NAEC-ASL 1114

U. S. NAVAL AIR ENGINEERING CENTERPHI LADELP HIA. PE NNSYLVAN IA

AERONAUTICAL STRUCTURES LABORATORY

Report No, NAEC-ASL-1114

June 1967

NEUBER'S RULE APPLIED TO FATIGUEOF NOTCHED SPECIMENS

by

T. H. Topper, R. M. Wetzel,, J. MorrowDepartment of Theoretical and Applied Mechanics

University of Illinois, Urbana

Contract No. N156-46083

Distribution of this documentI is unlimited

ko. ,,CC.9113,3 4 at V 1.$* 3 1 0LAr, NO. 11749

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NAEC-ASL -1114

FOREWORD

This investigation was conducted in the H. F. MooreFracture Research Laboratories of the Department ofTheoretical and Applied Mechanics, University of Illinois,in cooperation with the Aeronautical Structures Laboratoryof the NavE 1 Air Engineering Center.

This report covers work performed during the period1 February 1966 through 30 April 1967, and together withreport No. NAEC-ASL-1115 constitutes the final report onItem 2 of Contract N156-46083. Messrs. M. S. Rosenfeldand R. E. Vining acted as technical liaison for the Navyand Professor T. J. Dolan, Head of Theoretical and AppliedMechanics, furnished administrative and technical guidance.

iii

NAEC-ASL-1114

SUMMAR Y

A method is presented for predicting the fatigue life ofnotched members from smooth specimen fatigue data. Inelasticbehavior of the material at the notch root is treated using Neuber'srule which states that the theoretical stress concentration factoris equal to the geometric mean of the actual stress and strain con-centration factors. This provides indices of equal fatigue damagefor notched and unnotched members.

Experimental results for notched aluminum alloy platessubjected to one or two levels of completely reversed loadingare compared with predictions based on these indices, Measurednotched fatigue lives and lives predicted 'rom smooth specimensagree within a factor of two.

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NAEC-ASL-1114

TABLE OF CONTENTSPage

I. INTRODUCTION 1

II. ANALYSIS 1

III. DISCUSSION 2

IV. COMPARISON WITH EXPERIMENTAL RESULTS 4

V. CONCLUSIONS 5

VI. REFERENCES 6

Tables

NRi-a FATIGUE DAMAGE AT FAILURE FOR NOTCHED' 2024-T351 PLATES 8

I -b FATIGUE DAMAGE ATF FAILURE FOR NOTCHED7075-T651 PLATES 9

Figures

la Smooth Specimen Fatigue Data in a Form Suitable forPredicting Lives of Notched Members 10

lb Notched Fatigue Data Compared to the Life CurvePredicted from Smooth Specimen Data (2024-T3) 11

1c Notched Fatigue Data Compared to the Life CurvePredicted from Smooth Specimen Data (7075-T6) 12

2 Cyclic Stress-Strain Curves 13

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NAEC-ASL-1114

LIST OF SYMBOLS

E Modulus of elasticity

S Nominal stress on a notched member; axial load divided bynet area

e Nominal strain; strain which would occur in a smooth specimensubjected to S; equal to S/E when the nominal strain is elastic

a Actual stress at a point, frequently at a notch root

C Actual strain at a point, frequently at a notch root

A S, Ae, Au, AE Peak to peak change in the above quantities duringone reversal

K Theoretical stress concentration factort

Ka Stress concentration factor, Au divided by AS

K E Strain concentration factor, AE divided by Ae

Kf Fatigue strength reduction factor or effective "fatigue stressconcentration factor"

a Material constant (see Eq. 1)

r Notch root radius

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NAEC-ASL-1114

I. INTRODUCTION

Stowel- (1) and Neuber (2) have developed analyses which help describethe nonlinear stress-strain behavior of notches. Their work has recentlybeen applied to the notci fatigue problem by a number of authors (3 -6).These authors relate th cyclic load range on a notched member to the actualstress or strain range ai the notch root and then estimate the life of the notchedmember from stress vs Iife or strain vs life plots obtained from smoothspecimens.

An alternate approach is presented here which makes it unnecessaryto solve for the actual stress or strain at the notch root. Instead, Neuber'srule is used to convert the smooth specimen data for a given metal into amaster life pl3t which can , used to estimate the fatigue life of any notchedmember made of that particular metal.

U. ANALYSIS

The theort- cal stress, concentration factor, K. only applies when thematerial at the not ': root remains elastic. Neuber' (2) has proposed a rulewhich may be applie, .wn when the material at the notch root is strained intothe inelastic region. He state! that the theoretical stress concentration factoris equal to the geometric mean of the actual stress and strain concentrationfactors.

S t = (KaK )/2-------------------------- Neuber's Rule

That the product of Ka and K might be constant is intuitively reasonable7 because Ka deceases and K increases as yielding occurs.

It is well known that small notches have less effect in fatigue than isindicated by Kt . Several authors have suggested theoretical or empiricalexpressions for evaluating a "fatigue stress concentration factor." Kf) whichcorrects for size effect. In this paper we employ Kf factors based onPeterson's approach (7)

~K t -1i

Kf = + -------------------------- (1)- , + ar

where r is the root radius and "a" is a material constant determined fromlong life fatigue data for sharply notched specimens. For notches with largeradii K is nearly equal toK. For sharp notches, however, K isunnecestarily conservative and Kf should be used in preference to Kt .

NAEC-ASL-1114

To apply Neuber's rule to the notch fatigue problem, Kf will be usedin place of Kt and KC and K are written in terms of ranges of stress andstrain.

It is convenient to write the above equation in the following form:

Kf(AS Ze E)1 / 2 = (A AE E) 1/ 2 ................... (2)

where 6S and Ae are the nomiiial stress and strain ranges applied to anotched member, L'a and A are the local stress and strain ranges at thenotch root, and E is the elastic modulus.

Note that Eq. (2) reduces to the following simple form if the nominalstress and strain are limited to the elastic region.

1/2Kf AS = (A7 AE E)-- ------------------------- (2a)

This special case is important because it covers many problems of engineeringinterest.

At even longer lives and lower values of AS, the notch root remainsessentially elastic and Eq. (2) reduces to the familiar form

Kf LS = Aa ------------------------------- (2b)

This is the equation Ahich is frequently misused at shorter lives when thematerial near the notch behaves inelastically.

III. DISCUSSION

Equation (2) relates the nominal stress -strain behavior of a notchedmember to the actual stress -strain behavior at the critical location. It canalso be interpreted as furnishing indices of equal fatigue damage in notchedand unnotched specimens. ln completely reversed, constant amplitude tests,a notched specimen and a smooth s Decimen will form detectable cracks at thesame life pxvided Kf(AS neE) 1/h2 for the notched specimen is equal to(Aa Ac E)' for the smooth specimen. This means that life data fromnotched and unnotched specimens can be plotted on the same graph or thatsmooth specimen results can be used to produce master life plots for estimatingthe fatigue life of notched members.

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NAEC-ASL -1114

Figure la is an example of such a master plot of the quantity(Aa Ac E)1/ 2 vs life for two aluminum alloys using data reported by Endo andMorrow (8). Poin sepresent failure of smooth specimens for which thevalue of (6Af s E)" / was calculated from steady-state stress and s6tinranges. It is well documented (9) that the stress and strain ranges ofunnotched specimens approach a steady-state value after a small precentageof Ffe and Blatherwick and Olsen (10), and Crew and Hardrath (4) have shown

Y that the strain range at a notch root rapidly stabilizes. Recent results fromour laboratory (11) using the same metals shown in Fig. la, indicate thatrapid stabilization of the hysteresis ioup occurs following a step change instrain amplitude.

The life of a notched member can be predicted by entering the valueof Kf(ZAS e E)1/ 2 on the ordinate of smooth specimen curves of the typeshown in Fig. Ia. In the low life region, the loads may be large enough tocause yielding throughout the specimen. If this happens Ae must be deter-mined by entering LS on a cyclic stress -strain curve (Fig. 2). At longerlives there is no need for the cyclic stress -strain curve since the nomin~l,strains are essentially elastic. In this case, the quantity Kf(AS 6e E)-/reduces to Kf AS,

i Some of the limitations on the above approach to the notch fatigue

problem will now be discussed.

Crack Initiation and Propagation: The above method is limited to predictingcrack initiation or final failure where the crack propagation stage is negligible.This is usually the case for small unnotched specimens of the type used toobtain f-itigue lifc3 data.

In service applications, crack propagation may occupy a widelyvarying portion of the useful life of notched members and structures.Weight critical applications represent one extreme. The tendency is tosurround notches with a minimum of elastic material and to select a highstrength and therefore relatively brittle metal. In this case crack propa-gation may be a small part of the total life. On the other hand, heavystructures made of ductile metal may have relatively large flaws presentfrom the beginning and will occupy their entire life in propagating a crackto failure.

Effect of Mean and Residual Stress: The reader is reminded that the meanstress at the notch root has been assumed to be zero. Thus, the presentapproach is inadequate for predicting the effect of mean loads on the fatiguelife of notched members. Even if the loading is completely reversed, butthe level is changed during the test, the creation and relaxation of meanstress at notch roots may complicate the notch problem. Large tensileloads tend to induce compressive mean stresses for subsequent smaller

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NAEC-ASL-1 114

cycles while large compressive loads induce tensile mean stresses. Theensuing fatigue life may be greatly altered. The problem is furthercomplicated by the fact that mean stresses at the notch root will tend to relaxtoward zero in the presence of sufficient cyclic plastic strain (11).

Using Eq. (2) with the restrictions and limitations discussed above,it is possible to predict the lives of many types of notched specimens fromreadily available sf nn h specimen fatigue data. It should be noted thatcurves of (La &- E) vs life can be easily derived from any two of thefollowing cur-ves: stress vs life, total strain vs life, piastic strain vs life,and cyclic stress vs cyclic strain.

IV. COMPARISON WITH EXPERIMENTAL RESULTS

Two metals are considered, 2024 and 7075 aluminum alloys. Due tothe nearly identical fatigue properties of the T3, T351 and T4 conditions of2024 and T6 and T651 conditions of 7075, no distinction needs to be madebetween these various c. .iditions over the life region of interest here.

The smooth curves in Figs. lb and c are transferred from Fig. la.They represent the predicted lives of notched members of these metals.Points are from lUg's data for notched plates with Kt values of 2.0 and4.0 (12). Loading was completely reversed and therefore did not introducesignificant mean stress.

Values of Kf calculated from Eq. (1) are used in preference to Kt'The value of "a' for use in Eq. (1) was determined in the following manner:A value of K for Illg's sharply notched specimen was found directly bycomparison ck long life data for the sharply notched specimen with data forunnotched specimens. The Kf thus determined is 3. 0 for both materials;the value of Kt is 4. 0, and the root radius, r, is 0. 057 in. These valuesof K , K , and r were substituted into Eq. (1) and "a" was determined for usein cIlcurating K for notches .i other geometries. The value of "a" for both7075 and 2024 Jas found to be approximately 0. 028 in.

Agreement between life data and predictions is seen to be good for2024 and excellent for 7075. The relationship should be checked for othermaterials, particularly those with a yield point.

Step Tests: The curves in Fig. 1 were also used to perform a linear damagesummation for notched specimens subjected to two levels of reversed loadingas a part of this investigation. Damage is defined as the number ofreversals which occur at a given load level divided by the reversals to failurepredicted from Fig. 1. The results of these tests are given in Table 1.

4

NAEC-ASL-1114

Specimens are similar to those used by Blatherwick and Olson (10).The radii of the notches are 0. 25 in. or greater-, so that there is no significantdifference in Kt and Kf

Although only two amplitudes of loading were used in each test, theamplitude was frequently changed from one level to the other. Tests wereplanned so that nearly equal damage was done at each level. About 20changes in level were made in each test. Visible cracks were neverobserved until the last 20o of life and usually not until the last 107. Thetotal damage summations in Table 1 are remarkably close to 1. 0.

Even though the loading was completely reversed, there is a possi-bility of a mean stress effect depending upon how the amplitude is changedfrom the large to the small level. If the last peak reached at the higheramplitude is tensile. a beneficial compressive mean stress may be presentfor subsequent cycles at the lower amplitude. The effect may be detri -mental if the last peaks at the higher level are compressive. Only twospecimens were tested in a manner which could create compressive meanstresses and the results are inconclusive. However, for more severelynotched specimens subjected to a few large load cycles followed by manysmaller ones tius mean stress effect can be significant.

IV. CONCLUSION

The equation

Kf(nS Ae E) 1/2 (An Ao E)1/2

or for the case where nominal strains are essentially elastic.

1/2K f ,S =(tA E)1/

relates the behavior of notched spec,:,aens to readily available smooth speci-men data. Master plots of (Aa Ac E) 1/ 2 vs life based on smooth specimenfatigue results may be used to accurately predict fatigue of notcied aluminumalloy plates subjected to completely reversed loading.

5

NAEC-ASL-1114

VI. REFERENCES

1. Elbridge Z. Stowell, "Stress and Strain Concentration at a CircularHole in an Infinite Plate," National Advisory Committee for Aero-nautics, Technical Note 2073, April 1950,

2. i. Neuber, "Theory of Stress Concentration for Shear StrainedPrismatical Bodies with Arbitrary Non Linear Stress Strain Law,"Journal of Applied Mechanics, December 1961, pp. 544-550.

3. S.S. Manson and M. H. Hirschberg, "Crack Initiation and Propagationin Notched Fatigue Specimens," Proceedings of the First InternationalConference on Fracture, Vol. 1, Japanese Society for Strength andFracture of Materials, 1966, pp. 479-498.

4. J. H. Crews, Jr., and H. F. Hardrath, "A Sz'xdy of Cyclic PlasticStresses at a Notch Root," Experimental Mechanics, Vol. 6, No. 6,June 1966, pp. 313 -320.

5. T.J. Dolan, "Non-Linear Respo, nse Under Cyclic Loading Conditions,"to be published in Proceedings 9,h Midwest Mechanics Conference,paper presented at the University of Wisconsin, August 1965.

6. R. E. Peterson, "Fatigue of Metals in Engineering and Design,"Thirty-Sixth Edgar Marburg Lecture before the American Society forTesting and Materials, 1962, see also Materials Research andStandards, Vol. 3, No. 2, February 1963, pp. 122-139.

7. R. E. Peterson,"Notch-Sensitivity," Metal Fatigue, Chapter 13, Sinesand Waisman Editors, McGraw-Hill Book Company, Inc., 1959.

8. T. Endo and JoDean Morrow, "Cyclic Stress -Strain and Fatigue Be-havior of Representative Aircraft Metals, " paper to be presented atthe ASTM Summer Meeting, Boston, June 1967, see also Report No.NAEC-ASL-1105, U. S. Naval Air Engineering Center, Philadelphia,Pa., June 1966.

9. JoDean Morrow, "Cyclic PlastiL Otrain Energy and Fatigue of Metals,"Symposium on Internal Fraction, Damping and Cyclic Plasticity,American Society for Testing and Materials, Special TechnicalPublication No. 378, 1965, pp. 45-87.

10. A. A. Blatherwick and Byron K. Olson. "Stress Redistribution inNotched Specimens Under Cyclic Stress, " A. S. D. Technical Report61-451, Aero Systems Div., Wright-Patterson Air Force Base,Dayton, Ohio, 1961.

6

NAEC -ASL-1114

11. T. H. Topper, B. I. Sandor and JoDean Morrow, "Cumulative FatigueDamage Under Cyclic Strain Control, " papei presented at the ASTMSummer Meeting, Boston, June 1967, see also Report No. NAEC-ASL-1115, U. S. Naval Air Engineering Center, Philadelphia, Pa., June1967.

12. Walter 1lg, "Fatigue Tests on Notched and Unnotched Sheet Specimensof 2024 -T4 and 7075 -T6 Aluminum Alloy and of SAE 4130 Steel withSpecial Consideration of the Life Range from 2 to 10,000 Cycles,'National Advisory Committee for Aeronautics, Technical Note 3866,December 1956.,

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13-

UNCLASSIFIEDSecurity Classification

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University of Illinois UnclassifiedDepartment of Theoretical and Applied Mechanics Zb GROUP

Urbana, Illinois 61803 None3 REPORT TITLE

NEUBER'S RULE APPLIED TO FATIGUE OF NOTCHED SPECIMENS

4. DECCRIPTIVE NOTES (Type of report and inclueive datee)

Final Report

S. AUTHOR(S) (Last name, firet name, Initial)

Topper, T. H.Wetzel, R. M.Morrow, J.

S. REPORT DATE 70. TOTAL NO. OF PA$ZES 7b. NO. OF RapS

June, 1967 24 12

Be. CONTRACT OR GRWANT NO. 9.1 ORIGINATOR'S REPORT NUMBER(S)

N156-46083 NAEC-ASL-1114

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13 ABSTRACT

A method is presented for predicting the fatigue life of notched members fromsmooth specimen fatigue data. Inelastic behavior of the material at the notchroot is treated using Neuber's rule which states that the theoretical stressconcentratiln factor is equal to the geometric mean of the actual stress and strairconcentration factors. This provides indices of equal fatigue damage for notchedand unnotched members.

Experimental results for notched aluminum alloy plates subjected to one ortwo levels of completely reversed loading are compared with predictions based onthese indices. Measured notched fatigue lives and lives predicted from smoothspecimens agree within a factor of two.

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MetalsFatigueCyclic stress-strainControlled cyclic strainingStress concentrations

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