Calculations of biaxial fatigue limits with models using the experimental crack direction

International Journal of Fatigue 174 (2023) 107726 Available online 22 May 2023 0142-1123/© 2023 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/). Calculations of biaxial fatigue limits with models using the experimental crack direction V. Chaves a , * , J.A. Balbín a , b , A. Navarro a , L. Susmel c , D. Taylor d a Departamento de Ingeniería Mec´ anica y Fabricaci´ on, Escuela T´ ecnica Superior de Ingeniería, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain b Departamento de Ingeniería Minera, Mec´ anica, Energ´ etica y de la Construcci´ on, Escuela T´ ecnica Superior de Ingeniería, Universidad de Huelva, Campus Universitario de El Carmen, 21007 Huelva, Spain c Department of Civil and Structural Engineering, The University of Sheffield, Mappin Street, Sheffield S1 3JD, United Kingdom d Department of Mechanical Engineering, Trinity College, Dublin 2, Ireland ARTICLE INFO Keywords: Multiaxial fatigue Notch Fatigue limit Crack path Biaxial cyclic loading ABSTRACT Several models from the literature were used to predict the fatigue limit in notched components subjected to biaxial cyclic loading. The predictions of these models are based on the elastic stresses along a line which is considered to be representative of the crack direction in its initial part. The line used in the models changes considerably. For one of the studied models, the line direction corresponds to Mode I, while for another it is Mode II, and for the other two models considered the direction is between Mode I and Mode II. However, quite naturally, the experimental crack direction is unique. In recent years, a study of experimental fatigue limits and crack directions in its initial part for three materials was carried out in hollow cylindrical specimens with a circular hole subjected to cyclic axial, torsional and in-phase biaxial loading. The directions of the cracks that were measured experimentally are on average similar for the three materials and close to Mode I. The analysed models give, in general, good predictions of the experimental fatigue limits, although they use directions that are completely different and that they too differ markedly from the experimentally found ones. The predictions of the models using, in a forced way, the measured experimental directions are good in most cases, which reveals a surprising insensitivity of these models to the main hypotheses on which their own formulations are based. 1. Introduction Fatigue failure in real components generally occurs at stress concentrations, commonly referred to as notches. There are various models in the literature to predict the fatigue limit in notched components subjected to cyclic multiaxial loading. Three of them are analyzed in this document. Two of these models [1,2] combine the Critical Plane Approach for unnotched solids under multiaxial loading with Taylor’s Theory of Critical Distances (TCD) for notches under axial loading [3]. The third model is a short crack model that analyzes the interaction of the crack with the microstructural barriers of the material in the presence of a notch [4]. In the analysed models, the predictions are made based on the elastic stresses along a line whose length is of the order of the El Haddad small crack parameter, a 0 [5]. This line used in the models is considered representative of the experimental crack direction in its initial part. In the analyzed models, the direction of the line used varies considerably from one model to another: the Mode I direction, the Mode II direction and a mixed direction between that of Mode I and Mode II, depending on the type of material. Therefore, a great diversity of directions is used even though, evidently, the experimental crack direction is unique for a given geometry, loading and material. This diversity of directions used in the models might be due to the lack of a comprehensive experimental database on the direction of fatigue cracks growing from a notch for common industrial materials, especially in its initial part. There are several investigations in the literature on the matter, such as the classical investigations by Fenner et al. on mild-steel specimens containing sharp V-grooves under alternating axial loading [6], by Frost on non-propagating cracks from sharp V-notches in cylindrical specimens under reversed axial and rotating bending loading for mild steel and aluminum alloy [7] and by Kitagawa and Takahashi on very small surface notches under uniaxial cyclic loading [8]. More recently, Endo’s * Corresponding author. E-mail address: [email protected] (V. Chaves). Contents lists available at ScienceDirect International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue https://doi.org/10.1016/j.ijfatigue.2023.107726 Received 6 March 2023; Received in revised form 19 April 2023; Accepted 15 May 2023 International Journal of Fatigue 174 (2023) 107726 2 research on specimens containing small surface defects subjected to combined stress loading [9], Susmel and Taylor’s work on sharply notched specimens under in-phase Mode I and II loading [10], Meneghetti et al.’s study on U-notch steel specimens under axial loading [11], Tanaka et al.’s study on steel specimens with a hole subject to in-phase and out-of-phase axial and torsional loadings [12], Berto et al.’s tests on V-notched specimens made of hardened and tempered steel [13], Gates and Fatemi’s research on aluminum specimens with a circular hole subject to multiaxial loading [14], and Lorenzino and Navarro’s tests on aluminum alloy plates with a circular hole under axial loading [15]. All these experimental results from the literature present a great diversity of directions of the initial part of the crack, so that it would be very difficult to establish a clear criterion to define the crack direction in its initial part for a given material, load and notch geometry. In recent years, a comprehensive study of the experimental fatigue limits and crack directions in the initial part was carried out in hollow cylindrical specimens with a circular hole subjected to cyclic axial, torsional and in-phase biaxial loading. Three materials were tested, a stainless steel [16], a carbon steel [17] and an aluminum alloy [18]. All the details about the materials and the tests can be consulted in these previous documents [16–18]. These materials can be considered as brittle, intermediate ductile-brittle and ductile in fatigue, respectively, as will be seen later, which allows this study to analyze the effect of various material fatigue behaviors in the crack direction. The experimental crack directions were measured only for the broken specimens subjected to fatigue at a high number of cycles, with a life above 10 5 cycles. The present document shows an analysis of the measured experimental results. The fatigue limits predictions for the studied notches with the three multiaxial fatigue models for notches are also shown, and the average experimental crack directions are compared with the directions of the lines used by the models. As far as the authors are concerned, the exhaustive analysis of experimental crack directions for several materials from notches and the analysis of the models’ predictions from two points of view, the numerical fatigue limit prediction and the direction of the crack used in the models, are novelties in our scientific field. In addition, the models’ predictions using, in a forced way, as the line for the prediction the average experimental direction measured for each case are shown, constituting the second main novelty of this work. The objective is to analyze the adaptation capacity of each one of the models to the use of the experimental direction to make the predictions, which could be highly desirable in the near future, trusting that there will be a wide database of experimental cracks directions for notch multiaxial fatigue. 2. Materials and tests The three materials analyzed in this work are commercial AISI 304L stainless steel, S355 low carbon steel and 7075-T6 aluminum alloy. The main mechanical properties of the three materials are shown in Table 1, which are as follows: tensile strength σ UTS, yield strength σ YS, axial and torsion fatigue limits, σ FL and τ FL (for R σ = − 1), average grain size d and El Haddad short-crack parameter a0= (1/ π )(ΔKth/Δ σ FL)2 [5], where ΔKth is the threshold value of ΔK for fatigue crack growth and Δ σ FL the fatigue limit range. In the case of aluminum alloy, the values of σ FL and τ FL correspond to an estimated life of one million cycles, as aluminum does not have a defined fatigue limit. The ratio of the fatigue limits τ FL/ σ FL is also shown. This ratio has been used by several authors as an indicator of ductility in fatigue, suggesting that ductile materials in fatigue have a value close to 0.5 and brittle materials a value close to 1 [19,20]. In this case, the values of the ratio are 0.91, 0.76 and 0.58 for AISI 304L, S355 and 7075-T6, respectively. Thus, AISI 304L can be considered as a brittle material in fatigue, 7075-T6 as a ductile material in fatigue (with a value of a von Mises type material), and S355 as a material with an intermediate ductile-brittle behavior in fatigue. The chemical composition and other properties of these materials can be consulted in [16,17] and [18]. 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. Several hole radii R were studied, ranging from 0.4 to 1.7 mm. Fig. 1 shows the geometry of the specimen, in particular, the one with the hole radius R =0.5 mm (1 mm diameter). Three types of tests were performed with the notched specimens, in all cases under fully-reversed loading: axial tests, torsion test and inphase axial-torsion tests, of the type σ = τ , where σ is the normal stress amplitude and τ is the maximum shear stress amplitude, at the transverse cross-section, calculated with the equations of elasticity. The axial tests were made in a resonance machine, at a frequency of 100–150 Hz, and the torsional and biaxial tests in a servo-hydraulic axial-torsion load frame, at 6–10 Hz. The tests were stopped when a long crack of at least several mm was obtained or when a certain number of cycles were reached (run-outs). For AISI 304L stainless steel, the runout was established at 3.5×106 cycles, for S355 steel at 5 ×106 cycles and for 7075-T6 aluminum at 50 ×106 cycles for axial tests and at 5 × 106 cycles for torsion and biaxial tests. 3. Fatigue limits of the notched specimens The results of the fatigue tests of the notched specimens were used to construct the S-N curves and calculate the fatigue limits. Table 2 shows the experimental fatigue limits, expressed in term of stress amplitude, of the notched specimens, σ N FL and τ N FL, including the three materials, various hole radii R and the three types of loading, with a total of 15 values. For aluminum, the fatigue resistance values corresponding to one million cycles were taken again. Table 2 also includes the fatigue limit predictions with the models, which will be explained later in the document. 4. Experimental crack directions This section shows the experimental values of the crack initiation location and the crack direction in its initial part, both measured on the outer surface of the specimen. As mentioned in the introduction, it is important to determine the experimental crack direction in its initial part, since the most common models to calculate the fatigue limit in notches base their prediction on the elastic stresses in a line that is representative of the crack direction in this phase. Only specimens that failed after 10 5 cycles, within the so-called high cycle fatigue regime were considered. For the three materials all the cracks originated from the circular hole, and on both sides of the hole, as expected, since the hole is a stress concentrator. Fig. 2(a) shows the coordinate system OXYZ to define the crack direction, with its origin O located at the center of the circular hole on the outer surface of the specimen. 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 Table 1 Properties of the studied materials. Material σ UTS(MPa) σ YS(MPa) σ FL(MPa) τ FL(MPa) τ FL/ σ FL d, grain size (mm) a 0 (mm) AISI 304L stainless steel [16] 654 467 316 288 0.91 0.080 0.180 S355 low carbon steel [17] 586 412 275 208 0.76 0.033 0.158 (a) 7075-T6 aluminum alloy [18] 657 595 258 149 0.58 0.015 0.072 (a) (a) Estimated as a 0 =3.1 2 d/2 (see [21]). V. Chaves et al. International Journal of Fatigue 174 (2023) 107726 3 the axis of the hole. The location of the crack initiation point at the hole contour and the crack direction for a certain crack length a=  (Δx’)2+ (Δy’)2 √are indicated by the angles θ and θ 1 , respectively, whose geometrical definition are sketched in Fig. 2(b). Angles are measured counterclockwise. The crack initiation point angle θ is defined as tan−1(ΔY/ΔX). The crack direction angle θ 1 is measured using an additional coordinate system o’x’y’ whose origin coincides with the crack initiation point at the hole contour. The angle θ 1 is defined as tan−1(Δy’/Δx’). This crack direction angle θ 1 was measured experimentally for a crack length a equal to 0.50, 0.08 and 0.15 mm for the AISI 304L, S355 and 7075-T6 specimens, respectively. These crack length values are equal in length to 2.8a 0 , 0.5a 0 , and 2.1a 0 , respectively, or expressed as a function of average grain size, equal to 6.2d, 2.4d, and 10d, respectively. These values, of the order of magnitude of a 0 , were considered representative of the crack direction in its initial part. As will be seen later in the description of the models, they use a line whose length is of this order of magnitude to make the predictions. The choice of these lengths will be justified in the discussion section. The methodology used for measuring the crack initiation location and the crack direction angles is as follows: •The diameter of the circular hole is measured before the fatigue tests is carried out. •Once the fatigue test is complete, a high-quality picture of the hole with the crack is taken with an optical microscope. Before taking the picture, the specimen is carefully placed longitudinally under the microscope so that the y-axis of the picture coincides with the longitudinal axis of the specimen. On this picture, displayed on a large computer screen, the angles will be measured. •The presence of the crack divides the circular hole into two halves. One of the hole halves and its adjacent crack lines are selected for the measurements. •A circumference is drawn on the picture with a diameter equal to the diameter of the hole measured before the test. This circumference is made to coincide with the chosen half of the circular hole. The center of this circumference provides the center of the circular hole (origin of the OXYZ coordinate system defined in Fig. 2). The horizontal and vertical axes of the picture will be the X and Y axes of the coordinate system, respectively. Then, the OXY coordinate system is established in the picture. •The crack initiation point at the edge of the hole half is located and the crack initiation location angle θ is measured. •The additional coordinate system o’x’y’ is defined at the crack initiation point. •From the origin o’, an arc of radius a is drawn. Its intersection with the crack line defines a point. The line that joins this point with the origin o’ provides the angle θ 1 . A limitation of this measurement methodology is that the crack angle is defined by just two points and a straight line, so the possible zigzagging of the crack path between these two points is not taken into account. Fig. 3, Fig. 4 and Fig. 5 show representative pictures of the experimental cracks for the AISI 304L, S355 and 7075-T6, respectively. The calculated crack angles are also shown. Please note that for the sake of clarity the circumferences were not drawn in these pictures. For material S355 there are only axial tests. For this material, an enlarged picture of the left crack and another of the right crack have been added (Fig. 4(b) and Fig. 4(c)), since the angle θ 1 could not be clearly observed in the general picture (Fig. 4(a)), due to the fact that this material crack length a is very small. A tortuous crack path in these initial zones of the two cracks is observed. In some torsional tests, such as the one in Fig. 3(b), four cracks were observed. In this work only the two main cracks were analyzed. Table 3 shows a summary of the averages experimental crack angles θ and θ 1 for the studied materials, loading and hole radii. In addition, standard deviation values are included in parentheses. In total, the angles of 106 specimens were measured, 37 of AISI 304L, 22 of S355 and 47 of 7075-T6. Table 3 also includes the crack lines angles θ and θ 1 used by the models to make the predictions, which will be explained later in the document. For the three materials, on average, the experimental crack initiation point is close to the point of maximum principal stress at the hole contour, i.e. θ =0◦for axial load, θ =45◦for torsional load and θ =31.7◦for biaxial load. The experimental crack direction in its initial part, measured for a crack length a, is, for the three materials, on average, close to the direction of maximum principal stress, i.e. θ 1 = 0◦for axial load, θ 1 =45◦for torsional load and θ 1 =31.7◦for biaxial load. An effect of the type of material in the average crack initiation point and crack direction was not appreciated. Fig. 1. Geometry of the notched specimen. V. Chaves et al. International Journal of Fatigue 174 (2023) 107726 4 5. Brief description of three biaxial fatigue models for notches In this section a brief description of the following three multiaxial notch fatigue models is presented: the combined model of Susmel and Taylor (MWCM+PM), the combined model of Carpinteri et al. and the biaxial Navarro-Rios model. 5.1. The combined model of Susmel and Taylor (MWCM+PM) It consists of combining the Critical Plane Approach proposed by Susmel for solids without notches under multiaxial loading, called the Modified Wh¨ oler Curve Method (MWCM) [1], with Taylor’s Critical Distance Theory (TCD) for notches under axial loading [3]. The combined model consists of locating the point of maximum principal stress at the notch contour (the hot-spot). A straight line is then drawn from that point, representing the crack direction. There are two variants of the model: the Mode I variant, in which the line is drawn perpendicular to the notch contour at the hot-spot [22]; and the Mode II variant, in which the line is drawn at 45◦with respect to the line of the Mode I variant [23]. It is worth remembering also that, for the Mode I variant, the focus path is suggested as being taken coincident with the notch bisector when stress concentrators are modelled by imposing that the notch root radius equals zero. Within the TCD, if the point method PM is chosen, then the point to make the prediction is located at the chosen line and at a distance of a 0 /2 from the hot-spot. At this point, the critical plane is calculated as the one with the greatest value of the shear stress amplitude τ a . The stress normal to that plane at the chosen point is called σ n,max . From these two stresses, the fatigue limit of the notched component is reached if the following criterion is met: τ a+( τ FL − σ FL 2) σ n,max τ a = τ FL (1) Fig. 6(a) shows a sketch of the crack lines used for the MWCM+PM model for its two variants. The case of a plate with a hole under biaxial loading ( σ ∞ y, τ ∞) is represented. It is also shown the crack line used in the Carpinteri et al. model, which will be described below. 5.2. The combined model of Carpinteri et al. This model is qualitatively similar to the previous one. The prediction is based on the stresses at a point located at a 0 /2 from the hot-spot. In this case, the line where this point is located depends on the material and is defined by the angle δ with respect to the line normal to the notch contour at the hot-spot. The value of δ is calculated by the following Table 2 Experimental fatigue limits of the notched specimens and model’s predictions. Material R, hole radius (mm) Type of loading Notch fatigue limit, σ N FL or τ N FL (MPa) Experimental Model’s predictions MWCM+PM (Mode I) MWCM+PM (Mode II) Carpinteri et al. Biaxial N-R AISI 304L 0.5 Axial 148 125.6 124.7 149.1 145.0 1 Axial 154 118.4 116.0 128.6 118.7 1.5 Axial 151 114.7 112.6 121.5 110.3 0.5 Torsion 133 107.9 104.1 124.5 124.6 1 Torsion 130 94.5 91.2 102.1 96.0 0.5 Biaxial ( σ = τ ) 117 73.1 71.5 85.7 84.5 S355 0.4 Axial 175 127.2 121.8 134.1 142.4 0.75 Axial 152 112.4 107.8 117.2 117.9 1.7 Axial 138 101.2 98.7 105.8 102.4 7075-T6 0.5 Axial 95 105.1 100.1 96.6 119.6 1 Axial 94 95.5 92.8 91.2 110.0 1.5 Axial 85 92.3 90.5 89.5 107.3 0.5 Torsion 75 82.5 77.7 74.9 94.5 1 Torsion 67 73.3 70.8 69.6 83.8 0.5 Biaxial ( σ = τ )65 58.8 55.7 53.7 66.6 Average error (%) 17.5 % 18.1 % 12.1 % 18.7 % Fig. 2. (a) Sketch of the specimen with the axes. (b) Detail of the hole with the cracks and the variables used to define the crack direction. V. Chaves et al. International Journal of Fatigue 174 (2023) 107726 5 equation [2]: δ=3 π 8[1−( τ FL σ FL)2](2) For materials with brittle behavior in fatigue, i.e. τ FL/ σ FL =1, the line will coincide with that of the Mode I variant of the previous model, and for materials with ductile behavior in fatigue, with a value τ FL/ σ FL = 0.58, the line will coincide with that of the Mode II variant. Next, the normal stress N and shear stress C to the line at the critical point are calculated. An equivalent stress σ eq,a and the fatigue limit are defined by the following criterion: σ eq,a= N2+( σ FL τ FL )2 C2 √= σ FL (3) 5.3. Biaxial Navarro-Rios model (biaxial N-R model) The Navarro and de los Rios model (N-R model) [24,25] is a microstructural model that analyses the interaction of the crack and its associated plastic zone with microstructural barriers, such as grain boundaries. Mathematically, the crack, its plastic zone, and the microstructural barrier are modelled as a straight line by a distribution of dislocations. The remote applied stress necessary for the crack to overcome the successive barriers is calculated, the maximum of these values being the fatigue limit of the notched component. The equation to calculate the remote stress to overcome the i-th barrier, located at a distance of i grains from the crack initiation point, is the following [4]: σ i 3 m* σ i⋅ τ c + τ i 3 m* τ i⋅ τ c =1(4) The stresses σ i 3 and τ i 3 represent the shear and normal stresses in the ith barrier, calculated from the equilibrium of dislocations in the crack line, for the specific notched geometry and applied loads. The parameters m* σ i⋅ τ c and m* τ i⋅ τ c represent the normal and tangential stresses required to overcome the i-th barrier and are adjusted for each material, based on the axial and torsional fatigue limits, σ FL and τ FL, and the Kitagawa-Takahashi diagram of the material. A characteristic of this model is that the crack line is not established a priori, as in previous models, but is calculated as the one for which the applied stress required to overcome the successive barriers is a minimum. So the application of the model requires the repetition of the calculations for all possible initiation points and crack directions, defined by the crack initiation point (angle θ) and the crack line direction (angle θ 1 ). 6. Predictions with the models This section shows the notch fatigue limits predictions, σ N FL and τ N FL, with the described models for the three materials, the three types of loading and the various hole radii previously shown. The elastic stress fields to make the predictions with the models were calculated with the analytical expression of Kirsch for an infinite plate with a circular hole subjected to biaxial loading, available in elasticity books [26]. The simplification of using this analytical stress field for the studied geometry is reasonably justified, as discussed in [27]. For the MWCM+PM and Carpinteri et al. models, plane strain was assumed. Predictions with the two variants of the MWCM+PM, Mode I variant and Mode II variant, are presented. For the biaxial N-R model, the Kitawaga-Takahashi diagram was approximated with the equation proposed in [28], using f = 2.5. The number of algebraic equations was set to 200, the maximum crack length studied was set at 30 grains in length and the step of the θ and θ 1 angles for the study of the various directions was set at 1◦, all with Fig. 3. Examples of experimental crack angles for AISI 304L: (a) Axial loading, R =1 mm, σ =200 MPa, N =320,500 cycles. (b) Torsional loading, R =0.5 mm, τ = 156 MPa, N =385,700 cycles. (c) Biaxial loading, R =1 mm, σ = τ =130 MPa, N =178,400 cycles. V. Chaves et al. International Journal of Fatigue 174 (2023) 107726 6 the aim of achieving a high precision in the numerical solution without an excessive computational cost. To illustrate how the models were applied, the calculations with the various models for a specific case, material AISI 304L, R =0.5 mm and axial loading, are shown below. Beginning with the MWCM+PM model, Mode I variant, the hot-spot is for this case at θ =0◦and the crack direction at θ 1 =0◦. The stresses in the point located at the chosen line and at a distance of a 0 /2 =0.09 mm from the hot-spot, according to Kirsch’s expression, for an applied axial load of 1 MPa are as follows: σ rr = 0.3036 MPa, σ θθ =2.1328 MPa and τ τ θ =0 MPa. For this point, making simple calculations with the stress tensor, the maximum shear stress τ a =0.9146 MPa and its corresponding normal stress σ n,max =1.2182 MPa, are calculated. These stresses when introduced in the criterion (Eq. (1)) give the prediction σ N FL =125.6 MPa. For the Mode II variant of the MWCM+PM model, the crack direction changes to θ 1 = − 45◦. The stresses at a 0 /2 are in this case: σ rr =0.2561 MPa, σ θθ =2.2588 MPa and τ τ θ =0.0828 MPa, only slightly different from those of the Mode I variant. The stresses for the criterion are τ a =1.0048 MPa and σ n,max = 1.2575 MPa, giving the prediction σ N FL =124.7 MPa. Regarding the model of Carpinteri et al., θ =0◦as in the previous model, while θ 1 is calculated from the angle δ. Using Eq. (2) for this material, δ =0.1995 rad =11.43◦is obtained, providing θ 1 = − 11.43◦. Again, the stresses at a 0 /2 are: σ rr =0.3009 MPa, σ θθ =2.1403 MPa and τ τ θ =0.0267 MPa. The normal and shear stresses to the crack line in this point are, in modulus, N =2.0970 MPa, C =0.2802 MPa. When these stresses are introduced into the criterion (Eq. (3)), they provide the prediction σ N FL = 149.1 MPa. Regarding the biaxial N-R model, the calculated crack line for this case, the one for which the required applied stress to overcome the successive barriers is minimum, is defined by the angles θ =4◦and θ 1 =-25◦. In this line, the stresses of the biaxial N-R model at the first barrier, for an applied axial load of 1 MPa, are σ 1 3=10.77 MPa, τ 1 3= 3.25 MPa. For this material, the criterion parameters for the first barrier are m* σ 1⋅ τ c=1547 MPa and m* τ 1⋅ τ c=2240 MPa. If these stresses are introduced in Eq. (4), the applied axial stress required to overcome the first barrier is 118.82 MPa. The applied stresses required to overcome the following three barriers are 138.93 MPa, 145.04 MPa and 145.02 MPa, respectively. According to previous studies, the first local maximum is the global maximum [29]. Then, the predicted fatigue limit is calculated with the third barrier and the prediction is σ N FL =145.04 MPa. Table 2 shows the predictions of the notch fatigue limits for the 15 studied cases. Besides, an average error for each model is presented. The fatigue limit prediction error was calculated as follows: Error =Prediction-Experimental Experimental ⋅100 (%)(5) The absolute value of the errors was used to calculate the average error in order to prevent the positive and negative errors from compensating one another. In general, the models provide predictions close to the experimental ones, with an average error of less than 20% for all the models, which is considered a reasonable error in fatigue [3,1]. The smallest average error occurs for the Carpinteri et al. model, 12.1%. The other models give a very similar average error: 17.5%, 18.1% and 18.7%, for the MWCM+PM Mode I, MWCM+PM Mode II and biaxial N-R, respectively. In the case of the biaxial N-R model, the barrier that defined the notch fatigue limit was almost always within the first 10 grains. Regarding the crack line used in the models, Table 3 includes the crack lines angles θ and θ 1 used by the models to make the predictions. The MWCM+PM and Carpinteri et al. models use the angle θ =0◦for axial load, θ =45◦for torsional load and θ =31.7◦for biaxial load, i.e the point of maximum principal stress at the hole contour, which are Fig. 4. Example of experimental crack angles for S355: (a) Axial loading, R =0.75 mm, σ =158 MPa, N =876,500 cycles. (b) Detail of the angle θ 1 for the left crack. (c) Detail of the angle θ 1 for the right crack. V. Chaves et al. International Journal of Fatigue 174 (2023) 107726 7 very close to the average experimental values also shown in Table 3. For the biaxial N-R model, the values of the angles θ used to make the predictions, obtained by scanning directions, are also close to the experimental values for all the studied cases. Regarding the angle θ 1 , the MWCM+PM (Mode I variant) uses the Mode I direction, i.e θ 1 =0◦for axial load, θ 1 =45◦for torsional load and θ 1 =31.7◦for biaxial load, which are close to the average experimental values. However, the MWCM+PM (Mode II variant) uses the Mode II direction, i.e θ 1 =45◦for axial load, θ 1 =0◦for torsional load and θ 1 = − 13.3◦for biaxial load, which are clearly far away from the average experimental values. In the case of the Carpinteri et al. model, the angle θ 1 depends on the material, being relatively close to the experimental values for AISI 304L, further away for S355 and quite far away for Al 7075-T6, since for this last material the Mode II direction is used. For the biaxial N-R model, an intermediate value between the direction of Mode I and Mode II is generally obtained, and therefore far from the experimental values, which are close to that of Mode I. In summary, the MWCM+PM (Mode I variant) uses a crack line that is very similar to the experimental crack line, which is not the case for the rest of the models. Fig. 5. Examples of experimental crack angles for 7075-T6: (a) Axial loading, R =1 mm, σ =110 MPa, N =139,300 cycles. (b) Torsional loading, R =1 mm, τ =80 MPa, N =788,700 cycles. (c) Biaxial loading, R =1 mm, σ = τ =80 MPa, N =130,400 cycles. Table 3 Average experimental crack angles and crack lines angles used by the models. Material R, hole radius (mm) Type of loading Average experimental angles Angles used by the models MWCM+PM (Mode I) MWCM+PM (Mode II) Carpinteri et al. Biaxial N-R θ θ 1 θ θ 1 θ θ 1 θ θ 1 θ θ 1 AISI 304L 0.5 Axial −0.3◦(12.5◦) 4.4◦(8.5◦) 0.0◦0.0◦0.0◦45.0◦0.0◦11.4◦4.0◦−25.0◦ 1 Axial 1.3◦(9.8◦) 1.3◦(3.8◦) 0.0◦0.0◦0.0◦45.0◦0.0◦11.4◦2.0◦−20.0◦ 1.5 Axial 0.9◦(4.6◦) 0.9◦(7.8◦) 0.0◦0.0◦0.0◦45.0◦0.0◦11.4◦1.0◦−17.0◦ 0.5 Torsion 46.8◦(6.7◦) 45.2◦(2.1◦) 45.0◦45.0◦45.0◦0.0◦45.0◦33.6◦45.0◦−7.0◦ 1 Torsion 45.9◦(6.3◦) 42.2◦(2.8◦) 45.0◦45.0◦45.0◦0.0◦45.0◦33.6◦45.0◦16.0◦ 0.5 Biaxial 29.0◦(4.4◦) 32.0◦(3.1◦) 31.7◦31.7◦31.7◦−13.3◦31.7◦20.3◦35.0◦−6.0◦ S355 0.4 Axial 0.3◦(7.5◦) −1.0◦(14.1◦) 0.0◦0.0◦0.0◦45.0◦0.0◦28.7◦9.0◦−42.0◦ 0.75 Axial 0.4◦(4.5◦) 0.4◦(13.3◦) 0.0◦0.0◦0.0◦45.0◦0.0◦28.7◦4.0◦−35.0◦ 1.7 Axial −0.5◦(4.2◦) −5.1◦(7.8◦) 0.0◦0.0◦0.0◦45.0◦0.0◦28.7◦1.0◦−30.0◦ 7075-T6 0.5 Axial −1.5◦(8.5◦) 1.2◦(18.8◦) 0.0◦0.0◦0.0◦45.0◦0.0◦45.0◦5.0◦−29.0◦ 1 Axial −1.1◦(7.9◦) −0.4◦(15.8◦) 0.0◦0.0◦0.0◦45.0◦0.0◦45.0◦2.0◦–32.0◦ 1.5 Axial −0.1◦(10.2◦) 2.4◦(19.5◦) 0.0◦0.0◦0.0◦45.0◦0.0◦45.0◦1.0◦–33.0◦ 0.5 Torsion 42.4◦(10.4◦) 47.4◦(7.2◦) 45.0◦45.0◦45.0◦0.0◦45.0◦0.0◦45.0◦−7.0◦ 1 Torsion 45.2◦(8.0◦) 48.3◦(11◦) 45.0◦45.0◦45.0◦0.0◦45.0◦0.0◦45.0◦11.0◦ 0.5 Biaxial 29.2◦(7.6◦) 36.6◦(5.4◦) 31.7◦31.7◦31.7◦−13.3◦31.7◦−13.3◦38.0◦4.0◦ V. Chaves et al. International Journal of Fatigue 174 (2023) 107726 8 7. Predictions with the models using the experimental crack direction An interesting exercise is to carry out the models’ predictions imposing the crack direction measured experimentally, i.e using in the models the experimental angles θ and θ 1 , which in the studied cases have turned out to be, on average, close to the point of maximum principal stress and to the Mode I direction, respectively. This allows us to analyze whether the models could in the future be easily adapted to using the experimental crack direction to make the predictions. Table 4 shows a comparison of the models’ average errors using theoretical and average experimental angles θ and θ 1 . The results are differentiated for the three materials. Note that when using the experimental angles, the two variants of the MWCM+PM model merge into one. As previously explained, the theoretical θ angles of all the studied models are very similar to each other, and very similar to the experimental ones, all of them are equal or close to the angle defined by the hot-spot. However, the theoretical θ 1 angles of the models present great differences among them, varying between the direction of Mode I and Mode II. As seen in Table 4, the average errors of the MWCM+PM model using the average experimental angles are practically the same for the three materials as those obtained with the MWCM+PM model (Mode I) using the theoretical angles. It is reasonable, since as previously mentioned, the average experimental angles were very close to the Mode I direction for the studied cases. Furthermore, these errors are not far to those obtained with the MWCM+PM model (Mode II) using the theoretical angles. This result was not expected given that the average experimental angles were very far from the Mode II direction. This indicates that the MWCM+PM model is very insensitive to the angle θ 1 used for the predictions. The reason is that the stresses used in the model criterion, the maximum shear stress amplitude τ a and its normal stress σ n,max , change Fig. 6. Sketch of the crack lines used in the models: (a) MWCM+PM (Mode I and Mode II variant) and Carpinteri et al., (b) Biaxial N-R model. Table 4 Comparison of models’ errors using theoretical and average experimental angles θ and θ 1 . Material Average error using the theoretical crack angles (%) Average error using the average experimental crack angles (%) MWCM+PM (Mode I) MWCM+PM (Mode II) Carpinteri et al. Biaxial N-R MWCM+PM Carpinteri et al. Biaxial N-R AISI 304L 23.4 25.2 14.3 17.8 23.4 14.7 14.3 S355 26.7 29.3 23.2 22.3 26.7 25.2 5.7 7075-T6 8.3 6.1 5.2 20.4 8.3 6.5 175.7 Total average error (%): 18.0 18.4 12.4 19.8 18.0 13.5 77.1 V. Chaves et al. International Journal of Fatigue 174 (2023) 107726 9 very little if the angle θ 1 is varied from the Mode I to the Mode II direction, at least for this geometry, loads and materials. Regarding the model of Carpinteri et al. there is also little difference in the predictions, and therefore in the errors, when using the average experimental direction or the theoretical direction. In this case, the variation of θ 1 does cause considerable changes in the values of the criterion stresses, N and C (normal and shear stresses to the crack line). Generally, there is an increase in one of the stresses and a decrease in the another, but when introducing them into the criterion, the increase in one of them is offset by the decrease in the other, giving rise to a similar prediction. As an example, the case of material S355, R =1 mm and axial load (1 MPa), with θ =0◦(hot-spot) and θ 1 varying from 0◦(Mode I) to 45◦(Mode II) is analyzed. The stresses of the MWCM+PM model are τ a =1.18 MPa, σ n, max =1.36 MPa for θ 1 =0◦and τ a =1.25 MPa, σ n,max =1.39 MPa for θ 1 =45◦, which when introduced in the criteria give the predictions σ N FL = 107.6 MPa and σ N FL =103.8 MPa, respectively. That is, for this model the stresses and the predictions are very similar using the Mode I and Mode II directions. For the Carpinteri et al. model, the stresses are N =2.53 MPa, C =0 MPa for θ 1 =0◦and N =1.54 MPa, C =1.24 MPa for θ 1 = 45◦, which give the predictions σ N FL =108.5 MPa and σ N FL =122.4 MPa, respectively. In this case the stresses are very different for Mode I and Mode II directions, but the predictions are quite close, although not as much as for the MWCM+PM model. Regarding the biaxial N-R model, as seen in Table 4, the average errors for the AISI 304L and S355 materials with the average experimental angles are even smaller than those with the theoretical angles. However, for 7075-T6, the errors using the average experimental directions are extremely large. As it happens with the Carpinteri et al. model, the criterion stresses, in this case σ i 3 and τ i 3, change considerably as the angle θ 1 varies, so that generally one of them increases while the other decreases. The materials AISI 304L and S355, with brittle and intermediate ductile-brittle behavior, respectively, have values of the criterion parameters, m* σ i⋅ τ c and m* τ i⋅ τ c, of the same order of magnitude, so the increase of one of the stresses is partially compensated by the decrease of the other when they are introduced into the criterion, giving rise to predictions that are not very different. However, for the material 7075-T6, considered to be brittle in fatigue, the parameters m* σ i⋅ τ c and m* τ i⋅ τ c have a very different magnitude. So the changes of the stresses σ i 3 and τ i 3 with the variation of θ 1 are not compensated in any way when introduced into the criterion, giving rise to very different predictions. Lets analyze again the example of R =1 mm and axial load (1 MPa), with θ =0◦(hot-spot) and θ 1 varying from 0◦(Mode I) to 45◦(Mode II). For the material S355, the stresses of the biaxial N-R model at the first barrier are σ 1 3=14.3 MPa, τ 1 3=0 MPa for θ 1 =0◦and σ 1 3=9.0 MPa, τ 1 3=4.7 MPa for θ 1 =45◦. For this material, the criterion parameters for the first barrier are m* σ 1⋅ τ c=1785 MPa and m* τ 1⋅ τ c=1076 MPa. Assuming for simplicity that the fatigue limit is obtained with the first barrier, the predictions would be σ N FL =125.3 MPa and σ N FL =105.6 MPa, respectively, that is, not very different from each other. If the same analysis is done for material 7075-T6, the stresses of the model are σ 1 3= 14.5 MPa, τ 1 3=0 MPa for θ 1 =0◦and σ 1 3=9.1 MPa, τ 1 3=4.8 MPa for θ 1 =45◦, that is, quite similar to those obtained for the material S355. However, the criterion parameters for the first barrier are now m* σ 1⋅ τ c= 4247 MPa and m* τ 1⋅ τ c=666 MPa, that is, quite different in magnitude. The predictions would be now σ N FL =292.3 MPa and σ N FL =106.5 MPa, respectively, which means that difference between both predictions is extremely large. In summary, although the biaxial N-R predictions with the average experimental directions are the best for AISI 304L and S355, they turn out to be unacceptable for Al 7075-T6. 7.1. Analysis of the effect on the predictions of the experimental angles’ standard deviations As seen in Table 3, the average experimental angles θ are close to those defined with the hot-spot and θ 1 are close to Mode I direction. But the dispersion of these values are relatively large for all the materials, hole radii and loading, as seen in the standard deviation of these values, shown in parentheses in Table 3. This dispersion in the experimental crack angles is reasonable, since for the tested specimens there is obviously a dispersion in the surface finish quality at the notch surface and in the material microstructure (distribution of grains, pores, inclusions, etc.), which has an influence on where the crack initiates and in which direction, especially in high cycle fatigue. This section shows an analysis of the effect of this dispersion on the models’ predictions calculated with the experimental angles. Starting from the average experimental angles θ and θ 1 , whose predictions are called σ N FL(exp(θ,θ1)), a sweep of θ and θ 1 angles of ±1 standard deviation s of each of the variables was performed. The model’s predictions were made for all these directions, providing a range of predictions ( σ N FL(max), σ N FL(min)). Their dimensionless values, expressed as a percentage, were calculated using the following expressions: σ N FL(max,dim) = σ N FL(max) − σ N FL(exp(θ,θ1)) σ N FL(exp(θ,θ1)) ⋅100 (6) σ N FL(min,dim) = σ N FL(min) − σ N FL(exp(θ,θ1)) σ N FL(exp(θ,θ1)) ⋅100 (7) Note that Eq. (6) will always provide a positive value while Eq. (7) will always provide a negative value. Table 5 shows the average of these values for each material and model. Besides, a total average for each model is shown. As expected, after the analysis in the previous section, the dispersion of θ and θ 1 experimental angles generates very low dispersion in the MWCM+PM’s predictions for the three materials, and a bit more dispersion in the Carpinteri et al.’s predictions. Regarding the biaxial N-R model, there is a clear effect of the material, generating low dispersion for the AISI 304L, a higher dispersion for the S355 and a very high dispersion for the 7075-T6. These results indicate that the MWCM+PM’s and Carpinteri et al.’s predictions using the average experimental angles are highly representative of these models’ predictions for the entire range of experimental angles. The same is not the case with the biaxial N-R model, especially for the 7075-T6 material, for which it would be convenient to give a range of predictions and not just the value for the average experimental angles. 8. Discussion The experimental results shown in this work indicate that the fatigue crack initiates from a point of the notch contour close to the hot-spot and grows in its initial part close to the Mode I direction. These results were obtained after analyzing 106 specimens of three materials, with several hole radii and three types of loading, in the high cycle fatigue regime. This is a large study, although it is not comprehensive. Regarding the hole radii, values around 1 mm were analyzed, missing the study of very small radii, of the order of grain size, and very large radii, of the order of several mm. Recently, Lorenzino and Navarro tested an aluminum alloy with very large grains [15]. The specimens had a circular hole whose radius was of the order of the grain size and were subjected to cyclic axial loading. They observed a great dispersion of the crack initiation point and very irregular crack paths in their initial part, greatly affected Table 5 Range of predictions using the average experimental angles θ and θ 1 ±1 s. Material Range of predictions: ( σ N FL(max,dim), σ N FL(min,dim))(%) MWCM+PM Carpinteri et al. Biaxial N-R AISI 304L (3.5, −0.2) (4.8, −0.1) (1.8, −4.0) S355 (1.5, −0.6) (1.6, 0.0) (3.1, −15.9) 7075-T6 (4.8, −0.6) (4.6, −3.1) (25.9, −46.6) Total average: (3.6, ¡0.4) (4.1, ¡1.3) (11.7, –23.4) V. Chaves et al.