Microsoft Word - numero_44_art_5_AP M. Ciavarella et alii, Frattura ed Integrità Strutturale, 44 (2018) 49-63; DOI: 10.3221/IGF-ESIS.44.05 49 On notch and crack size effects in fatigue, Paris’ law and implications for Wöhler curves M. Ciavarella Politecnico di Bari, Department of Mechanics, Mathematics and Management, Viale Japigia 182, 70126 Bari, Italy mciava@poliba.it A. Papangelo Politecnico di Bari, Department of Mechanics, Mathematics and Management, Viale Japigia 182, 70126 Bari, Italy Hamburg University of Technology, Department of Mechanical Engineering, Am Schwarzenberg-Campus 1, 21073 Hamburg, Germany antonio.papangelo@poliba.it ABSTRACT. As often done in design practice, the Wöhler curve of a specimen, in the absence of more direct information, can be crudely retrieved by interpolating with a power-law curve between static strength at a given conventional low number of cycles N0 (of the order of 10-103), and the fatigue limit at a “infinite life”, also conventional, typically N∞=2·106 or N∞=107 cycles. These assumptions introduce some uncertainty, but otherwise both the static regime and the infinite life are relatively well known. Specifically, by elaborating on recent unified treatments of notch and crack effects on infinite life, and using similar concepts to the static failure cases, an interpolation procedure is suggested for the finite life region. Considering two ratios, i.e. toughness to fatigue threshold FK=KIc/Kth, and static strength to endurance limit, FRR0, qualitative trends are obtained for the finite life region. Paris’ and Wöhler’s coefficients fundamentally depend on these two ratios, which can be also defined “sensitivities” of materials to fatigue when cracked and uncracked, respectively: higher sensitivity means stringent need for design for fatigue. A generalized Wöhler coefficient, k’, is found as a function of the intrinsic Wöhler coefficient k of the material and the size of the crack or notch. We find that for a notched structure, k>m. Possible corrections would need to include the effect of plasticity at the crack tip, which effectively increases the size of the “equivalent crack”, but again this is not pursued in the present paper. The scope of the present paper is therefore to try to “unify” crudely various concepts for static and fatigue design, without any intention to give radically new methodologies, or empirical formulae, but with the simpler scope of examining various ranges of validity and overlap between the theories which often are treated separately, and with principally the suggestion to use interpolation between robust estimates of limit conditions and the use of all the material properties which are available, rather than extrapolation from a single methodology using a limited set of material properties, independently on how refined the methodology may appear to be. This is not necessarily limited to preliminary calculations, but also when there is possibility of some experimental investigations, as a simpler route for understanding of the behaviour in fatigue of a notched component. Ultimately, the core of the message becomes quite obvious to the engineer, and indeed it is the base of various standard procedures for specific fields, like for example the design guides of gears (see for example [13]): “interpolate” between limit conditions, using some knowledge of the notch size effect (in the lack of direct experimental data) as recently emerged more clearly at least for the infinite life region. In particular, the entire spectrum of possible behaviour can be described in a single diagram strength vs. notch/crack size. EMPIRICAL LAWS IN FATIGUE Wöhler curve mpirical laws have emerged in fatigue since when Wöhler was conducting his famous experiments of rotating bending fatigue in railways axles for the German State Railways in the 1860s. Various authors noticed empirically that it was convenient to plot SN data on a log/log (or a semi-log) diagram (for a detailed study of the old literature see the recent paper by Sendeckyj [14]). Since then, the so-called Wöhler SN diagram has been widely used. There is no fundamental reason to write the curve as a power-law, and indeed alternative equations have been suggested, but the power law between 2 given points is probably the simplest or most used form for the plain specimen, in the form (see Fig.1): E M. Ciavarella et alii, Frattura ed Integrità Strutturale, 44 (2018) 49-63; DOI: 10.3221/IGF-ESIS.44.05 52       k k R fN N N0 0 ,  fN N N0 (1) where ∆σ is the stress range (we assume at the moment for simplicity that amplitude and range coincide i.e. the load ratio R=0, although it is clear that in general it would perhaps be appropriate to rewrite Eq.(1) in terms of amplitude of the cycle σ) and the N0, and N are the number of cycles as defined in Fig.1. Clearly, Eq.1 also implies           r N k F k N0 Log Log   (2) and typically for steels considering N∞=107 and N0=103, for FRR0=2 we would have k=13.3, while for FR=3, k=8.4, in the typical range k=6-14 for Al or ferrous alloys. In strain-controlled fatigue, the fatigue curve is replaced by a sum of two power/law functions assuming the fatigue life to be dominated by plastic strain in the LCF regime, and elastic strains or stresses in the HCF. The resulting well know equation (Coffin/Manson) is expected to be more accurate (if anything because it has more degrees of freedom to reproduce the experimental SN curve) although there is still a need to introduce the cut-off thresholds on very low and very high number of cycles, particularly on the low number of cycles where it tends to have the wrong concavity. N R 0 Noo 0 tan( ) = k Figure 1: The simplified Wohler curve. Paris’ law The second important power law in fatigue is Paris’ law, giving the advancement of fatigue crack per cycle, va, as a function of the amplitude of stress intensity factor ΔK (see Fig.2)    ma da v C K dN ;    th IcK K K (3) where ΔKth is the “fatigue threshold”, and KIc the “fracture toughness” of the material. There is therefore no dependence on absolute dimension of the crack. The law is mostly valid in the range 10-5—10-3 mm/cycle, and in a simplified form it can be considered intersecting ΔKth and KIc at 10-6, 10-4 mm/cycle, respectively. This means that the constant C is not really arbitrary, since by writing the condition at the intersections,      m mth Ic C K K 6 410 10 . An alternative form can be obtained considering that Paris’ law is in general valid in the range 10-5—10-3 mm/cycle and hence instead of the constant C it is perhaps more elegant to define a constant ΔK-4, i.e. the range corresponding to a speed of propagation of 10-4 mm/cycle M. Ciavarella et alii, Frattura ed Integrità Strutturale, 44 (2018) 49-63; DOI: 10.3221/IGF-ESIS.44.05 53         m a K v C K 4 ' (4) where C’=10-4 mm/cycle by definition. In other words,         m C C K 4 1 ' (5) From the linearity in this range 10-5—10-3 mm/cycle in the log/log plot, Fleck et al [1] suggest to find the Paris exponent m as kF m 4 Log (6) and Fig.16 of their paper seems to confirm this assumption. More in general, it is possible to assume  c a k th a v m F v Log Log (7) where thav is a conventional velocity at the threshold, and c av at the critical conditions. A first obvious (and well known) link between the two curves (Wöhler and Paris) is obtained when considering the life of a distinctly cracked specimen having an initial crack size ai. Under the assumptions of constant remote stress and no geometrical effects, for m>2 the following is obtained (where the dependence on the final size of the crack af has been removed as relatively not influent)            m m mm fi m a C N 2 /22    2 (8) This is to be considered as a Wöhler curve of the cracked component and the Wöhler exponent turns out to be exactly equal to the Paris exponent, k’=m. It is interesting however to remark that the SN curve depends on the initial crack size, ai. Hence the threshold condition from Eq. (8) would tend not to coincide with that directly obtained from the threshold value which also depends on ai but with a different power      thlim th i K a ,  ; (9) In fact the two powers in Eqs. (8,9) coincide only if (m-2)/2m=1/2 which is only true for very high m, showing in fact that the Paris law should near the threshold have a vertical continuous slope, and the simplification of the Paris law corresponds to a bifurcation to the solution given by the two branches (the threshold, and the power-law regime). This is another example of the risk of using these equations for extrapolations, without considering also the other information we have on the material properties. So far, we have only dealt with the case of either completely uncracked or the distinctly cracked specimen. Most real cases would include notched specimen, or cracks of small size. We therefore need to introduce the theories on the effect of notches and cracks of varying size on fatigue life. Kitagawa and Atzori/Lazzarin diagrams For infinite life (or safe-life) design, Atzori & Lazzarin [5] have recently proposed a new diagram (a generalization of the celebrated Kitagawa diagram), which serves as a single “map” showing the fatigue limit reduction due to notch and cracks as a function of defect (or notch) size. For the interaction between fatigue limit and fatigue threshold for short cracks in the M. Ciavarella et alii, Frattura ed Integrità Strutturale, 44 (2018) 49-63; DOI: 10.3221/IGF-ESIS.44.05 54 Kitagawa diagram El Haddad et al. [15] had proposed the famous interpolating equation (concept of defect sensitivity): for a centred crack of size a, in terms of failure for a range Δσ f         th f K a a0 (10) where a0 is the intrinsic material size for infinite life, defined as          thKa 2 0 0 1 (11) where Δσ0 is fatigue limit and ΔKth is fatigue threshold of the material. In fact it is well known that cracks smaller than this size do not follow Paris law not even for ΔK>ΔKth, whereas the material is limited in this range by the fatigue limit, Δσ0. Figure 2: The Paris law. The denomination “intrinsic crack” is due to the fact that the fatigue limit from (11) is also      th K a 0 0 ) (12) and hence (10) is equivalent to (12) when the intrinsic crack is added. As originally proposed by Smith & Miller [4] any notch is practically equivalent to a crack up to a certain size, depending on the stress concentration factor, Kt. Hence, Atzori & Lazzarin [5] suggested to consider only (i) crack-like behaviour treatable with standard fracture mechanics (in particular, with Eq.(10)) and (ii) large blunt notches only, treatable with the simplest stress concentration factor approach. This is exemplified in the lines of Fig.3. For a constant size of the notch, this criterion can also be put in terms of a limit Kt, Kt* , beyond which fatigue limit is no further decreased, giving an area where cracks are supposed to initiate from the notch but not propagate, the so-called “non-propagating crack zone”. Notches with Kt>Kt* behave as defects of same dimension, i.e. are “crack-like tan( ) = m KTH KIC VC VTH M. Ciavarella et alii, Frattura ed Integrità Strutturale, 44 (2018) 49-63; DOI: 10.3221/IGF-ESIS.44.05 55 notches”. By defining instead of a transitional stress concentration factor, as transitional size of the notch, a*, as the intersection of the horizontal line  tK0 / with the long crack threshold, gives1  t a K a * 2 0 (13) For notches lager than this size a*, simply the peak stress condition can be written in terms of failure range Δσ f   f tK0  / (14) where tK is the stress concentration factor. It is natural to extend these concepts to the static failure case, drawing an El-Haddad “equivalent line” for the static case, and accordingly introduce the dimensions Sa0 analogous to (11) and depending this time by KIc, the toughness of the material and R its tensile strength as          S Ic R K a 2 0 1 (15) Figure 3: The Atzori-Lazzarin generalized diagram (Atzori & Lazzarin [16]). FATIGUE AND CRACK “SENSITIVITIES” AND OTHER MATERIAL PROPERTIES  n Fleck et al [1] and in Ashby [17, 18], a large number of material properties of interest are given, and of particular interest are the “intrinsic crack” sizes, a0, and a0S which can be retrieved qualitatively from some of the maps. They permit to classify “crack sensitivity” of the material, under static and fatigue load respectively (for example, a material 1 More precisely, the intersection should be defined with the El Haddad line not the long crack threshold. The difference can be neglected however, if the stress concentration is not too small. I M. Ciavarella et alii, Frattura ed Integrità Strutturale, 44 (2018) 49-63; DOI: 10.3221/IGF-ESIS.44.05 56 with high a0 will tend to be insensitive to cracks up to the size of the order of a0 in fatigue). Analogously, the two ratios   Ic K th K F K , and     R RF 0 define the “fatigue sensitivities”. Specifically, materials with high FK are fatigue sensitive when cracked, and those having large FR are fatigue sensitive when uncracked. In the former case, in the presence of a crack it is useful to design for fatigue crack propagation (like in the “damage tolerance” design approach), because the static limit is very high and the threshold condition is perhaps too strict, and there is margin to gain from a more elaborate design. Similarly, when FR>>1, it is convenient to design when uncracked for the fatigue limit, or perhaps to the finite life required. The opposite is true when FK, FR are both small and close to one, in which case it is generally sufficient to design statically. Finally, notice that as generally FK>>FR, a material sensitive to fatigue when uncracked is likely to be also sensitive to fatigue when cracked, whereas the vice versa is not true, a material sensitive to fatigue when cracked may not be sensitive to fatigue when uncracked. The two sensitivities (“crack sensitivity” and “fatigue sensitivity”) are not unrelated, as obviously a0S/a0=(FK/FR)2: when FK>>FR as it is usual, a0S/a0>>1 a fortiori. In other words, a material that is more sensitive to fatigue when uncracked than when cracked, then in terms of tolerance to crack sizes, is significantly more sensitive to cracks in fatigue than in static loading. Materials which are equally sensitive to fatigue when cracked or uncracked, would have equal sensitivity to cracks under fatigue or static loads. From the maps in Fleck et al [1] and in Ashby [17, 18], a large number of qualitative data can be retrieved on these material properties and their ratios, as well as the characteristic sizes a0 , Sa0 (which in turn for a given stress concentration factor can be put in terms of a*, Sa * ). For example, two maps are reproduced in Fig.4,5 here. In particular, Fig.4 gives the fatigue threshold vs the fatigue limit (in terms of amplitude endurance limit), and constant lines of         th e K 2 Δ1 4 , which can be put in relationship with the a0 defined in (11). For a0 we recognize values around 1 μm for some ceramic materials, up to few mm for some metallic alloys or polymers), whereas for the corresponding Sa0 we see the value for composite materials, whereas in this particular collection for metals and polymers the yield stress rather than the failure stress is given and hence the plastic radius can be estimated rather than our Sa0 , and finally for rocks and ceramics the compression failure stress is given. In all cases, we notice a certain correlation i.e. grouping around the diagonal line, corresponding to a tendency to have high values for properties at same time (however, within this general trend, there are remarkable exceptions, especially within single class of materials). However, it is seen that this holds more for uncracked properties, i.e. FR is relatively constant for materials (and for the definition of FR in Fleck et al [1] and in Ashby [17, 18] for some metals and polymers, we find FR>1). Vice versa, FK varies significantly more and more still Sa0 and a0 (particularly Sa0 ). In other words, as it is commonly known, to an increase of strength does correspond generally an increase of fatigue strength, but an increase of toughness does not always correspond to an increase of threshold. Moreover, to a greater threshold not always corresponds an increased fatigue limit, and even more the case that to an increase of toughness corresponds an increase of static strength. For example, for steel and metallic alloys, as is well known, to greater yield strength corresponds a reduced toughness, but this is not true for other classes of materials, such as composites, ceramics and cements. In general, Fk>>FR, and for metals typical values are 5-20, and 2, respectively, so that Sa0 is about 100 times greater than a0 . GENERAL WÖHLER CURVE he two Atzori-Lazzarin curves (static, and infinite life) permit some qualitative considerations on the intermediate, finite life, region of notched and cracked structures. The resulting map for general cracked or notched specimen is as shown in Fig.3, as first presented by Atzori-Lazzarin at a conference in Italy [16]. The shaded area corresponds to the “finite life region”. Given a material with Wöhler curve of exponent k, and of Paris exponent m, we expect that the limiting Wöhler curve exponent for a notched component will be the one for a large crack obeying Paris’ law, for which klim=m. Therefore, we can expect a notch of varying size and sharpness to cause a reduction of the Wöhler slope k a* whereas if Kt >FK/FR then a0S< a*.. We shall only consider FK>FR or as a limit case, FK=FR hence we have 3 cases: 1. Case (a) FK>FR and Kt< FK/FR (top of Fig.6 where we see a0< a* < a0S < aS* ) 2. Case (b) FK=FR and Kt >FK/FR=1 (bottom of Fig.6 where we see a0=a0S< a*= aS* ) 3. Case (c) FK>FR but Kt >FK/FR (Fig.7 where we see a0FK/FR in which case the limit ratios is obtained between the static and the fatigue limits, and consequently from (18)         R lim K k F k F Log Log (21) which is clearly the highest slope compatible to our criteria and the material properties ratios. The more general equation analogous to Eq. (20) could be obtained by using Kt>FK/FR in (20) or combining eqt(21) with Eq. (6-7), obtaining in any case                 lim c a th a N N k m v v 0 Log      Log (22) which clearly seems to link the limit generalized Wöhler slope to the Paris slope and the position of the key points in the Wöhler and Paris laws, as it is correct since the limit generalized Wöhler slope is indeed significant in the region where life would be mainly given by propagation. In fact, turning back to the standard assumptions for the key points ( thav , c av =10-6, 10-2 mm/cycle and, perhaps with less generality, N∞=107 and N0=103 cycles), we re-obtain the comforting result that the limiting Wöhler coefficient coincides numerically with the Paris coefficient:      lim k m m' 7 3 2 6 (23) as it was obtained independently from integrating Paris’ law in (8). Turning back to our classification, we have finally a 4th region, where a*FR but Kt FR and Kt< FK/FR and case (b) FK=FR and Kt>FK/FR=1. a0 a0 s a* as* R e tK KIC Kth e tK R 10 3 10 4 10 5 10 6 10 7 N e a0 a0 s a* as* a k Figure 7: The generalized Wohler slope as it results from an example interpolation procedure. Case (c) FK>FR but Kt>FK/FR. M. Ciavarella et alii, Frattura ed Integrità Strutturale, 44 (2018) 49-63; DOI: 10.3221/IGF-ESIS.44.05 61 These two curves (or better the inverse of these two curves) are reproduced in Fig.8,9, as a function of a/a0 for some example cases (typical steel and typical ceramic, where FK=15.5, FR=2.4, and FK=2, FR=1.5, respectively). Since the plots are given as a function of a/a0, the 1/Kf curve “bends” around x=1, whereas the corresponding 1/KS curve “bends” around x= a0S / a0 which in fact scales with the square of the ratio FK/FR = 15.5/2.4=6.5, and FK=2/1.5=1.33, and hence a0S / a0=41.7 and 1.7 respectively, since a0S/a0=(FK/FR)2. This El Haddad form is apparently more complicated, but in fact by repeating the same reasoning of the previous paragraph, we only need to distinguish 2 possible ranges: for aa*, the slope increases again,              R s Ic t Fk k K a a K 0 0 Log / Log Δ / a*as*. The resulting slopes are also indicated as ratio k’/k<1 in the Fig.8,9 for 3 example stress concentration factors Kt=2,5,10, showing how for steel the generalized Wöhler slope is already about 60% of the original one for notches slightly larger than a0 and with stress concentration factor only of about 2. The slope continues to decrease to about 40% when the notch is now significantly larger than a0 (specifically about 20 times larger than a0) and recollecting Eq. 6,2 for the estimate of the Paris and Wöhler slopes, respectively, we have about m=3.4, k=10.5 with the conclusion that the limit reduction of the generalized Wöhler slope is k’/k =32%, and hence with a stress concentration factor of about 5 we’re already very close to the limit slope. For the case of ceramic material in Fig.9, the estimates with eqts. 6,2 give m=13.3 and k=22.7 with the conclusion that the limit reduction of the generalized Wöhler slope is 59%. However, with the same concentration factors as the previous cases, i.e. Kt=2,5,10 we obtain that the decrease of the slope is already almost complete with a notch of the order of 2a0 and with M. Ciavarella et alii, Frattura ed Integrità Strutturale, 44 (2018) 49-63; DOI: 10.3221/IGF-ESIS.44.05 62 the smallest Kt=2. In this respect, the brittle ceramic material is more sensitive to small notches, and this is not entirely a surprise. Figure 8: The generalized Wohler slope as it results from an interpolation procedure using the El Haddad equation for both the static and the fatigue criteria, and for typical material constant ratios of steels. Figure 9: The generalized Wohler slope as it results from an interpolation procedure using the El Haddad equation for both the static and the fatigue criteria, and for typical material constant ratios of steels. CONCLUSIONS ften design is a process which starts from preliminary calculations, with limited degrees of knowledge of materials and their properties. In fact this is not always only a limit of preliminary design stages, since there is never enough knowledge in fatigue of a material, except when a real prototype test is conducted, which in fact is the case for some industries, despite the larger cost of such a test with respect to analytical or numerical “virtual” testing procedures. This paper assumes that the basic Wöhler curve of the unnotched material is known, as well as the basic Paris law of the O M. Ciavarella et alii, Frattura ed Integrità Strutturale, 44 (2018) 49-63; DOI: 10.3221/IGF-ESIS.44.05 63 cracked material. We then proceed to illustrate, from the Atzori and Lazzarin criteria [5, 16], some simple estimates for the generic Wöhler curve of notched specimen. REFERENCES [1] Fleck, N.A., Kang, K.J. and Asbhy, M.F., (1994). 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