A detailed study of short fatigue crack directions for carbon steel specimens with circular holes subject to cyclic biaxial loads

Depósito de Investigación de la Universidad de Sevilla https://idus.us.es/ This is an Accepted Manuscript of an article published by Elsevier in Theoretical and Applied Fracture Mechanics, Vol. 128, on December 2023, , available at: https://doi.org/10.1016/j.tafmec.2023.104153 © 2023 Elsevier. En idUS Licencia Creative Commons CC BY-NC-ND A detailed study of short fatigue crack directions for carbon steel specimens with circular holes subject to cyclic biaxial loads. J. A. Balb´ına,b, V. Chavesb, 1, A. Navarrob. aDepartamento de Ingenier´ıa Minera, Mec´anica, Energ´etica y de la Construcci´on, Escuela T´ecnica Superior de Ingenier´ıa, Universidad de Huelva, Campus Universitario de El Carmen, 21007 Huelva, Spain. bDepartamento de Ingenier´ıa Mec´anica y Fabricaci´on, Escuela Superior de Ingenier´ıa, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain Abstract The effect of notches and biaxial fatigue loading on the crack paths was studied in detail for thin-walled tube specimens with a passing-through hole, paying special attention to the short-crack period. The study was focused on the high cycle fatigue regime. The material was a carbon steel and the tests were under load control, at Rσ=−1. The crack initiation point on the notch surface and the crack direction were studied with an optical microscope on the specimen outer surface and with a scanning electron microscope and a non-contact 3D optical profilometer on the fracture surface. The crack direction was analyzed for several crack lengths, ranging from the length of one average grain to the length of twenty average grains, all lengths within the short-crack regime, in order to carefully observe the evolution of the crack direction in the Stage I and during the transition from Stage I to Stage Preprint submitted to Elsevier 30 October 2023 II. A statistical analysis of the crack initiation point and the experimental crack directions was carried out. In general, the crack initiation point was close to the maximum principal stress point. The crack direction during the first grains was approximately the Mode I direction. There was no initiation in Mode II. The crack continued in the Mode I direction as it got longer, within the short-crack period. The experimental fatigue limits were compared with the predictions calculated with two models from the literature. The direction of the straight lines used by the models to make the predictions were compared with the average crack directions measured experimentally. The goodness of the models, not only from the point of view of the fatigue limit value prediction but also from the closeness of the direction of the line used for the prediction to the experimental crack direction, was discussed. Key words: Biaxial loading, Notch, High cycle fatigue, Crack direction, Biaxial model 1Corresponding author. Tel: +34-954487311; fax: +34-954487295. E-mail address: chav[email protected] (V. Chaves) 2 1 Introduction Most engineering components have stress concentrators, such as holes, grooves and threads, in many cases subjected to a combined cyclic loading. The study of fatigue failure in notched components subjected to biaxial cyclic loading is of great importance for many engineering applications. The physics of this process is very complex and not yet well understood. Among other things, it remains to know precisely where the initiation of the fatigue crack (or cracks) will occur and its path, for a known notch geometry, material and load. Focusing on the study of the fatigue limit, the short-crack period, typically with a crack length of less than 1 mm, is especially important, since most of the component’s life will take place in this period. It is widely believed that fatigue cracks usually initiate approximately along the plane of maximum shear stress, in what is commonly called Mode II or sliding mode of crack surface displacement, and rotate to the plane of maximum tensile stress, called Mode I or opening mode of crack surface displacement, when they have reached a certain length [1]. These two processes of crack growth are known as Stage I and Stage II, respectively. At present, neither the precise lengths of these two stages, nor the main factors that affect them are fully known, although most researchers place this transition within the short-crack period [2–4]. A better knowledge of these two periods would lead to an improvement of the models used for the calculation of fatigue limits in components and the overall process of fatigue design of mechanical components. To attain this knowledge, it is essential to have a wide set of experimental results, including crack paths and directions, especially in the short-crack period, for a variety of notches, materials and load types. 3 Gough et al. [5] were among the pioneers in the study of multiaxial fatigue and did a very comprehensive experimental research. Focusing on their work on notches, they tested a V-notch geometry for 7 different materials and 7 biaxial load combinations, ranging from pure bending to pure torsional loading. They provided the S-N curves for all that set of tests, but unfortunately, they did not report any information on crack paths or crack directions for these experiments. Subsequent researchers began to report on fatigue crack paths and directions and not only on the value of the applied stress, aware of the scientific importance of this matter. Generally, these investigations consisted of studying the cracks on the outer surface of the specimen using an optical microscope. Endo [6] studied the fatigue limit of specimens containing small surface holes of diameters ranging from 40 to 500 µm, which acted as artificial defects. The tests were in-phase axial and torsional loading. The materials investigated were annealed carbon steel, quenched and tempered steel, high strength brass and nodular cast iron. Several optical micrographs of small nonpropagating cracks emanating from the holes are shown, although the exact value of the crack directions and the crack lengths are not specified. Tanaka et al. [7] studied thin-walled tubular specimens of medium-carbon steel with a through-thickness hole of 0.5 mm diameter. Several optical micrographs of propagating and non-propagating cracks growing from the hole for outof-phase axial-torsional loading are presented, taken on the outer specimen surface. According to the authors, the crack paths shown in these pictures are nearly straight following the radial direction of the hole in all but one case, for which it is slightly curved. As in Endo’s work, neither the numerical values of the crack directions nor the lengths in which they are being measured are provided. Gladskyi and Fatemi [8] analyzed the fatigue crack growth behaviour of tubular specimens with a through-thickness hole subjected to axial and tor4 sional loading. The hole diameter was 3.4 mm and the material a low-carbon steel. According to the authors, surface cracks initiated at the hole edge. They were located and oriented at about 0◦and 180◦to the axis of specimens for axial loading and at about ±45◦and ±135◦for torsional loading. The numerical values of the angles of the crack initiation location and the crack growth direction, for a crack length of 0.2 mm, are provided for 10 specimens, 5 for axial loading and 5 for torsional loading. In the case of specimens with circumferential notches, especially if the notch radius is small, the study of fatigue crack paths with optical microscopy is very complicated. The most common way to perform this study is to carry out a longitudinal section of the specimen and study the crack on the generated surface. This is what Atzori et al. [9] did in their study of specimens with circumferential V notches made of carbon steel subjected to combined tension and torsion loading. Several micrographs of propagating and non-propagating cracks in the longitudinal sections of the specimens are presented in this work. A qualitative analysis of the cracks propagation directions is done, although numerical values are not provided. Berto et al. [10] studied circumferentially V-notched specimens made of hardened and tempered steel subjected to combined tension and torsion loading and examined the paths of non-propagating cracks from some run-out specimens sectioned in the longitudinal section. Several micrographs of non-propagating cracks are shown in this document. Only the numerical values of the average lengths of these cracks for tension and torsion are provided. The analysis of the outer surface of the specimen may not be conclusive regarding the crack path in its initial part because the crack may not have initiated on the outer surface of the specimen but on an inner location of the 5 specimen, as suggested by Du Quesnay et al. [11]. In order to force cracks to initiate and grow on one of the specimen outer surfaces, Susmel and Taylor [12] machined V-notches on plates using a tool that allowed a notch root radius smaller on one of the outer surfaces compared to the other. The notches were manufactured with various inclinations with respect to the longitudinal axis of the specimen, to achieve 4 types of biaxial loading. The material was carbon steel and the thickness of the plates were 5 mm. The length and direction of non-propagating cracks were studied on the outer surface with the lowest notch radius with an optical microscope. Values of the length and the angles of the Stage I and Stage II of fatigue crack growth are presented for 3 nonpropagating cracks and 1 crack that failed in the high-cycle fatigue regime, 1 for each of the 4 types of biaxial loading. In summary, limited information on the numerical values of the crack initiation location and the crack direction during its initial part on notched components under multiaxial loading is available in the literature, and mostly obtained from the outer surface of the specimens. According to our knowledge, statistical studies of these variables are not available either, despite it is being well-known that the results in fatigue have a great dispersion, especially in the short-crack period, due to the considerable influence of the microstructure. In recent years, an extensive study of fatigue crack directions in the shortcrack period was carried out in 1.5 mm thick hollow cylindrical specimens with a through-thickness circular hole. The load was cyclic axial, torsional and in-phase biaxial loading. The tested materials were a stainless steel [13], an aluminum alloy [14] and a carbon steel [15]. The experimental crack paths were studied on the outer surface for 106 specimens in total. The mean value and the standard deviation of two experimentally measured angles were calcu6 lated. The first angle specified the crack initiation point location on the hole contour. The second angle specified the crack direction in its initial part, for a crack length of the order of the El Haddad short-crack parameter a0[16], assuming that the crack was a straight line. But, as previously mentioned, the crack could have initiated in an interior location of the specimen. During the last few years there has been a great progress in the field of microscopy. Specifically, noncontact 3D optical profilemeters have been greatly developed, which allow the accurate determination of the surface topography. This type of equipment was recently used to study the crack directions in internal planes for the last of the three materials analysed, the carbon steel [15]. Only the specimens subjected to axial load were studied, since current devices only allow the analysis of a surface placed perpendicular to the optical lens and approximately flat. The fracture surfaces obtained for axial loading were approximately perpendicular to the specimen axis and approximately flat, which allowed the analysis of a slice of the broken specimen in a simple way. The present document is a continuation of this last work. The tests were fully-reversed torsion and in-phase biaxial tension-torsion, under load control. As said before, the material was a carbon steel and the specimens were thinwalled tube specimens with a passing-through hole. The S-N curves were built for two hole radii, 0.75 mm and 1.7 mm. The crack path study was focused on the high-cycle fatigue regime and the short-crack period. The crack initiation point on the notch contour and the crack direction in its initial part were studied with an optical microscope on the specimen outer surface and with a non-contact 3D optical profilometer on the fracture surface. For this second study, as the fracture surfaces generated by the torsion and biaxial tests were inclined with respect to the specimen axis, it was necessary to design and 7 manufacture a new auxiliary piece to place the slice of the broken specimen on the optical profilometer and have the fracture surface approximately perpendicular to the optical lens. The crack direction was analyzed for several crack lengths, in order to carefully observe the evolution of the crack direction during the Stage I and the transition from Stage I to Stage II. The experimental fatigue limits were compared with the predictions calculated with various models from the literature. The direction of the straight lines used by the models to make the predictions were compared with the average crack directions obtained experimentally. The results of the present research can lead to a better understanding of the crack paths in notches, especially in the shortcrack period, and to a better selection of the crack lines used in the notch fatigue limit models. 2 Materials and tests The material is commercial low carbon steel (S355), with the following chemical composition (weight %): C 0.18, 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 specimens were machined from 20 mm diameter round bars. No heat treatment was applied after the machining of the specimens. The monotonic properties, as determined from 20 tensile tests, are as follows [15]: tensile strength σUT S = 586 MPa, yield strength σY S = 412 MPa, Youngs Modulus E= 208 GPa. Vickers hardness is 187.9 HV. The microstructure was ferrite-pearlite. The grain size was measured for the ferrite grains. The average grain size was calculated from the study of longitudinal and transverse sections of 16 specimens, following the ASTM standard E11296 [17], resulting in an average grain size of d= 33 µm. This material combines 8 did not initiate from a point close to the point of maximum principal stress. (Table 1 is presented here.) 5 Linear-elastic 3D finite element analysis of the entire specimen As discussed in the introduction, it is possible that the cracks did not start on the specimen’s outer surface, but from an inner location of the specimen, which is 1.5 mm thick. A linear-elastic 3D finite element analysis (commercial software ANSYS Mechanical APDL 18.0 [22]) was performed of the entire specimen to locate the point of the solid where the highest value of the first principal stress S1occurs, called the hot-spot, and check if it is located on the specimen outer surface or on an inner location. SOLID187 elements, which are 10-node and 3D tetrahedral structural solid elements were used. The mesh had more than two hundred thousand elements, with an element size in the region near the hole of less than 0.025 mm. Fig. 6 shows the detail of the mesh close the hole. For all models, a unitary applied stress was applied, based on the net area. (Fig. 6 is presented here.) A previous 2D linear-elastic analysis of an infinite plate with a circular hole indicates that the highest value of S1occurs at the hole contour, at θ= 45◦ for torsional loading and at θ= 31.7◦for biaxial loading (σ=τ). For the 3D model, the maximum value of S1in any section Z=constant continues to be located at the hole contour and the same angles, but there is an evolution 15 of the maximum S1value with Z, along the depth of the hole. Fig. 7 shows this evolution for the two studied load cases. For the sake of clarity, the stress S1is shown divided by the Ktof a hole in an infinite plate subjected to the corresponding load. For torsional load, S1is presented along the line (X= Rcos(45◦), Y =Rsin(45◦)), and for biaxial load S1is presented along the line (X=Rcos(31.7◦), Y =Rsin(31.7◦)). Z= 0 is the plane of the specimen outer surface and −Z= 1.5 mm is the plane of the specimen inner surface. The maximum of all these curves occurs always in an interior plane of the specimen. The position of these maximums (hot-spot) is shown in Table 2. The hot-spot is closer to the specimen inner surface (−Z= 1.5 mm) for the larger radius. 3D linear-elastic analysis of cylindrical specimens with other hole diameters and other types of load were made and it was concluded that the position of the hot-spot in a cylindrical specimen with a hole depends on many factors and it is difficult to know a priori. There is an influence of both the geometry (diameter and thickness of the specimen, and the diameter of the hole) and the type of load, and all these factors are coupled. Solid specimens with superficial holes, not through holes, were also analyzed, and in this case the hot-spot was very close to the outer surface and not on an inner plane. For the 4 cases analyzed in this study, Fig. 7 shows that the values at Z= 0 are clearly lower than the maximum values at the hot-spots, which are located in interior planes. It is therefore likely that the crack initiated in an interior location of the specimen and not on the outer surface. For this reason, in the next sections the crack paths in the interior planes will be analyzed, by studying the fracture surfaces. 16 (Fig. 7 is presented here.) (Table 2 is presented here.) 6 Analysis of the fracture surfaces In order to observe the fracture surfaces formed by fatigue, it was necessary to break the specimens into two parts by means of a tensile test. This allowed the study of the two fatigue surfaces formed at both sides of the hole in the case of biaxial loading. For the case of torsional loading, where 4 cracks were formed, only two of them could be studied, generally the surfaces of the two main opposite cracks, through which the specimen tended to break in the subsequent tensile test. The fracture surfaces at both sides of the hole were studied with an optical microscope and a scanning electron microscope (SEM). These microscopes allow one to acquire a general overview of the fracture surface and ascertaining whether one or several cracks were formed or whether the surface is rough or smooth, but they do not allow a crack path analysis on the desired Z=constant planes. To study the crack paths, a non-contact 3D optical profilometer (OP) was used. This type of microscopy, which has been developed remarkably in recent years, provides a 3D topographic map of the fracture surface. From the coordinates in a Z=constant plane, the crack trace on the plane is obtained. The resolution of the device is half a micron in the three coordinate axes and it allows measurement of high slope surfaces, up to 86◦. 17 Therefore, it is a very appropriate device to study the crack path during the short-crack period and, particularly, to study the crack direction during stages I and II of crack growth. Unfortunately, with this microscope only approximately flat surfaces and placed perpendicular to the optical lens can be analysed and the fracture surfaces in the biaxial and torsional loading cases were inclined with respect to the specimen axis. First of all, the half of broken specimen was cut by a plane perpendicular to the longitudinal axis of the specimen so as to generate a smaller piece that would fit in the microscope. Then, a support piece was designed to place the piece of broken specimen so that the fracture surface remained approximately perpendicular to the optical lens. Two support pieces were manufactured, one for the specimens tested under torsional load, with the inclined plane at 45◦and another for those tested under biaxial load, with the inclined plane at 31.7◦. These angles were the approximate inclinations of the fracture surfaces for each type of load, that is, close to the first principal direction for each load. An example of fractographs captured with the SEM and the OP are shown in Fig. 8. It corresponds to a specimen with R= 0.75 mm subjected to torsional loading, broken at 4086500 cycles. This crack was located in the third quadrant. The surface shown measures approximately 1.5 mm in the vertical direction of the picture (Z-axis), i.e. the full depth of the hole, and at least 0.7 mm in the horizontal direction of the picture, which is the direction of crack growth, equivalent to about 20 average grains at least, i.e. the maximum length of the paths studied in this document. This surface corresponds to the fatigue crack, as in all the studied specimens, the rapid fracture surface being outside the analyzed path. In the OP fractograph, red colors indicate 18 higher height values while blue colors refer to lower height values. The height scale is shown in the figure. In this case, the heights are between -18.85 µm (blue) and 18.85 µm (red), that is, there is a maximum height difference of about 40 µm. This indicates that this fracture surface is roughly horizontal, like, in general, all the studied fracture surfaces. It means that the experimental fatigue crack mostly follows the Mode I direction. Please note that new axis o′′x′′ was included. It will be explained in the next section. The second conclusion is that, since this surface is basically horizontal, the height difference between the crack initiation points on the outer surface (Z= 0) and on an inner Zplane at the hole surface will be very small. Therefore, the θangle measured on the outer surface and on an inner plane will be quite similar. It means that the crack initiation point, wherever it is along the hole depth, will be close to the ¯ θequiv values shown in Table 1, that is, roughly at θ= 45◦for torsional load and θ= 31.7◦for biaxial load. This conclusion can be, in general, extrapolated to most of the studied specimens. (Fig. 8 is presented here.) 6.1 Crack paths analysis on the plane that contain the hot-spot In this section, the crack paths analysis on the Z=constant plane that contains the hot-spot calculated with the 3D FEA is presented, assuming that the crack initiates from the hot-spot of the whole specimen. These Zvalues are shown in Table 2 for the 4 cases studied. The analysis will be carried out based on the 3D coordinates of the fracture surface points generated with 19 the OP. To simplify the analysis, the path’s profiles will be shown as they were directly obtained with the OP, without undoing the 45◦or 31.7◦rotation due to the assembly. Then, the horizontal direction will be the first principal stress direction, both for torsional and biaxial loading. A local coordinate system, o′′x′′y′′, whose origin coincides with the crack initiation point at the hole contour and with the x′′−axis parallel to the first principal stress direction and running in the radial direction, will be used (Fig. 9). This procedure allows carrying out the crack path analysis without concerning about the type of loading and the quadrant. The angle θ1measured in this local coordinate system will be called θ1L. The direction of crack growth will be deduced very easily: it will grow in Mode II if the path is inclined at ±45◦; it will grow in Mode I if it is at 0◦; or it will grow in a mixed mode, if it is inclined at an intermediate angle. If necessary, the real 3D profile of any crack path could be calculated just using the 3D coordinates obtained from the OP and undoing the rotation, of 45◦for torsion and of 31.7◦for biaxial, generated when the specimen was placed on the microscope support piece. (Fig. 9 is presented here.) Fig. 10 shows an example of the experimental crack path profile initiated at the hot-spot of the whole specimen, in this case at the plane Z=−590 µm. It corresponds to the same specimen shown in Fig. 8, with R= 0.75 mm subjected to torsional loading, and broken at 4086500 cycles. Fig. 10a shows a general view of the path, 700 µm long. The path is approximately horizontal, although with small zig-zags, indicating that the path is roughly in the Mode I direction. Fig. 10b shows the initial part of the path, 150 µm long, that is, approximately 5dlong, or approximately the length of a0. In this 20 length, some ups and downs can be seen, but in general the path keeps also quite horizontal. For this length the path therefore follows approximately the direction of Mode I. In the first 1-2 grains of length, a 45◦angle, typical of Mode II, is not observed either. In short, this path follows roughly the Mode I direction from the initiation along the short-crack period. (Fig. 10 is presented here.) The θ1Langles at the Zplane that contains the hot-spot were calculated for all the studied specimens. 27 specimens and 53 cracks were studied. The average values of all these angles, named ¯ θ1L, for each of the 4 studied cases and for all together, are presented in Table 3. Please note that the angles θ1Lat these planes were measured in local coordinates, o′′x′′y′′, as commented before. The experimental crack directions on these planes Z, measured for several crack lengths a, are, on average for all the results, close to the direction of maximum principal stress, i.e. ¯ θ1L= 0◦, even for very short cracks (2.5◦for a= 1d, 1.2◦ for a= 2d). The standard deviation is not large for very short cracks (9.8◦for a= 1d, 7.4◦for a= 2d), and approaches 0◦for longer cracks (1.5◦for a= 20d). For a crack length a=a0, the average and the standard deviation values are intermediate between those obtained for very short cracks (a= 1d, 2d) and longer cracks (a= 15d, 20d), as expected. The previous comments are extensible to each of the 4 cases studied - no appreciable differences have been observed among the values of these 4 cases. In summary, the crack paths at the hot-spot on average follow approximately the Mode I direction from its initiation. A careful analysis of each of the 53 paths shows that only one path initiated at an angle larger than 20◦, with θ1L= 31.5◦for a= 1d(biaxial, R= 1.7 mm). In this case the crack initiated in mixed-Mode, still away from 21 the Mode II direction. For a crack length a=a0, none of the 53 results exceeded the angle θ1L= 20◦. (Table 3 is presented here.) Also the angle θat this Zplane that contains the hot-spot was calculated. The θangle measured on the external surface at Z= 0 with the optical microscope served as the reference to calculate the value of this angle θ. This calculation consists on a relation using the height Ydifference between the crack initiation point at Z= 0 and at the chosen Z=constant plane. As indicated in the analysis of the outer surface, the angles θof quadrants II, III and IV were transformed to an equivalent angle of quadrant I, named ¯ θequiv. They are presented in Table 4 (please note that the angles ¯ θequiv measured on the planes Zthat contain the observed crack initiation zone, explained in the next section, are also shown). On average, the experimental crack initiation points at the Zplane that contains the hot-spot are close to the point of maximum principal stress at the hole surface, i.e. ¯ θequiv = 45◦for torsional loading and ¯ θequiv = 31.7◦for biaxial loading. The standard deviations are low, ranging between 4.0◦and 4.9◦. These average values of θare close to those of the outer surface, presented in Table 1, confirming that the fracture surfaces are, in general, approximately horizontal, as commented before. Regarding the detailed analysis of each of the cases, only in 3 cases the θangle was more than 10◦away from the angle of the point of maximum principal stress: 34.5◦ and 33.5◦for two cases of torsion and 19.8◦for one case of torsion. In these 3 cases, the crack did not initiate from a point close to the point of maximum principal stress, although it was not too far from it either. 22 (Table 4 is presented here.) 6.2 Crack paths analysis on planes that contain the observed crack initiation zone In this section, fracture surfaces are carefully analyzed in order to locate the crack initiation point at the hole contour and study the crack path initiating from this site. Figs. 11 and 12 shows an example of a fracture analysis, for a specimen with R= 0.75 mm subjected to biaxial loading, broken at 1036500 cycles. The fracture surface is the one at the right side of the hole (X > 0). The analysis of the fractographs (Fig. 11a) indicates that there is one main macrocrack. In the SEM fractography, radial lines pointing towards the initiation of the crack are observed (Fig. 11b), marked with a black arrow. The zone of possible crack initiation is located on the hole surface at approximately −Z∈[490 −540] µm, that is, close to the hot-spot, but not exactly at the hot-spot, located at −Z= 580 µm. Fig. 12a shows the experimental crack path profiles at Z=constant planes at the observed initiation zone. 6 paths are shown, at a distance of 10 µm from each other, sweeping the initiation zone, which is about 50 µm. The average path is also shown. The 6 paths are roughly parallel and horizontal, following approximately the direction of Mode I along the short-crack period. The average path follows an evolution very similar to the 6 paths. Fig. 12b shows the initial part of this path, 150 µm long. In this initial part the path is inclined upwards. Focusing on the first two average grains, the angle θ1Lfor the average path is 18.8◦for a= 1dand 14.7◦for a= 2d. This angle is not strictly close to the angle of the Mode I 23 direction, although it is far away from the Mode II direction at 45◦. (Fig. 11 is presented here.) (Fig. 12 is presented here.) Fig. 13 shows another example of fracture surface analysis, for a specimen with R= 0.75 mm subjected to biaxial loading, broken at 135237 cycles. The fracture surface is the one at the right side of the hole (X > 0). Also the 3D profile of the fracture surface obtained with the OP is shown. Two cracks are observed, one shown in blue and the other shown in red-yellow, abruptly connected in a zone shown in white. The connection zone of the two cracks in its initial part is at approximately −Z∈[400−500] µm. Then, the blue crack in its initial part is located at approximately −Z∈[0 −400] µm. Although the initiation zone of this crack was not clearly observed, it must necessarily be at −Z∈[0 −400] µm, that is, very far from the hot-spot, located at −Z= 580 µm. Therefore, this crack originates very far from the hot spot. Regarding the second observed crack, the one shown in red-yellow color, the possible initiation zone was located at −Z∈[710 −760] µm, as shown in Fig. 13b. This initiation zone again does not coincide with the hot-spot. Fig. 14a shows the experimental path profile of this crack at Z=constant at the observed initiation zone and Fig. 14b shows the initial part of this path profile. For the sake of simplicity, just the average path is shown. Small ups and downs are distinguished, but the overall trend is the horizontal direction. A 45◦inclination is not observed in the initial part, which is close to the horizontal direction. In summary, this crack follows the Mode I direction in 24 one. Regarding the N-R model, for the cases studied, θ1angles between Mode I and II were used for the predictions, and the decisive barrier that defined the fatigue limit of the notched component was located at a distance of 510 grains from the hot-spot. The experimental θ1angle for a crack length a= 5,10 grains was close to the Mode I direction. Then, the directions used for the predictions were far from the average experimental direction, although not as much as those of the Mode II variant of the MWCM+PM. 8 Discussion In this document, the crack direction during the short-crack period was studied in various cylindrical surfaces along the hole depth for each specimen: on the outer surface of the specimen, on a transverse plane to the hole that contained the point of maximum principal stress of the entire specimen and on a transverse plane to the hole that contained the observed crack initiation point. The crack paths were different on these 3 planes for each specimen, especially in the initial part. However, if an average analysis of all the specimens is made, the trends on these 3 planes are quite similar: the crack initiates at the hole contour from a point close to the maximum principal stress point of that plane. In its initial part, the crack follows approximately the Mode I direction. There is no initiation in Mode II. For a length equivalent to a0the crack grows close to Mode I. The crack continues in Mode I along the short-crack period. There is no abrupt transition from Stage I to Stage II, but rather a smooth evolution in the crack direction. The theoretical and experimental analysis carried out in this work indicates that the crack probably did not initiate on the outer surface but on an inner 31 plane. But since, on average, the crack directions measured on the outer surface were close to those measured on the inner planes where the crack probably initiated, it is reasonable to estimate the average crack directions directly from the study of the outer surface, which it is much simpler. Although this analysis based on the outer surface is not recommended if very few specimens are studied, since, as seen in this work for a large number of specimens, the dispersion of initiation directions was much greater on the outer plane than on the studied inner planes. It could happen that some of the few studied specimens initiated in mixed-Mode or even in Mode II on the outer surface, deducing from this that the cracks initiate in many cases in mixed-Mode or even in Mode II, when from the present study it is known that in the studied inner planes almost all cracks initiate in Mode I, except for very few cases that initiate in mixed-Mode. The results of the present work are complemented with those of a previous work focused on the same material and specimen’s geometry but for the case of axial loading [15]. The trend of the results is similar in both works, considering that for axial load the first principal direction is at 0◦while for the two load cases studied in the present work is at 45◦and at 31.7◦. In total, 49 specimens were analyzed. The high number of studied specimens, including three different types of load, give the results of the study a significant reliability. Focusing on the torsional tests presented in this document, generally 4 cracks formed from the hole, one in each quadrant. In addition, the analysis of the fracture surfaces showed that in some cases the main crack was actually formed by two cracks that initiated at different points and eventually joined. That is, it is probable that 6-8 fatigue cracks growing from the hole coexisted in the same specimen. The study of Stage I and Stage II of growth of a single crack 32 for the case of a specimen with a hole under torsional loading is probably very simplistic. It would be interesting to study to what extent all the coexisting cracks influence each other in their growth and whether there is an important coupling effect or not. Two models, the MWCM+PM and the N-R, the first with two variants, Mode I and Mode II, were used to predict the fatigue limit of the notched specimens under three types of loading. In general, the predictions of the models were good, with average errors well below 20%. In addition, the predictions of the MWCM+PM were also very similar to the predictions of the N-R model, despite the fact that these two models are considerably different: the N-R model is based on the study of the crack interaction with the microstructure, while the MWCM+PM is a combination of the Critical Plane Approach and the Critical Volume. Besides, the N-R model for this material, geometry and load, bases its predictions on the stresses at a line in the mixed-Mode direction. This leads one to wonder if there is any connection between the two models. It would be interesting to analyze this possible connection in the future. The present study about crack directions during the short-crack period was comprehensive. Even so, the question remains as to whether the Stage I of crack growth was really analyzed, since the analysis was limited to the study of broken specimens and no analysis was carried out during the test. Some kind of experimental technique should be applied during the test that would allow the detection of a very small crack, about the size of a grain. This technique should be applied to a large number of specimens, at least to 20-30 specimens, since the crack initiation process may vary considerably from one specimen to another, as it highly depends on the microstructure and surface finish. It would be very interesting to relate the crack initiation site with the microstructure, 33 and the crack initiation plane with the crystallographic orientation. In our opinion, this analysis is not currently feasible. We hope that in the coming years this type of equipment, as happened with 3D profilometers, becomes more accessible to research groups, allowing a more accurate analysis of the Stage I of fatigue crack growth in notched specimens, including the relationship between the crack initiation and the microstructure. In any case, we believe that the results presented in this work about the crack direction in notched solids represent a certain advance with respect to previous published works. 9 Conclusions In this work a crack path analysis during the short-crack period for a large batch of carbon steel specimens with a transverse circular hole subjected to cyclic biaxial loading was presented. Crack paths were studied on various planes, both on the specimen outer surface and on interior planes, including the plane with the maximum principal stress point of the specimen and the plane with the observed crack initiation point. Various types of microscopes were required for this type of study. On average, cracks initiated close to the point of maximum principal stress, located at the hole surface, followed the Mode I direction in the initial part, and continued in the Mode I direction as they became longer. There was no initiation in Mode II, the Stage I of crack growth was essentially in Mode I. There was also no abrupt transition between Stage I and Stage II of crack growth, but rather a smooth evolution. The dispersion of the measured directions decreased as crack length increased, being relatively low for crack lengths of the order or greater than a length equivalent to a0or 10 grains. Cracks probably did not initiate on the specimen outer surface but 34 from an inner plane. Although the average crack directions measured on the outer surface can be considered representative of the average crack directions on the interior plane where the crack probably initiated, as long as a large number of specimens are studied. The study of the crack direction on the outer surface with very few specimens could lead to erroneous conclusions, since the dispersion of directions is much greater in the outer plane than in the inner planes of probable initiation. Please note that these conclusions are only valid for this material, type of notch and type of load. To extend them to other cases, it will be necessary to carry out more tests in the future. In relation to the fatigue limit of the notched specimens for the studied types of loading and hole radii, the predictions with the analyzed models were reasonably good, with average errors clearly below 20%. The model predictions were close to each other, even though the models are quite different and base their predictions on elastic stresses along lines of very different direction. These low errors in the predictions of these models are not surprising, since previous studies about these models already published showed a similar trend. 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Notch sizze effect in multiaxial fatigue, in: Progettazione a fatica in presenza di multiassialit tensionali. L. Susmel, R. Tovo, editors; Ferrara, 2005. [28] Vallellano C, Navarro A, Dominguez J. Fatigue crack growth threshold conditions at notches. Part 1: theory. Fatigue Fract Engng Mater Struct 2000;23:113-121. 38 [29] Timoshenko S, Goodier JN. Theory of elasticity. McGraw-Hill; 1934. 39 Figure Captions Fig. 1. Geometry of the notched specimen (dimensions in mm). Fig. 2. S−Ncurves for the smooth and notched specimens: (a) smooth-axial, (b) smooth-torsion, (c) R= 0.75 mm and torsion, (d) R= 1.7 mm and torsion, (e) R= 0.75 mm and biaxial (σ=τ), (f) R= 1.7 mm and biaxial (σ=τ). Fig. 3. (a) Sketch of the notched specimen with the coordinate system. (b) Detail of the hole with four cracks and the variables used to define the crack direction. Fig. 4. Examples of experimental cracks on the outer surface. (a) Torsional loading, R= 1.7 mm, N= 515845 cycles. (b) biaxial loading, R= 1.7 mm, N= 762208 cycles. Fig. 5. Example of measured angles θand θ1on the outer surface. Fig. 6. Detail of the FE mesh near the hole. Fig. 7. First principal stress distribution along the 1500 µm hole depth. (a) R= 0.75 mm. (b) R= 1.7 mm. Fig. 8. Example of fractographs obtained with SEM and 3D profiler. Specimen with R= 0.75 mm subjected to torsional loading, broken at 4086500 cycles (X < 0). Fig. 9. Local axes o′′x′′y′′ used to analyse the crack paths on the fracture surfaces. Fig. 10. Example of a crack path profile initiating from the hot-spot of the whole specimen, at Z=−590 µm. Specimen with R= 0.75 mm subjected to torsional loading, broken at 4086500 cycles (X < 0). (a) Path 700 µm long. (b) Initial part of the path, 150 µm long. Fig. 11. Specimen with R= 0.75 mm subjected to biaxial loading, broken at 40 Fig. 6. Detail of the FE mesh near the hole. 47 0 0.5 1 1.5 -Z (mm) 0 0.2 0.4 0.6 0.8 1 1.2 SI/Kt(inf) (MPa) Torsion, R=0.75 mm Biaxial, R=0.75 mm (a) 0 0.5 1 1.5 -Z (mm) 0 0.2 0.4 0.6 0.8 1 1.2 SI/Kt(inf) (MPa) Torsion, R=1.7 mm Biaxial, R=1.7 mm (b) Fig. 7. First principal stress distribution along the 1500 µm hole depth. (a) R= 0.75 mm. (b) R= 1.7 mm. 48 18.85 mμ -18.85 mμ o’’ O x’’ Z 590 mμ o’’ x’’ 500 µm O Z 590 mμ Fig. 8. Example of fractographs obtained with SEM and 3D profiler. Specimen with R= 0.75 mm subjected to torsional loading, broken at 4086500 cycles (X < 0). 49 R y´´ x´´ X Y O o’’ IVIII III y´´ x´´ o’’ y´´ x´´ o’’ y´´ x´´ o’’ Fig. 9. Local axes o′′x′′y′′ used to analyse the crack paths on the fracture surfaces. . 50 0 100 200 300 400 500 600 700 -100 0 100 y'' ( m m) x'' ( m m) path through the hot-spot (a) 0 30 60 90 120 150 -30 0 30 y'' ( m m) x'' ( m m) path through the hot-spot (b) Fig. 10. Example of a crack path profile initiating from the hot-spot of the whole specimen, at Z=−590 µm. Specimen with R= 0.75 mm subjected to torsional loading, broken at 4086500 cycles (X < 0). (a) Path 700 µm long. (b) Initial part of the path, 150 µm long. 51 35.20 mμ -35.20 mμ 500 µm x’’ o’’ O Z O Z x’’ o’’ (a) 100 µm 500 µm (b) Fig. 11. Specimen with R= 0.75 mm subjected to biaxial loading, broken at 1036500 cycles (X > 0). (a) SEM and OP fractographies of the whole surface. (b) Detail of the zone of crack initiation with SEM. 52 10 m m-distanced paths Average path (a) 10 m m-distanced paths Average path (b) Fig. 12. Specimen with R= 0.75 mm subjected to biaxial loading, broken at 1036500 cycles (X > 0). (a) Crack path profiles initiating from the observed crack initiation zone, at Z∈[490 −540] µm. (b) Initial part of the path, 150 µm long. 53 Z O 113.61 mμ -113.61 mμ 500 µm 500 µm x’’ o’’ O Z x’’ o’’ O Z o’’ x’’ y’’ (a) 500 µm 500 µm 100 µm (b) Fig. 13. Specimen with R= 0.65 mm subjected to biaxial loading, broken at 135237 cycles (X > 0). (a) SEM and OP fractographies of the whole surface. (b) Detail of the initiation zone of one of the two main cracks with SEM. 54 0 100 200 300 400 500 600 700 -100 0 100 y'' ( m m) x'' ( m m) Average path (a) 0 30 60 90 120 150 0 30 60 y'' ( m m) x'' ( m m) Average path (b) Fig. 14. Specimen with R= 0.75 mm subjected to biaxial loading, broken at 135237 cycles (X > 0). (a) Crack path profile initiating from the observed crack initiation zone, at Z∈[710 −760] µm. (b) Initial part of the path, 150 µm long. 55 0 0.2 0.4 0.6 0.8 1 Distance (mm) 0 1 2 3 4 5 6 SI (MPa) Torsion, R=0.75 mm, analytical Torsion, R=0.75 mm, FE model Biaxial, R=1.7 mm, analytical Biaxial, R=1.7 mm, FE model Fig. 15. Stress gradients ahead of the notch, calculated analytically and with a 3D FE model. 56 Table 6 Experimental fatigue limits of the notched specimens and model’s predictions Loading RExperimental Model’s prediction MWCM+PM (Mode I) MWCM+PM (Mode II) N-R σN F L or τN F L σN F L or τN F L θ θ1σN F L or τN F L θ θ1σN F L or τN F L θ θ1 (mm) (MPa) (MPa) (MPa) (MPa) Axial 0.75 152 112.5 0◦0◦107.8 0◦45◦118.1 4◦−35◦ Axial 1.7 138 101.3 0◦0◦98.8 0◦45◦102.6 1◦−30◦ Torsion 0.75 100 90.4 45◦45◦85.4 45◦0◦97.0 45◦−6◦ Torsion 1.7 75 78.3 45◦45◦75.8 45◦0◦78.8 45◦12◦ Biaxial 0.75 78 63.6 31.7◦31.7◦60.5 31.7◦−13.3◦66.9 36◦−5◦ Biaxial 1.7 58 56.3 31.7◦31.7◦54.7 31.7◦−13.3◦56.8 33◦1◦ Average Error (%): 14.7 % 16.9 % 12.1 % 63