Effect of the notch-component relative size on the notch fatigue limit of AISI 304L specimens under push–pull tests

Depósito de Investigación de la Universidad de Sevilla https://idus.us.es/ This is an Accepted Manuscript of an article published by Sage: Balbín JA, Chaves V, Navarro A. Effect of the notch-component relative size on the notch fatigue limit of AISI 304L specimens under push–pull tests. The Journal of Strain Analysis for Engineering Design. 2019;54(7-8):379-389. doi:10.1177/0309324719858894 © IMechE 2019. En idUS Licencia Creative Commons CC BY-NC-ND Effect of the notch-component relative size on the notch fatigue limit of AISI 304L specimens under push-pull tests Journal of Strain Analysis 00(0):1–9 c The Author(s) 2019 Reprints and permission: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/ToBeAssigned www.sagepub.com/ SAGE J.A. Balb´ın, V. Chaves, A. Navarro Abstract Results of a large set of fatigue tests on AISI 304L stainless steel cylindrical round specimens with a circumferential notch of semicircular profile and subjected to uniaxial loading (R= -1) are reported. The outer diameter of the specimens has been kept fixed whereas the radius of the notch has been increased progressively. A whole range of configurations is studied, from combinations of notch radius and specimen diameter where the remaining ligament is much larger than the notch (semi-infinite problem) to configurations where the ligament is much smaller than the notch (finite problem) and where interaction effects have a noticeable influence on the shape of the stress gradient. An S-N curve has been obtained for each configuration. The experimentally obtained fatigue limits for the different notch radii have been compared with predictions made with several methods. However a strong divergence between the experimental values and the theoretical predictions, which were extremely conservative, has been found and this has been attributed to work-hardening and residual stress effects connected with the machining process. It has also been suggested that a geometrical notch strengthening effect, possibly connected with stress tri-axiality due to the notch, and the accompanying hydrostatic tensile stress, may also be a contributing factor. A simple practical engineering method to account for these effects has been proposed. Keywords Fatigue Limit, Notch, S-N Curve, Size Effect, Residual Stresses Introduction Mechanical elements usually contain geometric features such as section changes, grooves, holes, machining marks, etc., usually referred to by the generic term notches. These notches cause stress concentrations in their vicinity which may give rise, in turn, to early fatigue failure of the component when subjected to varying loads. A large number of authors have devoted efforts to the study of notches and the stress concentration in the presence of notches in an analytical or experimental way1–3. And several works have been devoted to fatigue at notches. One may refer, among many others, to the well known articles by Frost, who carried out tests on plates and cylindrical bars with V-notches4, Luk´as et al. who observed the behavior of cylindrical specimens with circumferential notches of semicircular profile5and El Haddad et al. who performed tests on plates with circular holes6. This work focuses on the effect of the relative size of the notch with respect to the overall size of the specimen. We are interested in what happens when the notch size increases to the point of being larger than the ligament and how it influences the fatigue limit. The aim is to show the differences between cases which may be regarded as finite sized and cases which can be seen as (semi)-infinite sized, all achieved through the parameterization of the notch radius. Cylindrical round specimens with a circumferential notch of semicircular profile have been used. The outer diameter of the specimens has been kept fixed whereas the radius of the notch has been increased progressively, giving rise to both configurations where the ligament is much larger than the Figure 1. Circumferentially notched specimens with increasingly smaller ligaments. notch (semi-infinite problem) and configurations where the ligament is much smaller than the notch (finite problem), see Figure 1. When the ligament is small, each side of the notch (in any longitudinal plane) acts as an effective “back surface” to the other side and this has strong implications for the shape of the stress profile throughwhich the fatigue crack must grow. This document provides analyses of the stress gradients at the notch roots as well as experimental results of fatigue tests. Specifically, the variation of the fatigue limit as a function of notch size is studied. The tests were carried out under axial cyclic load in mode I (R= -1) on stainless steel specimens. The experimental fatigue limits are compared Corresponding author: J.A. Balb´ın, Universidad de Sevilla, Escuela Superior de Ingenier´ıa, Departamento de Ingenier´ıa Mec´ anica y Fabricaci´ on, Camino de los Descubrimientos s/n, 41092, Sevilla, Espa˜ na. Email: [email protected] Prepared using sagej.cls [Version: 2017/01/17 v1.20] 2Journal of Strain Analysis 00(0) d D r x y r Figure 2. Detail of the notch for the machined specimens. with estimations made with different methods proposed in the literature. Material The material tested in this work is commercial stainless steel AISI 304L, with low carbon content and it had been characterized previously7. Its composition is (% weight): 0.021 C, 0.029 P, 0.024 S, 0.34 Si, 1.485 Mn, 18.227 Cr, 8.148 Ni, 0.215 Mo, 0.0005 Ti, 0.08 N and 0.39 Cu. The microstructure is formed by equiaxed austenitic grains with some delta ferrite bands and the average austenite grain size is 80 µm. The steel has the following mechanical properties: tensile strength SUT S = 654 MPa, yield strength Sy(0.2%) = 467 MPa and percent elongation 56%. Furthermore, the plain fatigue limit under push-pull tests (R= -1), SF L, is 316 MPa. Geometry The specimens are machined from 22 mm diameter cylindrical rods 4 m long. Each specimen has a circumferential notch with a semicircular profile which is located at the midpoint of the longitudinal axis of the specimen, as shown in Figure 2. Where Dis the outer diameter, or gross diameter, with a fixed value of 8 mm for all the manufactured specimens, dthe net diameter of the specimen in the notch area whose value is defined by the notch radius, r, which is equal to the notch depth. Up to six different notch radii have been used, namely 0.2, 0.6, 1.3, 2.0, 2.5 and 3.0 mm. Machining Specimens were manufactured by turning in a numerically control lathe in two basic steps. First, cylindrical blanks of 8 mm diameter were obtained from the 22 mm supplied rods. A constant spindle speed of 1000 rpm was selected in order to avoid unwanted vibrations. The turning operation was set with a radial depth of 0.5 mm and a feed rate of 0.15 mm/rev. The tools had a carbide coating. Next the notches were cut incrementally by a grooving operation, where the depth is increased in little steps. The machining parameter for this process are radial depth of 0.1 mm, feed rate of 0.12 mm/rev, and again a constant spindle speed of 1000 rpm. The smallest notches had to be manufactured in a different workshop, where the machining parameters were changed Figure 3. Roughness 3D profile extracted from the root of the notch. Table 1. Roughness values. Notch radius (mm) Ra(µm) Rq(µm) 0.60 0.1774 0.22747 1.30 0.2231 0.2786 2.00 0.2011 0.2522 3.00 0.1881 0.2597 slightly: radial depth of 0.7 mm and feed rate of 0.2 mm/rev for the first stage and radial depth of 0.1 mm and feed rate of 0.1 mm/rev for the grooving phase. The spindle speed was 1000 rpm all along. Finally, no heat treatment was applied after machining the specimens. Roughness Surface roughness measurements were taken at the notch root on specimens with notch radius of 0.6, 1.3, 2.0 and 3.0 mm. There was not enough space in the smallest notches to perform the measurements there. A non-contact 3D optical profilometer was used. The surface corresponding to the root of the notch is captured and a Gaussian filter is applied. After that, the curved surface is adjusted to remove the inclination and to obtain a leveled surface as seen in Fig. 3. As a result, values of mean roughness for the area of the captured surface, Ra, and root mean square roughness, Rq, are obtained, as indicated on ISO 25178 (See table 1). As can be seen, the surface finish is quite good in all cases and it should not affect fatigue strengths. Fatigue tests All tests were performed on a servo-hydraulic axial load frame at a frequencyof 8 Hz. The tests were conducted under fully-reversed loading (R= -1), using a sinusoidally varying load. The tests ended either by breakage of the specimen or by run-out, which was established at 3.5×106cycles. The S-N curves were constructed following the method proposed by the Japanese Society of Mechanical Engineers (JSME) that describes the technique to achieve a curve with only 14 tests8. The reason why it was decided to use this method was due to the small available quantity of stainless steel rods to get the specimens machined. The proposed method divides the S-N curve into two sections, an inclined section, Prepared using sagej.cls Balb´ın, Chaves and Navarro 3 consisting of 8 tests divided into four stress levels, and a horizontal section that represents the fatigue limit and it is defined by 6 tests in which the failures and run-outs are alternated. The division in that reduced number of tests for each part is based on obtaining a similar confidence interval in both parts. Sloping part An estimation of the stress at 5·104and 1·106cycles (S1and S2, respectively) is needed to start with this method. Initially, according to the usual recommendation in engineering design books9, the stress at 103cycles is estimated as 0.9·SUT S , and the stress at 106cycles as 0.5·0.85 ·SUT S . After that, with these two points the value of S1is interpolated. From this data, the stress step d1that separates the different stress levels forming this sloping part is calculated. d1=S1−S2 3(1) The first test is done at a stress S=S1−d1. Then, successive tests are carried out by subtracting the stress step d1until the first run-out is obtained (it should be obtained on the third or fourth test in order to not to waste too many specimens). The four stress levels above the stress at which the first run-out was obtained are defined by step d1. Finally, the goal is to get two breaks in each stress level until completing the 8 values needed for this line. Horizontal part The fatigue limit is obtained using the staircase technique in this part. The first run-out obtained in the sloping part, S(1), is taken as the first value of the total 6 required. If more runouts are obtained in the previous part, the maximum stress must betakenas S(1). It is also necessary to calculatea stress step, d2, by means of the product of S(1) and the coefficient of variation of the previously calculated sloping line µ(S). d2=S(1) ·µ(S)(2) Tests are performed following the staircase method, which consists on adding or subtracting the stress step d2if the previous test has been run-out or failure, respectively. Finally, the fatigue limit is calculated by making the arithmetic mean of the last six tests stresses. The S-N curves corresponding to each notch size of the machined specimens were constructed (Figures 4-9) following the ASTM E468 standard10. Fatigue limits are expressed in terms of net stress amplitude and calculated through the equation of elasticity as: Sa= 4F/πd2, where F is the applied force amplitude at the fatigue test. Failures are sketched using crosses and empty circles, and run-outs are printed as solid circles with arrows where the numbers denote the quantity of tests done. In addition, table 2shows the experimental values of notch fatigue limit, SN F L, and fatigue notch factor, Kf, which is defined as the quotient between the fatigue limit of unnotched specimen, SF L, and the fatigue limit of notched specimen: Kf=SF L/SN F L. The relevant thing here is that some of the notch fatigue limits are higher than the plain fatigue limit, which begs for some explanation. But before this, we must look at what the 103104105106107 N cycles 200 300 400 500 600 Nominal Stress amplitude S a (MPa) Run-outs Failures slope part Failures staircase part 2 Figure 4. S-N curve for notch radius 0.2 mm. 103104105106107 N cycles 200 300 400 500 600 Nominal Stress amplitude S a (MPa) Run-outs Failures slope part Failures staircase part Figure 5. S-N curve for notch radius 0.6 mm. 103104105106107 N cycles 200 300 400 500 600 Nominal Stress amplitude S a (MPa) Run-outs Failures slope part Failures staircase part 2 Figure 6. S-N curve for notch radius 1.3 mm. usual procedures for calculating notch fatigue limits tell us in the present case. Prepared using sagej.cls 4Journal of Strain Analysis 00(0) Table 2. Experimental values obtained for notched specimens. Notch radius, r(mm) 0.20 0.60 1.30 2.00 2.50 3.00 Notch fatigue limit, SN F L (MPa) 211.83 232.67 323.40 278.40 406.00 389.53 Kf1.50 1.80 1.58 1.31 1.17 1.08 103104105106107 N cycles 200 300 400 500 600 Nominal Stress amplitude S a (MPa) Run-outs Failures slope part Failures staircase part 2 Figure 7. S-N curve for notch radius 2.0 mm. 103104105106107 N cycles 200 300 400 500 600 Nominal Stress amplitude S a (MPa) Run-outs Failures slope part Failures staircase part 3 Figure 8. S-N curve for notch radius 2.5 mm. Table 3. Stress concentration factors. Notch radius (mm) Kt(FEM) Kt(Handbook3) 0.2 2.83 2.82 0.6 2.39 2.40 1.3 1.80 1.82 2.0 1.41 1.38 2.5 1.24 1.25 3.0 1.14 1.08 Stress gradient at the root of the notch To perform the numerical study of the stresses at the notch, the geometry of the specimen has been modeled by Finite Elements (commercial software Abaqus 6.1311), using the notch radius as a parametric variable. Although the real case corresponds to a 3D solid which represents the machined 103104105106107 N cycles 200 300 400 500 600 Nominal Stress amplitude S a (MPa) Run-outs Failures slope part Failures staircase part 2 Figure 9. S-N curve for notch radius 3.0 mm. specimens, since they are cylindrical, the analysis has been reduced here to a simpler axisymmetric model using CAX8R second-order reduced-integration elements in order to better capture the stress concentration on the curved notch surface and to get more accurate results. In this simplified model, the force Papplied to the upper and lower sections is such that the gross stress Sg= 4P/πD2has a value of 1 MPa. The stress distribution from the notch root to the longitudinal axis of symmetry of the specimen, σyy(X), is computed, as sketched in Figure 10, where σmax is the maximum stress at the notch root and σ(L/2) the stress at a distance L/2from the notch root, where Lcorresponds to the critical distance defined by Taylor12,13 (see below). The finite element model has been solved for a large numberof cases, specifically from notch radius r= 0.1 mm to r= 3.5 mm with a step of 0.1 mm. As is usual with this type of geometries, we define the nominal stress Snas the force in any circular cross-section divided by the ligament area (the area of the smallest section), Sn= 4P/πd2. The stress concentration factor, Kt, is the ratio between the maximum stress at the notch root, σmax, and the nominal applied stress Sn, Kt=σmax Sn (3) Figure 11showsthe stress σyy versus the distance from the notch root for the geometries of the manufactured specimens. The values are scaled so that the nominal stress Snis equal to 1 MPa in all cases, so that the stress concentration factors can be read off directly from the intersection of the curves with the vertical axis. The values compare pretty well with those obtained using Peterson’s stress concentration factors handbook3, see Table 3. It can be seen that the stress concentration factor decreases as the notch size increases. It is also worth noticing that the stress gradient is much more pronounced in the smaller Prepared using sagej.cls Balb´ın, Chaves and Navarro 5 Sg Sg L/2 X σmax σyy D/2 0 σ(L/2) r Figure 10. Sketch of the σyy stress gradient at the notch root. 0.00 0.03 0.06 L/2 = 0.09 0.12 0.15 0.18 1.0 1.5 2.0 2.5 3.0 r = 3.0 mm r = 2.5 mm r = 1 .3 m m r = 2.0 mm r = 0 .6 m m yy (M Pa ) X (mm) r = 0 .2 m m Figure 11. Stress gradient at the notch root for several notch radii. notches. The slope gets smaller and smaller as the notch size increases. It can be anticipated that this will play an important role later on when critical distance ideas are used to calculate fatigue strengths. Notch fatigue limit The techniques most frequently used nowadays to assess the fatigue strength of notched components are based on the idea of a critical volume or distance proposed by Neuber14 and Peterson15. The point or line methods introduced in recent decades by Taylor12,13 are also based on the same idea. These methods consider that the fatigue strength of a notched element does not depend only on the maximum stress at the notch root, but rather on some average of the stress in a certain volume of material located just in front of the notch. Failure of the component occurs if this average stress is higher than the fatigue limit of the material obtained in plain specimens. The notch fatigue limit is the value of the applied alternating nominal stress that results in fatigue failure of the specimen. The critical distance methods mentioned above are used here to estimate the notch fatigue limit for each specimen geometry. We also use the formulae based on the short crack growth model of Navarro and De los Rios16 (N-R model). Neuber’s method Neuber developed an empirical expression for this factor Kf under fully-reversed load conditions (R=−1). Kf= 1 + Kt−1 1 +pρ/r (4) Where ρis a characteristic length that depends on the material. The values of ρfor steels can be easily obtained since they are tabulated in a later work17. Peterson’s method Peterson defined another semi-empirical approximation3for Kf, Kf= 1 + Kt−1 1 + a/r (5) Where ais also a characteristic length of the material with its corresponding empirical equation18,19. a= 0.0254 ×2070 σUT S 1.8 (6) Taylor’s Point Method Taylor has extended immeasurably the critical distance methods by providing an explicit formula to calculate the characteristic length Lof the material using the Threshold value of the Stress Intensity factor and the plain fatigue limit. The details are well known. In his point method, the stress at a single point located at a distance L/2from the notch root is compared with the plain fatigue limit. For the AISI 304L stainless steel used in this study, a value L= 0.18 mm was obtained in our laboratory by Beretta20. N-R model for notches Departing from the critical distance paradigm, Chaves et al.21 describe a methodology which allows an easy calculation of the fatigue limit for any notched geometry. This technique combines the basic ideas of the N-R model, which explicitly represents the fatigue crack by a continuous distribution of dislocations, with Finite Elements computations of the un-crackedgeometry to obtain the stress gradient through which the crack must grow. Comparisons Figure 12 shows the values of the stress concentration factor, Kt, calculated by the finite elements model for the whole range of notch sizes. Also, the estimations of the fatigue notch factor Kfmade with all the methods referred to above are shown. Both factors are represented against the value of the dimensionless notch radius. It can be seen that while for very small notches the stress concentration factor Kttends to 3.065 (as expected, see the table and discussion on page 22 of Peterson3), the fatigue Prepared using sagej.cls 6Journal of Strain Analysis 00(0) 0.0 0.2 0.4 0.6 0.8 1.0 1.0 1.5 2.0 2.5 3.0 K t K f - Taylor K f - Neuber K f - Peterson K f - N-R model K f K t 2r/D Figure 12. Stress concentration factor Ktcalculated by finite elements and fatigue notch factor Kfobtained by several methods. notch factor Kftends to 1.0, for obviously a very small notch should not have any detrimental effect in fatigue. It is in the very small notch region where the different theories for the fatigue limit differ. For larger values of the notch size, all theories converge together and they also converge to the curve belonging to the stress concentration factor, indicating that notch sensitivity tends to one for large notch root radii, as is well know. Of course, the difference between the Ktand Kfcurves should also depend on the material, since there is also a scale factor that should arise when comparing the size of the notch and the size of the underlying microstructure of the material. This is born out by Figure 13, where fatigue notch factors Kfcalculated by the Point Method for different value of the critical length L(within a typical range of steels) are represented. We notice that as the notch becomes sufficiently large compared with each critical length, the corresponding Kf curve approaches closely the one corresponding to Kt. Doubtless, any other measure of the microstructural size, for example the grain size of a metal, could be used, and, in fact, similar calculations performed with N-R model varying the grain diameter lead to the same conclusion, namely, that for notches much larger than the typical microstructural length, the microctructure would be “seen” by the fatigue crack as just a continuum so that the difference between Kfand Kt should vanish. We come finally to an interesting feature in Figure 11 which has probably not gone unnoticed: if we check the intersections of the curves with the abscissa corresponding to the critical half-Lfor our AISI 304L stainless steel, L/2 = 0.09 mm, it would be noticed that the value for radius r= 0.2mm is lower than that for r= 0.6mm. Then the intersection for r= 1.3mm is noticeably smaller than for r= 0.6mm but not too different than for r= 0.2mm. Radii rlarger than 1.3 mm gives values progressively smaller. This implies that notched fatigue limits predicted should be pretty similar for r= 0.2mm and r= 1.3mm and both should be greater than that calculated for r= 0.6mm. In fact, if we plot the calculated notch fatigue limits against the size of 0.0 0.2 0.4 0.6 0.8 1.0 1.0 1.5 2.0 2.5 3.0 K t K f (L = 0.02 mm) K f (L = 0.06 mm) K f (L = 0.18 mm) K f (L = 0.24 mm) K f (L = 0.50 mm) K f 2r/D Figure 13. Ktand Kffactors calculated by finite elements for different critical lengths L. the notch, the resulting curve has a minimum around r= 0.6 mm, see Figure 15. Our experimental results tell otherwise, as it is apparent in the figure, which also shows that the experimental notch fatigue limits for notch radii of 2.5 and 3.0 mm turn out to be significantly higher than the plain unnotched fatigue limit, as has already been mentioned. For the 1.3 mm radius, the notch fatigue limit is just marginally higher than the plain fatigue limit. Except perhaps for the smallest notch, where in particular Taylor method is very accurate, the notch fatigue limits estimated with the different methods are very conservative. Discussion Although certainly striking, notch fatigue limits above the un-notched ones have been reported before for austenitic steels22. Frost et al.23 (p. 141) ascribe this to the fact that the machining process used to cut the notch may work-harden the material around the notch and introduce compressive residual stresses there too. They state “...The notched fatigue limit of specimens in which the material at the notch root has been heavily work-hardened and contains high-compressive residual stresses may well be higher than that obtained on carefully machined and stressrelieved specimens of the same geometry and material. Indeed, the conventional notched fatigue limit of heavily machined notched mild steel (or any material which is easily work-hardened) specimens may be double that of carefully machined and vacuum stress-relieved specimens of the same geometry ...”. This might indeed be the case here. It should be obvious that the analyses reported above for calculating notch fatigue limits, which are based on the plain fatigue limit of the un-hardenedand free of residual stresses material and the Ktvalue of the notch are doomed to failure. They must obviously be enhanced by somehow incorporating residual stresses and work-hardening of the material. There are, however,thorny theoretical issues involved even if the distribution of residual stresses and the work-hardening rate of the material in the notch root area were known, such as how to account for the possible relaxation of the residual stresses and how they interact with the work-hardening Prepared using sagej.cls Balb´ın, Chaves and Navarro 7 Figure 14. Fracture surfaces of an specimen broken in a tensile test. process. Moreover, hopes of calculating in a reliable manner the residual stresses and the degree of work-hardening introduced when manufacturing a particular component in practice are dim at best, given the uncertainties reported even for carefully controlled machining experiments24. And of course, measuring residual stresses is both time consuming and expensive,so that it can only be consideredfor the design of the most critical components. Other than fatigue testing “...Heavily machined notched specimens to obtain data representative of the behavior of a component having a notch or discontinuity machined in a similar manner ...”, as advised by Frost et al.23, we would like to see if an, admittedly crude, procedure based on simpler static testing may be employed to estimate fatigue strengths of notched components made out of materials that may exhibit this behavior. The idea is just to re-use the classical analyses but correcting the “plain fatigue limit” data employed for each notch, in recognition that the process of machining the notch has resulted in a somehow stronger material, different for each notch. Static tensile tests of the notched specimens were undertaken and the plain fatigue limit estimated by a customary rule. Table 4shows the results which are referred to net section, again. Unfortunately, at this stage of the investigation, we could only afford to test a single specimen for each notch size, so that no estimation of the scatter of the results can be given. For our particular geometry, the notch tensile strengths are seen to increase with the radius of the notch up to the two larger values, when it starts to fall, due probably to very small ligament for these geometries. All the notch tensile strengths are higher than the (plain) tensile strength. All fractures surfaces for the notched tensile tests were of the cup and cone type, displaying a definite ductile behaviour. An example can be seen on Fig. 14. We can see that for the “un-notched” material the ratio between the fatigue limit and the ultimate tensile strength is 316/654 = 0.48, pretty near the canonical value usually employed. Keeping this value fixed and using the tensile strengths in table 4, new reference “plain fatigue limit” data have been calculated for each notch size and the corresponding notch fatigue limits have been re-calculated. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 100 200 300 400 500 600 S FL = 316 MPa Fat igu e li mi t (M Pa) r Taylor Peterson Neuber N-R model Experimenta l Figure 15. Experimental notch fatigue limits and theoretical estimations. Only Taylor’s point method has been used for this exercise∗. The results are plotted in Fig. 16. Also, and just for the sake of comparison, the result of estimating an “average” plain fatigue limit data using the mean value of the notch tensile strength (865 MPa) for all notch radii is also depicted in the figure and compared with the new estimated plain fatigue limit which has been calculated applying 865 ·0.48 = 415.2 MPa. The simple method suggested seems to show some promise. A more practical alternative to assess the state of hardening in the notch area may be to measure hardness directly, rather than having to break the component in order to obtain an estimation of the notch tensile strength. Arguably the latter procedure would provide an averaged value through the notched section that may be thought to be more representative. The hardness value would give a local estimate. Perhaps taking several readings at different points along the notch profile and averaging them might be satisfactory. The chosen representative value could then be converted to an equivalentapproximatenotch tensile strength via conversion tables such as those given, for example, in the ISO 18265:2013 standard. Care must be exercised, though, when using any such conversions. Just to illustrate the feasibility of this procedure, Vickers hardness measurements were taken from a few specimens. It was only possible to measure specimens of notch radii 0.20, 2.50 and 3.00 mm, since the rest had all been already used for the fatigue and tensile tests. The HV10 values obtained were respectively 231, 293 and 266 for the mentioned three radii. Using Table A.1 from ISO 18265:2013,these values translate to 745, 942 and 855 MPa, which compare well with the respective notch tensile strengths values of 756, 890 and 857 (see table 4). We would like to stress that is just an example circumscribed to the present study. ∗We assume the critical distance does not change. However, if this were connected with a physically meaningful length, such as the grain size, it might happen that work-hardening is associated with a change (probably a decrease) of this length. This, obviously, has not been taken into account here. Prepared using sagej.cls 8Journal of Strain Analysis 00(0) Table 4. Experimental and re-calculated values for notch fatigue limit and ultimate tensile strength for each different notch radius. Notch radius (mm) – 0.20 0.60 1.30 2.00 2.50 3.00 Initial notch fatigue limit (MPa) – 217.01 232.66 323.45 278.42 406.01 389.53 Re-calculated notch fatigue limit (MPa) – 244.51 203.83 232.84 280.32 313.09 339.13 Ultimate tensile strength (MPa) 654.00 756.00 851.00 937.00 900.00 890.00 857.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 100 200 300 400 500 600 S FL = 415.2 MPa Fat igu e li mi t (M Pa) r Taylor Experimenta l Individual estimation Figure 16. Notch fatigue limits calculations based on the updated estimations of the plain fatigue limit. Comparison with experimental results.. Finally, we would like to close this discussion by thanking the two anonymous reviewers who have convinced us that mention should also be made of a further possibility to account for the strong divergence between the experimental values and the extremely conservative theoretical predictions: a geometrical notch strengthening effect, possibly connected with the stress tri-axiality due to the notch, and the accompanying hydrostatic tensile stress, may also be a contributing factor. We had discarded this explanation on two accounts. First, because fatigue strength is known to be negatively affected by the superposition of a hydrostatic tensile stress: it is thought that hydrostatic tensile stresses will promote the opening of fatigue cracks and thus reduce fatigue resistances. And this is exactly the opposite of what we are reporting here. Furthermore, while the strengthening effect of the notch in the tensile test has been recorded for many ductile materials, the realization of notched fatigue limits above the plain one has only been seldom reported. We have thus concluded that there must be something else at play. But having said this, we must nevertheless admit that perhaps there are some other effects upon fatigue associated with tri-axiality that we are not yet able to identify. For other materials or notch geometries, this can certainly be a important influence and must, therefore, be taken into careful consideration. The use of the notched tensile strength may still be a useful way of introducing in the fatigue criterion the influence of the tri-axial stress state caused by the presence of the notch. Conclusions •A large set of fatigue tests on AISI 304L stainless steel cylindrical round specimens with a circumferential notch of semicircular profile and subjected to uniaxial loading (R= -1) are reported. •The effect of the size of the notch relative to the overall specimen size has been analyzed, by keeping fixed the outer diameter of the specimens while progressively increasing the radius of the notches. •An S-N curvehas been obtained for each configuration and the experimentally obtained fatigue limits for the different notch radii have been compared with predictions made with several methods. •But a strong divergence has been found between the experimental values and the theoretical predictions which were extremely conservative, and this has been attributedto work-hardeningand residual stress effects connected with the machining process. •A simple practical engineering method to account for these effects has been proposed. Acknowledgements The authors would like to thank the Spanish Ministry of Education for its financial support through grant DPI2014-56904-P. References 1. Nowell D, Dini D and Du´o P. Stress analysis of V-notches with and without cracks, with application to foreign object damage. The Journal of Strain Analysis for Engineering Design 2003; 38(5): 429–441. 2. Filippi S, Lazzarin P and Tovo R. Developments of some explicit formulas useful to describe elastic stress fields ahead of notches in plates. International Journal of Solids and Structures 2002; 39(17): 4543–4565. 3. Peterson RE. Stress concentration factors. John Wiley and Sons, 1974. 4. Frost NE. A relation between the critical alternating propagation stress and crack length for mild steel. Proceedings of the Institution of Mechanicals Engineers 1959; 173(35): 811–827. 5. Lukas P, Kunz L, Weiss B et al. Non-damaging notches in fatigue. Fatigue and Fracture of Engineering Materials and Structures 1986; 9: 257–267. 6. El Haddad MH, Topper TH and Smith KN. Prediction of nonpropagating cracks. Engineering Fracture Mechanics 1979; 11: 573–584. 7. Chaves V, Navarro A and Madrigal C. Stage I crack directions under in-phase axial–torsion fatigue loading for AISI 304L stainless steel. International Journal of Fatigue 2015; 80: 10– 21. 8. Nakazawa H and Kodama S. Statistical S–N Testing Method with 14 Specimens: JSME standard method for determination of S–N curves. In Tanaka, T, Nishijima S and Ichikawa, M (ed.) Statistical Research on Fatigue and Fracture,Current Prepared using sagej.cls