soft inclusions in the local slip formation

International Journal of Fatigue 134 (2020) 105518

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3D crystal plasticity analyses on the role of hard/soft inclusions in the local slip formation

T

⁎

Riccardo Fincatoa, , Seiichiro Tsutsumia, Tatsuo Sakaib, Kenjiro Teradac a

Joining and Welding Research Institute, Osaka University, 11-1 Mihogaoka, Ibaraki, Osaka, Japan Department of Mechanical Engineering, College of Science and Engineering, Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu, Shiga 525-8577 Japan c Department of Civil Engineering, Tohoku University, Aza-Aoba 06, Aramaki, Aoba-ku, Sendai 980-8579, Japan b

A R T I C LE I N FO

A B S T R A C T

Keywords: Very high cycle fatigue Interior inclusion FEM analysis Stress distribution Local plastic deformation

The present work aims to investigate the phenomenon of local plastic slips formation in the surrounding of inclusions by means of 3D finite element analyses from a mesoscale point of view. Isotropic elastic spherical Al2O3 and MnS inclusions are modelled inside a cylindrical ferritic body subjected to tensile loading conditions with the superimposition of hydrostatic pressure. A crystal plasticity constitutive model within the framework of finite strains is adopted for the description of the matrix together with an anisotropic elasticity tensor. Several crystallographic orientations are considered, in relation to the loading direction, to observe the stress distribution and the formation of the plastic deformation in the surrounding of the inclusion. Two modelling strategies of the matrix-defect interface bond are investigated.

1. Introduction In civil, automotive, and aerospace industries, the ultimate goal is to improve the design of components to extend the service life and minimise the maintenance costs. In high-strength steels under high or very high cycle fatigue regime, the failure mechanism originates from internal defects in the form of micro-cracks and subsequently, it propagates until the formation of macro-cracks and the complete rupture of the material [1,2]. Several experimental studies focused on understanding the role played by the material heterogeneities or inclusions in crack formation [3–5]. Murakami’s exhaustive overview [4], and the recent works published by Sakai [2] and Karr et al. [6] contributed to clarifying the defect characteristics that mostly influence the fatigue life of metallic parts. In detail, crack formation in the surrounding of the inclusions or voids is attributed to the local stress distribution alteration in the matrix as a consequence of the type of defect (inclusion, pore, shrinkage, etc.), its shape (spherical, ellipsoidal, cubic, complex, etc.), its position inside the part (internal, surface), and the size of the defect. In very high cycles fatigue (VHCF) conditions or high cycles fatigue (HCF) problems, high-strength bearing steels tend to initiate the cracks at internal inclusions, especially if the surface is sufficiently smooth and does not contain relevant stress gradients [7]. The extensive VHCF experimental campaign conducted by Sakai et al. [2,8] confirmed that in most cases, regardless of the size or type of the inclusion, high⁎

strength steel subjected to fully reverse loading conditions gives internal failure initiation. Vincent et al. [9] offer an interesting study on the characterisation of fatigue around defects. This work follows the previous experimental campaign in [10], where the influence of spherical and ellipsoidal defects was analysed under tensile and torsional loading conditions. Besides, Vincent et al. tried extending the application of the defect stress gradient (DSG) [11], combining it with the equivalent inclusion method (EIM) by Eshelby [12]. The purpose of this research is to analytically compute the stress distribution around the inclusions, without the recourse to FE analyses. The results demonstrated a good predictive ability of the upgraded version of the DSG method. However, some discrepancies were also obtained due to local plasticisation in the neighbourhood of the ellipsoidal inclusions. The authors concluded that the role of plastic deformations induced around the defect is fundamental for the correct description of stress distribution. The work carried out by Alfredsson and Olsson [7] is worth mentioning. Alfredsson and Olsson used uniaxial experiments and numerical simulations to investigate the fatigue crack initiation at inclusions and the subsequent crack growth. A spherical aluminium oxide (Al2O3) inclusion was modelled by FE inside a cubic sample. Two different conditions were analysed: a perfect bond between the matrix and the defect and debonded bodies with different Coulomb friction coefficients. The authors concluded that the debonded case is more susceptible to fatigue.

Corresponding author. E-mail address: fi[email protected] (R. Fincato).

https://doi.org/10.1016/j.ijfatigue.2020.105518 Received 2 October 2019; Received in revised form 24 January 2020; Accepted 28 January 2020 Available online 30 January 2020 0142-1123/ © 2020 Elsevier Ltd. All rights reserved.

International Journal of Fatigue 134 (2020) 105518

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Cerullo and Tvergarard [13] proposed a micromechanical numerical study on the effects of inclusions in fatigue failure. Their work focuses on fatigue investigations for wind turbine bearings and considers two types of inclusions: an aluminium oxide and a titanium nitrate (TiN). The analyses are limited to 2D geometries. However, the study considers the size and shape effects of the inclusions, modelling circular Al2O3 and square TiN defects with various volume fractions. It was concluded that the damage factor, evaluated with a modified version [14] of the Dang Van criterion [15], is higher in the matrix material close to the inclusion, especially if the inclusion has sharp corners. Stiff and compliant particles are examined in this paper, which have come to be described as “hard” and “soft” in the literature, and as a consequence, this terminology is used throughout this paper. In detail; this paper aims to investigate the effect of spherical Al2O3 (hard) and MnS (soft) inclusions inside a ferritic matrix, which represent the most frequent crack initiation sites for high-strength steel [16]. The choice of the shape was motivated by experimental observations [1,2,7,17] on the geometry of the defects and to simplify the FE modelling of the problem. The inclusions are considered elastic with an isotropic elasticity tensor, whereas the surrounding ferritic matrix is modelled assuming an elasto-plastic behaviour within the framework of finite strain crystal plasticity. An anisotropic elasticity tensor is considered for the ferrite. The role of the plastic deformations in fatigue initiation has been underlined by the aforementioned authors [7,9,13]. The idea is to investigate the formation of plastic slips in the surrounding of the defect due to the difference in Young’s modulus between the inclusion and matrix for different orientations of the ferritic crystalline structure. Moreover, the interface between the two bodies is treated in two alternate ways: a perfect bond and a hard contact with friction. The numerical analyses are representative of internal inclusions, and they try to clarify the mechanism of the formation of local plastic slips at a nominal stress regime as a possible explanation for the crack nucleation. However, it should be highlighted that the fatigue initiation is not investigated at this stage, since no cyclic loading analyses were carried out. The study focuses the attention in understanding the influence of several factors (i.e. elastic anisotropy, crystallographic orientation, hydrostatic pressure, bond between the particle and the matrix) on the generation of irreversible deformations. Future work aims to extend the field of investigation to include the defect characteristics and different loading conditions (i.e. combination of tension/torsion, compression/torsion, shear, etc.) to offer a better characterization of the phenomenon.

Fig. 1. Schematic description of the deformation gradient decomposition.

deformation tensor d, the spin tensor w, and their elastic and plastic parts as:

d = sym [l] = de + d p = sym [l e] + sym [l p]; w = skw [l] = w e + w p = skw [l e] + skw [l p]

2.1. Flow rule for the crystallographic slip The conventional crystal plasticity models [18–20] assign a single crystal lattice structure, with several slip systems, to a single material point. The total amount of irreversible deformations for a single material point is represented by the sum of each plastic slip generated along one slip direction on the corresponding slip surface. In order to generate the single plastic slip, the stress – projected into the corresponding slip direction on a slip surface – must exceed the value of the so-called critical resolve shear stress or CRSS [21]. A generic slip system for the material point is indicated with α . The stress projection τ into a slip direction on a slip surface for the system α is called resolved shear stress (RSS) for the system α (hereafter indicated with τ (α ) ). The analytical definition of τ (α ) follows the standard approach proposed by Asaro and Lubarda, the reader is referred to [18] for a detailed description. Another important aspect to define is the slip rate γ̇(α ) , in the present work its definition has been assumed following Asaro [19]: n

γ ̇(α ) = ȧ |τ (α ) g (α ) |n sign (τ (α ) g (α ));

hαβ = qhαα

(4)

g (α )

is the slip resistance of the α where is the reference slip system, system (note that the initial slip resistance g0(α ) = τ0 ), where n controls the strain rate dependency, hαβ is the latent hardening modulus and hαα is the self-hardening function of the initial hardening modulus h 0 , q is the latent-hardening parameter and of τ0 and τs , the initial CRSS and the saturation stress respectively. At this point, it should be clarified that the numerical analyses in the Sections 3.4 and 3.5 do not aim to describe the elasto-plastic behaviour of the matrix. The idea of the work is to be able to investigate the influence of several factors on the initial local plasticisation of the material. Therefore, a proper characterisation of the hardening behaviour does not represent the goal of this paper. On the contrary, a proper definition of the material constants in Eq. (4) is fundamental to study the fatigue behaviour and the crack nucleation phenomenon.

The numerical analyses were conducted within the framework of finite strain crystal plasticity, adopting the theoretical approach proposed by Asaro and Lubarda [18]. The reader will be referred to the aforementioned literature for a complete explanation of the model. It is assumed that the total deformation gradient F can be multiplicatively decomposed into an elastic part F e , associated with the distortions of the crystal lattice, and into a plastic part Fp , associated with the crystallographic slips. Fig. 1 shows a sketch of the deformation gradient decomposition. The reference or initial configuration is one before the motion; the current configuration is one after the total motion and the intermediate configuration has been obtained without elastic deformations (by unloading from the current configuration or by applying irreversible deformation from the initial configuration). (1)

2.2. Constitutive law

(2)

The finite strain deformation of the crystal lattice is described by a hypo-elastic framework, in accordance with Asaro and Lubarda [18]:

Subsequently, the velocity gradient l can be expressed as:

̇ −1 = l e + l p l = FF

g (̇ α ) = ∑ hαβ γ ̇(β ) β=1

hαα = h 0sech2 |h 0 γ (τs − τ0)|;

2. Theoretical and computational framework

F = F e Fp

(3)

where le and lp are the elastic and plastic velocity gradient parts, respectively. At this point, it is possible to introduce the rate of

△ σ∗

2

+ σ tr [de] =  : de

(5)

International Journal of Fatigue 134 (2020) 105518

R. Fincato, et al. △

maximises the RSS. In case of material defects, Billaudeau et al. reported a series of SEM pictures displaying cracks around the inclusions just above the fatigue limit for tensile and torsional loading conditions. Fig. 3b shows the crack formed for a spherical defect under tensile loading conditions. The process that leads to the crack formation in Fig. 3b started in the slip system where the shear part of the stress tensor is maximum, usually around the top or bottom of the defect. However, the propagation of the cracks is usually observed to gradually shift along the equator of the defect and to propagate on a plane perpendicular to the loading direction. Billaudeau et al. attributed the initial formation of the cracks at the top and bottom of the particle to the stress distribution around the defect. The data on the plastic slip formation below the fatigue limit were not investigated. Thus, from an experimental point of view, it is unclear at what level of the macroscopically applied stress the matrix starts generating irreversible deformations that lead to crack formation. Therefore, it is important to study the effect of the defects on the matrix stress field and the local plastic field to understand the mechanism of crack nucleation. The following section introduces the assumptions and limitations of the present study.

where  represents the tensor of the elastic constants, σ ∗ is the corotational stress rate evaluated in the intermediate configuration and whose △

definition can be given using the Jaumann rate of the Cauchy stress σ as follows: △ σ∗

△

= σ + w p· σ − σ · w p;

△

σ = σ̇ − w · σ + σ · w

(6)

By combining equations (5) and (6), the Cauchy stress rate in the current configuration can be written as:

σ̇ =  : de − σ tr [de] + w e·σ − σ·w e

(7)

Last, recalling the definition of the plastic rate of deformation and plastic spin and the additive decomposition of the total rate of deformation and spin given in Eq. (3), Eq. (7) can be rewritten as:

σ̇ =  : [d − d p] − σ tr [d − d p] + [w − w p]·σ − σ·[w − w p]

(8)

3. Numerical analyses In general, the loading in HCF or VHCF testing is limited to stress regimes below the macroscopic yield stress. Therefore, most of the phenomenological plasticity models cannot detect the formation of plastic deformations under these conditions. Recently, the authors developed an elasto-plastic model based on an unconventional plasticity approach capable of describing a smooth development of irreversible deformation even below the macroscopic yield stress [22]. This approach has been used to model SM490 steel under HCF conditions. However, the description of the phenomenon is based on a macroscopic scale of observation and does not consider the presence of inclusions and defects. It has been experimentally observed that the potential sites for the micro-cracks formations often correspond with the generation of local plastic deformations in the matrix around inclusions or voids [23–25]. Therefore, the process must be described at a mesoscale level. Nondestructive techniques [26,27] can provide information about the formation of the cracks around inclusions and to monitor their propagations. It has been frequently observed that the crack initiation in the proximity of inclusions is accompanied by microstructural changes known as butterfly-wings (see Fig. 2). According to Vincent et al. [28], this phenomenon is strongly associated with the accumulation of plastic deformation in the matrix due to the presence of a defect with different mechanical properties. Billaudeau et al. [10] conducted several experiments in tension and torsion on C36 steel with defect-free samples and with specimens containing defects of a different shape and orientation. According to the results, the defect-free material showed the first slip band formation at the stress level of the fatigue limit (see Fig. 3a). The slip bands are tilted at 45° and take place on the slip systems with an orientation that

3.1. Numerical assumptions and limitations The constitutive equations introduced in the previous section were coded by a co-author’s research team [30] into a user-subroutine for the numerical commercial software Abaqus (version 6.14) and adapted for the investigation of the steel matrix by the authors of this work. The idea is to study alternatively the effect of small spherical aluminium oxide and manganese sulphide inclusions on the stress field of the surrounding matrix. Due to the large number of factors that contribute to the stress localisation and slip formation phenomenon [4,5], the following list states the aspects that were considered in the numerical analyses as well as the limitations of the present study.

• The difference in Young’s modulus between the inclusions and the

• •

•

•

Fig. 2. SEM image of a butterfly crack formed around Al2O3 inclusion indicates the WEA butterfly-wings having two boundaries: (a) containing a crack; (b) without a crack as observed by [29].

• 3

surrounding matrix has been regarded as one of the main factors that controls the fatigue initiations [31–33]. Therefore, the idea is to consider two types of inclusions: a hard (Al2O3) and a soft (MnS) defect. Al2O3 and MnS defects are commonly observed in highstrength steels [2,7,8,34]. The shape and size factors were not considered in the present study. Future work will focus on investigating the geometrical role of the inclusion. A previous work of the authors considered the shape effect of a circular and squared austenitic grain inside a linear elastic matrix [35]. Elastic isotropic behaviour has been considered for the two spherical inclusions. On the other hand, the matrix was modelled assuming an elasto-plastic behaviour with an anisotropic elasticity tensor. Previous works [7,10,13,32,33] neglected considering the matrix as anisotropic, assuming isotropic elasticity with different mechanical properties for both the matrix and the defect. The loading conditions are limited to uniaxial tensile tests and uniaxial tensile tests with the superimposition of a hydrostatic tensile/compressive pressure. However, several analyses were conducted varying the Euler’s angles of the ferritic body and observing the variation of the stress distribution in the surrounding of the matrix as a function of crystallographic orientations. The interface between the matrix and the inclusion is modelled with two different approaches: a perfect bond between the two bodies (i.e. PB hereafter) and two separate bodies with a contact interface (i.e. CF hereafter). The normal behaviour of the contact is modelled as ‘hard’ contact to avoid compenetration of different parts of the bodies through the contact surface under compressive conditions and not to allow the transfer of tensile stress across the interface. The tangential behaviour of the contact is modelled with friction. A last aspect that should be highlighted is that the present work

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Fig. 3. (a) Micro-plasticity below the fatigue limit in tension (σ = 240 MPa, N = 460,000 cycles), 45° tilted slip bands in a ferrite grain (Fig. 6a in [10]); (b) Crack propagation path for different defects: 160 μm spherical defect, tension (σ = 200 MPa, N = 2 × 106 cycles) (Fig. 8a in [10]).

central node pinned to prevent movements along x, z, and rotations. The lateral surface of the cylinder was free. A prescribed displacement boundary condition was applied on the top surface along the y-direction, giving a maximum nominal strain of 0.25% at a nominal straining rate of 10-3 [s−1]. The mesh consisted of 126,144 hexahedral elements with reduced integration (i.e. Abaqus element C3D8R) for 135,501 nodes. In detail, the geometry was divided into three different parts with different mesh refinement. A fine mesh was realised at the core of the cylindrical body since the purpose of the numerical analyses is to investigate the stress distribution and the formation of plastic slips surrounding the defect. Mesh 2 in Fig. 4b is extended up to twice the radius of the inclusion. The remaining part of the sample has a coarser mesh (i.e. mesh 3) since the stress is expected to be within the elastic domain. Table 1 displays detailed information of each mesh. The interface between mesh 2 and 3 was modelled with ties contact, which simulates a perfect bond between the two surfaces. The same solution was adopted for interface 1 in the set of analyses where a perfect bond between the inclusion and the matrix was assumed. Alternately, interface 1 was modelled with hard contact, with friction in the set of analyses where the two bodies were considered as separate. The value of the friction coefficient μ represents another variable of the problem. In the current study, μ = 0.1 is assumed, due to lack of information in the literature. Alfredsson and Olsson [7] conducted a parametric study of the friction coefficient, varying μ from 0.1 to 0.9. They concluded that the friction coefficient does not have a relevant influence on the fatigue. The inclusions were assumed with isotropic elastic behaviour and the values of the elastic constants were taken from the literature [39]. The body-centred cubic (BCC) structure of the matrix is assumed as

considers a single-crystal sample. The effect of grain boundaries is not considered as well as the influence of neighbour grains with different crystallographic orientations [36]. Future work will explore these aspects. 3.2. Mesh, geometry and material parameters A spherical defect, alternatively Al2O3 or MnS, was included inside a cylindrical single-crystal sample (see Fig. 4a). The ratio between the sphere and cylinder radii is 0.20, resulting in the inclusion volume fraction of 0.005. The choice of this numerical configuration aims to study the effect of a relatively small inclusion inside a ferritic grain. It is worth mentioning that, due to the wide range of sizes of Al2O3 or MnS inclusions, different configurations are also possible, for example, a large inclusion surrounded by grains with different crystallographic orientations. A previous work of Cerullo and Tvergaard [13] analysed the size effect of a circular Al2O3 and squared TiN inclusion inside a 2D representative volume element (RVE). According to their results, inclusions with a larger radius are less harmful than smaller ones. However, this aspect seems to be in contradiction with experimental results [37,38] and it can be due to the numerical implementation of the case study. The adoption of periodic boundary conditions and the hypothesis of plane strain may affect the stress distribution arising in the surrounding of the inclusion due to the fact that a bigger inclusion could reinforce the RVE reducing the stress peaks in the matrix. In this paper, the size of the inclusion corresponds to the smaller volume fraction value in [13]. A future parametric study is thus necessary to investigate the size effect. The cylinder was constrained at the base along the y-axis and the

Fig. 4. (a) Sketch of the cylindrical body and the spherical inclusion; (b) mesh. 4

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Table 1 Mesh information.

Mesh 1 Mesh 2 Mesh 3

Table 5 Numerical vs analytical comparison of the Young’s modulus. Number of elements

Number of nodes

BBC dir. || y-axis

Analytical [GPa]

Numerical [GPa]

Relative Err (%)

24,192 51,840 50,112

25,999 55,328 54,174

[1 0 0],[0 1 0],[0 0 1] [1 1 0],[1 0 1],[0 1 1] [1 1 1]

133.98 223.21 286.89

133.76 223.01 286.67

0.1045% 0.0851% 0.0767%

austenitic grain. An exact characterization of single-crystal behaviour would offer a better description of the real material behaviour. The other parameters for the definition of the slip rate in Eq. (4) are provided in Table 4.

Table 2 Elastic constants for the Al2O3 and MnS inclusions [39]. Type of inclusion

Young’s modulus [GPa]

Poisson’s ratio

Al2O3 MnS

389 138

0.25 0.30

3.3. Selection of the crystallographic orientations set A preliminary set of analyses was carried out to characterise the elastic anisotropic behaviour of the matrix and to justify the choice of the Euler’s angles sets adopted in the subsequent FE simulations. In fact, due to the high symmetry of the BCC structure, only a limited number of orientations can be taken into account. The initial set of numerical analyses consisted of a uniaxial extension of the cylindrical sample without the presence of the spherical defect. Uniaxial extension tests were carried out, varying the crystallographic orientation of the lattice. The Euler’s angles (φ , θ , ψ) for the BCC structure were considered according to the Bunge definition [44] with the so-called Z1XZ2 convention. In detail, the (0°,0°,0°) set implies the following alignments between the crystal axes and reference system axes: [1 0 0] || x-axis, [0 1 0] || y-axis, [0 0 1] || z-axis. Fig. 5 shows the Young’s modulus variation as a function of the first two Euler’s angles. The high symmetry of the BCC structure returns a limited variation of the Young’s modulus with the lattice orientation, from a minimum of 133 MPa (when the sides of the cubic structures are aligned with the pulling direction) to a maximum of 286 MPa (when the cubic diagonal is aligned with the y-axis). An additional check of the accuracy between the numerical and analytical values of the Young’s modulus was performed. The elastic modulus as a function of the mutual orientations between the lattice structure and the pulling direction are presented in Table 5, together with the relative errors. The previous analyses allowed the individuation of five representative crystallographic orientations that will be used in Sections 3.4 and 3.5 to evaluate the defect influence on the stress distribution in the ferritic matrix. For the purpose of simplicity, the symbol ‘°’ will be omitted in the following paragraphs when referring to Euler’s angle set.

Table 3 Elastic properties of ferrite [40].

Ferrite

C11 [GPa]

C12 [GPa]

C44 [GPa]

A = 2C44/(C11-C12)

233.5

135.5

118

2.41

Table 4 Crystal plasticity properties of the matrix for Eq. [40].

Ferrite

τ0 [MPa]

n

ȧ

q

g0(α )

100

100

0.001

0

τ0

anisotropic and defined by three elastic constants C11, C12, and C44. The values of the previous three constants were assumed according to Tjahjanto et al. [40] and based on the results of the x-ray diffraction measurement reported by Liu et al. [41]. C11, C12, and C44 characterise the matrix as mildly anisotropic with a Zener’s anisotropy factor [42] of 2.41. The material constants are reported in Tables 2 and 3. The nonlinear behaviour of the matrix has been assumed as perfectly elastoplastic, since the present work aims to observe the formation of the first plastic slip in the matrix depending on the local peaks of the stress. Under this assumption, the only relevant material parameter to be defined among all the material constants presented in Section 2.1 is the CRSS, which has been set to τ0 = 100 MPa. The value of τ0 derives from the calibration from experimental uniaxial stress–strain data of a dual phase poly-crystalline sample, mostly consisting of ferrite, carried out by Jacques et al. [43]. The approach used in the numerical simulations follows the idea carried out by Tjahjanto et al. [40], where the same CRSS was adopted for six ferritic grains surrounding a retained

3.4. Perfect bond between inclusion and matrix Before discussing the plastic slips formation in the surroundings of

Fig. 5. Young’s modulus distribution as a function of the (φ, θ ) pairs of Euler’s angles (values in GPa). 5

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Fig. 6. Hard inclusion: (a) peaks distribution as a function of the lattice orientation of the max Mises stress for the anisotropic sample normalised against the max Mises stress of the isotropic homogeneous solution (c) Mises stress contour field for the (0,0,0) orientation (legend in MPa). Soft inclusion: (b) peaks distribution as a function of the lattice orientation of the max Mises stress for the anisotropic sample normalised against the max Mises stress of the isotropic homogeneous solution d) Mises stress contour field for the (35.26,45,0) orientation (legend in MPa).

function of the first two Euler angles. The maps in red and green refer to the Al2O3 and MnS inclusions respectively. It can be concluded that the maximum values for the Al2O3 defects are found in correspondence with the minimum values of the Young’s modulus and vice versa (see Fig. 5). On the other hand, the investigations on the softer inclusion gave opposite results. The iso-lines of the Mises stress show, more or less, the same distribution of the Young’s modulus. In general, the normalised values of Mises stress are higher around the Al2O3, supporting Alfredsson and Olsson’s [7] conclusion that hard inclusions are more harmful than softer ones. Fig. 6a and 6b display only the peaks for the Mises stress without defining the location of the stress concentration in the surroundings of the inclusions. Fig. 6c and 6d depict the contour fields of the Mises stress for the (0,0,0) hard inclusion and the (35.26,45,0) soft inclusion respectively. These two configurations represent critical orientations for the aluminium oxide and manganese sulphide defects since, as will be clarified later, the generation of slips take place at the lower level of the nominal stress. As can be seen, the maximum values of the Mises stress for the hard inclusion are located almost on the top and bottom of the sphere, due to the shear component of the stress tensor. This might justify Billaudeau

Table 6 Crystallographic orientations selected for the numerical analyses in subsections 3.4 and 3.5. Orientation

φ [°]

θ [°]

ψ [°]

1 2 3 4 5

0 0 0 0 35.26

0 15 30 45 45

0 0 0 0 0

the defect, the elastic stress field around the inclusion has been discussed in the following paragraphs. A reference solution has been carried out by considering a homogeneous sample with isotropic elasticity (E = 208 GPa, υ = 0.287 [39]) and applying a uniformed pressure on the top of the cylinder along the y-axis up to 100 MPa, to avoid triggering any plastic deformations. Subsequently, the same boundary conditions have been applied to the sample containing the defect and with an anisotropic matrix. Fig. 6a and b depict the peak values of the max Mises stress normalised against the values obtained in the reference solution as a 6

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Fig. 7. (a) Isotropic behaviour of the ferrite without inclusion, (b) influence of the inclusions on the formation of plastic deformations.

seems to justify the fatigue failure from soft inclusions that can be observed experimentally. Furthermore, the boundary conditions were modified to consider a superimposed hydrostatic stress component by adding a compressive/ tensile pressure on the cylinder surfaces. Alternatively, four additional loading conditions were applied as a uniform pressure: −100 MPa, −50 MPa, +50 MPa, +100 MPa, along with a uniform displacement on the top of the cylinder as in the previous analyses. The idea behind this new set of investigations is to analyse the modification of the stress field and the generation of plastic deformation promoted by an additional mean stress component. Fig. 8 shows the stress-strain curves obtained considering the sample without the inclusion; the black solid line refers to the analyses without the effect of the hydrostatic pressure, the blue lines (dotted and dashed) consider a compressive pressure, and the purple lines (dotted and dashed) are representative of the material response under a tensile pressure. A first aspect to notice is that the addition of mean stress components modifies the elastic response of the material without changing the normal yield stress at which the first plastic slips take place. The curves show an increasing elastic stiffness from the +100 MPa condition to −100 MPa, without changing the nominal yield stress, marked with black dots. The variation in the elastic stiffness seems to be almost the same for different Euler’s angle orientations of the matrix. The red and green markers in Fig. 8 indicate the nominal stress and strain at which the first plastic slip occurs in the matrix due to the presence of the aluminium oxide and manganese sulphide defects. Fig. 8 demonstrates that the additional hydrostatic stress component does not change the general tendency obtained in the previous set of analyses without a superimposed pressure. The hard inclusion triggers the plastic slip at lower nominal stress regimes, except for the (35.26, 45,0) set of Euler’s angle. Fig. 9 analyses in detail the effect of the hydrostatic pressure in terms of the stress field in the surroundings of the inclusion. In detail, the two isolines maps depict the normalised nominal stress at the first plastic slip, for the two defects, against the nominal stress for the homogeneous elastic isotropic sample. Once again, it can be observed that the normalised axial stresses tend to increase from the (0,0,0) configuration to the (35.26, 45,0) for the Al2O3 defect, whereas they have the opposite trends for the MnS inclusion. It’s also interesting to observe the variation of the normalised axial stress for a fixed Euler’s angle set along the x-axis. The hard inclusion (see Fig. 9a) displays a moderate sensitivity to the variation of the superimposed pressure with lower values of the normalised stress in case of a tensile hydrostatic stress component. In particular, the (0,0,0) and the (35.26, 45,0) show higher increases, +4.8% and + 5.4% respectively, passing from the +100 MPa tensile to

et al.’s hypothesis that the crack is initially nucleated due to the maximum shear and leads to a debond, either from the top or bottom, that shifts the crack propagation on the plane along the equator of the defect and perpendicular to the loading direction. Instead, in the MnS case (Fig. 6d), the softer elastic modulus of the inclusion generates a stress localisation directly at the equator of the defects, where the crack will most likely nucleate. In the subsequent set of analyses, the formation of the plastic slips in the surroundings of the inclusions has been investigated. A displacement boundary condition was applied on the top of the cylinder up to 0.25% nominal strain to exceed the CRSS in one or more slip systems. A reference solution was carried out, once again considering the sample as homogeneous (i.e. without inclusion) and with isotropic elasticity. Five different uniaxial tensile tests were carried out considering the sets of Euler angles reported in Table 6. The results in terms of nominal stress and nominal strain have been shown in Fig. 7. The black dots in the graph mark the nominal strain at which the first plastic slip takes place. The different orientations of the crystallographic lattice induce the formation of plastic slips at different values of the nominal strain and stress. Early plastic slips are generated for the (0,15,0) and (0,30,0) orientations, whereas in case the BCC cubic diagonal is aligned with the pulling direction, the plastic slip takes place at higher values of nominal stress and strain. The same set of analyses were carried out considering the ferrite as elastically anisotropic (Fig. 7b). As shown, the elastic anisotropy varies the inclinations of the nominal stress-strain curves, which consequently modify the size of the elastic domain in the strain space. The graphs in Fig. 7 also show a polycrystalline solution, indicated by a dashed line. This curve was obtained considering the sample without inclusion and assigning a different crystallographic orientation to each element, in order to simulate a polycrystal stress-strain curve. The first plastic slip in the polycrystal analysis is indicated by a yellow marker on the curve. Fig. 7b reports the values of macroscopic nominal stress and strain, at which the first plastic slips are generated considering hard (red markers) and soft (green markers) defects. It must be noted that the stress-strain curves with and without the inclusions are not perfectly overlapping; however, to simplify, the markers have been reported on the curves for the homogeneous sample. By observing Fig. 7b, it can be concluded that the aluminium oxide defect is, in general, more harmful than the manganese sulphide, since it induces a plasticisation in the ferrite at a lower nominal stress regime. This tendency is valid for most of the set of Euler’s angles investigated, except for when the cubic diagonal of the lattice is parallel to the pulling direction. In this case, the MnS inclusion triggers the plastic slip ‘earlier’ than the Al2O3. This aspect is particularly important since it

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Fig. 8. PB interface. Nominal stress and nominal strain curves, including compressive/tensile pressure (a) (0,0,0) orientation, (b) (0,15,0) orientation, (c) (0,30,0) orientation, (d) (0,45,0) orientation, (e) (35.26,45,0) orientation.

the −100 MPa compressive condition. On the other hand, the soft inclusion (see Fig. 9b) displays a higher influence of the hydrostatic pressure in the (0,0,0) configuration with a decrease of almost 10% from the compressive to the tensile pressure boundary condition. This aspect can be also observed in Fig. 8a, where the green markers show a clear differentiation of the nominal stresses. Both the defects show monotonic decreases of the normalised nominal stresses passing from compressive to tensile conditions, which allow us to conclude that a tensile superimposed mean stress helps the plasticisation of the matrix in the surroundings of the defects. Moreover, it can be concluded that the MnS inclusion seems to be more sensitive to changes in mean stress, displaying the higher decrease in normalised axial stress.

3.5. Friction and contact between the inclusion and the matrix The bond at the interface between matrix and defects still lacks a proper characterisation, which would allow a better numerical modelling of the problem. An experimental study [45] showed a weak bonding force at the interface between Al2O3 inclusions and the surrounding matrix. Another interesting paper from Qiu et al. [46] analyses the different location of the crack formation as a function of the interface energy between the inclusion and the matrix. In the following set of analyses, the bond at the interface between the matrix and the inclusion was modelled with CF, imagining that the particle and the surrounding material are detached from the beginning. However, it must be noted that this new set of investigation also represents an abstraction. An intermediate configuration, where the defect and the inclusion are partially debonded, might be more 8

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Fig. 9. PB interface. Normalised nominal axial stress (σy, axial σy, axial − isotropic ) at the first plastic slip as a function of the hydrostatic pressure and the Euler’s angle sets. (a) hard inclusion, (b) soft inclusion.

where the normalised nominal stress at the first plastic slip has been displayed for all lattice orientations, inclusion type, and interface conditions. Moreover, an additional configuration is displayed by considering the spherical defect replaced by a spherical void at the centre of the sample. The results will be demonstrated in the following manner: black solid bars refer to the isotropic homogeneous cylinder without inclusions or void; red/green solid bars indicate the solution obtained with hard/soft inclusions and a PB between the matrix and the defect; red/green bars with diagonal stripes represent the friction with CF interface for the hard/soft inclusions; and white bars indicate the solution obtained with the spherical void. First, the results without the application of a superimposed hydrostatic pressure will be discussed in Fig. 12, followed by Fig. 13 that shows results obtained by applying tensile mean stress of +100 MPa. Fig. 14 shows results obtained with an applied compressive pressure of −100 MPa. The results obtained with the intermediate values of ± 50 MPa have not been discussed since they represent intermediate cases compared with the ± 100 MPa ones. The nominal stresses at which the first plastic slip is generated for the aluminium oxide with PB and CF shows maximum gap in the (35.26, 45, 0) lattice orientation. If CF is considered, the relative decrease with the PB solution at about 44% indicates that the interface condition plays a major role in the ferrite plasticisation around the inclusion. A relative decrease of the same order of magnitude is also applied for the manganese sulphide inclusion in the (0,0,0) orientation. If a PB is considered, the gap in nominal stress between the sample with the Al2O3 and the MnS inclusions is the highest in the (0,0,0) orientation, which corresponds to the minimum value of the elastic stiffness of the matrix, and it progressively reduces with the increase of the Young’s modulus. This phenomenon can be explained considering that for lower values of the Young’s modulus, the differences in stiffness between the aluminium oxide and the ferrite are higher than those of the manganese sulphide and the ferrite. On the contrary, when the Young’s modulus of the ferrite is maximised, the matrix and the Al2O3 are more ‘similar’ than the ferrite and MnS, and they tend to behave almost as a homogeneous body. On the other hand, if CF is considered, the deformation on top and bottom of the inclusion is free, and the sample tends to behave as if in the presence of a void. However, due to the presence of the defect, the lateral compression triggers the formation of plastic slips at a lower nominal stress regime compared to the void solutions. The importance of the correct modelling of the interface between the two bodies can be also observed in the different trends of the

representative of a real interaction between the two bodies. Future works aim to address this aspect more in detail. The choice of the friction coefficient represents another unknown in the problem. In the present work, a sensitivity study on the value of μ was not conducted, and a fixed value of μ = 0.1 was assumed in all the analyses, without distinction between static and dynamic friction coefficients. Future studies will address this aspect in detail. The boundary and loading conditions for the numerical analyses presented in this section are identical to the one discussed in Section 3.4, and the results are reported in Fig. 10. The red and green markers in Fig. 10 show a very different tendency as compared to the disposition of the same markers in Fig. 8. The first aspect to underline is that the gap in nominal stresses between the aluminium oxide and the manganese sulphide seems to be significantly small, especially for the (35.26, 45,0) crystallographic orientation. In general, Al2O3 defect is still more harmful than the MnS one in terms of plasticisation at a lower nominal stress regime for the same boundary conditions. However, depending on the imposed hydrostatic pressure, the MnS could represent the worst-case scenario within the same lattice orientation. Moreover, if the BCC solid diagonal (i.e. [1 1 1] direction) is aligned with the pulling direction, the differences between hard and soft inclusions tend to disappear, without showing the inversion in tendency that was observed in the PB interface. The effect of the hydrostatic mean stress is analysed in Fig. 11. A direct comparison between Figs. 9 and 11 reveals that the normalised nominal stress does not monotonically increase/decrease with the elastic stiffness as in the PB case. Both the hard and soft inclusions show an initial increase, followed by a decrease when the Euler’s angle set passes from (0,0,0) to (35.26, 45,0) with peaks in between the (0,15,0) and (0,30,0) lattice orientations. The range of variation of the normalised nominal stress in the Al2O3 is almost the same; whereas, higher decreases are registered for the manganese sulphide. Once again, the maximum variation is obtained for the MnS defect in correspondence with (0,0,0) angles set where the nominal stress increases 10.2% while passing from a tensile to compressive superimposed hydrostatic pressure. The general tendency that compressive mean stress delays the formation of plastic slip is confirmed in this set of simulation. 4. General discussion In this section, the results obtained in the Sections 3.4 and 3.5 will be discussed and compared by demonstrating the results in bar graphs 9

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Fig. 10. CF interface. Nominal stress and nominal strain curves, including compressive/tensile pressure (a) (0,0,0) orientation, (b) (0,15,0) orientation, (c) (0,30,0) orientation, (d) (0,45,0) orientation, (e) (35.26,45,0) orientation.

the PB is considered. The maximum gap in nominal stress between the hard and soft inclusion is obtained in the (0,0,0) orientation for the PB analyses, with a relative difference of about 34% between the two defects. The gap in nominal stress has been demonstrated through the striped bars in Fig. 12; it appears to have a less severe magnitude with a relative peak of 4.6% (obtained in the (0,15,0) orientation). It was concluded that the assumption of a PB is not conservative and could lead to an overestimation of the fatigue initiation. On the other hand, the CF analyses represent only the other extreme in modelling the bodies interface. However, it cannot be concluded that they represent the worst-case scenario, since intermediate configuration with partial bonding could enhance the stress localisation, and therefore the formations of plastic slips. Future work will investigate this aspect.

nominal stresses in different lattice orientations. The PB analyses show increasing values with the increase of the elastic stiffness for the Al2O3 defect, and vice versa for the MnS inclusion. A monotonic trend is not observed in the case of CF. The initial increasing values of the nominal stresses have been recorded for both the inclusions passing from the (0,0,0) configuration to the (0,15,0)~(0,30,0) orientation, with a subsequent decrease to the minimum values obtained when the cubic diagonal of the crystallographic structure is aligned with the pulling direction. A similar tendency is observed in the analyses of a spherical void. The aluminium oxide defect is always characterised to generate plastic slip at a lower nominal stress than the manganese sulphide in the CF analyses. The MnS could be responsible for the fatigue initiation in case

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Fig. 11. CF interface. Normalised nominal axial stress (σy, axial σy, axial − isotropic ) at the first plastic slip as a function of the hydrostatic pressure and the Euler’s angle sets. (a) hard inclusion, (b) soft inclusion.

Fig. 14. Summary of the uniaxial tests with a tensile hydrostatic pressure of −100 MPa. Normalised nominal axial stress at the first plastic slip.

Fig. 12. Summary of the uniaxial tests without hydrostatic pressure. Normalised nominal axial stress at the first plastic slip.

The configuration with the spherical void is more harmful than the presence of defect when a PB is considered. However, it does not appear to be the worst-case scenario if the interface between the particle and the matrix is completely detached. The effect of a superimposed hydrostatic pressure has been discussed by comparing Figs. 13 and 14. The difference in nominal stress between the hard and soft inclusion is maximised for the (0,0,0) Euler’s angle set for both the interface conditions and both the compressive/ tensile boundary conditions. However, relative gaps between the soft and hard inclusions with the same interface condition seem to be smaller if tensile mean stress is applied. Fig. 13 shows that the relative difference between the solid red and green bars is about 32% (in the (0,0,0) configuration), whereas it is about 2.7% between the striped red and green bars (in the (0,0,0) configuration). If compressive mean stress is applied (Fig. 14), the differences are significantly marked at about 37% and 8.2% respectively. The (35.26,45,0) lattice orientation with a tensile hydrostatic stress represents the worst-case scenario, where plastic slips are generated for the lowest normalised nominal stresses considering an aluminium oxide, a manganese sulphide defects and a spherical void. The difference in nominal stress between the two types of the inclusions or the void is almost negligible, and its magnitude is about 34% of the solution

Fig. 13. Summary of the uniaxial tests with a tensile hydrostatic pressure of + 100 MPa. Normalised nominal axial stress at the first plastic slip.

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Fig. 16. (a) Nominal stress and nominal strain curves for 10 sets of random Euler’s angles and CF interface. (b) Summary of the normalized nominal axial stress at the first plastic slip for 10 sets of random Euler’s angles and CF interface.

Fig. 15. (a) Nominal stress and nominal strain curves for 10 sets of random Euler’s angles and PB interface. (b) Summary of the normalized nominal axial stress at the first plastic slip for 10 sets of random Euler’s angles and PB interface.

Table 7 Euler’s angles for the ten random additional sets.

obtained with a isotropic homogeneous sample without imperfections. In case of compressive hydrostatic pressure, the (35.26,45,0) Euler’s angle set still represents the worst configuration. However, the differences in nominal stress among the Al2O3, MnS, and void are more significantly marked than in the tensile case. The hard inclusion gives the ‘earliest’ plastic slip formation. Additional analyses were carried out to verify the results obtained with the Euler’s angles sets defined in Section 3.3. Ten new sets of random crystallographic orientations were used to perform the same analyses in Sections 3.4 and 3.5, considering alternatively PB (Fig. 15a and b) and CF (Fig. 16a and b) conditions at the interface between the inclusion and the matrix, without the influence of a hydrostatic pressure. Table 7 reports the values of the Euler’s angles sets adopted. The crystallographic orientations in Fig. 15b and Fig. 16b are ordered considering increasing values of the elastic stiffness, and they include the (0,0,0) and the (35.26,45,0) Euler’s angles sets for sake of comparison. As it can be seen in Fig. 15b, the results show the same tendency observed in Fig. 12, where the minimum normalized nominal stress for the hard inclusion is still obtained when the crystallographic axes are aligned with the reference system (i.e. (0,0,0)). The minimum value for the MnS inclusion is obtained when the cubic diagonal is parallel to the pulling direction. For lower values of the elastic stiffness the Al2O3 inclusion induces plastic slips at lower nominal stress compared with the MnS inclusion, and vice versa for higher values of the elastic stiffness. In detail, the hard and soft inclusions have almost the same normalized nominal stress for the Euler’s angles set (150,50,65).

set

φ [°]

θ [°]

ψ [°]

set

φ [°]

θ [°]

ψ [°]

1 2 3 4 5

2 90 2 32 66

5 115 210 15 5

350 117 13 270 71

6 7 8 9 10

150 15 130 225 216

50 135 136 45 225

65 33 161 72 6

Fig. 16a and b display the results obtained with the CF as interface condition. Once again the differences between the aluminum oxide and manganese sulphide are smaller than in the PB case. Moreover, the worst scenario case is still represented by the (35.26,45,0) orientation. In conclusion, the additional analyses with the ten random crystallographic orientations confirmed the validity of the choice of the Euler’s angles done in Section 3.3. 5. Conclusions The present work aimed to investigate the effect of material inclusions inside a metallic matrix to understand the mechanism that leads to the formation of localised plastic deformations. According to the extensive experimental work conducted by various authors [1,5,7,8,10,34,45,47], several factors influence the generation of plastic deformation in the matrix surrounding the defect. The present work focused on investigating the role of the various magnitudes of the 12

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Future works will try to implement a different numerical approach to model the interface between the matrix and the defect. The shape effect is another aspect that must be included in future investigations since only spherical defects were considered in this study. The presence of sharp edges seems to enhance the stress localisation and therefore promote plastic slips [13]. The extension to torsional or the combination of tensile and torsional loading conditions will also enrich the information on the plasticisation in the defects’ surroundings.

Young’s modulus between the ferrite and two types of internal defects: aluminium oxide and manganese sulphide. In particular, the defects behaviour was assumed as elastic isotropic, whereas the ferrite behaviour was modelled with a finite strain crystal plasticity approach and an anisotropic elastic tensor. Moreover, the interface between the matrix and the defect was modelled considering two alternatives, a perfect bond, and two separate bodies, which interface considering CF properties. Additional analyses were also carried out to investigate the effect of a superimposed hydrostatic pressure (tensile and compressive) in the generation of irreversible deformations. Compared with the current literature the present work highlights several aspects that were not investigated before. In detail, most of the studies in the current literature provided a modelling of the matrix based on a phenomenological approach with elastic isotropy [7,10,13,48–50]. This implies that the effect of the different crystallographic orientations in relation with the anisotropy of the matrix are not accounted for. Moreover, the aforementioned literature considered a perfect bond between the defect and the surrounding material, not investigating the role played by the interface in the generation of plastic deformations. The role of the interface is considered only in Alfredsson and Olsson [7]. However, the study is limited to an isotropic Al2O3 inclusion with isotropic surrounding material and the role of the crystallographic orientation of the matrix is neglected. Moreover, none of the aforementioned literature considers the effect of the superimposition of a hydrostatic pressure nor offers a direct comparison with different configurations (i.e. hard/soft inclusion or void, tensile/compressive hydrostatic pressure, different interface condition). Therefore, the present paper aimed to clarify some aspects and the influence of several factors on the slip formation in the surrounding of hard/soft defects. The main findings have been summarised in the following points:

Declaration of competing interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] Sakai T. Review and prospects for current studies on very high cycle fatigue of metallic materials for machine structural use. J Solid Mech Mater Eng 2009;3:425–39. https://doi.org/10.1299/jmmp.3.425. [2] Sakai T, Nakagawa A, Oguma N, Nakamura Y, Ueno A, Kikuchi S, et al. A review on fatigue fracture modes of structural metallic materials in very high cycle regime. Int J Fatigue 2016;93:339–51. https://doi.org/10.1016/j.ijfatigue.2016.05.029. [3] Li SX. Effects of inclusions on very high cycle fatigue properties of high strength steels. Int Mater Rev 2012;57:92–114. https://doi.org/10.1179/1743280411Y. 0000000008. [4] Murakami Y. Metal fatigue, effects of small defects and nonmetallic inclusions. Oxford; 2002. doi: https://doi.org/10.1016/B978-008044064-4/50006-2. [5] Murakami Y. Effects of small defects and nonmetallic inclusions on the fatigue strength of metals. Key Eng Mater 2009. https://doi.org/10.4028/www.scientific. net/kem.51-52.37. [6] Karr U, Schuller R, Fitzka M, Schönbauer B, Tran D, Pennings B, et al. Influence of inclusion type on the very high cycle fatigue properties of 18Ni maraging steel. J Mater Sci 2017;52:5954–67. https://doi.org/10.1007/s10853-017-0831-1. [7] Alfredsson B, Olsson E. Multi-axial fatigue initiation at inclusions and subsequent crack growth in a bainitic high strength roller bearing steel at uniaxial experiments. Int J Fatigue 2012;41:130–9. https://doi.org/10.1016/j.ijfatigue.2011.11.006. [8] Sakai T, Sato Y, Oguma N. Characteristic S-N properties of high-carbon-chromiumbearing steel under axial loading in long-life fatigue. Fract Eng Mater Struct 2002;25:765–73. https://doi.org/10.1046/j.1460-2695.2002.00574.x. [9] Vincent M, Nadot-Martin C, Nadot Y, Dragon A. Fatigue from defect under multiaxial loading: Defect Stress Gradient (DSG) approach using ellipsoidal Equivalent Inclusion Method. Int J Fatigue 2014;59:176–87. https://doi.org/10.1016/j. ijfatigue.2013.08.027. [10] Billaudeau T, Nadot Y, Bezine G. Multiaxial fatigue limit for defective materials: mechanisms and experiments. Acta Mater 2004;52:3911–20. https://doi.org/10. 1016/j.actamat.2004.05.006. [11] Leopold G, Nadot Y. Fatigue from an Induced Defect: Experiments and Application of Different Multiaxial Fatigue Approaches. Fatigue Fract. Mech. 9th Int. Vol. 37th Natl. Vol., 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 194282959: ASTM International; n.d., p. 371-371–23. doi: 10.1520/STP49297S. [12] Eshelby JD. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Collect. Work. J. D. Eshelby 2007. https://doi.org/10.1007/14020-4499-2_18. [13] Cerullo M, Tvergaard V. Micromechanical study of the effect of inclusions on fatigue failure in a roller bearing. Int J Struct Integr 2015;6:124–41. https://doi.org/ 10.1108/IJSI-04-2014-0020. [14] Desimone H, Bernasconi A, Beretta S. On the application of Dang Van criterion to rolling contact fatigue. Wear 2006;260:567–72. https://doi.org/10.1016/j.wear. 2005.03.007. [15] Dang-Van K. Macro-micro approach in high-cycle multiaxial fatigue. Adv. Multiaxial Fatigue 2009. https://doi.org/10.1520/stp24799s. [16] Krupp U. Fatigue Crack Propagation in Metals and Alloys: Microstructural Aspects and Modelling Concepts; 2007. doi: 10.1002/9783527610686. [17] Lu J, Cheng G, Chen L, Xiong G, Wang L. Distribution and morphology of MnS inclusions in resulfurized non-quenched and tempered steel with Zr addition. ISIJ Int 2018;58:1307–15. https://doi.org/10.2355/isijinternational.ISIJINT-2018-081. [18] Asaro R, Lubarda V. Mechanics of solids and materials. 2006. doi: 10.1017/ CBO9780511755514. [19] Asaro RJ. Crystal plasticity. J Appl Mech 1983;50:921. https://doi.org/10.1115/1. 3167205. [20] de Souza Neto EA, Peric D, Owen DRJ. Comput. Method Plasticity 2008;55. https:// doi.org/10.1002/9780470694626. [21] Peirce D, Asaro RJ, Needleman A. An analysis of nonuniform and localized deformation in ductile single crystals. Acta Metall 1982;30:1087–119. https://doi. org/10.1016/0001-6160(82)90005-0. [22] Tsutsumi S, Fincato R. Cyclic plasticity model for fatigue with softening behaviour below macroscopic yielding. Mater Des 2018;107573. https://doi.org/10.1016/j. matdes.2018.107573. [23] Gillner K, Münstermann S. Numerically predicted high cycle fatigue properties

• The differences in Young’s modulus between the inclusion and the

•

•

•

•

matrix plays a fundamental role in the generation of plastic slip. This aspect is emphasized by the different values of the normalized nominal stress at which the first plastic slip occurs, in relation with the relative orientation of the crystallographic structure and the loading direction. Considering a PB between the matrix and the defect leads to the conclusion that the Al2O3 inclusion induces plastic slips at lower nominal stress compared with the MnS inclusion. This tendency varies with the Euler angles. In particular, when the cubic diagonal of the BCC structure is aligned with the pulling direction, the MnS particle triggers the formation of plastic deformations at a lower nominal stress compared with the Al2O3 inclusion (see Fig. 7b). Considering the interface between the defect and the matrix as CF shows lower values of the normalised nominal stress at the first plastic sleep compared with the PB analyses. Moreover, the results pointed out that the difference in nominal stress between a hard and soft inclusion are less significantly marked than the PB case (see Fig. 10 and Fig. 12). In general, the aluminium oxide inclusion is always more harmful than the MnS one in terms of fatigue initiation, which seems to confirm the findings of other authors [7,13]. It is important to underline that the modelling of the interface bond plays a fundamental role in the formation of local slips. This suggests that the interface behaviour should be considered as a main factor - together with the defect shape, size and position, etc. - for the possible interior crack formation since the generation of plastic deformation is strongly affected by this aspect. The MnS inclusion seems to be more sensitive to the variation of confining pressures. The analyses that consider CF interface show a higher influence on a superimposed pressure with the variation of the crystallographic orientation. In general, the addition of a tensile hydrostatic stress component facilitates the formation of plastic slips. Vice versa, the addition of a compressive hydrostatic stress component delays the formation of plastic slips (see Fig. 5). 13

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