# Non local multiaxial fatigue modeling of defects: A unified approach to interpret size and shape effects

Pierre Merot, Franck Morel, Camille Robert, Etienne Pessard, Linamaria

Gallegos Mayorga, Paul Buttin

► To cite this version:

Pierre Merot, Franck Morel, Camille Robert, Etienne Pessard, Linamaria Gallegos Mayorga, et al.  
Non local multiaxial fatigue modeling of defects: A unified approach to interpret size and shape effects.  
Theoretical and Applied Fracture Mechanics, 2024, 131, pp.104378. 10.1016/j.tafmec.2024.104378.  
hal-04604974

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# Non local multiaxial fatigue modeling of defects: A unified approach to interpret size and shape effects

Pierre Merot, Franck Morel, Camille Robert, Etienne Pessard, Linamaria

Gallegos Mayorga, Paul Buttin

► To cite this version:

Pierre Merot, Franck Morel, Camille Robert, Etienne Pessard, Linamaria Gallegos Mayorga, et al.  
Non local multiaxial fatigue modeling of defects: A unified approach to interpret size and shape effects.  
Theoretical and Applied Fracture Mechanics, 2024, 131, pp.104378. 10.1016/j.tafmec.2024.104378.  
hal-04604974

HAL Id: hal-04604974

https://hal.science/hal-04604974v1

Submitted on 7 Jun 2024

**HAL** is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

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**To cite this version :**

Pierre MEROT, Franck MOREL, Camille ROBERT, Etienne PESSARD, Linamaria GALLEGOS MAYORGA, Paul BUTTIN - Non local multiaxial fatigue modeling of defects: A unified approach to interpret size and shape effects - Theoretical and Applied Fracture Mechanics - Vol. 131, p.104378 - 2024
Science Arts & Métiers (SAM)

is an open access repository that collects the work of Arts et Métiers Institute of
Technology researchers and makes it freely available over the web where possible.

This is an author-deposited version published in: https://sam.ensam.eu
Handle ID: http://hdl.handle.net/null

<figure><img src="image_2.png" /></figure>

This document is available under CC BY license

**To cite this version :**

Pierre MEROT, Franck MOREL, Camille ROBERT, Etienne PESSARD, Linamaria GALLEGOS MAYORGA, Paul BUTTIN - Non local multiaxial fatigue modeling of defects: A unified approach to interpret size and shape effects - Theoretical and Applied Fracture Mechanics - Vol. 131, p.104378 - 2024
Non local multiaxial fatigue modelling of defects : a unified approach to
interpret size and shape effects

Pierre MEROT<sup>a,b</sup>, Franck MOREL<sup>a</sup>, Camille ROBERT<sup>a</sup>, Etienne PESSARD<sup>a</sup>, Linamaria GALLEGOS MAYORGA<sup>a</sup>, Paul BUTTIN<sup>b</sup>

<sup>a</sup>Arts et Metiers Institute of Technology, LAMPA, HESAM Université, Angers, France

<sup>b</sup>CEA, CEA Pays de la Loire, F-44340 Bouguenais, FRANCE

**Abstract**

This paper deals with the effect of defect size and shape under high cycle fatigue for metallic alloys. A large simulation campaign based on a multiaxial fatigue criterion and a non-local approach is presented. A relative defect size based on a ratio between the defect size and a characteristic length introduced by the non-local approach is defined. A normalized Kitagawa-Takahashi diagram is then obtained. A competition between the highly stressed volume size and the local maxima due to the defect is observed and seem dependent on the relative defect size. The effect of the loading mode (uniaxial and pure shear) and of the plasticity are discussed. Finally, a comparison of the simulation results with experimental data on a 316L L-PBF demonstrates the robustness of the proposed approach and explains the negligible effect of the defect morphology compared to its size.

Keywords: Finite Element, Non-local approach, Multiaxial fatigue criteria, Defects, High Cycle Fatigue

**1. Introduction**

High Cycle Fatigue (HCF) behaviour of metallic material with defects (cast alloys, Additively Manufactured (AM) alloys, ...) has been a critical issue for industrial applications in the past decades [11, 15, 36]. Such defects are in many cases identified as the root cause of crack initiation leading to failure and can significantly decrease a material's fatigue strength [3, 43, 49].

The origin of defects is nevertheless diverse and often unique to each industrial application. They may be induced by the manufacturing process (Lack of Fusion (LoF) pores, gas pores, microshrinkage pores,...) or by external environmental factors (scratches, corrosion pits, ...) [14, 22, 40]. Different types of defects usually lead to distinct defect populations in terms of shape and size. Depending on the type of coexisting defects, the variability in the defect population of a mechanical part can be significant.

In addition, the presence of defects in a material induces local variations on the stress field. This translates into higher than expected values as consequence of stress concentration, a multiaxial state, high value gradients and a complex 3D distribution. The consideration of the surrounding local microstructure, the texture, and the residual stresses, also related to the presence of defects, makes it difficult to understand which and how much each one of these parameters may alter the fatigue strength.

Historically, and under uniaxial loadings, Linear Elastic Fracture Mechanics (LEFM) models are a classic way to deal with defects. Only their size and position (external or internal) are taken into account and they are seen as pre-existing cracks. The crack is considered as propagating if the Stress Intensity Factor (SIF) induced by a mechanical loading exceeds a given threshold. The fatigue strength is considered as the minimal stress to make at least one of the defects to propagate without crack arrest. However, such models lead to the most conservative predictions (since defects are considered as cracks) and do not take into account the 3D defect's shape, which could play a significant role depending on the material and the defect size [4, 2, 41]. These models are also often difficult to apply for complex multiaxial loadings.

In contrast, a continuum mechanics approach, via the use of Finite Element Method (FEM) for instance, can also be envisioned but implies the use of relevant tools to take into account all the features of fatigue damage around defects. Crystal Plasticity Finite Element Method (CPFEM) is likely the most accurate method since both the effect of the defect and the local microstructure are taken into account [18, 20, 29].
Non local multiaxial fatigue modelling of defects : a unified approach to
interpret size and shape effects

Pierre MEROT<sup>a,b</sup>, Franck MOREL<sup>a</sup>, Camille ROBERT<sup>a</sup>, Etienne PESSARD<sup>a</sup>, Linamaria GALLEGOS MAYORGA<sup>a</sup>, Paul BUTTIN<sup>b</sup>

<sup>a</sup>Arts et Metiers Institute of Technology, LAMPA, HESAM Université, Angers, France

<sup>b</sup>CEA, CEA Pays de la Loire, F-44340 Bouguenais, FRANCE

**Abstract**

This paper deals with the effect of defect size and shape under high cycle fatigue for metallic alloys. A large simulation campaign based on a multiaxial fatigue criterion and a non-local approach is presented. A relative defect size based on a ratio between the defect size and a characteristic length introduced by the non-local approach is defined. A normalized Kitagawa-Takahashi diagram is then obtained. A competition between the highly stressed volume size and the local maxima due to the defect is observed and seem dependent on the relative defect size. The effect of the loading mode (uniaxial and pure shear) and of the plasticity are discussed. Finally, a comparison of the simulation results with experimental data on a 316L L-PBF demonstrates the robustness of the proposed approach and explains the negligible effect of the defect morphology compared to its size.

Keywords: Finite Element, Non-local approach, Multiaxial fatigue criteria, Defects, High Cycle Fatigue

**1. Introduction**

High Cycle Fatigue (HCF) behaviour of metallic material with defects (cast alloys, Additively Manufactured (AM) alloys, ...) has been a critical issue for industrial applications in the past decades [11, 15, 36]. Such defects are in many cases identified as the root cause of crack initiation leading to failure and can significantly decrease a material's fatigue strength [3, 43, 49].

The origin of defects is nevertheless diverse and often unique to each industrial application. They may be induced by the manufacturing process (Lack of Fusion (LoF) pores, gas pores, microshrinkage pores,...) or by external environmental factors (scratches, corrosion pits, ...) [14, 22, 40]. Different types of defects usually lead to distinct defect populations in terms of shape and size. Depending on the type of coexisting defects, the variability in the defect population of a mechanical part can be significant.

In addition, the presence of defects in a material induces local variations on the stress field. This translates into higher than expected values as consequence of stress concentration, a multiaxial state, high value gradients and a complex 3D distribution. The consideration of the surrounding local microstructure, the texture, and the residual stresses, also related to the presence of defects, makes it difficult to understand which and how much each one of these parameters may alter the fatigue strength.

Historically, and under uniaxial loadings, Linear Elastic Fracture Mechanics (LEFM) models are a classic way to deal with defects. Only their size and position (external or internal) are taken into account and they are seen as pre-existing cracks. The crack is considered as propagating if the Stress Intensity Factor (SIF) induced by a mechanical loading exceeds a given threshold. The fatigue strength is considered as the minimal stress to make at least one of the defects to propagate without crack arrest. However, such models lead to the most conservative predictions (since defects are considered as cracks) and do not take into account the 3D defect's shape, which could play a significant role depending on the material and the defect size [4, 2, 41]. These models are also often difficult to apply for complex multiaxial loadings.

In contrast, a continuum mechanics approach, via the use of Finite Element Method (FEM) for instance, can also be envisioned but implies the use of relevant tools to take into account all the features of fatigue damage around defects. Crystal Plasticity Finite Element Method (CPFEM) is likely the most accurate method since both the effect of the defect and the local microstructure are taken into account [18, 20, 29].
## Nomenclature

<table><tr><td>area</td><td>Projected area of the defect perpendicularly to the loading direction</td></tr><tr><td>area<sub>z</sub></td><td>Projected area of the defect perpendicularly to e<sub>z</sub></td></tr><tr><td>E, ν</td><td>Young's modulus and Poisson's coefficient</td></tr><tr><td>e<sub>x</sub>, e<sub>y</sub>, e<sub>z</sub></td><td>Axes of the orthonormal base</td></tr><tr><td>f</td><td>Fatigue test frequency</td></tr><tr><td>fr<sub>y</sub>, fr<sub>z</sub></td><td>Semi-ellipsoid aspect ratios</td></tr><tr><td>K<sub>DV</sub></td><td>Dang Van's stress concentration</td></tr><tr><td>K<sub>t</sub></td><td>Geometrical stress concentration</td></tr><tr><td>K<sub>zz</sub></td><td>Stress concentration for a load parallel to e<sub>z</sub></td></tr><tr><td>ΔK<sub>t</sub></td><td>Stress intensity factor, its threshold and long crack threshold</td></tr><tr><td>ΔK<sub>th,LC</sub></td><td></td></tr><tr><td>r<sub>x</sub>, r<sub>y</sub>, r<sub>z</sub></td><td>Semi-ellipsoid radii</td></tr><tr><td>R</td><td>Stress Ratio</td></tr><tr><td>R<sup>*</sup></td><td>Radius of the non-local sphere Ω</td></tr><tr><td>T<sub>a</sub></td><td>Macroscopic shear stress amplitude fatigue limit</td></tr><tr><td>U</td><td>Displacement vector</td></tr><tr><td>v<sub>i</sub></td><td>Volume of the element “i”</td></tr><tr><td>V</td><td>Volume of the non-local sphere Ω</td></tr><tr><td>α, β</td><td>Dang Van's hydrostatic parameter and Dang Van's stress criterion</td></tr><tr><td>σ</td><td>Mesoscopic stress tensor</td></tr><tr><td>Σ</td><td>Macroscopic stress tensor</td></tr><tr><td>σ<sub>DV</sub></td><td>Dang Van's stress</td></tr><tr><td>σ<sup>R*</sup><sub>DV</sub></td><td>Dang Van's stress mean on a non-local sphere with a R<sup>*</sup> radius</td></tr><tr><td>σ<sub>DV,i</sub></td><td>Dang Van's stress of the element “i”</td></tr><tr><td>σ<sup>R*</sup><sub>DV,i</sub></td><td>Mean Dang Van's stress of the element “i” on a non-local sphere with a R<sup>*</sup> radius</td></tr><tr><td>σ<sub>DV,0</sub></td><td>Plain Dang Van's stress (without defects)</td></tr><tr><td>σ<sub>DV,∞</sub></td><td>Dang Van's stress far from the defect</td></tr><tr><td>σ<sub>H</sub></td><td>Mesoscopic hydrostatic stress</td></tr><tr><td>σ<sup>R*</sup><sub>a</sub></td><td>Mesoscopic uniaxial stress amplitude fatigue limit with a non-local sphere with a R<sup>*</sup> radius</td></tr><tr><td>σ<sub>a,0</sub></td><td>Mesoscopic uniaxial plain stress amplitude fatigue limit (without defects)</td></tr><tr><td>σ<sub>∞</sub></td><td>Mesoscopic uniaxial plain stress amplitude far from the defect</td></tr><tr><td>Δσ, Δσ<sub>th,0</sub></td><td>Fatigue stress range and its plain threshold (without defect)</td></tr><tr><td>Σ<sub>a</sub></td><td>Macroscopic uniaxial stress amplitude fatigue limit</td></tr><tr><td>Ω</td><td>Non-local sphere domain</td></tr><tr><td>P</td><td>Plane maximizing the Dang Van stress in the defect vicinity</td></tr><tr><td>τ</td><td>Mesoscopic shear stress</td></tr><tr><td>τ<sup>R*</sup><sub>a</sub></td><td>Mesoscopic shear stress amplitude fatigue limit with a non-local sphere with a R<sup>*</sup> radius</td></tr><tr><td>τ<sub>a,0</sub></td><td>Mesoscopic shear plain stress amplitude fatigue limit (without defects)</td></tr></table>

## Acronyms

<table><tr><td>AM</td><td>Additive Manufacturing</td></tr><tr><td>CPFEM</td><td>Crystal Plasticity Finite Element Method</td></tr><tr><td>EBSD</td><td>Electron BackScatter Diffraction</td></tr><tr><td>FEM</td><td>Finite Element Method</td></tr><tr><td>FIP</td><td>Fatigue Indicators Parameter</td></tr><tr><td>HCF</td><td>High Cycle Fatigue</td></tr><tr><td>HSV</td><td>Highly Stressed Volume</td></tr><tr><td>LEFM</td><td>Linear Elastic Fracture Mechanics</td></tr><tr><td>L-PBF</td><td>Laser Powder Bed Fusion</td></tr><tr><td>LoF</td><td>Lack of Fusion</td></tr><tr><td>SEM</td><td>Scanning Electron Microscope</td></tr><tr><td>SIF</td><td>Stress Intensity Factor</td></tr></table>
## Nomenclature

<table><tr><td>area</td><td>Projected area of the defect perpendicularly to the loading direction</td></tr><tr><td>area<sub>z</sub></td><td>Projected area of the defect perpendicularly to e<sub>z</sub></td></tr><tr><td>E, ν</td><td>Young's modulus and Poisson's coefficient</td></tr><tr><td>e<sub>x</sub>, e<sub>y</sub>, e<sub>z</sub></td><td>Axes of the orthonormal base</td></tr><tr><td>f</td><td>Fatigue test frequency</td></tr><tr><td>fr<sub>y</sub>, fr<sub>z</sub></td><td>Semi-ellipsoid aspect ratios</td></tr><tr><td>K<sub>DV</sub></td><td>Dang Van's stress concentration</td></tr><tr><td>K<sub>t</sub></td><td>Geometrical stress concentration</td></tr><tr><td>K<sub>zz</sub></td><td>Stress concentration for a load parallel to e<sub>z</sub></td></tr><tr><td>ΔK<sub>t</sub></td><td>Stress intensity factor, its threshold and long crack threshold</td></tr><tr><td>ΔK<sub>th,LC</sub></td><td></td></tr><tr><td>r<sub>x</sub>, r<sub>y</sub>, r<sub>z</sub></td><td>Semi-ellipsoid radii</td></tr><tr><td>R</td><td>Stress Ratio</td></tr><tr><td>R*</td><td>Radius of the non-local sphere Ω</td></tr><tr><td>T<sub>a</sub></td><td>Macroscopic shear stress amplitude fatigue limit</td></tr><tr><td>U</td><td>Displacement vector</td></tr><tr><td>v<sub>i</sub></td><td>Volume of the element “i”</td></tr><tr><td>V</td><td>Volume of the non-local sphere Ω</td></tr><tr><td>α, β</td><td>Dang Van's hydrostatic parameter and Dang Van's stress criterion</td></tr><tr><td>σ</td><td>Mesoscopic stress tensor</td></tr><tr><td>Σ</td><td>Macroscopic stress tensor</td></tr><tr><td>σ<sub>DV</sub></td><td>Dang Van's stress</td></tr><tr><td>σ<sup>R*</sup><sub>DV</sub></td><td>Dang Van's stress mean on a non-local sphere with a R* radius</td></tr><tr><td>σ<sub>DV,i</sub></td><td>Dang Van's stress of the element “i”</td></tr><tr><td>σ<sup>R*</sup><sub>DV,i</sub></td><td>Mean Dang Van's stress of the element “i” on a non-local sphere with a R* radius</td></tr><tr><td>σ<sub>DV,0</sub></td><td>Plain Dang Van's stress (without defects)</td></tr><tr><td>σ<sub>DV,∞</sub></td><td>Dang Van's stress far from the defect</td></tr><tr><td>σ<sub>H</sub></td><td>Mesoscopic hydrostatic stress</td></tr><tr><td>σ<sup>R*</sup><sub>a</sub></td><td>Mesoscopic uniaxial stress amplitude fatigue limit with a non-local sphere with a R* radius</td></tr><tr><td>σ<sub>a,0</sub></td><td>Mesoscopic uniaxial plain stress amplitude fatigue limit (without defects)</td></tr><tr><td>σ<sub>∞</sub></td><td>Mesoscopic uniaxial plain stress amplitude far from the defect</td></tr><tr><td>Δσ, Δσ<sub>th,0</sub></td><td>Fatigue stress range and its plain threshold (without defect)</td></tr><tr><td>Σ<sub>a</sub></td><td>Macroscopic uniaxial stress amplitude fatigue limit</td></tr><tr><td>Ω</td><td>Non-local sphere domain</td></tr><tr><td>P</td><td>Plane maximizing the Dang Van stress in the defect vicinity</td></tr><tr><td>τ</td><td>Mesoscopic shear stress</td></tr><tr><td>τ<sup>R*</sup><sub>a</sub></td><td>Mesoscopic shear stress amplitude fatigue limit with a non-local sphere with a R* radius</td></tr><tr><td>τ<sub>a,0</sub></td><td>Mesoscopic shear plain stress amplitude fatigue limit (without defects)</td></tr></table>

## Acronyms

<table><tr><td>AM</td><td>Additive Manufacturing</td></tr><tr><td>CPFEM</td><td>Crystal Plasticity Finite Element Method</td></tr><tr><td>EBSD</td><td>Electron BackScatter Diffraction</td></tr><tr><td>FEM</td><td>Finite Element Method</td></tr><tr><td>FIP</td><td>Fatigue Indicators Parameter</td></tr><tr><td>HCF</td><td>High Cycle Fatigue</td></tr><tr><td>HSV</td><td>Highly Stressed Volume</td></tr><tr><td>LEFM</td><td>Linear Elastic Fracture Mechanics</td></tr><tr><td>L-PBF</td><td>Laser Powder Bed Fusion</td></tr><tr><td>LoF</td><td>Lack of Fusion</td></tr><tr><td>SEM</td><td>Scanning Electron Microscope</td></tr><tr><td>SIF</td><td>Stress Intensity Factor</td></tr></table>
However, CPFEM models can be difficult to use, firstly because of the large number of material characteri- zation prerequisites as model inputs (EBSD, material crystal behaviour, etc...), secondly, because of the high computational costs due to the problem complexity, particularly for 3D models.

In order to infer a fatigue strength based on the FEM resolution, multiaxial Fatigue Indicators Parameters (FIPs) must be computed. These parameters refer to mechanical quantities relevant for describing fatigue crack driving force (i.e. principal stress, Von Mises' stress, Crossland's criterion, ...). To avoid mesh artifacts and to consider FIPs values at microstructure features length scale (grain size, slip band width, etc...), applying a non-local regularization by averaging FIPs over a finite volume is a classic solution used in many recent works [10, 21, 23, 44, 48]. This regularization could be related to the critical distance theory introduced by Taylor [47, 17], which could be applied in 0D, 1D, 2D or 3D.

Previous research works have indeed tackled the “defect problem” through a CPFEM, multiaxial, non-local approach. Liang et al. [26] numerically investigated the effect of the microstructure, roughness and defects on the fatigue strength under uniaxial loading. The authors show that the effect of defect is predominant before both roughness and microstructure.

Similarly, Guerchais et al. [20] studied the effects of the size and shape of 2D defects on the fatigue strength under uniaxial loading. It was found that increasing the defect stress concentration (by increasing its acuity) leads to a decrease in terms of fatigue strength first, then to a slight increase. The bigger the defect, the more important is the role played by the defect.

In this work, the heterogeneous nature of the microstructure is not considered since the effect of defect seems predominant [26, 20]. Homogeneous material characteristics are introduced and the focus is put on the role of an individual defect. The objective of this paper is to propose a comprehensive study on the effect of defect shape and size on the fatigue strength through a multiaxial fatigue criterion and a non-local approach. The scale effect is not tackled. A large range of size and shape is investigated under fully alternate uniaxial (i.e. tension) and pure shear loadings. Since this paper aims to tackle the effect of defect from a general point of view (i.e. relevant for many alloys and processes), regular geometries are used. The influence of the characteristic length introduced by the non-local approach is discussed. The effect of the loading mode is also investigated and dealt with. Simulation results are then compared to experimental data for L-PBF 316L [31]. Finally, the effect of the plasticity on the predictions is discussed.

## 2. Finite element modeling

Defects studied in this paper are surface defects, which are known to be more detrimental than internal defects of the same size [43, 1, 34]. The effect of the defect geometry (size and morphology) under uniaxial (i.e. tension) or pure shear loading on the fatigue strength is investigated using a large number of finite element simulations. These cases are described in this section.

### 2.1. Geometrical defect modeling

In addition to the defect size, there are a lot of parameters which could be used to describe a defect morphology. For instance, the defect's depth, thickness, sphericity and orientation which lead to potential stress concentration might be used to represent realistic defects.

In order to observe the effect of critical defects morphologies, we chose to focus only on the effect of two main parameters: (i) the defect's depth and (ii) its thickness.

This choice is motivated by three principal reasons. First, to somewhat reduce the number of simulations by focusing primarily on parameters shown to affect significantly the fatigue strength in the literature [20, 42]. Second, to compare simulation and experimental results from a previous study [31], where different types of defects can be distinguished using these two parameters. Third, using a non-local method tends to make disappear the microgeometry effects. Then, features significantly smaller than the non-local characteristic length have no effects.

Defects were modeled by semi-ellipsoids with different aspect ratios and embedded in the volume as shown in figure 1. Each ellipsoid can be described by three radii ($r_x, r_y, r_z$) which correspond to the three half axes of the ellipsoid in the global coordinates system. The ellipsoid is not tilted so no angular parameters needed to be defined. Each ellipsoid can be thus described by the equation system 1.

$$ \left\{ \begin{array}{l} r_x \\ r_y = fr_y \times r_x \\ r_z = fr_z \times r_x \end{array} \right. \qquad (1) $$
However, CPFEM models can be difficult to use, firstly because of the large number of material characteri- zation prerequisites as model inputs (EBSD, material crystal behaviour, etc...), secondly, because of the high computational costs due to the problem complexity, particularly for 3D models.

In order to infer a fatigue strength based on the FEM resolution, multiaxial Fatigue Indicators Parameters (FIPs) must be computed. These parameters refer to mechanical quantities relevant for describing fatigue crack driving force (i.e. principal stress, Von Mises' stress, Crossland's criterion, ...). To avoid mesh artifacts and to consider FIPs values at microstructure features length scale (grain size, slip band width, etc...), applying a non-local regularization by averaging FIPs over a finite volume is a classic solution used in many recent works [10, 21, 23, 44, 48]. This regularization could be related to the critical distance theory introduced by Taylor [47, 17], which could be applied in 0D, 1D, 2D or 3D.

Previous research works have indeed tackled the “defect problem” through a CPFEM, multiaxial, non-local approach. Liang et al. [26] numerically investigated the effect of the microstructure, roughness and defects on the fatigue strength under uniaxial loading. The authors show that the effect of defect is predominant before both roughness and microstructure.

Similarly, Guerchais et al. [20] studied the effects of the size and shape of 2D defects on the fatigue strength under uniaxial loading. It was found that increasing the defect stress concentration (by increasing its acuity) leads to a decrease in terms of fatigue strength first, then to a slight increase. The bigger the defect, the more important is the role played by the defect.

In this work, the heterogeneous nature of the microstructure is not considered since the effect of defect seems predominant [26, 20]. Homogeneous material characteristics are introduced and the focus is put on the role of an individual defect. The objective of this paper is to propose a comprehensive study on the effect of defect shape and size on the fatigue strength through a multiaxial fatigue criterion and a non-local approach. The scale effect is not tackled. A large range of size and shape is investigated under fully alternate uniaxial (i.e. tension) and pure shear loadings. Since this paper aims to tackle the effect of defect from a general point of view (i.e. relevant for many alloys and processes), regular geometries are used. The influence of the characteristic length introduced by the non-local approach is discussed. The effect of the loading mode is also investigated and dealt with. Simulation results are then compared to experimental data for L-PBF 316L [31]. Finally, the effect of the plasticity on the predictions is discussed.

## 2. Finite element modeling

Defects studied in this paper are surface defects, which are known to be more detrimental than internal defects of the same size [43, 1, 34]. The effect of the defect geometry (size and morphology) under uniaxial (i.e. tension) or pure shear loading on the fatigue strength is investigated using a large number of finite element simulations. These cases are described in this section.

### 2.1. Geometrical defect modeling

In addition to the defect size, there are a lot of parameters which could be used to describe a defect morphology. For instance, the defect's depth, thickness, sphericity and orientation which lead to potential stress concentration might be used to represent realistic defects.

In order to observe the effect of critical defects morphologies, we chose to focus only on the effect of two main parameters: (i) the defect's depth and (ii) its thickness.

This choice is motivated by three principal reasons. First, to somewhat reduce the number of simulations by focusing primarily on parameters shown to affect significantly the fatigue strength in the literature [20, 42]. Second, to compare simulation and experimental results from a previous study [31], where different types of defects can be distinguished using these two parameters. Third, using a non-local method tends to make disappear the microgeometry effects. Then, features significantly smaller than the non-local characteristic length have no effects.

Defects were modeled by semi-ellipsoids with different aspect ratios and embedded in the volume as shown in figure 1. Each ellipsoid can be described by three radii ($r_x, r_y, r_z$) which correspond to the three half axes of the ellipsoid in the global coordinates system. The ellipsoid is not tilted so no angular parameters needed to be defined. Each ellipsoid can be thus described by the equation system 1.

$$ \left\{ \begin{array}{l} r_x \\ r_y = fr_y \times r_x \\ r_z = fr_z \times r_x \end{array} \right. \qquad (1) $$
Based on the two planar symmetries (0, $e_x$, $e_y$) and (0, $e_y$, $e_z$), the model was reduced to a quarter of the geometry under uniaxial loading. These symmetries are not kept under shear, the full 3D model was used in this case.

<figure><img src="image_4.png" /><figcaption>Figure 1: Ellipsoids geometry parameters.</figcaption></figure>

Different models were built using the open-source software FreeCAD and the parametric space employed is summarized in table 1. A set of 120 geometries for both uniaxial and shear cases was built. We chose to follow the Murakami's definition [34] to express the defect size as the square root of its projected area perpendicularly to the loading direction under a uniaxial load (along the direction $e_z$, see figure 2). The semi-ellipsoid area is given by the equation 2.

$$area_z = \frac{\pi r_x r_y}{2} \qquad (2)$$

This size was (with $fr_y$ and $fr_z$) one of the three inputs used to generate the models. A size range from 10 to 500 µm (see table 1) is fixed for comparison purpose later in the paper (subsection 4.3). The different defect shapes for the uniaxial mode are illustrated on figure 3. The same defect shapes population were introduced under shear loading mode.

<figure><img src="image_5.png" /><figcaption>Figure 2: Murakami's definition [34] of the defect size as the square root of the area of the defect projected on a plane perpendicular to the maximum tensile stress direction.</figcaption></figure>

It is worth noting that if this size definition may well represents the defect under uniaxial loads, this is not the case for the shear cases. In that case, this defect size was only used for parametric purposes. A more appropriate definition will be introduced in this paper and used in the analysis.

For all considered geometries, a mesh was generated using GMSH, a second open-source software. A minimal mesh size, equal to 1% of the defect size $\sqrt{area_z}$ (as defined in equation 2), was assigned to the nodes of the ellipsoid. Other nodes sizes were set to a maximal mesh size of 500 µm. First order tetrahedral elements were used. An example of a generated mesh is shown figure 4. It is worth noting that a twice finer
Based on the two planar symmetries ($0, e_x, e_y$) and ($0, e_y, e_z$), the model was reduced to a quarter of the geometry under uniaxial loading. These symmetries are not kept under shear, the full 3D model was used in this case.

<figure><img src="image_3.png" /><figcaption>Figure 1: Ellipsoids geometry parameters.</figcaption></figure>

Different models were built using the open-source software FreeCAD and the parametric space employed is summarized in table 1. A set of 120 geometries for both uniaxial and shear cases was built. We chose to follow the Murakami's definition [34] to express the defect size as the square root of its projected area perpendicularly to the loading direction under a uniaxial load (along the direction $e_z$, see figure 2). The semi-ellipsoid area is given by the equation 2.

$$area_z = \frac{\pi r_x r_y}{2} \qquad (2)$$

This size was (with $fr_y$ and $fr_z$) one of the three inputs used to generate the models. A size range from 10 to 500 µm (see table 1) is fixed for comparison purpose later in the paper (subsection 4.3). The different defect shapes for the uniaxial mode are illustrated on figure 3. The same defect shapes population were introduced under shear loading mode.

<figure><img src="image_4.png" /><figcaption>Figure 2: Murakami's definition [34] of the defect size as the square root of the area of the defect projected on a plane perpendicular to the maximum tensile stress direction.</figcaption></figure>

It is worth noting that if this size definition may well represents the defect under uniaxial loads, this is not the case for the shear cases. In that case, this defect size was only used for parametric purposes. A more appropriate definition will be introduced in this paper and used in the analysis.

For all considered geometries, a mesh was generated using GMSH, a second open-source software. A minimal mesh size, equal to 1% of the defect size $\sqrt{area_z}$ (as defined in equation 2), was assigned to the nodes of the ellipsoid. Other nodes sizes were set to a maximal mesh size of 500 µm. First order tetrahedral elements were used. An example of a generated mesh is shown figure 4. It is worth noting that a twice finer
<figure><img src="image_6.png" /><figcaption>Figure 3: The 12 different defect morphologies introduced in the simulation framework for the uniaxial loading mode. In this example, the defect size was set to 100 µm in terms of $\sqrt{area_z}$</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Figure 3: The 12 different defect morphologies introduced in the simulation framework for the uniaxial loading mode. In this example, the defect size was set to 100 µm in terms of $\sqrt{area_z}$</figcaption></figure>
<table><caption>Table 1: Geometric parametric space of the study</caption><thead><tr><th>Case</th><th>Parameter</th><th>Values</th></tr></thead><tbody><tr><td rowspan="3">Uniaxial</td><td>√<span style="text-decoration: overline;">area<sub>z</sub></span> [μm]</td><td>10, 20, 30, 50, 70, 100, 150, 200, 300, 500</td></tr><tr><td>fr<sub>y</sub></td><td>0.5, 1, 2</td></tr><tr><td>fr<sub>z</sub></td><td>0.15, 0.5, 1, 1.5</td></tr><tr><td rowspan="3">Shear</td><td>√<span style="text-decoration: overline;">area<sub>z</sub></span> [μm]</td><td>10, 20, 30, 50, 70, 100, 150, 200, 300, 500</td></tr><tr><td>fr<sub>y</sub></td><td>0.5, 1, 2</td></tr><tr><td>fr<sub>z</sub></td><td>0.15, 0.5, 1, 1.5</td></tr></tbody></table>

mesh leads to identical results, highlighting that the mesh size in the defect neighborhood is fine enough to provide stabilized and consistent results.

<figure><img src="image_7.png" /><figcaption>Figure 4: Example of a meshed geometry (shear, $\sqrt{area_z}$ =100 µm, $fr_y$ =1 and $fr_z$ =1.5).</figcaption></figure>

Once the geometry defined, the material behaviour was set.

## 2.2. Material behaviour properties

In the HCF regime, the material is macroscopically in its elastic domain (i.e. the stress level is lower than the yield stress). However, locally, the stress level around defects could be higher than the yield stress, then, plasticity is induced.

In this paper, we first consider the material as linearly elastic for sake of simplicity. This is supported by the fact that a non-local approach (which is detailed further in this paper) is applied. Still, a part of the discussion deals with this approximation and the effect of the plasticity is introduced. The Young's modulus was set to 225 GPa and is taken from a previous study [31]. The Poisson's ratio is set to 0.3, which is a typical value for steels.

To fully define the different models of the case study, boundary and loading conditions were applied as described as follow.

## 2.3. Boundaries and loading conditions

To discuss the effect of defects under different loading modes, two cases corresponding to uniaxial and shear loadings conditions were applied. Figure 5 illustrates these two loading modes.

For the uniaxial case, three symmetry conditions were used on three planes to have a fully defined problem. The loading was applied as a stress along $e_z$ on the surface opposite to the fracture surface.

For the shear case, a symmetry was applied on the opposite face of the free surface. Then, $U_x$ and $U_z$ displacements were equally applied on the top and right surfaces respectively. The displacements of the opposite surfaces were locked to apply pure shear.

We clarify that the fatigue criterion used in this work takes into account the stress state's path over a cycle and not only the maximal stress value. More precisely, for in phase loadings (which is the case for the simple loadings in this work), the stress state's bounds need to be defined in order to compute the mesoscopic shear stress which is an input of the fatigue criterion.

For the uniaxial case, the stress load was set from 0 to 1 first, and then to -1 since we chose to compare our model predictions with the experimental data at $R = -1$.
<table><caption>Table 1: Geometric parametric space of the study</caption><thead><tr><th>Case</th><th>Parameter</th><th>Values</th></tr></thead><tbody><tr><td rowspan="3">Uniaxial</td><td>√area<sub>z</sub> [µm]</td><td>10, 20, 30, 50, 70, 100, 150, 200, 300, 500</td></tr><tr><td>fr<sub>y</sub></td><td>0.5, 1, 2</td></tr><tr><td>fr<sub>z</sub></td><td>0.15, 0.5, 1, 1.5</td></tr><tr><td rowspan="3">Shear</td><td>√area<sub>z</sub> [µm]</td><td>10, 20, 30, 50, 70, 100, 150, 200, 300, 500</td></tr><tr><td>fr<sub>y</sub></td><td>0.5, 1, 2</td></tr><tr><td>fr<sub>z</sub></td><td>0.15, 0.5, 1, 1.5</td></tr></tbody></table>

mesh leads to identical results, highlighting that the mesh size in the defect neighborhood is fine enough to provide stabilized and consistent results.

<figure><img src="image_6.png" /><figcaption>Figure 4: Example of a meshed geometry (shear, $\sqrt{area_z}$ =100 µm, $fr_y$ =1 and $fr_z$ =1.5).</figcaption></figure>

Once the geometry defined, the material behaviour was set.

## 2.2. Material behaviour properties

In the HCF regime, the material is macroscopically in its elastic domain (i.e. the stress level is lower than the yield stress). However, locally, the stress level around defects could be higher than the yield stress, then, plasticity is induced.

In this paper, we first consider the material as linearly elastic for sake of simplicity. This is supported by the fact that a non-local approach (which is detailed further in this paper) is applied. Still, a part of the discussion deals with this approximation and the effect of the plasticity is introduced. The Young's modulus was set to 225 GPa and is taken from a previous study [31]. The Poisson's ratio is set to 0.3, which is a typical value for steels.

To fully define the different models of the case study, boundary and loading conditions were applied as described as follow.

## 2.3. Boundaries and loading conditions

To discuss the effect of defects under different loading modes, two cases corresponding to uniaxial and shear loadings conditions were applied. Figure 5 illustrates these two loading modes.

For the uniaxial case, three symmetry conditions were used on three planes to have a fully defined problem. The loading was applied as a stress along $e_z$ on the surface opposite to the fracture surface.

For the shear case, a symmetry was applied on the opposite face of the free surface. Then, $U_x$ and $U_z$ displacements were equally applied on the top and right surfaces respectively. The displacements of the opposite surfaces were locked to apply pure shear.

We clarify that the fatigue criterion used in this work takes into account the stress state's path over a cycle and not only the maximal stress value. More precisely, for in phase loadings (which is the case for the simple loadings in this work), the stress state's bounds need to be defined in order to compute the mesoscopic shear stress which is an input of the fatigue criterion.

For the uniaxial case, the stress load was set from 0 to 1 first, and then to -1 since we chose to compare our model predictions with the experimental data at $R = -1$.
<figure><img src="image_8.png" /><figcaption>Figure 5: Displacement and loading conditions applied to the geometry (a) under uniaxial load and (b) for pure shear.</figcaption></figure>

Since displacements of +1 mm then -1 mm were set for the shear case instead of a stress (and because the material behaviour is linear and elastic), a stress normalization was applied during the post-treatment ($\sigma_{xz} = \pm 1$ MPa).

At this point of the modeling, finite element simulations were carried out using Abaqus solver and the final stress-strain state was obtained. Results were then post-treated using a HCF fatigue criterion which is described in the next subsection.

## 2.4. Non-local fatigue criterion

Even in the uniaxial loading condition, the stress state at the defect vicinity is multiaxial. To take into account this complex stress field, we used the Dang Van criterion (see equation 3) which is particularly efficient in HCF [9].

$$ \sigma_{DV} = \max_{\underline{n}} \left\{ \max_{t \in T} \left[ \|\underline{\tau}(t, \underline{n})\| + \alpha \sigma_H(t) \right] \right\} \le \beta \quad (3) $$

$\alpha$ and $\beta$ are intrinsic material parameters, $\underline{\tau}(t, \underline{n})$ is the mesoscopic shear stress and $\sigma_H(t)$ is the mesoscopic hydrostatic stress. The $\alpha$ value was set to 0.4 which is relevant for steels [25]. The equivalent Dang Van stress $\sigma_{DV,i}$ was then computed for each element “i” of the FE model at the element centroid.

In fatigue, the crack initiation may occur if the stress field in a given volume is higher than a given critical level. Then, instead of dealing with the local maximal equivalent Dang Van stress, we chose to apply a non-local approach. For each element “i”, we computed its Dang Van mean value $\sigma_{DV,i}^*$ on a sphere $\Omega$ of a radius $R^*$ centered on its centroid (see equation 4).

$$ \left\{ \begin{array}{l} V = \sum_{\Omega} v_j \\ \sigma_{DV,i}^* = \frac{1}{V} \sum_{\Omega} v_j \sigma_{DV,j} \end{array} \right. \qquad (4) $$

with “j” the index of all the elements within the sphere $\Omega$ and $v_j$ the volume of the element “j”. It is worth noting that if the mesh size is smaller than $R^*$, results become independent of the mesh.

In this study, we chose to compute non-local Dang Van stresses for multiple values of $R^*$: 1, 2, 3, 4, 5, 10, 15, 20, 30, 40, 50, 60, 70, 80, 90, 100, 125, 150, 175, 200 and 250 $\mu m$ for the uniaxial and pure shear cases. This interval is chosen to contain typical microstructural features sizes for steels (such as the grain size or subgrains) which are characteristic lengths for these materials [6, 27, 37].

To sum up this section about the modeling, first, defects are modeled by surface simple semi-ellipsoids whose sizes, acuities and depths vary. Second, an elastic finite element computation is performed. Third, the Dang Van’s multiaxial fatigue criterion is computed. Fourth, a non-local method is applied, assigning to
<figure><img src="image_7.png" /><figcaption>Figure 5: Displacement and loading conditions applied to the geometry (a) under uniaxial load and (b) for pure shear.</figcaption></figure>

Since displacements of +1 mm then -1 mm were set for the shear case instead of a stress (and because the material behaviour is linear and elastic), a stress normalization was applied during the post-treatment ($\sigma_{xz} = \pm 1$ MPa).

At this point of the modeling, finite element simulations were carried out using Abaqus solver and the final stress-strain state was obtained. Results were then post-treated using a HCF fatigue criterion which is described in the next subsection.

## 2.4. Non-local fatigue criterion

Even in the uniaxial loading condition, the stress state at the defect vicinity is multiaxial. To take into account this complex stress field, we used the Dang Van criterion (see equation 3) which is particularly efficient in HCF [9].

$$ \sigma_{DV} = \max_{\underline{n}} \left\{ \max_{t \in T} \left[ \| \underline{\tau}(t, \underline{n}) \| + \alpha \sigma_H(t) \right] \right\} \le \beta \quad (3) $$

$\alpha$ and $\beta$ are intrinsic material parameters, $\underline{\tau}(t, \underline{n})$ is the mesoscopic shear stress and $\sigma_H(t)$ is the mesoscopic hydrostatic stress. The $\alpha$ value was set to 0.4 which is relevant for steels [25]. The equivalent Dang Van stress $\sigma_{DV,i}$ was then computed for each element “i” of the FE model at the element centroid.

In fatigue, the crack initiation may occur if the stress field in a given volume is higher than a given critical level. Then, instead of dealing with the local maximal equivalent Dang Van stress, we chose to apply a non-local approach. For each element “i”, we computed its Dang Van mean value $\sigma_{DV,i}^*$ on a sphere $\Omega$ of a radius $R^*$ centered on its centroid (see equation 4).

$$ \left\{ \begin{array}{l} V = \sum_{\Omega} v_j \\ \sigma_{DV,i}^{R*} = \frac{1}{V} \sum_{\Omega} v_j \sigma_{DV,j} \end{array} \right. \quad (4) $$

with “j” the index of all the elements within the sphere $\Omega$ and $v_j$ the volume of the element “j”. It is worth noting that if the mesh size is smaller than $R^*$, results become independent of the mesh.

In this study, we chose to compute non-local Dang Van stresses for multiple values of $R^*$: 1, 2, 3, 4, 5, 10, 15, 20, 30, 40, 50, 60, 70, 80, 90, 100, 125, 150, 175, 200 and 250 $\mu m$ for the uniaxial and pure shear cases. This interval is chosen to contain typical microstructural features sizes for steels (such as the grain size or subgrains) which are characteristic lengths for these materials [6, 27, 37].

To sum up this section about the modeling, first, defects are modeled by surface simple semi-ellipsoids whose sizes, acuities and depths vary. Second, an elastic finite element computation is performed. Third, the Dang Van’s multiaxial fatigue criterion is computed. Fourth, a non-local method is applied, assigning to
every element the mean Dang Van's stress over a sphere $\Omega$ centered on it. It is worth noting that this type of approach is known to be efficient for fatigue applications [5, 20, 48]. Still, the implementation of such models could be difficult to handle, time consuming, and different choices of fatigue criterion or non-local method could be done.

Once this post-treatment was achieved, results were analyzed as described in the next section.

### 3. Fatigue strength assessments

In this section, results from the parametric study presented previously are shown. The process to convert the non-local fatigue criteria results is presented, then, the effects of a critical defect and the loading mode are highlighted.

#### 3.1. Evaluation of the fatigue strength

In order to evaluate the fatigue strength and because the loading stress magnitude was normalized, the maximal value of the Dang Van equivalent stress was compared to its theoretical value for a geometry without defect for each case. Since the theoretical value depends on the loading mode, the two cases are investigated in the two following parts.

##### 3.1.1. Uniaxial (R=-1)

If $R = -1$ and the loading is uniaxial, the macroscopic stress tensor in the volume could be written as :

$$ \underline{\underline{\Sigma}}(t) = \Sigma_a \sin(2\pi ft) \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} \qquad (5) $$

with $\Sigma_a$ the loading stress amplitude and $f$ the frequency. Then, for each instant of the cycle, equation 6 gives the values of mesoscopic hydrostatic and maximal shear stresses.

$$ \left\{ \begin{array}{l} \sigma_H(t) = \frac{1}{3} \Sigma_a \sin(2\pi ft) \\ \max_{\underline{n}} \| \underline{\underline{\tau}}(t, \underline{n}) \| = \frac{1}{2} \Sigma_a \sin(2\pi ft) \end{array} \right. \qquad (6) $$

$\Sigma_a$ is set to 1 for normalization purposes. Finally, the theoretical equivalent plain Dang Van stress could be written as :

$$ \sigma_{DV,0} = \left( \frac{1}{2} + \frac{\alpha}{3} \right) \qquad (7) $$

We can then predict different normalized fatigue limits for each cases with the following equation :

$$ \frac{\sigma_a^{R*}}{\sigma_{a,0}} = \left( \frac{\sigma_{DV}^{R*}}{\sigma_{DV,0}} \right) \qquad (8) $$

with $\sigma_{a,0}$, the fatigue strength of the defect-free material in fully reversed uniaxial mode, $\sigma_{DV}^{R*}$ is the maximal Dang Van's stress computed with a non-local sphere $\Omega$ of radius $R^*$ and $\sigma_a^{R*}$ correspond to the fatigue strength amplitude associated to the non-local Dang Van's stress.

##### 3.1.2. Pure shear (R=-1)

For pure shear, the same method is used. The macroscopic stress tensor in the volume could be written as :

$$ \underline{\underline{\Sigma}}(t) = T_a \sin(2\pi ft) \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{bmatrix} \qquad (9) $$

with $T_a$ the loading shear amplitude. The values of mesoscopic hydrostatic and maximal shear stresses are :

$$ \left\{ \begin{array}{l} \sigma_H(t) = 0 \\ \max_{\underline{n}} \| \underline{\underline{\tau}}(t, \underline{n}) \| = T_a \sin(2\pi ft) \end{array} \right. \qquad (10) $$
every element the mean Dang Van's stress over a sphere $\Omega$ centered on it. It is worth noting that this type of approach is known to be efficient for fatigue applications [5, 20, 48]. Still, the implementation of such models could be difficult to handle, time consuming, and different choices of fatigue criterion or non-local method could be done.

Once this post-treatment was achieved, results were analyzed as described in the next section.

### 3. Fatigue strength assessments

In this section, results from the parametric study presented previously are shown. The process to convert the non-local fatigue criteria results is presented, then, the effects of a critical defect and the loading mode are highlighted.

#### 3.1. Evaluation of the fatigue strength

In order to evaluate the fatigue strength and because the loading stress magnitude was normalized, the maximal value of the Dang Van equivalent stress was compared to its theoretical value for a geometry without defect for each case. Since the theoretical value depends on the loading mode, the two cases are investigated in the two following parts.

##### 3.1.1. Uniaxial ($R=-1$)

If $R = -1$ and the loading is uniaxial, the macroscopic stress tensor in the volume could be written as :

$$ \underline{\underline{\Sigma}}(t) = \Sigma_a \sin(2\pi ft) \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} \qquad (5) $$

with $\Sigma_a$ the loading stress amplitude and $f$ the frequency. Then, for each instant of the cycle, equation 6 gives the values of mesoscopic hydrostatic and maximal shear stresses.

$$ \left\{ \begin{aligned} \sigma_H(t) &= \frac{1}{3} \Sigma_a \sin(2\pi ft) \\ \max_{\underline{n}} \|\underline{\tau}(t, \underline{n})\| &= \frac{1}{2} \Sigma_a \sin(2\pi ft) \end{aligned} \right. \qquad (6) $$

$\Sigma_a$ is set to 1 for normalization purposes. Finally, the theoretical equivalent plain Dang Van stress could be written as :

$$ \sigma_{DV,0} = \left( \frac{1}{2} + \frac{\alpha}{3} \right) \qquad (7) $$

We can then predict different normalized fatigue limits for each cases with the following equation :

$$ \frac{\sigma_a^{R*}}{\sigma_{a,0}} = \left( \frac{\sigma_{DV}^{R*}}{\sigma_{DV,0}} \right) \qquad (8) $$

with $\sigma_{a,0}$, the fatigue strength of the defect-free material in fully reversed uniaxial mode, $\sigma_{DV}^{R*}$ is the maximal Dang Van's stress computed with a non-local sphere $\Omega$ of radius $R^*$ and $\sigma_a^{R*}$ correspond to the fatigue strength amplitude associated to the non-local Dang Van's stress.

##### 3.1.2. Pure shear ($R=-1$)

For pure shear, the same method is used. The macroscopic stress tensor in the volume could be written as :

$$ \underline{\underline{\Sigma}}(t) = T_a \sin(2\pi ft) \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{bmatrix} \qquad (9) $$

with $T_a$ the loading shear amplitude. The values of mesoscopic hydrostatic and maximal shear stresses are :

$$ \left\{ \begin{aligned} \sigma_H(t) &= 0 \\ \max_{\underline{n}} \|\underline{\tau}(t, \underline{n})\| &= T_a \sin(2\pi ft) \end{aligned} \right. \qquad (10) $$
with $T_a = 1$. The theoretical equivalent plain Dang Van stress $\sigma_{DV,0}$ is equal to 1 in pure shear mode (without defect).

The normalized fatigue limit is defined by :

$$ \frac{\tau_a^{R*}}{\tau_{a,0}} = \left( \frac{\sigma_{DV}^{R*}}{\sigma_{DV,0}} \right) \qquad (11) $$

with $\tau_{a,0}$, the fatigue strength of the defect-free material in fully reversed pure shear mode, $\tau_a^{R*}$ is the shear fatigue strength computed from the non-local approach (with a sphere of radius $R^*$).

In the following subsections, the presented results are all normalized in terms of stress.

## 3.2. Effect of the defect (Kitagawa-Takahashi diagram)

In HCF, the effect of surface defects is well known to be detrimental to mechanical parts [15, 43, 45]. Particularly, the defect size is observed to be the first order parameter influencing its criticality [16, 36, 39]. The results in this subsection are then represented in Kitagawa-Takahashi diagrams (fatigue limit vs defect size).

### 3.2.1. Definition of the defect size

As discussed in the subsection 2.1, the definition of the defect size given by Murakami [34] is not an obvious choice for the pure shear case. We chose to define the size for pure shear as the square root of the area of the defect projected on a plane maximizing the Dang Van criteria in the defect vicinity ($\mathcal{P}$). It is important to notice that this choice is arbitrarily but the authors believe that this indicator is relevant. More specifically, this indicator takes into account the multiaxial stress state near a defect, is defined non-locally (maximization over a plane) and seems adapted for complex defects' geometries.

The plane $\mathcal{P}$ was constrained to contain both the defect center and the $e_y$ direction (normal to the free surface). $\theta$ is the angle between $e_x$ and $\mathcal{P}$, $n(\theta)$ is the plane's normal and $OE_{i,c}$ the vector from the defect center to the element "i" centroid. Only the elements in the domain $\mathcal{S}$ are considered (i.e. in the defect vicinity). The equation 12 defines the problem solved to determine $\theta$.

$$ \left\{ \begin{array}{l} i \in \mathcal{S} \Leftrightarrow ||OE_{i,c}|| < 1.5 \times \max(r_x, r_y, r_z) \\ n(\theta) = [-\sin(\theta), 0, \cos(\theta)] \\ w_i = \exp[-|OE_{i,c} \cdot n(\theta)|] \\ \theta \mid \max_{\theta \in [0, \pi]} \{\sum_{i \in \mathcal{S}} w_i \sigma_{DV,i}\} \end{array} \right. \qquad (12) $$

The defect vicinity domain $\mathcal{S}$ is a sphere centered on the defect with a radius equal to 1.5 times the maximum radii (between $r_x$, $r_y$ and $r_z$). The exponential decrease applied on the distance between the element "i" and the plane $\mathcal{P}$ (computed by the scalar product between $n(\theta)$ and $OE_{i,c}$) in the equation was used to weigh the effect of the elements. The larger the distance, the smaller the attribute weight $w_i$. Finally, the sum $\sum_{i \in \mathcal{S}} w_i \sigma_{DV,i}$ is computed for $\theta \in [0, \pi]$ and the $\theta$ maximizing this score is used to compute the projected defect size. The figure 6 illustrates the position of the plane $\mathcal{P}$.

<figure><img src="image_9.png" /><figcaption>Figure 6: Maximal plane in terms of Dang Van criterion for pure shear as defined by equation 12.</figcaption></figure>

In the rest of this article, the defect size $\sqrt{area}$ refers to the square root of the projected defect area on the plane $\mathcal{P}$. The computation was only performed for pure shear since $\sqrt{area} = \sqrt{area_z}$ under uniaxial loading.
with $T_a = 1$. The theoretical equivalent plain Dang Van stress $\sigma_{DV,0}$ is equal to 1 in pure shear mode (without defect).

The normalized fatigue limit is defined by:

$$ \frac{\tau_a^{R*}}{\tau_{a,0}} = \left( \frac{\sigma_{DV}^{R*}}{\sigma_{DV,0}} \right) \qquad (11) $$

with $\tau_{a,0}$, the fatigue strength of the defect-free material in fully reversed pure shear mode, $\tau_a^{R*}$ is the shear fatigue strength computed from the non-local approach (with a sphere of radius $R^*$).

In the following subsections, the presented results are all normalized in terms of stress.

## 3.2. Effect of the defect (Kitagawa-Takahashi diagram)

In HCF, the effect of surface defects is well known to be detrimental to mechanical parts [15, 43, 45]. Particularly, the defect size is observed to be the first order parameter influencing its criticality [16, 36, 39]. The results in this subsection are then represented in Kitagawa-Takahashi diagrams (fatigue limit vs defect size).

### 3.2.1. Definition of the defect size

As discussed in the subsection 2.1, the definition of the defect size given by Murakami [34] is not an obvious choice for the pure shear case. We chose to define the size for pure shear as the square root of the area of the defect projected on a plane maximizing the Dang Van criteria in the defect vicinity ($P$). It is important to notice that this choice is arbitrarily but the authors believe that this indicator is relevant. More specifically, this indicator takes into account the multiaxial stress state near a defect, is defined non-locally (maximization over a plane) and seems adapted for complex defects' geometries.

The plane $P$ was constrained to contain both the defect center and the $e_y$ direction (normal to the free surface). $\theta$ is the angle between $e_x$ and $P$, $n(\theta)$ is the plane's normal and $OE_{i,c}$ the vector from the defect center to the element "i" centroid. Only the elements in the domain $S$ are considered (i.e. in the defect vicinity). The equation 12 defines the problem solved to determine $\theta$.

$$ \left\{ \begin{array}{l} i \in S \Leftrightarrow ||OE_{i,c}|| < 1.5 \times \max(r_x, r_y, r_z) \\ n(\theta) = [-\sin(\theta), 0, \cos(\theta)] \\ w_i = \exp[-|OE_{i,c} \cdot n(\theta)|] \\ \theta = \max_{\theta \in [0, \pi]} \{\sum_{i \in S} w_i \sigma_{DV,i}\} \end{array} \right. \qquad (12) $$

The defect vicinity domain $S$ is a sphere centered on the defect with a radius equal to 1.5 times the maximum radii (between $r_x$, $r_y$ and $r_z$). The exponential decrease applied on the distance between the element "i" and the plane $P$ (computed by the scalar product between $n(\theta)$ and $OE_{i,c}$) in the equation was used to weigh the effect of the elements. The larger the distance, the smaller the attribute weight $w_i$. Finally, the sum $\sum_{i \in S} w_i \sigma_{DV,i}$ is computed for $\theta \in [0, \pi]$ and the $\theta$ maximizing this score is used to compute the projected defect size. The figure 6 illustrates the position of the plane $P$.

<figure><img src="image_8.png" /><figcaption>Figure 6: Maximal plane in terms of Dang Van criterion for pure shear as defined by equation 12.</figcaption></figure>

In the rest of this article, the defect size $\sqrt{area}$ refers to the square root of the projected defect area on the plane $P$. The computation was only performed for pure shear since $\sqrt{area} = \sqrt{area_z}$ under uniaxial loading.
### 3.2.2. The depth effect

Figure 7 represents the effect of the defect morphology on the fatigue strength under uniaxial loading through different Kitagawa-Takahashi diagrams. Each of these 12 diagrams represents a unique defect morphology. The depth of the defect is controlled by $f_{ry}$ parameter. The higher $f_{ry}$, the deeper the defect.

Table 2 gives the value of the stress concentrations associated to the defect geometries under uniaxial loading. Stress concentrations are given following the load direction ($K_{zz}$) and in terms of Dang Van ($K_{DV}$). Equation 13 defines the value of these stress concentration indicators as follows:

$$ \begin{cases} K_{zz} = \max_i(\sigma_{zz,i})/\sigma_{zz,\infty} \\ K_{DV} = \max_i(\sigma_{DV,i})/\sigma_{DV,\infty} \end{cases} \qquad (13) $$

with "i" for all the elements, and "∞" referring to an element far from the defect. $\sigma_{zz,\infty} = 1$ for a unitary loading. $\sigma_{DV,\infty} = \alpha/3 + 1/2 = 0.633$ under uniaxial loading and $\sigma_{DV,\infty} = 1$ under pure shear. We can notice that $K_{zz}$ and $K_{DV}$ are close in value in table 2. This means that the degree of triaxiality under uniaxial loading is low.

<table><caption>Table 2: Stress concentrations associated to the different defect geometries under uniaxial loading.</caption><thead><tr><th>f<sub>ry</sub> = 0.5</th><th>f<sub>ry</sub> = 1</th><th>f<sub>ry</sub> = 2</th><th></th></tr></thead><tbody><tr><td>K<sub>zz</sub> = 6.84</td><td>K<sub>zz</sub> = 9.45</td><td>K<sub>zz</sub> = 11.60</td><td rowspan="2">f<sub>rz</sub> = 0.15</td></tr><tr><td>K<sub>DV</sub> = 6.88</td><td>K<sub>DV</sub> = 9.16</td><td>K<sub>DV</sub> = 11.46</td></tr><tr><td>K<sub>zz</sub> = 2.50</td><td>K<sub>zz</sub> = 3.46</td><td>K<sub>zz</sub> = 4.36</td><td rowspan="2">f<sub>rz</sub> = 0.5</td></tr><tr><td>K<sub>DV</sub> = 2.54</td><td>K<sub>DV</sub> = 3.49</td><td>K<sub>DV</sub> = 4.42</td></tr><tr><td>K<sub>zz</sub> = 1.65</td><td>K<sub>zz</sub> = 2.04</td><td>K<sub>zz</sub> = 2.48</td><td rowspan="2">f<sub>rz</sub> = 1</td></tr><tr><td>K<sub>DV</sub> = 1.67</td><td>K<sub>DV</sub> = 2.07</td><td>K<sub>DV</sub> = 2.51</td></tr><tr><td>K<sub>zz</sub> = 1.39</td><td>K<sub>zz</sub> = 1.63</td><td>K<sub>zz</sub> = 1.91</td><td rowspan="2">f<sub>rz</sub> = 1.5</td></tr><tr><td>K<sub>DV</sub> = 1.41</td><td>K<sub>DV</sub> = 1.64</td><td>K<sub>DV</sub> = 1.93</td></tr></tbody></table>

We can observe that globally, a defect is more detrimental in terms of fatigue strength when it is deeper than shallow. It could be explained by the stress concentration resulting of the geometry (see table 2).

We can also observe that the bigger the mean sphere $\Omega$ (controlled by $R^*$), the lower the effects of the depth and size of the defect (see figure 7). This is due to the mean (see equation 4) applied in the non-local approach which decreases the effect of the stress concentration and takes into account the stress gradient near the defect (i.e. the effect of the Highly Stressed Volume HSV [13]).

A finer analysis of the figure 7 reveals that a deeper defect (resulting in a higher stress concentration) is not always more detrimental than a shallow one. For instance, for $f_{rz} = 1.0$, $R^* = 100 \ \mu m$ and a defect size equal to 100 $\mu m$, the normalized fatigue strength is equal to 0.9 for $f_{ry} = 2.0$ while it is equal to 0.8 for $f_{ry} = 0.5$. This is due to the competition between the effect of stress concentration (local property) and the stress gradient (non-local property).

### 3.2.3. The thickness effect

Figure 7 also highlights the effect of the defect thickness on the fatigue strength. The defect thickness decreases when the parameter $f_{rz}$ drops (from top to bottom in the figure). The thicker the defect, the lower the stress concentration in terms of $K_{zz}$ and $K_{DV}$ (see table 2).

The same conclusions as for the effect of the defect depth could be drawn. The competition between the effect of the stress gradient and the stress concentration is observable. The stress gradient contribution is zero if $R^* = 0$, and it gets higher when $R^*$ increases. The effect of the defect acuity is then higher than the stress gradient for low $R^*$ and vice versa for high $R^*$.

Figure 8 represents the stress field in the defect vicinity for two defects thickness configurations and a given defect size (100 $\mu m$). The stress fields are shown for six different $R^*$. It can be noticed that a thin defect which creates a high stress concentration is more critical for small mean volume (figure 8.b.1) than a hemispherical one (figure 8.a.1). For bigger $R^*$, the opposite is observed: the hemispherical defect (figure 8.a.6) is more dangerous than the thin one (figure 8.b.6).

If the defect is small compared to $R^*$, its local maximal value (i.e. acuity) is smoothened by the mean applied in the non-local approach adopted here. On the contrary, if the defect is large compared to $R^*$, the mean is only applied on highly stressed elements, which is equivalent to classic local approach were the only parameter driving the defect criticality is the stress concentration (i.e. no effect of the defect size).
### 3.2.2. The depth effect

Figure 7 represents the effect of the defect morphology on the fatigue strength under uniaxial loading through different Kitagawa-Takahashi diagrams. Each of these 12 diagrams represents a unique defect morphology. The depth of the defect is controlled by $f_{ry}$ parameter. The higher $f_{ry}$, the deeper the defect.

Table 2 gives the value of the stress concentrations associated to the defect geometries under uniaxial loading. Stress concentrations are given following the load direction ($K_{zz}$) and in terms of Dang Van ($K_{DV}$). Equation 13 defines the value of these stress concentration indicators as follows:

$$ egin{cases} K_{zz} = \max_i(\sigma_{zz,i})/\sigma_{zz,\infty} \\ K_{DV} = \max_i(\sigma_{DV,i})/\sigma_{DV,\infty} \end{cases} \qquad (13) $$

with "i" for all the elements, and "∞" referring to an element far from the defect. $\sigma_{zz,\infty} = 1$ for a unitary loading. $\sigma_{DV,\infty} = \alpha/3 + 1/2 = 0.633$ under uniaxial loading and $\sigma_{DV,\infty} = 1$ under pure shear. We can notice that $K_{zz}$ and $K_{DV}$ are close in value in table 2. This means that the degree of triaxiality under uniaxial loading is low.

<table><caption>Table 2: Stress concentrations associated to the different defect geometries under uniaxial loading.</caption><thead><tr><th>f<sub>ry</sub> = 0.5</th><th>f<sub>ry</sub> = 1</th><th>f<sub>ry</sub> = 2</th><th></th></tr></thead><tbody><tr><td>K<sub>zz</sub> = 6.84</td><td>K<sub>zz</sub> = 9.45</td><td>K<sub>zz</sub> = 11.60</td><td rowspan="2">f<sub>rz</sub> = 0.15</td></tr><tr><td>K<sub>DV</sub> = 6.88</td><td>K<sub>DV</sub> = 9.16</td><td>K<sub>DV</sub> = 11.46</td></tr><tr><td>K<sub>zz</sub> = 2.50</td><td>K<sub>zz</sub> = 3.46</td><td>K<sub>zz</sub> = 4.36</td><td rowspan="2">f<sub>rz</sub> = 0.5</td></tr><tr><td>K<sub>DV</sub> = 2.54</td><td>K<sub>DV</sub> = 3.49</td><td>K<sub>DV</sub> = 4.42</td></tr><tr><td>K<sub>zz</sub> = 1.65</td><td>K<sub>zz</sub> = 2.04</td><td>K<sub>zz</sub> = 2.48</td><td rowspan="2">f<sub>rz</sub> = 1</td></tr><tr><td>K<sub>DV</sub> = 1.67</td><td>K<sub>DV</sub> = 2.07</td><td>K<sub>DV</sub> = 2.51</td></tr><tr><td>K<sub>zz</sub> = 1.39</td><td>K<sub>zz</sub> = 1.63</td><td>K<sub>zz</sub> = 1.91</td><td rowspan="2">f<sub>rz</sub> = 1.5</td></tr><tr><td>K<sub>DV</sub> = 1.41</td><td>K<sub>DV</sub> = 1.64</td><td>K<sub>DV</sub> = 1.93</td></tr></tbody></table>

We can observe that globally, a defect is more detrimental in terms of fatigue strength when it is deeper than shallow. It could be explained by the stress concentration resulting of the geometry (see table 2).

We can also observe that the bigger the mean sphere $\Omega$ (controlled by $R^*$), the lower the effects of the depth and size of the defect (see figure 7). This is due to the mean (see equation 4) applied in the non-local approach which decreases the effect of the stress concentration and takes into account the stress gradient near the defect (i.e. the effect of the Highly Stressed Volume HSV [13]).

A finer analysis of the figure 7 reveals that a deeper defect (resulting in a higher stress concentration) is not always more detrimental than a shallow one. For instance, for $f_{rz} = 1.0$, $R^* = 100 \ \mu m$ and a defect size equal to 100 $\mu m$, the normalized fatigue strength is equal to 0.9 for $f_{ry} = 2.0$ while it is equal to 0.8 for $f_{ry} = 0.5$. This is due to the competition between the effect of stress concentration (local property) and the stress gradient (non-local property).

### 3.2.3. The thickness effect

Figure 7 also highlights the effect of the defect thickness on the fatigue strength. The defect thickness decreases when the parameter $f_{rz}$ drops (from top to bottom in the figure). The thicker the defect, the lower the stress concentration in terms of $K_{zz}$ and $K_{DV}$ (see table 2).

The same conclusions as for the effect of the defect depth could be drawn. The competition between the effect of the stress gradient and the stress concentration is observable. The stress gradient contribution is zero if $R^* = 0$, and it gets higher when $R^*$ increases. The effect of the defect acuity is then higher than the stress gradient for low $R^*$ and vice versa for high $R^*$.

Figure 8 represents the stress field in the defect vicinity for two defects thickness configurations and a given defect size (100 $\mu m$). The stress fields are shown for six different $R^*$. It can be noticed that a thin defect which creates a high stress concentration is more critical for small mean volume (figure 8.b.1) than a hemispherical one (figure 8.a.1). For bigger $R^*$, the opposite is observed: the hemispherical defect (figure 8.a.6) is more dangerous than the thin one (figure 8.b.6).

If the defect is small compared to $R^*$, its local maximal value (i.e. acuity) is smoothened by the mean applied in the non-local approach adopted here. On the contrary, if the defect is large compared to $R^*$, the mean is only applied on highly stressed elements, which is equivalent to classic local approach were the only parameter driving the defect criticality is the stress concentration (i.e. no effect of the defect size).
<figure><img src="image_10.png" /><figcaption>Figure 7: Effect of the defect morphology (size and shape) and the $R^*$ on the normalized fatigue strength assessment under uniaxial loading. The 12 subplots correspond to the 12 defect shapes introduced in the simulation framework. The columns represent the effect of the defect’s depth ($f_{ry}$ equal to 0.5, 1, 2), the rows correspond to the effect of the thickness ($f_{rz}$ equal to 0.15, 0.5, 1, 1.5). The different colors of the curves show the effect of $R^*$.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Figure 7: Effect of the defect morphology (size and shape) and the $R^*$ on the normalized fatigue strength assessment under uniaxial loading. The 12 subplots correspond to the 12 defect shapes introduced in the simulation framework. The columns represent the effect of the defect’s depth ($f_{ry}$ equal to 0.5, 1, 2), the rows correspond to the effect of the thickness ($f_{rz}$ equal to 0.15, 0.5, 1, 1.5). The different colors of the curves show the effect of $R^*$.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Figure 8: Effect of the defect shape and the R\* value under uniaxial loading on the stress gradient (in terms of Dang Van $\sigma_{DV}^{R\*}$) in the defect vicinity. Two defects with identical size ($\sqrt{area} = 100 \ \mu m$) are represented in two columns : (a) is an hemispherical defect ($f_{ry} = f_{rz} = 1$), (b) is a low thickness ellipsoid ($f_{ry} = 1, f_{rz} = 0.15$). The different rows represent the evolution of the stress gradient as R\* increases (0, 5, 10, 20, 40 and 100 $\ \mu m$).</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Figure 8: Effect of the defect shape and the R\* value under uniaxial loading on the stress gradient (in terms of Dang Van $\sigma_{DV}^{R\*}$) in the defect vicinity. Two defects with identical size ($\sqrt{area} = 100 \ \mu m$) are represented in two columns : (a) is an hemispherical defect ($f_{ry} = f_{rz} = 1$), (b) is a low thickness ellipsoid ($f_{ry} = 1, f_{rz} = 0.15$). The different rows represent the evolution of the stress gradient as R\* increases (0, 5, 10, 20, 40 and 100 $\ \mu m$).</figcaption></figure>
This is the behaviour observed on the figure 7 : for R* = 100 µm, small defects do not affect the fatigue strength and for R* = 5 µm, the fatigue strength remains constant for large defects. For a given geometry, all the curves (different R*) converge to a horizontal asymptote.

<figure><img src="image_12.png" /><figcaption>Figure 9: Evolution of the stress (in terms of Dang Van $\sigma_{DV}^*/\sigma_{DV,\infty}$) from the defect hot spot to the material core under uniaxial loading. The values taken into account are the elements crossed by the line passing through the points (0,0,0) and the element centroid with the maximal $\sigma_{DV}^*/\sigma_{DV,\infty}$ value. The three sublots correspond to the 1D gradients plotted for three extreme cases (described in the legend and illustrated below) in terms of defect shape for identical sizes ($\sqrt{area}=100$ µm). The effect of R* is also represented by plotting the gradient for six R* values (see global legend).</figcaption></figure>

Figure 9 represents the stress gradient along one direction for three extreme cases of defect geometry.

The defect size $\sqrt{area}$ is equal to 100 µm. The presented values are the elements crossed by a line from the defect center (0,0,0) and the element centroid with the maximal $\sigma_{DV}^*/\sigma_{DV,\infty}$ value. The distance is measured from the defect surface. The results for six values of R* are displayed : 0, 5, 10, 20, 50 and 100 µm.

We can observe that for R* = 0 µm (local approach), the maximal stress increases with the stress concentration due to the defect (see table 2). Still, the stress decreases fast in the defect vicinity, resulting in a high stress gradient intensity. The HSV is bigger when the acuity drops.

We can observe that the mean effect of the characteristic length R* leads to a modification of the order of defect criticality. When R* increases, the gradient is smoothened. The higher the stress concentration, the stronger the effect of the non-local approach. For R* = 100 µm, the highest stress corresponds to the lowest stress concentration. This means that the HSV weight becomes preponderant. The R* value is the tuning parameter of the competition between the HSV and the local maximal stress.

In the following sections, the three extreme cases in terms of defect morphology presented in the previous paragraph are kept for comparison purposes. The next section focuses on the effect of the loading mode.

### 3.3. Effect of the loading mode

For material containing defects, the defect criticality depends on the loading mode applied. The two next sections deal with R=-1 uniaxial and pure shear loadings.

#### 3.3.1. Uniaxial (R=-1)

Figure 10 represents the effect of the defect morphology (three extreme cases) under uniaxial loading for different R* values in a Kitagawa-Takahashi diagram.

It can be noticed that for R* ≥ 50 µm, a typical Kitagawa diagram (in terms of shape) is obtained, as observed by Taylor [46] using different non-local approaches. As observed in the previous section, for all R*
This is the behaviour observed on the figure 7 : for R* = 100 µm, small defects do not affect the fatigue strength and for R* = 5 µm, the fatigue strength remains constant for large defects. For a given geometry, all the curves (different R*) converge to a horizontal asymptote.

<figure><img src="image_11.png" /><figcaption>Figure 9: Evolution of the stress (in terms of Dang Van $\sigma_{DV}^*/\sigma_{DV,\infty}$) from the defect hot spot to the material core under uniaxial loading. The values taken into account are the elements crossed by the line passing through the points (0,0,0) and the element centroid with the maximal $\sigma_{DV}^*/\sigma_{DV,\infty}$ value. The three sublots correspond to the 1D gradients plotted for three extreme cases (described in the legend and illustrated below) in terms of defect shape for identical sizes ($\sqrt{area}=100$ µm). The effect of R* is also represented by plotting the gradient for six R* values (see global legend).</figcaption></figure>

Figure 9 represents the stress gradient along one direction for three extreme cases of defect geometry.

The defect size $\sqrt{area}$ is equal to 100 µm. The presented values are the elements crossed by a line from the defect center (0,0,0) and the element centroid with the maximal $\sigma_{DV}^*/\sigma_{DV,\infty}$ value. The distance is measured from the defect surface. The results for six values of R* are displayed : 0, 5, 10, 20, 50 and 100 µm.

We can observe that for R* = 0 µm (local approach), the maximal stress increases with the stress concentration due to the defect (see table 2). Still, the stress decreases fast in the defect vicinity, resulting in a high stress gradient intensity. The HSV is bigger when the acuity drops.

We can observe that the mean effect of the characteristic length R* leads to a modification of the order of defect criticality. When R* increases, the gradient is smoothened. The higher the stress concentration, the stronger the effect of the non-local approach. For R* = 100 µm, the highest stress corresponds to the lowest stress concentration. This means that the HSV weight becomes preponderant. The R* value is the tuning parameter of the competition between the HSV and the local maximal stress.

In the following sections, the three extreme cases in terms of defect morphology presented in the previous paragraph are kept for comparison purposes. The next section focuses on the effect of the loading mode.

### 3.3. Effect of the loading mode

For material containing defects, the defect criticality depends on the loading mode applied. The two next sections deal with R=-1 uniaxial and pure shear loadings.

#### 3.3.1. Uniaxial (R=-1)

Figure 10 represents the effect of the defect morphology (three extreme cases) under uniaxial loading for different R* values in a Kitagawa-Takahashi diagram.

It can be noticed that for R* ≥ 50 µm, a typical Kitagawa diagram (in terms of shape) is obtained, as observed by Taylor [46] using different non-local approaches. As observed in the previous section, for all R*
values, the effect of the stress gradient (i.e. the HSV) is preponderant for small defects and the effect of the sharpness becomes the first order parameter for large defects.

The largest differences between geometries can be observed for large defects with $R^* = 5$ µm : $\sigma_a/\sigma_{a,0} \approx 0.7$ with $f_{ry} = 0.5$, $f_{rz} = 1.5$ (low acuity) and $\sigma_a/\sigma_{a,0} \approx 0.1$ with $f_{ry} = 2$, $f_{rz} = 0.15$ (high acuity).

<figure><img src="image_13.png" /><figcaption>Figure 10: Effect of the defect morphology (extreme cases) on the normalized fatigue strength assessment under uniaxial loading. The 4 subplots correspond to 4 values of $R^*$ equal to 5, 20, 50 and 100 µm. The color and shape of the points correspond to three extreme cases described in the legend.</figcaption></figure>

### 3.3.2. *Pure shear (R=-1)*

Figure 11 represents the effect of three extreme defect shapes on the fatigue strength under pure shear loading for different $R^*$ values. The geometries and the $R^*$ values are the same used in figure 10 for comparison purposes.

Differences in terms of sizes between the different morphologies with identical $\sqrt{area_z}$ are due to the link between the rotation of the plane maximizing the Dang Van criterion and the geometries shapes under pure shear (see figure 6). In other words, the orientation of the maximizing plane depends on the defect morphology.

As observed under uniaxial loading, the defect shape and size also play a role under pure shear. However, the effect of defect seems lower in this configuration than under uniaxial loading. Particularly, differences linked to the defect morphology are not as important as for uniaxial loading. For instance, the largest
values, the effect of the stress gradient (i.e. the HSV) is preponderant for small defects and the effect of the sharpness becomes the first order parameter for large defects.

The largest differences between geometries can be observed for large defects with $R^* = 5$ µm : $\sigma_a/\sigma_{a,0} \approx 0.7$ with $f_{ry} = 0.5$, $f_{rz} = 1.5$ (low acuity) and $\sigma_a/\sigma_{a,0} \approx 0.1$ with $f_{ry} = 2$, $f_{rz} = 0.15$ (high acuity).

<figure><img src="image_12.png" /><figcaption>Figure 10: Effect of the defect morphology (extreme cases) on the normalized fatigue strength assessment under uniaxial loading. The 4 subplots correspond to 4 values of $R^*$ equal to 5, 20, 50 and 100 µm. The color and shape of the points correspond to three extreme cases described in the legend.</figcaption></figure>

### 3.3.2. *Pure shear (R=-1)*

Figure 11 represents the effect of three extreme defect shapes on the fatigue strength under pure shear loading for different $R^*$ values. The geometries and the $R^*$ values are the same used in figure 10 for comparison purposes.

Differences in terms of sizes between the different morphologies with identical $\sqrt{area_z}$ are due to the link between the rotation of the plane maximizing the Dang Van criterion and the geometries shapes under pure shear (see figure 6). In other words, the orientation of the maximizing plane depends on the defect morphology.

As observed under uniaxial loading, the defect shape and size also play a role under pure shear. However, the effect of defect seems lower in this configuration than under uniaxial loading. Particularly, differences linked to the defect morphology are not as important as for uniaxial loading. For instance, the largest
<figure><img src="image_14.png" /><figcaption>Figure 11: Effect of the defect morphology (extreme cases) on the normalized fatigue strength assessment under pure shear loading. The 4 subplots correspond to 4 values of R* equal to 5, 20, 50 and 100 µm. The color and shape of the points correspond to three extreme cases described in the legend.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>Figure 11: Effect of the defect morphology (extreme cases) on the normalized fatigue strength assessment under pure shear loading. The 4 subplots correspond to 4 values of R* equal to 5, 20, 50 and 100 µm. The color and shape of the points correspond to three extreme cases described in the legend.</figcaption></figure>
difference observed between the low acuity defect and high acuity one was 60% under uniaxial loading. This
difference becomes less than 50% under pure shear.

The comparison between these two loading modes is discussed further in the paper.

**4. Discussion**

Simulation results and direct observations are presented in the previous section. In this part, a discussion about the parametric study results is proposed. Then, the effect of plasticity is analyzed. Finally, a comparison with experimental results is carried out.

4.1. Normalized Kitagawa-Takahashi

In section 3, the results of the parametric simulation framework are presented. This subsection proposes further analyses by introducing a normalized Kitagawa-Takahashi diagram.

4.1.1. Normalized defect size

As introduced in §3.2, the effect of the critical defect size seems to be the main parameter influencing the fatigue strength of materials. However, several works highlighted the subjectivity of the killing defect size, which should be compared to microstructural features of the material such as the grain size [6, 37, 38]. A surface defect is considered as potentially critical when its size is bigger than the strongest microstructural feature which can hinder crack propagation.

In this paper, the characteristic length $R^*$ introduced with the non-local approach may be correlated to
a material microstructure feature. Based on this idea, figure 12 could be obtained by dividing the defect size
$\sqrt{area}$ by the characteristic length $R^*$.

<figure><img src="image_15.png" /><figcaption>Figure 12: Effect of the defect morphology (extreme cases) on the normalized fatigue strength assessment under uniaxial and pure shear loading. The 2 subplots correspond to 2 loading modes. The abscissa axes are the normalized size of the defect by the $R^*$ value. The color and shape of the points correspond to three extreme cases described in the legend.</figcaption></figure>

For a given geometry, identical $\sqrt{area}/R^*$ ratios (with different $\sqrt{area}$ and $R^*$ values) lead to the same
normalized fatigue limit ($\sigma_a/\sigma_{a,0}$ or $\tau_a/\tau_{a,0}$ depending on the loading mode). For instance, the predicted
fatigue limit for a given geometry shape with $\sqrt{area} = 100~\mu\text{m}$ and $R^* = 10~\mu\text{m}$ is the same with $\sqrt{area} =
200~\mu\text{m}$ and $R^* = 20~\mu\text{m}$. We can notice that for each defect geometry, 10 defect sizes are generated, with
$21~R^*$ computed. Then, each of the three shape cases presented in figure 12 are described by 210 points.
difference observed between the low acuity defect and high acuity one was 60% under uniaxial loading. This
difference becomes less than 50% under pure shear.

The comparison between these two loading modes is discussed further in the paper.

**4. Discussion**

Simulation results and direct observations are presented in the previous section. In this part, a discussion about the parametric study results is proposed. Then, the effect of plasticity is analyzed. Finally, a comparison with experimental results is carried out.

4.1. Normalized Kitagawa-Takahashi

In section 3, the results of the parametric simulation framework are presented. This subsection proposes further analyses by introducing a normalized Kitagawa-Takahashi diagram.

4.1.1. Normalized defect size

As introduced in §3.2, the effect of the critical defect size seems to be the main parameter influencing the fatigue strength of materials. However, several works highlighted the subjectivity of the killing defect size, which should be compared to microstructural features of the material such as the grain size [6, 37, 38]. A surface defect is considered as potentially critical when its size is bigger than the strongest microstructural feature which can hinder crack propagation.

In this paper, the characteristic length $R^*$ introduced with the non-local approach may be correlated to
a material microstructure feature. Based on this idea, figure 12 could be obtained by dividing the defect size
$\sqrt{area}$ by the characteristic length $R^*$.

<figure><img src="image_14.png" /><figcaption>Figure 12: Effect of the defect morphology (extreme cases) on the normalized fatigue strength assessment under uniaxial and pure shear loading. The 2 subplots correspond to 2 loading modes. The abscissa axes are the normalized size of the defect by the $R^*$ value. The color and shape of the points correspond to three extreme cases described in the legend.</figcaption></figure>

For a given geometry, identical $\sqrt{area}/R^*$ ratios (with different $\sqrt{area}$ and $R^*$ values) lead to the same
normalized fatigue limit ($\sigma_a/\sigma_{a,0}$ or $\tau_a/\tau_{a,0}$ depending on the loading mode). For instance, the predicted
fatigue limit for a given geometry shape with $\sqrt{area} = 100~\mu\text{m}$ and $R^* = 10~\mu\text{m}$ is the same with $\sqrt{area} =
200~\mu\text{m}$ and $R^* = 20~\mu\text{m}$. We can notice that for each defect geometry, 10 defect sizes are generated, with
$21~R^*$ computed. Then, each of the three shape cases presented in figure 12 are described by 210 points.
Under uniaxial and pure shear loading conditions, small relative defect sizes ($\sqrt{area}/R^* < 5 \times 10^{-1}$) lead to a small reduction in terms of normalized fatigue strength (< 5) The HSV is small compared to $R^*$ and so is its effect. For medium scale relative defect size ($5 \times 10^{-1} < \sqrt{area}/R^* < 10^1$), the normalized fatigue strength decreases quickly and the effect of the morphology increases with $\sqrt{area}/R^*$. The competition between the defect acuity and the HSV is important, with an increasing effect of the stress concentration. For large relative defect size ($10^1 < \sqrt{area}/R^*$), the normalized fatigue strength decreases more and more slowly. The effect of the acuity is preponderant and the fatigue strength can be readily reached by a local approach (independent of the defect size) based on the $K_{DV}$ value (i.e. a horizontal asymptote in the normalized Kitagawa-Takahashi diagram). The value of the asymptote is equal to $1/K_{DV}$.

Finally, the non-local approach allows a continuous description of the defect effect, from the preponderance of the HSV for small defects to become acuity driven for large ones, and captures important trends of fatigue behaviour.

### 4.1.2. Uniaxial vs pure shear

To compare the effect of uniaxial and pure shear loadings in the presence of a defect, figure 13 represents the ratio between the fatigue strength for pure shear and uniaxial loadings as a function of the defect relative size. The relation between the defect free fatigue limits under pure shear ($\tau_{a,0}$) and uniaxial ($\sigma_{a,0}$) loading is given by equation 14.

$$ \tau_{a,0} = \sigma_{a,0} \left( \frac{1}{2} + \frac{\alpha}{3} \right) \qquad (14) $$

For $\alpha = 0.4$, $\tau_{a,0} \approx 0.633 \times \sigma_{a,0}$.

The $\tau_a/\sigma_a$ ratio is almost always higher than 0.633. This is due to a more detrimental effect of defects under uniaxial loading than pure shear. This is consistent with observations in the literature, where defects under torsion are less critical than under uniaxial loading [19, 33, 38, 40].

The ratio continues to increase when the relative defect size increases, particularly for high acuity defects where the ratio becomes even higher than 1. Still, for the $f_{ry} = 0.5$, $f_{rz} = 1.5$ geometry, the ratio seems to tend to a value lower than 0.633. Then, for this geometry, the defect is more critical under shear than uniaxial loading.

The comparison of the results between uniaxial and pure shear loadings for identical geometries could be discussed. Indeed, depending on the loading mode, the $K_{DV}$ of defect are not identical. A comparison in terms of stress concentration may be more relevant when dealing with different loading modes. This is discussed in the following section.

### 4.1.3. Stress concentration effect - Frost diagram

As introduced previously, it is possible to distinguish cases by using $K_{DV}$, corresponding to the maximal stress concentration due to the defects in terms of Dang Van criterion. First, this allows to reduce the shape parameters $f_{ry}$ and $f_{rz}$ into a single indicator. For more complex defect geometry, this is also a relevant indicator for comparison. Second, this representation could be more pertinent to compare different loading modes such as uniaxial and pure shear.

Figure 14(a) illustrates schematically a typical Frost's diagram for metals. This diagram represents the effect of the stress concentration $K_t$ on the fatigue limit $\sigma_a$ for a given defect size. The black decreasing curve corresponds to the crack initiation stress threshold. The higher the stress concentration, the lower the stress initiation threshold.

Below a certain stress (which could be related to the long crack propagation threshold), the crack cannot propagate independently to the $K_t$ value. This is due to a driving force too low to propagate the crack till fracture [8]. It can be mentioned that the discussed model in this paper only deals with crack initiation. The fatigue limits are directly linked to the crack initiation stress threshold, which provides a conservative criterion since crack arrest could occur.

Figure 14(b) represents the Frost's diagrams under uniaxial and pure shear loadings. The normalized fatigue strengths are plotted as a function of the $K_{DV}$ values (corresponding to one defect shape for each loading mode). The colors are associated to the defect relative size.

For readability purposes, only five sizes are shown in figure 14(b). Plotted ratios $\sqrt{area}/R^*$ are equal to 0.1, 2, 5, 20, 300. Continuous and derivable functions whose form is given by equation 15 are fitted on the
Under uniaxial and pure shear loading conditions, small relative defect sizes ($\sqrt{area}/R^* < 5 \times 10^{-1}$) lead to a small reduction in terms of normalized fatigue strength (< 5) The HSV is small compared to $R^*$ and so is its effect. For medium scale relative defect size ($5 \times 10^{-1} < \sqrt{area}/R^* < 10^1$), the normalized fatigue strength decreases quickly and the effect of the morphology increases with $\sqrt{area}/R^*$. The competition between the defect acuity and the HSV is important, with an increasing effect of the stress concentration. For large relative defect size ($10^1 < \sqrt{area}/R^*$), the normalized fatigue strength decreases more and more slowly. The effect of the acuity is preponderant and the fatigue strength can be readily reached by a local approach (independent of the defect size) based on the $K_{DV}$ value (i.e. a horizontal asymptote in the normalized Kitagawa-Takahashi diagram). The value of the asymptote is equal to $1/K_{DV}$.

Finally, the non-local approach allows a continuous description of the defect effect, from the preponderance of the HSV for small defects to become acuity driven for large ones, and captures important trends of fatigue behaviour.

### 4.1.2. Uniaxial vs pure shear

To compare the effect of uniaxial and pure shear loadings in the presence of a defect, figure 13 represents the ratio between the fatigue strength for pure shear and uniaxial loadings as a function of the defect relative size. The relation between the defect free fatigue limits under pure shear ($\tau_{a,0}$) and uniaxial ($\sigma_{a,0}$) loading is given by equation 14.

$$ \tau_{a,0} = \sigma_{a,0} \left( \frac{1}{2} + \frac{\alpha}{3} \right) \qquad (14) $$

For $\alpha = 0.4$, $\tau_{a,0} \approx 0.633 \times \sigma_{a,0}$.

The $\tau_a/\sigma_a$ ratio is almost always higher than 0.633. This is due to a more detrimental effect of defects under uniaxial loading than pure shear. This is consistent with observations in the literature, where defects under torsion are less critical than under uniaxial loading [19, 33, 38, 40].

The ratio continues to increase when the relative defect size increases, particularly for high acuity defects where the ratio becomes even higher than 1. Still, for the $f_{ry} = 0.5$, $f_{rz} = 1.5$ geometry, the ratio seems to tend to a value lower than 0.633. Then, for this geometry, the defect is more critical under shear than uniaxial loading.

The comparison of the results between uniaxial and pure shear loadings for identical geometries could be discussed. Indeed, depending on the loading mode, the $K_{DV}$ of defect are not identical. A comparison in terms of stress concentration may be more relevant when dealing with different loading modes. This is discussed in the following section.

### 4.1.3. Stress concentration effect - Frost diagram

As introduced previously, it is possible to distinguish cases by using $K_{DV}$, corresponding to the maximal stress concentration due to the defects in terms of Dang Van criterion. First, this allows to reduce the shape parameters $f_{ry}$ and $f_{rz}$ into a single indicator. For more complex defect geometry, this is also a relevant indicator for comparison. Second, this representation could be more pertinent to compare different loading modes such as uniaxial and pure shear.

Figure 14(a) illustrates schematically a typical Frost's diagram for metals. This diagram represents the effect of the stress concentration $K_t$ on the fatigue limit $\sigma_a$ for a given defect size. The black decreasing curve corresponds to the crack initiation stress threshold. The higher the stress concentration, the lower the stress initiation threshold.

Below a certain stress (which could be related to the long crack propagation threshold), the crack cannot propagate independently to the $K_t$ value. This is due to a driving force too low to propagate the crack till fracture [8]. It can be mentioned that the discussed model in this paper only deals with crack initiation. The fatigue limits are directly linked to the crack initiation stress threshold, which provides a conservative criterion since crack arrest could occur.

Figure 14(b) represents the Frost's diagrams under uniaxial and pure shear loadings. The normalized fatigue strengths are plotted as a function of the $K_{DV}$ values (corresponding to one defect shape for each loading mode). The colors are associated to the defect relative size.

For readability purposes, only five sizes are shown in figure 14(b). Plotted ratios $\sqrt{area}/R^*$ are equal to 0.1, 2, 5, 20, 300. Continuous and derivable functions whose form is given by equation 15 are fitted on the
<figure><img src="image_16.png" /><figcaption>Figure 13: Comparison between shear and uniaxial loading modes for three extreme cases. The abscissa axis is the normalized size of the defect by the R* value, the ordinate axis is the ratio between the fatigue strength for pure shear and uniaxial loadings. The color and shape of the points correspond to three extreme cases described in the legend.</figcaption></figure>

dataset.

$$ \begin{cases} f(x) = y_1 + b_1 \exp[-a_1 * (x - x_1)] + y_2 + b_2 \arctan[a_2 * (x - x_2)] \\ f(1) = 1 \end{cases} \qquad (15) $$

These functions are plotted for readability purposes and their form is arbitrarily chosen.

It can be noticed that for identical geometries in the simulation framework, the range in terms of K<sub>DV</sub> is bigger under uniaxial than in pure shear loading. This observation is probably the main reason explaining the higher effect of the defect shape for uniaxial loading. However, defects in this study are not the most critical cases in terms of orientation for pure shear. Other configurations could lead to bigger range for K<sub>DV</sub>.

For identical size and K<sub>DV</sub>, it seems that the normalized fatigue limit is higher for pure shear than uniaxial loading. Again, this observation shows a more detrimental effect of defects under uniaxial loading than shear. This is even more pronounced if the shear fatigue limit without defect τ<sub>a,0</sub> is lower than under uniaxial loading σ<sub>a,0</sub>, as observed in the experimental studies cited previously [19, 38, 40].

For high relative defect size, the fatigue strength decreases monotonically when K<sub>DV</sub> increases as described in figure 14(a). As observed by Guerchais et al. [19], for lower relative size of the defect, it seems that counter-intuitively, after a decrease of the fatigue strength due to the stress concentration, the fatigue strength increases with K<sub>DV</sub>. It may be a consequence of the competition between the HSV size and the local stress concentration. Indeed, the lower the relative size √area/R*, the higher the effect of the HSV. Since the HSV size decreases with K<sub>DV</sub>, the fatigue strength may increase with K<sub>DV</sub>.

In this study, the material mechanical behaviour was modeled by linear elasticity. The next subsection suggests a comparison between fatigue strength predictions for purely alternate uniaxial loading with linear elastic and elastoplastic materials.

## 4.2. Role of the plasticity

The elastoplastic behaviour of the material was modeled by the Lemaitre and Chaboche [24] non-linear isotropic and kinematic hardening model. The model could be written by the equations :

$$ \begin{cases} \sigma^0 = \sigma_0 + Q_{\text{inf}} [1 - \exp(-b\epsilon^{pl})] \\ \dot{\alpha} = C \dot{\epsilon}^{pl} \frac{1}{\sigma^0} (\sigma - \alpha) - \gamma \alpha \dot{\epsilon}^{pl} \end{cases} \qquad (16) $$
<figure><img src="image_15.png" /><figcaption>Figure 13: Comparison between shear and uniaxial loading modes for three extreme cases. The abscissa axis is the normalized size of the defect by the R* value, the ordinate axis is the ratio between the fatigue strength for pure shear and uniaxial loadings. The color and shape of the points correspond to three extreme cases described in the legend.</figcaption></figure>

dataset.

$$ \begin{cases} f(x) = y_1 + b_1 \exp[-a_1 * (x - x_1)] + y_2 + b_2 \arctan[a_2 * (x - x_2)] \\ f(1) = 1 \end{cases} \qquad (15) $$

These functions are plotted for readability purposes and their form is arbitrarily chosen.

It can be noticed that for identical geometries in the simulation framework, the range in terms of K_DV is bigger under uniaxial than in pure shear loading. This observation is probably the main reason explaining the higher effect of the defect shape for uniaxial loading. However, defects in this study are not the most critical cases in terms of orientation for pure shear. Other configurations could lead to bigger range for K_DV.

For identical size and K_DV, it seems that the normalized fatigue limit is higher for pure shear than uniaxial loading. Again, this observation shows a more detrimental effect of defects under uniaxial loading than shear. This is even more pronounced if the shear fatigue limit without defect τa,0 is lower than under uniaxial loading σa,0, as observed in the experimental studies cited previously [19, 38, 40].

For high relative defect size, the fatigue strength decreases monotonically when K_DV increases as described in figure 14(a). As observed by Guerchais et al. [19], for lower relative size of the defect, it seems that counter-intuitively, after a decrease of the fatigue strength due to the stress concentration, the fatigue strength increases with K_DV. It may be a consequence of the competition between the HSV size and the local stress concentration. Indeed, the lower the relative size $\sqrt{area}/R^*$, the higher the effect of the HSV. Since the HSV size decreases with K_DV, the fatigue strength may increase with K_DV.

In this study, the material mechanical behaviour was modeled by linear elasticity. The next subsection suggests a comparison between fatigue strength predictions for purely alternate uniaxial loading with linear elastic and elastoplastic materials.

## 4.2. Role of the plasticity

The elastoplastic behaviour of the material was modeled by the Lemaitre and Chaboche [24] non-linear isotropic and kinematic hardening model. The model could be written by the equations :

$$ \begin{cases} \sigma^0 = \sigma_0 + Q_{\text{inf}} [1 - \exp(-b\epsilon^{pl})] \\ \dot{\alpha} = C \dot{\epsilon}^{pl} \frac{1}{\sigma^0} (\sigma - \alpha) - \gamma \alpha \dot{\epsilon}^{pl} \end{cases} \qquad (16) $$
<figure><img src="image_17.png" /><figcaption>Figure 14: (a) Schematic typical Frost's diagram for metallic materials representing the fatigue strength as a function of the stress concentration $K_t$ for identical defect size (inspired by [8]). (b) Comparison between shear and uniaxial loading modes in Frost's diagrams. The two diagrams represent the two loading modes. The abscissa axes are the $K_{DV}$ values (see equation 13), the ordinate axes are the normalized fatigue strength for uniaxial and pure shear loadings. The colors correspond to five different defect relative sizes $\sqrt{\text{area}}/R^*$. Functions are fitted between the simulated points and are represented by five colored curves on each graph.</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>Figure 14: (a) Schematic typical Frost's diagram for metallic materials representing the fatigue strength as a function of the stress concentration $K_t$ for identical defect size (inspired by [8]). (b) Comparison between shear and uniaxial loading modes in Frost's diagrams. The two diagrams represent the two loading modes. The abscissa axes are the $K_{DV}$ values (see equation 13), the ordinate axes are the normalized fatigue strength for uniaxial and pure shear loadings. The colors correspond to five different defect relative sizes $\sqrt{\text{area}}/R^*$. Functions are fitted between the simulated points and are represented by five colored curves on each graph.</figcaption></figure>
with $\sigma^0$ the size of the yield surface, $\sigma_0$ is the yield surface size at zero plastic strain, $\alpha$ is the kinematic shift, $Q_{\text{inf}}$, $b$, $C$ and $\gamma$ are material parameters.

The parameters $\sigma_0$, $Q_{\text{inf}}$, $b$, $C$ and $\gamma$ were fitted on experimental data for a 316L L-PBF from Liang study [25]. Purely alternate uniaxial cyclic loading in total deformation control was applied at four different amplitudes (0.30%, 0.45%, 0.60% and 0.75%). The identification was performed on half-life hysteresis loops.

The elastic behaviour was modelled by a Poisson's coefficient equal to 0.3 and a Young modulus $E$ which was used as an optimization variable. Without this identification of the Young modulus, the model didn't fit well the experimental data.

The optimization was done on a unique element. The six variables to be fitted were used as inputs. Five cycles were simulated, results were analysed on the last one. The cost function for each deformation amplitude was defined by the equation 17.

$$F(E, \sigma_0, Q_{\text{inf}}, b, C, \gamma) = \frac{1}{n} \sum_{i=1}^{n} \left| \frac{\sigma_{\text{exp},i} - \sigma_{\text{sim},i}}{\sigma_{\text{exp},i}} \right| \quad (17)$$

The global objective function to minimize is the sum of the four cost functions for each deformation level.

The minimization was achieved following two steps. First, a global optimization is realized. The optimum obtained is then injected as the input of a local minimization algorithm (gradient descent).

Finally, the obtained parameters are $E = 108$ GPa, $\sigma_0 = 192$ MPa, $Q_{\text{inf}} = 37$ MPa, $b = 6.16$, $C = 190$ GPa and $\gamma = 1100$. We can notice that the identified Young's modulus on Liang's data [25] is twice bellow the 225 GPa taken from a previous paper [31]. It can be due to different experimental setup and material elastic properties (due to different crystallographic textures).

Figure 15 represents experimental and simulated data in a stress/deformation plot. Even if the model does not fit perfectly the experimental data (due to non-symmetric hardening behaviour) the identified elastoplastic model describes well the experimental data in terms of stress and deformation peaks.

<figure><img src="image_18.png" /><figcaption>Figure 15: Comparison between experimental and simulated data.</figcaption></figure>

Simulations were performed on the cases summarized in the table 3. Three defect sizes were used with 21 $R^*$ combinations ($R^* = 0$ µm set apart) which lead to 63 different normalized sizes from 0.04 to 500. The 12 morphologies introduced in figure 3 were modelled. Since the material is modelled by an elastoplastic
with $\sigma^0$ the size of the yield surface, $\sigma_0$ is the yield surface size at zero plastic strain, $\alpha$ is the kinematic shift, $Q_{\text{inf}}$, $b$, $C$ and $\gamma$ are material parameters.

The parameters $\sigma_0$, $Q_{\text{inf}}$, $b$, $C$ and $\gamma$ were fitted on experimental data for a 316L L-PBF from Liang study [25]. Purely alternate uniaxial cyclic loading in total deformation control was applied at four different amplitudes (0.30%, 0.45%, 0.60% and 0.75%). The identification was performed on half-life hysteresis loops.

The elastic behaviour was modelled by a Poisson's coefficient equal to 0.3 and a Young modulus $E$ which was used as an optimization variable. Without this identification of the Young modulus, the model didn't fit well the experimental data.

The optimization was done on a unique element. The six variables to be fitted were used as inputs. Five cycles were simulated, results were analysed on the last one. The cost function for each deformation amplitude was defined by the equation 17.

$$F(E, \sigma_0, Q_{\text{inf}}, b, C, \gamma) = \frac{1}{n} \sum_{i=1}^{n} \left| \frac{\sigma_{\text{exp},i} - \sigma_{\text{sim},i}}{\sigma_{\text{exp},i}} \right| \quad (17)$$

The global objective function to minimize is the sum of the four cost functions for each deformation level.

The minimization was achieved following two steps. First, a global optimization is realized. The optimum obtained is then injected as the input of a local minimization algorithm (gradient descent).

Finally, the obtained parameters are $E = 108$ GPa, $\sigma_0 = 192$ MPa, $Q_{\text{inf}} = 37$ MPa, $b = 6.16$, $C = 190$ GPa and $\gamma = 1100$. We can notice that the identified Young's modulus on Liang's data [25] is twice bellow the 225 GPa taken from a previous paper [31]. It can be due to different experimental setup and material elastic properties (due to different crystallographic textures).

Figure 15 represents experimental and simulated data in a stress/deformation plot. Even if the model does not fit perfectly the experimental data (due to non-symmetric hardening behaviour) the identified elastoplastic model describes well the experimental data in terms of stress and deformation peaks.

<figure><img src="image_17.png" /><figcaption>Figure 15: Comparison between experimental and simulated data.</figcaption></figure>

Simulations were performed on the cases summarized in the table 3. Three defect sizes were used with 21 $R^*$ combinations ($R^* = 0$ µm set apart) which lead to 63 different normalized sizes from 0.04 to 500. The 12 morphologies introduced in figure 3 were modelled. Since the material is modelled by an elastoplastic
<table><caption>Table 3: Parametric space investigated in elastoplasticity.</caption><thead><tr><th>Parameter</th><th>Values</th></tr></thead><tbody><tr><td>√area<sub>z</sub> [μm]</td><td>10,100,500</td></tr><tr><td>f<sub>ry</sub></td><td>0.5, 1, 2</td></tr><tr><td>f<sub>rz</sub></td><td>0.15, 0.5, 1, 1.5</td></tr><tr><td>R<sup>*</sup> [μm]</td><td>0, 1, 2, 3, 4, 5, 10, 15, 20, 30, 40, 50, 60, 70, 80, 90, 100, 125, 150, 175, 200, 250</td></tr><tr><td>σ<sub>a</sub> [MPa]</td><td>75, 100, 125, 150, 175, 200, 225, 250, 275, 300, 325, 350, 375, 400</td></tr><tr><td>R</td><td>-1</td></tr><tr><td>N [cycles]</td><td>20</td></tr></tbody></table>

behaviour, 14 values of stresses were tested. The uniaxial purely alternate loading was applied over 20 cycles which leads to a stabilized state over the geometry in terms of stress field. Results are discussed over the last cycle.

To compute the value of $\sigma_a$, an affine interpolation was perfomed between the different stress levels on the couple ($\sigma_{DV}^R$, $\sigma_a$). Values of $\sigma_a$ out of the interval [50, 425] MPa are not displayed.

Figure 16 shows a normalized Kitagawa-Takahashi diagram to compare elastoplastic and linear elastic results. Again, even in elastoplasticity, the normalized size seems to be the driving parameter since the results from three very different defect sizes (10, 100 and 500 µm) lead to one master curve per defect shape once divided by $R^*$.

It seems that elastoplastic simulations lead to similar trends compared to pure elastic results. However, elastic results are more conservative than elastoplastic ones. This is due to the smoothing effect of the plasticity. Indeed, the mean applied in the non-local approach and the elastoplastic material behaviour both decrease the stress level in the geometry.

<figure><img src="image_19.png" /><figcaption>Figure 16: Comparison between the elasticity and elastoplasticity simulation results for three extreme cases under uniaxial loading. The abscissa axis is the normalized size of the defect by the $R^*$ value, the ordinate axis is the ratio between the normalized fatigue strength. Plain large markers indicate elastoplastic results, transparent smaller ones represent elastic results (upper legend). The color and shape of the points correspond to three extreme cases described in the lower legend.</figcaption></figure>

The effect of the plasticity seems to increase with the relative defect size $\sqrt{area}/R^*$. The higher the stress level, the larger the difference between elasticity and elastoplasticity. Since the non-local approach smoothens local values, a decrease of $R^*$ leads to higher stress levels, then to more pronounced effect of the plasticity.

It is worth noting that the effect of the plasticity does not increase with the stress concentration $K_{DV}$. Indeed, the plasticity seems to affect $f_{ry} = f_{rz} = 1$ ($K_{DV} = 2.07$) more than $f_{ry} = 0.5$, $f_{ry} = 1.5$ ($K_{DV} =$
<table><caption>Table 3: Parametric space investigated in elastoplasticity.</caption><thead><tr><th>Parameter</th><th>Values</th></tr></thead><tbody><tr><td>√area<sub>z</sub> [μm]</td><td>10,100,500</td></tr><tr><td>f<sub>ry</sub></td><td>0.5, 1, 2</td></tr><tr><td>f<sub>rz</sub></td><td>0.15, 0.5, 1, 1.5</td></tr><tr><td>R<sup>*</sup> [μm]</td><td>0, 1, 2, 3, 4, 5, 10, 15, 20, 30, 40, 50, 60, 70, 80, 90, 100, 125, 150, 175, 200, 250</td></tr><tr><td>σ<sub>a</sub> [MPa]</td><td>75, 100, 125, 150, 175, 200, 225, 250, 275, 300, 325, 350, 375, 400</td></tr><tr><td>R</td><td>-1</td></tr><tr><td>N [cycles]</td><td>20</td></tr></tbody></table>

behaviour, 14 values of stresses were tested. The uniaxial purely alternate loading was applied over 20 cycles which leads to a stabilized state over the geometry in terms of stress field. Results are discussed over the last cycle.

To compute the value of $\sigma_a$, an affine interpolation was perfomed between the different stress levels on the couple ($\sigma_{DV}^R$, $\sigma_a$). Values of $\sigma_a$ out of the interval [50, 425] MPa are not displayed.

Figure 16 shows a normalized Kitagawa-Takahashi diagram to compare elastoplastic and linear elastic results. Again, even in elastoplasticity, the normalized size seems to be the driving parameter since the results from three very different defect sizes (10, 100 and 500 µm) lead to one master curve per defect shape once divided by $R^*$.

It seems that elastoplastic simulations lead to similar trends compared to pure elastic results. However, elastic results are more conservative than elastoplastic ones. This is due to the smoothing effect of the plasticity. Indeed, the mean applied in the non-local approach and the elastoplastic material behaviour both decrease the stress level in the geometry.

<figure><img src="image_18.png" /><figcaption>Figure 16: Comparison between the elasticity and elastoplasticity simulation results for three extreme cases under uniaxial loading. The abscissa axis is the normalized size of the defect by the $R^*$ value, the ordinate axis is the ratio between the normalized fatigue strength. Plain large markers indicate elastoplastic results, transparent smaller ones represent elastic results (upper legend). The color and shape of the points correspond to three extreme cases described in the lower legend.</figcaption></figure>

The effect of the plasticity seems to increase with the relative defect size $\sqrt{area}/R^*$. The higher the stress level, the larger the difference between elasticity and elastoplasticity. Since the non-local approach smoothens local values, a decrease of $R^*$ leads to higher stress levels, then to more pronounced effect of the plasticity.

It is worth noting that the effect of the plasticity does not increase with the stress concentration $K_{DV}$. Indeed, the plasticity seems to affect $f_{ry} = f_{rz} = 1$ ($K_{DV} = 2.07$) more than $f_{ry} = 0.5$, $f_{ry} = 1.5$ ($K_{DV} =$
1.39) and $f_{ry} = 2, f_{ry} = 0.15$ ($K_{DV} = 11.46$). It could be due to the effect of the stress gradient magnitude which leads to very different level of plasticity between element near the hot spot. Again, the competition between the local maxima and the HSV is at the origin of these differences.

As the discussed models in this paper show consistent results regarding some well-known fatigue behaviours, a confrontation with experimental data is proposed in the next subsection. The beneficial effect of taking into account the plastic behaviour of the material for fatigue strength prediction is discussed regarding the experimental data.

### 4.3. Comparison with uniaxial experimental data

In this part, experimental data collected for 316L stainless steel manufactured by Laser Powder Bed Fusion (L-PBF) under alternate symmetrical (R=-1) uniaxial loading are presented [31, 30]. The non-local approach is then compared with these data and with Linear Elastic Fracture Mechanics (LEFM) models which were the object of a previous paper [32].

#### 4.3.1. Material and experiments

The investigated material is a 316L grade stainless steel manufactured by L-PBF. The alloy was supplied as powder by TLS Technik. A Farsoon FS271M machine was used to build fatigue specimens vertically. An identical optimal manufacturing strategy was applied for each of the 5 plates produced for the experimental campaign. The material density achieved was near 99.9%.

After L-PBF processing, raw specimens were machined up to their final geometry and polished. Neither thermal nor chemical treatments were applied, the material bulk was characterized in its as-built state. More details about the fabrication of the fatigue specimens, the material static behaviour and its microstructure can be found in previous papers [31, 32]. It is worth mentioning that residual stresses were measured using X-rays diffraction and were found insignificant.

In the current study, three batches of fatigue specimens corresponding to three surface conditions are discussed: (1) polished (no other step after fabrication), (2) with one Electric Discharge Machined (EDM) hemispherical defect and (3) pre-corroded specimens. These surface states are supposed to represent typical defects due to the manufacturing process or environmental exposure in realistic service conditions. Experimental data are accessible on an open-source repository [30].

The pre-corrosion was achieved by electrochemical way in an aqueous neutral saline solution. A more precise description of the pre-corrosion setup was given in a previously published work [31].

Fatigue tests were carried under uniaxial loading with $R = -1, f = 15$ Hz and at room temperature. A step by step method was used, by increasing the stress amplitude $\sigma_a$ by 25 MPa every $10^6$ cycles. As described in a previous paper [32], the fatigue strength is finally evaluated using Maxwell and Nicholas correction [28].

In the next subsection, simulation results are confronted to the experimental data through a Kitagawa-Takahashi diagram.

#### 4.3.2. Comparison between experimental data and models

In this experimental campaign, all the fracture surfaces were observed using a Scanning Electron Microscope (SEM). On each fractography, a unique critical defect was identified as the root cause of the crack initiation and its size $\sqrt{area}$ was measured. Three types of killer defects were identified: Lack of Fusion (LoF) defects related to the L-PBF process, corrosion pits due to the pre-corrosion and EDM defects (see figure 17).

By testing specimens with blunt notches (showing a reduced HSV), the defect free fatigue strength $\sigma_{a,0}$ was estimated equal to 450 MPa. It is worth mentioning that the fatigue strength is close to the yield stress of the material (485 MPa). Several zones of the material might plastify under such loading.

The figure 18 shows the corresponding Kitagawa-Takahashi diagram. Experimental data of 316L L-PBF material is represented by colored markers. The El-Haddad's prediction is taken from a previous paper [32] and is discussed in the next sub-section.

A large variability in terms of killer defect morphology was observed. Lack of fusion pores can be approximated by $f_{ry} = 2, f_{rz} = 0.15$ geometry (high acuity) while EDM defects could be represented by hemispherical defect $f_{ry} = f_{rz} = 1$. Corrosion pits morphology are more difficult to model since some are high stress concentrators and other are rather smooth and shallow [31]. It seems acceptable to consider their shape varying between $f_{ry} = 2, f_{rz} = 0.15$ and $f_{ry} = f_{rz} = 1$.
1.39) and $f_{ry} = 2, f_{ry} = 0.15$ ($K_{DV} = 11.46$). It could be due to the effect of the stress gradient magnitude which leads to very different level of plasticity between element near the hot spot. Again, the competition between the local maxima and the HSV is at the origin of these differences.

As the discussed models in this paper show consistent results regarding some well-known fatigue behaviours, a confrontation with experimental data is proposed in the next subsection. The beneficial effect of taking into account the plastic behaviour of the material for fatigue strength prediction is discussed regarding the experimental data.

### 4.3. Comparison with uniaxial experimental data

In this part, experimental data collected for 316L stainless steel manufactured by Laser Powder Bed Fusion (L-PBF) under alternate symmetrical (R=-1) uniaxial loading are presented [31, 30]. The non-local approach is then compared with these data and with Linear Elastic Fracture Mechanics (LEFM) models which were the object of a previous paper [32].

#### 4.3.1. Material and experiments

The investigated material is a 316L grade stainless steel manufactured by L-PBF. The alloy was supplied as powder by TLS Technik. A Farsoon FS271M machine was used to build fatigue specimens vertically. An identical optimal manufacturing strategy was applied for each of the 5 plates produced for the experimental campaign. The material density achieved was near 99.9%.

After L-PBF processing, raw specimens were machined up to their final geometry and polished. Neither thermal nor chemical treatments were applied, the material bulk was characterized in its as-built state. More details about the fabrication of the fatigue specimens, the material static behaviour and its microstructure can be found in previous papers [31, 32]. It is worth mentioning that residual stresses were measured using X-rays diffraction and were found insignificant.

In the current study, three batches of fatigue specimens corresponding to three surface conditions are discussed: (1) polished (no other step after fabrication), (2) with one Electric Discharge Machined (EDM) hemispherical defect and (3) pre-corroded specimens. These surface states are supposed to represent typical defects due to the manufacturing process or environmental exposure in realistic service conditions. Experimental data are accessible on an open-source repository [30].

The pre-corrosion was achieved by electrochemical way in an aqueous neutral saline solution. A more precise description of the pre-corrosion setup was given in a previously published work [31].

Fatigue tests were carried under uniaxial loading with $R = -1, f = 15$ Hz and at room temperature. A step by step method was used, by increasing the stress amplitude $\sigma_a$ by 25 MPa every $10^6$ cycles. As described in a previous paper [32], the fatigue strength is finally evaluated using Maxwell and Nicholas correction [28].

In the next subsection, simulation results are confronted to the experimental data through a Kitagawa-Takahashi diagram.

#### 4.3.2. Comparison between experimental data and models

In this experimental campaign, all the fracture surfaces were observed using a Scanning Electron Microscope (SEM). On each fractography, a unique critical defect was identified as the root cause of the crack initiation and its size $\sqrt{area}$ was measured. Three types of killer defects were identified: Lack of Fusion (LoF) defects related to the L-PBF process, corrosion pits due to the pre-corrosion and EDM defects (see figure 17).

By testing specimens with blunt notches (showing a reduced HSV), the defect free fatigue strength $\sigma_{a,0}$ was estimated equal to 450 MPa. It is worth mentioning that the fatigue strength is close to the yield stress of the material (485 MPa). Several zones of the material might plastify under such loading.

The figure 18 shows the corresponding Kitagawa-Takahashi diagram. Experimental data of 316L L-PBF material is represented by colored markers. The El-Haddad's prediction is taken from a previous paper [32] and is discussed in the next sub-section.

A large variability in terms of killer defect morphology was observed. Lack of fusion pores can be approximated by $f_{ry} = 2, f_{rz} = 0.15$ geometry (high acuity) while EDM defects could be represented by hemispherical defect $f_{ry} = f_{rz} = 1$. Corrosion pits morphology are more difficult to model since some are high stress concentrators and other are rather smooth and shallow [31]. It seems acceptable to consider their shape varying between $f_{ry} = 2, f_{rz} = 0.15$ and $f_{ry} = f_{rz} = 1$.
<figure><img src="image_20.png" /><figcaption>Figure 17: Typical killer defects observed on fracture surfaces with (a) a lack of fusion defect, (b) a corrosion pit and (c) an EDM defect. Fatigue tests are performed under fully alternate uniaxial loading.</figcaption></figure>

For model prediction, the defect free fatigue limit introduced previously in this subsection is multiplied by the normalized fatigue strength (450 MPa).

Three model types are applied on the two extreme geometries introduced previously (leading to six models predictions). The three model types correspond to: elastic material behaviour with a non-local regularization ($R^* = 20 \ \mu m$), elastoplastic material behaviour ($R^* = 0 \ \mu m$) and elastoplastic material behaviour with a non-local regularization ($R^* = 10 \ \mu m$). The non-local regularization volume (through $R^*$ parameter) is manually chosen.

The elastoplastic model without non-local approach is not adapted to take into account the effect of defect in HCF domain. Indeed, as observed for elastic behaviour, the stress concentration seems of first order compared to the defect size, which is the opposite of what is seen for L-PBF 316L. It seems that taking into account the plasticity alone (without non-local approach) is not sufficient for HCF predictions.

The elastic and elastoplastic non-local models seem adapted to represent the effect of defect of the L-PBF 316L. Predictions could be made based on the defect free fatigue strength $\sigma_{a,0}$ and a characteristic length $R^*$ depending on the material. The best mean volume size is obtained for $R^*$ equal to 10 $\mu m$ for elastoplastic material and 20 $\mu m$ for elastic material. It worth noting that both plasticity and the non-local approach lead to a lowering of the stress level at the hot spot. This is the reason of a smaller non-local sphere for the elastoplastic material.

Comparing the elastic and elastoplastic behaviours with the experimental dataset does not show any improvement by taking into account the plasticity. Modeling the material behaviour with elasticity is the most adequate solution when using non-local approaches.

If $\sigma_{a,0}$ could be physically measured, the link between $R^*$ and the material microstructure is not trivial.

Figure 19 shows quality patterns images obtained from EBSD analyses carried on 316L L-PBF samples. Colors in the quality patterns are used only to differentiate grains without any link with crystalline orientations. The grain size is computed as the mean value of the square root of the area of each grain in the image 19(a) (crack propagation plane) and is approximately 12 $\mu m$. This value is close to the $R^*$ value giving the best predictions for elastoplastic (10 $\mu m$) and elastic (20 $\mu m$) materials. Then, the grain size may be a good indicator of the characteristic length $R^*$ to apply in the non-local approach.

However, the L-PBF material also contains subgrains features, such as solidification cells (around 500 nm) or group of cells, which make it difficult to reduce the microstructure into a single parameter. As suggested by Maierhofer *et al.* [27], it could be interesting to take into account more than a unique characteristic length. Indeed, the role of microstructural features acting as damage barriers might be seen as a weighted sum of their respective contributions depending on the defect size.

The $R^*$ may also be influenced by the texture, the material behavior (such as plasticity), but also depends on the operation associated to the non-local approach (a mean over a sphere in this work).

The main conclusion of the paper dealing with the analysis of these experimental results was that the defect morphology (except its size) does not seem to influence the fatigue strength of the material [31]. The current paper gives some argument to explain this observation.
<figure><img src="image_19.png" /><figcaption>Figure 17: Typical killer defects observed on fracture surfaces with (a) a lack of fusion defect, (b) a corrosion pit and (c) an EDM defect. Fatigue tests are performed under fully alternate uniaxial loading.</figcaption></figure>

For model prediction, the defect free fatigue limit introduced previously in this subsection is multiplied by the normalized fatigue strength (450 MPa).

Three model types are applied on the two extreme geometries introduced previously (leading to six models predictions). The three model types correspond to: elastic material behaviour with a non-local regularization ($R^* = 20 \ \mu m$), elastoplastic material behaviour ($R^* = 0 \ \mu m$) and elastoplastic material behaviour with a non-local regularization ($R^* = 10 \ \mu m$). The non-local regularization volume (through $R^*$ parameter) is manually chosen.

The elastoplastic model without non-local approach is not adapted to take into account the effect of defect in HCF domain. Indeed, as observed for elastic behaviour, the stress concentration seems of first order compared to the defect size, which is the opposite of what is seen for L-PBF 316L. It seems that taking into account the plasticity alone (without non-local approach) is not sufficient for HCF predictions.

The elastic and elastoplastic non-local models seem adapted to represent the effect of defect of the L-PBF 316L. Predictions could be made based on the defect free fatigue strength $\sigma_{a,0}$ and a characteristic length $R^*$ depending on the material. The best mean volume size is obtained for $R^*$ equal to 10 $\mu m$ for elastoplastic material and 20 $\mu m$ for elastic material. It worth noting that both plasticity and the non-local approach lead to a lowering of the stress level at the hot spot. This is the reason of a smaller non-local sphere for the elastoplastic material.

Comparing the elastic and elastoplastic behaviours with the experimental dataset does not show any improvement by taking into account the plasticity. Modeling the material behaviour with elasticity is the most adequate solution when using non-local approaches.

If $\sigma_{a,0}$ could be physically measured, the link between $R^*$ and the material microstructure is not trivial.

Figure 19 shows quality patterns images obtained from EBSD analyses carried on 316L L-PBF samples. Colors in the quality patterns are used only to differentiate grains without any link with crystalline orientations. The grain size is computed as the mean value of the square root of the area of each grain in the image 19(a) (crack propagation plane) and is approximately 12 $\mu m$. This value is close to the $R^*$ value giving the best predictions for elastoplastic (10 $\mu m$) and elastic (20 $\mu m$) materials. Then, the grain size may be a good indicator of the characteristic length $R^*$ to apply in the non-local approach.

However, the L-PBF material also contains subgrains features, such as solidification cells (around 500 nm) or group of cells, which make it difficult to reduce the microstructure into a single parameter. As suggested by Maierhofer *et al.* [27], it could be interesting to take into account more than a unique characteristic length. Indeed, the role of microstructural features acting as damage barriers might be seen as a weighted sum of their respective contributions depending on the defect size.

The $R^*$ may also be influenced by the texture, the material behavior (such as plasticity), but also depends on the operation associated to the non-local approach (a mean over a sphere in this work).

The main conclusion of the paper dealing with the analysis of these experimental results was that the defect morphology (except its size) does not seem to influence the fatigue strength of the material [31]. The current paper gives some argument to explain this observation.
<figure><img src="image_21.png" /><figcaption>Figure 18: Comparison between the models for extreme cases and experimental values in terms of fatigue strength for L-PBF 316L. The marker color represents the batch of the specimen (polished, EDM defect, pre-corroded), its shape the killer defect type (LoF, corrosion pit, EDM). Six model predictions are shown. The green lines correspond to hemispherical defect ($f_{ry} = f_{rz} = 1$), the red one to a high acuity defect ($f_{ry} = 2, f_{rz} = 0.15$). Plain green and red lines correspond to an elastic material (elast) with a non-local regularization ($\sigma_{a,0} = 450$ MPa, $R^* = 20$ µm). Dotted green and red lines correspond to an elastoplastic material (plast) without non-local regularization. Dashed green and red lines correspond to an elastoplastic material (plast) with a non-local regularization ($\sigma_{a,0} = 450$ MPa, $R^* = 10$ µm). The El-Haddad's model predictions are represented with a black dotted line for comparison purpose.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>Figure 19: Quality patterns images (from EBSD analyses) of the 316L L-PBF in this work (a) perpendicularly to the building direction (BD) and (b) parallely to BD.</figcaption></figure>
<figure><img src="image_20.png" /><figcaption>Figure 18: Comparison between the models for extreme cases and experimental values in terms of fatigue strength for L-PBF 316L. The marker color represents the batch of the specimen (polished, EDM defect, pre-corroded), its shape the killer defect type (LoF, corrosion pit, EDM). Six model predictions are shown. The green lines correspond to hemispherical defect ($f_{ry} = f_{rz} = 1$), the red one to a high acuity defect ($f_{ry} = 2, f_{rz} = 0.15$). Plain green and red lines correspond to an elastic material (elast) with a non-local regularization ($\sigma_{a,0} = 450$ MPa, $R^* = 20$ µm). Dotted green and red lines correspond to an elastoplastic material (plast) without non-local regularization. Dashed green and red lines correspond to an elastoplastic material (plast) with a non-local regularization ($\sigma_{a,0} = 450$ MPa, $R^* = 10$ µm). The El-Haddad's model predictions are represented with a black dotted line for comparison purpose.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Figure 19: Quality patterns images (from EBSD analyses) of the 316L L-PBF in this work (a) perpendicularly to the building direction (BD) and (b) parallely to BD.</figcaption></figure>
First, for small defect size compared to $R^*$, the effect of the morphology is insignificant in terms of fatigue strength variation. It is then possible that defects of the experimental campaign were not large enough relatively to $R^*$ to observe an effect of the morphology (the value of $R^* = 20 \ \mu m$ was set manually in figure 18).

Second, morphology parameters analysed in the paper focusing on the experimental study were 2D parameters [31], which may be not representative of the 3D defect shape.

Finally, fatigue damage is essentially a stochastic phenomena because of the high number of parameters affecting it (local material behavior, alignment of testing machine and specimens, defect population,...) [35]. Since these factors may influence one another and are not always accessible, it may be difficult to measure precisely the effect of a parameter without carrying a very large amount of fatigue tests to be representative of the material.

In the current paper, the fatigue crack initiation was modeled using a non-local approach based on finite element simulations and a multiaxial fatigue criteria. In a previous paper [32], three Linear Elastic Fracture Mechanics (LEFM) models were used and compared to experimental data. The next subsection proposes a comparison between these two categories of models.

### 4.3.3. Comparison with LEFM models

LEFM models are based on the computation of the Stress Intensity Factor (SIF) indicator $\Delta K$, which is considered as the driving force of the fatigue crack. The SIF could be computed as :

$$ \Delta K = f \Delta \sigma \sqrt{\pi a} \qquad (18) $$

with $\Delta \sigma$ the fatigue stress range, $f$ the crack geometric correction and $a$ the crack length. As proposed by Murakami [33], if $\sqrt{area}$ is used instead of the crack length $a$, $f$ is approximated to 0.65 for semi-circular cracks (surface intersecting crack) and to 0.5 for internal circular cracks.

Crack is considered as propagating if the SIF is higher than a threshold value ($\Delta K > \Delta K_{th}$). This threshold was modeled in the previous paper using (1) Murakami's model [34], (2) El-Haddad's model [12] and (3) Chapetti's model [6]. These three models gave globally similar results which were in good agreement with experimental data [32].

Compared to the non-local approach, LEFM models only take into account the defect size and position. No influence of its morphology is considered. This is consistent with the application domain of these models, which are supposed to address the relatively small defect sizes [7, 33].

On a previous article [32], three classical LEFM based models were implemented and then identified (fitted) on experimental data. One of these models was the El-Haddad's model, which could be written as follow :

$$ \left\{ \begin{array}{l} \Delta K_{th} = \Delta K_{th,LC} \sqrt{\frac{a}{a+a_0}} \\ a_0 = \frac{1}{\pi} \left[ \frac{\Delta K_{th,LC}}{f \Delta \sigma_{th,0}} \right]^2 \end{array} \right. \qquad (19) $$

with $\Delta K_{th,LC} = 9.04 \ MPa\sqrt{m}$ and $\Delta \sigma_{th,0} = 900 \ MPa$. The predictions in terms of fatigue strength based on this model are plotted in figure 18.

Results obtained with the non-local approach are then consistent with LEFM predictions for small defects relatively to the $R^*$ value. The bigger the defect, the bigger the difference between models. It is worth noting that LEFM models tend to 0 when the defect size tend to $+\infty$ instead of $\sigma_{a,0}/K_{DV}$.

LEFM models have the advantage to be fast and easy to implement while the non-local approach implied FEM and data post processing. The non-local approach is an interesting tool to explain mechanically the low effect of the morphology for small defect size depending on the material.

Furthermore, results obtained by Schonbauer and Mayer [41] for martensitic stainless steels which show a detrimental effect of increasing the defect acuity on the fatigue strength for large defects is also consistent with non-local approach predictions. Classical LEFM models could not describe this behaviour.

## 5. Conclusion

The effect of defects on the fatigue strength of metallic alloys has been investigated through the use of both a non-local approach and a multiaxial fatigue criterion. Finite element simulations allow to get the local mechanical fields on which the approach is applied. A large range of defect size and shape is investigated. The main results of this study are :
First, for small defect size compared to $R^*$, the effect of the morphology is insignificant in terms of fatigue strength variation. It is then possible that defects of the experimental campaign were not large enough relatively to $R^*$ to observe an effect of the morphology (the value of $R^* = 20 \ \mu m$ was set manually in figure 18).

Second, morphology parameters analysed in the paper focusing on the experimental study were 2D parameters [31], which may be not representative of the 3D defect shape.

Finally, fatigue damage is essentially a stochastic phenomena because of the high number of parameters affecting it (local material behavior, alignment of testing machine and specimens, defect population,...) [35]. Since these factors may influence one another and are not always accessible, it may be difficult to measure precisely the effect of a parameter without carrying a very large amount of fatigue tests to be representative of the material.

In the current paper, the fatigue crack initiation was modeled using a non-local approach based on finite element simulations and a multiaxial fatigue criteria. In a previous paper [32], three Linear Elastic Fracture Mechanics (LEFM) models were used and compared to experimental data. The next subsection proposes a comparison between these two categories of models.

### 4.3.3. Comparison with LEFM models

LEFM models are based on the computation of the Stress Intensity Factor (SIF) indicator $\Delta K$, which is considered as the driving force of the fatigue crack. The SIF could be computed as :

$$ \Delta K = f \Delta \sigma \sqrt{\pi a} \qquad (18) $$

with $\Delta \sigma$ the fatigue stress range, $f$ the crack geometric correction and $a$ the crack length. As proposed by Murakami [33], if $\sqrt{area}$ is used instead of the crack length $a$, $f$ is approximated to 0.65 for semi-circular cracks (surface intersecting crack) and to 0.5 for internal circular cracks.

Crack is considered as propagating if the SIF is higher than a threshold value ($\Delta K > \Delta K_{th}$). This threshold was modeled in the previous paper using (1) Murakami's model [34], (2) El-Haddad's model [12] and (3) Chapetti's model [6]. These three models gave globally similar results which were in good agreement with experimental data [32].

Compared to the non-local approach, LEFM models only take into account the defect size and position. No influence of its morphology is considered. This is consistent with the application domain of these models, which are supposed to address the relatively small defect sizes [7, 33].

On a previous article [32], three classical LEFM based models were implemented and then identified (fitted) on experimental data. One of these models was the El-Haddad's model, which could be written as follow :

$$ \left\{ \begin{array}{l} \Delta K_{th} = \Delta K_{th,LC} \sqrt{\frac{a}{a+a_0}} \\ a_0 = \frac{1}{\pi} \left[ \frac{\Delta K_{th,LC}}{f \Delta \sigma_{th,0}} \right]^2 \end{array} \right. \qquad (19) $$

with $\Delta K_{th,LC} = 9.04 \ MPa\sqrt{m}$ and $\Delta \sigma_{th,0} = 900 \ MPa$. The predictions in terms of fatigue strength based on this model are plotted in figure 18.

Results obtained with the non-local approach are then consistent with LEFM predictions for small defects relatively to the $R^*$ value. The bigger the defect, the bigger the difference between models. It is worth noting that LEFM models tend to 0 when the defect size tend to $+\infty$ instead of $\sigma_{a,0}/K_{DV}$.

LEFM models have the advantage to be fast and easy to implement while the non-local approach implied FEM and data post processing. The non-local approach is an interesting tool to explain mechanically the low effect of the morphology for small defect size depending on the material.

Furthermore, results obtained by Schonbauer and Mayer [41] for martensitic stainless steels which show a detrimental effect of increasing the defect acuity on the fatigue strength for large defects is also consistent with non-local approach predictions. Classical LEFM models could not describe this behaviour.

## 5. Conclusion

The effect of defects on the fatigue strength of metallic alloys has been investigated through the use of both a non-local approach and a multiaxial fatigue criterion. Finite element simulations allow to get the local mechanical fields on which the approach is applied. A large range of defect size and shape is investigated. The main results of this study are :
* Typical Kitagawa effect is observed using a volumic non-local approach as described by Taylor [46].

* A normalized Kitagawa-Takahashi is obtained by normalizing the fatigue strength by the defect free fatigue strength and the defect size by the non-local characteristic length $R^*$ (named relative size).

* For small relative sizes, the fatigue strength is governed by the Highly Stressed Volume (HSV) and the defect size. For high relative sizes, the defect shape (through the induced stress concentration) becomes the main controlling parameter. A competition between the HSV (i.e. the stress gradient) and the local maxima (i.e. the geometrical stress concentration) is highlighted.

* Under uniaxial loading, the effect of a defect seems more critical than under pure shear.

* A normalized Frost diagram is built from simulation results. It seems that increasing the acuity of the defect leads first to a decrease of the fatigue strength until a minimum is reached. A slight increase is observed after this minimum. These results are in good agreement with Guerchais et al. results [20].

* The role of the plasticity is found more significant for large than for small defects. Elastic simulations provide comparable but more conservative trends.

* Simulation results are in good agreement with laser powder bed fused 316L experimental dataset discussed in a previous paper [31]. Taking into account the plasticity does not improve model predictions. Modeling the material with simple elastic behaviour, multiaxial fatigue criterion and a non-local approach seems to be the most appropriate solution.

* For relatively small defects under uniaxial loading, simulation results are consistent with linear elastic fracture mechanics' models predictions.

The introduction of the relative size (ratio between the defect size and the characteristic length $R^*$) helps to better understood the effect of the $R^*$ parameter. The determination of this material parameter is crucial to apply the non-local approach for fatigue strength predictions. The physical link between this parameter and microstructural features is still to be clarified, even if the grain size seems to be good first approximation. Once such model's parameters fully identified, (a multiaxial fatigue criterion and a non-local approach), the fatigue strength of mechanical parts, with or without defects, under complex stress state, could be evaluated.

# 6. Acknowledgments

This research was funded by a PhD grant from the French region Pays de la Loire. The authors would like to thank the Cassiopée HPC team for the cluster management and maintenance. All the simulations discussed in this paper were run on this platform.

# References

[1] Andreau, O. (2020). *Nocivité en fatigue et contrôle de défauts produits par fabrication additive*. PhD thesis.

[2] Ben Ahmed, A., Nasr, A., Bahloul, A., and Fathallah, R. (2017). The impact of defect morphology, defect size, and SDAS on the HCF response of A356-T6 alloy. *The International Journal of Advanced Manufacturing Technology*, 92(1-4):1113–1125.

[3] Blinn, B., Klein, M., Gläßner, C., Smaga, M., Aurich, J., and Beck, T. (2018). An Investigation of the Microstructure and Fatigue Behavior of Additively Manufactured AISI 316L Stainless Steel with Regard to the Influence of Heat Treatment. *Metals*, 8(4):220.

[4] Bonneric, M., Brugger, C., Saintier, N., Moreno, A. C., and Tranchand, B. (2022). Contribution of the introduction of artificial defects by additive manufacturing to the determination of the Kitagawa diagram of Al-Si alloys. *Procedia Structural Integrity*, 38:141–148.
* Typical Kitagawa effect is observed using a volumic non-local approach as described by Taylor [46].

* A normalized Kitagawa-Takahashi is obtained by normalizing the fatigue strength by the defect free fatigue strength and the defect size by the non-local characteristic length $R^*$ (named relative size).

* For small relative sizes, the fatigue strength is governed by the Highly Stressed Volume (HSV) and the defect size. For high relative sizes, the defect shape (through the induced stress concentration) becomes the main controlling parameter. A competition between the HSV (i.e. the stress gradient) and the local maxima (i.e. the geometrical stress concentration) is highlighted.

* Under uniaxial loading, the effect of a defect seems more critical than under pure shear.

* A normalized Frost diagram is built from simulation results. It seems that increasing the acuity of the defect leads first to a decrease of the fatigue strength until a minimum is reached. A slight increase is observed after this minimum. These results are in good agreement with Guerchais et al. results [20].

* The role of the plasticity is found more significant for large than for small defects. Elastic simulations provide comparable but more conservative trends.

* Simulation results are in good agreement with laser powder bed fused 316L experimental dataset discussed in a previous paper [31]. Taking into account the plasticity does not improve model predictions. Modeling the material with simple elastic behaviour, multiaxial fatigue criterion and a non-local approach seems to be the most appropriate solution.

* For relatively small defects under uniaxial loading, simulation results are consistent with linear elastic fracture mechanics' models predictions.

The introduction of the relative size (ratio between the defect size and the characteristic length $R^*$) helps to better understood the effect of the $R^*$ parameter. The determination of this material parameter is crucial to apply the non-local approach for fatigue strength predictions. The physical link between this parameter and microstructural features is still to be clarified, even if the grain size seems to be good first approximation. Once such model's parameters fully identified, (a multiaxial fatigue criterion and a non-local approach), the fatigue strength of mechanical parts, with or without defects, under complex stress state, could be evaluated.

## 6. Acknowledgments

This research was funded by a PhD grant from the French region Pays de la Loire. The authors would like to thank the Cassiopée HPC team for the cluster management and maintenance. All the simulations discussed in this paper were run on this platform.

## References

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[3] Blinn, B., Klein, M., Gläßner, C., Smaga, M., Aurich, J., and Beck, T. (2018). An Investigation of the Microstructure and Fatigue Behavior of Additively Manufactured AISI 316L Stainless Steel with Regard to the Influence of Heat Treatment. *Metals*, 8(4):220.

[4] Bonneric, M., Brugger, C., Saintier, N., Moreno, A. C., and Tranchand, B. (2022). Contribution of the introduction of artificial defects by additive manufacturing to the determination of the Kitagawa diagram of Al-Si alloys. *Procedia Structural Integrity*, 38:141–148.
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[23] Le, V.-D., Saintier, N., Morel, F., Bellett, D., and Osmond, P. (2018). Investigation of the effect of porosity on the high cycle fatigue behaviour of cast Al-Si alloy by X-ray micro-tomography. *International Journal of Fatigue*, 106:24–37.

[24] Lemaitre, J., Chaboche, J., Benallal, A., and Desmorat, R. (2009). *Mécanique Des Matériaux Solides - 3ème Édition*. Physique. Dunod.

[25] Liang, X. (2020). *High Cycle Fatigue Behavior of Additive Manufactured Stainless Steel 316L: Free Surface Effect and Microstructural Heterogeneity*. PhD thesis, École Nationale Supérieure d'Arts et Métiers.

[26] Liang, X., Robert, C., Hor, A., and Morel, F. (2021). Numerical investigation of the surface and microstructure effects on the high cycle fatigue performance of additive manufactured stainless steel 316L. *International Journal of Fatigue*, 149:106273.

[27] Maierhofer, J., Gänser, H.-P., and Pippan, R. (2015). Modified Kitagawa–Takahashi diagram accounting for finite notch depths. *International Journal of Fatigue*, 70:503–509.

[28] Maxwell, DC. and Nicholas, T. (1999). A Rapid Method for Generation of a High Cycle Fatigue. In Panontin, TL. and Sheppard, SD., editors, *Fatigue and Fracture Mechanics: 29th Volume*, pages 626–641. ASTM International, West Conshohocken, PA.

[29] McDowell, D. and Dunne, F. (2010). Microstructure-sensitive computational modeling of fatigue crack formation. *International Journal of Fatigue*, 32(9):1521–1542.

[30] Merot, P., Gallegos Mayorga, L., Morel, F., Pessard, E., Buttin, P., and Baffie, T. (2022a). 316L L-PBF fatigue dataset.

[31] Merot, P., Morel, F., Gallegos Mayorga, L., Pessard, E., Buttin, P., and Baffie, T. (2021). Observations on the influence of process and corrosion related defects on the fatigue strength of 316L stainless steel manufactured by Laser Powder Bed Fusion (L-PBF). *International Journal of Fatigue*, 155:106552.

[32] Merot, P., Morel, F., Pessard, E., Gallegos Mayorga, L., Buttin, P., and Baffie, T. (2022b). Fatigue strength and life assessment of L-PBF 316L stainless steel showing process and corrosion related defects. *Engineering Fracture Mechanics*, 276:108883.

[33] Murakami, Y. (2002). *Metal Fatigue: Effects of Small Defects and Nonmetallic Inclusions*. Elsevier.

[34] Murakami, Y. and Endo, M. (1994). Effects of defects, inclusions and inhomogeneities on fatigue strength. *International Journal of Fatigue*, 16(3):163–182.

[35] Murakami, Y., Takagi, T., Wada, K., and Matsunaga, H. (2021). Essential structure of S-N curve: Prediction of fatigue life and fatigue limit of defective materials and nature of scatter. *International Journal of Fatigue*, 146:106138.

[36] Nadot, Mendez, Ranganathan, and Beranger (1999). Fatigue life assessment of nodular cast iron containing casting defects. *Fatigue & Fracture of Engineering Materials & Structures*, 22(4):289–300.

[37] Nadot, Y., Nadot-Martín, C., Dragon, A., and Vincent, M. (2017). Competition between surface defect and grain size under fatigue loading - ARMCO iron. *Procedia Structural Integrity*, 7:530–535.

[38] Pessard, E., Bellett, D., Morel, F., and Koutiri, I. (2013). A mechanistic approach to the Kitagawa-Takahashi diagram using a multiaxial probabilistic framework. *Engineering Fracture Mechanics*, 109:89–104.

[39] Pessard, E., Lavialle, M., Laheurte, P., Didier, P., and Brochu, M. (2021). High-cycle fatigue behavior of a laser powder bed fusion additive manufactured Ti-6Al-4V titanium: Effect of pores and tested volume size. *International Journal of Fatigue*, 149:106206.
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[43] Serrano-Munoz, I., Buffiere, J.-Y., and Verdu, C. (2018). Casting defects in structural components: Are they all dangerous? A 3D study. *International Journal of Fatigue*, 117:471–484.

[44] Stopka, K. S. and McDowell, D. L. (2020). Microstructure-sensitive computational multiaxial fatigue of Al 7075-T6 and duplex Ti-6Al-4V. *International Journal of Fatigue*, 133:105460.

[45] Tanaka, K. and Akiniwa, Y. (1988). Resistance-curve method for predicting propagation threshold of short fatigue crack at notches. *Engineering Fracture Mechanics*, 30(6):863–876.

[46] Taylor, D. (1999). Geometrical effects in fatigue: A unifying theoretical model. *International Journal of Fatigue*, 21:413–420.

[47] Taylor, D. (2008). The theory of critical distances. *Engineering Fracture Mechanics*, 75(7):1696–1705.

[48] Vayssette, B., Saintier, N., Brugger, C., El May, M., and Pessard, E. (2019). Ti-6Al-4V obtained by additive manufacturing. *International Journal of Fatigue*, pages 180–195.

[49] Zerbst, U., Madia, M., Klinger, C., Bettge, D., and Murakami, Y. (2019). Defects as a root cause of fatigue failure of metallic components. III: Cavities, dents, corrosion pits, scratches. *Engineering Failure Analysis*, 97:759–776.
[40] Sausto, F., Carrion, P., Shamsaei, N., and Beretta, S. (2022). Fatigue failure mechanisms for AlSi10Mg manufactured by L-PBF under axial and torsional loads: The role of defects and residual stresses. *International Journal of Fatigue*, 162:106903.

[41] Schönbauer, B. M. and Mayer, H. (2019). Effect of small defects on the fatigue strength of martensitic stainless steels. *International Journal of Fatigue*, 127:362–375.

[42] Schönbauer, B. M., Yanase, K., and Endo, M. (2017). The influence of various types of small defects on the fatigue limit of precipitation-hardened 17-4PH stainless steel. *Theoretical and Applied Fracture Mechanics*, 87:35–49.

[43] Serrano-Munoz, I., Buffiere, J.-Y., and Verdu, C. (2018). Casting defects in structural components: Are they all dangerous? A 3D study. *International Journal of Fatigue*, 117:471–484.

[44] Stopka, K. S. and McDowell, D. L. (2020). Microstructure-sensitive computational multiaxial fatigue of Al 7075-T6 and duplex Ti-6Al-4V. *International Journal of Fatigue*, 133:105460.

[45] Tanaka, K. and Akiniwa, Y. (1988). Resistance-curve method for predicting propagation threshold of short fatigue crack at notches. *Engineering Fracture Mechanics*, 30(6):863–876.

[46] Taylor, D. (1999). Geometrical effects in fatigue: A unifying theoretical model. *International Journal of Fatigue*, 21:413–420.

[47] Taylor, D. (2008). The theory of critical distances. *Engineering Fracture Mechanics*, 75(7):1696–1705.

[48] Vayssette, B., Saintier, N., Brugger, C., El May, M., and Pessard, E. (2019). Ti-6Al-4V obtained by additive manufacturing. *International Journal of Fatigue*, pages 180–195.

[49] Zerbst, U., Madia, M., Klinger, C., Bettge, D., and Murakami, Y. (2019). Defects as a root cause of fatigue failure of metallic components. III: Cavities, dents, corrosion pits, scratches. *Engineering Failure Analysis*, 97:759–776.