m "^
TOHNISCHE HOGFSfHOOI
T H E C O L L E G E O F A E R O N A U T I C S
C R A N F I E L D
AN IMPROVED LAW OF CUMULATIVE DAMAGE
IN METAL FATIGUE
by
August. 1961
THE C O L L E G E OF AERONAUTICS
CRANFIELD
An Improved Law of Cumulative Damage in Metal Fatigue b y
-S. R. Swanson, B . A . S c , D . C . A e .
SUMMARY
After a brief account of t h e o r i e s c u r r e n t l y put forward on damage accumulation, a modification of the best of t h e s e , the linear rule of damage accumulation, i s p r e s e n t e d . This theory is based on a new approach to the problem wherein the m a t e r i a l behaviour in each of t h r e e different s t r e s s regions is analysed,in contrast to c u r r e n t approaches which endeavour to fit fatigue behaviour over the whole s t r e s s range into a single fundamental relation.
In the t h r e e - r e g i o n approach, evidence is given to show that the linear rule is effective in the intermediate s t r e s s levels of the S - log N relation, and that this region a p p e a r s linear due to an exponential crack initiation theory, a s opposed to present power function t h e o r i e s , which i s based on p r i o r history effects in plastic yielding. Non-linear damage activity Is seen to modify this fundamental behaviour outside this s t r e s s region.
To substantiate these t h e o r i e s , an experimental p r o g r a m m e was undertaken using plain L.65 b a r s p e c i m e n s . After establishing the S - N relation, a ' t r a v e r s e ' of the alternating s t r e s s range was made, cycling specimens at eight ' p r e s t r e s s ' l e v e l s , each conforming in size to an envelope spectrum for a projected a i r l i n e r , and then cycled at a datum level in the i n t e r m e d i a t e - s t r e s s region. F r o m these r e s u l t s , the non-linear functions were e s t i m a t e d , and the final modified linear law developed. Specimen life under the spectrum was then calculated and compared with unmodified linear rule calculations.
The r e s u l t s of the experimental t e s t s , the background of the statistical methods used in assembling the data, and an a s s e s s m e n t of the relative m e r i t s of three fillet designs a r e presented in Appendices.
Page Summary
Notation
Introduction 1 Existing Theories of Cumulative Damage 1
Improving the Linear Rule 7
Load Spectrum 15 Simplification of the Linear Rule 23
References 29 Appendix 1 - Experimental Test Data 33
Appendix II - The Statistical Treatment
of Test Data 52 Appendix III - Photoelastlc Investigation
of Three Fillet Shapes 55 Table 1 - Basic Gust Spectrum by Difference 57
Table 2 - Reduction of Data from Partial
Damage Tests 58 Figures
n Number of cycles experienced at a given level of s t r e s s N Number of cycles to failure, at a given level of s t r e s s
R Ratio of cycles experienced, to cycles to failure, at a given level of s t r e s s o r load, called the cycle ratio
S Alternating s t r e s s , o r s t r e s s amplitude used in a fatigue cycle. When the word " s t r e s s " i s used alone, it denotes alternating s t r e s s
S Mean s t r e s s of the fatigue cycle
S S , The maximum, minimum value of s t r e s s in a fatigue cycle m a x , min * •' D The damage quantity, a function of cycle ratio R defined further
where it a p p e a r s
Ig Unaccelerated level flight condition where unit acceleration (32.2 feet/ s e c . / s e c . ) exists everywhere in the region concerned
U Gust velocity in feet p e r second equivalent airspeed (Br, definition) F Cycle-based factor used in life a s s e s s m e n t calculations
h Crack depth
E Endurance limit, (in t e r m s of s t r e s s amplitude)
y Gust velocity function for z e r o log cycles; i . e . for one cycle of fatigue a The relative a i r density
b , c , C , k , m , A , K Constants used in this work
S-N o r SN The general relation between alternating s t r e s s level and endurance for a m a t e r i a l tested under constant amplitude fatigue cycles
P a r t i a l Damage - Two-level constant amplitude fatigue t e s t s where the first s t r e s s level is run continuously to its completion, and the specimen tested to destruction at the second level of alternating s t r e s s Element - A single non-redundant s t r u c t u r a l m e m b e r which fails when it is
fractured in fatigue, a s opposed to a redundant s t r u c t u r e which is only weakened when one of its elements fails. This is the only type of s t r u c t u r e having a fatigue "life" in the usual sense
The Static Axis - The i cycle line on a n o r m a l plot of the S-N relation
In g e n e r a l , where special notation is used in a single p a r t i c u l a r section of the r e p o r t , the t e r m s a r e defined in that section a s they a r e used.
1. Introduction
The simple linear law of cumulative damage in fatigue fails to take account of healing and damaging effects. Various investigators have introduced m o r e complex relationships in an attempt to include such effects, but unfortunately the r e s u l t s a r e no m o r e r e p r e s e n t a t i v e of what happens to an aircraft in s e r v i c e than a r e those of the linear law.
In this work, a new approach has been made in that the behaviour of the m a t e r i a l under repeated s t r e s s is shown to fall into three different s t r e s s r e g i o n s . In the
intermediate s t r e s s region, it a p p e a r s that cumulative damage does follow a linear law. Outside this intermediate region, non-linear damage modifies the fundamental behaviour, and from an analysis of experimental r e s u l t s the nonlinear functions have been e s t i -mated. The modified linear law is then applied to the determination of the fatigue behaviour of a p a r t i c u l a r aircraft project.
2. Existing Theories of Cumulative Damage The Linear Cumulative Damage Law
This simple, widely known and used hypothesis owes its origins to a number of i n v e s t i g a t o r s . Miner, however, i s probably chiefly responsible for bringing it to the attention of the aeronautical world.
The r u l e , a s he stated it in Ref. 1, i s :
"Damage could be expressed in t e r m s of the number of cycles applied, divided by the number to produce failure at a given s t r e s s level. When the summation of these increments of damage at several s t r e s s levels became unity, failure o c c u r r e d " .
Thus, if W is the net work absorbed at failure at N^ cycles by a specimen loaded sinusoidally at constant amplitude at a given s t r e s s level S , and if w is the work done in n, cycles, then, assuming work is absorbed ( i . e . damage is accumulated) at a rate linearly proportional to the number of cycles applied:
w n
w; =
N7
(This law will henceforth be called the linear rule due to its linear relation with cycle ratio).
If damage accumulation is independent of s t r e s s level, then summing over the s t r e s s levels:
If no work hardening o c c u r s , let the total work that can be absorbed by the specimen be tantamount to failure.
I . e . 2 ^ W 1 at failure
Thus Z — = 1 denotes failure of the part
This rule can be evolved, without the work assumptions, m e r e l y from the s t a t e m e n t s :
(1) Damage depends upon a single p a r a m e t e r , the cycle ratio
(2) The fatigue p r o c e s s is relatively the same at every s t r e s s level, a s has been done in Ref. 2.
By summing i n c r e m e n t s of t r a v e l along two affine fatigue crack growth c u r v e s , F . R.Shanley in Ref. 3 a r r i v e d at the linear rule in an indirect manner. The d e s t i n -ation of his t r a i n of thought was inevitable, however, for while he attached great Importance to the p a r t i c u l a r form of his crack growth curve, it can be shown that it is almost i m m a t e r i a l what form the curve may t a k e . As long a s the curves a r e non-dimensionally Invariant with s t r e s s , ( i . e . affine), and proportional in some way to the cycle r a t i o (as this work is) the linear rule will follow. The following figure shows how the accumulation of damage is proposed to take place:
A s t r e s s S, is applied for 6 n, cycles, after which s t r e s s S^ is applied for A n^ cycles. This load spectrum is repeated until failure - denoted by the attainment of an a r b i t r a r y crack depth - o c c u r s at N c y c l e s .
Arbitrary
Dapth FIG. A.
An, CycUt
The actual form of the final relation is given in t e r m s of a reduced s t r e s s , S r x •
Z An s I I An
where x is the r e c i p r o c a l of the slope of the best straight line*through the*S-N curve when plotted on log s t r e s s - log cycles paper.
Deficiencies
An examination of a great number of r e p o r t s r e v e a l s the following four main deficiencies of the beautifully simple linear r u l e .
(1) The r a t e of damage accumulation i s not a simple linear function of cycle ratio, but a complex relation e x i s t s which is dependent on s t r e s s level, sequence of loads, and type of loading.
(2) P r e v i o u s h i s t o r y h a s an enormous effect on the summation. This is not accounted for, and values have ranged "from 0.18 to 23.0 with only a small portion giving the assumed value of unity". (Ref. 4).
(3) T h e r e is no provision for beneficial effects a s can be obtained from certain applications of single preloads (Ref. 5) o r of large numbers of ultra-low s t r e s s level cycles which have actually improved on the single-level lives in many studies, including the p r e s e n t one.
(4) As the rule is dependent on cycle ratio a s obtained from the S-N c u r v e , those s t r e s s levels below the level at which the SN curve flattens out, a r e i n adequately r e p r e s e n t e d . As D r . G a s s n e r points out in Ref. 6, "In c o n t r a -distinction to the view expressed in British l i t e r a t u r e , (these levels) play an active part in the formation of fatigue f r a c t u r e s " . Many r e s e a r c h e r s , such a s Ref. 7, have shown v e r y appreciable differences in lives obtained by
including and then excluding low level cycles which predominate in numbers in the gust t r a c e s taken from actual flights.
An interesting and quite popular notion has grown out of the lack of appreciation of the final point, in the idea of "most damaging s t r e s s l e v e l s " .
The mixture of both healing and damaging effects is evident from the fact that averaging of many summation values over, say, the s t r e s s range often r e s u l t s in an answer close to unity. This s e e m s to indicate that the linear law is truly part of the s t o r y , but that there exist non-linear effects which a r e not accounted for.
To conclude, it has become increasingly evident that the widely used linear theory does not provide a sufficiently close approximation to the r e a l r a t h e r complex and probably non-monotonic p r o g r e s s of fatigue damage under varying s t r e s s amplitudes. How could it, when it is defined by only a single p a r a m e t e r like the cycle ratio ? Cumulative Damage by E m p i r i c a l Two-Level Relations
A great number of investigators (Refs. 8 - 1 2 ) have tried to c l e a r up the m y s t e r y of damage accumulation by studying carefully the effect of a given s t r e s s level upon a second level, by partially running the specimen at constant load amplitude at the first level and then running the specimen to failure at the second level. By defining
100% damage a s given by the utilization of unity in total cycle r a t i o , other quantities of damage can be found by failure at other values of cycle r a t i o , from relation of the
example, a specimen tested to 0.5 cycle ratio at the first s t r e s s level, which then fails half-way to the single-level cycles-to-failure at the second level, h a s endured a total of 0.75 cycle ratio and accumulated 0.50 of the nominal damage.
Some examples of the general r e s u l t s a r e given in the following diagram :
l O
'°-(^V»,+ÖS-),^«;
(")»...example of 3 4 S - T 3 «hcct from R « f 3 - I 2 near Endurance limit
2 " " 2
FIG. B.
In general the interaction of two levels r e s u l t s in a power relation between the fraction of cycle r a t i o used and the fraction of nominal (single level) damage obtained. One such relation is given by the finite power s e r i e s :
N 1
for two level t e s t s with L . 6 5 from Ref. 11. H e r e , the questionable ordinate " P e r c e n t Damage" was replaced by the m o r e honest label "Subsequent cycle ratio for F a i l u r e " . D . L . H e n r e y in Ref. 13 has been able to develop this approach to adequately account
for t w o - l e v e l behaviour of the endurance l i m i t with s t e e l .
T h i s approach a v o i d s the use of affine c u r v e s but instead u s e s a questionable quantity c a l l e d P e r c e n t D a m a g e to obtain non-affine c u r v e s from which to form a cumulative d a m a g e s u m m a t i o n technique b a s e d on s u m m i n g the p r o j e c t i o n s of t h e s e d a m a g e c u r v e s . Unfortunately t h i s method i s r e s t r i c t e d to the p a r t i c u l a r s t r e s s l e v e l s and c y c l e r a t i o s i n v e s t i g a t e d a s t h e r e s e e m to be no o b s e r v a t i o n s thus far of a g e n e r a l nature.
A s t h i s method i s an e m p i r i c a l study of the i n t e r a c t i o n s of s t r e s s l e v e l s . a prohibitive amount of t e s t i n g i s n e c e s s a r y to c o v e r all the c o m b i n a t i o n s of s t r e s s l e v e l s that w i l l be m e t in s e r v i c e . The m a j o r difficulty, h o w e v e r , i s that t w o - l e v e l fatigue i s a l m o s t n e v e r m e t with in s e r v i c e . Studies r e v e a l that t h e r e i s often no c o r r e l a t i o n between t w o - s t e p t e s t s of t h i s t y p e , w h e r e all the c y c l e s at the f i r s t l e v e l a r e c o m p l e t e d b e f o r e m o v i n g to the s e c o n d l e v e l , and t w o - l e v e l programime t e s t s , w h e r e the l e v e l s a r e alternated but with the s a m e equivalent c y c l e r a t i o s at f a i l u r e . T h i s t w o - l e v e l approach, like the o t h e r s d i s c u s s e d s o f a r , actually r e l i e s on a s i n g l e p a r a m e t e r , the r e m a i n i n g endurance at the second s t r e s s l e v e l . It i s t h e r e f o r e I m p o s s i b l e to define contributions f r o m one l e v e l to another at s t r e s s e s below the endurance l i m i t e v e n though they have b e e n proved to contribute to f a i l u r e in m u l t i -l e v e -l t e s t s . Extending t h i s approach to the p r a c t i c a -l c a s e s of many s t r e s s -l e v e -l s i s an a l m o s t i m p o s s i b l e c o m p l i c a t i o n . Even e x t e n s i o n to three s t r e s s l e v e l s r e s u l t s in f a m i l i e s of i r r e g u l a r s u r f a c e s in a t h r e e - d i m e n s i o n a l d a m a g e plot (Ref. 12). In s h o r t , t h i s approach t e n d s to be d e s c r i p t i v e , but not v e r y explanatory in nature.
Cumulative D a m a g e by P o w e r - W e i g h t i n g the S t r e s s L e v e l s
(a) When P r o f e s s o r F . R. Shanley o b s e r v e d that the affine c u r v e r e l a t i o n obtained the l i n e a r r u l e , he studied t e s t r e s u l t s and concluded that the l i n e a r rule tended to u n d e r -e s t i m a t -e th-e d a m a g -e contribution at high s t r -e s s -e s . R -e - -e x a m i n i n g th-e -e q u a t i o n s , h-e obtained a new rule c a l l e d the 2 x m e t h o d , which m e r e l y p o w e r w e i g h t s the c o n t r i b u -t i o n s a c c o r d i n g -to s -t r e s s l e v e l in a uniform m a n n e r . T h i s h a s been found -to be -too s e v e r e a 'solution' and at any rate i t s s o u r c e i s quite o b s c u r e , for Schijve w r i t e s in Ref. 14:
"The d e r i v a t i o n given in (Ref. 3) i s i n c o m p l e t e and it m u s t be concluded that it e i t h e r contains an e r r o r , or i s based on s o m e additional a s s u m p t i o n which has not been mentioned".
An a l g e b r a i c check by the author using the r e f e r e n c e d i r e c t l y c o n f i r m s this
viewpoint. The final form of the 2 x equation i s a l s o given in t e r m s of a reduced s t r e s s S s i m i l a r to the affine r e l a t i o n . r 1_ 2x S r 2x EAn s EAn
F r o m a detailed study of damage behaviour, it appears that overall c o r r e c t i v e m e a s u r e s such a s this add little to our knowledge of fatigue. A consideration of the deficiencies of the linear rule shows that such crude and severe m e a s u r e s only tend to obscure the actions of the various contributory effects.
(b) Marco and Starkey in Ref. 10 studied two-level fatigue r e s u l t s using 75 S-T
Aluminium Alloy. They concluded that the dependence of damage on s t r e s s level could be given by the cycle r a t i o , a s in the linear r u l e , above a certain c r i t i c a l s t r e s s level marking the change in fatigue crack formation from a single crack at low s t r e s s levels to many c r a c k s at high s t r e s s l e v e l s , and by the square of the cycle ratio below this c r i t i c a l s t r e s s . This c r i t i c a l s t r e s s level lies in the intermediate range of the S-N curve for this m a t e r i a l . P r o f e s s o r Miles, in Ref. 15, studied random loading using this c r i t e r i o n , to compare it with linear rule r e s u l t s , and found relatively small
changes "compared with experimental s c a t t e r " . He considers therefore that the l i n e a r rule is adequate for random loading. T h i s , however, can be generally refuted. This r u l e , then, s e e m s to be inadequate to account for the deficiencies of the linear m e r e l y by dividing the whole S-N curve into two p a r t s . It does, however, r e p r e s e n t a slight improvement over the Shanley 2 x method in that it recognizes that the damage function i s not uniform in i t s dependence on the s t r e s s level.
Cumulative Damage with S t r e s s Interaction F a c t o r s
P r o f e s s o r A. M. Freudenthal, in Ref. 16, has developed a generalized damage relation based on the summation of c y c l e - r a t i o functions raised to a power which v a r i e s in a general way with s t r e s s level. F r a c t u r e is defined in the usual way; i . e . when the summation i s unity.
He then extends this to include s t r e s s interaction effects. This can only be done, of c o u r s e , if the p r o c e s s is truly random a s the damage is a non-linear function depending on s t r e s s level, and any well-defined sequence will give its own particular r e s u l t . He then introduces a factor which wiUbe a function of s t r e s s levels and of the relative frequency of s t r e s s which i s then applied to the general equation. He suggests, from experimental evidence, that the interaction factor will be n e a r unity for high s t r e s s e s , and n e a r ten for s t r e s s e s close to the endurance limit. Because of these powerful sequential effects, he s t a t e s "that s t r e s s interaction effects of such magnitude reduce to p r a c t i c a l insignificance the possible effects of non-linearity of the damage r a t e " .
The approach adopted by P r o f e s s o r Freudenthal is undoubtedly the most promising yet encountered, for it deals adequately with non-linear damage accumulation, and perhaps adequately with sequence effects. However, it suffers from a lack of simplicity which may be remedied l a t e r when m o r e random test r e s u l t s a r e available.
It is c l e a r , t h e r e f o r e , from these alternative theories that the linear rule is very useful a s it i s , and the author is convinced that no radical alternative will be found. While tne four deficiencies of the linear rule must be adequately remedied, everything should be done to retain a s much a s possible of its simplicity.
3. Improving the Linear Rule C h a r a c t e r i s t i c s of the S-N Relation
In Ref. 17, Weibull has shown that, under certain conditions, the whole S-N relation plotted on a semilog b a s i s can be transformed into a single straight line on a plot of s t r e s s v e r s u s (cycles)"*^ i . e . a r e c i p r o c a l plot of cycles. This o c c u r s when the p a r a m e t e r s match in the equations.
N = k(S - E)""^ for the low s t r e s s levels (1) N + B = KS for high s t r e s s cycles (2) where B is a p a r a m e t e r which must be taken into account when dealing with the high
s t r e s s region of the S-N curve. Lundberg in Ref. 18, omits this p a r a m e t e r and combines the equations (1) and (2) in the form
N = o(S - E ) " ^ (3) where a and ^ a r e constants, stating that :
" F o r the fatigue of aircraft s t r u c t u r e s due to gust loads, the damage caused by the high loads corresponding to the lefthand part of the endurance curve is usually negligibly s m a l l , and then the s i m p l e r form of the equation can be used".
Equation (3) can be expressed in the form:
S = b N " ^ + E (4) While the above argument may be t r u e where s t r e s s concentrations a r e b e n e
-ficially affected by high l o a d s , it has not been found t r u e with plain s p e c i m e n s , a s the test r e s u l t s of Appendix I show. On the c o n t r a r y , most r e s e a r c h so far confirms that high loads play a v e r y significant part in the accumulation of fatigue damage.
The above simplification has s e r i o u s shortcomings.
(1) The matching of p a r a m e t e r s is quite r a r e . Weibull states that a s a rule the two curves a r e not amenable to such 'integration'. The author has c a r r i e d out this plotting for the test data and a narrow S-N curve r e s u l t s .
(2) T h e r e is nothing to account for the effects which contribute to damage accumulation below the endurance l i m i t .
(3) T h e r e is evidence that crack initiation follows an exponential law and not a re ci proc a l law. While this does not appear significant at first sight, it will be seen shortly that the crack growth equation yields the fundamental relation
There is always a certain amount of a r b i t r a r y smoothing out of the curve joining endurance means at different s t r e s s levels. With this latitude in drawing S-N c u r v e s , the assumption has tacitly grown over the y e a r s that the S-N relation is essentially made up of two p a r t s , as given in the equations (1) and (2), which join at an inflection point at an intermediate s t r e s s level. In fact, life a s s e s s m e n t formulae have been built upon an S-N equation fitting the lower half of the curve only, coupled with the linear rule (Ref. 18).
One of the most common complaints has been that the linear rule underestimates damage effects at high s t r e s s levels. This complaint and the deficiencies of the linear rule enumerated in the previous section have led the author to suspect the above
assumption, and to examine this aspect of endurance curves in detail. Choice of Plot
Some workers argue that the S-N relation should be plotted linearly in both directions. This gives the t r u e s t picture of the actual nature of the curve, with the enormous increase in endurance and in scatter that takes place at low s t r e s s levels. The plot is not satisfactory, for to be practical the endurance scale must be condensed and the behaviour at s t r e s s e s above the endurance limit is then difficult to a s s e s s .
The best plot for the S-N curve is that which makes the fundamental p r o c e s s stand out clearly, and in a manner amenable to study. Toward this end, P r o f e s s o r F . R. Shanley, in Ref. 3, suggests the log - log plot which also appears to resolve the S-N curve into a single straight line, and he has drawn attention to fatigue crack initiation as the process to be made amenable to study. He has developed an exponential crack growth equation which leads to a linear relation for the log - log plot. This then leads to a damage accumulation law which would apply to the whole S-N relation, since many data make it appear linear in this plot. Unfortunately his relation has been shown to be an indirect form of the linear r u l e , and no p r o g r e s s seems possible past this point at present.
The semilog plot, on the other hand, appears promising to the author for the following r e a s o n s :
(a) Exponential crack growth equations can lead to a semilog plot of s t r e s s - l o g cycles, as well as the log stress-log cycles plot, as will be shown shortly.
(b) There is evidence pointing to a division of the S-N relation into three a r e a s of behaviour wherein the middle region has 'ideal' c h a r a c t e r i s t i c s . With the concept of three a r e a s in mind, nearly all the S^SJ data can be plotted in three sections of which the centre section is linear, in a semilog plot.
(c) The relative frequency of random forces can be accurately represented by a straight line on a semilog plot. A really significant gain in simplicity should then be possible if the fundamental behaviour of fatigue also consists of a straight line relation in this plot.
(d) F r o m a purely p r a c t i c a l standpoint, the vast majority of S-N data a r e plotted on this form of s c a l e .
The following diagram shows the notation which will be used to identify the t h r e e a r e a s of behaviour: stress Amplitude or Gust Velocity Cycles
Region (1): This region, of high alternating s t r e s s e s , is characterized by the convex-upwards section of the curve. Reports on plain specimens seem to agree that this is an a r e a where the r a t e of change of damage activity with s t r e s s is s e v e r e and n o n - l i n e a r . Multiplicity of crack nuclei and other c h a r a c t e r i s t i c s of high s t r e s s level fatigue fracture all point to the accumulation of damage in a s e v e r e m a n n e r . F r o m the c h a r a c t e r i s t i c gradual curving of the S-N relation it a p p e a r s that the non-linear effect i s a gradual one i n c r e a s i n g in frequency with s t r e s s level, until nearly all specimens fail at v e r y low endurance just below the static ultimate. This region will be called the region of non-linear damage.
Region (2): This region c o m p r i s e s that part of the finite-endurance S - Log N relation in which the t e s t data appear to fall in a straight line. This is the intermediate s t r e s s level region commonly considered the region around the point of inflection in p r e s e n t - d a y fatigue r e s e a r c h . This region i s characterized then by a constant change in the damage r a t e with s t r e s s level, and will be called the linear region.
Region (3): This l o w - s t r e s s level region i s characterized by a slackening off of the damage activity in which a random healing effect allows m o r e and m o r e specimens to withstand the fatigue indefinitely, a s the s t r e s s level is lowered. This region will be called the healing region.
Evidence from Two Level Damage T e s t s
The figure above shows what a r e perhaps the only general features of the r e s u l t s from two-level damage t e s t s .
(a) If S > S , damage, defined in degree by the remaining endurance at S^, proceeds rapidly at f i r s t , and then slowly as the cycle ratio i n c r e a s e s . The damage function thus follows a curve of the family D = R^ where x < 1, and the g r e a t e r S^ is in
comparison with S^ the further the resulting curve of r e s u l t s deviates from the straight line given by the straight line D = R. ( i . e . x d e c r e a s e s toward z e r o ) .
(b) If S < S damage proceeds slowly, then rapidly a s cycle ratio i n c r e a s e s . The damage function follows a curve of the family D = R''*' where x > 1 and the lower S, is
compared with S^ the further the curve moves from the straight line D = R (and x i n c r e a s e s rapidly). Also in n e a r l y all c a s e s where a beneficial healing effect is encountered (i. e. where the run at S^ actually improves on the single level endurance at S ), the effect is a deviation of this curve to points below the D = 0 a x i s . (Ref. 4). (c) If S^ 7 Sj the damage relation follows the straight line D = R and t h e r e is excellent agreement with the l i n e a r rule of damage accumulation.
The s t r e s s level S , in most w o r k s , l i e s in Region (2) which is fortuitous, a s will be seen l a t e r , but quite understandable a s this is the region of most clearly-defined e n d u r a n c e s .
Evidence from Statistical Behaviour
F I G . D.
Weibull in Ref. 19 has shown how the statistical distribution of S-N test points cycle-wise changes with s t r e s s level
|ncr««iA9 «tcftt ltv«l
FIG. E.
(a) The curve (1) is seen to deviate from normal distribution in a manner suggesting that the distribution i s being affected in the low probability of failure region so a s to r e s u l t in a distribution skewed thus:
FIG. F.
(b) Curve (2), almost l i n e a r , r e p r e s e n t s the intermediate s t r e s s levels and p r e s e n t s an ideal n o r m a l population in most c a s e s .
FIG.G.
(c) Cuirve (3) deviates due to effects in the high failure probability region, resulting in a distribution skewed, and then truncated at s t r e s s e s n e a r the so-called endurance l i m i t .
FIG. H.
In line with the c u r r e n t two-region "inflection point" concept, Weibull s t a t e s , "The t e s t points may follow a straight line r a t h e r closely
with some specific v e r y narrow range of test level, giving the Impression of log-normal distribution. It is however Impossible to fit the same distribution to data from significantly different test l e v e l s " .
The statistical study thus emphasizes the complementary nature of the p r o c e s s e s in regions (1) and (3). Rather than replacing the fundamental p r o c e s s exhibited in region (2), the secondary p r o c e s s e s m e r e l y act in addition and finally swamp the fundamental p r o c e s s altogether at extreme values of s t r e s s .
Evidence from Physical Behaviour
A survey of a large amount of literature reveals the following interesting division of physical phenomena.
(a) High s t r e s s level fractures a r e characterized by a multiplicity of crack nuclei, i . e . something is accelerating the crack initiation p r o c e s s .
(b) Intermediate (and low s t r e s s levels in the finite endurance range) fatigue fractures are characterized by the propagation of a few cracks and general crack initiation is unaccelerated.
(c) Below the endurance limit, crack propagation has been wholly counterbalanced by some mechanism, but there is firm evidence that fatigue p r o c e s s e s are still at work in this region as shown by their contributory effects on finite endurances. P e r h a p s the p r o c e s s of crack initiation still e x i s t s .
The Crack Initiation P r o c e s s
There a r e two distinct phases in the growth of a fatigue crack. They a r e : 1. The crack initiation phase, - this is the development of a microcrack; i . e . the
growth of a crack of microscopic dimensions. This phase has been proved to exist from almost the first cycle of fatigue, and in most types of fatigue t e s t , it prevails for about 95% of the endurance. This stage is apparently completed when the crack a s s u m e s visual dimensions.
2. The crack propagation period. This phase covers the microscopic growth of the crack past the 'localized' phase, as its growth is now governed by the effect of its presence on the s t r u c t u r e .
Constant-load testing of small non-redundant elements greatly a c c e l e r a t e s this second phase, as has been pointed out by Weibull in Ref. 20. In fact, visual o b s e r v -ations by the author during the t e s t s currently made with plain specimens has shown that at intermediate or high s t r e s s levels, the second phase can be neglected in
conventional testing. The attainnaent of the crack length corresponding to the completior of the first phase may then be taken as the criterion of failure in these c a s e s . This is the most important assumption in the development of the crack theory for region (2).
In this section, the first phase is called 'crack growth', while crack propagation will only refer to the second phase. While the first phase is considered to be of fundamental importance in cumulative damage theory, the second is of tremendous practical importance in aircraft s t r u c t u r e s , for redundant s t r u c t u r e s of large c r o s s -sectional a r e a fail under the action of crack propagation. Ref. 21 emphasizes, however, the distinction which must be made between the two phases.
Because of the overshadowing economical importance of its brother, crack initiation has received very little attention, even though it is probably the key to
damage accumulation. If this is t r u e , it can be readily seen how the second stage would tend to obscure any study of cumulative damage on composite s t r u c t u r e s . Possible Crack Growth Relations
In 1946, J . A. Bennett obtained m e a s u r e m e n t s of crack growth a s given by the a r c length of a m i c r o c r a c k on a specimen under rotating bending fatigue, using a stroboscope and a graduated-lens m i c r o s c o p e . F r o m this he obtained the relation (Ref. 8).
log (L - C) = 7 N (5) i . e . h = Ae'''^ (6) where L is the circumferential length of the crack; C and 7 a r e constants, being
dependent on the s t r e s s .
This then gives the equation, from experiment, for crack growth. Its exponential nature satisfies all the fundamental c h a r a c t e r i s t i c s , for these m i c r o c r a c k s have often been reported to:
(a) exist from a v e r y e a r l y point in the life of the specimen (b) to grow exceedingly slowly for a v e r y long period, and
(c) then to rapidly grow at a r a t e which s e e m s to i n c r e a s e with the depth of crack attained.
If the attainment of a crack depth of a certain magnitude h is equivalent to failure, the relation then becomes
h = Ae'>'^ (7) o
o r log h = log A + 7N
7N = log h - log A = K^ , a constant
and 7 = _ i (8) N
This t e r m 7 is the only p a r a m e t e r dependent on s t r e s s according to Ref. 8 and 3. Therefore its form is of fundamental importance in the development of a cumulative damage relation between s t r e s s levels.
A Power Function as the P a r a m e t e r of S t r e s s
To obtain a function to describe this s t r e s s p a r a m e t e r , P r o f e s s o r Shanley in Ref. 3 considers that the relation is connected in some way with plastic behaviour in a static tensile t e s t . A power function has been derived to fit the plastic strain part
of this curve for aluminium. This was done by W. Ramberg and W. R. Osgood in describing s t r e s s - s t r a i n curves by t h r e e p a r a m e t e r s . (Ref. 22). Shanley took this function in its general form for the p a r a m e t e r
7 = cs^ (9)
where x is obtained from the s t r e s s - s t r a i n curve, fitting in the plastic range.
'^'^"^ s^ = ^ <10) N
which is a linear relation on log-log paper. It should be added that this expression also follows for a fatigue equation based on a constant rate of crack growth:
h = As^N (11) o
which is certainly not c h a r a c t e r i s t i c of crack growth. Shanley then applied this to the whole of the S-N relation without regard to any corrections at different s t r e s s levels. An Exponential Function a s the S t r e s s P a r a m e t e r
The s t r e s s p a r a m e t e r is probably intimately connected with crystal behaviour in the plastic s t r e s s range. This behaviour is thought by the author to be slightly
different in some way from conventional behaviour under static loading. P e r h a p s the static plastic behaviour considered by P r o f e s s o r Shanley is that for aggregate
components, and not the same for single c r y s t a l s acting a s part of the aggregate. If the degree of yeilding in the plastic range is not simply given by the particular conditions at the t i m e , but is also governed by the previous s t r a i n history, it may be that this general mode of action, i . e . p r o g r e s s according to p r i o r effect, is sufficient to describe the behaviour. Mathematical formulation of this particular property is given by:
P- =
C7(12)
ds
This states that the plastic state p a r a m e t e r changes with s t r e s s at a rate proportional to its p r i o r magnitude. Solving:
7 = ke''^ (13) When substituted in (8)
cs K
(14)
Thus: s = K^ - - log N (15) This is the equation of a straight line on a plot of s t r e s s v e r s u s log cycles.
E. S. Machlin (Ref. 23) developed the dislocation theory of physical metallurgy to account for the behaviour of "submicroscopic" cracks formed by fatigue. He proved that fatigue failure depends on plastic deformation, showed how multiplicity of crack initiation accounts for the brittle-like failure of metals in fatigue, and
developed a comprehensive relation linking endurance with s t r e s s level, t e m p e r a t u r e , frequency, and m a t e r i a l p a r a m e t e r s . F r o m this equation, the relation between
s t r e s s and log cycles with other p a r a m e t e r s fixed, is also linear in form. Conversion to the Linear Rule
The final form of equation (6) i s : cs
h = Ae ^ " (16) o r for an a r b i t r a r y crack length denoting failure,
h = Ae o Combining (16) and (17)
h e^V-N)
h = ^ oSubstituting equation (17) on the right side of this equation ^ - 1
u / h \N
•r • ( x ) . <•«•'
oPlotting these relations on a scale of crack depth v e r s u s cycle r a t i o , the curves will all have the same shape, i . e . the crack growth curves for different s t r e s s levels a r e affine. This simple c h a r a c t e r i s t i c means that the relation will follow the linear rule from one level of s t r e s s to another.
4. Load Spectrum
Before a simplified expression for the linear rule can be derived, it is n e c e s s a r y to consider the load spectrum.
The trans-Atlantic airliner project described in Ref. 24 is taken as an example, only the atmospheric gust spectrum being considered h e r e , mainly because it is usually the most significant contributor to 'dangerous' fatigue damage. In addition the following data a r e also relevant:
(17)
Average passenger load factor 80% Average fuel load at mid-flight 40% Assumed cruising height 30,000 ft.
Approx. weight at mid-flight 240,000 1b. Design service life 30,000 h r s . Cruising speed (0.8 Mach) 543 mph T. A. S. Total m i l e s travelled 15 ,000,000 miles (based on 500 mph)
At 30,000 ft:
Wing lift slope 4.77 per degree Speed of sound 995 ft. / s e c .
o^ 0.612 Gust Response. A satisfactory estimate of nominal gust response can be made using Airworthiness Requirements (B. C. A. R.) as described in Ref. 25. Equation 26 of this reference was used to obtain the loading due to a sharp-edged gust, taking into account tailplane and elevator effects. Using Fig. 5 of Ref. 24 (variation of lift curve slopes of surfaces and controls with Mach number) compressibility effects were also included. In the requirements for civil aircraft the static design gust velocity, 50 f t . / s e c . E . A . S . at 25,000 ft. drops lijiearly to 46 f t . / s e c . at the cruising altitude of 30,000 ft. This gust velocity was used to obtain the variation in response with altitude and weight and the resulting curves appear in Fig. 1.
Frequency of O c c u r r e n c e . It has generally been found (Ref. 26) that:
(a) The relative frequency of gusts is essentially constant r e g a r d l e s s of speed, altitude, o r aircraft c h a r a c t e r i s t i c s .
(b) The absolute frequency v a r i e s markedly with altitude, but is insensitive to speed of flight. F o r a given height, therefore, the absolute frequency, given by miles per gust, is largely independent of the aircraft. This is confirmed in Ref. 27. Because of these points, it is possible to use the invaluable gust data based on Comet operations. The speeds, powerplant and altitudes for the projected jet a i r -liner make the Comet data the only logical source for this analysis.
The Climb and Descent. F o r the Comet, Fig. 2 from Ref. 28 gives the variation in gust intensity with height in the climb and descent. The gusts per 1,000 miles g r e a t e r than or equal to 10 ft. / s e c . have been plotted. The absolute frequency of gusts is seen
to v a r y enormously with height. Assuming that a gust gives the same final damage r e g a r d l e s s of height, the effective fatigue height for climb and descent i s :
30,000
H (gusts at height h) dh
F Gusts at sea level
Fig. 3 shows that the number of gusts g r e a t e r or equal to 10 ft. / s e c . E. A . S . at sea level is 1,000 in 1,000 m i l e s . By graphical integration and using the above relation:
H.^ = 3,000 feet F
In this work the height is taken above sea level which is appropriate to t r a n s -Atlantic s e r v i c e . Routes over ground r e q u i r e careful study a s there is much evidence that gust intensity is governed by height above ground. Also the t e r r a i n has a marked effect on absolute frequency, especially at low or moderate altitudes.
The Effective Fatigue Height. To obtain the overall effective fatigue height, the gust intensities a r e weighted according to the time spent in each condition of flight.
Assuming 10% of the time is spent in climb and descent and 90% of the time is spent at cruising altitude:
^ H ^ = 0.9 G + 0.1 G ,. ^
F cruise climb
where G = overall gusts per 1,000 miles at effective fatigue height.
" F G = gusts per 1,000 m i l e s > 10 f t . / s e c . at 30,000 ft. = 2.8 cruise G ,, ^ = gusts per 1,000 m i l e s > 10 f t . / s e c . at 3,000 ft. = 208 c l i m b O f ' from Fig. 2. Therefore G „ = 0.9 x 2.8 + 0.1 x 208 " F = 23.32 gusts > 10 f t . / s e c . E . A . S . per 1,000 m i l e s
= 425 miles per gust > 10 f t . / s e c . E . A . S . F r o m Fig. 2 this corresponds to
H ^ = 12,000 ft. F
Knowing the relative frequency of occurrence of gusts of different magnitudes, which is considered to be Invariant with altitude (Ref. R . A e . S . Data Sheet L . 0 1 . 0 1 ) , the number of miles per gust exceeding other values of E . A . S . can be determined. The line through these points, shown in Fig. 4, is then the required gust spectrum.
It is seen that the line coincides with the 5,200 ft. spectrum and not with the position which would be obtained by d i r e c t interpolation in the altitude variation of the relative frequency c u r v e s . This is because the gust intensity curve of climb and descent is significantly different from the one obtained from data on piston engined a i r c r a f t , which is used to show the altitude effect in Fig. 4.
The final gust load spectrum has thus been obtained and is shown in F i g . 4 . It i s thus evaluated in t e r m s of distance which is the fundamental unit in gust r e s e a r c h ("gust encountered" is a d i r e c t function of the distance covered).
Ref. 29 points out the danger of separating the relative frequency of gusts from the total number of gusts, a s has been done in Ref. 30. Ref. 29 says "in doing this with data with widely different ' t h r e s h o l d s ' below which no data a r e available, assumptions have to be m a d e " in this r e s p e c t . In the linear method of life a s s e s s m e n t , the
threshold is probably the most important single factor in the calculation. The use of •miles per gust' o v e r c o m e s the problem a s it combines relative frequency with absolute frequency.
In tabulating this spectrum in Table 1, use is made of the linear extrapolation to U = 0 ft. / s e c . gust velocity to d e t e r m i n e the intercept value log N . This is standard procedure in Swedish work, and is implicit in many other r e p o r t s , e . g . the region n e a r the Ig line is for " r e a s o n s inherent in the technique of m e a s u r e m e n t never v e r y
a c c u r a t e l y a s c e r t a i n a b l e . However, for p r a c t i c a l r e q u i r e m e n t s it should be sufficiently well defined by l i n e a r extrapolation". (Dr. G a s s n e r in Ref. 6).
Certain a i r c r a f t p a r a m e t e r s , such a s centre of gravity position, short-period damping and wing bending flexibility all tend to give a wide variation in response throughout the s t r u c t u r e . (Refs. 3 1 , 32). T h e r e f o r e , the incremental load factor v a r i e s with a given gust velocity due to variations in r e s p o n s e . The s t r e s s level also v a r i e s with a given load factor due to eccentricity of loading, distance of the detail from the neutral axis of the component and the a r e a of m a t e r i a l used.
P e r h a p s the best method of dealing with these two variations, is to set up an envelope load spectrum covering all the probable response c a s e s . Fig. 1 shows that the worst value for the nominal load factor increment is about 2.5. This value should then c a t e r for the weight, a e r o e l a s t i c load amplification and the other factors
affecting r e s p o n s e . The final level chosen should be checked with the maximum static limit load factor which Is usually of the s a m e o r d e r of magnitude. n = 1 + 2.5 = 3.5 which o c c u r s once in ten y e a r s (Ref. 3 3 . ) . n,. , = 3.6 for the aircraft wing. Gust Spectrum Theory. D r . G a s s n e r , in Ref. 6, has stated that the load spectrum follows a logarithmic binomial distribution. He has obtained good agreement with the e m p i r i c a l data accumulated by Taylor (Refs. 27, 32). These experimental r e s u l t s have been represented by a linear plot of the gust frequency on semilogarlthmic paper in common with many other data from the other c e n t r e s of r e s e a r c h .
The equation of a straight line i s given by 1
n o
^ m
where y is a function of the gust velocity.
This equation is then the equation of the gust summation curve, thus for any quantity y t h e r e i s a value log N which r e p r e s e n t s the total number of the quantity of y above and at that level. It i s therefore an integration curve for all values taken from the m a x i m u m , y .
•'m
The decision a s to what value y should take depends on considerations of integration of the load spectrum with static r e q u i r e m e n t s , i . e . the value of y to settle upon^becomes r e s t r i c t e d when it i s related to the m a t e r i a l p r o p e r t i e s , for the minimum rational value of Log N i s that corresponding to \ cycle. This can be called the Static Axis, for all static failures r e p r e s e n t the completion of j cycle of fatigue.
A dove-tailing of static and fatigue loads i s quite possible h e r e , for the static gust c a s e s have evolved through the y e a r s due to the frequency of occurrence of these g u s t s . It i s not s u r p r i s i n g , t h e r e f o r e , that the spectrum t i e s in closely with the single o c c u r r e n c e of a gust at about the limit load (actually 49 ft. / s e c , when extrapolated) in the 30,000 hour life chosen.
In the a s s e s s m e n t of aircraft life, the cycle ratio a p p e a r s to be a fundamental unit. Because of t h i s , it i s n e c e s s a r y to d e t e r m i n e not the sum of all cycles at and above the given level, but only the number of cycles at the p a r t i c u l a r level.
Since the summation i s a continuous c u r v e , the required number of cycles i s obtained from a differentiation of the summation equation thus:
dn W'^'^") cld-y/yj
' ^ d(i - y / y ^ ) ^y = N . log N . — o e o y "'m = log N e o . n . ^m summationThe exponential differentiation r e s u l t s in a factor on the original equation. Thus the differentiated spectrum i s a straight line parallel to the summation line. I t s intercept on the y = 0 axis i s :
(N ) d i f f e r e n t i a t e d = N o o l o g N e o m N s u m m a t i o n In t h e s u m m i a t i o n c u r v e of F i g . 4 u s e d t o d r a w up T a b l e 1: T h e n T h e r e f o r log^ e N o N Ï o logg = 8,340,000 = 15.91 N o ^ m s u m m a t i o n N = 0.325 15.91 49 = 0.325 N if y = (U ) m m a x s , z e r o . log a x i s = 0.325 X 8,340,000 o s 49 ft. / s e c . = 2 , 7 2 0 , 0 0 0 c y c l e s . T h u s a l i n e p a r a l l e l t o t h e s u m m a t i o n l i n e , but b e g i n n i n g at 2,720,000 c y c l e s at t h e U = 0 a x i s r e p r e s e n t s t h e s p e c t r u m in unit g u s t v e l o c i t y i n t e r v a l s . T a b l e 1 when p l o t t e d g i v e s t h e s a m e l i n e but l e s s a c c u r a t e l y . It i n v o l v e s the t e d i o u s p r o c e s s , c o m m o n in t h i s w o r k , of talking d i f f e r e n c e s f r o m t h e s u m m a t i o n c u r v e . T h e q u a n t i t y y i s i n t e n t i o n a l l y m a d e a g e n e r a l function in o r d e r t o avoid c e r t a i n p i t f a l l s in t h e u s e of g u s t s p e c t r a . It i s defined b y t h e e q u a t i o n y = f(U) w h e r e U i s g u s t v e l o c i t y . If y i s u s e d t o d e n o t e s t r e s s , it i s a s s u m e d t o be a l i n e a r function of t h e g u s t v e l o c i t y . F o r a g i v e n s t r e s s l e v e l , t h e c y c l e r a t i o i s t h e q u a n t i t y of I n t e r e s t , and a g e n e r a l e x p r e s s i o n f o r it f o l l o w s . T h e c y c l e r a t i o d e p e n d s on: l o g N ^^ " y ' ^ m > (a) T h e i n t e r v a l of d i f f e r e n t i a t i o n : N ^ e o ^, s o , = S . N d o y s ' m s (b) T h e s t r e s s - g u s t v e l o c i t y r e l a t i o n : n = N (1 - y / y ^ ) m T h u s l o g N *e o ra N (1 - y / y ^ ) s s 1 (1 - y / y ^ ) d
B e c a u s e of t h e two f a c t o r s (a) and ( b ) , t h e final e x p r e s s i o n f o r the c y c l e n u m b e r n i s t h e q u a d r a t i c :
where
i°g,o "d = <^ - ^y^ <^ - y ^ v ^
d A = log /log N N+ log N 10 oand log N
^10 o
m
A and B a r e then fixed once the summation curve has been chosen. The simple summation afforded by using unit intervals of gust velocity in the differentiation is a strong argument for its u s e . This would standardize the present position where i n t e r v a l s of 0 . 1 , 0.5, and even 3.0 fps (4.5 fps - Ref. 30) have been taken without explanation. What happens when these different intervals a r e taken is shown below:
M9h
respont trial S-N Curve
RG. J.
N Cycles Decrcasinq interval Increasing interval
The Take -Off and Landing Cycle. F o r the project aircraft, (Ref. 24) it has been conservatively assumed that 1,000 flights a r e made each y e a r .
By a r b i t r a r i l y defining the m i d - c r u i s e bending moment a s the datum ( i . e . the Ig level of s t r e s s ) the effective s t r e s s levels a r e :
(1) Mid c r u i s e : (2) P r e take-off:
I g f o r 2.496 x 10 lb. ft. 2.730
(3) P o s t take-off: (4) P r e landing: (5) P o s t landing: 1.877 2.496 2.206 2.496 0.797 2.496
0.75 g due to fuel relief. 0.88 g due to low weight.
- 0.32 g due to wing weight alone.
These r e s u l t s a r e plotted graphically in Fig. 5. The wing has a "cycle" of t Ig about mean s t r e s s from this periodic change in loads.
Ref. 27 s t a t e s that 99% of all gust peaks away from level flight (Ig) r e t u r n to Ig before moving away again. This statement does not refer to all the gust peaks on the t r a c e but to a smoothed out substitute t r a c e , where all changes of O.lg or l e s s have been ignored. A cycle of fatigue i s defined a s one positive gust and one negative gust of equal magnitude.
The spectrum finally used can be in the form of a bar chart where the gusts p e r interval a r e obtained by differences, o r by the differentiation technique. It is quite important to Indicate the c l a s s Interval used in such differentiation a s it is impossible to use the data properly without the knowledge for the final Integration of damage. If one u s e s the Swedish work based on the linear r u l e , the usual method of numerical integration can be avoided.
The use of envelope spectrum i s definitely recommended, for gust response o r sensitivity v a r i e s so much that it is virtually impossible to cater for every change in the response while trying to consider the frequency of occurrence as well.
It i s important to note that the American and British gust definitions a r e different, but the alleviation factors a r e defined in different ways so that the two r e q u i r e m e n t s a r e virtually the s a m e . However without the alleviating factors the relation in units between the two definitions in gust velocity is given by U . = 0 . 6 U Q . . , .
u
fps 0 2.5 5 10 15 20 25 30 35 40 45 50 Simplification Miles per < ( i . e . 1 + & ] 1.8 4 9 44 230 1,000 5,000 28,000 140,000 780,000 4,000,000 20,000,000 of the Linear RuleCycles from the Envelope Spectrum
'cle > U Cycles per 15 million 1 - gust) m i l e s > U 8.340,000 3,750,000 1,665,000 341,000 65,300 15,000 3,000 535 107 30 3.75 0.75 5 .
This section will derive a simplified expression for the linear rule where the load spectrum and the corresponding m a t e r i a l S-N relation may be represented by two straight lines on a semilog plot. The S-N relation i s here shown in t e r m s of the equivalent gust velocity.
FIG.K.
L 0 9 Cycles
LoqN;
The general equation of a straight line given by its axes intercepts i s : 1 - H _
The general expression for the cycle ratio is: n 1 n 2 (1 N 1 (1 N, - ^ ) ", - Ü ) Us N 1 N 2 1 N "= / 2
{ 1
\ -N / . U \ / (20)Often Uj i s difficult to obtain d i r e c t l y . Its value can be obtained from the v a l u e n, at the l e v e l of u^, i . e . u, = 7Ï-J r— (21) a (log n ) 2 "2 l o g N . -t N
Let 'a' represent the constant _ L and let 'r' represent the constant in the brackets in Na
u
equation (20). Thus the cycle ratio R = ar . This shows that cycle ratios merely form a geometric progression. If the interval of gust velocity taken is unit feet per second, the linear rule i s simply the sum of this procession, i . e .
u " l
' «r." - a(r ' - 1) - „ .
S ar = — (22) u=0
If the linear rule summation is desired over, say, the interval u = u to u = u^ the expression i s :
"3 u a "3 "1
2 ar'' = -^^ (r ' - r * - 1) (23) u
4
This simple expression then performs the numerical integration of cycle ratios for the Ideal process following the linear rule.
Cumulative Damage Behaviour in Regions (1) and (3)
Two sources of data were available to a s s e s s the corrections necessary in these areas of nonlinearity. They are:
(a) The 0.5 Probability S-N curve of endurance with the plain specimens tested as indicated in the appendices, shown in Fig. 5 and
(b) The results of the partial damage tests carried out with the same specimen type described in Appendix I, and shown in Table 2. The general layout of these tests
In o r d e r to c o r r e l a t e at least qualitatively the r e s u l t s from both of these s o u r c e s , the curve of the S-N relation was rotated, to present the straight line portion of
Region (2) a s a region of constant damage activity. No attempt is made to define this activity, it is m e r e l y used to form a qualitative starting point. The r e s u l t a p p e a r s in F i g . 7.
Once this h a s been done, the boundaries of each region were decided and drawn a s shown. Then the r e s u l t s of the reduction of the partial damage data given in Table 2 w e r e superimposed.
F r o m the partial damage r e s u l t s in Region (1) the acceleration of the fundamental p r o c e s s i s quite evident, a s the relation a p p e a r s to be approximately:
D = R ' / ^ °
(24)
The limiting case for Region (1), given by the static test r e s u l t s , can be r e -presented by the equation:
D = R ° (25) When the static r e s u l t s a r e considered together with the fatigue r e s u l t s , the
non-l i n e a r S-N curve in Region (1) suggests the renon-lation:
D = R* (26) where x moves from unity in Region (2) to 0.1 in the a r e a of the p a r t i a l damage t e s t s
to z e r o at the static a x i s . The total damage for Region (1) in the example has been evaluated and i s shown in Table 2.
In Region (2), the r e s u l t s indicate that the damage was so small from the first level a s to not survive the effects of s c a t t e r ; i . e . the mean endurance at the second level fell within the confidence l i m i t s of the mean of the endurance from single level testing, a s shown in F i g . 8. This s e e m s unfortunate, but in view of the significant r e s u l t s obtained in Region (1), it is obvious that the damage function must be close to being linear in this region, for the cycle r a t i o s used a r e l e s s than one per cent. They therefore could cause comparatively little damage if the linear rule were valid h e r e . F o r the example: Thus N, = N^ = a = 1 u^ = 42 ft. u = 96 ft.
(i
r = N " . N = / s e c . / s e c .- i )
" i 3,000,000 cycles (Fig. 6) Hence r = N "* "i = N " ° ° ^ ^ = 0.844Summing from equation (22)
^y . " 1 - r^^ 0.990
E a r = •; = n ,..„ = 6.35 (27) 1 - r 0.156
o
In Region (3) the partial damage test data indicated that healing was encountered at the lower l e v e l s . The behaviour in this region suggests the relation:
D = R - R^ (28) i . e . the fundamental p r o c e s s is being balanced by a function dependent on s t r e s s level
and cycle r a t i o . In the limiting c a s e , u = 0, no damage could be possible.
i . e . D = R - R^ = 0 (29) The healing function s e e m s to prevent actual propagation, and is responsible for the
endurance limit effect. At the lowest level of the p a r t i a l damage t e s t s , D = R - R^ = - 0.60 from Table 2.
If the linear rule prevailed a s in Region (2) D = R = 0.37
Thus the healing function at this level is R^ = 0.60 + 0.37 = 1.0 Thus X = 0 in this region.
F r o m the r e s u l t s at the other two levels in Region (3), the healing function rapidly dies away some distance below the endurance limit s t r e s s . These r e s t r i c t i o n s on the
healing function led the author to the following form:
R^ = R (30) where u is a level of maximum healing action. This function satisfies all the
r e s t r i c t i o n s above, and the summation for the example is given in Table 2. The Final F o r m of the Corrected Rule
In its corrected form the rule states that failure will occur when, D + D + D = 1
(U - 1 ) 2
In the example,
48 0 1 42 10 ( | - 1 ) 2
L R + E R - E R =• 2.765 + 6,350 - 8.555
43 o o
= 0.56
The summation of damage s t a t e s that 56% of the life has been used up. The actual life of the element based on the 30,000 hours load spectrum is then 53,500 h o u r s .
While this corrected form r e q u i r e s m o r e than just the simple S-N curve and the appropriate s p e c t r u m , further work on the correlation between two level t e s t s and S-N non-linearity may ultimately allow such simplification, m e r e l y by carefully constructing the S-N relation. The method will then be a s quick and almost a s simple a s the present basic linear r u l e . It may then be extended so that correction p a r a -m e t e r s -may be evaluated for practical aircraft -m a t e r i a l s relative to so-me basic load s p e c t r u m .
The damage curve for using the uncorrected linear rule on the spectrum and S-N curve of the example is shown in F i g . 9. The total damage by this method was 0.235, i . e . 23,5% of the life was used up. This would give an uncorrected lifetime of
128,000 h o u r s . The damage curve using the corrected rule is shown in F i g . 10. The p a r t i a l damage r e s u l t s have been superimposed. The general forms of the non-linear components in Equation 31 a r e shown to follow the experimental r e s u l t s faithfully. Conclusions
1. The non-linear effects a s evaluated in the example each a p p t . r to be of the same o r d e r a s the basic linear damage. They thus r e p r e s e n t good examples of the significance of non-linear damage accumulation. P a r t of the severity of Region (1) may be a d i r e c t result of the use of a precipitation-treated aluminium alloy.
Regardless of the p a r t i c u l a r r e s u l t s of this data, the general method of performing a 'partial damage t r a v e r s e ' of the s t r e s s levels met in service is
suggested a s a good procedure to gauge damage activity qualitatively. Comparison with data of other r e s e a r c h works was not possible, a s the p a r t i c u l a r conditions of m a t e r i a l , mode of testing, mean s t r e s s , statistical accuracy and machine s c a t t e r factor have not been found in agreement with the present work.
2. Much work should be done to study the c h a r a c t e r i s t i c s of crack initiation and the transition to the p r o c e s s of crack propagation. Little is really known about this part of crack growth. It would be of great use to know whether the assumption of affine crack growth with s t r e s s can be relied upon for the ideal fatigue p r o c e s s . 3. It is unfortunate that the principles developed in this method cannot be readily
put to a test which exactly simulates service conditions, and is governed by the same spectrum of loads. Such a random load test would quickly prove the worth of any cumulative daniage theory. Until such tests a r e possible, m o r e
sophisticated sequence p r o g r a m m e s for loading a r e not of any m o r e value than the two-level testing procedure used h e r e , a s all these a r b i t r a r y modes of testing will probably r e q u i r e their own p a r t i c u l a r c o r r e c t i o n s for random loading.
4. Many w o r k e r s in this field have turned to non-affine crack growth equations and other difficult relations in their efforts to obtain one relation which will fit the overall p i c t u r e . This is the important point on which the present work differs. The chief difference between the procedure proposed h e r e , and the work of other r e s e a r c h c e n t r e s l i e s not in fundamental formulation, but in its application.
While the continuing lack of s u c c e s s of these other t h e o r i e s does not mean that they a r e l e s s promising in comparison, the evidence given here s e e m s to show that p r a c t i c a l use can be made of this new method. F r o m the substantial degree of s u c c e s s the basic linear rule has had in the p a s t , the author is convinced that only a procedure fundamentally based on it will have any m e a s u r e of s u c c e s s in p r a c t i c a l engineering.
References MINER, M. SCHIJVE, J . , JACOBS, F . SHANLEY, F . R . HEYWOOD, R . B . GASSNER, E. WALLGREEN, G. BENNETT, J . A. NEWMARK, N . M . MARCO and STARKEY WILKINS. E . W . C .
Cumulative Damage in Fatigue.
J n l . Applied Mechanics, September, 1945. Fatigue T e s t s in Notched and Unnotched Clad 2 4 . S . T . Sheet Specimens to Verify the Cumulative Damage Hypothesis.
N . L . L . Report M. 1982, April, 1955.
A Theory of Fatigue Based on Unbonding during Reversed Slip.
Rand Publication P . 350, May. 1953. The Fatigue of Metals and S t r u c t u r e s .
Navaer 00 - 25-53 , U . S . Government Printing Office, 1954.
The Influence of Preloading on the Fatigue Life of Aircraft Components and S t r u c t u r e s .
R . A . E . Report S t r u c t u r e s 182.
The Problem of Fatigue Strength in Aircraft S t r u c t u r e s .
Translated by T. Haas, Aircraft Engineering, July, 1956.
Fatigue T e s t s with S t r e s s Cycles of Varying Amplitude.
F . F . A. Report 28. » A Study of the Damaging Effect of Fatigue
S t r e s s i n g on X4130 Steel. A . S . T . M . P r o c e e d i n g s , 1946.
A Review of Cumulative Damage in Fatigue, in "Fatigue and F r a c t u r e of M e t a l s " , edited by W . M . M u r r a y . Wiley, 1952.
A Concept of Fatigue Damage. T r a n s . A . S . M . E . 76, 1954. Cumulative Damage in Fatigue.
International Colloquium on Fatigue, Stockholm, 1955.
LEVY, J . C . Cumulative Damage in Fatigue - A Design Method based on the S-N Curve.
References (Continued) 13. HENREY, D . L . 1 4 . 1 5 . 16. 17. 1 8 . 19. 20. 2 1 . 22. 23. 24. SCHIJVE, J . , JACOBS. F . A . MILES, J . W . FREUDENTHAL . A . M . WEIBULL, W. LUNDBERG, B. WEIBULL, W. WEIBULL, W. BENNETT. J . A. RAMBERG, W. , OSGOOD. W.R. MACHLIN, E . S .
A Theory of Fatigue Damage Accumulation in Steel.
T r a n s . A . S . M . E . , v o l . 7 7 , August, 1955. R e s e a r c h on Cumulative Damage in the Fatigue of Riveted Aluminium Alloy J o i n t s .
N . L . L . Report M.1999.
On Structural Fatigue under Random Loading. J n l . A e r o . Scs, , November, 1954.
P h y s i c a l and Statistical Aspects of Cumulative Damage.
Stockholm Colloquium on Fatigue, May, 1955. A New Method for the Statistical T r e a t m e n t of Fatigue Data.
SAAB T . N . 30, May, 1954.
Fatigue Life of Aircraft S t r u c t u r e s . F . F , A. Report 60.
The Scatter of Fatigue Life and Fatigue Strength of Aircraft Structural Materials and P a r t s . F . F . A . Report 73, November, 1956.
Effect of Crack Length and S t r e s s Amplitude on the Growth of Fatigue C r a c k s .
F . F . A . Report 65, May, 1956.
The Distinction between Initiation and Propagation of a Fatigue Crack.
Session 6, Proceedings of the International Conference on Fatigue of Metals, 1956. Description of S t r e s s - S t r a i n Curves by Three P a r a m e t e r s .
N . A . C . A . Tech. Note No. 902. July. 1943. Dislocation Theory of the Fatigue of Metals. N . A . C . A . Tech. Note No. 1489.
1957 P r o j e c t - Trans-Atlantic A i r l i n e r . Design Note D e s / 6 5 . Department of Aircraft Design, The College of Aeronautics.
References (Continued)
26.
27.
28.
29.
30.
31.
32.
33.
34.
TYE, W.
TAYLOR, J.
TAYLOR. J.
KLOOS, J.
RHODE, R.V. ,
DONELY, P.
PRESS. H. ,
HOUBOLT, J.C.
TAYLOR, J.
CHEVERTON, H.
WEIBULL, W. 35. RANSOM, J . T .Re qui r em e n t s. Loading and Stressing Cases ( B . C . A . R . ) .
L e c t u r e Supplement D e s / 4 2 , Department of Aircraft Design, The College of Aeronautics. The Outlook on Airframe Fatigue.
J . R . A e . S . , May, 1953.
Measurement of Gust Loads in Aircraft. J . R . A e . S . , F e b r u a r y , 1953.
Fatigue Loads and t h e i r Effect on Aircraft S t r u c t u r e s .
Appendix 1, Third Conference of the International Committee on Aeronautical Fatigue, Cranfield, J a n u a r y , 1955.
Gust Loads,
Appendix 8, Second Conference of the Internation-al Committee on AeronauticInternation-al F a t i g u e ,
Stockholm, 1953.
Frequency of Occurrence of Atmospheric Gusts and of Related Loads on Airplane S t r u c t u r e s . N . A . C . A . , A . R . R . L . 4 1 2 1 , 1944.
Applying Generalized Harmonic Analysis to Aircraft Gust Loads.
J n l . A e r o . Scs. J a n u a r y , 1955. Guslts and their M e a s u r e m e n t . J . R . A e . S . , D e c e m b e r , 1954.
The Effect of Fatigue on Aircraft Design. E x t e r n a l Design L e c t u r e , F e b r u a r y , 1958, The College of A e r o n a u t i c s .
Scatter in Fatigue T e s t s .
Minutes of the Second Conference of the International Committee on Aeronautical Fatigue. Appendix 2, September, 1953. Discussion by W. Weibull, Statistical Methods for Fatigue.
References (Continued) 36. WEIBULL, W. 37. 38. WEIBULL, W. BENDER. A. , HAMM, A.
Static Strength and Fatigue P r o p e r t i e s of Unnotched Circular 75 S-T Specimen under Repeated Tension.
F . F . A . Report 68. June, 1956.
Statistical Handling of Fatigue Data and the Planning of Small T e s t S e r i e s .
F . F . A . Report 69, October, 1956.
The Application of Probability P a p e r to Life on Fatigue Testing.
Engineering Dept. , Delco Remy. General Motors Corporation.
