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Verification of various methods for fatigue notch effect estimations in case of aircraft materials
VERIFICATION OF VARIOUS METHODS FOR FATIGUE NOTCH EFFECT ESTIMATIONS IN CASE OF AIRCRAFT MATERIALS ALFRED BUCH Dqartmeat of AeronauticalEngineering,Tecbion, IsraelInstituteof Technology,Haifa,Israel AIMmt-lke is as yet no generalagreementregardiigthe propermethodof analyzingsituationsinvolvingstressraisers,in &?4m~tion with combined static Md altemptipgloading!&In this investigatkmfatigue tests were performedin In&?&g tension and tah-aqmesdw on intemally-notchedsheet specimens (& = 246) for cmparison of the fatigue notch f&or K, andnotch smitivity index K,/K, of ahnninum-alloysheetm~taislswithandwithoutcMding,andforveaiUcation dvuiolumcthods~notebcdtect~timstion,espeehllyintheweofpulutia?twrrion.ThcntiorK,/K=rbowcd,forJ1 ipvertbrttdrpecimanr,
scdhhty
index q -f(r). An analytical form-
of the fur&h K,/K, = f(r) pcmitted sopmate can&b&m ~~~~~~ten~.Themethodu~baeismoreexlla~thecommoaly~ ch~sical methodof N&m, eapeci&yin the case of puht& tensioa
iUV@Sti#WiCMSOften
NOTATION Neuber’smaterialconstant lImehI umstantof tbc two-mter formula gza-id.i8lt @iticd iayu thicklless)
StfeSS8lnpihk llEMlStIGSS nwtimDmSWSS tk&UllUStnss
V*
R
ntbOftlElllihl~StfCS,tOtbemutimumcrtnss
fat@limitincaseR--1 frtiaclidthClWR=0 f8ti#lBclimitofllotchedrpedmens fatigue notch factor,ratiov,/v,. for idmtkal R and N stress comxotratkm factor semi&y ;yl, ,ym (2 r lMK+ - 1) 4 2: KPI --=l=e*) KFO notch factor for R = 0 (for pulsating-te~~sion) t&@b&hgtest(R=-1) R.B. T.C. kiowcoqireasion test (R = - 1) P.T. pulsating-tension test (R - 0) V./V, ratio of !lighlystressedvolumesof umlotchedand notchedspecimens Vdr, @dO %I
2
INTRODUCTION THEPERTINENT literature contains a wealth of information on the fatigue properties of aluminium
alloys used in aircraft: What is lacking, however, is complete data on the fatigue notch factors of aircraft materials for different stress concentrations, types of notches and stress ratios. The evaluation of the fatigue strength of a structure with the aid of the exact theoretical stress concentration factor and handbook fatigue data for smooth specimens of a&raft materials is not sufficiently exact. Serious errors are also likely in applying the results for small specimens to larger structures, if the general relations governing the notch-size effects are not appreciated and the effect of the static mean stress is not taken into account. The object of this report is to present some results permitting a better estimation of the notch effect and of the irdhrence of the mean stress in the case of fatigue limit of various high strength ahrminium alloys. 661
662
ALFREDBUCH
The Neuber theory and other known notch sensitivity theories do not take into account the influenceof the stress R = o,,&r,, on the notch effect. In this investigationresults for pulsating tension and tension-compression fatigue loading were compared in order to achieve a better understanding of the influence of the stress ratio R.
NOTCH EFFECT iN TEE CASE OF COMPLETELYREYERSEDLOADING Many authors assumed that the notch sensitivity factor q = (I& - l)/(& - 1) is a regular itweagit~~ function of the notch radius. This permitted estimation of the fatigue strength of a notched structure according to the formula
with the function q =f(r) estimated analytically or graphkaUy[l,2]. Figure 1 shows the dependence of the factor q on the notch radius r in the case of axially loaded (R = - 1)2024-T3 specimens. The average curve is widely used[l, 21for estimating the notch sensitivity factor q of various hi&Wren& aluminium alloys, in spite of the large scatter of the test points and the difbrence in the notch sensitivity between flat and round specimens. Another method[4,3] involves the use of Neuber’s notch sensitivity formula
by relating empirically the maUrial constant a to the tensile strength of the alum&m alloys. Comparisonof the experimentaland theoreticalresults for round 707%T6specimenswith grooves (Pii 2) shows that Neuber’sformula (2)fails to give good agreement with test results and that the notch sensitivity factor q = (KF - l)/(& - 1) is not always an increasing function of the notch radius r. The author assumed[6,7’jthat the dependence KF/KT = f(r) is a better increas& function of thcnotchradiuuttmq ~e(r)~~btttCtav~OftlWtCIlt~.PiOun3~ruch results for 2024-T3Alclad and 6061-T4 sheet tkpdlWStiVM~Mr,strass
~frctoasK*andstressrrrtiosR.Ascrrnbesscn,theratioKIK=~iPgtaarrl
withthewtchrodiui,r.Th-curvesinFig.3aretheoreticallineswhich~a two-parameter formula derived[7-91 as an extension of Peterson’s[2] formula Kp = Krll -(at/r)1 (3)
Notch
rodlus
r
Fig.1.Notchsensitivityindexq of 2024-Taiuminiumalloy.Completelyreversedaxialloading[2].
663
Notch
rodiua r,
mm
?&W-T3
Notch mdius r, mm pi
3. meisof notcbradiusr on theratioK,/Kr
ALCLAD
Notch radius r.
for ahhimn
mm
OUOY sbcetwcimens.
whereCi2*1aMiroa6*3h/(3_A)iacascaf~~ofin~~sbsd~ (assuming KFIKT = lb for r - 0). A = (KTIK~),>,- = const.andi==const.aretwospeoimen sixt wqandent paramctcrBt* Pi 4 and Tabk 1 present results for round externally notched 2024-Tspecimens tested in rot&& bending or tension-compression (R = - 1). As can be seen the two-parameter formula (curve 1) yields much better agrccmcnt with the expe&nental results than Neubcr’s classical method. It can also be noted that Neubcr’sline is not conservative in this case and that this formula (2) does not di&rcntiate between flat and round specimens. Comparison of Figs. 4 and 3 (and of curves 2 and 3 in Fig. 1)shows that for a similarnotch radius the notch sensitivity is higher in the case of round specimaus, which can be explained by the @iaxialstate of stress in this case. Another extension of Peterson’s formula is a following equation of Raske[l4]
ALTRED BUCH
Notch radius r.
RB. 6ov
430
370
6.74
-
1 &
MO ‘g
1 1.56 l-4 @625 1.8
1 1.32 l-62
1% 1H) 122
1 21.5 0.46 2.55 0*12s 4.4 0.07s 5.5
1 1.4 2.2 2.8 3.2
190 136.5 87.5 67.9 593
:.94 odl6 0.64 0.60
& o-77 0.53 027
7.62
1 0-7s 2 O*OS 5
1 <168 1*85 91 2.3 73.5
1 0.93 Od6
0.85 0.32
6.25
0.25 -
z
1 :
mm
0+6 515
469
3%
294
476
322
560
441
7.62
8*89
0.90
HOI 0% 0.77 r1u
2.6 I
i-36
207 87.5
i-91
0%
:.4
f.7
175 101‘5
1 0.5
0.29
O-25 -
3.4 1
:.7
189 70
A.79
OG
12.5 0-U 5%
468
IO.1
600 4%
438 363
7.5 7.5
0; 1.5
:.2s :.77 1 L 2.25
‘Z ‘6a:
k79 I t+73
0<2 O‘fl
510
340
10.1
g 0.25
1 2.4 I.6 3.4
1.56 2.08 2.5
1:: 84 70
L75 0.87 0-73s
G3 677 0.62
1.6 3 :.79
1 I.35 2.3 I.69
161 119 70 189 112
1 O&5 0.76 I 0943
0.58 0.65 0.87
5254069 45v R.B.
042
:.w
515
395
-1 t-6 0.25 7.5 OG
TECHNION 1974 own meQ8.
1121
r131
Fat&
665
notch et&t estimations
Similar to Peterson’s formula this equation does not fulfil the boundary condition for small and large notch radii.
NOTCH RFFECT FOR PULSATING TENSION COMPAREDWlTE TENSION-COMPRESSION Intbecase of an aircraft sheet structure (e.g.a wing)the dynamic loads (e.g.those due to gusts)
act in conjunction with static loads. Therefore, it is important to know the notch factor values not only for completely reversed loadings (R = - 1)but also for some other stress ratios R, especidly for pulsating tension (R z 0), where the fatigue limit amplitude ranges are smaller than in the case OfR = - 1. The author uses the usually accepted definition of the notch factor Kp KI7=
cr- for unnotched specimens os, for notched specimens at identical R and N
The notch-factor values in Fiis. 1,2 and 4 and Table 1 are given for N = 10’cycles and R = - 1. Table 2 gives some results for various cycle numbers and stress ratios R, gtdhered together from various sources. The following conclusions may be derived from the data in Table 2: (1) The corresponding notch factors Kp for the same stress ratio R and stress concentration factor K7 increase with the munber of cycks. They remain nearly constant only for specimens with small stress concentration factors (Kr = 1.5 and 2). (2) The corresponding notch factors Kp for the same stress concentration factor KT and fatigue life N are in general smalkr for pulsating tension than for tension-compression. Exceptions are probably the result of test scatter, KF being a quotient of two fatigue strengths determined with limited accuracy. (3) The corresponding notch factors Kp are sometimesclose for 2026T3 and 7075-T6barsheet specimens, especiatly for smalkr stress concentration factors (KT < 3). For high K,(4 and 5) and high cycle numbers N = lo’, KP is larger in case of 707%T6. Theauthorinvestigated[17)aluminium alloysheet spec&enswithvariousstressconcentration factors for stress ratios R = - 1 and R -0. The comparative results for is&m&y-notched specimens (Reynolds aircraft quality sheets, with thicknesses t = 1.25; 1.32 and 1~6mm)are presented in brief in Table 3. The following conclusions may be derived from the data in Tabk 3: (1) The corresponding notch factors Kp (for the same R, KT and t) are sometimes not close for both investigated Alclad materials.In the case of smaller KT (2.07and 2.52)they are mostly higher for 7075-T6Aclad specimens. It must be mentioned here that the cladding alloys were not the same in both materials (7072in the cause of 7075, and 1050in the case of 2024). T&k 2. Notch factors for various stress ratios R and stress concentration factors KT (bar sheet Ipscimms, N = lo’, lo’and 10’) Kr for 2024-T3 0 0.2
N
KT
l(r
1.5 2 4 5 1.5 2 4 5 I.5 2 4 5 2.43 2.83’
::: 3.4 1% 2.10
1.3 1.8 3-o 3.3 1.60 1.97
2.12 2.83
1.72 2.19
1.43 2.10
lo-’
IO’
10’ 10’
BFMVd7.No.4-D
R=-1 1.3 1.7 2.8 2.9 1.3 2.1 3.4 3.1 I.3
1.2 1.3 :f I.5 1.7 ;::
I.1 1.2 2.2 2.4 1.3 1.6 3.4 3.4 1.4 1.6 :::
R=-I
Kp for 7075-T6 0.2 0
1.3 1.7 3.2 3.3 1.3 1.8 3.2 3.5 1.8 1.9 4.0 5.0 1.68 2.16
1.3 1.6 3.2 3.8 1.2 1.6 ::: 1.5 1.6 4.1 4.5 1.61 1.96
1.71 2.28
1.52 2.08
1.3 1.4 2.8 3.3 1.3 1.7 3.4 4.0 1.5 1.7 4.3 4.5
0.5
and kind of notch
1.3 2.2 2.5 -
WI cdSenotcb
::; 3.9 1.8 4.6 4.7 I161 centlmlhok d =3*2mm cL?ntlalhok d-64mm
666
ALFRED BUCH Tabk 3. Notch factors for various stress ratios R and stress concentration factors Kr (sheet spccimen~, N = 10’)
Type of specimen Notch shape r(mm) K, Hok ii&. IMe Hok Hole
o-5 2&l 4.6 : 3.6 4 2.52 U-5 2.07 2.07 20
2024.T3-AJckd K: =ip;Kr 1.87 3.23 244 1.a 1.70 l-67
0.6s 0.70 O-68 0.65 0.82 0.80
7075-T6-Al&d KrR i:,Kr
KpR 3KJ),Kr K: =i;:Kr 1.25 2.78 2.37 l-S4 1.58 1.54
0.43 0.60 O-66 0.61 0.76 0.74
144 3-02 l-95 1.92 2.10 1.90
O-SO O-66 0.54 O-76 1.01 0.92
6061.T4
1.33 3.62 2-M 2.13 2.00 I.95
046 0.79 0.57 O-84 0.96 0.94
KpR‘i-jK= 1.84 2.97 260 I.84 1.73 1.68
0639 O-646 0.722 0.730 0.837 O-810
Kr
R=O KFIKT
1.15 248 la3 1.38 1.33 1.41
0.399 0.539 OJ22 o-s48 043 oa1
(2) The corresponding notch factors (for the same RT and t) are smallerfor pulsating tension than for tension-compression for 2024-T3-Akladand 6061-T4ahuninium alloys. (3) The notch factors are dependent not only on the stress concentration factors RT but also on thenotchradiirandsttessratiosR.The~~of Krisstr~thanthatoftandR for specimens with & = const. (4) The ratio &/I& is in general an increasing function of the notch radius both for tension-compression and for pulsating tension. The decrea~ of the notch effect with decreasing the basic fatigue life N (Tabk 2) may be expkined by the relatively higher than in smooth specimens cyclical strain hardening of the material at the notch root under the higher tensik stress corresponding to the smalkr fatigue life. SimikrQ,thekwernotcht&ctiuthecaseofB *OcompmdwithR=-lismwitlrtbs smaller a0 ad rclativdy lower cyclical strain hardening in the second case. Therefor& the material constant A in formula (3),which is equal to &I& for r + h, should d&r between the twocasesR =-l~R-ObecauscofthtdiflenncesintheKIvaiuesinbothcares. Table 4 presents the vahres of the material constants chosen for the investigated aircraft materials with a view of optimal approximation of the functions K&G -f(r). The method of choice of these constants has been explained elsewhere(81.As can be seen from Tabk 4, the parameter A is (withthe exception of 7075.TMlckd) alwayshigher for pukating tension than for tensiobcomplession, and the difference between the values of A for R - 0 aad R = - 1 is especially evident for the medium-strength ahoy 6061-T4for which the cyclical hardening is stronger than for the high-strength Al-alloys 7075-T6and 2024T3. T~4.VllusllofthccoMtrntrAmd~oftheformuh(3)foredculrtDaof tbcfat@notcllfactor Material
Tewioa-comprcssioa (R = - 1) A Mm)
hlsatiag tension (R - 0) A h (mm)
6061-T4 2024-T3 7075-T6
1.20
0.15
1.55
0.15
165
0.23
1.10
0.25
2024T3 Alclad
1.20
0.20
1.35
o-20
7075-T6 Al&d
leoo
040
140
o-50
a. -WI, LINES FOR AIRCRAFT AL-ALLOYS The effect of the mean stress amplitude is often represented by a straight Goodman line. The results of our investigationpermitted verificationof the theoretical Goodmanline and presentation of the experimental oh - u,,, relationship for N = 10’cycles in the case of aircraft materials.The CK, - a,,, relationshipsare presented in Figs. 5 and 6 for the investigatedAlclad sheet specimens, in Figs. 7-9for bar sheet specimens.The lines were constructed using the experimentalpoints for the fatigue limits in the cases R = - 1and R = 0 and the tensile strength values (UTS).As can be seen, the broken lines are sometimes close to the straight lines of Goodman (e.g. 6061-T4,7075-T6and 202CT3).In the case of 2024-T3-Alckdand 7V75-T6-Alckdthe experimental points for R = 0 are always below the corresponding theoretical Goodman line. The experimental a. - u,,, lines for
COIiiP~N OF THE ACCURACYOF TEB TWO-P~~ANI)QF~~~~F~~~~~X=--I~W=O Tsbft 5 prusentsthe error in cacao the fatigue n&ch factor & according to foal (3) and acwrdhg to Ncuber’sformula. As can be seen. the errors arc mostly hi@cr when Neuber’s
method is used. In this case the caicukted restits are closer ta the cxperim~~ for R = - I than for R a 0. This is connected with the additional error of the ~~~~ti~n that &a notcti factor Km for R = 0 is eqW to the notch factor I&, for R - - 1, tince Neubw’s constant was found fur R = - 1(4,3$ when Neuba% m&hodis used the maximum errors for the investigated materials are as high as 37 per cent for R ~-landdlpcrctntforR~O.Inthiscajetheerrorsf~rthe various types of notch& specimens were mostly ORthe safe side, Calculations accord&g to formula (3)resulted in avatxge errors between 2.0 otad167 per cent for R = - I and between 4.3 and 189 per test for R = 0, The errors were most@ much below M per cent. The caI&ated resti& were not always on the conservative side, and therefore the use of a &=I& value WI5 higbcr should be recommended. In this case the method is conservative (Fig. 3) and (as proved by ~~~a~~~~ still more exact than Neuber’s mctktd,
606t-T4 bar uT!s-310
:
4.6
--Q
-_; +9 t9.5 +2 6.8
+9 +31 +44*5 t39 33.7
+2 t1 +6 -3 3.5
-5 t7 +6.5 6.2
-‘; -Id +12-s ZLxi ii.5 & -6 Averagepcrcmf of squareerror 8.1
+ 12-S +I5 +43-s +20 25.9
-iis +3*5 + 15.5 -1.5 9.5
-GS t 12.5 t37 t 11.5 20.4
- 12.5 +8 +1 +49 24.9
-11 +30 t6.5 -7-S 16.7
+5 t14 t12 t23.5 15.1
-_; -0.5 -3.5 t7.s 4.3
+55 t4 -6 +16-S 9.4
t3 to.5 +2 - 1.5 2.0
t 11.5
-_;*s t7.J +8-S 46
+7*5 t I2*5 + 13.5 ll*S
t3 t3 t1.5 26
t7.5 t2.s t 1.5 4.7
::; i.5 2.07 Averplepercentof squareerror 2@4-T3 Alclad boa
i2
7075.T6 AlChd
i-6 3.6
1 ;_
-27.5
:: t23 SidCS 2.5 -8.5 uTs=wl 1:s 297 -9 Averagepercentof squareerror 18.9 both
2
2024m
1 G $1434 3.2 E 6.35 243 12.7 2.12 Avenge percentof agusmam 707~lx bar ?lTS=5@
i-k :Sa 3.2 2.88 12.7 2-u Avessgepercentof squrucemr
-: t1.5 6.3
(4 IlKam a coaservative resutf (-1 an tmconservative.
ESTIMATION OF TEE FATIGUE LIMIT
OF NOTSPECIMENS WEflcI u. >O FROM THE Km DATA FOR ALTERNATING TENSION-COMPRESSION There is as yei no general agreement regarding the proper method of ana&& sbations invobing stress raisers in conjunction with combinfd mean and a&erMkg stress. One nwthod (I) applies the notch factor only to the al&mating stress. In this case the fatigue limit of the notched specimen for N = 10’ cycles is
where G, is the al&mat&g stress component of the fatigue limit for the not&d specimen, Q=is the alternating stress component of the fatigue limit of the unnotchcd specimen, in the case of the same nominal mean stress an as for the notched specimen,
KFPZ
isthenotchfactorforR
= -1.
Another method (II) assumes that KFi is applied to the maximal stress and Km should be equal to KF,
talc. ace.ICalc. act. II %of %or
Notch dills luIulfrl
Mwmmd values
r(lllm) KT
&o
ki,,
emrt
err04
fore;,
forU&
-?
t 16.5 +n*s
~N~~~ 6061.T4 E&=31*
:
A.5 Average pcrceat of
1
1
4.6
2.48 l-88
;I; 2.07
+3*5 +6 +6 5.8
1.33 1.38
squareerror
zoWT3
1
T:d i.78 2.31 E sides : ::: 1.54 m-46@ 12.5 2.07 1.58 Avenge percmt of squarewror
:::: 1.65 169
7075.T6 A&d
1
4.6 1
:.62
z-z : 3.6 UTS-560 1:.5 fz7 Ava;llsparaptofmet=
E 2.00
2024-n lmr UTS-460
1 1.6 2.90 :.11 3.2 2.66 2.11 6.35 243 2.13 12.7 2.12 1.72 Avcmgc perceat of squue ellof
F*T6
76 1 Lo 3.2 ::: 2.08 12.7 2.12 1.71 ptrmt of squareerror
UTS=58@ Avm
:*Oo 2.19 2.10 1*97
ho 2.28 1.91
71 25.5 30 46.2 45
z.5 55.5 55 68
‘i538.5 59
72
-_; -10 Ii.5 6
t26 t23 23.6 - 14
tf t6.5 t65 8.5
G.5 +14 -15 -8 16.8
-Tr; -5 -11
-10 -5 -7-S +3+5 7
G.5
t4 t&S t1*5 2.5
t43 11.9
t3.5 -1.5 t 13.5 7.5 7
t9 t 10.5 8.0
where oh is the fatigue limit bf notched specimen for R = 0 and @dois the fatigue limit of the mater&l for R - 0. Metlmd (I) was used for calculating Q= for R = 0. The alternating stress compoamt a, of the unnotched specimen in the case of the same mean stress as forthe notched specimen (a,,* Oh,,) was found with the aid of the experimental u‘ - a,,, lines ia Figs. S-9. The measured and calculated values of aon are compared in Table 6, which also includes the pertenta(lcerrorsforthefatieuelimitscalculaMaccordiagtoboth~~s.Ascanbese~nfrom Tabk 6, method I is evidently more accurate than method II in the case of the ductile material 606bT4, and nearly equally accurate for the other investigated materials. McEvbfijl9J derived a foIlowing rektknship between Kpt and GO, U&I the OO* ~1~~~~s:
Calculationsproved that this formula is in good agreement with experimental results in Table 3. The accuracy of formula (7) is especially high for 606LT4. CONCLUSIONS (1) The ratio Kp/Kr is a better characteristic of the notch sensitivity for flat and round, internally and ~x~m~i~notched aircraft material specimens than the wideiy-used notch
Fatiguenotch effect estimations
671
sensitivity factor q, because-of the better regularity of the dependence on the notch radius in the former case. (2) With the same notch radius is the notch sensitivity relatively higher for round than for flat specimens. (3) The notch effect is in general stronger in tension-compression than in pulsating tension. (4) The intluence of the stress concentration factor (at IL C 5) on the fatigue notch factor Kp is in general stronger than of the notch radius r and the stress ratio R for specimens with KT = const. REFERENCES [l] A&al Handbook, Suppl. Am. Sot. Met. Cleveland (1954). [21 G. Sines and J. L Waisman (Eds.), Metal F&M McGraw-Hill, New York (1959). [3] R. C. Juvinall, EngineeringConsidem~ionsof S&ss, &ah ond stnngth. McGraw-Hill, New York (1967). [4] P. Kuhn, SAE-AShfE Paper 843 G (1964). [A W. S. Hyler d aL, NACA TN 3291 (1954). [6] A. Buch, Materi&@ 13.6 (1971). m A. Buch, ZZZZnt. Conf. on Z+act#re, V-431 B, Munich (1973). [s] A. Buch, Met Sci. Engng 15, (1974). I91 A. Buch, Z W&&#&A 5.4 (1974). 1101J. A. Bmmet ~IKII. G. Wai&eg, J. Ru. Nat. EIUWMStand 52.5 (1954). 1111R. B. Heywood, higaing Ag&sl Foliguc Chapman and Hall, London (1%2). [12] H. J. Grova et eL, Fatigue of Metals and stnrcturcs. U. S. Gov. Priot OK Washiion (1960). (131 Aircmlfl FarigueHandbook,ARIC-W-76, Aircraft Ind. Ass. Caliiornia (1957). [14] D. T. Raske, 1. Tcrt. Eu&urion 1, 5 (1973). [Is) J. !Jchijve, NACA TM 19395 (1954). [16] M. B. ti and H. F. Hardrath, NACA TN 3631 (1956). 117.lA. Buch, TECHNION TAE Report 184 (1973). [Is] W. Ill& NACA TN 3866 (1956). [19] A. J. McEvdi, 3rd ZIU.Ci& oa rhe Srm@ oflUe@le& AUvys, clmbidp Prper (1973).
(Receiced 2 February 1974)
scdhhty
index q -f(r). An analytical form-
of the fur&h K,/K, = f(r) pcmitted sopmate can&b&m ~~~~~~ten~.Themethodu~baeismoreexlla~thecommoaly~ ch~sical methodof N&m, eapeci&yin the case of puht& tensioa
iUV@Sti#WiCMSOften
NOTATION Neuber’smaterialconstant lImehI umstantof tbc two-mter formula gza-id.i8lt @iticd iayu thicklless)
StfeSS8lnpihk llEMlStIGSS nwtimDmSWSS tk&UllUStnss
V*
R
ntbOftlElllihl~StfCS,tOtbemutimumcrtnss
fat@limitincaseR--1 frtiaclidthClWR=0 f8ti#lBclimitofllotchedrpedmens fatigue notch factor,ratiov,/v,. for idmtkal R and N stress comxotratkm factor semi&y ;yl, ,ym (2 r lMK+ - 1) 4 2: KPI --=l=e*) KFO notch factor for R = 0 (for pulsating-te~~sion) t&@b&hgtest(R=-1) R.B. T.C. kiowcoqireasion test (R = - 1) P.T. pulsating-tension test (R - 0) V./V, ratio of !lighlystressedvolumesof umlotchedand notchedspecimens Vdr, @dO %I
2
INTRODUCTION THEPERTINENT literature contains a wealth of information on the fatigue properties of aluminium
alloys used in aircraft: What is lacking, however, is complete data on the fatigue notch factors of aircraft materials for different stress concentrations, types of notches and stress ratios. The evaluation of the fatigue strength of a structure with the aid of the exact theoretical stress concentration factor and handbook fatigue data for smooth specimens of a&raft materials is not sufficiently exact. Serious errors are also likely in applying the results for small specimens to larger structures, if the general relations governing the notch-size effects are not appreciated and the effect of the static mean stress is not taken into account. The object of this report is to present some results permitting a better estimation of the notch effect and of the irdhrence of the mean stress in the case of fatigue limit of various high strength ahrminium alloys. 661
662
ALFREDBUCH
The Neuber theory and other known notch sensitivity theories do not take into account the influenceof the stress R = o,,&r,, on the notch effect. In this investigationresults for pulsating tension and tension-compression fatigue loading were compared in order to achieve a better understanding of the influence of the stress ratio R.
NOTCH EFFECT iN TEE CASE OF COMPLETELYREYERSEDLOADING Many authors assumed that the notch sensitivity factor q = (I& - l)/(& - 1) is a regular itweagit~~ function of the notch radius. This permitted estimation of the fatigue strength of a notched structure according to the formula
with the function q =f(r) estimated analytically or graphkaUy[l,2]. Figure 1 shows the dependence of the factor q on the notch radius r in the case of axially loaded (R = - 1)2024-T3 specimens. The average curve is widely used[l, 21for estimating the notch sensitivity factor q of various hi&Wren& aluminium alloys, in spite of the large scatter of the test points and the difbrence in the notch sensitivity between flat and round specimens. Another method[4,3] involves the use of Neuber’s notch sensitivity formula
by relating empirically the maUrial constant a to the tensile strength of the alum&m alloys. Comparisonof the experimentaland theoreticalresults for round 707%T6specimenswith grooves (Pii 2) shows that Neuber’sformula (2)fails to give good agreement with test results and that the notch sensitivity factor q = (KF - l)/(& - 1) is not always an increasing function of the notch radius r. The author assumed[6,7’jthat the dependence KF/KT = f(r) is a better increas& function of thcnotchradiuuttmq ~e(r)~~btttCtav~OftlWtCIlt~.PiOun3~ruch results for 2024-T3Alclad and 6061-T4 sheet tkpdlWStiVM~Mr,strass
~frctoasK*andstressrrrtiosR.Ascrrnbesscn,theratioKIK=~iPgtaarrl
withthewtchrodiui,r.Th-curvesinFig.3aretheoreticallineswhich~a two-parameter formula derived[7-91 as an extension of Peterson’s[2] formula Kp = Krll -(at/r)1 (3)
Notch
rodlus
r
Fig.1.Notchsensitivityindexq of 2024-Taiuminiumalloy.Completelyreversedaxialloading[2].
663
Notch
rodiua r,
mm
?&W-T3
Notch mdius r, mm pi
3. meisof notcbradiusr on theratioK,/Kr
ALCLAD
Notch radius r.
for ahhimn
mm
OUOY sbcetwcimens.
whereCi2*1aMiroa6*3h/(3_A)iacascaf~~ofin~~sbsd~ (assuming KFIKT = lb for r - 0). A = (KTIK~),>,- = const.andi==const.aretwospeoimen sixt wqandent paramctcrBt* Pi 4 and Tabk 1 present results for round externally notched 2024-Tspecimens tested in rot&& bending or tension-compression (R = - 1). As can be seen the two-parameter formula (curve 1) yields much better agrccmcnt with the expe&nental results than Neubcr’s classical method. It can also be noted that Neubcr’sline is not conservative in this case and that this formula (2) does not di&rcntiate between flat and round specimens. Comparison of Figs. 4 and 3 (and of curves 2 and 3 in Fig. 1)shows that for a similarnotch radius the notch sensitivity is higher in the case of round specimaus, which can be explained by the @iaxialstate of stress in this case. Another extension of Peterson’s formula is a following equation of Raske[l4]
ALTRED BUCH
Notch radius r.
RB. 6ov
430
370
6.74
-
1 &
MO ‘g
1 1.56 l-4 @625 1.8
1 1.32 l-62
1% 1H) 122
1 21.5 0.46 2.55 0*12s 4.4 0.07s 5.5
1 1.4 2.2 2.8 3.2
190 136.5 87.5 67.9 593
:.94 odl6 0.64 0.60
& o-77 0.53 027
7.62
1 0-7s 2 O*OS 5
1 <168 1*85 91 2.3 73.5
1 0.93 Od6
0.85 0.32
6.25
0.25 -
z
1 :
mm
0+6 515
469
3%
294
476
322
560
441
7.62
8*89
0.90
HOI 0% 0.77 r1u
2.6 I
i-36
207 87.5
i-91
0%
:.4
f.7
175 101‘5
1 0.5
0.29
O-25 -
3.4 1
:.7
189 70
A.79
OG
12.5 0-U 5%
468
IO.1
600 4%
438 363
7.5 7.5
0; 1.5
:.2s :.77 1 L 2.25
‘Z ‘6a:
k79 I t+73
0<2 O‘fl
510
340
10.1
g 0.25
1 2.4 I.6 3.4
1.56 2.08 2.5
1:: 84 70
L75 0.87 0-73s
G3 677 0.62
1.6 3 :.79
1 I.35 2.3 I.69
161 119 70 189 112
1 O&5 0.76 I 0943
0.58 0.65 0.87
5254069 45v R.B.
042
:.w
515
395
-1 t-6 0.25 7.5 OG
TECHNION 1974 own meQ8.
1121
r131
Fat&
665
notch et&t estimations
Similar to Peterson’s formula this equation does not fulfil the boundary condition for small and large notch radii.
NOTCH RFFECT FOR PULSATING TENSION COMPAREDWlTE TENSION-COMPRESSION Intbecase of an aircraft sheet structure (e.g.a wing)the dynamic loads (e.g.those due to gusts)
act in conjunction with static loads. Therefore, it is important to know the notch factor values not only for completely reversed loadings (R = - 1)but also for some other stress ratios R, especidly for pulsating tension (R z 0), where the fatigue limit amplitude ranges are smaller than in the case OfR = - 1. The author uses the usually accepted definition of the notch factor Kp KI7=
cr- for unnotched specimens os, for notched specimens at identical R and N
The notch-factor values in Fiis. 1,2 and 4 and Table 1 are given for N = 10’cycles and R = - 1. Table 2 gives some results for various cycle numbers and stress ratios R, gtdhered together from various sources. The following conclusions may be derived from the data in Table 2: (1) The corresponding notch factors Kp for the same stress ratio R and stress concentration factor K7 increase with the munber of cycks. They remain nearly constant only for specimens with small stress concentration factors (Kr = 1.5 and 2). (2) The corresponding notch factors Kp for the same stress concentration factor KT and fatigue life N are in general smalkr for pulsating tension than for tension-compression. Exceptions are probably the result of test scatter, KF being a quotient of two fatigue strengths determined with limited accuracy. (3) The corresponding notch factors Kp are sometimesclose for 2026T3 and 7075-T6barsheet specimens, especiatly for smalkr stress concentration factors (KT < 3). For high K,(4 and 5) and high cycle numbers N = lo’, KP is larger in case of 707%T6. Theauthorinvestigated[17)aluminium alloysheet spec&enswithvariousstressconcentration factors for stress ratios R = - 1 and R -0. The comparative results for is&m&y-notched specimens (Reynolds aircraft quality sheets, with thicknesses t = 1.25; 1.32 and 1~6mm)are presented in brief in Table 3. The following conclusions may be derived from the data in Tabk 3: (1) The corresponding notch factors Kp (for the same R, KT and t) are sometimes not close for both investigated Alclad materials.In the case of smaller KT (2.07and 2.52)they are mostly higher for 7075-T6Aclad specimens. It must be mentioned here that the cladding alloys were not the same in both materials (7072in the cause of 7075, and 1050in the case of 2024). T&k 2. Notch factors for various stress ratios R and stress concentration factors KT (bar sheet Ipscimms, N = lo’, lo’and 10’) Kr for 2024-T3 0 0.2
N
KT
l(r
1.5 2 4 5 1.5 2 4 5 I.5 2 4 5 2.43 2.83’
::: 3.4 1% 2.10
1.3 1.8 3-o 3.3 1.60 1.97
2.12 2.83
1.72 2.19
1.43 2.10
lo-’
IO’
10’ 10’
BFMVd7.No.4-D
R=-1 1.3 1.7 2.8 2.9 1.3 2.1 3.4 3.1 I.3
1.2 1.3 :f I.5 1.7 ;::
I.1 1.2 2.2 2.4 1.3 1.6 3.4 3.4 1.4 1.6 :::
R=-I
Kp for 7075-T6 0.2 0
1.3 1.7 3.2 3.3 1.3 1.8 3.2 3.5 1.8 1.9 4.0 5.0 1.68 2.16
1.3 1.6 3.2 3.8 1.2 1.6 ::: 1.5 1.6 4.1 4.5 1.61 1.96
1.71 2.28
1.52 2.08
1.3 1.4 2.8 3.3 1.3 1.7 3.4 4.0 1.5 1.7 4.3 4.5
0.5
and kind of notch
1.3 2.2 2.5 -
WI cdSenotcb
::; 3.9 1.8 4.6 4.7 I161 centlmlhok d =3*2mm cL?ntlalhok d-64mm
666
ALFRED BUCH Tabk 3. Notch factors for various stress ratios R and stress concentration factors Kr (sheet spccimen~, N = 10’)
Type of specimen Notch shape r(mm) K, Hok ii&. IMe Hok Hole
o-5 2&l 4.6 : 3.6 4 2.52 U-5 2.07 2.07 20
2024.T3-AJckd K: =ip;Kr 1.87 3.23 244 1.a 1.70 l-67
0.6s 0.70 O-68 0.65 0.82 0.80
7075-T6-Al&d KrR i:,Kr
KpR 3KJ),Kr K: =i;:Kr 1.25 2.78 2.37 l-S4 1.58 1.54
0.43 0.60 O-66 0.61 0.76 0.74
144 3-02 l-95 1.92 2.10 1.90
O-SO O-66 0.54 O-76 1.01 0.92
6061.T4
1.33 3.62 2-M 2.13 2.00 I.95
046 0.79 0.57 O-84 0.96 0.94
KpR‘i-jK= 1.84 2.97 260 I.84 1.73 1.68
0639 O-646 0.722 0.730 0.837 O-810
Kr
R=O KFIKT
1.15 248 la3 1.38 1.33 1.41
0.399 0.539 OJ22 o-s48 043 oa1
(2) The corresponding notch factors (for the same RT and t) are smallerfor pulsating tension than for tension-compression for 2024-T3-Akladand 6061-T4ahuninium alloys. (3) The notch factors are dependent not only on the stress concentration factors RT but also on thenotchradiirandsttessratiosR.The~~of Krisstr~thanthatoftandR for specimens with & = const. (4) The ratio &/I& is in general an increasing function of the notch radius both for tension-compression and for pulsating tension. The decrea~ of the notch effect with decreasing the basic fatigue life N (Tabk 2) may be expkined by the relatively higher than in smooth specimens cyclical strain hardening of the material at the notch root under the higher tensik stress corresponding to the smalkr fatigue life. SimikrQ,thekwernotcht&ctiuthecaseofB *OcompmdwithR=-lismwitlrtbs smaller a0 ad rclativdy lower cyclical strain hardening in the second case. Therefor& the material constant A in formula (3),which is equal to &I& for r + h, should d&r between the twocasesR =-l~R-ObecauscofthtdiflenncesintheKIvaiuesinbothcares. Table 4 presents the vahres of the material constants chosen for the investigated aircraft materials with a view of optimal approximation of the functions K&G -f(r). The method of choice of these constants has been explained elsewhere(81.As can be seen from Tabk 4, the parameter A is (withthe exception of 7075.TMlckd) alwayshigher for pukating tension than for tensiobcomplession, and the difference between the values of A for R - 0 aad R = - 1 is especially evident for the medium-strength ahoy 6061-T4for which the cyclical hardening is stronger than for the high-strength Al-alloys 7075-T6and 2024T3. T~4.VllusllofthccoMtrntrAmd~oftheformuh(3)foredculrtDaof tbcfat@notcllfactor Material
Tewioa-comprcssioa (R = - 1) A Mm)
hlsatiag tension (R - 0) A h (mm)
6061-T4 2024-T3 7075-T6
1.20
0.15
1.55
0.15
165
0.23
1.10
0.25
2024T3 Alclad
1.20
0.20
1.35
o-20
7075-T6 Al&d
leoo
040
140
o-50
a. -WI, LINES FOR AIRCRAFT AL-ALLOYS The effect of the mean stress amplitude is often represented by a straight Goodman line. The results of our investigationpermitted verificationof the theoretical Goodmanline and presentation of the experimental oh - u,,, relationship for N = 10’cycles in the case of aircraft materials.The CK, - a,,, relationshipsare presented in Figs. 5 and 6 for the investigatedAlclad sheet specimens, in Figs. 7-9for bar sheet specimens.The lines were constructed using the experimentalpoints for the fatigue limits in the cases R = - 1and R = 0 and the tensile strength values (UTS).As can be seen, the broken lines are sometimes close to the straight lines of Goodman (e.g. 6061-T4,7075-T6and 202CT3).In the case of 2024-T3-Alckdand 7V75-T6-Alckdthe experimental points for R = 0 are always below the corresponding theoretical Goodman line. The experimental a. - u,,, lines for
COIiiP~N OF THE ACCURACYOF TEB TWO-P~~ANI)QF~~~~F~~~~~X=--I~W=O Tsbft 5 prusentsthe error in cacao the fatigue n&ch factor & according to foal (3) and acwrdhg to Ncuber’sformula. As can be seen. the errors arc mostly hi@cr when Neuber’s
method is used. In this case the caicukted restits are closer ta the cxperim~~ for R = - I than for R a 0. This is connected with the additional error of the ~~~~ti~n that &a notcti factor Km for R = 0 is eqW to the notch factor I&, for R - - 1, tince Neubw’s constant was found fur R = - 1(4,3$ when Neuba% m&hodis used the maximum errors for the investigated materials are as high as 37 per cent for R ~-landdlpcrctntforR~O.Inthiscajetheerrorsf~rthe various types of notch& specimens were mostly ORthe safe side, Calculations accord&g to formula (3)resulted in avatxge errors between 2.0 otad167 per cent for R = - I and between 4.3 and 189 per test for R = 0, The errors were most@ much below M per cent. The caI&ated resti& were not always on the conservative side, and therefore the use of a &=I& value WI5 higbcr should be recommended. In this case the method is conservative (Fig. 3) and (as proved by ~~~a~~~~ still more exact than Neuber’s mctktd,
606t-T4 bar uT!s-310
:
4.6
--Q
-_; +9 t9.5 +2 6.8
+9 +31 +44*5 t39 33.7
+2 t1 +6 -3 3.5
-5 t7 +6.5 6.2
-‘; -Id +12-s ZLxi ii.5 & -6 Averagepcrcmf of squareerror 8.1
+ 12-S +I5 +43-s +20 25.9
-iis +3*5 + 15.5 -1.5 9.5
-GS t 12.5 t37 t 11.5 20.4
- 12.5 +8 +1 +49 24.9
-11 +30 t6.5 -7-S 16.7
+5 t14 t12 t23.5 15.1
-_; -0.5 -3.5 t7.s 4.3
+55 t4 -6 +16-S 9.4
t3 to.5 +2 - 1.5 2.0
t 11.5
-_;*s t7.J +8-S 46
+7*5 t I2*5 + 13.5 ll*S
t3 t3 t1.5 26
t7.5 t2.s t 1.5 4.7
::; i.5 2.07 Averplepercentof squareerror 2@4-T3 Alclad boa
i2
7075.T6 AlChd
i-6 3.6
1 ;_
-27.5
:: t23 SidCS 2.5 -8.5 uTs=wl 1:s 297 -9 Averagepercentof squareerror 18.9 both
2
2024m
1 G $1434 3.2 E 6.35 243 12.7 2.12 Avenge percentof agusmam 707~lx bar ?lTS=5@
i-k :Sa 3.2 2.88 12.7 2-u Avessgepercentof squrucemr
-: t1.5 6.3
(4 IlKam a coaservative resutf (-1 an tmconservative.
ESTIMATION OF TEE FATIGUE LIMIT
OF NOTSPECIMENS WEflcI u. >O FROM THE Km DATA FOR ALTERNATING TENSION-COMPRESSION There is as yei no general agreement regarding the proper method of ana&& sbations invobing stress raisers in conjunction with combinfd mean and a&erMkg stress. One nwthod (I) applies the notch factor only to the al&mating stress. In this case the fatigue limit of the notched specimen for N = 10’ cycles is
where G, is the al&mat&g stress component of the fatigue limit for the not&d specimen, Q=is the alternating stress component of the fatigue limit of the unnotchcd specimen, in the case of the same nominal mean stress an as for the notched specimen,
KFPZ
isthenotchfactorforR
= -1.
Another method (II) assumes that KFi is applied to the maximal stress and Km should be equal to KF,
talc. ace.ICalc. act. II %of %or
Notch dills luIulfrl
Mwmmd values
r(lllm) KT
&o
ki,,
emrt
err04
fore;,
forU&
-?
t 16.5 +n*s
~N~~~ 6061.T4 E&=31*
:
A.5 Average pcrceat of
1
1
4.6
2.48 l-88
;I; 2.07
+3*5 +6 +6 5.8
1.33 1.38
squareerror
zoWT3
1
T:d i.78 2.31 E sides : ::: 1.54 m-46@ 12.5 2.07 1.58 Avenge percmt of squarewror
:::: 1.65 169
7075.T6 A&d
1
4.6 1
:.62
z-z : 3.6 UTS-560 1:.5 fz7 Ava;llsparaptofmet=
E 2.00
2024-n lmr UTS-460
1 1.6 2.90 :.11 3.2 2.66 2.11 6.35 243 2.13 12.7 2.12 1.72 Avcmgc perceat of squue ellof
F*T6
76 1 Lo 3.2 ::: 2.08 12.7 2.12 1.71 ptrmt of squareerror
UTS=58@ Avm
:*Oo 2.19 2.10 1*97
ho 2.28 1.91
71 25.5 30 46.2 45
z.5 55.5 55 68
‘i538.5 59
72
-_; -10 Ii.5 6
t26 t23 23.6 - 14
tf t6.5 t65 8.5
G.5 +14 -15 -8 16.8
-Tr; -5 -11
-10 -5 -7-S +3+5 7
G.5
t4 t&S t1*5 2.5
t43 11.9
t3.5 -1.5 t 13.5 7.5 7
t9 t 10.5 8.0
where oh is the fatigue limit bf notched specimen for R = 0 and @dois the fatigue limit of the mater&l for R - 0. Metlmd (I) was used for calculating Q= for R = 0. The alternating stress compoamt a, of the unnotched specimen in the case of the same mean stress as forthe notched specimen (a,,* Oh,,) was found with the aid of the experimental u‘ - a,,, lines ia Figs. S-9. The measured and calculated values of aon are compared in Table 6, which also includes the pertenta(lcerrorsforthefatieuelimitscalculaMaccordiagtoboth~~s.Ascanbese~nfrom Tabk 6, method I is evidently more accurate than method II in the case of the ductile material 606bT4, and nearly equally accurate for the other investigated materials. McEvbfijl9J derived a foIlowing rektknship between Kpt and GO, U&I the OO* ~1~~~~s:
Calculationsproved that this formula is in good agreement with experimental results in Table 3. The accuracy of formula (7) is especially high for 606LT4. CONCLUSIONS (1) The ratio Kp/Kr is a better characteristic of the notch sensitivity for flat and round, internally and ~x~m~i~notched aircraft material specimens than the wideiy-used notch
Fatiguenotch effect estimations
671
sensitivity factor q, because-of the better regularity of the dependence on the notch radius in the former case. (2) With the same notch radius is the notch sensitivity relatively higher for round than for flat specimens. (3) The notch effect is in general stronger in tension-compression than in pulsating tension. (4) The intluence of the stress concentration factor (at IL C 5) on the fatigue notch factor Kp is in general stronger than of the notch radius r and the stress ratio R for specimens with KT = const. REFERENCES [l] A&al Handbook, Suppl. Am. Sot. Met. Cleveland (1954). [21 G. Sines and J. L Waisman (Eds.), Metal F&M McGraw-Hill, New York (1959). [3] R. C. Juvinall, EngineeringConsidem~ionsof S&ss, &ah ond stnngth. McGraw-Hill, New York (1967). [4] P. Kuhn, SAE-AShfE Paper 843 G (1964). [A W. S. Hyler d aL, NACA TN 3291 (1954). [6] A. Buch, Materi&@ 13.6 (1971). m A. Buch, ZZZZnt. Conf. on Z+act#re, V-431 B, Munich (1973). [s] A. Buch, Met Sci. Engng 15, (1974). I91 A. Buch, Z W&&#&A 5.4 (1974). 1101J. A. Bmmet ~IKII. G. Wai&eg, J. Ru. Nat. EIUWMStand 52.5 (1954). 1111R. B. Heywood, higaing Ag&sl Foliguc Chapman and Hall, London (1%2). [12] H. J. Grova et eL, Fatigue of Metals and stnrcturcs. U. S. Gov. Priot OK Washiion (1960). (131 Aircmlfl FarigueHandbook,ARIC-W-76, Aircraft Ind. Ass. Caliiornia (1957). [14] D. T. Raske, 1. Tcrt. Eu&urion 1, 5 (1973). [Is) J. !Jchijve, NACA TM 19395 (1954). [16] M. B. ti and H. F. Hardrath, NACA TN 3631 (1956). 117.lA. Buch, TECHNION TAE Report 184 (1973). [Is] W. Ill& NACA TN 3866 (1956). [19] A. J. McEvdi, 3rd ZIU.Ci& oa rhe Srm@ oflUe@le& AUvys, clmbidp Prper (1973).
(Receiced 2 February 1974)