CASE III RESIDUAL STRESS TENDING TOWARD ZERO
5. Accurate fatigue life predictions for spectrum loaded structures are possible only when the cycle-dependent residual stress behavior is included
References
[1] Heywood, R. B. in Colloquium on Fatigue, Springer-Verlag, 1956, pp. 92-102.
[2] Mordfin, L. and Halsey, N. in Fatigue Tests of Aircraft Structures: Low-Cycle, Full-Scale, and Helicopters. A S T M STP 338, American Society for Testing and Materials, 1963, pp.
25 t-272.
[3] Breyan, W. in Effects of Environment and Complex Load History on Fatigue Life, A S T M STP 462, American Society for Testing and Materials, 1970, pp. 127-166.
[4] Kessler, A. F. and Hillberry, B. M., "The Effect of Periodic Overstress on the Fatigue Life of Notched Aluminum Specimens," SAE Paper 710599, Society of Automotive Engineers, New York, June 1971.
[5] Neuber, H., Transactions of the ASME, Journal of Applied Mechanics, Dec. 1961, p. 544.
[6] Wetzel, R. M., Journal of Materials, Vol. 3, No. 3, Sept. 1968, p. 646.
[7] Wetzel, R. M., Morrow, JoDean, and Topper, T. H., "Fatigue of Notched Parts with Emphasis on Local Stresses and Strains," NADC-ST-6818, Naval Air Development Center, Johnsville, Pa., Sept. 1968.
[8] Potter, J. M., "A Simple Graphical Procedure for Determining Cyclic Elasto-Plastic Notch Stresses," AIAA Journal, American Institute of Aeronautics and Astronautics, Vol.
10, No. 10, Oct. 1972, p. 1304.
[9] Potter, J. M., "A General Fatigue Prediction Method Based on Neuber Notch Stresses and Strains," AFFDL-TR-70-161, Air Force Flight Dynamics Laboratory, (AD 723631) Dayton, Ohio, Feb. 1971.
[10] Crews, J. H., Jr., "Local Plastic Stresses in Sheet Aluminum Alloy Specimens with Stress Concentration Factor of 2 Under Constant-Amplitude Loading," NASA TN D-3152, Dec. 1965.
[1 / ] Crews, J. H., Jr., "'Elasto Plastic Stress-Strain Behavior at Notch Root in Sheet Specimens Under Constant-Amplitude Loading," NASA TN D-5253, June 1969.
[12] Peterson, R. E., Stress Concentration Design Factors, Wiley, New York, 1953.
[13] Metallic Materials and Elements for Aerospace Vehicle Structures, MIL-HDBK-5A, Department of Defense, Washington, D. C., Feb. 1966.
[14] Stadnick, S. J., "Simulation of Overload Effect in Fatigue Based on Neuber's Analysis,"
Report 325, Department of Theoretical and Applied Mechanics, University of Illinois, Urbana, Ill., 1969.
[15] Morrow, JoDean, Ross, A. S., and Sinclair, G. M., Transactions, Society of Automotive Engineers, Vol. 68, 1960, pp. 40-48.
[16] Rosenthal, D. and Sines, G., Proceedings, American Society for Testing and Materials, Vol. 51, 1951, p. 593.
[17] Impellizzeri, L. F. in Effects of Environment and Complex Load History on Fatigue Life, A S T M STP 462, American Society for Testing and Materials, 1970, pp. 40-68.
B. N. Leis ~, C. V. B. Gowda ~, and T. H. Topper ~
Cyclic Inelastic Deformation and the Fatigue Notch Factor
REFERENCE: Leis, B. N., Gowda, C. V. B., and Topper, T. H., "Cyclic Inelastic Deformation and the Fatigue Notch Factor," Cyclic Stress-Strain Behavior--Analysis, Experimentation, and Failure Prediction, A S T M S T P 519, American Society for Testing and Materials, 1973, pp. 133-150.
ABSTRACT: Cyclic inelastic deformation response and the fatigue notch factor are presented and discussed for two metals of contrasting stress-strain behavior (2024-T351 aluminum alloy and SAE 1015 mild steel). Thin circular or elliptical notch plates fabricated from these metals were subjected to cyclic straining in a servo-controlled testing machine.
Experimental results obtained from the deformation studies indicate that for small scale confined plasticity, the transient notch response had little effect on nominal defor- mation resistance. However, as the size of the zone increased or the degree of confine- ment decreased, transient notch response was reflected in the overall deformation response. These results are discussed in terms of cyclic stress and strain concentration factors and related to notch plate deformation through Neuber's rule. In the low en- durance region, the results indicate that the deformation analysis employed, rather than a material "size effect," is responsible for values of Kf less than Kt 9
KEY WORDS: aluminum, steels, test specimens, stress distribution, strain distribution, notch constraint, fatigue (materials), fatigue notch factor, predictions
Nomenclature
g t K~
K~
@, E S, e S, e N Ao', mE
Theoretical stress concentration factor, defined on Fig. 3a Actual stress concentration factor
Actual strain concentration factor Fatigue notch factor
Stress, strain at a notch root, respectively Stress, strain in a smooth specimen, respectively Nominal stress, nominal strain in notched plates, respectively
Peak to peak change in the corresponding quantities during one cycle
1Research assistant, assistant professor, and professor, respectively, Dept. of Civil Engineering, University of Waterloo, Waterloo, Ontario, Canada.
134 CYCLIC STRESS-STRAIN BEHAVIOR
As, Ae Peak to peak change in the corresponding quantities during one cycle
AS, /ke N Peak to peak change in the corresponding quantities during one cycle
~, Load level - - actual load divided by the maximum load N Number of cycles
N i Number of cycles to crack initiation E Modulus of elasticity
r Notch root radius
x Distance from the center of a plate
Structural components and machine parts invariably contain stress raisers introduced either by assembly processes or design requirements. The presence of these stress raisers imposes severe limitations on service life and load levels and presents the design engineer with large unpredictable gradients of stress and strain, particularly when plastic strains exist at notch roots.
An inelastic deformation analysis for stresses and strains at discon- tinuities is thus essential for accurate fatigue predictions. Several analyses have been attempted [1,2,3] 2 of which that of Neuber [3] appears most suitable [4]. This analysis which equates the geometric mean of the stress and strain concentration factors to the theoretical stress concentration fac- tor has been verified for centrally notched plates when limits on notch plasticity for the material were imposed [5]. Experimental results [6,7,8] also indicated that its validity was confined to small scale plasticity in sharply yielding materials. In addition, several studies have shown the Neuber for- mulation to be suitable for cyclic deformation when concentration factors are redefined in terms of stress and strain ranges. Fatigue life predictions for notched specimens, once the stresses and strains at the stress raiser are deter- mined, are based on fatigue data obtained from strain controlled smooth samples [4,9-12]. Implied in this approach is the assumption that both smooth and notched specimens will have the same life to fatigue crack initia- tion if both have the same stress-strain history at the crack initiation site.
Some doubt exists, however, that the foregoing assumption is accurate, because sharp notches are often less severe in fatigue than indicated by deformation analyses. A fatigue notch factor which is less than the theoretical stress concentration factor is introduced to account for this dis- crepancy. Several authors [13-17] have developed equations relating this fatigue notch factor to the theoretical concentration factor based on a material "size effect" or "notch sensitivity" which causes the small amount of highly stressed material at a sharp stress raiser to be more resistant to fatigue than the much larger amount of material in smooth samples.
An alternative explanation for a reduced concentration factor in fatigue is that the deformation theory from which the concentration factor is obtained
2The italic numbers in brackets refer to the list of references appended to this paper.
LEIS ET AL ON DEFORMATION AND THE FATIGUE NOTCH FACTOR 135
does not accurately determine inelastic notch root deformations. Some evidence [7,8] indeed indicates that the geometric mean of concentration factors varies from the elastic value of K t for cyclic inelastic straining.
Similarly, recent studies of long life fatigue in ductile metals [18-20] show in- elastic strains at life levels previously assumed to be elastic which would ac- count for stress errors in concentration factors based on conventional elastic computations.
The purpose of this investigation is to assess to what extent each of material size effect and inaccuracies in the Neuber solution for notch defor- mations account for discrepancies between the theoretical stress concentra- tion factor and the observed fatigue notch factor. Results are interpreted in the light of their implications for fatigue prediction techniques applied to notched components.
Experimental Details
Materials, Specimens and Instrumentation
The materials used in this investigation were a 2024-T351 aluminum alloy, and an SAE 1015 mild steel. A description of these materials is provided in Table 1, while the various mechanical properties are given in Table 2. The aluminum alloy was used as supplied, but the mild steel
TABLE 1--Description of materials.
Designation Condition
2024-T351 aluminum alloy IA in. rolled sheet T351 heat treatment, prestretched at mill 1015 mild steel 2~ by 1 in. rolled bar
semi-kiUed, annealed
TABLE 2--Properties of materials.
Designation
Modulus of Ultimate
Elasticity (E)~ Yield Strength Strength
ksi (S o, ), ksi (Su), ksi
Strain Hardening Exponent (n)
2024 T351 10 100 52.0 a 70.0 0.07
aluminum alloy
1015 mild steel 29 500 45.0 b 60.0 0.26
a Yield stress corresponding to 0.2 percent offset.
h Yield stress corresponding to upper yield point.
136 CYCLIC STRESS-STRAIN BEHAVIOR
specimens were normalized just prior to final machining. Monotonic and cyclic stress strain response of the aluminum alloy and the steel, are presented in Figs. 1 and 2, respectively. It can be observed that the mild steel
(:3 g
-- 0.1 0 5 1.0 5 . 0
NOMINAL STRAIN AMPLITUDE, .-_~__, % FIG. l--Cyclic stress-strain response o f 2024-T351 aluminum.
,ooo/
O0 0.5 1.0 1.5
NOMINAL STRAIN AMPLITUDE , ~ , %
Z
FIG. 2--Cyclic stress-strain response o f S A E 1015 mild steel.
LEIS ET AL ON DEFORMATION AND THE FATIGUE NOTCH FACTOR 137
softens at cyclic strain amplitudes less than 0.3 percent, is approximately stable between 0.3 and 0.4 percent, and hardens at strain amplitudes greater than 0.4 percent, whereas the aluminum alloy cyclically hardens at strain amplitudes greater than 0.5 percent.
Circular and elliptical centrally notched plates used in this investigation have been described elsewhere [8], but some features are worthy of repeti- tion. Specimen dimensions were chosen giving careful consideration to availability of material, machine capacity, critical buckling load, and stress diffusion from the grip end to test section. The notch size and specimen width were selected so as to minimize the biaxial stress field in the specimen but still allow testing at reasonable load levels. The ratio of notch size to specimen thickness was chosen such that conditions of plane stress were ap- proximated.
Measurement of notch root and surface strains was accomplished through electrical resistance gages bonded in the notch roots and across the transverse net section of the plate. The gages were selected and applied in ac- cordance with the results of recent studies [7] as well as a calibration program conducted concurrent to this work [22]. Hence, gage results proved highly reliable and free of many problems commonly associated with the use of strain gages in the measurement of cyclic strains at large amplitudes.
Under monotonic loading, failure of gages was generally due to insufficient bond or gage element cracking, while under cyclic loading, gages failed by the formation of microcracks in the gage element resulting in supersensitive response before complete failure. As a result, data are reported for only the first 80 to 90 percent of gage life. It is expected that the gage errors will be less than 10 percent since all possible means of exacting close control over gage response were exercised (see Appendix 1 of Ref 7).
Apparatus and Procedure
All experiments were conducted in a closed-loop, servo-controlled elec- trohydraulic testing machine under deformation control. Axial strains at the edge of the plate were controlled by means of a clip-on extensometer, while load was monitored by means of a commercially available load cell. A pic- torial view of the experimental set up is shown in Fig. 4 of Ref 8.
During the first quarter cycle (all tests were started in tension), a digital voltmeter reading of nominal load was recorded at regular increments of edge strain. In addition, continuous records of notch strain versus cycles, notch strain versus edge strain, and edge strain versus nominal load were recorded on strip charts and X-Y graphs. The continuous record of notch root strain was achieved by conditioning each notch gage signal on separate Wheatstone bridges from which the output was fed to a calibrated X-Y plotter and a strip chart recorder. Periodic monitoring of the surface strain distribution was achieved through a commercially available data logger.
138 CYCLIC STRESS-STRAIN BEAVIOR
Records of surface strain were made at zero, one-quarter, one-half, three- quarters, and full cycle states.
The inducement of residual mounting stresses was avoided by means of a molten metal grip located between the lower grip and ram.
The experimental program consisted of subjecting each specimen to a monotonic incremental loading to a given level of edge strain, then cycling at constant amplitudes under fully reversed conditions.
Deformation and Fatigue Analysis
Because thin plate notch root stress in fatigue applications is always deter- mined on the basis that material in this location exhibits the same stress- strain response as a uniaxial sample, strain is the only independent variable.
Evaluation of the degree to which size effect alters the fatigue resistance of the material at the root of sharp stress raisers from that of the same material in a smooth sample may, therefore, be based on strains only. In other words, if both smooth and notched specimens have identical strain ranges at their respective crack initiation sites at the same life to fatigue crack initiation, the material shows no size effect, and discrepancies between theoretical and fatigue concentration factors are wholly accounted for by the deformation analysis.
Cyclic Deformation Analysis
As applied in cyclic deformation analysis, Neuber's rule is rewritten in the following form [4,11].
(Ae 9 Ae 9 E) 1/2
Kt ( A S 9 Ae N 9 E) 1/2 (1)
where AS and A e N are ranges of nominal (net section) stress and strain, and Act and AxE are ranges of notch root stress and strain. All ranges of stress and strain are measured from one reversal point to the next in each half cy- cle of loading. In practice, relationships between AS and AelV , and Ar and AXE, are derived from material deformation response by strain controlled testing of smooth specimens [4,9]. In cases such as the present investigation, where appreciable hardening or softening takes place both at the notch root and throughout the plate [4], both nominal and notch root material proper- ties are estimated from curves relating stress amplitude to strain amplitude at a specific number of cycles such as those given in Figs. 1 and 2. In the pre- sent investigation, nominal strain was obtained from measured nominal stress as shown in Fig. 3a. Similarly, the method of obtaining notch root and surface stresses from measured strains is illustrated in this figure. Cyclic stress and strain concentration factors, Ka and K E , are then determined as
LEIS ET AL ON DEFORMATION AND THE FATIGUE NOTCH FACTOR 139
SHOOTH SAMPLE CYCLIC S-e CURVES viz. F i g l . 1 & 2
Ri s
START
%
LOAD v s . LOCAL STRAIN f r o m n o t c h e d s p e c / m e n t e s t s
/
r
SMOOTH SAMPLE CYCLIC S-e CURVES. viz. FIGS. I & 2
s
t'eNl Fi(j,~cl DETEPC:IXATION OF NOTCH STRESS, NOMINAL STRAIN ~ND CONCENTRATION FACTORS
9
K = S'
F ~ b E V A L U A T I O N OF Kf FROM FATIGUE DATA
- - N 1
Ar 1 5CI
K e = /le2,l,T I K = ~S-~ : DEFINITIONS OF Kt, Kr a K = (~e'A's'E)% :
~ ~ [ (~eN' ~S'E) % DEFINITION OF Kf
N" "
N 1 N l
F I G . 3--Graphical presentation of definitions used in the deformation analysis.
the ratios of notch root stress to nominal stress and notch root strain to nominal strain, respectively.
Fatigue Evaluation of Deformation Analysis
The applicability of this analysis to fatigue results is most conveniently evaluated from a plot of the form shown in Fig. 3b [4] where notch specimen nominal deformation as measured by (AS .Ae N .E) 1/2, and smooth specimen deformation measured by (As 9 Ae .E) 1/2 are plotted against life.
In this application, Eq 1 takes the form:
( A s 9 A e " E ) 1/2
Kt = (la)
( A S . Ae g 9 E ) 1/2
where As and Ae are ranges of smooth specimen stress and strain.
Here the assumption that smooth and notched specimens form cracks at the same life for equal deformations at the critical crack initiation location is invoked.
Consequently, intersecting both curves with a vertical line, we should ob- tain corresponding nominal and notch root values of the ordinate for the
140 CYCLIC STRESS-STRAIN BEAVIOR
notched plate. However, the actual value of the right hand side of Eq la usually is somewhat less than K t , and Eq la is replaced by
(As 9 Ae 9 E) 1/2
(AS - Ae N . E) 1/2 (lb)
where K f is an experimentally determined variable defined by Eq lb and termed herein "the fatigue notch factor."
We may conveniently compare our measured notch root deformations with those derived from fatigue data in terms of a parameter P.
(As 9 A e " E ) 1/2
P = K t ( A S - Ae N 9 E) 1/2 (2)
Substituting from Eq 1 for K t in Eq 2 and simplifying gives
_ K f (As 9 A e " E ) 1/2
e - =
K t (Ao 9 Ae 9 E) 1/2 (2a)
Equation la indicates that if the N e u b e r deformation theory is accurate, P will have a value o f one, while Eq lb indicates that P determined from fatigue results in the m a n n e r suggested above will have a value K f / K t . Thus, if the value of K f / K t determined from fatigue results and the value of P determined from deformation studies are coincident, then the deforma- tion analysis rather than a material size effect is responsible for values of K f less than K t .
E x p e r i m e n t a l R e s u l t s
Before proceeding to examine the results of this study, it is worthwhile to evaluate the accuracy o f the experimental techniques on which the results depend.
S t r a i n Sensing E l e m e n t s - - W h i l e some of the problems inherent in the use of strain gages result in gage response deviating from actual strain response, the techniques employed suppress or correct these variations so that errors do not exceed 10 percent (see Ref 7, Appendix 1 and R e f 22).
E f f e c t s o f B i a x i a l S t r a i n S t a t e in the Transverse N e t S e c t i o n - - I n deter- mining surface stress distributions, the procedure outlined in the analysis section assumes the existence o f a uniaxial strain state in the plate where a multiaxial state is known to exist. T o estimate the magnitude of this error,
LEIS ET AL O N D E F O R M A T I O N A N D THE FATIGUE N O T C H F A C T O R 141
the same assumption is made in determining the stress distribution at ap- propriate load levels for the results presented in R e f 21. A comparison of this stress distribution with that obtained from an iterative solution of the Prandtl-Reuss stress-strain relations [21] is presented in Fig. 4 and shows less than 7 percent error when the effect o f biaxial strain is neglected.
Biaxial Stress State at the Notch Root--The effect of biaxial stresses at notch roots due to thickness effect are considered small for the geometries investigated and hence are neglected.
30-0[ DATA FROM REF. 2 1 , K t =2.56
/
/ - - - - DISTRIBUTION INCLUDING
/ BIAXlAL INFLUENCE
/ . . . . PREDICTED FROM STRESS
/ STRAIN RESPONSE
o x = 0 - 4 0
2 o o 0 5 0
I V K " , o 0-60
.q ' --
I 0 . 0 - 1-- 03
o.oo I 1 I I I
I-0 2.0 3.0 4.0 5.0 6.0 7.0
x DISTANCE
~ - , DISTANCE FROM CENTER, NOTCH RADIUS
FIG. 4--Comparison o f the derived stress distribution with that calculated considering biaxial strains.
Equilibrium Check on Calculated Stress Distribution--Comparison o f measured nominal stress (from load cell records) with the stress obtained from an integration o f the calculated stress distribution across the net sec- tion o f the plate, in no case indicates errors greater than 10 percent. Since load cell calibration is accurate within 1 percent, most o f this error must be attributed to the strain sensing elements.
With these considerations in mind, the results are now presented.
Differential transient deformation response across the elliptical notched
142 CYCLIC STRESS-STRAIN BEHAVIOR
aluminum plates is portrayed in terms of surface stress and strain dis- tributions and shown in Fig. 5, while the transient variation in cyclic con-
2 0
~ , 5 2
~ , o
z
Eo5
O3
0 0 _
s
7OO
. 6 0 0 bJ c~
5 0 0
co 4 0 0 03 bJ c~ 3 0 0
2 0 2 4 T351 ALUMINUM
AeN 4
K t ~ 2 6 8 T = 0 3 5 %
0 FO
i 1
0 5 0 ~ O 0
2024 T351 A L U M I N U M
o 5
0 Io
0.50 I O0
DISTANCE FROM C E N T E R , INCHES 50
t 0 ~e N
K t = 4 6 0 T ; o 2o*/,
0 8 o N = I
0 6 v 2
0 I0
'~ 0 4 @ I00
Z
0 2 o 0
0 0 % 0150 iioo
6 0 0
~ 5 0 0
i.~40 o
< 3 0 0
bJ
~ 2 0 0 I 0 0
D I S T A N C E FROM CENTER, INCHES 2024 T351 ALUMINUM
= ~ : 0 20%
0 N= I
0 I0
IO0
I
0150 1.00
5b
F I G . 5--Stress and strain distributions for notched aluminum plates.
centration factors is shown in Fig. 6. Figure 7a shows stable notch root strain as a function of smooth specimen strain for equal life to crack initia- tion while Fig. 10 presents the concentration factors shown in Fig. 6 in terms of the Neuber parameter defined in the analysis section. Typical results for the remainder of the notch plates are shown in Figs. 7, 8, 9, and 10. Strain controlled smooth specimen response is shown in Fig. 1 for the aluminum alloy and Fig. 2 for the mild steel.
Discussion Aluminum Plates
Material Deformation Response--The cyclic stress-strain response for the aluminum alloy is presented in the form of stress amplitude-strain amplitude curves in Fig. 1. This figure reveals that elastic response at strain
LEIS ET AL O N DEFORMATION A N D THE FATIGUE NOTCH FACTOR 143 b l 0
~"9
8
0 7
z 6
o
r r Z
U z
8
-A ( J
2024 T351 ALUMINUM, K t--4"60 K s
_ - Ko- 0 - ~ = 0 " 1 0 0 %
O ; O. 200%
= 0 . 2 2 2 % - - ~ - - ' - I ] ~ ' - ~ - M : ] . . ~ ~ = 0 - 2 8 1 %
~ " --U--I-II~ t] = 0 . 3 4 5 % _ ~ - - -'~"~-~,~..A___~. m___A. ___A_.
4 (
3 _ zS~z__:_,~,Tz~ = ~ ; _ _ - - ~ , _ ~ - -._.-~__.. _-.--~..--e~ ~ -~v---
2 : ~ T . ~ .. - - - - E r - " --'-Er" " l ' - - - "
I I I
0 I
l0 D 102
CYCLES, N
-o - o
F I G . 6--Concentration factors f o r aluminum plates with K t = 4.60.
amplitudes below 0.5 percent is followed by hardening which becomes more rapid with increasing strain level. Stable material response which follows the hardening period is achieved at rapidly decreasing cycle numbers as strain is increased.
Strain and Stress Distributions--Typical strain and stress distributions in aluminum plates are given in Fig. 5. Strains change rapidly at first, then stabilize: the strains in the region nearest the notch root decrease, while those a little further out increase slightly, and the remainder show no measurable change. Examination of material stress response (Fig. 1) at the measured strain levels indicates that the period in which transient changes in strain distributions are observed coincides with the interval in which tran- sient material hardening is expected.
The effect of geometry on strain distribution becomes evident if dis- tributions for elliptically and circularly notched plates are compared on the basis of equal notch root strains. Strain gradients are much sharper and zones of inelastic strains much more restricted for the sharper elliptical notch.
The stress distributions of Fig. 5 which were obtained from measured plate strains and the transient cyclic stress-strain curves of Fig. 1 show an in- crease in stresses adjacent to the notch root coupled with a decrease in the immediately adjacent region while the remainder of the plate which is at elastic strain levels shows no change. Stable stress distributions, available