Analysis of biaxial fatigue limit models for cases with circular notches

International Journal of Fatigue 162 (2022) 106981 Available online 11 May 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/). Analysis of biaxial fatigue limit models for cases with circular notches V. Chaves a , * , A. Navarro a , L. Susmel b , D. Taylor c 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 Department of Civil and Structural Engineering, The University of Sheffield, Mappin Street, Sheffield S1 3JD, United Kingdom c Department of Mechanical Engineering, Trinity College, Dublin 2, Ireland ARTICLE INFO Keywords: Multiaxial fatigue Notch High cycle fatigue Fatigue limit Biaxial cyclic loading ABSTRACT This work shows an analysis of several models of multiaxial fatigue for notches: Navarro-Rios’ model, which analyses the interaction between the crack and its associated plastic zone with the material microstructural barriers, and three models that combine a critical volume method for notches with a critical plane model for multiaxial fatigue in unnotched solids. Specifically, the application of these models for the prediction of the fatigue limit for a plate with a circular hole subjected to axial, shear and in-phase biaxial cyclic loading is studied. The effects of two parameters are analysed: the radius of the hole and the relationship between the torsional and axial fatigue limits. For all the analysed models, cases are observed in which an increase in the hole radius produces an increase in the predicted fatigue limit, that is, the evolution of the fatigue limit with an increasing hole radius is not always monotonically decreasing, as would be expected. These effects, which we have called “humps” because of their appearance on the prediction graphs, mainly occur in shear loading. No humps were observed in the studied experimental results, but the number of available experimental results is too small to assure this tendency. The results shown in the work indicate that a greater knowledge of the physics of multiaxial fatigue in notches is necessary to achieve models that are capable of providing increasingly accurate predictions. 1. Introduction It is well known that fatigue failure occurs very frequently from stress concentrations or notches. Various models for predicting fatigue failure in the presence of notches have been published since the 1950s, with the so-called critical volume models being probably most widely used. These models include the classic Neuber [1] and Peterson models [2], and, more recently, the Theory of Critical Distances (TCD) [3]. They are based on assessing the stress in a certain volume of material ahead of the notch root. Fatigue failure occurs if the average stress in this volume reaches a certain critical value. The size of the critical volume is material dependent. Within the Critical Distance Methods of Taylor, the point method (PM) stands out for its simplicity and good predictions. In the PM, fatigue failure is predicted from the elastic stress at a single point, located at a distance L/2 from the notch edge, where L is El Haddad’s short crack parameter of the material. Failure will occur if the maximum principal stress at the distance L/2 equals the material endurance limit or fatigue limit. From a different point of view, some short-crack growth models, such as Navarro-Rios’ model, have been developed to predict fatigue failure at notches [4,5,6]. Navarro-Rios’ model analyses the capability of the crack, which is formed at the notch root, and its associated plastic zone, to overcome successive microstructural barriers such as grain boundaries. It is assumed that the plastic displacement takes place in rectilinear slip bands across grains in the solid. The microcrack nucleates in the most favourable grain and within this grain the plastic zone spreads from the crack tip to the grain boundary, where it is arrested. The crack, the plastic zone and the barrier are modelled by means of a continuous distribution of dislocations. This allows a correct representation of these three zones and the obtaining of the local stress at the barrier as a function of the applied stress. If the applied stress is high enough, plastic slip will be promoted into the next grain. Crack propagation and blocking at the grain boundary would occur in each successive grain. Fatigue failure occurs if the crack spreads through the successive grains. All the presented models were developed to study cases of axial cyclic loading. However, components are frequently subject to multiaxial cyclic loading. The study of plain specimens under multiaxial cyclic loading has received the attention of many researchers for quite some time, and several failure prediction models have been proposed, including the so-called “critical plane” models, such as the model of * 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.2022.106981 Received 18 March 2022; Received in revised form 29 April 2022; Accepted 2 May 2022 International Journal of Fatigue 162 (2022) 106981 2 Findley [7] and the model of Brown and Miller [8]. These models are based on the evidence that the fatigue crack grows on a certain plane and correlate fatigue failure with reaching a certain value of stresses or strains in the critical plane. Despite the long time devoted by the scientific community to the prediction of failure in multiaxial fatigue, the results provided by the most widespread models are very different from one another even for relatively simple cases such as the case of a proportional biaxial stress [9], which indicates that it is necessary to continue improving the multiaxial fatigue models. The combination of the presence of a notch and the application of multiaxial cyclic loading implies a problem of even greater complexity. Experimental studies in this combined field are not so extensive and existing models are not so consolidated. Focusing on high-cycle fatigue, in the experimental field, the extensive classical work of Gough et al. in specimens containing complex stress raisers made of various materials and subjected to in-phase bending-torsion loading is remarkable [10]. More recently, Endo and Murakami tested cylindrical specimens with small superficial circular holes under bending and torsion [11], Susmel and Taylor tested notched cylindrical samples under in- and out-of- phase biaxial situations, including non-zero mean values and different frequencies [12], and plates with an inclined V-notch subject to in-phase mixed Mode I/II loadings [13]. Fatemi et al. [14] and Chaves et al. [15,16] tested cylindrical specimens with a circular hole under biaxial loading. Regarding the models in this field, most are relatively recent. Proposals that combine a critical volume model for notches with a critical plane model for multiaxial fatigue stand out, such as the combined model proposed by Taylor and Susmel [17]. With a different philosophy, there is a version for notched solids under multiaxial loading of Navarro-Rios’ short-crack model [18]. Comparing the predictions obtained with various notch multiaxial fatigue models for simple geometry and load cases can provide great insight into the degree of accuracy and reliability of the model predictions. This information can be supplemented by comparing some of these model predictions with some experimental cases. As for smooth specimens, this type of study can help to determine the current models’ degree of maturity for notches in multiaxial fatigue and if an evolution, or certain modifications, are necessary to obtain more accurate predictions. It could also indicate the need for a specific test campaign to verify whether or not certain trends in the model predictions are correct. This paper shows an analysis of various multiaxial fatigue models for notches. The study is focused on high cycle fatigue and the study of the endurance limit, or the fatigue limit, for axial, torsional and in-phase biaxial cyclic loading in specimens with a circular notch. The analysis is conducted based on two parameters: the radius of the circular hole R and the ratio between the torsional and axial endurance limits, or fatigue limits, τ FL/ σ FL, which is an indicator of the ductile or brittle behaviour of the material in fatigue (for simplicity, only the term “fatigue limit” will be used from now on). Finally, a comparison between the experimental results from the literature and predictions with the models is presented. 2. Description of the models In this section a brief description of several models of multiaxial notch fatigue is presented. The models are as follows: the model resulting from the combination of Taylor’s Theory of Critical Distances and Susmel’s Critical Plane Method (TCD +MWCM) [17], the multiaxial critical plane criterion combined with the PM proposed by Carpinteri et al. [19] and the microstructural model of Navarro and de los Rios for biaxial loading in notches (Biaxial N-R model) [18]. 2.1. The combined model of Taylor and Susmel (TCD +MWCM) Taylor and Susmel proposed using a combination of one of Taylor’s methods of critical distance for notches under axial loading (TCD) with the critical plane method proposed by Susmel for multiaxial fatigue in unnotched components, the Modified Wh¨ oler Curve Method (MWCM) [3,17], to study the effect of notches under multiaxial fatigue loading. If the PM is chosen as the critical distance method, then the combined method for calculating the fatigue limit in a notched component consists of locating the point of maximum stress at the edge of the notch (the hotspot) and determining the critical plane at a point located at a distance of L/2, where L is the critical distance of the material for TCD and is defined as L= (1/ π )(ΔKth/Δ σ FL)2, where ΔK th is the threshold value of ΔK for fatigue crack growth and Δ σ FL is the fatigue limit range of the material, at R σ = − 1. Then, L is equal to the material length a 0 used by El Haddad et al. [20]. The direction along which this distance is measured gives rise to two variants discussed below. The critical plane is chosen as that plane passing through the point located at L/2 from the hot-spot which has the largest value of the amplitude of the shear stress τ a. The stress normal to this critical plane is denoted as σ n,max. Once both stresses are known, the fatigue limit is reached if the following criterion is met: τ a+( τ FL − σ FL 2) σ n,max τ a= τ FL (1) Where τ FL is the torsional fatigue limit. There are two variants of the method, depending on where the point at L/2 is located, which we will call Mode I variant and Mode II variant. Mode I variant In the Mode I variant, the point at a distance L/2 from the hot-spot is located on a straight line emanating from the assumed crack initiation location and perpendicular to the component surface at the hot spot itself [21,17]. The direction of the line can be considered as the direction corresponding to a crack formed by normal stresses, i.e. growing in Mode I. A scheme of the crack line for this variant for a plate with a Fig. 1. Crack lines for the MWCM +PM and Carpinteri et al. models. V. Chaves et al. International Journal of Fatigue 162 (2022) 106981 3 circular notch subjected to biaxial loading, σ ∞ y and τ ∞, is shown in Fig. 1. It is worth remembering here also that 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 [17]. Mode II variant In the Mode II variant, the point at a distance L/2 from the hot-spot is on a line forming 45◦to the line normal to the notch surface at the hotspot; that is, in the direction of the maximum shear stress at the hot-spot [22,3]. This would be the direction corresponding to a crack formed by shear stresses; that is, a crack growing in Mode II (see Fig. 1). 2.2. The combined model of Carpinteri et al. This model is qualitatively similar to the previous one, since it is based on extending a multiaxial fatigue critical plane model to the case of notches by incorporating a critical distance method, specifically the PM [19]. As in the previous model, the stresses are analysed at a point located at L/2 from the maximum stress point or hot spot, which is considered the crack initiation point. The direction of the line where the point is located depends on the material and is defined by the angle δ with respect to the line normal to the surface at the hot spot: δ=3 π 8[1−( τ FL σ FL)2](2) The crack line for this model lies between the crack lines for the two MWCM +PM variants, as sketched in Fig. 1. For a material completely brittle in fatigue, with τ FL/ σ FL =1, the value of δ is 0; therefore, the line chosen is that of the Mode I variant, while for a material with τ FL/ σ FL = 1/ 3 √the line is at 45◦, i.e. Mode II variant. For intermediate values of the relation τ FL/ σ FL intermediate directions are also obtained. Once the direction of the crack line and the position of the point are defined, the maximum normal stress N max and the shear stress C in the critical plane at the critical point are evaluated, with the normal to the critical plane defined as the line perpendicular to the crack line. An equivalent stress and the fatigue limit criterion are defined by the following equation: σ eq,a= N2 max +( σ FL τ FL )2 C2 a √= σ FL (3) Please note that a different notation for normal and shear stress has been used in Eqs. (1) and (3). This has been done to facilitate as much as possible the comparisons with the original works. 2.3. Biaxial Navarro-Rios’ model (biaxial N-R model) The Navarro and de los Rios microstructural model (N-R model) [4,5] studies the interaction of the crack and its associated plastic zone with microstructural barriers, such as grain boundaries. The authors modelled the crack as a straight line. Mathematically, the crack, its associated plastic zone and the microstructural barrier are modelled by a distribution of dislocations. The remote stress required for the crack to overcome each successive barrier is then calculated. The maximum of that succession of calculated remote stresses will be the fatigue limit of the component. According to [23], this maximum value corresponds to the first local maximum. The model has been applied to predict the fatigue limit of notched components subject to fatigue axial loading [6,23]. The model has also been extended to in-phase biaxial loading [24]. For the case of a notched component under in-phase biaxial loading, two sets of dislocations are employed to account for the climb and glide components of displacements. The conditions for activation of dislocations sources in the neighbouring grain are established and this leads to a biaxial criterion, which is as follows (for a detailed description see [18]): σ i 3 m* σ i⋅ τ c+ τ i 3 m* τ i⋅ τ c=1(4) The parameters σ i 3 and τ i 3 represent the stresses at the i-th barrier, obtained from the equilibrium of dislocations in the crack line for the specific geometry and applied loads. This equilibrium involves the forces between dislocations, which depend on the geometry of the notch, and the normal and shear linear elastic stresses at the crack line, obtained from an analysis of the un-cracked solid, as in the Bueckner’s superposition technique. Mathematically it gives rise to singular integral equations, which are transformed into a system of algebraic equations by means of quadratures. The parameters m* σ i and m* τ i are orientation factors that project the stresses at the i-th barrier onto the sliding plane and the sliding direction of the source of dislocations, and τ c is the critical stress needed to activate the dislocation sources in the (i +1)-th grain. Their values are adjusted for the specific material from its axial and torsional fatigue limits, τ FL and σ FL, respectively, from the average grain size d and from the Kitagawa-Takahashi diagram of the plain material. The values of m* σ 1⋅ τ c and m* τ 1⋅ τ c, which are the constants of the criterion for the first grain, are as follows (the values for the following grains are based on the values for the first grain and the Kitagawa- Takahashi diagram): m* σ 1⋅ τ c= π 2⋅cos−1n• σ FL (2− σ FL τ FL )(5) m* τ 1⋅ τ c= π 2⋅cos−1n• σ FL  σ FL τ FL −1 √(6) where n is the ratio between the crack length and the crack +barrier length. For a plate with a circular notch subjected to in-phase biaxial cyclic loading, σ ∞ y and τ ∞, the calculation of the biaxial loading necessary to overcome the successive barriers along the crack line is repeated for all possible directions, defined by the crack initiation point (angle θ) and the angle of the direction of the crack (angle θ 1 ) (see Fig. 2). The direction for which overcoming successive barriers requires the least remote biaxial applied stress will provide the sought biaxial fatigue limit. In this direction, the barrier that requires the highest applied stress to be overcome will be the so-called “decisive barrier”, which defines the value of the notch fatigue limit. 3. Predictions using the models Fatigue limit predictions in the case of an infinite plate with a circular hole subject to in-phase biaxial cyclic loading were made with the previously presented models. The material properties of a specific stainless steel (AISI 304L) tested in our laboratory were used for the predictions with the models in the parametric study of this section: average grain size d =0.08 mm, El Haddad length a 0 =L =0.18 mm, and axial fatigue limit σ FL =316 MPa (R σ = − 1) [15]. For calculating the linear elastic stresses in the torsional and axial fatigue limits, the analytical expression of Kirsch, available in elasticity books [25], was used. For the radius R, a wide range of values, from very small to very large, was studied. To analyze the influence of the parameter τ FL/ σ FL in the models’ predictions, virtual materials with the properties stated above and ratios of τ FL/ σ FL that varied between 0.58 and 1 were studied. Specifically, three values were considered: 1/ 3 √≅0.58 (von Misestype material), 0.75 and 1, which allowed us to analyse behaviour ranging from a ductile material in fatigue ( τ FL/ σ FL =0.58) to a brittle behaviour in fatigue ( τ FL/ σ FL =1), as well as an intermediate value. Predictions of the fatigue limit of the notched component σ N FL were obtained. For the biaxial N-R model, 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 V. Chaves et al. International Journal of Fatigue 162 (2022) 106981 4 directions was set at 1◦, all with the aim of achieving a high precision in the numerical solution without an excessive computational cost. The Kitawaga-Takahashi diagram was approximated with the equation proposed in [26], using f =2.5. For the models of Taylor and Susmel, and Carpinteri et al., plane strain was assumed. Three load cases were studied: pure axial, pure torsional and inphase axial–torsional ( σ ∞ y= τ ∞), all of them fully reversed. The hotspot for the three cases was located at the maximum principal stress points, i.e., θ =0◦, θ =45◦and θ =32◦, respectively. The positions of the hot-spots agree with the crack initiation points observed experimentally for this type of notched specimen and loading [15,16]. The predictions obtained with the different models for axial cyclic loading are shown in Fig. 3. Model predictions converge to approximately σ FL for very small radii and to σ FL/Kt= σ FL/3 for large radii, which are the expected results. However, for intermediate radii the differences between the model predictions are quite noticeable, which is surprising for a case as simple as that of pure axial loading. The “type” of material, as per the τ FL/ σ FL ratio, has an important influence, especially in the MWCM +PM model (Mode I variant), where the differences in the material-dependent predictions are quite large for R/a 0 values of approximately 1 (Fig. 3(a)). For the MWCM +PM model (Mode I variant) and brittle material in fatigue ( τ FL/ σ FL =1), it is observed that the curve is not monotonically decreasing, but that there is a small upward zone near R/a 0 =2. This material influence result contrasts with the predictions that would be obtained with an axial fatigue notch model, such as Taylor’s PM [3] or the axial N-R model [6], which would give a single curve regardless of the value of the torsional fatigue limit τ FL. Fig. 4 shows the predictions of the four models for a material with von Mises-type fatigue behaviour ( τ FL/ σ FL =0.58) which is a representative behaviour of ductile materials such as carbon steels, subjected to axial cyclic loading. Predictions with the PM and monoaxial N-R model have been added. There are considerable differences between the methods, especially in the vicinity of R/a 0 =0.3, with differences greater than 20%. The predictions of the four biaxial models for shear loading are shown in Fig. 5. For radii tending to zero, the model predictions approximately converge to the torsional fatigue limit, which is correct. For radii tending to infinity, the models approximately converge to τ N FL = σ FL/Kt= σ FL/4, which is the expected result. It can be seen again that not all curves are monotonically decreasing, but that some of them have a rising and falling area, which we have called humps. These humps are particularly noticeable in the MWCM +PM model (Mode I). Furthermore, for the MWCM +PM model (Mode I) and a material with τ FL/ σ FL =1, the curve shows values above 1 (for the case of R/a 0 =0.28, it is predicted that τ N FL/ σ FL =1.06), that is, the fatigue limit of the notched component τ N FL is predicted to be above the fatigue limit of the plain component σ FL, which is quite an unexpected result. Fig. 6 shows the predictions of the four biaxial models for cyclic shear loading for a material with von Mises-type behaviour ( τ FL/ σ FL = 0.58). The biaxial N-R and MWCM +PM (mode II) models give fairly close predictions, while the MWCM +PM (mode I) model shows more distant predictions. The differences between the model’s predictions are greater than 20% in the vicinity of R/a 0 =0.5. Furthermore, for all four models, the so-called humps are observed near R/a 0 =0.5, the largest being that of the MWCM +PM model (Mode I). For in-phase cyclic biaxial loading, the predictions of the models present an intermediate trend with respect to those shown previously for axial loading and shear loading. Fig. 7 shows the predictions of each of the four models for a biaxial in-phase load of the type σ ∞ y= τ ∞. For radii tending to infinity, the models converge to approximately σ FL/Kt= σ FL/5.47 =0.18, as expected, while for radii tending to zero the models give slightly more varied predictions, due to the different influences of the shear and normal stresses in each of the four fatigue limit criteria. The humps are clearly smaller than those in the case of shear loading, and, as shown in Fig. 8 for a von Mises material, the differences between the predictions of the four models are also smaller than in the case of shear loading. 4. Analysis of the humps by an elastic stress study near a circular hole In some of the graphs in the previous section, primarily in the shear cases (Fig. 5), it has been shown that the evolution of the notched fatigue limit prediction curves is not monotonically decreasing, but has a rising and falling area that we call a hump. The reason for the existence of these humps may be related to the evolution along the crack line of some of the elastic stresses used in the criteria, which also present rises and falls. The effect of the elastic stresses evolution on the fatigue predictions evolution with the root radius for each of the four models is analysed next. 4.1. Elastic stress analysis for the MWCM +PM (Mode I) model Fig. 9 shows the evolution of the elastic stresses in the line θ =θ 1 = 45◦as a function of the distance r to the centre of the hole divided by the hole radius R for an infinite plate with a circular hole subject to a remote applied stress of τ ∞ = − 1 MPa, i.e. shear loading. This line is the crack path used for predictions with the MWCM +PM (mode I variant) model for this type of applied load. Note that with the stress sign criterion adopted in this research, it is required to apply a stress τ ∞=− 1 MPa to have the hot spot and the focus path in the first quadrant. It can be seen that the stress σ θθ starts from a value of 4 MPa and monotonically decreases to a value of 1 MPa at far distances. The shear stress τ rθ is 0 for any distance r. The stress σ rr starts from a value of 0 MPa to meet the boundary condition in the hole contour, rises to a maximum value of Fig. 2. Scheme of the crack initiation point and crack direction for the biaxial N-R model. V. Chaves et al. International Journal of Fatigue 162 (2022) 106981 5 0.33 for r/R =1.22, and then decreases to a value of −1 away from the hole. For this particular case, since the shear stress τ rθ is zero, the two stresses σ θθ and σ rr are principal stresses. For plane strain, the third principal stress is σ zz = ν ( σ θθ + σ rr), which has an intermediate evolution between the other two principal stresses for a considered value of υ = 0.3, as shown in Fig. 9. Therefore, the stresses σ θθ and σ rr are the principal stresses I and III, respectively. The particular case of the prediction with the MWCM +PM model (Mode I variant) for a material with τ FL/ σ FL =0.5 subjected to cyclic shear loading will now be studied, a case that allows a relatively simple analytical study. In the case τ FL/ σ FL =0.5, the fatigue limit criterion, defined with Eq. (1), is greatly simplified, since τ FL − σ FL/2 =0, leaving the following expression: τ a= τ FL (7) Therefore, in this particular case, the stress σ n,max is not used in the criterion, only the maximum shear stress τ a, and the criterion is fulfilled when the stress τ a reaches the value of the torsional fatigue limit τ FL. As seen previously, since the stresses σ θθ and σ rr are the principal stresses I and III respectively, the maximum shear stress τ a is: τ a= σ θθ− σ rr 2(8) The analytical expression for the shear stress τ a, for θ =θ 1 =45◦, is taken directly from the expressions for σ θθ and σ rr for the shear case [27]: τ a= σ θθ − σ rr 2= − τ ∞ 2(2+6R4 r4−4R2 r2)(9) Fig. 9 shows its evolution (for τ ∞ =− 1 MPa) as a function of r/R. It is not monotonically decreasing. Instead, it has an initial downward part and a later upward part. It begins with a value of 2 for r/R =1, continues downward, reaches a minimum, and then continues upward, tending to 1 as r/R tends to infinity. This non-monotonic evolution of τ a clearly (a) MWCM+PM (Mode I) (b) MWCM+PM (Mode II) (c) Carpinteri et al. model (d) Biaxial N-R model Fig. 3. Fatigue limit predictions for an infinite plate with a circular hole subjected to axial cyclic loading with several models: (a) MWCM +PM (Mode I) model, (b) MWCM +PM (Mode II) model, (c) Carpinteri et al. model and (d) Biaxial N-R model. Fig. 4. Fatigue limit predictions of various models for a material with τ FL/ σ FL =0.58 for an infinite plate with a circular hole subjected to axial cyclic loading. V. Chaves et al. International Journal of Fatigue 162 (2022) 106981 6 arises from the non-monotonic evolution of the stress σ rr. Deriving the expression of τ a in Eq.Eqn 9 and setting it to zero, the minimum of τ a is obtained, which is located at r/R =1.73 and has a value of τ a=0.67. From the value of τ a at the critical distance a 0 /2 from the hot spot for a unitary applied load, which we call τ a(a 0 /2), the notch fatigue limit in torsion τ N FL will be the value that multiplied by τ a(a 0 /2) gives a value equal to the plain fatigue limit in torsion τ FL: τ a(a0 2)⋅ τ N FL = τ FL→ τ N FL = τ FL τ a(a0 2)→ τ N FL σ FL =0.5 τ a(a0 2)(10) According to this equation, the evolution of the dimensionless notch fatigue limit τ N FL/ σ FL is inverse to the evolution of τ a(a 0 /2). The evolution of τ N FL/ σ FL with respect to the dimensionless hole radius R/a 0 is shown in Fig. 10 for the case of shear and τ FL/ σ FL =0.5. The curve begins from a value of 0.5 as R/a 0 tends to zero. For a typical value of a 0 of a few hundred microns, R/a 0 tending to zero implies that R also tends to zero, and a 0 /2R tends to infinity. Therefore, the prediction of τ N FL/ σ FL as R/a 0 tends to zero is made with the value of τ a corresponding to r/R tending to infinity, which, according to Fig. 9, is τ a=1. Applying Eq. Eqn 10, τ N FL/ σ FL is equal to 0.5/1 =0.5, which is precisely the value of τ N FL/ σ FL indicated above. Continuing with the analysis of the evolution of τ N FL/ σ FL, the curve rises from 0.5 to a maximum value of 0.75 for R/a 0 =0.68; that is, for a 0 /2 =0.73R. This maximum value is obtained using the value of τ a corresponding to r/R =1.73, i.e. 0.67, which is precisely the minimum of the evolution of τ a, as seen in Fig. 9. Again, if Eq. Eqn 10 is used, τ N FL/ σ FL is equal to 0.5/0.67 =0.75, as shown in Fig. 10. Therefore, the ascending evolution of τ N FL/ σ FL from 0.5 to 0.75, for R/a 0 from 0 to 0.68 corresponds to the descending evolution of τ a, for r/R from infinity to 1.73. The increase in the variable R/a 0 used in the graph of Fig. 10 implies a decrease in the variable r/R used in the graph of Fig. 9. Finally, the curve of τ N FL/ σ FL decreases from the maximum value of 0.75, tending to 0.25 as R/a 0 tends to infinity. This descending part of the τ N FL/ σ FL curve up to a value of 0.25 corresponds to the part of the τ a curve from r/R = 1.73 to r/R =1, where it reaches a value of τ a=2. Again, if Eq. Eqn 10 is (a) MWCM+PM (Mode I) (b) MWCM+PM (Mode II) (c) Carpinteri et al. model (d) Biaxial N-R model Fig. 5. Fatigue limit predictions for an infinite plate with a circular hole subjected to shear cyclic loading with several models: (a) MWCM +PM (Mode I) model, (b) MWCM +PM (Mode II) model, (c) Carpinteri et al. model and (d) Biaxial N-R model. Fig. 6. Fatigue limit predictions of several models for a material with τ FL/ σ FL = 0.58 for an infinite plate with a circular hole subjected to shear cyclic loading. V. Chaves et al. International Journal of Fatigue 162 (2022) 106981 7 used, a value of τ N FL/ σ FL =0.5/2=0.25 is obtained, as expected. In summary, it is the rise and fall on the evolution of the stress σ rr along the line θ =θ 1 =45◦that produces the fall and rise of the stress τ a along the same line, which causes the rise and fall, i.e. the hump, in the evolution of the MWCM +PM (Mode I) predictions for the case of shear and a material with τ FL/ σ FL =0.5, as shown in Fig. 10. For the same model and the same loading (shear), but for materials with τ FL/ σ FL ∕= 0.5, the situation changes since the second term, which we call term2, on the left side of the fatigue criterion (Eq. (1)), i.e. ( τ FL − σ FL/2)•( σ n,max/ τ a), is no longer zero. Again, if we know the values of τ a and σ n,max at the critical distance a 0 /2 from the hot-spot for a (a) MWCM+PM (Mode I) (b) MWCM+PM (Mode )I) (c) Carpinteri et al. model (d) Biaxial N-R model Fig. 7. Fatigue limit predictions for an infinite plate with a circular hole subjected to in-phase biaxial cyclic loading ( σ ∞ y= τ ∞). with several models: (a) MWCM + PM (Mode I) model, (b) MWCM +PM (Mode II) model, (c) Carpinteri et al. model and (d) Biaxial N-R model. Fig. 8. Fatigue limit predictions of several models for a material with τ FL/ σ FL = 0.58 for an infinite plate with a circular hole subjected to in-phase biaxial cyclic loading ( σ ∞ y= τ ∞). Fig. 9. Stresses evolution over the line θ =θ 1 =45◦, for shear loading τ ∞=− 1 MPa. V. Chaves et al. International Journal of Fatigue 162 (2022) 106981 8 unitary applied load, which we call τ a(a 0 /2) and σ n,max(a0/2), and the term2 formed with these two stresses, which we call term2(a 0 /2), then the notch fatigue limit will be the value that multiplied by τ a(a 0 /2) and σ n,max(a0/2)fulfills the criterion of Eq. Eqn 1, that is: τ a(a0 2)⋅ τ N FL +( τ FL − σ FL 2) σ n,max(a0 2)⋅ τ N FL τ a(a0 2)⋅ τ N FL = τ FL⇒ ⇒ τ N FL = τ FL − ( τ FL − σ FL 2) σ n,max(a0 2) τ a(a0 2) τ a(a0 2)⇒ τ N FL σ FL = τ FL −term2(a0 2) τ a(a0 2)⋅ σ FL (11) The combined stress term τ a(a0 2)• σ FL τ FL−term2(a0 2)of Eq. Eqn 11 is called τ comb (a 0 / 2). According to Eq. Eqn 11, the evolution of the dimensionless notch fatigue limit τ N FL/ σ FL for materials with τ FL/ σ FL ∕= 0.5 is inverse to the evolution of τ comb , which is similar to the relationship between the notch fatigue limit τ N FL/ σ FL and τ a for the materials with τ FL/ σ FL =0.5 shown in Eq. Eqn 10. Fig. 11 shows the evolution of τ comb (for τ FL/ σ FL =0.5,0.58, 0.75,1), τ a and σ n,max for an applied stress τ ∞ = − 1 MPa on the line θ = θ 1 =45◦as a function of r/R. The 4 curves of τ comb begin with a value of 4 for r/R =1; they then decrease to a minimum, and finally tend towards constant values of 1/0.5, 1/0.58, 1/0.75 and 1/1, respectively, i.e. 1/( τ FL/ σ FL), as r/R tends to infinity. According to Eq. Eqn 11, and repeating the previous reasoning used for Eq. Eqn 10, the value of τ comb =4 as r/R =1, implies that τ N FL/ σ FL =1/4=0.25 as R/a 0 tends to infinity, as shown in Fig. 5(a). The value of τ comb =1/ τ FL as r/R tends to infinity, implies that τ N FL/ σ FL = τ FL as R/a 0 tends to 0, as shown in Fig. 5 (a). As explained above for τ a, the falling and rising of these τ comb curves correspond to the rising and falling in the curves of the model predictions, as shown in Fig. 5(a). The deepest trough of τ comb occurs for a material with τ FL/ σ FL =0.5, and becomes smoother as the τ FL/ σ FL value increases. The explanation must be sought in the evolution of τ a and σ n,max shown in Fig. 11, the first having a well-defined minimum while the second is monotonically decreasing. As the value of τ FL/ σ FL increases, the weight of σ n,max increases with respect to τ a in the combined stress τ comb , which causes the valley of τ comb to become shallower and almost disappear for materials with τ FL/ σ FL =1. For this type of material the hump in the model predictions is very small, as shown in Fig. 5(a). In summary, the non-monotonic evolution of τ a causes the humps of the MWCM +PM model (Mode I) prediction curves for shear loading. For higher values of τ FL/ σ FL, the weight of τ a versus σ n,max decreases in the prediction, causing the size of the hump to decrease. Fig. 12 shows the evolution of the stresses σ θθ, σ rr, τ a, σ n,max and τ comb in the line θ =θ 1 =0◦as a function of r/R, for the case of axial loading ( σ ∞ y =1 MPa). This line is the crack path used for predictions with the MWCM +PM model (mode I) for this loading. The evolution of σ θθ is similar to the case of shear (Fig. 9), which is monotonically decreasing. The great difference with respect to shear is in σ rr: it increases from 0 MPa to 0.37 MPa, similar to the case of shear, which increases from 0 MPa to 0.33 MPa. However, the descent from this maximum is much smoother in the axial case than in the shear case, tending to 0 MPa as r/R tends to infinity, instead of to −1 MPa as it occurs in the shear case. Due to the gentler evolution of σ rr, τ a is almost a monotonically decreasing function, with a very smooth valley. For this reason, the hump for τ FL/ σ FL =0.58 is much smaller for axial loading (3(a)) than for shear loading (5(a)). For higher values of τ FL/ σ FL, such as τ FL/ σ FL =0.75, the hump disappears completely due to the influence of σ n,max, which makes the τ comb monotonically decrease. For very high τ FL/ σ FL, such as τ FL/ σ FL = 1, a greater influence of σ n,max even produces a hump in the evolution of τ comb . The τ comb hump produces the valley observed in the evolution of the model prediction for τ FL/ σ FL =1, as shown in Fig. 3(a). In summary, the smoother evolution of the stress σ rr along the line θ =θ 1 =0◦, for axial loading, as compared to its evolution along the line θ =θ 1 =45◦for shear loading, essentially to fulfill the boundary conditions of the problem at infinity, produces a smoother evolution of the stress τ a, which causes less pronounced humps in the evolution of the predictions Fig 10. Notch fatigue limit evolution for a plate with a circular hole subjected to cyclic shear loading and a material with τ FL/ σ FL =0.5 with the MWCM +PM (Mode I) model. Fig. 11. Combined stresses evolution used in the MWCM +PM (Mode I) model for several materials over the line θ =θ 1 =45◦, for shear loading ( τ ∞= −1 MPa). Fig. 12. Stresses evolution used in the MWCM +PM (Mode I) model criterion for several materials over the line θ =θ 1 =0◦, for axial loading ( σ ∞ y=1 MPa). V. Chaves et al. International Journal of Fatigue 162 (2022) 106981 9 with the MWCM +PM (Mode I) model, as compared to the shear case. For in-phase biaxial cyclic loading ( σ ∞ y= τ ∞), the evolution of the predictions of the MWCM +PM (Mode I) model (Fig. 7(a)) is intermediate between the evolution shown for axial loading (Fig. 3(a)) and shear loading (Fig. 5(a)), with small humps for τ FL/ σ FL =0.58,0.75 and no valley for τ FL/ σ FL =1. Again the explanation of this evolution lies in the stresses along the path used for the predictions, in this case the line θ = θ 1 =32◦, as shown in Fig. 13, for σ ∞ y= − τ ∞=1. σ rr presents an intermediate evolution with respect to the axial and shear case. As an example, observe its trend as r/R tends to infinity, tending to −0.62 MPa, which is an intermediate value between 0 MPa (value for the axial case) and −1 MPa (value for the shear case). This generates an intermediate evolution of τ a, between the axial and shear evolution, with a smooth valley that is less pronounced than in shear, and very smooth valleys in the evolution of τ comb , without having any humps as in the axial case and τ FL/ σ FL =1. 4.2. Elastic stress analysis for the MWCM +PM (Mode II) model The evolution of the MWCM +PM (Mode II) model predictions, shown in Figs. 3(b), 5(b) and 7(b), presents fewer humps and is smaller than that of the previously analysed MWCM +PM (Mode I). The curves in Figs. 3(b), 5(b) and 7(b) monotonically decrease, except for the material τ FL/ σ FL =0.58 and the cases of shear loading (Fig. 5(b)) and inphase biaxial loading ( σ ∞ y= τ ∞) (Fig. 7(b)), with the rise and fall being almost imperceptible in this second case. The explanation for the evolution of the predictions is found, again, in the evolution of the elastic stresses used in the predictions. In the Mode II variant, the point at L/2 is on the line that forms 45◦with the line normal to the surface at the hot-spot. Fig. 14 shows the prediction point using the Mode II variant for the case of shear, defined with polar coordinates (r, β). The stresses at this prediction point calculated with the Kirsch expressions include a nonzero shear stress term, except for a hole radius R tending to infinity, for which the prediction point has polar coordinates r =R and β =45◦. This is an important difference with respect to the Mode I variant, which, as we saw previously, has the prediction point at the line β =45◦ for any hole radius R, and always has a zero value of the shear stress. Fig. 15(a) shows the evolution of the elastic stresses used for the predictions with Mode II variant versus the position of the prediction point. For hole radii R between 0 and infinity, a shear load τ ∞= − 1 MPa and the same value of a 0 used previously (a 0 =0.18 mm). For each value of R, the polar coordinates (r, β) of the prediction point were calculated and its stresses were obtained from the Kirsch expressions. To facilitate a comparison with Mode I variant, the position of the prediction point was expressed by the dimensionless variable r/R. Please note that the r value of the prediction point will be different for the Mode I variant and the Mode II variant, since the prediction point of each variant lies on different lines. For values of R/a 0 tending to zero the prediction point is at r/R tending to infinity and for values of R/a 0 tending to infinity the prediction point is at r/R tending to 1. As seen in the graph, σ θθ has a decreasing evolution, from 4 MPa to 0 MPa, instead of to 1 MPa, as was the case for Mode I variant. σ rr has a non-monotonic evolution, with a maximum of 0.31 MPa and tends to 0 at both ends, not to 0 and 1 at the ends, as was the case for Mode I variant. Finally, τ rθ is different from zero and progresses from 0 to 1 MPa. Despite the clear difference in the evolution of these stresses, the evolutions of the stresses used in the criterion, τ a and σ n,max, are quite similar in both variants, with τ a going from 2 MPa to 1 MPa with a valley in between, and σ n,max monotonically decreasing from 2 to 0 MPa. As seen in Fig. 5(a) and (b), the predictions with the two variants tend to the same values at the extremes; however, they present differences in the central part, in the vicinity of R/a 0 =1, with larger humps for Mode I variant. This difference in the central part is essentially due to the difference in the τ a valley, which is smoother for Mode II variant. As seen previously, a smooth valley in τ a causes little hump for materials with τ FL/ σ FL close to 0.5 and no humps for materials with τ FL/ σ FL close to 1. Fig. 15(b) shows the evolution of the elastic stresses used for predictions with Mode II variant for axial load σ ∞ y=1 MPa. Again, there are differences in the evolution of the elastic stresses with respect to the evolution for Mode I variant: σ θθ goes from 3 to 0.5 MPa instead of from 3 to 1 MPa, σ rr goes from 0 to 0.5 MPa instead of from 0 to 0 MPa and τ rθ is different from 0. Despite this, the stresses used for the criterion, τ a and σ n,max, have the same values at the extremes for the two variants, 1.5 and 0.5 MPa, so the predictions of the two models converge to the same values at the extremes (Fig. 3(a) and (b)). Again, there is a certain difference in the predictions in the vicinity of R/a 0 =1: for Mode II variant, all the curves are monotonically decreasing and relatively close to one another, while for Mode I variant, one of the curves has a hump and Fig. 13. Stresses evolution used in the MWCM +PM (Mode I) model criterion for several materials over the line θ =θ 1 =32◦, for biaxial loading ( σ ∞ y= − τ ∞=1 MPa). Fig. 14. Crack line for the the MWCM +PM (Mode II) model for shear cyclic loading ( τ ∞=− 1 MPa). V. Chaves et al. International Journal of Fatigue 162 (2022) 106981 16 important role, adding uncertainty to the predictions for these cases. For the case of axial loading, predictions with specific models for this type of loading were included, such as the PM and the axial N-R model, providing different predictions with respect to those given by the biaxial models (see Fig. 4). A fundamental difference between these two types of models was that the predictions with the biaxial models depend on the torsional fatigue limit of the material, whereas this parameter does not affect the predictions with the axial models. A legitimate question is: which is more convenient to use for an axial load case, an axial model or a biaxial model? As seen in the prediction curves of the biaxial models in Fig. 4, the value of τ FL clearly influences the predictions obtained with the MWCM +PM models, especially in the area of R/a 0 =1. The influence of τ FL is much less for Carpinteri et al. and the biaxial N-R models, and is obviously zero for the axial models. Conceptually, the question arises as to whether the torsional fatigue limit has an effect on the fatigue behaviour of a notched solid subject to fatigue axial loading. In addition, humps are not predicted with the axial models, since only the stress σ θθ is used for the predictions, which has a monotonically decreasing evolution with respect to the hole radius. The average errors of the models’ predictions for axial loading, with 13 cases studied, are shown in Table 3. The smallest average error occurs for an axial model, i. e. the PM (7.0%), but there is a biaxial model, that of Carpinteri et al., which also has a very low error (7.6%). The other studied axial model, the axial N-R, has the highest average error (13.1%). Thus, the analysed cases do not show a clear trend of the convenience of using an axial or biaxial model for axial loads. In any case, the number of hole radii and materials analysed are too few to be able to draw reliable conclusions. Again, it would be very convenient to conduct a campaign of fatigue axial tests in notched solids with several root radii made of materials with different τ FL values. This would allow for the questions that have been raised to be answered, specifically, the effect (or lack thereof) of τ FL and the existence (or lack thereof) of humps for the axial case. Additionally, the diversity of directions used by the various models to make predictions is striking, from the Mode I direction to the Mode II direction, passing through intermediate values. This does not prevent the models from giving similar results for small and large radii, since they are essentially adjusted with the same parameters, the plain fatigue limits of the material. Again, it would be necessary to perform a comprehensive experimental analysis for various hole radii and various materials of the crack initiation point and the crack direction in its initial part in order to establish a single crack line to make predictions with all the models. In recent years, work has been performed in this field [15,16] but much work remains. Current knowledge indicates that the Fig. 24. Experimental results by Chaves et al. [16] and model predictions for the Al 7075: (a) axial loading), (b) torsional loading, and (c) in-phase biaxial loading τ ∞= σ ∞ y. Table 3 Statistics of the predictions with the models for the axial loading cases. Model Average Error (%) MWCM +PM (Mode I) 12.7 MWCM +PM (Mode II) 8.5 Carpinteri et al. 7.6 Biaxial N-R 12.0 PM 7.0 Axial N-R 13.1 V. Chaves et al. International Journal of Fatigue 162 (2022) 106981 17 experimental crack line is not straight, but instead zigzagged and changes direction when passing from Stage 1 to Stage 2. This implies that to improve the predictions, it would probably be necessary for current models to evolve towards the use of non-straight crack lines for prediction, which is closer to the experimental crack lines. Though it is not possible to confirm the existence of these humps by reference to experimental data, it is interesting to speculate whether their existence is reasonable, considering the physical mechanisms involved. As is well known, fatigue proceeds in two stages, Stage 1 and Stage 2, which are loosely described as crack initiation and crack propagation. Previously, Susmel [21] showed that the length of the Stage-1 crack is similar to the critical distance in steel, which justifies the use of the TCD (combined with a critical plane theory) as a way of capturing the stresses acting during the initiation stage. We show in the present work that the stresses in this region, and especially the shear stress on the critical plane, vary in a complex way with distance from the hole, implying the existence of a hump in the fatigue limit predictions. From this we conclude that the existence of the hump could be consistent with the physical mechanisms of fatigue. It is perhaps also worth noting that a similar effect has already been established in relation to the variation of strength with notch radius. Considering the variation of fracture strength (or measured fracture toughness) with root radius for notches of constant length, TCD predictions using the PM show a small decrease in strength with increasing radius for radii similar in magnitude to L. This counter-intuitive result has been shown to occur also in the experimental data for brittle fracture of some steels [3] and, recently, carbon fibre composite material [34]. It arises, in a similar way to that shown in the present work, due to a complex variation of stress with distance near the notch. One striking thing the N-R model does not get right is the obvious difference it reports for pure axial loading between the biaxial formulation and the monoaxial formulation for the larger notches, as depicted in Fig. 4. It is clear the predictions with the biaxial formulation are wrong since they converge to a value σ N FL/ σ FL of 0.4 rather than to the expected value around 1/3. The reason for this is the much simplified geometry the model uses for the slip band. In its present formulation, the crack growth direction is fixed from the initiation point onwards and does not change once it has been selected as the direction that maximizes the “pressure” upon the first microstructural barriers. The initiation direction is therefore set and taken to be the same for propagation until failure. This is to say that at present the model does not have the capability of reproducing the switch between stage I, mode II growth typically observed at initiation and stage II, mode I displayed later on. Current work is being done to incorporate the possibility of changes of direction in the propagation path, but the mathematical and numerical difficulties are great and no satisfactory strategy to come around these difficulties has yet been found. Finally, the prediction curves of the N-R model are less smooth than those of the other models (see Figs. 3, 5 and 7). There are two reasons that can justify this behaviour of the N-R model. First, it is a model in which the equations are solved numerically; therefore there will always be a small numerical error when calculating the fatigue limit. Second, the N-R model is a discrete model, which bases its predictions on the analysis of a discrete set of crack directions and crack lengths. In this study an angular step of 1◦was used for the crack directions. The discrete set of crack lengths corresponds to crack tips located at the successive grain boundaries. 7. Summary and conclusions This work presented an analysis of the fatigue limit predictions for a plate with a circular hole subjected to cyclic axial, shear and in-phase biaxial loading through several biaxial models. There were clear differences between the models in terms of the crack initiation point and the crack line used for making the predictions. For small and large radii all the models gave similar predictions, tending to the plain fatigue limit and the plain fatigue limit divided by K t as the hole radii tended to zero and infinity, respectively. For intermediate radii, there were considerable differences in the values of the predictions obtained with the models, despite the fact that this was a simple geometry. In some cases, and for all the models, increases in the notch fatigue limit were predicted with an increasing hole radius, especially in torsion, which is something that has not been observed experimentally, nor does it seem logical from a physical point of view. This non-monotonic evolution of the predictions with an increasing hole radius, called a hump, is also due to the non-monotonic evolution of the σ rr and τ rθ stresses along the crack lines used in the models. It would be very interesting to perform an exhaustive experimental work involving the fatigue testing of specimens with circular holes under several types of loading and including several materials. The tests should be focused on hole radii for which the models predict a nonmonotonic evolution with the hole size, the so-called humps. For the tests, it would be very interesting to experimentally obtain the crack initiation point and the crack path in the initial part of the crack. Later, it would be convenient to adapt the crack lines used for the prediction in the current models to fit them as closely as possible to the experimentally observed crack paths. Additionally, the model criteria were adjusted to provide an evolution of the predictions with the hole radius close to that observed experimentally, either with or without humps. All this work would probably result in predictions of fatigue limits for notched specimens subjected to biaxial loading that are closer to the experimental values and, ultimately, provide the industry with better tools for predicting fatigue failure. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The authors would like to thank the European Union, the Spanish Government and the Junta de Andalucía, Spain, for its financial support through grants DPI2017-84788-P and PID2020-117407 GB-I00 (FEDER/Ministerio de Ciencia e Innovaci´ on - Agencia Estatal de Investigaci´ on) and P18-FR-4306 (‘‘Fondo Europeo de Desarrollo Regional (FEDER) y Consejería de Economía, Conocimiento, Empresas y Universidad de la Junta de Andalucía, dentro del Programa Operativo FEDER 2014–2020’’). V. Chaves et al. International Journal of Fatigue 162 (2022) 106981 18 Appendix Experimental notch fatigue limits and predictions with the models. References [1] Neuber H, Kerbspannungslehre. Theory of notches [Edwards JW, Eng Trans]. Ann Arbor MI: Springer-Verlag; 1946. [2] Peterson RE. Notch sensitivity. In: Sines G, Waisman JL, editors. Metal fatigue. New York; 1959. [3] Taylor D. The theory of critical distances: a new perspective in fracture mechanics. Elsevier; 2007. [4] Navarro A, de los Rios ER. An alternative model of the blocking of dislocations at grain boundaries. Philos Mag A 1988;57:37–42. [5] Navarro A, de los Rios ER. Short and long fatigue crack growth: a unified model. Philos Mag A 1988;57:12–36. [6] Chaves V, Navarro A. Application of a microstructural model for predicting notch fatigue limits under mode I loading. Int J Fatigue 2009;31(5):943–51. [7] Findley WN. A theory for the effect of mean stress on fatigue of metals under combined torsion and axial load or bending. J Eng Ind 1959;81(4):301–6. [8] Brown MW, Miller KJ. A theory for fatigue under multiaxial stress-strain condition. Proc Inst Mech Eng 1973;187:745–56. [9] Koutiri I, Bellett D, Morel F. The effect of mean stress and stress biaxiality in highcycle fatigue. Fatigue Fract Engng Mater Struct 2018;41(2):440–55. [10] Gough HJ, Pollard HV, Clenshaw WJ. Some experiments on the resistance of metals to fatigue under combined stresses. In: Ministry of supply, aeronautical research council reports and memoranda. London: His Majesty’s Stationary Office; 1951. p. 1–141. [11] Murakami Y. Metal fatigue: effect of small defects and non-metallic inclusions. Elsevier; 2002. [12] Susmel L, Tayor D. A critical distance/plane method to estimate finite life of notched components. Int J Fatigue 2012;38:7–24. [13] Susmel L, Taylor D. Two methods for predicting the multiaxial fatigue limit of sharp notches. Fatigue Fract Eng Mater Struct 2003;26:821–33. [14] Fatemi A, Gates N, Socie DF, Pham N. Fatigue crack growth behaviour of tubular aluminium specimens with a circular hole under axial and torsion loadings. Eng Fract Mech 2014;123:137–47. [15] Chaves V, Beretta G, Navarro A. Biaxial fatigue limits and crack directions for stainless steel specimens with circular holes. Eng Fract Mech 2017;174:139–54. [16] Chaves V, Beretta G, Balbin JA, Navarro A. Fatigue life and crack growth direction in 7075–T6 aluminium alloy specimens with a circular hole under biaxial loading. Int J Fatigue 2019;125:222–36. [17] Susmel L. Multiaxial notch fatigue: from nominal to local stress/strain quantities. Woodhead Publishing; 2009. V. Chaves et al. International Journal of Fatigue 162 (2022) 106981 19 [18] Chaves V, Navarro A, Beretta G, Madrigal C. Microstructural model for predicting high cycle fatigue strength in the presence of holes under proportional biaxial loading. Theor Appl Fract Mec 2014;73:27–38. [19] Carpinteri A, Spagnoli A, Vantadori S, Viappani D. A multiaxial criterion for notch high cycle fatigue using a critical-point method. Eng Fract Mech 2008;75:1864–74. [20] El Haddad MH, Topper TH, Smith KN. Prediction of non propagating cracks. Eng Fract Mech 1979;11(3):573–84. [21] Susmel L. A unifying approach to estimate the high-cycle fatigue strength of notched components subjected to both uniaxial and multiaxial cyclic loadings. Fatigue Fract Eng Mater Struct 2004;27(5):391–411. [22] Taylor D. Notch size effect in multiaxial fatigue. In: Susmel L, Tovo R, editors. Progettazione a fatica in presenza di multiassialit´ a tensionali. Ferrara; 2005. [23] Chaves V, Navarro A. Fatigue limits for notches of arbitrary profile. Int J Fatigue 2013;48:68–79. [24] Navarro A, Vallellano C, Chaves V, Madrigal C. A microstructural model for biaxial fatigue conditions. Int J Fatigue 2011;33:1048–54. [25] Timoshenko S, Goodier JN. Theory of elasticity. McGraw-Hill; 1934. [26] Vallellano C, Navarro A, Dominguez J. Fatigue crack growth threshold conditions at notches. Part I: Theory. Fatigue Fract Eng Mater Struct 2000;23:113–21. [27] Gao Y, Ricoeur A. The refined theory of plane problems for one-dimensional quasicrystalline bodies. J Appl Mech 2012;79:011004. [28] Endo M. Effects of small defects on the fatigue strengths of steel and ductile iron under combined axial/torsional loading. In: Ravichandran KS, Ritchie RO, Murakami Y, editors. Small fatigue cracks: mechanics mechanisms and applications. Elsevier; 1999. [29] Endo M. The multiaxial fatigue strength of specimens containing small defects. In: Carpinteri A, de Freitas M, Spagnoli A, editors. Biaxial/multiaxial fatigue and fracture ESIS publication. Elsevier; 2003. p. 243–64. [30] Murakami Y, Endo M. Quantitative evaluation of fatigue strength of metals containing various small defects or cracks. Eng Fract Mech 1983;17(1):1–15. [31] Navarro A, Vallellano C, de los Rios ER, Xin XJ. Notch sensitivity and size effects by a short crack propagation model. In: Engineering Against Fatigue, Proc Int Conf. Rotterdam: AA Balkema Publishers; 1997, Sheffield, 1999. [32] Beretta G. PhD Thesis. Fatiga en componentes con concentradores de tension bajo carga biaxial. Universidad de Sevilla (Espa˜ na; 2016. [33] Navarro A, Chaves V, Balbín JA. Variations on a critical distance theme. Int J Fatigue 2021;152:106453. [34] Morgan D, Quinlan S, Taylor D. Using the theory of critical distances to predict notch effects in fibre composites. Theor Appl Fract Mec 2022;118:103285. V. Chaves et al.