Fatigue Probabilistic Approach of Notch Sensitivity of 51CrV4 Leaf Spring Steel Based on the Theory of Critical Distances

Abstract

The mechanical and structural design of railway vehicles is heavily influenced by their lifetime. Because fatigue is a significant factor that impacts the longevity of railway components, it is imperative that the fatigue resistance properties of crucial components, like leaf springs, be thoroughly investigated. This research investigates the fatigue resistance of 51CrV4 steel under bending and axial tension, considering different stress ratios across low-cycle fatigue (LCF), high-cycle fatigue (HCF), and very-high-cycle fatigue (VHCF) regimes, using experimental data collected from this work and prior research. Data included fractographic analyses aiming to help in understanding some of failures for different loads. The presence of geometric discontinuities, such as notches, amplifies stress concentrations, requiring a probabilistic approach to fatigue assessment. To address notch effects, the theory of critical distances (TCD) was employed to evaluate fatigue strength. TCD model was integrated in fatigue statistical models, such as the Walker model (WSN) and the Castillo–Fernández-Cantelli model adapted for mean stress effects (ACFC). Extending the application of the TCD theory, this research provides an improved probabilistic fatigue model that integrates notch sensitivity, mean stress effects, and fatigue regimes, contributing to more reliable design approaches of railway leaf springs or other components produced in 51CrV4 steel.

1. Introduction

The design of railway vehicles must thoroughly account for their longevity. Essential components for their functionality and safety should be engineered to ensure sustained performance over the years, requiring only regular inspection and maintenance [1]. Such essential components designed to operate for a large number of cycles in the railway sector are, for example, bogie structures, leaf springs, and axles [2,3,4]. Regarding the leaf springs of railway-wagon axle suspensions, these components have been analysed in terms of strength, more specifically fatigue strength, over several years under different loading conditions [5,6,7,8,9,10,11,12,13,14]. Regardless of the chemical composition, spring steel, in general, exhibits a high fatigue strength [15,16,17,18,19,20,21,22,23]; however, in the presence of surface defects, this strength can be drastically affected. The presence of surface defects may come from poor production handling defects, producing scratches, among others, on the surface [18,24,25], from optimized component geometry [26,27,28,29,30], or even from production processes inherent to the leaf spring that lead to high surface roughness (see Figure 1) [31].

1.1. Notch-Effect Analysis in Fatigue Strength of Steels

Over the years, different approaches to characterising the fatigue strength of metals have been released [32]. The analysis methods can be characterised as stress-based models, strain-based models, or energy-based models [33]. Within the group of stress-based models, there are the average-stress methods, fracture mechanics methods, and stress field intensity methods [33]. The average-stress methods are based on the assumption that the parameters that control the fatigue failure are the averaged stresses. The averaged stress may be computed over a line, area, or volume. In other words, these models consider that fatigue failure occurs if a critical volume is subjected to a certain critical stress. This method was widely used to determine the dynamic stress concentration factor parameter, $K_f$, by Kuhn et al. [33,34], Neuber [35], Peterson [36,37], Heywood [38], Siebel [33,39], Buch [40], and Wang [33,41], who related the static stress concentration factor, $K_t$, to material constants and notch geometric characteristics. Contrary to the average-stress methods, methods based on stress field intensity assume that the fatigue failure is induced by the damage accumulation in the vicinity of the notch tip [42]. The damage depends on the peak stress and the stress field intensity in the damage region, whose dimension is of the order of several grains. In this method, the dynamic stress concentration, $K_f$, is determined based on the stress field intensity function which depends on the volume corresponding to the fatigue damage region, an equivalent stress function, a weight function, and the stress gradient [33,43,44]. In methods based on the fracture mechanics theory, $K_f$ is defined by several factors, including the intrinsic crack length, $a_o$ [43,45,46]. The intrinsic crack length is usually described by the functions well represented in the Kitagawa–Takahashi diagram [47]. The physical meaning of the intrinsic crack length is that for crack lengths smaller than $a_o$, the Linear Elastic Fracture Mechanics (LEFM) is no longer valid, since $a<a_o$ are considered small crack defects. According to this assumption, the intrinsic crack length is related to the fatigue limit stress range, $\mathsf{\Delta }{\sigma }_o$, and the threshold stress-intensity factor range, $\mathsf{\Delta }{K}_{th}$ by Equation (1) [48], such that

$$a_o={\displaystyle \frac{1}{\pi }}{\left({\displaystyle \frac{\mathsf{\Delta }{K}_{th}}{\mathsf{\Delta }{\sigma }_o}}\right)}^2.$$

(1)

Thus, Equation (1) states that for small cracks with $a<a_o$, their presence may be innocuous compared to long cracks $a>a_o$. Joining the averaged-stress methods, the theory of critical distances (TCD) method is also a stress-averaged method, where the characteristic material dimension, $D_m$, is based on fracture mechanics, as described previously. TCD has been applied as a method for fatigue analysis of notched components due to its practicality and simplicity, and this method has been widely used and probably improved [43,44,49,50]. TCD considers an average effective stress, ${\sigma }_{eff}$, obtained from the elastic stress distribution in front of the notch root, over $D_m$. This is the reason why the TCD method is also considered as a averaged-stress method [33].

As the applied stress level grows up, the local stresses in the vicinity of notches also increase until the local stresses reach the yield strength of the material. Under these conditions, Neuber [51] detected that the elastic stress concentration factor, $K_t$, is not longer representative of fatigue analysis, since plasticity consideration needs to be accounted for. At sites with high plasticity, Neuber’s theory [51] with the respective improvements [33,52,53,54] should be considered. In similarity to the stress-based models, a local stress–strain intensity model, was also developed on the local strain field intensity [33,55].

To overcome the difficulties exhibited by stress-based models and strain-based models, models based on the strain-energy density have been developed in order to facilitate the consideration of the stress concentration factor in the presence of plasticity at the notch root. These models verified that under the localised plasticity around the notch tip, the distribution of the strain-energy density in the plastic zone is approximately equal to the strain-energy density for a linear elastic material behaviour [56]. This method has the great advantage that the strain-energy density can be computed considering only the elastic stress distribution, even in the presence of localised plasticity around the notch root, and provides better results, the smaller the plastic region is in an elastic matrix [57,58]. The method suggested by Glinka, the equivalent strain-energy density method, ESED, has been improved in order to improve the prediction of fatigue strength from a relationship of the stress and deformation range levels, and $K_t$ with the plastic and elastic strain-energy per cycle, which becomes quite useful in the presence of an elasto-plastic behavior [33,59,60].

1.2. Outline of the Article

This paper presents the probabilistic modelling of the notch effect in spring steels for parabolic leaf springs when subjected to in-phase cyclic loading. Although leaf spring steels are very often subjected to severe external environmental conditions, for example, corrosion, this investigation follows a material characterisation campaign under controlled fatigue conditions to provide a better understanding of the 51CrV4 steel fatigue behaviour, since in the literature, no extensive data discusses the resistance of this spring steel grade.

Since in most engineering design projects the stress field is the parameter normally considered in components that require high fatigue performance, stress-based methods are considered. Furthermore, considering the applicability of the TCD method, it is considered for the assessment of the spring steel fatigue strength.

Initially, the fatigue-modeling process for notched components using the TCD approach is approached. Briefly, the first section describes the procedure and assumptions of the TCD approach, the methodologies that improve the application of TCD, such as the cyclic plastic zone, CPZ, and how TCD can be implemented along with fatigue-modeling methods, accounting for different conditions, such as smooth components, components subjected to different stress ratios and different types of loading.

In this paper, the theory of critical distances is described and applied in order to determine an effective stress that allows us to relate it to the size of the intrinsic defect, taking into account the results obtained in fracture mechanics tests in [61] with the fatigue resistance of the material for smooth specimens [62,63]. Since in geometric discontinuities, the stress and strain states can be multiaxial, the von Mises criterion is considered to correlate the fatigue dataset obtained in notched and smooth specimens. The fatigue models considered for the modeling are the Walker model (WSN) and the Castilo and Ferancdez-Cantelli model adapted for mean stresses (ACFC) [63].

Since the test specimens are tested under rotating–bending (fully-reversed load condition), axial tension (positive stress ratios), and 3-point in-plane bending (positive stress ratios) and are merged with literature fatigue data for the same material but in the absence of notches (from low-cycle fatigue, LCF, to very-high-cycle fatigue, VHCF), the equivalent effective strain, ${\epsilon }_{a,VM}$, is considered as a fatigue damage parameter for WSN and ACFC fatigue models. Additionally, due to the existence of some scatter in the fatigue dataset in relation to the modeling curve, fatigue dataset are further corrected by applying the cyclic plasticity zone concept, CPZ.

Moreover, a qualitative analysis of the fracture surfaces of the specimens and the type of fracture for the different fatigue test regions and loading conditions are analysed, highlighting the zones of appearance of multiple cracks, single cracks on the surface, and cracks initiated in the sub-surface region or in the interior through the non-metallic inclusions or voids.

The main result obtained in this article is the calibration of a fatigue prediction model, taking into account the effect of mean stress and the stress concentration effect due to geometric discontinuities, from the low-cycle fatigue region to the very-high-cycle fatigue region. The development of the probabilistic model $\epsilon -N_f$ can be considered in the future analysis of fatigue damage of leaf springs when subjected to real load spectra.

2. Fatigue Modelling of Notched Components via the Theory of the Critical Distances

2.1. Effective Stress/Strain Component

The theory of the critical distances, TCD, considers an effective averaged stress, ${\sigma }_{eff}$, determined from the elastic stress distribution in front of the notch root and over a characteristic material dimension $D_m$. Then, the TCD method is considered as an average-stress method [33], whose effective averaged stress relies on the calculation method considered. Currently, the TCD methodology offers four methods to estimate the averaged stress: point method (PM) (Equation (2a)), line method (LM) (Equation (2b)), area method (AM) ((Equation (2c)), and volume method (VM) ((Equation (2d)), written as

$${\sigma }_{eff}=\sigma \left(D_{PM}\right),$$

(2a)

$${\sigma }_{eff}={\displaystyle \frac{1}{D_{LM}}}{\int }_0^{D_{LM}}\sigma \left(r\right)\mathrm{d}r,$$

(2b)

$${\sigma }_{eff}={\displaystyle \frac{2}{\pi D_{AM}^2}}{\int }_{-\pi /2}^{\pi /2}{\int }_0^{D_{AM}}\sigma (r,\theta )\mathrm{d}r\mathrm{d}\theta ,$$

(2c)

and

$${\sigma }_{eff}={\displaystyle \frac{3}{2\pi D_{VM}^3}}{\int }_0^{\pi }{\int }_{-\pi /2}^{\pi /2}{\int }_0^{D_{VM}}\sigma (r,\theta ,\phi )r^{2}sin\theta \mathrm{d}r\mathrm{d}\theta \mathrm{d}\phi ,$$

(2d)

with r, $\theta$, and $\phi$ denoting the spherical coordinates. The PM, LM, and AM calculation methods are illustrated in Figure 2 [64]. Notice that the usage of both point and line methods provides very similar results. Additionally, results obtained from these two methods are sufficient to describe the effective stress at the notch root, and hence, the area and volume methods do not present a significant increase in the accuracy of the obtained results.

TCD states that fatigue failure occurs when the effective stress at the notch components exceeds the fatigue limit of smooth components. According this assumption, the critical distance, $D_m$, is calculated by the El Haddad’s formula (Equation (3)) such that the critical distance for each method is $D_{PM}=a_0/2$, $D_{LM}=2a_0$, $D_{AM}=1.32a_0$ and $D_{VM}=1.54a_0$ [43]. Notice that in these conditions, the critical distance is assumed to be given by the intrinsic crack length, $a_0$, which is determined by $\mathsf{\Delta }{K}_{th}$ and $\mathsf{\Delta }{\sigma }_0$ (Equation (1)), rewritten here using Equation (3), such that

$$a_o={\displaystyle \frac{1}{\pi }}{\left({\displaystyle \frac{\mathsf{\Delta }{K}_{th}}{\mathsf{\Delta }{\sigma }_o}}\right)}^2,$$

(3)

where the propagation threshold value, $\mathsf{\Delta }{K}_{th}$, is obtained considering Walker’s formulation (Equation (4)) [33,61], such that

$$\mathsf{\Delta }{K}_{th}=\mathsf{\Delta }{K}_{th(R=0.0)}{\left(1-R_{\sigma }\right)}^{1-\gamma },$$

(4)

where $\gamma$ and $\mathsf{\Delta }{K}_{th(R=0.0)}$ are material parameters fitted by experimental data [65]. Regarding the fatigue limit range, $\mathsf{\Delta }{\sigma }_o$ is determined for the specific stress ratio. For steels, the Goodman criterion [66] can be considered as

$$\mathsf{\Delta }{\sigma }_0\left(R_{\sigma }\right)=\mathsf{\Delta }{\sigma }_{0_{\left(R_{\sigma }=-1\right)}}\left(1-{\displaystyle \frac{{\sigma }_m}{{\sigma }_{uts}}}\right),$$

(5)

with $\mathsf{\Delta }{\sigma }_{0_{\left(R_{\sigma }=-1\right)}}$ denoting the fully-reversed fatigue limit range, $\mathsf{\Delta }{\sigma }_0=2{\sigma }_f$ [65].

2.2. Cyclic Plastic Zone, CPZ

Fatigue occurs from the continuous accumulation of plastic strains in the vicinity of the notch root due to the cyclic loadings. According to the investigations carried out to explain the development and dynamics of the plastic zone, it was verified that the plastic zone is dependent on loading conditions [67]. Figure 3a illustrates the differences among the plastic zones developed under cyclic and monotonic loading conditions. Notice that both a cyclic plastic zone (CPZ) and a monotonic plastic zone (MPZ) are developed during cyclic loading. Despite being generated under cyclic loading conditions, CPZ and MPZ present some differences, such as the size. According [68,69], for steels, MPZ is usually 1/4 wider than CPZ. Additionally, MPZ is formed during a maximum stress phase of the cyclic loading, which is associated with the primary plastic zone, PPZ, created in the first instance of plastic damage. This zone is usually denominated as a persistent plastic zone, since its size remains unchanged during the following cycles.

In contrast, CPZ is a dynamic zone located in the front of the notch root, where the elastic stress distribution exceeds the yield strength. In addition, CPZ varies its size in accordance with the plastic deformation direction. In other words, under compression state, the cyclic plastic zone contracts (backwards CPZ) and in tension, the cyclic plastic zone expands (forward CPZ) [70]. The cyclic plastic zone has been shown to have an important role in controlling the crack growth and the crack-propagation driving force [71].

The presence of a cyclic plastic zone at the notch root will extend the initial size due to the plastic deformation. The higher the level of plastic deformation at the notch root, the lesser valid the critical distance methods described with the form of Equations (2a)–(2d). Thus, a correction in the critical distance methods described in Equations (2a)–(2d) should be made considering the effect of the cyclic plastic zone. For components under plane-stress and plane-strain conditions, the plastic zone radius, $r_{CPZ}$, is determined as [70]

$$r_{CPZ}={\displaystyle \frac{1}{8\pi }}{\left({\displaystyle \frac{\mathsf{\Delta }K}{{\sigma }_y}}\right)}^2$$

(6)

and

$$r_{CPZ}={\displaystyle \frac{1}{24\pi }}{\left({\displaystyle \frac{\mathsf{\Delta }K}{{\sigma }_y}}\right)}^2,$$

(7)

respectively. In [70], the effect of the plastic zone radius was considered in the application of the point and line methods and compared with the classic TCD Equations (2a)–(2d) and the updated TCD applied for low- and medium-fatigue lives [72], showing a better agreement between the prediction model and experimental data. Following these results, it is suggested that the critical distance for the point and line methods be corrected to the following expressions

$${\sigma }_{eff}=\sigma \left(D_{PM}+{\displaystyle \frac{1}{12\pi }}{\left({\displaystyle \frac{\mathsf{\Delta }K}{{\sigma }_y}}\right)}^2\right)$$

(8a)

and

$${\sigma }_{eff}={\displaystyle \frac{1}{\left[D_{LM}+{\displaystyle \frac{1}{12\pi }}{\left({\displaystyle \frac{\mathsf{\Delta }K}{{\sigma }_y}}\right)}^2\right]}}{\int }_0^{\left[D_{LM}+{\displaystyle \frac{1}{12\pi }}{\left({\displaystyle \frac{\mathsf{\Delta }K}{{\sigma }_y}}\right)}^2\right]}\sigma \left(r\right)\mathrm{d}r,$$

(8b)

with ${\sigma }_y$ denoting the yield strength and $\mathsf{\Delta }K$ denoting the stress-intensity factor range.

However, if it is necessary to use computational methods to determine the CPZ, cyclic plasticity models are required. In steels, the combined Chaboche model is widely used to model the cyclic behaviour of these materials [73]. This model is based on the von Mises performance criterion (Equation (9)), such that

$$F=\sqrt{({\mathit{\sigma }}^{\prime }-{\mathit{\alpha }}^{\prime }):({\mathit{\sigma }}^{\prime }-{\mathit{\alpha }}^{\prime })}-({\sigma }_y+R)=0,$$

(9)

and on associative flow rule (Equation (10)), which follows the principle of normality, such that

$${\dot{\mathit{\epsilon }}}^p=\dot{\mathit{\lambda }}{\displaystyle \frac{\partial F}{\partial \sigma }}.$$

(10)

In Equations (9) and (10), ${\sigma }_y$ denotes the initial size of the elasticity limit surface, and R its size variation. ${\mathit{\alpha }}^{\prime }$ and ${\mathit{\sigma }}^{\prime }$ are the deviatoric part of the back-stress tensor, $\mathit{\alpha }$ and stress tensor, $\mathit{\sigma }$, respectively. ${\dot{\mathit{\epsilon }}}^p$ and $\dot{\lambda }$ represent, respectively, the rate of plastic strain tensor and the plastic multiplier (determined from the consistency condition).

The combined Chaboche model [73] is composed of the isotropic hardening component, which represents the change in the size of the elasticity limit surface, $\dot{R}$, and the kinematic hardening component, which represents the translation of the elasticity limit surface over the stress space. In isotropic hardening, $\dot{R}$, is given by

$$\dot{R}=b_{\infty }\left(R_{\infty }-R\right)\dot{p},$$

(11)

with the size in each cycle given as

$$R=R_{\infty }\left[1-exp\left(-b_{\infty }p\right)\right],$$

(12)

where $R_{\infty }$ and $b_{\infty }$ are material-dependent parameters, with $b_{\infty }$ indicating the rate saturation of the isotropic model.

On the other hand, in Chaboche’s kinematic hardening model, the translation of the elasticity limit surface is given by a sum of the i-th component of the back-stress tensor, ${\mathit{\alpha }}_i$, such that

$$\mathit{\alpha }=\sum _{i=1}^N{\mathit{\alpha }}_i,$$

(13)

with the variation of each individual back-stress tensor, ${\dot{\mathit{\alpha }}}_i$, expressed as

$${\dot{\mathit{\alpha }}}_i={\displaystyle \frac{2}{3}}{C}_i\dot{{\mathit{\epsilon }}^p}-{\gamma }_i{\mathit{\alpha }}_i\dot{p}.$$

(14)

According to research carried out, a minimum of three back-stress components are required for the kinematic model to obtain a good prediction of ratcheting behaviour [73,74].

2.3. Multiaxial Fatigue Modeling

In the presence of geometrical discontinuities, the component presents a multiaxial stress state at the notch, even under loading conditions producing a global uniaxial stress distribution. Additionally, the stress state is essentially multiaxial with a non-uniform stress distribution at the notch vicinity. Thereby, the application of fatigue criteria that take into account the multiaxial state around the notch, should be considered in order to obtain better fatigue-life predictions.

For several years, different multiaxial fatigue-life prediction models have been developed. They are usually classified as invariant tensor methods and critical plane methods. Critical plane methods are formulated in order to predict the fatigue life based on the crack-propagation orientation or on the failure plane, being categorised by their maximum shear or maximum tensile stress/strain planes. Some of these best-known models are the model proposed by Findley [75], based on the shear stress amplitude and the normal stress of the shear stress plane, the model proposed by Brown and Miller [76], based on the maximum shear–strain amplitude and normal stress to the maximum shear plane), the Fatemi and Socie model [77] considering the maximum shear–strain amplitude and the normal stress acting at the plane of the maximum shear–strain amplitude normalised by the monotonic yield strength, and the model proposed by Smith, Watson, and Topper, SWT [78], which considers as damage parameters the maximum principal strain range and maximum normal stress acting at principal strain amplitude plane. The SWT model was further modified by adding the shear–strain-energy component [79,80], which makes the model capable of predicting different cracking modes. More recently, critical plane models based on strain-energy methods have also been developed. These models consider specific planes experiencing the maximum amount of fatigue damage, where the generalized tensile fatigue damage, GSE, including both the normal strain and shear–strain-energy terms in a plane experiencing the maximum generalized tidal energy, or the maximum generalized strain amplitude, GSA, are highlighted [81].

In the case of invariant tensor methods, these are an extension of the equivalent stress and equivalent strain to fatigue performance analysis. For a multiaxial fully-reversed cyclic loading acting proportionally, the most applicable criteria are the maximum principal stress (Equation (15a)), the maximum shear stress (Equation (15b)), and the octahedral shear stress (Equation (15c)). An equivalent nominal stress, ${\sigma }_a^e$, may be obtained according to mentioned criteria:

$${\sigma }_a^e={\sigma }_{a1},$$

(15a)

$${\sigma }_a^e={\sigma }_{a1}-{\sigma }_{a3},$$

(15b)

and

$${\sigma }_a^e={\displaystyle \frac{1}{\sqrt{2}}}\sqrt{{\left({\sigma }_{a1}-{\sigma }_{a2}\right)}^2+{\left({\sigma }_{a2}-{\sigma }_{a3}\right)}^2+{\left({\sigma }_{a3}-{\sigma }_{a1}\right)}^2},$$

(15c)

where ${\sigma }_{a1}$, ${\sigma }_{a2}$, and ${\sigma }_{a3}$ are, respectively, the principal alternating nominal stresses, with ${\sigma }_{a1}>{\sigma }_{a2}>{\sigma }_{a3}$. Once ${\sigma }_a^e$ is calculated, the stress state is represented by a uniaxial equivalent stress state. Equation (15a) is more useful for brittle materials, while Equation (15b) is adequate for ductile ones. In the presence of non-null mean or residual stresses, its equivalent effect, ${\sigma }_m^e$, may be considered by Equation (16a) or (16b).

$${\sigma }_m^e={\displaystyle \frac{1}{\sqrt{2}}}\sqrt{{\left({\sigma }_{m1}-{\sigma }_{m2}\right)}^2+{\left({\sigma }_{m2}-{\sigma }_{m3}\right)}^2+{\left({\sigma }_{m3}-{\sigma }_{m1}\right)}^2}$$

(16a)

and

$${\sigma }_m^e={\sigma }_{m1}+{\sigma }_{m2}+{\sigma }_{m3}={\sigma }_{m,xx}+{\sigma }_{m,yy}+{\sigma }_{m,zz},$$

(16b)

where ${\sigma }_{m1}$, ${\sigma }_{m2}$, and ${\sigma }_{m3}$ are, respectively, the principal nominal mean stresses, and ${\sigma }_{m,xx}+{\sigma }_{m,yy}+{\sigma }_{m,zz}$ is a stress invariant. Equation (16b) is preferable because fatigue has been shown to be sensitive to hydrostatic stress, and using criteria based on Equation (16a) will always lead to non-negative equivalent stress. Although Equation (16b) is better, it also permits that effects of tensile mean stresses may be cancelled out by compressive stress acting at different directions, which is not in accordance with experiments [32]. To overcome this issue, the effects of mean stress should be determined using uniaxial fatigue criteria such as Goodman (for ductile materials), considering the respective equivalent stresses, ${\sigma }_m^e$ and ${\sigma }_a^e$. This relationship may be used further in Basquin’s model [82] to determine the number of cycles to failure.

The assumptions applied to stress-state approaches are also extensible to a strain-life approach, such that

$${\epsilon }_a^e={\epsilon }_{a1},$$

(17a)

$${\epsilon }_a^e={\displaystyle \frac{{\epsilon }_{a1}-{\epsilon }_{a3}}{1+\nu }},$$

(17b)

and

$${\epsilon }_a^e={\displaystyle \frac{1}{\sqrt{2\left(1+\nu \right)}}}\sqrt{{\left({\epsilon }_{a1}-{\epsilon }_{a2}\right)}^2+{\left({\epsilon }_{a2}-{\epsilon }_{a3}\right)}^2+{\left({\epsilon }_{a3}-{\epsilon }_{a1}\right)}^2},$$

(17c)

where ${\epsilon }_{a1}$, ${\epsilon }_{a2}$, and ${\epsilon }_{a3}$ are, respectively, the principal alternating strains, with ${\epsilon }_{a1}>{\epsilon }_{a2}>{\epsilon }_{a3}$. The effect of mean stress is determined based on equivalent mean stresses, ${\sigma }_m^e$, as aforementioned. After determining ${\epsilon }_a^e$, and ${\sigma }_m^e$, the fatigue life is obtained using the strain–life curve with a respective mean stress effect for the uniaxial state.

Another more complex multiaxial fatigue criterion was proposed by Sines [83,84]. The Sines criterion is a stress-based criterion that considers the octahedral shear stress (von Mises plasticity criterion, ${\sigma }_{vM}$) and the hydrostatic stress (maximum hydrostatic pressure) in the calculation of the equivalent stress amplitude. Also, Crossland [84,85] also included in his criterion the effect of the von Mises equivalent stress. However, Crossland’s criterion considers the effect of the mean hydrostatic pressure, instead of the maximum hydrostatic pressure. Although these approaches have demonstrated simplicity, the correlation with multiaxial fatigue has shown wide limitations for non-proportionality and some materials.

2.4. Equivalent Fatigue Models for Notch Analysis

The prediction of fatigue resistance is made from average curves that relate stress amplitude level, ${\sigma }_a$, or strain amplitude level, ${\epsilon }_a$, with the number of cycles to failure, $N_f$. The most useful technique for uniaxial fatigue life prediction is the log–log straight-line SN model proposed by Basquin [82,86]. According to Branco [33], for notched specimens, fatigue life can be also correlated by the Basquin model, considering the equivalent stress amplitude given by the von Mises equivalent stress, ${\sigma }_{a,vM}$, instead. Thus, the Basquin model for notched components is written as

$${\sigma }_{a,vM}=C_b{\left(N_f\right)}^{n_b},$$

(18)

with $C_b$ denoting the strength coefficient, and $n_b$, the strength exponent.

Equation (18) is only applicable for a certain stress ratio, $R_{\sigma }$, not contemplating the mean stress effect correction. To overcome this issue, Morrow [87] or SWT [78] criteria should be considered for fatigue-life prediction [33]. According to previous investigations, the Walker method (Equation (19)) provides better predictions for estimating the parameter $\gamma$ [88,89,90,91]. Additionally, WSN model is fits to the material’s sensitivity to mean stress, including the notch effect. WSN model [92] is used to assess the mean stress effect on fatigue life behaviour with an equivalent von Mises strain amplitude (Walker fatigue damage parameter), ${\epsilon }_{a,w,vM}$, such that

$${\epsilon }_{a,w,vM}={\epsilon }_{a,vM}{\left({\displaystyle \frac{2}{1-R_{\sigma }}}\right)}^{\gamma }=C_b{N}_f^{n_b},$$

(19)

where $\gamma$, $C_b$, and $n_b$ are determined by multilinear regression [93], considering the dataset of failure specimens. Equation (19) is also applicable in stress field, by considering in Equation (19), the equivalent von Mises stress amplitudes ${\sigma }_{a,w,vM}$ and ${\sigma }_{a,vM}$, instead of ${\epsilon }_{a,w,vM}$ and ${\epsilon }_{a,vM}$. Notice that, if $\gamma$ assumes the value 0.5, Equation (19) becomes the SWT fatigue model [78].

The determination of parameters, $C_b$, $n_b$, and $\gamma$ is performed according to the ordinary least-squares method, OLS, for a linearised model of the Equation (19), with the independent variables given by $log\left({\epsilon }_{a,vM,i}\right)$ and $log\left(2/\left(1-R_{\sigma ,i}\right)\right)$, and the dependent variable given by $log\left(N_{f,i}\right)$, with i denoting the respective data point [88], such that

$$log\left(N_f\right)={\beta }_0+{\beta }_{1}log\left({\epsilon }_{a,vM}\right)+{\beta }_{2}log\left({\displaystyle \frac{2}{1-R_{\sigma }}}\right).$$

(20)

Equation (20) is solved using multi-linear regression methods [93]. Notice that Equation (20) is in accordance with the standardised procedure in ASTM E732-9 standard [94], if the mean stress effect is not considered. According to the linear regression model, the points around the average curve are assumed to have a Gaussian probability distribution function with the failure probability, $P_f$, defined as

$$P_f={\displaystyle \frac{1}{2}}\left[1+\mathrm{erf}{\displaystyle \frac{\left(N_f^{*}-{\beta }_0+{\sum }_{i=1}^2{\beta }_i{\mu }_{{\epsilon }_{a,vM,i}^{*}}\right)}{\sqrt{2}{\sigma }_{N_f^{*}}}}\right],$$

(21)

where $\mathrm{erf}\left[·\right]$ denotes the error function, ${\mu }_{{\epsilon }_{a,vM,i}^{*}}$ denotes the average value of the independent variables, and ${\sigma }_{N_f^{*}}$ denotes the standard deviation of the dependent variable.

Apetre and Castillo–Fernández-Cantelli Fatigue Model

The Apetre and Castillo–Fernández-Cantelli, ACFC, fatigue prediction model can be considered for a more accurate representation of fatigue behaviour regimes for VHCF regimes. ACFC fatigue models consider the statistical distribution model given by Weibull [62,86,95,96,97,98,99,100,101,102,103], and are in agreement with several fatigue analyses [86,104,105,106,107]. Similar to the research brought in [63] for specimens with zero and non-zero stress, and the assumptions of applying the WSN models to notched specimens, the ACFC model [108,109] is then written as

$${\psi }_{\epsilon ,vM}={\epsilon }_{a,vM}{\left({\displaystyle \frac{2}{1-R_{\sigma }}}\right)}^{\left(1-{\gamma }_w\right)}=exp\left\{B+{\displaystyle \frac{\lambda +\delta {\left[-log\left(1-P_f\right)\right]}^{1/\beta }}{log{N}_f-C}}\right\},$$

(22)

where ${\psi }_{\epsilon ,vM}$ denotes the equivalent strain fatigue parameter and ${\gamma }_w$, the Walker parameter fitted from Dowling’s model. $\lambda$, $\beta$, and $\delta$ are the location, the shape, and the scale parameters of Weibull’s distribution, respectively, whereas B and C represent, respectively, the logarithm of the threshold value for life $N_f$ and the logarithm of the threshold of the fatigue life for a fatigue limit, ${\sigma }_f$. Equation (22) results in the probability failure function written as

$$P_f=1-exp\left[-{\left({\displaystyle \frac{\left(log{N}_f-B\right)\left(log{\psi }_{\epsilon ,vM}-C\right)-\lambda }{\delta }}\right)}^{\beta }\right],$$

(23)

where the Weibull random variable is

$$V=\left(log{N}_f-B\right)\left(log{\psi }_{\epsilon ,vM}-C\right).$$

(24)

Probabilistic model parameters, B, C, and Weibull parameters ($\beta$, $\lambda$, and $\delta$) are then determined via E-M algorithm methodology [86,110], after ${\gamma }_w$ is known.

3. Material and Procedures for the Fatigue Modeling for Notched Details

3.1. Material, Microstructure, and Chemical Properties

The steel that is being examined is 51CrV4, a chromium–vanadium alloy, which has an average carbon content of about 0.50%, as shown in

Table 1. Chemical composition in % weight of 51CrV4 steel grade.

Material	C	Fe	Si	Mn	Cr	V	S	Pb
DIN 51CrV4 (1.815)	0.47–0.55	96.45–97.38	≤0.40	0.70–1.10	0.90–1.20	≤0.10–0.25	≤0.025	≤0.025

. This steel grade is standardised to be quenched at 850 °C for 40 min in an oil bath and then tempered at 450 °C for 90 min, which gives it the tempered martensite microstructure as illustrated in Figure 4 [111].

Regarding mechanical strength properties under monotonic and cyclic loading,

Table 2. Elasto-plastic parameters that define monotonic and cyclic behaviour of the 51CrV4 steel grade [112,113].

E	$\mathit{\nu }$	${\mathit{\sigma }}_{\mathit{y}}$	${\mathit{R}}_{\mathit{\infty }}$	${\mathit{b}}_{\mathit{\infty }}$	${\mathit{\sigma }}_{\mathit{y}}^{\prime }$	${\mathit{C}}_1$	${\mathit{\gamma }}_1$	${\mathit{C}}_2$	${\mathit{\gamma }}_2$	${\mathit{C}}_3$	${\mathit{\gamma }}_3$	${\mathit{C}}_4$
[GPa]		[MPa]	[ MPa]		[ MPa]	[MPa]		[MPa]		[MPa]		[MPa]
202.5	0.29	1089.2	−232.2	0.3705	676.7	80,097.1	648.4	20,486.2	164.9	5650.1	40.64	2302.7

summarises the material properties of 51CrV4. The properties were obtained from previous investigations [112,113] in suitable tests, following ISO 6892-1 standard [114] and ASTM E606 [115]. The 51CrV4 steel exhibits a modulus of elasticity, E, of 200.54 MPa, a Poisson’s ratio, $\nu$ = 0.29, and a yield strength of ${\sigma }_y$ of 1271.48 MPa. Regarding the isotropic hardening behaviour, 51CrV4 exhibits an $R_{\infty }$ of −232.2 MPa and a $b_{\infty }$ of 0.3705, with a cyclic yield strength, ${\sigma }_y^{\prime }$ = 676.7 MPa. Regarding the kinematic hardening properties, 51CrV4 presents a 6-parameter nonlinear model ($C_1=\mathrm{20,486.2}$ MPa, ${\gamma }_1=648.4$, $C_2=\mathrm{20,486.2}$ MPa, ${\gamma }_2=164.9$, $C_3=5650.1$ MPa, and ${\gamma }_3=40.64$) plus one linear parameter ($C_4=2302.7$ MPa).

Regarding the fatigue strength for crack propagation and total fatigue-life properties, necessary for the development of the investigation, these properties were obtained in proper tests presented in [62] and in [61], respectively. Thus, under mode I crack-propagation conditions for $R_{\sigma }=0.0$, the propagation threshold value $\mathsf{\Delta }{K}_{th(R=0.0)}$ is 7.0578 MPa$\sqrt{\mathrm{m}}$, and $\gamma$ is equal to 0.5767. Regarding fatigue limits, ${\sigma }_f$, for $R_{\sigma }$ = −1, for bending loadings ${\sigma }_f$ = 660.80 MPa and for tension/compression loadings, ${\sigma }_f$ = 649.51 MPa.

3.2. Fatigue-Failure Dataset Collection for 51CrV4 Steel

In order to extend the model from the LCF regime to the VHCF regime under fully reversed loading and non-zero mean stress, the fatigue strength data of 51CrV4 steel were gathered from the literature at [112] for the LCF regime and [62,63] for the HCF and VHCF regimes.

For shorter lifetimes, within the LCF regime [112], 7 fatigue tests were conducted under constant controlled uniaxial amplitude loading conditions on an Instron 8801 servo-hydraulic machine with a 100 kN load cell and an Instron 2620-202 dynamic clip gauge at a strain ratio of $R_{\epsilon }=0.0$, following the ASTM E606 standard [115].

For medium lifetimes, 13 uniaxial tension/compression and 43 rotating bending tests were conducted for $R_{\sigma }=-1$ (in the HCF fatigue regime (approximately ${10}^3$ to ${10}^7$ cycles [112])), according to the fatigue standard [116]. Uniaxial tension/compression fatigue tests were performed on a Rumul Testronic HCF machine operating under load control conditions with a 150 kN load cell, where the rotating–bending specimens were tested on a home-made machine using the single-point loading configurations outlined in the ISO 1413 standard [117], with a test frequency of 25 Hz.

To evaluate the non-null mean stress effect in the HCF fatigue regime, 36 and 39 fatigue tests under 3-point in-plane bending conditions and tension conditions were made, respectively. In-plane bending fatigue tests, following the ISO 7438 standard [118], were performed under displacement-controlled conditions on a home-made electromechanical machine (up to $5\times {10}^6$ cycles), with two load cells (maximum load capacity of 30 kN), operating at 23 Hz, supported by a data acquisition system, SPIDER 8, and acquisition software, Catman (v4.5). The length between supports was maintained at 150 mm. On the other hand, uniaxial tension fatigue tests were carried out under force-controlled constant amplitude conditions at various stress ratios ($R_{\sigma }$ of 0.1, 0.3, and 0.5), following ASTM 4662-1 [116], on two MTS servo-hydraulic testing machines (up to $3\times {10}^6$ cycles) with 100 kN and 250 kN load cells, operating at a maximum frequency of 12 Hz.

The extension of the fatigue curve to longer service life (in the VHCF region, ≈${10}^9$ cycles) [112], the uniaxial tension/compression fatigue tests ($R_{\sigma }=-1$) were performed using the USF-2000 fatigue machine from Shimadzu manufacturer that operates at a resonance frequency of 20 kHz under displacement control. The experimental campaign resulted in 26 fatigue tests.

3.3. Specimen Geometry and Testing Setup for Fatigue Experiments

The notched specimens were analysed in the HCF fatigue regime under rotating–bending [62], 3-point in-plane bending [63], and axial tension [63] loading conditions. The test machines and setups used for fatigue testing and test standards were the same as those described in Section 3.2.

The geometry of notched specimens tested under fatigue rotating–bending loading are illustrated in Figure 5, which shows the detail of the notch manufactured in the reduced zone, and the respective description of the dimensions for the definition of the specimen geometry. The value of the notch radius, $r_{nt}$, and the inside diameter, $d_0$, are measured indirectly from the measurement of the dimension $z_{nt}$, $w_{nt}$, and $D_0$. The detail A in Figure 5 shows the notch surface of one of the test specimens. The notches on the test specimens were sanded with 800-grit sandpaper to eliminate possible irregularities arising from the manufacturing processes.

In order to analyse the effect of different notch radii, two batches of 22 test specimens were produced, for $r_{nt}$ of 0.81 and 1.074 mm, respectively. The notch radii, $r_{nt}$, were obtained by measuring the parameters of notch depth, $z_{nt}$, notch width, $w_{nt}$, notch inner diameter, $d_0$, and notch outer diameter, $D_0$.

Table 3. Average dimensions of the notched specimens used in rotating–bending fatigue tests.

${\mathit{r}}_{\mathit{nt}}$	${\mathit{z}}_{\mathit{nt}}$	${\mathit{w}}_{\mathit{nt}}$	${\mathit{d}}_0$	${\mathit{D}}_0$	${\mathit{z}}_0$	L	${\mathit{L}}_0$	${\mathit{L}}_{\mathit{f}}$	D	${\mathit{D}}_{\mathit{b}}$	${\mathit{L}}_{\mathit{b}}$	M
[mm]	[mm]	[mm]	[mm]	[mm]	[mm]	[mm]	[mm]	[mm]	[mm]	[mm]	[mm]	[mm]
0.81	0.319	1.283	5.149	5.782	20.8	100.01	58.58	36.35	12	10	8	M5
± 0.04	± 0.143	± 0.060	± 0.096	± 0.093	± 0.77	± 0.24	± 0.49	± 0.02
1.074	0.990	2.136	5.268	7.248	20.61	99.34	58.15	36.34	12	10	8	M5
± 0.058	± 0.051	± 0.106	± 0.234	± 0.190	± 0.70	± 0.71	± 0.64	± 0.01

presents the average dimensions for the notched specimens used in rotating–bending fatigue tests.

Regarding the geometry of notched specimens tested under fatigue 3-point in-plane bending loading, it is illustrated in Figure 6. Figure 6 also illustrates the geometry of the analysed notch types, Dt A1 and Dt A2. On the right side of the Figure 6 the respective nomenclature for defining the geometry of the specimen is shown. The notch-effect analysis was performed using two batches, one with 14 and another with 15 test specimens for $r_{nt}$ of 0.763 and 1.228 mm, respectively notch geometry (Dt. A2 and Dt. A1), respectively. The notch radius was also determined indirectly by measuring the parameters $z_{nt}$ and $w_{nt}$.

Table 4. Average dimensions of notched specimens used in the 3-point in-plane bending fatigue tests.

${\mathit{r}}_{\mathit{nt}}$ [mm]	${\mathit{z}}_{\mathit{nt}}$ [mm]	${\mathit{w}}_{\mathit{nt}}$ [mm]	${\mathit{h}}_0$ [mm]	${\mathit{W}}_0$ [mm]	${\mathit{H}}_0$ [mm]	${\mathit{L}}_0$ [mm]	L [mm]	R [mm]
0.763	0.583	0.909	6.062	19.95	6.55	150	220	4
± 0.128	± 0.261	± 0.025	± 0.090	± 0.00
1.228	1.552	3.00	4.06	20.67	6.55	150	220	4
± 0.023	± 0.042	± 0.03	± 0.17	± 0.94

presents the summary of the average dimensions of notched specimens used in the 3-point in-plane bending fatigue tests, where $h_0$ denotes the height of the cross-section in the notched zone, $W_0$, denotes the width of the unnotched cross-section and $H_0$, the height of the unnotched cross-section. The tests were carried out for stress ratios of 0.0, 0.1, and 0.3.

For notch specimens tested under uniaxial tensile-loading conditions, Figure 7 presents the geometry of the notched fatigue specimen for axial tensile alternating loading with the detail of the notch produced on the specimen (Dt.1) and the rendered image of the CAD model showing the dimensions of the specimen geometry. The notches on the test specimens were sanded with 800-grit sandpaper to eliminate possible irregularities arising from the manufacturing processes.

In similarity with fatigue tests performed in notched specimens under rotating and in-plane bending loadings, the notch-effect analysis was carried out using a batch of 20 specimens. The notch radius, $r_{nt}$ and inner diameter, $d_0$, were also measured indirectly by measuring the parameters $z_{nt}$, $w_{nt}$, and $D_0$. Fatigue tests were performed for stress ratio, $R_{\sigma }$, of 0.1 and 0.3.

Table 5. Average dimensions of notched specimens used in tensile fatigue tests, taking as reference the ASTM E466-21 standard [116].

${\mathit{r}}_{\mathit{nt}}$	${\mathit{z}}_{\mathit{nt}}$	${\mathit{w}}_{\mathit{nt}}$	${\mathit{d}}_0$	${\mathit{A}}_0$	${\mathit{D}}_0$	${\mathit{L}}_{\mathit{d}}$	R	Thread	L	${\mathit{L}}_{\mathit{M}}$
[mm]	[mm]	[mm]	[mm]	[mm2]	[mm]	[mm]	[mm]	[mm]	[mm]	[mm]
1.119	0.318	1.558	4.197	13.930	4.833	10.28	85	M12	92	20
± 0.080	± 0.022	± 0.039	± 0.070	± 0.590	± 0.066

presents the summary of the average dimensions of the notched specimens used in axial tensile fatigue tests.

Scanning electron microscopes, JEOL JSM 6301F (JEOL Ltd., Akishima, Tokyo, Japan)/Ozford INCA Energy 350 (Oxford Instruments plc, Abingdon, UK) and FEI Quanta 400 FEG ESEM (Thermo Fisher Scientific (FEI), Waltham, MA, USA)/EDAX Genesis X4M (AMETEK (EDAX brand) Mahwah, NJ, USA) were used in fracture surface analysis of the fractured specimens.

3.4. Computational Approach for Notch-Effect Analysis

Numerical analyses based on the finite-element method, using ANSYS 2023 R2 software were performed to determine the stress profile generated in the notched specimens with the geometries shown in the Figure 5, Figure 6 and Figure 7. The linear equations system considered for the analysis is

$${\mathbf{K}}_t^{\left(n\right)\left(i\right)}\mathsf{\Delta }{\mathbf{u}}^{\left(n\right)\left(i\right)}={\mathbf{F}}_{ext}^{\left(n\right)\left(i\right)}-{\mathbf{F}}_{int}^{\left(n\right)\left(i\right)},$$

(25)

where ${\mathbf{K}}_t$ is the tangent stiffness matrix, $\mathsf{\Delta }\mathbf{u}$ is the applied increment of displacement, ${\mathbf{F}}_{ext}$ the nodal force vector, and ${\mathbf{F}}_{int}$ denotes the restoring force vector. i and n denote the number of the iteration and the increment number of the time step, respectively. Brick iso-parametric elements formulated in the displacement field with quadratic shape functions and a uniform reduced integration scheme are considered. The dimension of the finite elements was defined by mesh convergence analysis, considering as a selection criterion, the relative difference between the stresses measured for the different mesh sizes, of 0.1 %. Equation (25) is solved following the Newton–Raphson scheme procedure using the LDLT factorisation in the Sparse direct method, assuming the $L2$-Norm as the convergence criterion with tolerances of 0.001.

Two different cases are considered in the analysis: (i) a total linear elastic material model, and (ii) a multi-material contemplating a global linear elastic material behaviour and a cyclic elasto-plastic material close to the notch. In case (i), the elastic behaviour is considered to assess the stress distribution profile in front of the notch root; however, since localised plastic deformation can occur around the notch, elasto-plastic behaviour is considered in order to analyse the cyclic plastic zone, CPZ, and monotonic plastic zone, MPZ.

Therefore, the constitutive matrix $\mathbf{C}$ is assumed to be both a constitutive linear elastic material matrix and a constitutive elasto-plastic consistent tangent material matrix for Chaboche’s combined hardening model with elastic constants of Young’s modulus, E = 202.5 GPa, and the Poisson’s ratio, $\nu$ = 0.29, and the parameters corresponding to isotropic and kinematic hardening model as presented in Table 2. Still concerning case (ii), in the Equation (25), the discretisation of the constitutive equation is made following an Euler Backward implicit strategy, using the radial return algorithm for a small deformation problem [112,119,120,121,122].

Due to the geometric and physical geometry of the experimental model, boundary conditions are proposed for 1/4 of the model of the original geometry.

Due to the geometric and physical geometry of the rotating–bending problem, the kinematic and boundary conditions of the numerical model are represented by a 1/4 of the model. The YZ symmetric plane and the XZ antisymmetric plane are considered as shown in Figure 8.

Notice the model presents a notched specimen under conditions of a static cantilever beam, where only the exposed length is considered in the simulation. At one end, displacement boundary conditions are applied in the z and y directions a null displacement for specimen fixation. On the opposite side, the imposed load is applied over the area corresponding to the length $L_b$ (see Figure 5). Figure 8 also illustrates the type of mesh considered for both the two notch details, NR0.3RB and NR1.0RB. In general, in the area close to the notch, elements were modelled with approximately a regular size of 0.1 mm, while elements in the vicinity tend to be larger by imposing an increase in their aspect ratio as they move away from the notch zone.

Contrarily, in the case of notched specimens under axial loading conditions, due to geometric and loading conditions, 1/8 of the model was considered. Thus, all planes of symmetry YZ, XY, and XZ were considered as shown in Figure 9. The thread zone has been omitted in the numerical model, having been replaced by a uniformly distributed load at the end. Figure 9 also shows the mesh considered in the notch analysis zone. The same approach considered in Figure 8 was considered, such that the elements in the notch zone have an approximately regular dimension of 0.1 mm.

On the other hand, in the double-supported rectangular beam model, only 1/4 of the model was considered, having applied the YZ and XY planes of symmetry as represented in Figure 10. In the simulation of the bending of the specimen, the contact forces were replaced by a load distributed along the nodes for a contact width verified after each test of approximately 3 mm. A vertical load distributed along the face is applied to the end of the specimen. Note that only the length of the specimen corresponding to $L_s$ = 150 mm was considered, and the remaining length was not considered in the analysis. Figure 10 also shows the notch details for the nominal radii of 1.5 and 0.5 mm. In both models, a structured mesh was assumed with a minimum element dimension of 0.075 mm around the notch radius.

4. Results and Discussion

4.1. Experimental Data Pool

The data obtained by fatigue tests on notched specimens are merged with the smooth data pool obtained from [62,63,112], for rotating bending, in-plane bending (Figure 11), and uniaxial tension/compression (Figure 12). The stress amplitude levels considered for notched specimens take into account nominal stress lower than the elastic limit of the material. Collected data is identified by the initials ”NR” for notched specimens, in both Figure 11 and Figure 12.

4.2. Stress Distribution in Front of Notch Root

Following the assumption already referenced in [33], the fatigue life of notched specimens defined by the WSN model (Equation (19)) and ACFC model (Equation (22)), is determined using the equivalent strain amplitude, ${\epsilon }_{a,w,vM}$, determined for the critical distance (line method), $D_{LM}$, according to Equation (2b). In the first analysis performed (rotating–bending loading condition), only $R_{\sigma }$ = −1 is considered, then the Basquin model (Equation (18)) is used.

The von Mises stress distribution, ${\sigma }_{vM}$, in front of the notch root is determined using the computational method already described and converted to the von Mises strain distribution, ${\epsilon }_{vM}$. When the equivalent strains, ${\epsilon }_{vM}$, exceed the yield strain of the material, the cyclic plastic zone, CPZ, approaches are used to correct the effective ${\epsilon }_{a,w,vM}$ to improve fatigue-life prediction models. Recall that the effective von Mises equivalent strains using the line method are calculated using the crack-propagation properties and Goodman’s criterion mentioned in Section 3.1.

Figure 13 shows the stress profile given by the linear elasticity model of the material, normalised by the radius of the nominal section of the specimen. A rational 3-order polynomial function is considered to approximate the points collected from the stress profile. Notice that in accordance with the information provided from the tensile stress profile in Figure 13c), at least three specimens have the entire cross-section area with a stress above the yield strength. These high peak level stresses above the yield strength are more often observed in the specimens under in-plane bending loading with a notch depth of 0.3 mm, as shown in Figure 13b).

4.3. WSN Fatigue Model Based on TCD Approach

4.3.1. Notch Effect Under Rotating–Bending Loading

Regarding the notched specimens subjected to rotating–bending, Figure 14 shows the multiaxial stress state of the simulated specimen under the conditions described in Figure 8. It is observed that there is a biaxial stress state on the surface of the notch and a triaxial stress state in front of the notch root.

Combining only the results obtained from simulations of notched specimens under rotating–bending using the critical distance line method (Equation (2b)), with a critical distance $D_{LM}$ = 0.0402 mm (Equation (3) with $D_{LM}=2a_0$), with the data from smooth specimens under rotating–bending conditions ($R_{\sigma }=-1$), via Equation (18), results in the regression model presented in Figure 15.

In general, the application of the $J_2$-criterion and the critical distance, $D_{LM}$ = 0.0404 mm (Equation (3) with $D_{LM}=2a_0$), considering the equivalent von Mises strain, resulted in an overlapping of the fatigue data of notch and smooth specimens. Close to the fatigue limit, the superposition of the characteristic data scatter in this region is verified. The statistical analysis of the $\epsilon -N_f$ curve results in a coefficient of determination, $R^2$ = 0.5002, a coefficient $C_b$ of 0.0230 and a $n_b$ of −0.1776. Table 6—column 1 presents the statistical data obtained for the linear regression of the model shown in Figure 15.

4.3.2. Mean Stress and Notch Effects in Rotating and In-Plane Bending Loads

Following the analysis of the effect of mean stress and notch effect on bending loading, the in-plane bending failure data are analysed together with the rotating–bending data. Figure 16 shows that the von Mises stress distribution is almost constant from the comparison between the two specimen geometries, with notch detail NR0.4IPB and NR1.5IPB specimens. Although the distribution is quite similar, the highest level of the von Mises equivalent stress occurs in the centre of the specimen, and therefore the stress analysis will be carried out in the central zone of the specimen.

Considering WSN model adapted with equivalent strain amplitude, ${\epsilon }_{a,w,vM}$, (Equation (19)), with ${\epsilon }_{a,vM}$ determined with critical distances of $D_{LM}$ = 0.0404 mm for both notched specimens NR0.3RB and NR1.0RB, $D_{LM}$ = 0.0382 ± 0.0072 mm (Equation (3) with $D_{LM}=2a_0$), whose maximum value was 0.0504 mm and the minimum value was 0.0295 mm, for notched specimens NR0.5IPB, and $D_{LM}$ = 0.0273 ± 0.0074 mm, whose maximum value was 0.0408 mm and the minimum value was 0.0201 mm, for notched specimens NR1.5IPB, a $C_b$ = 0.0292, a $n_b=-0.1987$, and a $\gamma =0.1201$ are obtained, with a coefficent of determination of $R^2$ = 0.5012. Despite the considerable increase in the number of points, $R^2$ is approximately equal to the one obtained for the regression presented in Figure 15. Furthermore, the value of $\gamma =0.1201$ highlights that the parameter $\gamma$ will also be dependent on the type of loading applied, since it did not vary with the introduction of failure data for notched specimens in the smooth specimen data pool (see Figure 17).

Table 6. A summary of the regression models considering the notch effect and mean stress in specimens subjected to rotating–bending (SRB + NR0.3RB + NR1.0RB), in-plane bending (SIPB + NR0.5IPB + NR1.5IPB), and axial tension/compression (S-SAT + U-SAT + S-NR0.3AT) conditions in the fatigue strength curve considering the equivalent strain computed by the Walker parameter and of the critical distance based on the line method. (Note: RB = SRB + NR0.3RB + NR1.0RB; IPB = SIPB + NR0.5IPB + NR1.5IPB; AT = S-SAT + U-SAT + S-NR0.3AT)).

	Bending (RB)	Bending (RB + IPB)	Tension/Comp. (AT)
Strength Coefficient, $C_b$	0.0230	0.0293	0.0160
Strength Exponent, $n_b$	−0.1776	−0.1987	−0.1185
Coefficient of Determination, $R^2$	0.5002	0.2529	0.5314
Walker Parameter, $\gamma$		0.1201	0.8522
Mean Square Error, $MSE$	0.1742	0.5012	0.8676
Coefficient of Interception, ${\beta }_0$	−9.226	−7.727	−15.16
Coefficient of Slope 1, ${\beta }_1$	−5.630	−5.034	−8.439
Coefficient of Slope 2, ${\beta }_2$		−0.6044	−7.322
Average Ind. Var1, $log{\sigma }_a$,	−2.485	−2.508	−2.598
Std.Ind. Var1, $log{\sigma }_a$,	0.0736	0.1035	0.1210
Average Ind. Var2, $log2/\left(1-R_{\sigma }\right)$,		0.1281	0.211
Std.Ind. Var2, $log2/\left(1-R_{\sigma }\right)$,		0.1723	0.1947
Average Dep. Var, $log{N}_f$,	4.770	4.821	5.220
Std. Dep. Var, $log{N}_f$,	0.4145	0.5315	1.753

Table 6. A summary of the regression models considering the notch effect and mean stress in specimens subjected to rotating–bending (SRB + NR0.3RB + NR1.0RB), in-plane bending (SIPB + NR0.5IPB + NR1.5IPB), and axial tension/compression (S-SAT + U-SAT + S-NR0.3AT) conditions in the fatigue strength curve considering the equivalent strain computed by the Walker parameter and of the critical distance based on the line method. (Note: RB = SRB + NR0.3RB + NR1.0RB; IPB = SIPB + NR0.5IPB + NR1.5IPB; AT = S-SAT + U-SAT + S-NR0.3AT)).

	Bending (RB)	Bending (RB + IPB)	Tension/Comp. (AT)
Strength Coefficient, $C_b$	0.0230	0.0293	0.0160
Strength Exponent, $n_b$	−0.1776	−0.1987	−0.1185
Coefficient of Determination, $R^2$	0.5002	0.2529	0.5314
Walker Parameter, $\gamma$		0.1201	0.8522
Mean Square Error, $MSE$	0.1742	0.5012	0.8676
Coefficient of Interception, ${\beta }_0$	−9.226	−7.727	−15.16
Coefficient of Slope 1, ${\beta }_1$	−5.630	−5.034	−8.439
Coefficient of Slope 2, ${\beta }_2$		−0.6044	−7.322
Average Ind. Var1, $log{\sigma }_a$,	−2.485	−2.508	−2.598
Std.Ind. Var1, $log{\sigma }_a$,	0.0736	0.1035	0.1210
Average Ind. Var2, $log2/\left(1-R_{\sigma }\right)$,		0.1281	0.211
Std.Ind. Var2, $log2/\left(1-R_{\sigma }\right)$,		0.1723	0.1947
Average Dep. Var, $log{N}_f$,	4.770	4.821	5.220
Std. Dep. Var, $log{N}_f$,	0.4145	0.5315	1.753

This statement is justified by the investigations carried out in [63], where the WSN model obtained for smooth specimens tested for rotating–bending and 3-point in-plane bending resulted in a $\gamma$ of approximately 0.12. Comparing the notched specimens tested under conditions of rotating–bending and in-plane bending, it is verified that failures occur for values of equivalent strain greater than the average fatigue curve for notches of 0.4 mm and for lower values in the case of notches of 1.5 mm. This scatter may be due to the fact that the model only considers elastic material behaviour. Table 6—column 2 presents the statistical data obtained for the multilinear regression of the model shown in Figure 17.

4.3.3. Mean Stress and Notch Effects in Tension/Compression Loadings

Likewise, the effect of mean stress on tension/compression load specimens was evaluated. Figure 18 shows the representation of a von Mises stress distribution for all specimen geometries tested, over 1/8 of the analysed model for an arbitrary uniaxial load. The fatigue model considered for stress/strain data obtained from Figure 18 was based on the equivalent mean strain parameter, ${\epsilon }_{a,w,vM}$, considering Walker’s model and the critical distance-based line method.

Using Equation (19) with a critical distance of $D_{LM}$ = 0.0490 ± 0.0353 mm (with maximum value of 0.1428 mm and minimum value of 0.0231 mm) for notched specimens S-NR0.3AT, the effect of the type of loading on the value of $\gamma$ is observed again with a coefficient of determination, $R^2=$ 0.5314, $C_b$ = 0.0156 and $n_b$ = −0.1185. Under these conditions, $\gamma$ results in 0.8676, which is very close to the value obtained for tests on smooth specimens [63]. In general, the failure data relates to smooth specimens are consistent with the average curve ${\epsilon }_{a,w,vM}-N_f$, except for failure data of specimens tested in the LCF regime, as visualised in Figure 19. In fact, the strain data considered for the determination of the parameter ${\epsilon }_{a,w,vM}$ are assumed to be elastic, which translate to a lower value of ${\epsilon }_{a,w,vM}$ for shorter lifetimes.

In addition to these specimens, the notch specimens present a trend different from that obtained by the curve ${\epsilon }_{a,w,vM}-N_f$. This difference is even more noticeable for strains, ${\epsilon }_{a,w,vM}>0.6$%. For this equivalent strain level and consulting data in Figure 12, it is verified that for a load ratio of 0.3 and a stress amplitude of approximately 500 MPa, the value of ${\sigma }_{max}$ is greater than the yield strength of the material. Thus, a correction of the data according to the cyclic plastic zone should be considered. Due to this large scatter, the failure curves with lower and higher probabilities are far apart. Table 6—column 3 presents the statistical data obtained from the multilinear regression for the construction of the failure probability curves shown in Figure 19.

4.3.4. Mean Stress and Notch Effects in Combined Loading: Bending and Tension/Compression

Similarly to the analyses carried out in previous investigations [62,63], the loading effect with the respective effects of mean stress and the notch effect were also analysed together. Figure 20 presents the dataset obtained after computing the regression parameters via multilinear regression.

It is worth mentioning that the value of Walker’s parameter was 0.5845, which is consistent with the average value obtained in the investigation [63], with the combined effect between uniaxial tension and bending. Regarding the multilinear regression parameters, these can be consulted in Table 7—column 1. According to the multilinear model, $C_b$ = 0.0330 and $n_b$ = −0.1944 with a $R^2$ = 0.2951. The value of the coefficient of determination obtained is representative of the verified data scatter concerning the giga-cycle fatigue and low-cycle fatigue, and specimens tested under in-plane bending conditions.

The damage parameter, considering only the von Mises elastic strain, did not show good agreement with the prediction made for the data associated with the low-cycle fatigue regime. Thus, the total strain is taken into consideration when predicting the fatigue model instead. Figure 21 presents a multiaxial regression model considering the dataset of notched and smooth specimens under tension and bending loadings (rotating and in-plane), applying the critical distance approach based on the line method and considering the equivalent total strain.

From the value of the coefficient of determination, $R^2$ = 0.3464, an improvement in the distribution of data in relation to the prediction model is observed. The increase in $R^2$ is essentially due to the consideration of total strain rather than elastic strain in the low-cycle regime fatigue data. Furthermore, it is easily visible in Figure 21A that there is a good trend shown by the failure dataset with fatigue model from very short lives to the transition zone (from low cycle to high cycle, ≈${10}^4$ cycles). In the high-cycle fatigue regime, it is shown that the failure probability region from 25 to 75 % contains most of the failure data. The fatigue model in Figure 21 is modelled considering the data in Table 7—column 2, with $C_b$ of 0.0413 and $n_b$ = −0.2105. Regarding the $\gamma$ parameter, fitted from the least-squares minimization process, a value of 0.6497 was determined, which is slightly higher than that determined from failure dataset of rotating–bending and in-plane bending, as illustrated in Figure 20 and WSN combined fatigue model, of $\gamma =0.58$, as suggested in [63]. This increase indicates that the $\gamma$ parameter tends to be affected by the presence of plastic strain.

4.4. Monotonic and Cyclic Plastic Zones: MPZ and CPZ Radius

Despite the improvement observed in the prediction model by considering equivalent total strain instead of the equivalent elastic strain (as illustrated in Figure 21), the failure data with respect to the in-plane bending tests was shown to be within the trend of the average fatigue curve, but with an equivalent strain level that was too high. Since the value of the parameter ${\epsilon }_{a,w,vM}$ is higher than the yield strength of the material, then there is the occurrence of localised plasticity in the front of the notch root. As previously described, in the presence of plasticity in front of the notch root, the critical distance, $D_{LM}$ must be corrected with the radius of the cyclic plastic zone, CPZ. Following these statements, the elastic stress profile presented in Figure 13 is used to evaluate the occurrence of local plasticity in each of the load conditions analysed. As an explanatory example, Figure 22 illustrates an analysis of the cyclic behaviour of the material in front of the notch root for four locations more or less equidistant. The combined Chaboche model with parameters in Table 2 was considered, such that the occurrence of hysteresis loops is visible up to a distance of 0.013 mm. After that, the stabilisation of cyclic behaviour is verified such that the level of relaxation of the mean stress will depend only on the applied load. In this case, the diameter of the cyclic plastic zone would be 0.013 mm.

Table 7. Summary of the regression models considering the notch effect and mean stress in specimens subjected to rotating–bending (RB), in-plane bending (IPB), and axial tension (AT) conditions in the fatigue strength curve considering the equivalent strain computed by the Walker parameter and the critical distance based on the line method with and without data correction by using the cyclic plastic zone (CPZ) approach. (Note: RB = SRB + NR0.3RB + NR1.0RB; IPB = SIPB + NR0.5IPB + NR1.5IPB; AT = S-SAT + U-SAT + S-NR0.3AT).

	RB & IPB & AT		RB & IPB	RB & IPB & AT
	Elastic	Total	Total Strain	Total Strain
	Strain	Strain	& CPZ	& CPZ
Strength Coefficient, $C_b$	0.0330	0.0413	0.0235	0.0340
Strength Exponent, $n_b$	−0.1944	−0.2105	−0.1789	−0.1943
Coeff. of Determination, $R^2$	0.2951	0.3644	0.5097	0.3534
Walker Parameter, $\gamma$	0.5845	0.6497	0.1716	0.6508
Mean Square Error, $MSE$	0.7602	0.7048	0.2486	0.6973
Coeff. of Interception, ${\beta }_0$	−7.618	−6.575	−9.107	−7.558
Coeff. of Slope 1, ${\beta }_1$	−5.144	−4.751	−5.590	−5.1455
Coeff. of Slope 2, ${\beta }_2$	−3.007	−3.087	−0.9594	−3.349
Avg. Ind. Var1, $log{\sigma }_a$,	−2.564	−2.545	−2.514	−2.549
Std.Ind. Var1, $log{\sigma }_a$,	0.1196	0.1259	0.0978	0.1221
Avg. Ind. Var2, $log2/\left(1-R_{\sigma }\right)$,	0.1630	0.1704	0.1281	0.1704
Std.Ind. Var2, $log2/\left(1-R_{\sigma }\right)$,	0.1861	0.1865	0.1723	0.1865
Avg. Dep. Var, $log{N}_f$,	4.989	4.989	4.821	4.989
Std. Dep. Var, $log{N}_f$,	0.8314	0.8303	0.5712	0.8860

Table 7. Summary of the regression models considering the notch effect and mean stress in specimens subjected to rotating–bending (RB), in-plane bending (IPB), and axial tension (AT) conditions in the fatigue strength curve considering the equivalent strain computed by the Walker parameter and the critical distance based on the line method with and without data correction by using the cyclic plastic zone (CPZ) approach. (Note: RB = SRB + NR0.3RB + NR1.0RB; IPB = SIPB + NR0.5IPB + NR1.5IPB; AT = S-SAT + U-SAT + S-NR0.3AT).

	RB & IPB & AT		RB & IPB	RB & IPB & AT
	Elastic	Total	Total Strain	Total Strain
	Strain	Strain	& CPZ	& CPZ
Strength Coefficient, $C_b$	0.0330	0.0413	0.0235	0.0340
Strength Exponent, $n_b$	−0.1944	−0.2105	−0.1789	−0.1943
Coeff. of Determination, $R^2$	0.2951	0.3644	0.5097	0.3534
Walker Parameter, $\gamma$	0.5845	0.6497	0.1716	0.6508
Mean Square Error, $MSE$	0.7602	0.7048	0.2486	0.6973
Coeff. of Interception, ${\beta }_0$	−7.618	−6.575	−9.107	−7.558
Coeff. of Slope 1, ${\beta }_1$	−5.144	−4.751	−5.590	−5.1455
Coeff. of Slope 2, ${\beta }_2$	−3.007	−3.087	−0.9594	−3.349
Avg. Ind. Var1, $log{\sigma }_a$,	−2.564	−2.545	−2.514	−2.549
Std.Ind. Var1, $log{\sigma }_a$,	0.1196	0.1259	0.0978	0.1221
Avg. Ind. Var2, $log2/\left(1-R_{\sigma }\right)$,	0.1630	0.1704	0.1281	0.1704
Std.Ind. Var2, $log2/\left(1-R_{\sigma }\right)$,	0.1861	0.1865	0.1723	0.1865
Avg. Dep. Var, $log{N}_f$,	4.989	4.989	4.821	4.989
Std. Dep. Var, $log{N}_f$,	0.8314	0.8303	0.5712	0.8860

Going into greater detail in the elastic analysis at the notch roots, a comparison is made between the von Mises stress profiles for the linear pure elastic behavior and the elastic–plastic behavior, at the moment of loading and unloading, as performed by the Irwin method of elastic unloading and plastic reloading, near the crack tip [69]. Note that for a yield strength of 1089.2 MPa, there is a stress saturation until a value of approximately 0.6 mm. This value corresponds to the diameter of the MPZ zone as exemplified in Figure 23b). Taking into account the assumption that the cyclic plastic radius is usually 1/4 of the monotonic plastic radius (see the red circle in Figure 23b which represents the size of the CPZ zone).

Comparing the CPZ size with the stress distribution in loading and unloading, it is verified that this zone corresponds to the moment of lowest equivalent stress in the loading step and the inversion of the gradient of equivalent stresses in the moment of unloading. Since the criterion $r_{CPZ}=1/4$$r_{MPZ}$ is perfectly valid, this criterion will be considered for the determination of $r_{CPZ}$ and subsequent $D_{LM}$ from Equation (2b), rewritten here in the equivalent strain amplitude form as

$${\epsilon }_{a,vM}={\displaystyle \frac{1}{\left[D_{LM}+2r_{CPZ}\right]}}{\int }_0^{\left[D_{LM}+2r_{CPZ}\right]}\epsilon \left(r\right)\mathrm{d}r.$$

(26)

4.5. WSN Fatigue Modeling Based on TCD and CPZ Approaches

4.5.1. CPZ Correction in WSN Fatigue Model: Rotating and In-Plane Bending Loads

Applying the Equation (26) to the data collected in fatigue tests, with a plastic radius, $r_{CPZ}$ = 0.0012 mm for only one notched specimen under rotating–bending, and $r_{CPZ}$ = 0.0177 ± 0.0144 mm (maximum of 0.0503 mm and minimum 0.0009 mm) for notched specimens under in-plane bending, results in the WSN average curve and WSN percentile curves as shown in Figure 24. Note that when comparing Figure 24 with Figure 17, there is a decrease in the value of ${\epsilon }_{a,w,vM}$, such that they approach the WSN average curve. The correction effect using the CPZ approach is only notable at the most extreme points due to the fact that only these data points have a plastic region in front of the notch root. CPZ correction has little effect on $R^2$, remaining at 0.5. Furthermore, this fitting by considering local plasticity effects led to the change in the value of $\gamma$ from 0.1201 to 0.1716. Regarding the regression parameters, a $C_b$ value of 0.0235 and an $n_b$ of −0.1789 were obtained, which are slightly lower (0.0292 and −0.1987, respectively).

After correction with the introduction of the cyclic plastic radius for the determination of the effective value, it is verified that the largest number of failures are found between the 0.25 and 0.75 quartiles and practically all the failures are between the 0.5 and 0.95 quartiles. Note that the closest failures to the 5 % failure percentile curve correspond to in-plane specimens with a notch radius of 1.5 mm, such that the size effect may have some influence on the strength of the component. Table 7—column 3 presents the summary of the parameters of the fatigue model and the statistical model considered.

4.5.2. CPZ Correction in WSN Fatigue Model: Bending and Tension/Compression Loads

Applying the CPZ method’s corrections to failure dataset, considering the cyclic plastic radius for rotating–bending and in-plane bending mentioned previously and $r_{CPZ}$ = 0.0311 ± 0.0425 mm (maximum of 0.1501 mm and minimum 0.0013 mm) for notched specimens under axial tension/compression, results in the fatigue curves shown in Figure 25. Comparing Figure 25 with Figure 21, it is observed that $\gamma$ remains practically unchanged ($\gamma =0.6508$), with a slight reduction of the parameters of $C_b$ and $n_b$ to 0.0340 and −0.1943, respectively. In addition, the value of $R^2$ also remains unchanged ($R^2$ = 0.3534); however, the failure dataset corresponding to the in-plane bending tests approached the average curve, with most of data points remaining within the quartiles of 0.25 and 0.75. Notice that for lower loads, in-plane bending fatigue-failure data are outside this zone due to the fact that the data are in a fatigue limit zone for their respective load ratio, and consequently, the occurrence of outliers increases. Regarding the failure data of specimens in uniaxial tension/compression, specimens with the highest value of ${\epsilon }_{a,w,vM}$, after correction, such as $r_{CPZ}$, remains on the failure probability curve of 25 % and data related to failures in LCF specimens. Table 7—column 4, shows the regression parameters used to build the fatigue prediction model.

4.5.3. ACFC Fatigue Model for Combining Bending and Tension/Compression Loads

Introducing the data regarding failures obtained for tension/compression loading conditions in notch and smooth specimens, into the bending failure dataset, illustrated in Figure 26, the extended hyperbolic fatigue field (from LCF to VHCF fatigue regime) results in Figure 27. It is interesting to note that in the low-cycle fatigue regime, the failure data, despite being far from the median curve, have the same trend as the 25th percentile curve. This deviation may be due to the presence of a larger amount of failure data for ${\psi }_{\epsilon ,w,VM}$ between 0.6 and 0.8%. However, the variation tendency for smooth specimens as the number of cycles increases follows a failure probability of 5%. Regarding failures in notched specimens, the greater number of failures occurs between probabilities of 5 and 25% for longer lives, but changes when the value of ${\psi }_{\epsilon ,w,VM}$ increases to values greater than 0.5%. In fact, failures in notched specimens for values greater than 0.5% under tension and in-plane conditions occurred with the combined effect of mean stress and localised plasticity in front of the notch root. The same deviation of failure data from in-plane specimens was observed in the analysis performed in Section 4.5, due to the generalised $\gamma$ value of 0.65 for 51CrV4 steel. The parameters of the statistical model are presented in

Table 8. The summary of the estimators for the three-parameter Weibull distribution on smooth and notched specimens under bending and tensile conditions considering Walker’s fatigue parameter (Equation (22)). (Note: RB = SRB + NR0.3RB + NR1.0RB; IPB = SIPB + NR0.5IPB + NR1.5IPB; AT = S-SAT + U-SAT + S-NR0.3AT).

Testing	Elasto-Plastic	Elasto-Plastic
Conditions	Bending (RB + IPB)	Tensile & Bending (RB + IPB + AT)
Walker Par., ${\gamma }_w$	0.83	0.35
Vert. Asymptote, B	0.00 (1 [cycle])	0.00 (1 [cycle])
Horiz. Asymptote, C	−2.52 (0.08 [%])	−2.58 (0.08 [%])
Shape Par., $\beta$	1.81	1.72
Scale Par., $\delta$	6.42	9.02
Location Par., $\lambda$	10.91	11.66
Avg. Random Var. v, ${\mu }_v$	16.62	19.70
Std. Random Var. v, ${\sigma }_v$	10.66	23.21
Quantile $(P_f=50\%)$	16.15	18.95

—column 2.

4.6. Fracture Surface Analysis

4.6.1. Fracture Surfaces of Notched Specimens Under Rotating–Bending Loading

Regarding fracture surfaces obtained from failure specimens under rotating–bending loading, Figure 28 presents the fracture surfaces obtained for five distinct nominal stress amplitudes and notch radii (0.3 and 1.0 mm). Figure 28A,B refer to specimens with a notch radius of 1 mm, while Figure 28C–E refer to specimens with a notch radius of 0.3 mm. It is possible to observe that for the highest stress amplitude levels, in both geometries of Figure 28A,D, initiation of multiple cracks is shown around the circumference, typically for notches subjected to high local stress concentration, as previously verified in smooth specimens [62]. For the fracture surface illustrated in Figure 28A, their multiple cracks give rise to a main crack surrounding the entire perimeter of the specimen section, but the crack propagation becomes greater in the upper part of the sample. On the other hand, in the fracture surface presented in Figure 28D, the same propagation mechanism does not occur, and therefore, the propagation of one of the cracks becomes greater than the rest. In both stress amplitude levels, one can observe that the critical crack length is approximately the same, around 1.8 mm.

Figure 29 illustrates in greater detail the three regions associated with crack propagation, initiation zone, stable propagation, and unstable propagation. It is interesting to note the effect that rotating–bending has on the surface in the crack initiation zone. According to Figure 29 in macro-scale, each crack initiates in different topographic planes. Regarding the propagation stable zone, the presence of polished facets resulting from the crack closure generated in the fatigue cycles is verified.

For nominal stress amplitudes less than 455 MPa, inclusive, single crack initiation is observed in both notched geometries. The critical crack length is approximately half of the internal diameter, $d_0$, which corresponds to 2.6 mm for failure illustrated in Figure 28B, and 2.5 mm for failures in Figure 28D,E.

4.6.2. Fracture Surfaces of Notched Specimens Under In-Plane Bending Loading

Regarding fracture surfaces of failure specimens subjected to 3-point in-plane bending loads, one fracture surface for each notch geometry type was analysed. The samples were defined considering the highest stress amplitude levels for each test. Figure 30 illustrates the fracture surface for a geometry with a notch radius of 0.4 mm, subject to ${\sigma }_{a,n}$ = 480 MPa, ${\sigma }_{m,n}$ = 560 MPa with the detail in the initiation zone (I) and in the transition zone from stable propagation to unstable propagation. Note that this fracture surface can be characterised with an initiation crack length of the order of 0.6–0.7 mm (detail (I)), and a critical crack length of 3.76 mm. The crack width at the surface is extended over almost the entire width of the specimen, 20.65 mm. As regards the fracture surface of specimens with a notch radius of 1.5 mm illustrated in Figure 31, the identification of the different propagation zones is clear. Zone (I) depicts a crack with a width of 20.46 mm, an initiation length of 0.4 mm and a critical crack length of 1.42 mm.

4.6.3. Fracture Surfaces of Notched Specimens Under Axial Tension Loading

The SEM samples of the failure specimens subjected to tension/compression loading were analysed for two different load cases. In Figure 32A, ${\sigma }_{a,n}$ = 540 MPa and ${\sigma }_{m,n}$ = 660 MPa illustrate the fracture surface of specimens subjected to the highest nominal stress amplitude level tested, with $R_{\sigma }=0.1$ and a notch radius of $z_{nt}=0.3$ mm, whereas Figure 32B represents the typical fracture surface for the lowest nominal stress amplitude levels tested (${\sigma }_{a,n}$ = 235 MPa and ${\sigma }_{m,n}$ = 290 MPa, for $R_{\sigma }=0.1$ and the notch radius, $z_{nt}=0.3$ mm).

Comparing the critical fatigue crack sizes at fracture, we can verify that for load case (A) (Figure 32A), a critical size of 600 $\mathsf{\mu }$m was observed, while for load case (B) ((Figure 32B), the crack grew to about a length of 900 $\mathsf{\mu }$m. In load case (A), the size of crack propagation is not very large, and the initiation phase is not very easy to identify. However, there were some regions where the initiation crack length appears to have a length of around 150 $\mathsf{\mu }$m. On the other hand, in case (B), a minimum initiation crack length can be identified, such that the crack has a length of 250 $\mathsf{\mu }$m.

Concerning the highest loading level (load case A), it is observed at the initiation and propagation zones that the fracture micro-mechanisms are essentially micro-cleavage, but the appearance of very small ductile dimples is also observed. At the crack-propagation transition zone from stable propagation to unstable propagation, the appearance of dimples is superior and exists in larger sizes. In both zones, (I) and (II), the appearance of cleavage micro-cracks is also observed. In contrast, for lower loading levels (load case B), the appearance of ductile dimples is not clear. At the zone close to the sample surface, the fracture micro-mechanisms are shown to be micro-cleavage with the appearance of some large cleavage facets, in the order of 30 $\mathsf{\mu }$m. In (II), the same type of failure micro-mechanism is observed, and likewise, the appearance of large cleavage facets.

5. Conclusions

This investigation analysed the notch effect in 51CrV4 spring steel with application in parabolic leaf springs for railway freight wagons. The analysis consisted of modeling fatigue behaviour using the Walker model (WSN) and the Apetre and Castillo–Fernandez-Cantelli model (ACFC), combined with fatigue analysis approaches for notches based on the theory of critical distances (TCD), namely the line method. The model was based on FEM-based elastic stress profiles in front of the notch root for all specimen geometries tested under rotating–bending, 3-point in-plane bending, and tension/compression loads at different stress ratios. For specimens where localised plastic deformation occurred in front of the notch root, the correction of $D_{LM}$ was applied using the cyclic plastic zone (CPZ) approach. From the carried-out analyses, it was found that the CPZ size is practically 1/4 of the radius of the monotonic plastic zone, MPZ, and then this relation was assumed to calculate the radius of CPZ, and consequently update $D_{LM}$.

Failure data from notched specimens were combined with failure data from smooth specimens (data obtained from the literature for 51CrV4 steel with the same material characteristics) to determine a master curve based on an effective fatigue damage parameter. For both the WSN and the ACFC (3-parameter Weibull distribution) models, the effective fatigue damage parameter defined from the equivalent von Mises strain amplitude, ${\epsilon }_{a,w,vM}$, was used. In the specific case of the ACFC fatigue model, the damage parameter, ${\psi }_{\epsilon ,w,vM}$, is used.

Under non-null mean bending stress conditions, the computed WSN model based on von Mises effective strains showed good agreement between fatigue-failure data of smooth and notched specimens, since a superposition of most data points was observed on the average curve. Addionally, the found $\gamma$ parameter was similar to that found in the regression for only smooth specimens ($\gamma =0.12$). On the other hand, under uniaxial stress conditions, a value of $\gamma =0.86$ was obtained, slightly higher than that obtained in the regression found for smooth specimens tested under tension/compression loads. Despite this increase, it is still verified the sensitivity effect of mean stress and type of loading in the fatigue model.

The data gathered in bending and tension/compression for smooth and notch specimens were merged into a single master fatigue strength curve from the low-cycle to giga-cycle regime. The consideration of von Mises elastic strain, only applying the Walker model, showed a poor data fitting for short lives; however, incorporating the plastic strain component resulted in a slight improvement of WSN regression, such that $\gamma$ was 0.6497, $C_b$ was 0.0413, and $n_b$ was -0.2105, with a coefficient of determination $R^2$ = 0.3464 (representing a large scatter). Taking into account the importance of plastic deformation to obtain a better-fit model, the cyclic plastic zone concept was considered in determining the effective damage parameter. The results showed better agreement than those that did not consider the effect of the cyclic plastic radius. The data that were further away from the mean curve became closer without any significant change in the regression parameters, such that it resulted in $\gamma =0.1716$, $C_b=0.0235$, and $n_b=-0.1789$ for bending conditions and $\gamma =0.6508$, $C_b=0.0340$, and $n_b=-0.1943$ for bending and axial tensile conditions. The increase in the value of $\gamma$ with the incorporation of the effect of localised plastic deformation shows that this parameter will be sensitive to the presence of both generalised and local plasticity.

Regarding the analysis of fatigue behaviour using the ACFC model, the damage parameter ${\psi }_{\epsilon ,w,vM}$ allowed us to obtain a good correlation between the failure data and the prediction model for bending conditions, with ${\gamma }_w$ = 0.83, B = 0.00, C = −2.52, and Weibull’s distribution parameters, $\beta$ = 1.81, $\delta$ = 6.42, and $\lambda$ = 10.91. Most failure data were located in the 5th and 75th percentile curves from shorter to medium lifetimes, above ${\psi }_{\epsilon ,w,vM}$ > 0.3%. By combining fatigue-failure data from rotating and bending loads with a set of failures (specific smooth and notched) under uniaxial tension/compression loads, it was possible to create a probabilistic model extended from short lives ($2.0\times {10}^2$ cycles) to lives that reached the giga-cycle regime ($\approx 1.0\times {10}^9$ cycles) with a good representation among probability-of-failure curves and the observed failures. The ACFC fatigue model that calibrates this number of failures has a ${\gamma }_w$ = 0.35, a B = 0.00, and a C = −2.58, and its Weibull distribution parameters are $\beta$ = 1.72, $\delta$ = 9.02, and $\lambda$ = 11.66. In general, the ACFC model showed that for shorter fatigue lives, most of the observed failures are between the 25th and 50–75th percentile curves, while for longer lifetimes, due to the large scatter inherent to the VHCF regime, the greater concentration of failure points is located between the 0.05 and 0.95 quartiles.

Concerning the fracture surfaces observed, for fatigue specimens tested under rotating–bending, multiple initiation cracks were found in the specimens subjected to higher loading, which might merge during propagation. For lower loads, single crack initiation occurs. In the case of specimens subjected to in-plane bending loads, higher loads showed different propagation zones on the fracture surface, with critical crack lengths of about more or less half of the nominal cross-section area. For specimens under axial fatigue tensile conditions, a single crack initiation occurs; however, the detection of initiation and propagation zones is not clear. The SEM analysis allowed us to identify that under tensile-loading conditions in notch specimens, the fatigue occurs essentially with flat micro-cleavage, being visible in some ductile dimples despite their size and quantity being relatively small. Moreover, a qualitative analysis of the fracture surfaces of the specimens and the type of fracture for the different fatigue test regions and loading conditions are analysed, highlighting the zones of appearance of multiple cracks, single cracks on the surface, and cracks initiated in the sub-surface region or in the interior, through the non-metallic inclusions or voids.

In summary, the research performed and presented in this scientific article aims to provide fatigue prediction models calibrated for several geometric and loading conditions, including the presence of notches in components, non-null mean stress, and loading conditions that cause uniaxial normal stress (bending and tensile stress). Since the WSN and ACFC fatigue models are calibrated based on a strain-based fatigue damage model ($\epsilon -N_f$ fatigue model), they can be suitable for elastoplastic loading conditions. Furthermore, the probabilistic models can also be considered in further analyses of the fatigue damage of components made of 51CrV4 martensitic steel, in addition to parabolic leaf springs. Additionally, since the fatigue behaviour under controlled environmental conditions was determined, the gathered fatigue data is suitable for future exploratory fatigue analysis in severe environmental conditions.