order to roundout this paper, an empirical formula has b
38、een devised which has givenreasonable accuracy in predicting the tensile strength of notched sheetspecimens. (The formula is not applicable if the cross section of thespecimen approaches a square or circle.) It consists of a correctionfactor applied to the factor Ku given by formula (4) and results
39、inthe corrected factorKu*- (6)_Y1 + Bm tanh p_u Pwhere B is a function of ET given by the graph in figure 9. It maybe noted that the correction is most important for rather mild notches,having a maximum value at ET_ 5- For very sharp notches or cracks,on the one hand, or very mild notches on the oth
40、er hand, it becomesnegligible.The second, or buckling, correction applies to sheet tension speci-mens containing an internal transverse slot or crack, such as shown infigure l(b). Examination of the stresses in such a specimen shows thatthere are compressive stresses in the transverse direction, app
41、roxi-mately parallel to and close to the boundaries of the slot. If thesheet is thin, these compressive stresses cause buckling of the lipsof the slot out of the original plane of the sheet and thus causestresses which add to those due to the stress-raising action of theslot in the unbuckled sheet.
42、If desired, these additional stresses dueto buckling can be eliminated in tests by using guide plates on bothsides of the sheet. The use of such guide plates is fairly standardProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-16practice in fatigue test
43、s, but has not been general practice in (static)tensile tests. When no guide plates are used, a correction should bemade for the buckling effect.Since the distribution of the stresses around the slot is highlynonuniform, a theory for the buckling stress would be rather difficultto derive, and the th
44、eory for the stresses arising from the postbucklingdeformation would be even more difficult. No attempt appears to havebeen made to develop such a theory. Experimental evidence has shownthat the buckling effect is important only in rather extreme cases -thin sheets with long slots or cracks. Consequ
45、ently, it is felt thatas an interim measure, a simple empirical correction may be regarded asacceptable. The correction is incorporated in the formulaooo ) (7)where Su* denotes the predicted net-seetion failing stress when buck-ling is not prevented, while Su denotes the predicted stress whenbucklin
46、g is prevented. (Su equals Gu/Ku or _u/Ku*, but the lattercase is unlikely to occur in practice.)I174EXPERIMENTAL EVIDENCEGeneral DiscussionThe methods of predicting fatigue and static notch factors pre-sented in this paper rest on a theory of size effect. This theory isan “engineering“ or “working“
47、 theory, terms which are meant to implythat no claim is made for great depth of the physical foundations. Asa matter of principle, a theory of this type should be substantiated bya volume of test data commensurate with the scope claimed for the theory -a principle unfortunately often violated in the
48、 literature. In an attemptto honor this principle, a rather large set of comparisons between testresults and predictions is presented in the following sections.In some types of tests, it is possible to achieve test accuraciesof a few percent and to demonstrate that the repeatability of the testsis o
49、f the same order. In such circumstances, differences between theoryand tests which amount to, say, 20 percent can and must be attributedessentially to weakness of the theory. However, in the field of notchstrength (fatigue or static), the test accuracy and repeatability arenot within a few percent at present. While there has been a vast amountof discussion on the subject of scatter in fatigue _ there has bee
