Requirements for stress gradient-based fatigue assessment of notched structures according to theory of critical distance

Notches, local stress raisers within structural components, are one of the most important locations for fatigue crack initiation. It is well known that fatigue is governed by the effective stresses in the vicinity of notches. Within this study, differences in prediction accuracy between different types of theory of critical distance methods, that is, point and line methods, are systematically investigated in conjunction with a sensitivity study regarding mesh refinement, assumed strength hypothesis and material behaviour. For this purpose, a finite element analysis parameter study on notched structures is performed and recommendations for the application of stress gradient methods are presented. Difference in effective stress of up to 30%, and hence a significant difference in fatigue life (e.g., 185% for a slope of S-N curve of k = 4), is found for typical notch shapes, for example, in welded joints.


| INTRODUCTION
Notches, a form of geometric discontinuity within structural components, are one of the most significant locations for fatigue crack initiation due to local stress and strain concentrations. Thus, fatigue damage often originates at notches. The reliable assessment of fatigue strength is thus very important for the purpose of cost reduction and risk assessment.
As notches lead to high local stresses and a multiaxial stress/strain state, they represent an area which is of great interest for fatigue limit state design of structures; this is especially true because their fatigue life is significantly reduced compared with unnotched structures. In fatigue assessment of notched structures, it is well known that the fatigue strength is not governed by the linear elastic maximum stress at the notch tip, but rather by the stress field surrounding the notch. 1 The main reasons are the high stress field gradients at sharp notches, the local plastic zone and initial damage in the form of microscopic cracks. 2 Moreover, assessment based on the elastic peak stresses fails to predict size effects. 3 Since the beginning of fatigue strength assessment, many different approaches to determine effective stress parameters for the stress fields surrounding notches have been developed. Later, these approaches have been extended and applied to welded joints. 4 All of them are based on the idea of taking into account the support effect of the material surrounding a local stress raiser by mapping the elastic stress field by a suitable functional 5 (cf. Berto et al. 6 or textbooks 7 regarding analytical functionals for the stress field).
A way of distinguishing different effective stress approaches is by considering dimensional space (e.g., point, line, area or volume). Because of increased computational power, such methods are more easily applicable nowadays. The point (PM) and line (LM) methods-based on the gradient of the stress acting perpendicular to the expected crack path-have gained especial interest, judging by the variety of investigated applications.  The main advantage of gradient methods, compared with other fatigue assessment methods, is the possibility of taking thickness effects directly into account; see Hobbacher 30 regarding limitations to assess thickness effects with different fatigue assessment methods. Moreover, recent research results have shown that such methods can be applied to notches with significantly different proportions or strength grades, such as welded and post-weld treated joints at the same time 11,13 ; in many technical fields, however, application of such methods is currently not possible because of guidelines and design codes that are limited to nominal, hot-spot, notch stress and fracture mechanics approaches. [31][32][33] Typically, as with every emerging method, general applicability has first to be proven by case studies. Moreover, increased prediction accuracy has to outweigh higher preparation efforts.
Of all effective stress methods, two versions of stress gradient methods have been popularly applied in the recent years. One of these approaches averages the stress over a certain microstructural support length ρ * proposed by Neuber. 1 The underlying assumption is that crack initiation is controlled by the averaged stress in a small material volume (defined by the averaging length ρ * ) at the point of maximum stress. 34 A different idea, which was proposed shortly afterwards, defines the effective stress σ eff as the value at a certain distance a from the root of a notch. 35 The idea behind this approach is that crack initiation is related to the effective stress evaluated at a material dependent critical depth a below the notch root surface. 34 Thus, the fatigue strength at notches is described by the fatigue notch factor K f , which is related to the elastic notch stress concentration factor K t with where r is the notch radius and K t the ratio between the maximum stress at the notch root and the nominal stress. For design, however, it is more common to calculate fatigue life from effective stresses in conjunction with appropriate stress-life (S-N) curves than to calculate the fatigue notch factor K f ; see Radaj et al. 36 Today, these methods are sometimes summarized under the term theory of critical distances (TCD). 2 In addition to PM, based on Peterson, 35 and LM, based on Neuber, 1,37 a number of other approaches take into account area or volume. [38][39][40] For typical notched structures like machinery components, stress gradient methods are regularly applied for fatigue assessment; however, gradient methods based on the TCD have successfully been applied to welded joints in recent years as well. 9,13,15 Interestingly, both PM and LM are assumed to result in equal stress values for certain distance of evaluation ratios. 41 A ratio of a = ρ * /4 is expected to yield equal results, although differences in prediction accuracy between both have already been addressed. 3,17,20,34,42 In high cycle fatigue assessment (N > 10 4 ), the lifetime N can by calculated by where Δσ eff is the effective stress range and N R is the reference number of cycles to failure with the corresponding reference fatigue strength Δσ R . By raising the stress to the power of the slope exponent k of a S-N or Wöhler curve, small differences in effective stress may lead to high deviation in estimated lifetime. This study aims to contribute to the applicability of TCD for fatigue assessment by analysing the effect of notch geometry on fatigue strength, using stress gradient methods. In this context, the assumed ratio a = ρ * /4 between PM and LM, expected to result in equal effective stresses, is analysed. For this investigation, finite element analysis is applied, starting with a parameter study regarding mesh requirements.
In literature, many cases report differences in prediction accuracy between different types of TCD methods. 3,9,17,20,42 To the author's knowledge, there has been no systematic investigation in conjunction with a sensitivity study for mesh refinement and assumed strength hypothesis. In order to reach a level of applicability in international standards and guidelines, meshing recommendations and clarity for differences in prediction accuracy are required. Hence, this study seeks to solve this problem by numerically analysing notched structures with varying geometry parameters, strength hypothesis and material behaviour (plane stress and plane strain).

| NOTCH PARAMETERS
Notches, as typical engineering stress raisers, can occur in many different shapes. For the scope of this paper, however, only u-type and v-type notches (with variable notch radius) are analysed in context of practical applications. In order to define the geometry of a notch, multiple parameters such as notch radius, notch depth and the opening angle 7 are required. Two of these parameters, the notch radius r and the opening angle ω, are shown in Figure 1A. The importance of notches in structural components is illustrated by engineering examples like welded joints ( Figure 1B,C), additive manufactured parts ( Figure 1D) and medical implants ( Figure 1E).

| GRADIENT METHODS
As mentioned in Section 1, different forms of the TCD are currently used to determine the effective stress in order to evaluate the lifetime of a structural component (see Figure 2). One form of the TCD is the point method; this is based on the critical distance approach by Peterson 35 and defines effective stress as the stress value perpendicular to the root of the notch at a fixed distance. The effective stress σ eff for the PM can thus be calculated by where σ y is the stress at a certain distance perpendicular to the root of the notch (i.e., along the notch bisector) and θ and s are polar coordinates. Here, L is a material characteristic length dependent on the material, the number of cycles to failure 42 and the type of loading. 2,43 For the PM, it is assumed that failure occurs, if the effective stress range Δσ eff at L/2 is equal or exceeds the fatigue limit Δσ 0 . 2 On the basis of fracture mechanics, it can be shown that the material characteristic length L is related to the fatigue crack propagation threshold ΔK th and the fatigue limit with The material characteristic length L is thus directly related to the transition point between short and long crack growth.
A different form of the TCD to determine the effective stress is the LM. This approach, as opposed to the PM, averages the stress perpendicular to the root of notch over the material characteristic length to determine the effective stress. 1 The resulting effective stress can be calculated as follows 2 : The distance L to evaluate the stress henceforth refers to Neuber's 1 and Peterson's 35 initial propositions. In case of PM, the notation a (a = L/2), and for LM, ρ * (ρ * = 2L), will be used to distinguish both methods.
For some applications, values for different materials can be found in textbooks or publications 34,37 ; however, for every new application material characteristic length, values have to be found by solving the inverse problem (i.e., finding the length parameter that describes a dataset best; see Santus et al. 17 ). Typical values for welded joints

| DEFINITION OF NOTCHED GEOMETRY
The commercial software ANSYS is used to perform finite element analysis for the parameter study and the comparison of gradient methods. In this study, the two main parameters varied are the notch radius r and the opening angle ω. The depth of the notch is kept constant at 10 mm. Two opening angles ω are investigated (0 and 135 ), both of which are frequently found in welded joints and notched structures. Notches at weld roots often approach opening angles close to 0 (i.e., almost crack like), whereas v-type notches with approximately 135 opening angles are typical for weld toes (see Figure 1). For the notch radius r, values of 0.05 and 0.5 mm (representing a sharp, almost crack-like and a blunt notch) are used within this study. Moreover, both values enable comparability with other studies. 7,44-46 An overview of the parameter combinations investigated are listed in Table 1.
The investigation of notch stress, in context of the aims described in Section 1, does not demand a particularly complex model. In order to investigate the aforementioned effects, a two-dimensional model is sufficient (see Baumgartner and Bruder 44 and Leitner et al. 47 ); this also leads to a smaller computational effort. For the finite element analysis, typical material properties for steel are applied, which are summarized in Table 2.
A sketch of the half model is shown in Figure 3. The symmetry about the neutral axis is used to minimize computational effort. Parameters used as input data for the geometry of the model are marked in red, whereas blue indicates parameters that are dependent on one or more of the input parameters.
The radius of the notch r and the opening angle ω are the main parameters varied during the analysis (see Table 1). The depth of the notch d is set to 10 mm, as previously mentioned. Furthermore, the distance g between the root of the notch and the neutral axis is set to 10 mm, thus yielding a notch depth to height ratio of d/h = 0.5. The load, which is applied on the right end of the specimen, is specified as a unit stress (1 N mm −2 ).
The number of elements n, which represent the circular arc of the notch, is defined as number of elements over 360 (full circle). Values for n are investigated as part of the parameter study. Although a finer mesh generally results in more accurate stress values, it is also more computationally expensive. As can be seen from Figure 3, the specimen is divided into two areas (one global area and one local area around the root of the notch) with the size of the area around the root of the notch defined by two parameters. One of them, the depth of the area r 1 , is kept constant at 5 mm. The other, the angle φ, is a function of the opening angle ω (φ = 180 − ω). Within this area, the element size e is

Material properties
Young's modulus dependent on two parameters: n (the number of elements over 360 ) and the aspect ratio of the elements. In order to obtain accurate results with minimum preparation, the aspect ratio is set to one, although elements with an aspect ratio up to two (shorter in depth-direction) might reduce the number of elements required. 44 The global element size e g is a function of the element size e 1 at the outer radius of the area around the root of the notch. This ensures the elements in the transition area remain well shaped, leading to a precise analysis. The element type for the analysis is a 2D eight-node structural solid (plane183 in ANSYS) with plane strain condition. Figure 4 depicts an example of the performed simulation for r = 0.05 mm, ω = 0 and n = 72, with first principal nodal stress results.
First principal and von Mises nodal stress results are extracted along the notch bisector for each simulation. The evaluation of the stress gradient is subsequently performed using scripts in Matlab. Because polynomial curve fitting does not lead to a satisfying evaluation of the raw stress data, piecewise cubic hermite interpolating polynomial (PCHIP) curve fitting is applied. The PCHIP curve fitting is shape preserving, meaning that slopes at the evaluation points are chosen such that the interpolant preserves the shape of the data to be fitted. Numerical integration (trapezoidal rule) with 5Á10 −4 mm increments is applied for averaging the stress gradient according to LM.

| MESHING REQUIREMENTS FOR STRESS GRADIENT METHODS
As mentioned in Section 1, the first part of the finite element analysis is a parameter study to determine the required number of elements n to calculate consistent stress results for application of stress gradient methods. This is achieved by determining values of n that yield an effective stress result with less than 1% deviation from the finest analysed mesh; these results will be presented later. On the basis of the assumption that fixed microstructural support length ρ * = 0.4 mm for a cast iron-like microstructure of weld metal, Radaj 4 developed the effective notch stress concept widely applied for fatigue assessment of welded joints nowadays; because of its significance, this value will also be used as a reference in the present study.
The mesh generation, and thereby the required number of elements per 360 , is governed by the need for an element border at the notch bisector (see Figure 2). Given the definition of n, different elements are required to model the circular arc (i.e., n/2 for 0 and n/8 for 135 ). Table 3 lists the range of values for n in combination with the parameters for the notch radius r and the opening angle ω of Table 1. Resulting meshes for r = 0.05 mm and ω = 0 are presented in Figure 5.
The results of the parameter study are shown in Figure 6 using the example of r = 0.5 mm and ω = 0 . Each of the coloured lines in the five subfigures represent a different number of elements n over 360 . The three numbers in the notation of the simulations signify the following: mesh, element side ratio and number of elements over 360 . Figure 6A presents the maximum principal stresses σ 1 . In order to differentiate between the curves, a more detailed view of the stress is depicted over the first 0.5 mm in Figure 6B showing a clear difference between stress gradients for varying numbers of elements n. Notably, the curve for n = 8 deviates more than the other curves. Unsurprisingly, having a low number of elements inaccurately represents stress gradients. The effective stress σ eff derived with LM is presented in Figure 6C. For comparison, the effective stress is normalized by the maximum effective stress for the first averaging length ρ * = 0.05 mm with σ eff,norm = σ eff /σ eff,max in Figure 6D. On the basis of the graphs of Figure 6, the deviation between stress gradients for varying numbers of elements is hardly discernible. Even the most detailed plot, Figure 6B, indicates very similar trends for all curves. A more detailed presentation of the variation of normalized effective stress for averaging length ρ * = 0.4 mm is given in Figure 7.
The number of elements n over 360 is on the horizontal axis and effective stress normalized by the maximum effective stress (σ eff,norm = σ eff /σ eff,max ) is on the vertical axis. The maximum effective stress is the value for the highest number of n, where convergence is expected based on the asymptotic behaviour of the graphs. The converged stress values are assumed to be found for n = 288 and n = 312, respectively. This definition of the normalized effective stress allows a direct comparison based on the percent deviation between the stress values and the converged stress. Within Figure 7, the trends of all four investigated parameter combinations (outlined in Table 1) are presented. Contrary to the impression of Figure 6, an error of up to approximately 6% is found. In order to reach an error less than 1% for effective stress values, the required number of quadratic elements n is 32 for ω = 0. In contrast, the required number of elements for v-type notches with ω = 135 is only 24. These conclusions hold true for radii of both 0.05 and 0.5 mm.

-
The finding that n = 32 yields a sufficient result for ω = 0 aligns with recommendations for other local fatigue assessment methods found in literature, including the effective notch stress method. 44,47,48 For the opening angle ω = 135 , only 24 quadratic elements per 360 are required because of the circular arc being modelled with four elements; this was also found to be sufficient for the effective notch stress method. 49,50 Thus, on the basis of the aim of less than 1% deviation for effective stress from the converged stress, n = 24 and n = 32 actually lead to the same number of elements representing the notch root. A deviation of 1% is considered small compared with the general scatter found in fatigue assessment. Moreover, mesh refinement equal to those in other methods, in conjunction with the benefit of incorporation of size effects, underlines the potential for stress gradient methods to be generally applied for notched structures.
In order to facilitate the application readiness of often-applied versions of TCD, that is, PM and LM, the previous findings of mesh refinement for effective stress calculation are subsequently applied for comparison of both methods. For this purpose, the previously determined mesh refinements, n = 24 and n = 32, are applied for the two analysed opening angles.

| COMPARISON OF GRADIENT METHODS
Following the investigation on required mesh refinement, this section describes the comparison of gradient methods to discuss the assumption of PM and LM yielding the same results for a ratio of a = ρ * /4. For this purpose, the previously defined parameter combinations of Table 1 are investigated in conjunction with the findings of the parameter study. Figure 8A shows the first principal stress evaluation of PM and LM, where r = 0.05 mm and ω = 0 . This figure includes two x-axes scaled with the above-mentioned ratio between a and ρ * , according to the assumption of Taylor and Hoey. 41 The resulting stresses indicate a clear deviation. The deviation decreases for larger distances to the root of the notch, especially in the case of sharp notches, which are related to the high stress gradient in depth direction. The parameter combination of r = 0.05 mm and ω = 135 ( Figure 8B) shows the same general trend of the stress gradient. On the other hand, it is interesting to note that r = 0.5 mm and ω = 0 ( Figure 8D) yield almost equal stress values for small evaluation lengths of ρ * = 0.05 mm and a = 0.0125 mm. This is followed by a brief increase then subsequent decrease in deviation. This will be revisited later for further explanation.
The results of the von Mises stresses in Figure 9 also support the hypothesis of inequality for the two methods. Moreover, the diagrams present a similar trend for the same radii and opening angles compared with the first principal stress graphs in Figure 8.
In order to further illustrate the deviation between the two methods, Figure 10 presents the percent deviation (defined as deviation = 1 − σ eff,PM /σ eff,LM ) between both methods for first principal stress ( Figure 10A) and von Mises stress ( Figure 10B). In none of these evaluated cases does the deviation approach zero for long material characteristic lengths ρ * or a. For first principal stress, the parameter combination with minimal deviation (1.5%) is r = 0.05 mm and ω = 0 , at ρ * = 1.5 mm and a = 0.375. At the same time, this parameter combination leads to the biggest deviation of 19.9% at ρ * = 0.1 mm and a = 0.025 mm. It is also worth noting that the curves for the same radii show similar trends, which can be explained by the radius of the notch being the main parameter governing notch sharpness. Interestingly, the graphs show a declining deviation for increasing material characteristic length for r = 0.05 mm. This is due to the small radius of the notch creating steep stress gradient. The graphs for r = 0.5 mm, however, show a slight initial deviation that first increases before it subsequently decreases again.
Within the range of the investigated parameters, a = ρ * /4 never leads to the same stress values for the PM and LM, thus supporting the hypothesis that there is no constant ratio a = ρ * /4 between both methods. The different trends for sharp and blunt notches can be explained by the varying degree of notch acuity, and thereby initial steepness of the stress gradient. As can be seen from Figures 8 and 9, the stress gradients are far steeper for r = 0.05 mm compared with r = 0.5 mm. This, in conjunction with the idea behind the LM of averaging the stress over a certain length, causes the different general trends in Figure 10A,B.
For sharp notches and von Mises stress results, the highest deviation is found to be close to 30% when ρ * = 0.05 mm. The highest deviation for blunt notches (approximately 27.5%) is found at ρ * = 0.5 mm and a = 0.125 when r = 0.5 mm and ω = 0 . On the other hand, r = 0.05 mm and ω = 0 results in almost no deviation (0.0015%) when ρ * = 0.25 mm and a = 0.0625. Thus, both the largest and smallest deviations are found for von Mises stress criterion. Depending on radius r and opening angle ω, as well as distance a and ρ * , even higher deviations are theoretically possible. Nearly all cases show deviations between PM and LM; the sole exception to this occurs when ρ * = 0.25 mm and a = 0.0625 mm for von Mises stress criterion with r = 0.05 mm and ω = 0 . On the basis of the assumed ratio of a = ρ * /4, PM usually results in higher stresses calculations than LM.

| DISCUSSION
The calculated difference is highest for von Mises stress criterion and blunt notches, with up to 27.5%; this is interesting to note because literature data based on fatigue tests do not always show such a clear trend. 17,20,42 Notch acuity levels, material, and specimen geometry may be responsible for this difference. Moreover, as this study has shown, a certain degree of mesh refinement is required to allow accurate calculation of effective stresses based on stress gradient methods. Consequently, different numerical modelling strategies might have also affected the outcome of other studies-for example, Susmel and Taylor 42 and Louks and Susmel 20 reported error intervals of about 20%.
For a slope of S-N curve of k = 4-as proposed by Baumgartner et al. 9 -a 20% difference in effective stress assessed by the TCD results in more than a 100% difference in the predicted lifetime of welded joints. This, however, leads to question which ratio (q = a/ρ * ) is required to obtain the same stress values for both methods when considering the investigated distances from the notch and whether there is a general relation between the two (i.e., equal stress results of the PM and LM). These questions are investigated in Figure 11, where the required ratio q to obtain the same stress values for PM and LM (σ eff,PM = σ eff,LM ) is presented over the averaging length ρ * of the LM. The ratio q never reaches a value of 1/4 within the investigated range for first principal stress, which confirms the findings from the previous investigations. For r = 0.05 mm and ω = 0 at ρ * = 1.5 mm, however, a ratio of q = 0.257 is found; this is relatively close to the assumed ratio. It is also worth noting that, in all cases, the biggest deviation occurs closest to the root of the notch. Thereafter, a steady decline can be observed. Furthermore, for both opening angles when r = 0.05 mm, an almost constant ratio q between PM and LM is observed for long material characteristic lengths L (q = 0.257 for ω = 0 and q = 0.3 for ω = 135 ). In contrast, the trends for r = 0.5 mm show a continuously decreasing ratio q at ρ * = 1.5 mm. Moreover, the ratio q for short and long L (ρ * = 0.05 mm and ρ * = 1.5 mm) is almost equal for both opening angles. Between these two points, the deviation between the curves initially increases before approaching each other again. On the other hand, the trends for r = 0.05 mm are initially similar but subsequently diverge. In general, the difference between solutions for the two investigated opening angles is higher for the smaller notch radius of r = 0.05 mm. The observations on first principal stress described so far can also be seen in the trends of von Mises stress curves in Figure 11B. As opposed to first principal stress, the ratio yielding the same results for both methods seem to converge when r = 0.05 mm for lengths longer than ρ * = 0.5 mm for ω = 135 and ρ * = 1.5 mm for ω = 0 .
First principal stress results for plane stress material behaviour (e.g., thin plates) are identical to plane strain material behaviour; however, a smaller difference between PM and LM for von Mises stress results is observed throughout the analysed range of material characteristic lengths, as can be seen in Figure 12.
On the basis of Figure 11 and Figure 12, no general conclusion can be drawn regarding the required ratio q for the parameter a in order to achieve the same effective stress value by PM and LM. A converged value can only be found in a few cases within the investigated range. Furthermore, the ratios q differ for the same radii as well as opening angles. Therefore, the only connection observed is the similarity of curves for the same radii. From the observed trend in Figure 11A,B, it seems questionable whether the results of both methods can be directly compared on the basis of a simple conversion (as assumed by Taylor and Hoey 41 ); however, it is important to mention that there are reported results where their hypothesis was confirmed based on fatigue test data on seam welded joints. 9 In order to analyse the applicability of TCD methods for fatigue assessment, the determined requirements for stress gradient based fatigue assessment of notched structures according to TCD need to be applied to fatigue test data. For this purpose, fatigue test data obtained for buttwelded joints with artificial notches obtained by Fischer et al. 45 and fatigue test data at low temperatures 51 are currently assessed and will be published in the near future.
Finally, it is important to understand that the changing ratio q over the distance of up to 1.5 mm has a significant effect on all kinds of applications. Materials with high static strength, such as metals, usually possess rather short microstructural support lengths (i.e., below ρ * = 0.5 mm 37 ), and the magnitude of microstructural support lengths for adhesive bonds can reach values in the range of millimetres. 8

| CONCLUSIONS
In this study, the requirements for fatigue assessment by means of stress gradient methods according to the TCD were investigated with regard to mesh requirements for numerical simulation and differences in prediction accuracy between the two most commonly applied versions of the TCD-PM and LM. The following conclusions can be drawn from the investigations: • On the basis of a parameter study, it was found that 32 elements over 360 are sufficient to calculate effective stresses that deviate less than 1% from the converged stress value for sharp notches. For blunt notches, 24 eight-node elements over 360 are sufficient with an aspect ratio of one. The difference from the converged stress is considered small compared with the differences in calculated effective stress between both analysed methods. • In order to calculate effective stresses from steep stress gradients, PCHIP curve fitting is recommended to preserve the shape of the stress data curve. • Furthermore, a difference in effective stress of up to 30%, and hence a significant difference in fatigue life (e.g., 185% for a slope of S-N curve of k = 4), can be observed for PM and LM results with the assumed ratio of a = ρ * /4. For plane strain material behaviour, the difference between the methods is higher for von Mises stress than for first principal stress. • A smaller difference between the methods is observed for plane stress material behaviour. • Within the investigated range of material characteristic lengths, up to 1.5 mm, the required ratio between a and ρ * is constantly changing and only converges in a few cases. Thus, it does not seem possible to harmonize both methods; different S-N curves are thus required for the two methods.