Directions of high cycle fatigue cracks emanating from circular notches studied by optical profilometry

International Journal of Fatigue 165 (2022) 107117 Available online 13 July 2022 0142-1123/© 2022 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/bync-nd/4.0/). Contents lists available at ScienceDirect International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue Directions of high cycle fatigue cracks emanating from circular notches studied by optical profilometry J.A. Balbín a,b,∗, V. Chaves a, N.O. Larrosa c, C. Madrigala, A. Navarro a aDepartamento de Ingeniería Mecánica y Fabricación, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain bDepartamento de Ingeniería Minera, Mecánica, Energética y de la Construcción, Escuela Técnica Superior de Ingeniería, Universidad de Huelva, Campus Universitario de El Carmen, 21007 Huelva, Spain cDepartment of Mechanical Engineering, University of Bristol, Bristol BS8 1TR, England, United Kingdom ARTICLE INFO Keywords: High cycle fatigue Notch Crack path Crack initiation point Crack direction ABSTRACT Current models for predicting the fatigue endurance of notched solids use the stresses along a straight line, beginning at the notch root, as a simplification of the real crack propagation path. In this work, the experimental crack paths for hollow notched samples were analysed through different microscopy techniques, with the objective of establishing high cycle fatigue crack growth directions in a mild steel. Fully reversed tension–compression fatigue tests (𝑅= −1) of thin-walled tube specimens with a passing-through hole were carried out. The crack paths observed in the outer cylindrical surface were studied in each case, with special attention to the crack initiation point and the crack direction along the first grains. Moreover, the analysis of the fracture surfaces allowed the same analysis to be performed to determine the internal crack paths. It was observed that the crack initiation point was close to the maximum principal stress point at the hole contour as obtained from linear elastic finite element analysis, and the crack direction in its initiation was generally close to Mode I direction, contrary to the conventionally accepted 45◦crack growth direction. 1. Introduction In polycrystalline materials, such as metals, fatigue crack initiation generally occurs near the plane of maximum shear stress, i.e. in the mode II direction, and then crack propagation carries on near the plane perpendicular to the maximum tensile stress, i.e. in the mode I direction. These two phases of crack growth are commonly referred to as stage I and stage II of fatigue crack growth, respectively, which were analysed in depth by Forsyth [1,2]. In [1], Forsyth shows an excellent photograph for a pure cold rolled aluminium alloy, in which these two phases are clearly observed. The stage I to stage II transition is not yet well established. According to Forsyth, stage I ceases when the crack meets a slip obstacle such as a grain boundary, so the crack commonly changes to normal growth when the tip of the crack reaches the first grain boundary [1]. Miller indicates that the transition occurs usually at a crack depth of 3-to-5 grains [3]. However, reality does not seem so simple: Forsyth himself indicates that fractographic evidences suggest that during stage I the crack may deviate from the maximum shear plane, following structural inhomogeneities [2], and that under certain conditions, stage I may not occur in a detectable way, as it happened ∗Corresponding author at: Departamento de Ingeniería Mecánica y Fabricación, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain. E-mail address: [email protected] (J.A. Balbín). with an aluminium–zinc–magnesium alloy, in which the iron aluminide particles inhibited the sliding of dislocations and caused the crack not to start at 45◦but normal to the tensile stress [1]. The presence of a notch, characterised by a high stress region close to the notch root, from where the crack generally initiates, might modify stages I and II of fatigue crack growth. According to Forsyth, the combined effect of a notch and high mean stress usually encourages stage II, with no presence of stage I [2]. There are several experimental studies in the literature about the crack propagation direction during the initial period of crack growth in notched solids, and they are inconclusive: Frost studied non-propagating cracks from circumferential sharp V-notches in cylindrical specimens under reversed axial and rotating bending loading [4,5] for mild steel and aluminium alloy. Unbroken specimens were sectioned in the longitudinal direction and the non-propagating cracks were examined at the notch root with an optical microscope. Of the 19 cracks analysed and shown in the tables and figures of these articles, most were formed at 90◦with respect to the load direction, that is, in the mode I direction, typical of stage II. Although there were also several cracks that were formed at 45◦ https://doi.org/10.1016/j.ijfatigue.2022.107117 Received 1 April 2022; Received in revised form 14 June 2022; Accepted 30 June 2022 International Journal of Fatigue 165 (2022) 107117 2 J.A. Balbín et al. (in mode II), 2 at 120◦and 1 at 30◦. Meneghetti et al. studied low carbon steel plates weakened by U-notches [6]. The specimens were tested under axial cyclic loading and the crack paths were inspected with a microscope on the two surfaces of each specimen. The angle of crack initiation, measured with respect to the notch bisector, ranged from 16◦to 30◦and the average value was equal to about 25◦. Tanaka et al. studied thin-walled tubular specimens of mediumcarbon steel with a hole drilled in the middle of the specimen, subject to in-phase and out-of-phase combinations of cyclic torsion and axial loadings [7]. The cracks, observed with an optical microscope on the specimen surface, propagated almost straight for all cases examined. Berto et al. [8] tested circumferentially V-notched specimens made of hardened and tempered steel. As in the case of Frost, some run-out specimens were sectioned in the longitudinal direction and the paths of non-propagating cracks were examined. As seen in two pictures of the article, for two axial load tests, the cracks had approximately the mode I direction from their beginning. Gates and Fatemi conducted multiaxial fatigue tests on thin-walled aluminium tubular specimens with a circular transverse hole and observed cracks on the specimen surface with an optical microscope [9]. The cracks nucleated and grew on planes of maximum shear for, at least, 100 μm before turning to grow on the maximum principal stress plane. Lorenzino and Navarro tested Al1050 aluminium alloy plates with a circular hole under axial loading [10]. The material had a large grain size, with the notch size of the same order or even smaller than the grain size. They studied the cracks growing from the hole on the two specimen surfaces with two optical microscopes and found that the crack paths were very irregular in their initial part, corresponding to the first grains, being highly affected by the microstructure of the material. Based on all these results from the literature, it is difficult to establish what stages I and II are like in notches, with cases of cracks growing from the beginning in mode I, others with cracks initiating in mode II, and a few cases with cracks initiating in mixed mode direction. This may be just as well, for this great experimental diversity of stage I directions from notches could justify the use of very different crack paths in current notch multiaxial fatigue limit models: mode I direction [11], mode II direction [12] or a mixed-mode direction [13–15]. In addition, these experimental studies from the literature analysed the crack initiation direction on the external surface of the specimens [6,7,9,10] and, in the case of the works by Frost [4,5] and Berto et al. [8], in a longitudinal section of the cylindrical specimens. However, it is likely that the crack started somewhere inside the specimen and not on the external surface or, in the case of the specimens analysed by Frost and Berto et al. in a different section from the one studied. This implies that the directions studied in these works [4–10] may not be exactly representative of the crack initiation directions as the cracks could have initiated in a plane different from the studied ones. In recent years, there has been a great advance in the field of microscopy. Specifically, 3D profilometers have been developed, which allow the determination of the surface topography. This type of equipment has been used in a recent experimental campaign to analyse the fracture surfaces of notched specimens subjected to cyclic loading and to record the evolution of the crack direction in the internal planes of the specimens, and not only in the plane of the external surface of the specimen. It allows a more precise knowledge of stages I and II of fatigue crack growth from notches. This experimental campaign began with a study on AISI 304L stainless specimens [16] and continued with 7075 aluminium alloy specimens [17]. The present document reports the experimental study on a third material, a mild steel. The geometry was a thin-walled tube with a passing-through hole under axial loading. The crack paths were studied on the external surface of the specimen and also on the fracture surfaces. The results of the present research can lead to a better understanding of the crack paths in notches and to a better selection of the crack lines in notched fatigue limit models so that they are truly representative of the experimental crack paths. Fig. 1. Granulometry of S355 low carbon steel. Fig. 2. Geometry of the notched specimen. 2. Material and tests The material used in this work is commercial low carbon S355 steel (also known as St52). Its chemical composition (weight %) is: 0.18 C, Mn 1.28, Si 0.30, P 0.03, S 0.02, Cr 0.18, Ni 0.06, Al 0.025, Mo 0.01. The average grain size is 33 μm (see Fig. 1). No heat treatment was applied after the machining of the specimens. The monotonic properties, as determined from 20 tensile tests, are as follows: tensile strength 𝜎𝑈𝑇 𝑆 =586 MPa, yield strength 𝜎𝑌 𝑆 =412 MPa, Young’s Modulus 𝐸=208 GPa. Vickers hardness equals to 187.9 HV. This material combines good fatigue behaviour with low environmental impact for non-transport applications [18]. Firstly, fatigue tests under fully-reversed axial loading (𝑅= −1) were conducted on smooth cylindrical specimens in a servo-hydraulic axial load frame at 6–8 Hz. Then, notched specimens were manufactured for fatigue tests. The geometry of the notched specimen was a thin-walled tube of 1.5 mm thickness with a passing-through hole in the central section. Three different hole diameters 𝑑=0.8, 1.5 and 3.4 mm were done, in order to check whether there was a hole size effect on the experimental crack paths or not. As for the smallest hole size, a diameter 𝑑=0.8 mm was chosen. This diameter fulfilled the basic desired conditions: to be machined with a conventional drilling International Journal of Fatigue 165 (2022) 107117 3 J.A. Balbín et al. Fig. 3. Example of 3D captured hole surface from which surface roughness was measured. machine and to meet the tolerances proposed in the drawings. Regarding the largest size, its choice was based on achieving a geometry which had an elastic stress field in a plane perpendicular to the hole similar to that of an infinite plate with a hole of the same size. This would greatly facilitate future parametric predictions with multiaxial notch fatigue models, as the Kirsch’s equations [19, p. 91] could be used. It is the effect of the curvature of the cylinder that makes the stress field in the vicinity of the hole different from that of a hole in an infinite plate. Although if the cylinder is large enough and the transverse circular hole is small enough, then the difference in stresses in both cases becomes very small. For the cylindrical specimens tested in the present work, with an outside diameter of 16 mm, elastic studies for axial, torsional and biaxial loads (𝜎∞ 𝑦=𝜏∞) indicated that for a circular hole with a diameter not greater than 3.5 mm, the stress gradient ahead of the hole was very similar to the case of an infinite plate with a hole, the difference in 𝐾𝑡being less than 5% [20]. As a result, a maximum hole diameter of 3.4 mm was chosen to achieve a similar stress field to that in the infinite plate configuration. Finally we chose an intermediate diameter of 1.5 mm. Fig. 2 shows the geometry of the specimen. These notched specimens were tested in a resonance fatigue testing machine, at approximately 150 Hz for fully-reversed axial loading (𝑅= −1). Surface roughness measurements were taken at the drilled hole to assess the surface finishing of this stress concentrator element. A noncontact 3D optical profilometer was used. First, the internal surface of the hole was captured (see Fig. 3), and then a Gaussian filter was applied. After that, the curved surface was adjusted to remove the inclination and to obtain a levelled surface. Three specimens were chosen from each of the hole sizes 𝑑= 0.8 mm and 𝑑= 1.5 mm and three roughness measurements were taken at the bottom of the hole on each sample. Values of mean roughness for the area of the captured surface, 𝑅𝑎, root mean square roughness, 𝑅𝑞, and maximum roughness, 𝑅𝑧, were obtained as indicated in ISO 25 178. Mean roughness and maximum roughness presented an average value of 0.66 μm and 9.49 μm, respectively. 3. Experimental results Fatigue tests results were used to construct the S-N curves in accordance with ASTM E739 [21], assuming a linear relationship between the variables 𝑙𝑜𝑔(𝜎)and 𝑙𝑜𝑔(𝑁). Fatigue tests results are plotted in Fig. 4 where stress values are referred to the net section. Hollow circles refer to broken specimens and solid squares refer to run-out tests. The stress is expressed in terms of stress amplitude, calculated by the expression 𝜎=𝐹∕𝑠, where 𝐹is the applied force amplitude and 𝑠is the area of the cross section of the notched hollow specimen. All S-N curves shown in Fig. 4 have the same scale in order to be able to clearly appreciate the differences among cases. Tables summarising the tests loadings and fatigue lives are given in Appendix A. As expected, Fig. 4 shows that the S-N curve for the smooth specimen is above the curves for the notched specimens, and with respect to the notched specimens, the curves for those with a smaller diameter are above the ones for those with a larger diameter. As a matter of fact, the endurance limit at 106cycles was 𝜎 =275 MPa for the smooth specimen and 𝜎=175, 152, 138 MPa for the notched specimens with 𝑑=0.8, 1.5 and 3.4 mm, respectively. 4. External crack paths In this work, special attention has been paid to the crack initiation location and the crack direction path, especially during the first period of crack growth, as some models base their evaluations on stresses along a line that is supposed to be representative of the crack path. This work deals with the experimental measurement of the crack initiation point and the crack direction not only at the external surface of the specimen but also at the specimen’s internal planes. Measurements of the angle of the crack initiation point and crack direction have been performed only in specimens falling in the high cycle fatigue regime, that is, specimens broken beyond 105cycles. Fig. 5 shows the axis layout reference framework used, where the origin of the coordinate system OXYZ is located on the specimen’s external surface in the centre of the hole. The 𝑋-axis is in the central transverse section of the specimen, the 𝑌-axis runs along the longitudinal dimension of the specimen and the 𝑍-axis coincides with the axis of the hole. The location of the crack initiation point at the hole contour and the crack direction at some specific distances from the crack initiation point are indicated by the angles 𝜃and 𝜃1, respectively, whose geometrical definition is sketched in Fig. 6. Angles are measured counterclockwise. The crack initiation point angle 𝜃is defined as tan−1 (𝛥𝑌 )∕(𝛥𝑋). The crack direction angle 𝜃1is measured using an additional coordinate system o′𝑥′𝑦′whose origin coincides with the crack initiation point at the hole contour. The angle 𝜃1is defined as tan−1 (𝛥𝑦′)∕(𝛥𝑥′). This crack direction angle 𝜃1has been studied at several crack lengths, 𝑎≈√(𝛥𝑥′)2+ (𝛥𝑦′)2, equal to some multiples of the averaged grain size, 𝐷, and also to some values based on the El-Haddad short-crack parameter 𝐿[22], that is 1, 2, 5, 10, 15 and 20 grains, and also 𝐿∕2,𝐿and 2𝐿. The reason is that several wellknown notched fatigue models based their estimations on the elastic stresses calculated on lines whose lengths are multiples of the grain size 𝐷, such as the Navarro–Rios model [23], and lengths that are based on the parameter 𝐿, such as Taylor’s Point and Line Methods [12]. In this work, a value of 𝐿= 158 μm was estimated for the studied material through the value of 𝐾𝑡ℎ obtained using the microstructural equation for the Kitagawa–Takahashi diagram 𝐷=2 𝜋(𝐾𝑡ℎ 𝜎𝐹 𝐿 )2(𝑚∗ 1 𝑚∗ ∞)2(see [24]), where 𝑚∗ 1is the grain orientation factor when the crack spans a single grain, 𝐾𝑡ℎ is the fatigue stress intensity factor threshold, 𝜎𝐹 𝐿 is the plain fatigue limit, and the term 𝑚∗ ∞is the orientation factor for truly polycrystalline behaviour when the crack is large enough to span International Journal of Fatigue 165 (2022) 107117 4 J.A. Balbín et al. Fig. 4. Fatigue tests conducted on S355 steel specimens. Fig. 5. Axes layout. several grains. Tables summarising crack initiation point angles (𝜃) and crack direction angles (𝜃1) for the several crack lengths measured are given in Appendices B and C. Regarding the crack at the external surface of the specimen (Z =0, data included in Appendix B and Table C.1 of Appendix C), a total of 22 specimens have been analysed for the three circular notch Fig. 6. Scheme of crack initiation point and crack direction angles. diameters, with two opposite cracks emerging from the hole in all cases. Fig. 7 shows two cases of the experimental measurement of the crack initiation point. Crack initiation point angles in Fig. 7a are close to zero, that is, 𝜃= −3.9◦for the crack growing in the left side of the hole (𝑋 < 0), and 𝜃= −1.8◦for the crack growing in the right side of the hole (𝑋 > 0). Likewise, Fig. 7b also shows angles close to zero, that is, 𝜃= 0.7◦(𝑋 < 0) and 𝜃= −5.8◦(𝑋 > 0). In the same way, Fig. 8 shows how the crack direction angle 𝜃1 was measured in two different specimens (𝑋 > 0). In Fig. 8a, angle 𝜃1= −19.4◦is found for a crack length of two averaged grain sizes, and in Fig. 8b angle 𝜃1= −0.2◦is found for a crack length of five averaged grain sizes. Once all external cracks were evaluated, the analysis was made using the absolute values of the 𝜃and 𝜃1angles, to prevent positive and negative values from cancelling each other when averaging results. Thus, the crack initiation point angle, 𝜃, presented a average value of 4.2◦with an standard deviation of 2.7◦. That is, the crack initiation point is close to the maximum principal stress point at the hole contour International Journal of Fatigue 165 (2022) 107117 5 J.A. Balbín et al. Fig. 7. Crack initiation point angles, 𝜃, measured over the external surface of the specimen. Fig. 8. Crack direction angles, 𝜃1, measured over the external surface of the specimen. Fig. 9. (a) Specimen model and mesh used in FEM analysis. Detail of the mesh around the hole. (b) First principal stress 𝑆11 along the 1500 μm hole depth (line 𝑋=𝑑∕2) for axial loading and 𝑑= 1 mm. International Journal of Fatigue 165 (2022) 107117 6 J.A. Balbín et al. Fig. 10. (a) Fracture surface and mid-thickness plane (Z= −750 μm) and (b) heights of the crack path points. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.). Fig. 11. (a) SEM (b) 2D profile (OP) and (c) 3D profile (OP) fractographs of a fracture surface. Specimen broken after 169000 cycles and hole diameter 𝑑= 1.50 mm. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.). Fig. 12. Average path at Z= −478 μm. Specimen broken after 169 000 cycles. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.). (𝜃= 0◦). Similarly, an average crack direction angle of 𝜃1= 21.8◦was found for a crack length of one averaged grain size, 1𝐷, and a 𝜃1= 14.7◦ angle for a crack length of two averaged grain sizes, 2𝐷. Furthermore, this crack direction angle 𝜃1presented an average value of 9.4◦for a crack length of 𝐿∕2, and 6.0◦angle for a crack length of 𝐿. These values certainly differ from Mode I and Mode II directions. However, the crack direction angles measured at larger crack lengths approach the Mode I direction. 5. Crack path analysis on the fracture surfaces In a previous work [17], the point of maximum principal stress of a whole specimen, similar to those tested in this work, was calculated through a finite element analysis (commercial software ANSYS v15.0 [25]). SOLID187 elements, which are 10-node and 3D tethrahedral structural solid elements, and different mesh sizes were used in a linear elastic analysis. The models were re-meshed to ensure that the difference between the stresses of a model and the previous one were practically negligible. The final mesh had an element size of approximately 2 mm, being refined at the hole area, up to an element size of 0.025 mm. Fig. 9a shows the model geometry and mesh, and the detail of the mesh around the hole, for a 𝑑= 1 mm case. As for the International Journal of Fatigue 165 (2022) 107117 7 J.A. Balbín et al. Fig. 13. (a) SEM and (b) 2D profile (OP) fractographs of a fracture surface. (c) Average path at Z= −649 μm. Specimen broken after 315 600 cycles and hole diameter 𝑑= 3.40 mm. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Fig. 14. (a) SEM and (b) 2D profile (OP) fractographs of a fracture surface. Specimen broken after 491 200 cycles. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.). boundary conditions, the specimen was clamped at the bottom surface and a unitary stress was applied on the upper surface. Fig. 9b shows the maximum principal stress 𝑆11 along the line 𝑋=𝑑∕2 = 0.5 mm, i.e. the line at the hole surface at 𝜃= 0◦, where the maximum principal stress is located. This line has the length of the hole, i.e., 1500 μm. As a result of the low out-of-plane constraint conditions at the sample surface (i.e. reduced stress triaxiality), the maximum principal stress point is not located at the external surface (Z=0) but at Z= −750 μm, i.e., at the mid-thickness plane of the specimen. However, the stress profile is pretty flat in the central part of the hole. Because of that, the crack initiation point and the crack direction were subsequently evaluated at this Z= −750 μm internal plane, assuming that this would be the most likely plane in which fatigue crack initiation would occur. To do that, an analysis of the fracture surfaces at both sides of the hole was carried out using an optical microscope, a scanning electron microscope (SEM) and a noncontact 3D optical profilometer (OP). The 3D topography of the fracture surfaces was captured using the OP through the Focus Variation technology, which is useful for measuring the shape of large rough surfaces. It also allows measurements including high slope surfaces (up to 86◦), high measurement speed and large vertical range. The resolution is half a micron in the three coordinate axes. An example of the fracture surface captured with the OP is presented in Fig. 10a. Only one of the two fracture surfaces generated from the hole is shown, in this case the one located at 𝑋 < 0(following the coordinate system defined in Fig. 5). The points of the fracture surface with higher Yvalues are shown in red, blue points refer to lower Y values and white areas indicate steep slope. The intersection of the fracture surface with the mid-thickness plane (Z= −750 μm) results on the mid-thickness crack path shown in Fig. 10b, plotted as a solid black line in Fig. 10a. The path in the mid-thickness plane has a similar trend to the path studied on the outer surface, that is, the initiation point (at 𝑋+𝑑∕2 = 0 μm) is close to the point of maximum principal stress and the direction of the crack is close to Mode I (horizontal direction) from the initiation. The angle 𝜃measured on the external surface of the specimen (Z=0) serves as the reference to obtain the value of this angle 𝜃in the Z= −750 μm crack path. This calculation consists on a relation between the Yheights of the crack initiation point at Z= 0 and the crack initiation point at Z= −750 μm. The average value of the angle 𝜃for all the Z= −750 μm crack paths was 4.9◦, which is very similar to average value 4.2◦reached on the external surface of the specimen (Z=0). For the Z= −750 μm crack path, an average crack direction angle of 𝜃1= 14.2◦was found for a crack length of one averaged grain size, 1𝐷, and a 𝜃1=9.0◦angle for a crack length of two averaged grain sizes, 2𝐷. These values also differ from Mode II direction. Crack direction angles measured at larger crack lengths approach to Mode I direction again. An example is a 𝜃1=2.5◦angle obtained for a crack length of ten averaged grain sizes, 10𝐷. A more careful analysis of the fracture surfaces showed that the cracks did not initiate exactly from the mid-thickness plane (Z= −750 μm). Radial lines pointing to the origin of the crack were observed in the SEM and OP fractographs, indicating that the cracks’ origins were scattered along the central segment of the hole. The most likely reason for this large variability in the initiation point location may be the low variation of the maximum stress throughout the 1500 μm depth of the hole. The first principal stress profile shown in Fig. 9b indicates that its value changes very little over the 1500 μm studied, except for the two end zones, corresponding to the external and internal surface of the hollow specimen. Hence, if cracks usually initiate from the point where the maximum principal stress is reached and it remains almost constant in the studied case, the cracks could randomly initiate at any location within this region that is, the cracks could have initiated from almost any 𝑍point along the planes 𝑋= ±𝑑∕2 and not exactly from the midthickness plane (Z= −750 μm). For this reason, the fracture surfaces were carefully analysed with the SEM and OP equipments, trying to International Journal of Fatigue 165 (2022) 107117 8 J.A. Balbín et al. Fig. 15. (a) SEM, (b) 2D profile fractographs of a fracture surface, (c) External crack path (Z=0), (d) Average path around the initiation point and (e) midplane thickness crack path (Z= −750 μm). Specimen broken after 831 100 cycles. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.). locate the fatigue cracks and their initiation points, to subsequently study the corresponding crack paths. Fig. 11 shows the fractographic examination of a fracture surface (𝑋 > 0). The SEM and 2D/3D topography captured profiles (OP) are provided. In all these fractographic images, several planes are observed that probably correspond to different cracks. The smooth fatigue surface crosses the entire thickness of the specimen, i.e. 1500 μm. Paying attention to the area adjacent to the hole, three macrocracks are observed, which in the OP fractographs correspond to three zones of different colour: red–orange, blue–green–yellow and blue areas observed in the downward direction. Each of these macrocracks are relatively horizontal, although located at different 𝑌heights, then connecting abruptly where they meet, as can be seen in the 3D fractography. The two areas of abrupt connection are in white, corresponding to zones of steep slope. Ratchet marks when the three cracks intersects, can also be seen in the SEM picture. Radial lines pointing to the crack initiation zone can be observed in the SEM and OP fractographs. The initiation zones of two of the three cracks are indicated with black arrows, none of them located in the mid-thickness plane (Z= −750 μm). The possible crack initiations are located around Z= −478 μm and Z = −1078 μm in this case. The crack paths corresponding to the planes where each crack initiated were studied. As it was not possible to locate the exact initiation points of the cracks, but rather an initiation zone, which was estimated to be 50 μm in size, paths in several parallel planes at a International Journal of Fatigue 165 (2022) 107117 9 J.A. Balbín et al. Table 1 Crack initiation point angle, 𝜃, and crack direction angle, 𝜃1, for specimen broken at 831 100 cycles, showed in Fig. 15. Crack path location 𝜃 𝜃1 1D 2D L/2 2L External surface Z= 0 μm 3.6◦14.1◦−0.6◦−10.0◦−2.8◦ Mid-thickness plane Z= −750 μm −5.8◦−18.1◦−1.9◦−2.4◦1.6◦ Localised crack initiation Z= −559 μm −5.9◦16.6◦12.8◦7.1◦1.4◦ distance of Z= 5 μm from each other, and sweeping the 50 μm of the initiation zone, were studied. Thus, 11 parallel paths were studied for each initiation zone. Fig. 12 shows the heights of these 11 paths for the crack that starts at Z= −478 μm located on the specimen shown in previous Fig. 11. It is observed that the 𝑌heights of these paths run approximately parallel to each other. The average path of these 11 paths, shown in blue colour, was taken as representative of the path of this crack. It is observed that the direction of this average path is very close to 0◦, even at the initial period of crack growth, contrary to the commonly assumed 45◦crack initiation direction, which would correspond to Mode II direction. The crack paths have been only evaluated along a 700 μm length since this distance approximately represents a crack that spans twenty grains of the material used in this work. In contrast to what has just been described, other surfaces analysed show a single plane with a main crack occupying a large area of the specimen’s section, as represented in (a) SEM and (b) OP fractographs included in Fig. 13. In this case (𝑋 < 0), the yellow area is predominant over almost the entire surface and the crack initiation could be located at approximately where the black arrow is placed (Z= −649 μm), that is the red little area adjacent to the hole to which the radial lines converge. Furthermore, a mixed-Mode crack initiation is observed in its corresponding average path, included in Fig. 13c. In this average path, the crack direction angle is 𝜃1= 21.8◦for a crack length of one averaged grain size and 𝜃1= 10.8◦for a crack length of two averaged grain sizes. A total of ten clearly detectable crack initiation locations were observed among all the fracture surfaces analysed microscopically. Additionally, other cases have been observed where the fracture surface seems to be uniform but no clear crack initiation is visible. An example of this situation is shown in Fig. 14, where an almost homogeneous fracture surface is observed. The area adjacent to the hole is practically red–yellow coloured and almost the entire blue surface indicates that is a uniform surface. A possible explanation for not finding a clear crack initiation location is that a large number of small cracks might be initiated in different locations. To make a final analysis of the results obtained, a comparison between the external path (Z=0), the mid-thickness plane path (Z= −750 μm) and the localised crack initiation average path (Z= −559 μm) is done, all corresponding to the fracture surface presented in Fig. 15 which belongs to a specimen broken after 831100 cycles and hole diameter 𝑑= 1.50 mm. Please note that in the plotted paths, the 𝑋-axis has been reversed. As can be seen, there are clear differences among the crack paths in this case. The external crack path initiates almost horizontally and just then decreases slightly in contrast to average path Z= −559 μm which shows a clear mixed-Mode crack direction at the beginning and then stays approximately horizontal. However, the three paths continue horizontally at larger crack lengths. The Z= −750 μm path seems to be almost horizontal for the whole plotted path. These differences are more noticeable in Table 1 where a brief summary of the angles 𝜃and 𝜃1are provided for the three crack paths of this case. For the sake of simplicity to make the comparison, the real angle values are provided instead of absolute values. It is observed that the external crack path might not be representative of the crack which grows inside. As a summary, Table 2 shows the average values of angles 𝜃and 𝜃1 measured at different locations of the crack path for all the evaluated specimens in the present work. In addition, standard deviation values are included in parentheses. This Table 2 includes data obtained from the 22 fatigue-tested specimens broken beyond 105cycles. The crack initiation point is close to the maximum principal stress point (𝜃= 0◦) since 𝜃angle does not exceed the value of six degrees in the three analysed locations of the crack path. In general, the crack direction angles 𝜃1indicate that mixed-Mode direction is found at the initial period of crack growth on the external surface of the specimen (Z=0) as it is close to nineteen degrees. Regarding the crack initiations that could be located on the fracture surfaces, the average path indicates that the crack direction is close to Mode I at the beginning (6.6◦) but it is important to note that these results are based on a sample of ten localised cracks. Thus, for the material studied and the loads applied, a certain discrepancy has been observed between the cracks directions studied on the external surface of the specimen and those studied in the internal sections where it originates. Crack initiation in Mode II direction is not representative of the samples studied in this work. Additionally, the influence of the hole diameter 𝑑on the crack path was analysed. This analysis was based on the ten crack paths for which the crack initiation location was clearly detectable, that is, eight paths for 𝑑=1.5 mm and two paths for 𝑑=3.4 mm (Appendix C, Table C.3). Unfortunately, there were no clearly detectable crack initiations for 𝑑 =0.8 mm, so, the analysis was focused only on the other two hole diameters. Regarding the crack initiation angle 𝜃, the average values were 𝜃=4.4◦and 𝜃=3.7◦for 𝑑=1.5 and 3.4 mm, respectively. The average values for the crack propagation angle 𝜃1for crack lengths 1, 5 and 15 averaged grain sizes (1𝐷, 5𝐷and 15𝐷) were 𝜃1=4.8◦(1𝐷), 𝜃1 =2.8◦(5𝐷) and 𝜃1=1.2◦(15𝐷) for 𝑑=1.5 mm, and 𝜃1=13.7◦(1𝐷), 𝜃1=4.0◦(5𝐷) and 𝜃1=1.3◦(15𝐷) for 𝑑=3.4 mm. These results indicate that the average values of the angles 𝜃and 𝜃1for the two studied diameters are relatively close, with the exception of the value of 𝜃1for the first grain (1𝐷), whose values are 4.8◦and 13.7◦for 𝑑= 1.5 and 3.4 mm, respectively. In any case, the trend of the crack path is basically similar for the two diameters, that is, the crack initiates from a point close to the maximum principal stress point, its direction is close to that of Mode I in the first grains, converging to the Mode I direction as its length increases. 6. Discussion In this work, a study of the crack directions for S355 steel tubular specimens with a circular hole subjected to cyclic axial loading was carried out. As expected, the cracks started from the circular notch near the point of maximum principal stress. Fracture surface analysis indicated that the cracks initiated from an internal plane and that the crack direction from its initiation was close to the direction perpendicular to the applied load, i.e., in the Mode I direction. A Mode II crack initiation was not observed. The experimental results, although limited to a material and to specific test conditions, are contrary to the usual idea established in the fatigue community that the crack initiates in Mode II and then propagates in Mode I. The theoretical and experimental study of the two stages of fatigue crack growth by Forsyth was mainly carried out for unnotched solids [2]. The presence of a notch, with its effect of generating a stress gradient, might reduce the size of the Mode II initiation stage or even completely inhibit it. Thus, more theoretical and experimental work on notches is needed to determine whether Mode II initiation stage actually occurs or not and which factors influence it.