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Notch geometry effects in fatigue: A conservative design approach
Engineering FailureAnalysis, Vol 1, No. 4 pp. 275-287, 1994 Copyright© 1995ElsevierScienceLtd Printed in Great Britain.All rightsreserved 1350-6307/94$7.00 + 0.00
Pergamon
1350-6307(94)00022-0
NOTCH GEOMETRY CONSERVATIVE
EFFECTS DESIGN
IN FATIGUE:
A
APPROACH
DAVID TAYLOR a n d MANUS O ' D O N N E L L Department of Mechanical and Manufacturing Engineering, Trinity College, Dublin 2, Ireland
(Received 4 July 1994) Abstract--The understanding of stress concentrators (notches) is an important element in the prevention of failure in components and in the analysis of failures when they occur. This paper examines critically the methods currently used to predict the fatigue strength of components containing notches, with particular attention to the need for a conservative design approach. Current methods, if correctly applied, are shown to give conservative predictions of fatigue limit and high-cycle fatigue strength, and it is shown that the same philosophy can be extended to cover very small notches, including surface roughness. The presentation of these predictions in the form of a "mechanism map" for notch fatigue is advocated as a useful tool for designers. The problems of extending the approach to other types of stress concentrator, such as fillet radii, are discussed.
1. INTRODUCTION Geometric stress concentrators such as notches exert a pronounced effect on fatigue strength; failures of components almost invariably occur from stress concentrators, making their behaviour critical to the design process. Fatigue strength (i.e. the stress range needed to cause failure after some specified number of cycles) is reduced by some factor, Kf, when the notch is introduced. In some cases Kf is numerically equal to the elastic stress concentration factor for the notch, K t , but in many cases KI < K t , so the use of K t leads to a conservative prediction: one in which the predicted fatigue strength is lower than its true value. Whilst this is good in principle, the degree of conservatism is excessive in some cases, including sharp notches, low-cycle fatigue and low-strength materials. The present paper considers the problem of the fatigue strength of notches as follows. (a) Previous work is reviewed, beginning with that of Frost and co-workers [1, 2]. It is shown that the theoretical approach developed is implicitly conservative. The degree of conservatism is discovered by reference to experimental data. (b) The approach is extended to cover the case of very small notches, using modifications which retain the conservatism of the original methods. Experimental data are used to validate this method. (c) A new method of representation of the data is proposed, in which data from a wide range of geometries are presented on the same diagram, using the "mechanism map" approach. (d) Brief consideration is given to the case of stress concentrators which are not notches, examining data from fillet radii.
1.1. Previous work: notch geometry effects in high-cycle fatigue It has been known for some time that notches can be divided quite distinctly into two categories, displaying different behaviour as regards their tendency to reduce fatigue strength. Figure 1 summarises the effect, showing fatigue strength as a function of K t for the case in which g t is varied by changing notch root radius (p) whilst keeping notch depth (D) constant. A clear division can be made between 275
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DAVID TAYLOR and MANUS O'DONNELL
"blunt" notches, for which fatigue strength decreases as Kt is increased, and "sharp" notches, for which fatigue strength remains constant despite changes to Kt. A similar transition is apparent if one plots the fatigue strength as a function of notch depth for constant K t (Fig. 2); when expressed in this form it is commonly referred to as a "size effect" for the notch, because the behaviour of small notches differs from that of large ones. The transition point, K* (Fig. 1), is found to be a function not only of material properties but also of notch geometry and number of cycles to failure. This behaviour was reported by Frost and co-workers in a number of papers in the late 1950s (e.g. [1, 2]) and summarised by Frost et al. in 1974 [3]. They argued that blunt notches (Kt < Kt*) could be analysed by dividing the plain fatigue limit by Kt, and that the behaviour of sharp notches (Kt > Kt~') was independent of K t but controlled by D, according to the formula 03D -----constant.
(1)
Smith and Miller [4] reported similar results in 1978 and showed that the sharp-notch condition could be described using a fracture mechanics approach. If the notch is imagined to be a crack of the same depth, D, then the condition for the propagation of this crack to failure becomes AK = Agcrit ,
(2)
where A K is the range of stress intensity applied and A Kcrit is the value required for propagation to failure within the specified number of cycles. In the case of the fatigue limit, AKcrit becomes equal to the threshold value, AKth. For the case of a surface notch the condition becomes 1.12Aa~
= constant.
(3)
Smith and Miller expressed this as 2 A o ' { D = constant,
'o c os
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STRESS CONCENTRATION FACTOR, K t FIG. 1. Schematic variation of fatigue strength with stress concentration factor, Kt, for a series of specimens in which the notch depth, D, remains constant.
FATIGUE
~i
nstantKt
STRENGTH
NOTCH DEPTH, D FIG. 2. Schematic variation of fatigue strength with notch depth, D, for a series of specimens with constant gt, showing the notch size effect.
(4)
Notch geometry effects in fatigue: a conservative design approach
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which is numerically similar to Eqn (3). The underlying conclusion is that there is a different mechanism at work in the fatigue failure of blunt notches from that of sharp notches. Blunt notches produce a region of high stress but do not alter the processes of crack initiation, short crack growth and long crack growth which precede eventual failure. Sharp notches, however, act as if they were pre-existing cracks, effectively bypassing the initiation and (unless we are dealing with micro-notches) the short-crack growth stages. Their behaviour is thus governed by the quite different laws of fracture mechanics; this gives rise to, for example, a different mean-stress effect. It is interesting to note from a historical point of view that Frost's original equation for sharp notches [Eqn (1)] was derived before the advent of fracture mechanics in fatigue. In his later work in 1974 [3], the use of AK is considered (Pards-law and threshold data were then available for a range of materials), but rejected on the grounds that the (T3~vf-Dparameter gave a better fit to experimental data, even to data on sharp cracks. This seems surprising until we realise that some of these data were in the range of crack lengths which we would now term "short cracks"; for lengths less than 0.5 mm, Eqn (4) gives poor predictions and so Eqn (1), though purely empirical, was retained.
1.2. Conservative nature of the prediction The above analyses can be viewed in a slightly different light, as shown in Fig. 3. It can be noted that, for any given notch, there will always be two predictions of fatigue strength: one, which will be termed the NOTCH prediction, obtained by dividing the plain-specimen fatigue strength by Kt, and an alternative, termed the CRACK prediction, obtained by assuming that the notch is a crack. It is important to realise that both of these predictions are conservative. The CRACK prediction is conservative because it assumes that the notch is a sharp crack (i.e. that it has zero root radius); the N O T C H prediction is conservative because it ignores local plasticity and the effects of stressed volume and local stress-gradient. It is precisely because both predictions are conservative that we are justified in using the higher of the two in all cases. In fact the experimental data are expected to fall as shown in Fig. 3, coinciding with the predicted values at the two extremes of the plot and lying somewhat above the highest prediction for all intermediate values. This approach has a number of consequences: (a) Its value for designers is greatly enhanced by the knowledge that the method is conservative. In practice many designers are still reluctant to use fracture mechanics for sharp notches, even though Kt is so conservative as to be impractical; the knowledge that they are using a method which is still conservative is of great value. (b) It now becomes important to know the degree to which the predictions are conservative. Intuitively one would expect that conservatism would be highest for Kt values close to K*, where both predictions are at their least valid.
FATIGUE STRENGTH
Constant N o t c h D e p t h , D
~RESULTS PREDICTION CRACK ~- NOTCH
STRESS CONCENTRATION FACTOR, K t FIG. 3. Comparison between typical experimental results (thick line) and theoretical predictions using the NOTCH and CRACK approaches.
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(c) A corollary to the above is that the predictions will be improved by using alternative equations which predict higher fatigue strengths provided these alternative methods are also conservative. This indicates a method for refining the approach and, more importantly, for extending it into difficult areas such as micro-notches and low-cycle fatigue.
2. E X P E R I M E N T A L DETAILS 2.1. Materials Two materials were chosen for study: a mild steel (En3b, normalised) and a low-alloy steel (Enl9, quenched and tempered). Tables 1 and 2 give details of composition and mechanical properties, including the stress-intensity threshold obtained from standard load-reduction tests on long cracks. 2.2. Specimens In order to investigate fatigue strengths in the high-cycle region, through-thickness surface notches were machined into bars of length 120 mm, width (in the notch depth direction) 20mm and thickness 12.5 mm. The notches were of uniform depth (2.5 mm) and varying root radius so as to produce g t values of 1.0 (plain bar), 3.0, 4.5, 5.8 and 10.2; this included values below, equal to and above the calculated value of K* Specimens were also produced from this material in which the stress concentrator was a fillet radius instead of a notch (Fig. 4); K t values from 3.9 to 6.9 were obtained by varying the fillet radius only. The Enl9 alloy steel was machined to varying degrees of roughness; rectangular bars were used, of geometry similar to those described above for the mild steel. Different machining methods were used in order to vary roughness. Profile analysis using a Talysurf showed that the depths of the largest notches on each surface varied from 3 to 390 ~m, depending on the machining method used. This work has already been reported in detail elsewhere [5]. All specimens were fully stress-relieved before testing, by annealing in a vacuum at 600 °C for three hours. 2.3. Testing All specimens were tested in four-point bending, using a servo-hydraulic testing machine in load control. The R ratio was kept constant at 0.1; frequencies were in the range 20-100 Hz. Failure was defined as the appearance of a crack at least 2 mm long. Data were collected at various stress ranges with two aims: (a) to define the fatigue limit, and (b) to define the fatigue strength for failure after 1 million cycles. TABLE 1. Composition of materials tested (wt%, balance Fe)
En19 En3b
C
Si
Mn
Ni
Cr
Mo
Cu
Nb
Sn
Co
Al
0.39 0.14
0.27 0.21
0.96 0.83
0.25
0.95
0.30
0.23
0.01
0.02
0.03
0.04
TABLE 2. Mechanical properties of materials tested
En19 En3b
Yield strength (MPa)
UTS (MPa)
Threshold AKth (R = 0.1) (MPa~mm)
980 556
1360 670
5.5 7.0
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Notch geometry effects in fatigue: a conservative design approach
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FIG. 4. Specimen geometry used to investigate the effect of fillet radius. All dimensions in mm. Thickness 12.5 mm.
3. RESULTS AND DISCUSSION 3.1. The degree of conservatism
Figure 5 shows the results from the mild steel, in which the fatigue strength (at 1 million cycles to failure) is plotted as a function of K t for constant D = 2.5 mm. It is evident that the results follow the scheme outlined above (Fig. 3). The results are all conservative; if we define the degree of conservatism as the difference between the measured fatigue strength and the predicted value, expressing this as a percentage of the measured value, the results range from 8 to 25%. Figure 6 shows the degree of conservatism plotted against K t for the present work, and also for various other studies [2, 3, 6]. Data spanning a wide range of K t values were obtained from Tanaka and Akiniwa [6] and from Frost and Dugdale [2], both using steel alloys; in the latter case a value for the threshold of the material was found by consulting the later publication of Frost et al. [3]. Many other datasets appear in the literature on notch fatigue strengths, but in many cases there is no accompanying threshold value. A prediction can still be made, however, for data in the blunt notch region. This enabled data from a number of other studies to be added to Fig. 6, for K t values less than 2.5. These data points, which are labelled "Various", include nine different studies covering ferritic steels, stainless steels, aluminium alloys, copper alloys and zinc alloys, all of which can be found in Frost et al. [3]. It is evident that some degree of conservatism is the general rule; from a total of 70
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DAVID TAYLOR and MANUS O'DONNELL
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FiG. 6. Degree of conservatism in predicted fatigue strength, taken from previous work [2, 3, 6] and the present investigation (Fig. 5).
results quoted here, only two are non-conservative (by 4 and 10%, respectively) and a further six show 0% conservatism. Most of the results fall within a band in the range 8-20%, which is a very acceptable level of safety from a designer's point of view. There does not seem to be any trend towards an increased conservatism around K* as suggested above (Fig. 3), but this may be masked by measurement errors.
4. SMALL NOTCHES 4.1. Theoreticalapproaches It was noted above (Fig. 2) that a size effect--a variation of fatigue strength with D at constant Kt--is implied by the transition from blunt to sharp-notch behaviour. In addition to this, a quite separate effect will come into play when very small notches are considered. For the case of crack-like notches, the so-called "short-crack effect" has been well characterised by many reports (see, for example, conference proceedings [7, 8]); for cracks less than some critical length, fracture mechanics predictions become non-conservative. Data are commonly represented on a "Kitagawa plot" in which fatigue limit is presented as a function of crack length; Fig. 7 shows one such plot in schematic form. In the present study the requirement is for a means of representing such data in a conservative fashion, in order to allow the CRACK prediction of Fig. 3 to be made when D is small. Critical values of D, below which this correction becomes important, vary greatly from material to material, but can be as high as 2 mm. Thus, this correction is required for all cases of surface roughness, for many manufacturing defects and even for certain design features such as grooves and holes.
FATIGUE STRENGTH OF CRACKED SPECIMEN
EHA VIOUR
(log.scale)
CRACK LENGTH (log.scale) FIo. 7. Schematic illustration of the "Kitagawa" plot showing effect of crack length on measured fatigue strength.
Notch geometry effects in fatigue: a conservative design approach
281
The most commonly-used equation for this short-crack region is that due to E1 Haddad et al. [9], which takes the form Act =
AKth . (5) 1.12~/rr(a + ao) The essence of this approach is that the actual crack length, a, is augmented by some notional value, a o, which is defined so that, when a - - 0 , Act becomes equal to the fatigue limit. Thus, ao is a constant, at least for a given material and crack geometry. It has no physical significance, but its use is convenient for modelling the short-crack situation. In essence the equation proposes that a short crack behaves as if its length was augmented by ao and the laws of L E F M were applicable. This equation is useful in the present study because it can be shown to be conservative when compared to experimental data. The equation must by its nature be conservative for very low crack lengths and for lengths approaching the intersection with the long-crack line. This is because stress values predicted by Eqn (5) will always be lower than the fatigue limit (for finite crack lengths) and also always lower than the stress predicted using the long-crack threshold (though this difference will become increasingly small as crack length increases). It is only by examining experimental data that the conservative nature of the equation can be demonstrated for the mid range of length values. Figure 8 indicates the degree of conservatism achieved by using Eqn (5), when applied to data from a range of short-crack studies [3, 10-12], which cover five different materials, including high- and low-strength steels and copper. In this case the degree of conservatism (defined as above for notch predictions) is plotted as a function of crack length, normalised with respect to ao so as to place all data in the same position relative to the Kitagawa plot. It can be seen that the predictions are conservative for all but two cases, both of which refer to a steel of unusually low strength (try--180 MPa) tested by Usami [10]. The use of Eqn (5) is therefore appropriate in the present study. It should be noted that one set of data, due to Frost et al. [3], was omitted from this plot. These results, obtained using an alloy steel, gave rise to a non-conservative prediction from Eqn (5); however, the problem in this case does not seem to be caused by the choice of short-crack analysis, because the non-conservatism persisted even for large crack lengths of the order of 3 mm. Since ao in this case was equal to 0.042 mm, its effect will be insignificant when considering a 3 mm crack. It seems that these particular data do not conform to a normal fracture-mechanics prediction, despite the fact that sharp cracks were used and that similar studies by the same authors on different materials conformed well. Having established a suitable method for making the C R A C K prediction, we also
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FIG. 8. Degree of conservatism in predicted fatigue strength of cracked specimens, using the El Haddad correction for short cracks [Eqn. (5)]; data from various sources [3, 10-12].
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DAVID TAYLORand MANUSO'DONNELI.
require a conservative N O T C H prediction for small notches. This problem has not been studied to the same extent as the short-crack problem; small, blunt notches are normally dealt with by modifying K t according to some empirical equation based on the root radius of the notch. Typical equations have been proposed by Peterson [13] and Mitchell [14]. The underlying approach to which these equations can be related is Neuber's concept of elementary structural volumes [15[. This approach assumes that failure will occur if the average stress over some volume of material ahead of the notch exceeds some critical value. This leads to an equation of the form Kf = 1 +
Kt - 1
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(6)
where p is the root radius of the notch and p' is a material constant expressing the size of an elementary volume in terms of a distance ahead of the notch. Peterson [13] and Mitchell [14] used a variation of Eqn (6), thus: Kf =
1 +
K t -- 1
1 + p'/p
.
(7)
Experimental results were used to optimise the choice of p', which was found to be related to the UTS for a range of steels. Neuber also modified his original equation to give gt
Kf-
(1 +
sp'/p) 1/2'
(8)
where s is a constant depending on specimen and stress type, typically equal to 2.0. These approaches have been criticised in the light of more recent knowledge that fatigue is a process of crack initiation and propagation, but current ideas about short cracks may suggest an alternative interpretation. Since it is known that short cracks of length similar to a microstructural unit (e.g. the grain size) will initiate readily even at stresses below the fatigue limit, then the fatigue strength of any notch may be governed by whether there is sufficient stress on this microcrack to enable it to grow. The microcrack size may be equated with Neuber's elementary volume. This exercise was carried out by Klesnil and Lucas [16] who, by considering the threshold condition for a crack of length ao at the notch root, derived an equation as follows: gt
Kf = (1 +
4.5ao/P) 1/~"
(9)
This is exactly the same as Neuber's modified equation except that the elementary volume constant p' is replaced by the El Haddad crack-length constant ao and a factor of 4.5 is added. Interestingly, for the E n l 9 steel used in this work for short-notch examinations, the ratio between ao and p' is 4.22--very close to this factor of 4.5. For our purposes, two difficulties are presented by this type of equation. Firstly, it is clearly inappropriate for sharp notches, where p is small compared to D; in the extreme case of a long, sharp crack of length a, the very small value of p would lead one to predict no reduction in fatigue strength, which is clearly not the case. This problem arises because of the assumption that the stress concentration will decrease to negligible values as one moves away from the notch tip by a distance of the order of p; this is true for blunt notches, but for sharp notches it is more appropriate to say that the stress concentration exists over a distance of the order of D. Equations (6)-(9) can be modified in various ways, the simplest of which is to replace p by D whenever D > p. In practice this is not a serious problem because in the case of sharp notches the C R A C K solution will generally dominate over the N O T C H solution anyway, but it may become a problem for very small notches.
Notch geometry effects in fatigue: a conservative design approach
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The second problem associated with any of Eqns (6)-(9) is to demonstrate that a conservative prediction is being made, as required by the present approach. The simplest way to be conservative is not to use these equations at all, simply reverting to the use of the full K t factor for all notches, however small. In practice this will lead to the C R A C K equation [Eqn (5)] being the dominant one for most cases of small notches. If a choice is to be made between these four equations, it can be noted that Eqn (6) will be more conservative than Eqn (7) provided that either D or p is greater than p', which in practice is invariably the case for designed notches (including surface roughness) and will almost always be the case for defects, inclusions, etc. Equations (8) and (9) are not usable for very small values of p and D because they lead to Kf < 1. 4.2.
Experimental data: surface roughness
The most stringent test of the predictive equations for small notches is the case of surface roughness. Typical roughness produced by manufacturing operations consists of a series of notches which have a very low Dip ratio, leading to K t values in the range 1.2-1.4, with values of D which are equal to, and often less than, ao, depending on the material. Figure 9 shows data from fatigue limit measurements on Enl9 steel; further details are available elsewhere [5]. Four different machining processes were used to vary D; the data are presented in the same form as the Kitagawa diagram for short cracks, replacing crack length by Dmax (which is the size of the largest surface roughness feature for each surface tested), measured using a stylus instrument. The CRACK and NOTCH predictions are shown, along with corrections for the case of small cracks [using Eqn (5)] and small notches [using Eqn (6), which was found to be the most conservative of Eqns (6)-(9)]. Two N O T C H predictions were required: one for the various machined surfaces (polished, ground and milled) and one for the rougher surfaces which were created using a shaping tool, which had a higher value of the ratio Dmax/pa n d thus a higher value of g t. Figure 10 summarises the results of Fig. 9, showing the degree of conservatism for each prediction. All results are conservative, to a degree that varies from 0.4% (for one of the shaped surfaces) to 16%. Considering the inherent difficulty of defining notch size and shape in this situation, the results are very encouraging. With the sole exception of the polished surface, the NOTCH prediction [Eqn (6)] dominated over the CRACK prediction. The change in predicted fatigue strength obtained by using Eqn (6) rather than the Kt value was small--of the order of 1%--so in this case the simpler, and more conservative, use of Kt is recommended.
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FIG. 10. Degree of conservatism of results from Fig. 9.
5. A MECHANISM MAP FOR NOTCHES It is evident from the above discussion that the behaviour of sharp notches differs fundamentally from that of blunt notches. Miller [17] has expressed this by saying that sharp notches are controlled by LEFM, whereas blunt notches (and plain specimens) display behaviour which he terms "Microstructural Fracture Mechanics (MFM)" to emphasise the degree of control exerted by the microstructure. One can state that the mechanism of fatigue is distinctly different for these two types of notches. If we accept that fatigue proceeds through three definite (though overlapping) mechanisms of crack initiation, short-crack growth and long-crack growth, then, in the case of blunt notches, all three mechanisms will occur just as they do in plain specimens. However, for large, sharp notches the first two mechanisms disappear, leaving only long-crack growth. Or, at least, it has been shown that these notches can be analysed as if long-crack growth is the only mechanism operating. Also, as has been shown above, there will be a category of small notches for which short-crack growth will occur first, followed of course by long-crack growth. The concept of a mechanism map was first introduced by Ashby to describe creep behaviour and has since been extended into other areas. The approach is useful in a situation where a failure process (e.g. creep) can proceed through two or more different mechanisms. The dominant mechanism will be a function of two independent parameters in the case of a 2D map (e.g. temperature and applied stress); an area on the map outlines the possible combinations of parameters for which a given mechanism occurs. Furthermore, contours representing some deduced variable (e.g. creep strain rate) can be plotted on the map. The value of the map lies not only in theoretical development of the subject but also in practical use by designers, since it indicates which equation(s) should be used to predict behaviour for a given set of operating parameters. In principle, maps can be plotted with more than two parameters, but visualisation becomes difficult. It is proposed that the mechanism map concept can be used in the area of notch fatigue strength, because it possesses the same ingredients as described above. Figure 11 shows one possible realisation of such a map, using values relevant to the mild steel material tested in this study. In this case the two parameters chosen for the axes relate to notch size and shape. Since the present map covers all semi-elliptic surface notches, the two axes chosen are D and p; for notches of a different basic shape other parameters will be needed, but in many cases two parameters will be a sufficient number. The map is then divided up into two basic regions: NOTCH and CRACK, indicating that we are dealing with a blunt notch or a sharp notch, respectively. Contours can now be plotted of fatigue strength (defined for any chosen number of cycles to failure).
285
Notch geometry effects in fatigue: a conservative design approach F a t i g u e Strength (MPa)
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300
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1001~m '
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FI6.11. Mechanism map for notch fatigue effects in En3b; fatigue strength here is the stress range for 1 million cycles to failure at R = 0.I.
The advantage of this type of map for designers is that it provides a ready visualisation of the effect of notch geometry, presenting a wide range of possible sizes and shapes on the same plot. It also assists in theoretical developments, in which regard two features of the map are worth mentioning. Firstly, the value of p at which the transition occurs from NOTCH to CRACK regions is almost constant for all large notches. This fact is not evident from the equations until one examines them in detail. The implication is that there is a critical value of p which may be a material constant, and so may be a more appropriate way to define the transition point than the parameter K* Secondly, the behaviour of the plot at very low values of D is interesting. Under the present approach, where Eqn (6) is used but modified when D > p as outlined above, then the CRACK region tends to dominate at low D for the mild steel (Fig. 11). However, the plot for the Enl9 alloy steel (Fig. 12) on which the surface roughness experiments were performed shows a much smaller CRACK region at low D, though in this case the iso-fatigue-strength lines are very steep in parts of the NOTCH region, indicating that predictions from Eqns (5) and (6) will be quite similar. Comparison of Figs 11 and 12 also shows the effect of material properties. At first sight it seems contradictory that for the alloy steel, which is expected to be more F a t i g u e Strength (MPa)
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F~o. 12. Mechanism map for notch fatigue effects in Enl9; fatigue strength here is the fatigue limit stress range at R = 0.1.
DAVID TAYLOR and MANUS O'DONNELL
286
"notch sensitive", the CRACK region is smaller. Crack-like behaviour is indeed more restricted in the case of materials which have higher fatigue strengths or lower threshold values; notch sensitivity is reflected in the fact that there is a larger region of the map over which fatigue strength is sensitive to the value of/9.
6. STRESS CONCENTRATORS WHICH ARE NOT NOTCHES The use of a fracture mechanics analysis assumes the presence of a notch with a definable D value, which can be modelled as a crack of the same depth. However, in practice, many components suffer fatigue failures from stress concentrators which are not notches; examples are fillets, corners and keyways. A detailed discussion of this problem is beyond the scope of the present paper; it is, however, pertinent to present a small number of results generated using a stress concentrator from this group. Figure 13 shows fatigue S/N results for the specimen geometry shown in Fig. 4. The material and test method were similar to those used for the notch results reported above (Figs 5 and 9), the only difference being that the test bars were machined with a fillet instead of a notch, the only variable being the root radius of the fillet. It can be seen that, with the possible exception of the data point at K t = 5.5, all results fall onto a common S/~I curve despite the fact that K t varies from 3.9 to 6.9, indicating that all notches tested lie within the CRACK region. A fracture mechanics prediction cannot be made because there is no D value definable from the geometry; the value of crack length which would best fit the data is small [of the order of 0.5 mm (making no correction for short-crack behaviour)]; this can be regarded as a notional crack length which may be useful but has no physical meaning in terms of the geometry of the stress concentrator.
7. RESIDUAL STRESS AND R-RATIO EFFECTS It should be emphasised that this paper has considered only the effects of notch
geometry, ignoring many other aspects of notches in manufactured components. One feature which should be mentioned is residual stress. Any manufacturing operation will produce residual stress at the notch root; for example, most machining operations result in compressive residual stresses, which commonly take values of several hundred MPa and which vary greatly according to machining parameters. Residual stress can have a profound effect on fatigue behaviour, which will be similar to the effect of an imposed mean stress, though not identical because residual stresses tend Nominal 8trees Range (MPa) 500
5.0 O
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[] 6.9 D
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Number of Cycles to Failure
FiG. 13. S/N curve for fillet-radius specimens of En3b, showing the point.
Kt
factor for each data
Notch geometry effects in fatigue: a conservative design approach
287
to fade away under the effects of cyclic plasticity. Studies of notch geometry should always be carried out using fully stress-relieved specimens in order to avoid this complication. Regarding R-ratio effects in general, most of the data on notches have been obtained at R = - 1 or close to R = 0. In principle the method of analysis advocated here can be used to describe behaviour at any R ratio, provided the appropriate material properties are available (plain fatigue limit, threshold, etc.) at the required R value.
8. CONCLUSIONS (a) Previous research on the prediction of fatigue strength for notched components can be expressed in terms of a choice between two conservative predictions, termed the N O T C H and CRACK predictions. (b) Since these two predictions are both conservative, the most accurate one is always the highest value, and the resulting estimate is always conservative, typically by 10% when compared with experimental data. (c) The method can be extended to cover the case of very small notches, by choosing modifications to the CRACK and N O T C H predictions which are also known to be conservative. Predictions compare favourably with data on surface roughness effects and microscopic notches. (d) This approach can be usefully represented in the form of a mechanism map, which provides a convenient method of assessment of stress concentrators for designers and failure analysts. Acknowledgements--The continuing support of Rover Group (UK) and GEC ALSTHOM (UK) is acknowledged. This project was also partially funded through the EC-BRITE/EURAM programme as part of the " E N D D U R E " initiative.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
N. E. Frost, Proc. Inst. Mech. Engrs 173,811 (1959). N. E. Frost and D. S. Dugdale, J. Mech. Phys. Solids 5,182 (1957). N. E. Frost, K. J. Marsh and L. P. Pook, Metal Fatigue, Oxford University Press, Oxford (1974). R. A. Smith and K. J. Miller, Int. J. Mech. Sci. 20,201 (1978). D. Taylor and O. M. Clancy, Fatigue Fract. Engng Mater. Struct. 14,329 (1991). K. Tanaka and Y. Akiniwa, Fatigue 87, p. 739, EMAS, Warley (1988). The Behaviour of Short Fatigue Cracks (EGF1) (Edited by K. J. Miller and E. R. de los Rios), MEP, London (1986). Short Fatigue Cracks (ESIS Publication 13) (Edited by K. J. Miller and E. R. de los Rios), MEP, London (1992). M. H. E1 Haddad, N. F. Dowling, T. H. Topper and K. N. Smith, Int. J. Fract. 16, 15 (1980). S. Usami, Soc. Mater. Sci. Japan 101 (1985). J. Lankford, Int. J. Fatigue 16, 7 (1980). M. N. James, PhD thesis, University of Cambridge (1984). R. E. Peterson, in Metal Fatigue (Edited by Sines and Waisman), McGraw-Hill, New York (1959). M. R. Mitchell, SAE/SP-79/448, Society of Automotive Engineers (1979). H. Neuber, Kerbspannungslehre, Springer-Verlag, Berlin (1958). M. K. Klesnil and P. L. Lucas, Fatigue of Metallic Materials, Elsevier, Amsterdam (1980). K. J. Miller, Mater. Sci. Technol. 9,453 (1993).
Pergamon
1350-6307(94)00022-0
NOTCH GEOMETRY CONSERVATIVE
EFFECTS DESIGN
IN FATIGUE:
A
APPROACH
DAVID TAYLOR a n d MANUS O ' D O N N E L L Department of Mechanical and Manufacturing Engineering, Trinity College, Dublin 2, Ireland
(Received 4 July 1994) Abstract--The understanding of stress concentrators (notches) is an important element in the prevention of failure in components and in the analysis of failures when they occur. This paper examines critically the methods currently used to predict the fatigue strength of components containing notches, with particular attention to the need for a conservative design approach. Current methods, if correctly applied, are shown to give conservative predictions of fatigue limit and high-cycle fatigue strength, and it is shown that the same philosophy can be extended to cover very small notches, including surface roughness. The presentation of these predictions in the form of a "mechanism map" for notch fatigue is advocated as a useful tool for designers. The problems of extending the approach to other types of stress concentrator, such as fillet radii, are discussed.
1. INTRODUCTION Geometric stress concentrators such as notches exert a pronounced effect on fatigue strength; failures of components almost invariably occur from stress concentrators, making their behaviour critical to the design process. Fatigue strength (i.e. the stress range needed to cause failure after some specified number of cycles) is reduced by some factor, Kf, when the notch is introduced. In some cases Kf is numerically equal to the elastic stress concentration factor for the notch, K t , but in many cases KI < K t , so the use of K t leads to a conservative prediction: one in which the predicted fatigue strength is lower than its true value. Whilst this is good in principle, the degree of conservatism is excessive in some cases, including sharp notches, low-cycle fatigue and low-strength materials. The present paper considers the problem of the fatigue strength of notches as follows. (a) Previous work is reviewed, beginning with that of Frost and co-workers [1, 2]. It is shown that the theoretical approach developed is implicitly conservative. The degree of conservatism is discovered by reference to experimental data. (b) The approach is extended to cover the case of very small notches, using modifications which retain the conservatism of the original methods. Experimental data are used to validate this method. (c) A new method of representation of the data is proposed, in which data from a wide range of geometries are presented on the same diagram, using the "mechanism map" approach. (d) Brief consideration is given to the case of stress concentrators which are not notches, examining data from fillet radii.
1.1. Previous work: notch geometry effects in high-cycle fatigue It has been known for some time that notches can be divided quite distinctly into two categories, displaying different behaviour as regards their tendency to reduce fatigue strength. Figure 1 summarises the effect, showing fatigue strength as a function of K t for the case in which g t is varied by changing notch root radius (p) whilst keeping notch depth (D) constant. A clear division can be made between 275
276
DAVID TAYLOR and MANUS O'DONNELL
"blunt" notches, for which fatigue strength decreases as Kt is increased, and "sharp" notches, for which fatigue strength remains constant despite changes to Kt. A similar transition is apparent if one plots the fatigue strength as a function of notch depth for constant K t (Fig. 2); when expressed in this form it is commonly referred to as a "size effect" for the notch, because the behaviour of small notches differs from that of large ones. The transition point, K* (Fig. 1), is found to be a function not only of material properties but also of notch geometry and number of cycles to failure. This behaviour was reported by Frost and co-workers in a number of papers in the late 1950s (e.g. [1, 2]) and summarised by Frost et al. in 1974 [3]. They argued that blunt notches (Kt < Kt*) could be analysed by dividing the plain fatigue limit by Kt, and that the behaviour of sharp notches (Kt > Kt~') was independent of K t but controlled by D, according to the formula 03D -----constant.
(1)
Smith and Miller [4] reported similar results in 1978 and showed that the sharp-notch condition could be described using a fracture mechanics approach. If the notch is imagined to be a crack of the same depth, D, then the condition for the propagation of this crack to failure becomes AK = Agcrit ,
(2)
where A K is the range of stress intensity applied and A Kcrit is the value required for propagation to failure within the specified number of cycles. In the case of the fatigue limit, AKcrit becomes equal to the threshold value, AKth. For the case of a surface notch the condition becomes 1.12Aa~
= constant.
(3)
Smith and Miller expressed this as 2 A o ' { D = constant,
'o c os
sh,r. No,c,os
STRESS CONCENTRATION FACTOR, K t FIG. 1. Schematic variation of fatigue strength with stress concentration factor, Kt, for a series of specimens in which the notch depth, D, remains constant.
FATIGUE
~i
nstantKt
STRENGTH
NOTCH DEPTH, D FIG. 2. Schematic variation of fatigue strength with notch depth, D, for a series of specimens with constant gt, showing the notch size effect.
(4)
Notch geometry effects in fatigue: a conservative design approach
277
which is numerically similar to Eqn (3). The underlying conclusion is that there is a different mechanism at work in the fatigue failure of blunt notches from that of sharp notches. Blunt notches produce a region of high stress but do not alter the processes of crack initiation, short crack growth and long crack growth which precede eventual failure. Sharp notches, however, act as if they were pre-existing cracks, effectively bypassing the initiation and (unless we are dealing with micro-notches) the short-crack growth stages. Their behaviour is thus governed by the quite different laws of fracture mechanics; this gives rise to, for example, a different mean-stress effect. It is interesting to note from a historical point of view that Frost's original equation for sharp notches [Eqn (1)] was derived before the advent of fracture mechanics in fatigue. In his later work in 1974 [3], the use of AK is considered (Pards-law and threshold data were then available for a range of materials), but rejected on the grounds that the (T3~vf-Dparameter gave a better fit to experimental data, even to data on sharp cracks. This seems surprising until we realise that some of these data were in the range of crack lengths which we would now term "short cracks"; for lengths less than 0.5 mm, Eqn (4) gives poor predictions and so Eqn (1), though purely empirical, was retained.
1.2. Conservative nature of the prediction The above analyses can be viewed in a slightly different light, as shown in Fig. 3. It can be noted that, for any given notch, there will always be two predictions of fatigue strength: one, which will be termed the NOTCH prediction, obtained by dividing the plain-specimen fatigue strength by Kt, and an alternative, termed the CRACK prediction, obtained by assuming that the notch is a crack. It is important to realise that both of these predictions are conservative. The CRACK prediction is conservative because it assumes that the notch is a sharp crack (i.e. that it has zero root radius); the N O T C H prediction is conservative because it ignores local plasticity and the effects of stressed volume and local stress-gradient. It is precisely because both predictions are conservative that we are justified in using the higher of the two in all cases. In fact the experimental data are expected to fall as shown in Fig. 3, coinciding with the predicted values at the two extremes of the plot and lying somewhat above the highest prediction for all intermediate values. This approach has a number of consequences: (a) Its value for designers is greatly enhanced by the knowledge that the method is conservative. In practice many designers are still reluctant to use fracture mechanics for sharp notches, even though Kt is so conservative as to be impractical; the knowledge that they are using a method which is still conservative is of great value. (b) It now becomes important to know the degree to which the predictions are conservative. Intuitively one would expect that conservatism would be highest for Kt values close to K*, where both predictions are at their least valid.
FATIGUE STRENGTH
Constant N o t c h D e p t h , D
~RESULTS PREDICTION CRACK ~- NOTCH
STRESS CONCENTRATION FACTOR, K t FIG. 3. Comparison between typical experimental results (thick line) and theoretical predictions using the NOTCH and CRACK approaches.
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DAVID TAYLOR and MANUS O'DONNELL
(c) A corollary to the above is that the predictions will be improved by using alternative equations which predict higher fatigue strengths provided these alternative methods are also conservative. This indicates a method for refining the approach and, more importantly, for extending it into difficult areas such as micro-notches and low-cycle fatigue.
2. E X P E R I M E N T A L DETAILS 2.1. Materials Two materials were chosen for study: a mild steel (En3b, normalised) and a low-alloy steel (Enl9, quenched and tempered). Tables 1 and 2 give details of composition and mechanical properties, including the stress-intensity threshold obtained from standard load-reduction tests on long cracks. 2.2. Specimens In order to investigate fatigue strengths in the high-cycle region, through-thickness surface notches were machined into bars of length 120 mm, width (in the notch depth direction) 20mm and thickness 12.5 mm. The notches were of uniform depth (2.5 mm) and varying root radius so as to produce g t values of 1.0 (plain bar), 3.0, 4.5, 5.8 and 10.2; this included values below, equal to and above the calculated value of K* Specimens were also produced from this material in which the stress concentrator was a fillet radius instead of a notch (Fig. 4); K t values from 3.9 to 6.9 were obtained by varying the fillet radius only. The Enl9 alloy steel was machined to varying degrees of roughness; rectangular bars were used, of geometry similar to those described above for the mild steel. Different machining methods were used in order to vary roughness. Profile analysis using a Talysurf showed that the depths of the largest notches on each surface varied from 3 to 390 ~m, depending on the machining method used. This work has already been reported in detail elsewhere [5]. All specimens were fully stress-relieved before testing, by annealing in a vacuum at 600 °C for three hours. 2.3. Testing All specimens were tested in four-point bending, using a servo-hydraulic testing machine in load control. The R ratio was kept constant at 0.1; frequencies were in the range 20-100 Hz. Failure was defined as the appearance of a crack at least 2 mm long. Data were collected at various stress ranges with two aims: (a) to define the fatigue limit, and (b) to define the fatigue strength for failure after 1 million cycles. TABLE 1. Composition of materials tested (wt%, balance Fe)
En19 En3b
C
Si
Mn
Ni
Cr
Mo
Cu
Nb
Sn
Co
Al
0.39 0.14
0.27 0.21
0.96 0.83
0.25
0.95
0.30
0.23
0.01
0.02
0.03
0.04
TABLE 2. Mechanical properties of materials tested
En19 En3b
Yield strength (MPa)
UTS (MPa)
Threshold AKth (R = 0.1) (MPa~mm)
980 556
1360 670
5.5 7.0
279
Notch geometry effects in fatigue: a conservative design approach
P/2
P/2
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)
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>
T OT 1)/2
P/2
FIG. 4. Specimen geometry used to investigate the effect of fillet radius. All dimensions in mm. Thickness 12.5 mm.
3. RESULTS AND DISCUSSION 3.1. The degree of conservatism
Figure 5 shows the results from the mild steel, in which the fatigue strength (at 1 million cycles to failure) is plotted as a function of K t for constant D = 2.5 mm. It is evident that the results follow the scheme outlined above (Fig. 3). The results are all conservative; if we define the degree of conservatism as the difference between the measured fatigue strength and the predicted value, expressing this as a percentage of the measured value, the results range from 8 to 25%. Figure 6 shows the degree of conservatism plotted against K t for the present work, and also for various other studies [2, 3, 6]. Data spanning a wide range of K t values were obtained from Tanaka and Akiniwa [6] and from Frost and Dugdale [2], both using steel alloys; in the latter case a value for the threshold of the material was found by consulting the later publication of Frost et al. [3]. Many other datasets appear in the literature on notch fatigue strengths, but in many cases there is no accompanying threshold value. A prediction can still be made, however, for data in the blunt notch region. This enabled data from a number of other studies to be added to Fig. 6, for K t values less than 2.5. These data points, which are labelled "Various", include nine different studies covering ferritic steels, stainless steels, aluminium alloys, copper alloys and zinc alloys, all of which can be found in Frost et al. [3]. It is evident that some degree of conservatism is the general rule; from a total of 70
Constant Notch Depth, D Fatigue Strength (MPa) 500 f 400 300 2O0 17
100
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Stress Concentration Factor Kt FIG. 5. Experimental results for En3b mild steel, showing the effect of Kt on fatigue strength (for 1 million cycles to failure), along with theoretical predictions made using the NOTCH and CRACK methods.
DAVID TAYLOR and MANUS O'DONNELL
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FiG. 6. Degree of conservatism in predicted fatigue strength, taken from previous work [2, 3, 6] and the present investigation (Fig. 5).
results quoted here, only two are non-conservative (by 4 and 10%, respectively) and a further six show 0% conservatism. Most of the results fall within a band in the range 8-20%, which is a very acceptable level of safety from a designer's point of view. There does not seem to be any trend towards an increased conservatism around K* as suggested above (Fig. 3), but this may be masked by measurement errors.
4. SMALL NOTCHES 4.1. Theoreticalapproaches It was noted above (Fig. 2) that a size effect--a variation of fatigue strength with D at constant Kt--is implied by the transition from blunt to sharp-notch behaviour. In addition to this, a quite separate effect will come into play when very small notches are considered. For the case of crack-like notches, the so-called "short-crack effect" has been well characterised by many reports (see, for example, conference proceedings [7, 8]); for cracks less than some critical length, fracture mechanics predictions become non-conservative. Data are commonly represented on a "Kitagawa plot" in which fatigue limit is presented as a function of crack length; Fig. 7 shows one such plot in schematic form. In the present study the requirement is for a means of representing such data in a conservative fashion, in order to allow the CRACK prediction of Fig. 3 to be made when D is small. Critical values of D, below which this correction becomes important, vary greatly from material to material, but can be as high as 2 mm. Thus, this correction is required for all cases of surface roughness, for many manufacturing defects and even for certain design features such as grooves and holes.
FATIGUE STRENGTH OF CRACKED SPECIMEN
EHA VIOUR
(log.scale)
CRACK LENGTH (log.scale) FIo. 7. Schematic illustration of the "Kitagawa" plot showing effect of crack length on measured fatigue strength.
Notch geometry effects in fatigue: a conservative design approach
281
The most commonly-used equation for this short-crack region is that due to E1 Haddad et al. [9], which takes the form Act =
AKth . (5) 1.12~/rr(a + ao) The essence of this approach is that the actual crack length, a, is augmented by some notional value, a o, which is defined so that, when a - - 0 , Act becomes equal to the fatigue limit. Thus, ao is a constant, at least for a given material and crack geometry. It has no physical significance, but its use is convenient for modelling the short-crack situation. In essence the equation proposes that a short crack behaves as if its length was augmented by ao and the laws of L E F M were applicable. This equation is useful in the present study because it can be shown to be conservative when compared to experimental data. The equation must by its nature be conservative for very low crack lengths and for lengths approaching the intersection with the long-crack line. This is because stress values predicted by Eqn (5) will always be lower than the fatigue limit (for finite crack lengths) and also always lower than the stress predicted using the long-crack threshold (though this difference will become increasingly small as crack length increases). It is only by examining experimental data that the conservative nature of the equation can be demonstrated for the mid range of length values. Figure 8 indicates the degree of conservatism achieved by using Eqn (5), when applied to data from a range of short-crack studies [3, 10-12], which cover five different materials, including high- and low-strength steels and copper. In this case the degree of conservatism (defined as above for notch predictions) is plotted as a function of crack length, normalised with respect to ao so as to place all data in the same position relative to the Kitagawa plot. It can be seen that the predictions are conservative for all but two cases, both of which refer to a steel of unusually low strength (try--180 MPa) tested by Usami [10]. The use of Eqn (5) is therefore appropriate in the present study. It should be noted that one set of data, due to Frost et al. [3], was omitted from this plot. These results, obtained using an alloy steel, gave rise to a non-conservative prediction from Eqn (5); however, the problem in this case does not seem to be caused by the choice of short-crack analysis, because the non-conservatism persisted even for large crack lengths of the order of 3 mm. Since ao in this case was equal to 0.042 mm, its effect will be insignificant when considering a 3 mm crack. It seems that these particular data do not conform to a normal fracture-mechanics prediction, despite the fact that sharp cracks were used and that similar studies by the same authors on different materials conformed well. Having established a suitable method for making the C R A C K prediction, we also
%age Conservative 50 40 30 20
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High Strength Steels
+
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~
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FIG. 8. Degree of conservatism in predicted fatigue strength of cracked specimens, using the El Haddad correction for short cracks [Eqn. (5)]; data from various sources [3, 10-12].
282
DAVID TAYLORand MANUSO'DONNELI.
require a conservative N O T C H prediction for small notches. This problem has not been studied to the same extent as the short-crack problem; small, blunt notches are normally dealt with by modifying K t according to some empirical equation based on the root radius of the notch. Typical equations have been proposed by Peterson [13] and Mitchell [14]. The underlying approach to which these equations can be related is Neuber's concept of elementary structural volumes [15[. This approach assumes that failure will occur if the average stress over some volume of material ahead of the notch exceeds some critical value. This leads to an equation of the form Kf = 1 +
Kt - 1
1 + (p,/p)l/2'
(6)
where p is the root radius of the notch and p' is a material constant expressing the size of an elementary volume in terms of a distance ahead of the notch. Peterson [13] and Mitchell [14] used a variation of Eqn (6), thus: Kf =
1 +
K t -- 1
1 + p'/p
.
(7)
Experimental results were used to optimise the choice of p', which was found to be related to the UTS for a range of steels. Neuber also modified his original equation to give gt
Kf-
(1 +
sp'/p) 1/2'
(8)
where s is a constant depending on specimen and stress type, typically equal to 2.0. These approaches have been criticised in the light of more recent knowledge that fatigue is a process of crack initiation and propagation, but current ideas about short cracks may suggest an alternative interpretation. Since it is known that short cracks of length similar to a microstructural unit (e.g. the grain size) will initiate readily even at stresses below the fatigue limit, then the fatigue strength of any notch may be governed by whether there is sufficient stress on this microcrack to enable it to grow. The microcrack size may be equated with Neuber's elementary volume. This exercise was carried out by Klesnil and Lucas [16] who, by considering the threshold condition for a crack of length ao at the notch root, derived an equation as follows: gt
Kf = (1 +
4.5ao/P) 1/~"
(9)
This is exactly the same as Neuber's modified equation except that the elementary volume constant p' is replaced by the El Haddad crack-length constant ao and a factor of 4.5 is added. Interestingly, for the E n l 9 steel used in this work for short-notch examinations, the ratio between ao and p' is 4.22--very close to this factor of 4.5. For our purposes, two difficulties are presented by this type of equation. Firstly, it is clearly inappropriate for sharp notches, where p is small compared to D; in the extreme case of a long, sharp crack of length a, the very small value of p would lead one to predict no reduction in fatigue strength, which is clearly not the case. This problem arises because of the assumption that the stress concentration will decrease to negligible values as one moves away from the notch tip by a distance of the order of p; this is true for blunt notches, but for sharp notches it is more appropriate to say that the stress concentration exists over a distance of the order of D. Equations (6)-(9) can be modified in various ways, the simplest of which is to replace p by D whenever D > p. In practice this is not a serious problem because in the case of sharp notches the C R A C K solution will generally dominate over the N O T C H solution anyway, but it may become a problem for very small notches.
Notch geometry effects in fatigue: a conservative design approach
283
The second problem associated with any of Eqns (6)-(9) is to demonstrate that a conservative prediction is being made, as required by the present approach. The simplest way to be conservative is not to use these equations at all, simply reverting to the use of the full K t factor for all notches, however small. In practice this will lead to the C R A C K equation [Eqn (5)] being the dominant one for most cases of small notches. If a choice is to be made between these four equations, it can be noted that Eqn (6) will be more conservative than Eqn (7) provided that either D or p is greater than p', which in practice is invariably the case for designed notches (including surface roughness) and will almost always be the case for defects, inclusions, etc. Equations (8) and (9) are not usable for very small values of p and D because they lead to Kf < 1. 4.2.
Experimental data: surface roughness
The most stringent test of the predictive equations for small notches is the case of surface roughness. Typical roughness produced by manufacturing operations consists of a series of notches which have a very low Dip ratio, leading to K t values in the range 1.2-1.4, with values of D which are equal to, and often less than, ao, depending on the material. Figure 9 shows data from fatigue limit measurements on Enl9 steel; further details are available elsewhere [5]. Four different machining processes were used to vary D; the data are presented in the same form as the Kitagawa diagram for short cracks, replacing crack length by Dmax (which is the size of the largest surface roughness feature for each surface tested), measured using a stylus instrument. The CRACK and NOTCH predictions are shown, along with corrections for the case of small cracks [using Eqn (5)] and small notches [using Eqn (6), which was found to be the most conservative of Eqns (6)-(9)]. Two N O T C H predictions were required: one for the various machined surfaces (polished, ground and milled) and one for the rougher surfaces which were created using a shaping tool, which had a higher value of the ratio Dmax/pa n d thus a higher value of g t. Figure 10 summarises the results of Fig. 9, showing the degree of conservatism for each prediction. All results are conservative, to a degree that varies from 0.4% (for one of the shaped surfaces) to 16%. Considering the inherent difficulty of defining notch size and shape in this situation, the results are very encouraging. With the sole exception of the polished surface, the NOTCH prediction [Eqn (6)] dominated over the CRACK prediction. The change in predicted fatigue strength obtained by using Eqn (6) rather than the Kt value was small--of the order of 1%--so in this case the simpler, and more conservative, use of Kt is recommended.
Fatigue Limit (MPa)
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Fic. 9. Fatigue limit of Enl9 as a function of notch depth, Dmax, created by varying surface roughness: experimental data and predictions.
284
DAVID TAYLOR and MANUS O'DONNELL %age Conservative 5O 40 30 20
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Shaped Surfaces
FIG. 10. Degree of conservatism of results from Fig. 9.
5. A MECHANISM MAP FOR NOTCHES It is evident from the above discussion that the behaviour of sharp notches differs fundamentally from that of blunt notches. Miller [17] has expressed this by saying that sharp notches are controlled by LEFM, whereas blunt notches (and plain specimens) display behaviour which he terms "Microstructural Fracture Mechanics (MFM)" to emphasise the degree of control exerted by the microstructure. One can state that the mechanism of fatigue is distinctly different for these two types of notches. If we accept that fatigue proceeds through three definite (though overlapping) mechanisms of crack initiation, short-crack growth and long-crack growth, then, in the case of blunt notches, all three mechanisms will occur just as they do in plain specimens. However, for large, sharp notches the first two mechanisms disappear, leaving only long-crack growth. Or, at least, it has been shown that these notches can be analysed as if long-crack growth is the only mechanism operating. Also, as has been shown above, there will be a category of small notches for which short-crack growth will occur first, followed of course by long-crack growth. The concept of a mechanism map was first introduced by Ashby to describe creep behaviour and has since been extended into other areas. The approach is useful in a situation where a failure process (e.g. creep) can proceed through two or more different mechanisms. The dominant mechanism will be a function of two independent parameters in the case of a 2D map (e.g. temperature and applied stress); an area on the map outlines the possible combinations of parameters for which a given mechanism occurs. Furthermore, contours representing some deduced variable (e.g. creep strain rate) can be plotted on the map. The value of the map lies not only in theoretical development of the subject but also in practical use by designers, since it indicates which equation(s) should be used to predict behaviour for a given set of operating parameters. In principle, maps can be plotted with more than two parameters, but visualisation becomes difficult. It is proposed that the mechanism map concept can be used in the area of notch fatigue strength, because it possesses the same ingredients as described above. Figure 11 shows one possible realisation of such a map, using values relevant to the mild steel material tested in this study. In this case the two parameters chosen for the axes relate to notch size and shape. Since the present map covers all semi-elliptic surface notches, the two axes chosen are D and p; for notches of a different basic shape other parameters will be needed, but in many cases two parameters will be a sufficient number. The map is then divided up into two basic regions: NOTCH and CRACK, indicating that we are dealing with a blunt notch or a sharp notch, respectively. Contours can now be plotted of fatigue strength (defined for any chosen number of cycles to failure).
285
Notch geometry effects in fatigue: a conservative design approach F a t i g u e Strength (MPa)
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FI6.11. Mechanism map for notch fatigue effects in En3b; fatigue strength here is the stress range for 1 million cycles to failure at R = 0.I.
The advantage of this type of map for designers is that it provides a ready visualisation of the effect of notch geometry, presenting a wide range of possible sizes and shapes on the same plot. It also assists in theoretical developments, in which regard two features of the map are worth mentioning. Firstly, the value of p at which the transition occurs from NOTCH to CRACK regions is almost constant for all large notches. This fact is not evident from the equations until one examines them in detail. The implication is that there is a critical value of p which may be a material constant, and so may be a more appropriate way to define the transition point than the parameter K* Secondly, the behaviour of the plot at very low values of D is interesting. Under the present approach, where Eqn (6) is used but modified when D > p as outlined above, then the CRACK region tends to dominate at low D for the mild steel (Fig. 11). However, the plot for the Enl9 alloy steel (Fig. 12) on which the surface roughness experiments were performed shows a much smaller CRACK region at low D, though in this case the iso-fatigue-strength lines are very steep in parts of the NOTCH region, indicating that predictions from Eqns (5) and (6) will be quite similar. Comparison of Figs 11 and 12 also shows the effect of material properties. At first sight it seems contradictory that for the alloy steel, which is expected to be more F a t i g u e Strength (MPa)
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F~o. 12. Mechanism map for notch fatigue effects in Enl9; fatigue strength here is the fatigue limit stress range at R = 0.1.
DAVID TAYLOR and MANUS O'DONNELL
286
"notch sensitive", the CRACK region is smaller. Crack-like behaviour is indeed more restricted in the case of materials which have higher fatigue strengths or lower threshold values; notch sensitivity is reflected in the fact that there is a larger region of the map over which fatigue strength is sensitive to the value of/9.
6. STRESS CONCENTRATORS WHICH ARE NOT NOTCHES The use of a fracture mechanics analysis assumes the presence of a notch with a definable D value, which can be modelled as a crack of the same depth. However, in practice, many components suffer fatigue failures from stress concentrators which are not notches; examples are fillets, corners and keyways. A detailed discussion of this problem is beyond the scope of the present paper; it is, however, pertinent to present a small number of results generated using a stress concentrator from this group. Figure 13 shows fatigue S/N results for the specimen geometry shown in Fig. 4. The material and test method were similar to those used for the notch results reported above (Figs 5 and 9), the only difference being that the test bars were machined with a fillet instead of a notch, the only variable being the root radius of the fillet. It can be seen that, with the possible exception of the data point at K t = 5.5, all results fall onto a common S/~I curve despite the fact that K t varies from 3.9 to 6.9, indicating that all notches tested lie within the CRACK region. A fracture mechanics prediction cannot be made because there is no D value definable from the geometry; the value of crack length which would best fit the data is small [of the order of 0.5 mm (making no correction for short-crack behaviour)]; this can be regarded as a notional crack length which may be useful but has no physical meaning in terms of the geometry of the stress concentrator.
7. RESIDUAL STRESS AND R-RATIO EFFECTS It should be emphasised that this paper has considered only the effects of notch
geometry, ignoring many other aspects of notches in manufactured components. One feature which should be mentioned is residual stress. Any manufacturing operation will produce residual stress at the notch root; for example, most machining operations result in compressive residual stresses, which commonly take values of several hundred MPa and which vary greatly according to machining parameters. Residual stress can have a profound effect on fatigue behaviour, which will be similar to the effect of an imposed mean stress, though not identical because residual stresses tend Nominal 8trees Range (MPa) 500
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400
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FiG. 13. S/N curve for fillet-radius specimens of En3b, showing the point.
Kt
factor for each data
Notch geometry effects in fatigue: a conservative design approach
287
to fade away under the effects of cyclic plasticity. Studies of notch geometry should always be carried out using fully stress-relieved specimens in order to avoid this complication. Regarding R-ratio effects in general, most of the data on notches have been obtained at R = - 1 or close to R = 0. In principle the method of analysis advocated here can be used to describe behaviour at any R ratio, provided the appropriate material properties are available (plain fatigue limit, threshold, etc.) at the required R value.
8. CONCLUSIONS (a) Previous research on the prediction of fatigue strength for notched components can be expressed in terms of a choice between two conservative predictions, termed the N O T C H and CRACK predictions. (b) Since these two predictions are both conservative, the most accurate one is always the highest value, and the resulting estimate is always conservative, typically by 10% when compared with experimental data. (c) The method can be extended to cover the case of very small notches, by choosing modifications to the CRACK and N O T C H predictions which are also known to be conservative. Predictions compare favourably with data on surface roughness effects and microscopic notches. (d) This approach can be usefully represented in the form of a mechanism map, which provides a convenient method of assessment of stress concentrators for designers and failure analysts. Acknowledgements--The continuing support of Rover Group (UK) and GEC ALSTHOM (UK) is acknowledged. This project was also partially funded through the EC-BRITE/EURAM programme as part of the " E N D D U R E " initiative.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
N. E. Frost, Proc. Inst. Mech. Engrs 173,811 (1959). N. E. Frost and D. S. Dugdale, J. Mech. Phys. Solids 5,182 (1957). N. E. Frost, K. J. Marsh and L. P. Pook, Metal Fatigue, Oxford University Press, Oxford (1974). R. A. Smith and K. J. Miller, Int. J. Mech. Sci. 20,201 (1978). D. Taylor and O. M. Clancy, Fatigue Fract. Engng Mater. Struct. 14,329 (1991). K. Tanaka and Y. Akiniwa, Fatigue 87, p. 739, EMAS, Warley (1988). The Behaviour of Short Fatigue Cracks (EGF1) (Edited by K. J. Miller and E. R. de los Rios), MEP, London (1986). Short Fatigue Cracks (ESIS Publication 13) (Edited by K. J. Miller and E. R. de los Rios), MEP, London (1992). M. H. E1 Haddad, N. F. Dowling, T. H. Topper and K. N. Smith, Int. J. Fract. 16, 15 (1980). S. Usami, Soc. Mater. Sci. Japan 101 (1985). J. Lankford, Int. J. Fatigue 16, 7 (1980). M. N. James, PhD thesis, University of Cambridge (1984). R. E. Peterson, in Metal Fatigue (Edited by Sines and Waisman), McGraw-Hill, New York (1959). M. R. Mitchell, SAE/SP-79/448, Society of Automotive Engineers (1979). H. Neuber, Kerbspannungslehre, Springer-Verlag, Berlin (1958). M. K. Klesnil and P. L. Lucas, Fatigue of Metallic Materials, Elsevier, Amsterdam (1980). K. J. Miller, Mater. Sci. Technol. 9,453 (1993).