# Analytical Modeling of an Oblique Edge Crack in Rolling Contact Fatigue

Authors: Francesca Di Puccio
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

Surface cracks represent a frequent cause of damage and even failure in rolling contacts, observed in gears, cams, rails, and so on. In the literature, different approaches have been applied to describe the crack behaviour by means of Fracture Mechanics parameters, such as the stress intensity factors (SIFs) and theJ-integral. In this paper, a general procedure for dealing with plane problems is presented, which is based on Linear Elastic Fracture Mechanics hypotheses. It combines the Weight Function Method for evaluating the SIFs in a loading cycle with the Kolosov-Muskhelishvili complex variable approach for estimating the nominal stress field. In this way, a completely analytical procedure can be applied for a general loading condition, assuming that the real geometry can be simplified in a half-plane with an oblique edge crack. As test case, a travelling load has been considered representing a combination of three contributions: Hertzian pressure distribution, traction force due to friction, and pressurization of the crack faces. A comparison with literature results proved that the proposed approach can be an efficient tool for SIFs estimation and crack growth description.


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## Body

## 1. Introduction

It is well known that rolling contact fatigue may be a primary cause of failure in many mechanical applications, such as cams, gears, and railway tracks, as already observed by Tallian in 1969 [1]. Damage processes as pitting, spalling, fretting [2], or even catastrophic failures [3] are frequently attributed to fatigue phenomena. This topic has attracted many researchers since the 1980s, for example, [4–7], but a renewed interest has characterized the last decade, for example, [8–11]. In most of the studies in the literature, the rolling contact between two bodies is simplified as a two-dimensional problem of a half-space with a travelling pressure distribution over its surface. Repeated loading, that is, repeated passages of such pressure distribution, causes initiation of fatigue cracks from defects or inclusions below or directly from the surface. Experimental results show cases of fatigue cracks propagating in the load movement direction with a small angle with respect to the surface (α ≈ 20°–30°) [5]. However, when only pressure distribution is considered, crack propagation should occur in a compressive field that does not actually explain experimental observations. Thus, other actions have to be taken into account, starting from a “tractive” tangential force along the surface. Bower in [7] pointed out two additional mechanisms influencing crack propagation in wet cases: the lubricant action on the crack faces inducing a “hydraulic pressure mechanism” and the presence of trapped fluid inside the crack. All these conditions complicate the theoretical investigation of the phenomenon.In [8], a survey of some approaches applied to the two-dimensional problem is presented, focused on the estimation of the stress intensity factors (SIFs), which represent the main parameters for describing the crack behaviour in Linear Elastic Fracture Mechanics [12, 13]. Keer and Bryant initially proposed the application of the dislocation method [4] subsequently followed by Bower [7] and more recently by Datsyshyn and Marchenko [8]. Murakami and Nemat-Nasser introduced the so-called “body force method” in 1983 also applied to a 3D case [6]. After Bogdański et al. in 1996 [14], the finite element method has been frequently applied in these problems, as [11, 15].The present paper describes a procedure based on the application of the Weight Function (WF) Method to evaluate the SIFs in rolling contact plane problems. The approach was initially applied by the author in 1999 [16] to compare the effects of Hertzian and Elasto-Hydrodynamic pressure distribution on the SIFs in a loading cycle. More recently, the use of WFs has been reproposed by Beghini et al. in [9, 17]. They combined the WFs with a discretization approach to estimate the nominal stress field while, in this study as in [16], the Kolosov-Muskhelishvili complex variable method is considered.The aim of this study is to present the theoretical foundations and the implementation features of the proposed approach since it represents a powerful tool for investigating this type of problems and can be easily extended to different loading conditions.


## 2. Theoretical Background

In Linear Elastic Fracture Mechanics, it is assumed that the behaviour of a crack is governed by the elastic stress field at its tip, which is characterized by the stress intensity factors (SIFs), commonly denoted asK. This means that the effect of the loading conditions on the crack propagation may be evaluated in terms of K.In a general plane problem, two types of fracture modes can be distinguished: the opening of the crack faces (mode I) and their relative sliding (mode II), shown in Figure1(a). For completeness, we recall that a third mode is present in 3D cases. For each mode, a SIF is defined, labelled as KI or KII, respectively. In a loading cycle, the behaviour of a plane crack depends on the time history of the two SIFs that represents a measure for assessing rolling contact fatigue. Thus, SIF evaluation is a key issue in these problems.Figure 1
(a) Crack fracture modes in plane; (b) left: couple of unitary forces opening the crack; right: general traction distributions on the crack faces.
(a) (b)


### 2.1. The Weight Functions Method

The WF Method offers a powerful tool for evaluating the SIFs since it requires a rather simple stress analysis and, obviously, the knowledge of the WFs that for many cases are available in the literature [20, 21].In order to give a brief overview of the physical meaning of the WFs, let us consider a simple geometry shown in Figure1(b) (left), representing a panel with a crack of length a loaded by a couple of unitary forces F at a distance x∗ from the crack mouth. Since this is a symmetrical opening action for the crack, the crack behaviour depends only on KI that can be estimated as KI=hI(x∗,a), where hI(x,a) is the WF of this specific case. Conversely, hI(x,a) represents the SIF for the geometry under examination produced by a couple of unitary forces applied in x. Due to the linearity of the problem, the superposition principle can be applied to evaluate the SIF for a general distribution ϕ(x) of normal actions on the crack faces shown in Figure 1(b) (right); that is,(1)KI=∫0ahIx,aϕxdx.However, the WFs Method can be applied also for a general system of forces, not limited to tractions on the crack faces. The idea is described in Figure2(a), representing a double edge cracked panel in tension (again mode I). According to the superposition principle, the SIF produced in the cracked panel may be considered as the sum of the one produced in the uncracked panel by the external applied loads (no crack implies KI=0) and the one caused by a stress distribution acting on the crack lips. It can be observed that such a stress distribution is opposite to the one acting in the integer panel (nominal stress field), as the crack faces in the cracked panel are free. Combining this result with the above described meaning of the WFs, the SIF for the cracked panel can be calculated as(2)KI=∫0ahIx,aσnxdx.In this way, once the WF is known for the geometry under examination, it is sufficient to evaluate the nominal normal stresses σn(x) in the integer body for determining the SIF. The extension to mode II loading is straightforward, KII being connected, in a symmetrical geometry, to the nominal shear stresses τn(x); that is,(3)KII=∫0ahIIx,aτnxdx.Two points about the sign of the SIFs are worth being underlined. Positive nominal normal stresses σn(x) in the uncracked panel correspond to tensile stresses and consequently to compressive actions on the crack lips, as in Figure 2(a) (right). These compressive actions produce a positive KI. Thus, in (2), the sign of nominal stresses (plus) is maintained and not reversed. Conversely, compressive nominal normal stresses mean tractive forces on the crack faces that would close the crack causing contact between the lips. Such contact actions guarantee that, in any case, KI≥0. Moreover, the sign of KII is arbitrary as the sign of shear stresses τn and they are chosen coherently so that when τn>0 also KII>0 and vice versa.Figure 2
(a) Superposition principle for the SIF in a cracked panel. (b) Panel with an oblique crack in tension.
(a) (b)When the geometry is not symmetrical as in the previous example, for example, when the crack is oblique as in Figure2(b), a coupled influence of normal and shear stresses on KI and KII has to be taken into account and the complete SIF expressions become(4)KI=∫0ahI,σx,aσnx+hI,τx,aτnxdx,KII=∫0ahII,σx,aσnx+hII,τx,aτnxdx.It must be underlined that, for these more complex cases, the WFs have been reported rather recently in the literature and only for few geometries, as a cracked half-plane [18, 19].


### 2.2. Stress Field in a Half-Plane

Let us consider a homogenous, isotropic linear elastic half-plane, lying in the party≤0 of a Cartesian x-y frame. Neglecting volume force distributions, the stress field can be described using the so-called Airy’s function A(x,y) [22] as(5)σxx=∂2Ax,y∂y2,σyy=∂2Ax,y∂x2,σxy=∂2Ax,y∂x∂yholding in both plane stress and plane strain conditions. In these cases, the first invariant of the stress tensor must satisfy the following relationship:(6)∂2∂x2+∂2∂y2σxx+σyy=0orΔσxx+σyy=0so that Airy’s function must be of class C4 and biharmonic; that is,(7)ΔΔAx,y=0.Introducing the complex variable z=x+iy, with z- its complex conjugate, A(z) can be written starting from the general expression of a biharmonic function as(8)ΔΔAz=0⟶2Az=Rez¯φz+χz,where φ(z) and χ(z) are holomorphic functions. Accordingly, the stress components are related through the following equations:(9)σxx+σyy=4Reφ′z=4ReΦz,σyy-σxx+2iσxy=2z¯φ′′z+χ′z=2z¯Φ′z+Ψz,where Φz=φ′(z) and Ψz=χ′(z), known as Muskhelishvili’s potential functions, are conveniently introduced [23].Given the normalN and shear T load distributions on the boundary (i.e., along the x-axis), the two potentials can be calculated as(10)Φz=-12πi∫-∞∞Nl-iTll-zdl,Ψz=-12πi∫-∞∞Nl+iTll-zdl-Φz-zΦ′z.Both N(l) and T(l) have to be continuous and tend to zero (at least as 1/l) when l→∞. When only normal actions N are applied, the stress tensor components are(11)σyy=2ReΦz+2yImΦ′zσxx=2ReΦz-2yImΦ′zσxy=-2yReΦ′z→at  the  boundaryy=0σxx=σyy=2ReΦx+i0σxy=0.On the other hand, for only T we obtain(12)σyy=Rez¯-zΦ′zσxx=Re4Φz-z¯-zΦ′zσxy=Imz¯-zΦ′z-2Φz→at  the  boundaryy=0σyy=0σxx=4ReΦx+i0σxy=-2ImΦx+i0.


## 2.1. The Weight Functions Method

The WF Method offers a powerful tool for evaluating the SIFs since it requires a rather simple stress analysis and, obviously, the knowledge of the WFs that for many cases are available in the literature [20, 21].In order to give a brief overview of the physical meaning of the WFs, let us consider a simple geometry shown in Figure1(b) (left), representing a panel with a crack of length a loaded by a couple of unitary forces F at a distance x∗ from the crack mouth. Since this is a symmetrical opening action for the crack, the crack behaviour depends only on KI that can be estimated as KI=hI(x∗,a), where hI(x,a) is the WF of this specific case. Conversely, hI(x,a) represents the SIF for the geometry under examination produced by a couple of unitary forces applied in x. Due to the linearity of the problem, the superposition principle can be applied to evaluate the SIF for a general distribution ϕ(x) of normal actions on the crack faces shown in Figure 1(b) (right); that is,(1)KI=∫0ahIx,aϕxdx.However, the WFs Method can be applied also for a general system of forces, not limited to tractions on the crack faces. The idea is described in Figure2(a), representing a double edge cracked panel in tension (again mode I). According to the superposition principle, the SIF produced in the cracked panel may be considered as the sum of the one produced in the uncracked panel by the external applied loads (no crack implies KI=0) and the one caused by a stress distribution acting on the crack lips. It can be observed that such a stress distribution is opposite to the one acting in the integer panel (nominal stress field), as the crack faces in the cracked panel are free. Combining this result with the above described meaning of the WFs, the SIF for the cracked panel can be calculated as(2)KI=∫0ahIx,aσnxdx.In this way, once the WF is known for the geometry under examination, it is sufficient to evaluate the nominal normal stresses σn(x) in the integer body for determining the SIF. The extension to mode II loading is straightforward, KII being connected, in a symmetrical geometry, to the nominal shear stresses τn(x); that is,(3)KII=∫0ahIIx,aτnxdx.Two points about the sign of the SIFs are worth being underlined. Positive nominal normal stresses σn(x) in the uncracked panel correspond to tensile stresses and consequently to compressive actions on the crack lips, as in Figure 2(a) (right). These compressive actions produce a positive KI. Thus, in (2), the sign of nominal stresses (plus) is maintained and not reversed. Conversely, compressive nominal normal stresses mean tractive forces on the crack faces that would close the crack causing contact between the lips. Such contact actions guarantee that, in any case, KI≥0. Moreover, the sign of KII is arbitrary as the sign of shear stresses τn and they are chosen coherently so that when τn>0 also KII>0 and vice versa.Figure 2
(a) Superposition principle for the SIF in a cracked panel. (b) Panel with an oblique crack in tension.
(a) (b)When the geometry is not symmetrical as in the previous example, for example, when the crack is oblique as in Figure2(b), a coupled influence of normal and shear stresses on KI and KII has to be taken into account and the complete SIF expressions become(4)KI=∫0ahI,σx,aσnx+hI,τx,aτnxdx,KII=∫0ahII,σx,aσnx+hII,τx,aτnxdx.It must be underlined that, for these more complex cases, the WFs have been reported rather recently in the literature and only for few geometries, as a cracked half-plane [18, 19].


## 2.2. Stress Field in a Half-Plane

Let us consider a homogenous, isotropic linear elastic half-plane, lying in the party≤0 of a Cartesian x-y frame. Neglecting volume force distributions, the stress field can be described using the so-called Airy’s function A(x,y) [22] as(5)σxx=∂2Ax,y∂y2,σyy=∂2Ax,y∂x2,σxy=∂2Ax,y∂x∂yholding in both plane stress and plane strain conditions. In these cases, the first invariant of the stress tensor must satisfy the following relationship:(6)∂2∂x2+∂2∂y2σxx+σyy=0orΔσxx+σyy=0so that Airy’s function must be of class C4 and biharmonic; that is,(7)ΔΔAx,y=0.Introducing the complex variable z=x+iy, with z- its complex conjugate, A(z) can be written starting from the general expression of a biharmonic function as(8)ΔΔAz=0⟶2Az=Rez¯φz+χz,where φ(z) and χ(z) are holomorphic functions. Accordingly, the stress components are related through the following equations:(9)σxx+σyy=4Reφ′z=4ReΦz,σyy-σxx+2iσxy=2z¯φ′′z+χ′z=2z¯Φ′z+Ψz,where Φz=φ′(z) and Ψz=χ′(z), known as Muskhelishvili’s potential functions, are conveniently introduced [23].Given the normalN and shear T load distributions on the boundary (i.e., along the x-axis), the two potentials can be calculated as(10)Φz=-12πi∫-∞∞Nl-iTll-zdl,Ψz=-12πi∫-∞∞Nl+iTll-zdl-Φz-zΦ′z.Both N(l) and T(l) have to be continuous and tend to zero (at least as 1/l) when l→∞. When only normal actions N are applied, the stress tensor components are(11)σyy=2ReΦz+2yImΦ′zσxx=2ReΦz-2yImΦ′zσxy=-2yReΦ′z→at  the  boundaryy=0σxx=σyy=2ReΦx+i0σxy=0.On the other hand, for only T we obtain(12)σyy=Rez¯-zΦ′zσxx=Re4Φz-z¯-zΦ′zσxy=Imz¯-zΦ′z-2Φz→at  the  boundaryy=0σyy=0σxx=4ReΦx+i0σxy=-2ImΦx+i0.


## 3. Materials and Methods

### 3.1. Geometrical Assumptions

Rolling contact fatigue affects many mechanical components, from gears to roller bearings. Each case is characterized by its own geometry and working conditions. However, with some simplifying assumptions, the presented approach can be applied to most of them.As far as the geometry is concerned, we will assume that the crack lengtha is much smaller than the main dimension of the problem (e.g., radius of curvature r≫a) so that the crack region can be described as a half-plane with a surface crack, as shown in Figure 3(a). A reference frame S on the half-plane is assumed, with origin at the crack mouth (Figure 3(b)), x-axis along the boundary, and the body extending over y≤0, as for (10)–(12).A surface crack usually initiates from the free surface and grows through mode I and II mechanisms at an angle α with respect to the surface. Typical values of α are in the range 20°–30° with the tip of the crack shifted towards the rolling direction with respect to its mouth.Figure 3
(a) Scheme of the geometry: from the real problem to a cracked half-plane. (b) Reference frames and other symbols.
(a) (b)


### 3.2. Weight Functions

For an oblique edge crack in a half-plane, two sources for the WFs can be found: [18, 19]. In the first study, a solution is provided for 20°≤α≤90° through 80 coefficients of power series expansions, while in the second case 8 coefficients are given, but only for some discrete values of α (15°, 30°, 45°, 60°, and 90°) (see Appendix for complete details). Such functions are compared in Figure 4 for two angles of interest, that is, α = 15° and α = 30°, for a unitary crack length. It can be observed that the four functions are very similar apart from the mixed term hI,τx,a in the range 0≤x/a≤0.6. Actually, they are also proposed assuming a different positive direction for the tangential actions τn (and thus for KII), as detailed in Appendix, so care should be taken when choosing the WFs.Figure 4
Comparison of the WFs from [18, 19] for an oblique edge crack in a half-plane for α = 15° and α = 30°, a=1. (solid line [18], dashed line [19]).


### 3.3. Loading Conditions

In this simplified geometry, fatigue actions are described as travelling loads that move along thex-axis. In rolling contact fatigue, travelling loads are usually distinguished in(a)
normal loads (N-type): contact pressures, typically approximated with Hertzian contact formulas,(b)
tractive loads (T-type): tangential actions, representing frictional forces, generally expressed as f|N|, with f coefficient of friction. A sign is attributed to f to simulate driving (i.e., in the motion direction, f>0) or driven (i.e., opposite to the motion direction, f<0) conditions.We will assume that these loads simply translate along thex-axis maintaining their characteristics. Thus, a second reference frame St{Ω,xt,yt} is introduced where they can be easily defined and remain constant. Accordingly, the normal load due to Hertzian pressure can be expressed as(13)Nxt=-p01-xtaH2for-aH≤xt≤aH,where p0 is the maximum pressure and aH the contact half-width. The minus sign indicates a compressive action. Therefore, the tangential load is(14)Txt=fp01-xtaH2.In dry conditions, these are the external loads. Additionally, contact actionsσc between the crack lips can arise to avoid their overlapping as a consequence of compressive forces. Actually, the crack can be totally or partially closed, depending also on the crack size and on the load position (in particular on the ratio a/aH). A correct formulation of the problem can be found, for example, in [8, 11], and introduces important complexities to the solution passing through the estimation of the displacement field (not only at the crack tip). Thus, most frequently general cases are studied with finite element analyses, as in [11]. In this paper, a simplification proposed in [9, 16] is adopted, which can be useful for dealing with short cracks, where the crack is mainly open or totally closed. The simplification assumes a proportionality between those contact actions σc on the crack lips and the nominal normal tractions σn. The scale factor between σn and σc is calculated imposing KI=0, as described in Section 3.4 (see (20)-(21)). Contact actions can imply further frictional tangential tractions between the crack lips that can be in stick or slip condition [7, 24].In the so-called wet conditions, hydraulic loads due to the presence of fluid (lubricant) within the crack must also be considered. They can affect crack propagation in a double way, as shown in Figure5: direct “hydraulic pressure mechanism” and entrapment of the fluid inside the crack as stated by Bower in [7].Figure 5
Actions on the crack faces due to the presence of fluid.The description of the first effect requires an assumption on the value of fluid pressure within the crack. Generally, it is considered uniform and equal to the pressure at the crack mouth (as in [7]), or varying linearly from the mouth to the tip where it becomes zero (as in [25]). Both cases can be easily considered with the WFs.On the other side, fluid entrapment is quite difficult to be evaluated even if it represents a very critical condition for defects propagation. It takes into account the fact that as the contact passes over the crack mouth, a local closure of the faces may be observed in rather long cracks, that is, withaH<a. In this case, for some time, the lubricant may remain trapped inside the crack though the lips are closed at the mouth. A volume conservation for the lubricant is usually considered in order to evaluate the actions at the crack tip. As already stated, dealing with contact between the crack lips results in important complications, particularly when the conservation of volume must be guaranteed. However, the condition of fluid entrapment can be considered less dangerous than the crack pressurization as it is assumed that pressure in the trapped fluid is lower than the external one.


### 3.4. Model Implementation

The model of the oblique surface crack has been written in a completely symbolic form in Mathcad. Equations (10)–(14) have been used to estimate the nominal stress field that, for a given load, will be written as a function of xt and yt. Then, the stress tensor components inS are simply obtained replacing xt→x-d and yt=y, obtaining the symbolic functions(15)σyy=σyyx,y,d,σxx=σxxx,y,d,σxy=σxyx,y,d.Then, since the nominal normal and shear stresses are required along the crack faces for applying (4), the stress tensor T must be expressed in a rotated reference frame Sc{O,xc,yc} with O, at the crack mouth and xc along the crack (Figure 3(b)). That can be done as(16)TSc=RTSRT,where, for positive α,(17)R=cos⁡α-sin⁡αsin⁡αcos⁡α.Then, the nominal normal and shear stresses on the crack lip (σn and τn) having normal yc can be written as a function of xc and d, replacing x=xccos⁡αy=-xcsin⁡α, obtaining(18)σnx,y,d=σxxx,y,dsin2⁡α+σyyx,y,dcos2⁡α+2σxyx,y,dcos⁡αsin⁡ασnxc,d⟷σnxccos⁡α,-xcsin⁡α,dτnx,y,d=σxxx,y,d-σyyx,y,dcos⁡αsin⁡α+σxyx,y,dcos2⁡α-sin2⁡ατnxc,d⟷τnxccos⁡α,-xcsin⁡α,d.It can be noticed that positive values ofσn and τn mean stresses in agreement with the direction of yc and xc, respectively. At this point, the SIFs can be calculated by means of (4), changing the sign of τn if the WFs from [19] are chosen, that is, as anticipated in Section 3.2 and detailed in the Appendix.In order to reduce the computational time, it is convenient to select some discrete values ofd, for example, for -3aH≤d≤3aH, and of xc, for 0≤xc<a. Consequently, the integrands in (4) can be numerically estimated, fitted with a spline in xc and integrated for every value of d.At this point, a check of the obtained SIFs has to be done since, as already stated,KI must be nonnegative; that is,(19)KId=∫0ahI,σxc,aσnxc,d+hI,τxc,aτnxc,ddxc≥0.When (19) is not satisfied, it means that the crack is closed and contact actions σc on the crack lips must be introduced. According to the simplification introduced in the previous section, such contact actions are estimated as(20)σcxc,d=λdσnxc,d,where λ(d) is calculated imposing KI=0; that is,(21)∫0ahI,σxc,aσnxc,d+hI,τxc,aτnxc,ddxc+λ∫0ahI,σxc,aσnxc,ddxc=0⟶λ=-∫0ahI,σxc,aσnxc,d+hI,τxc,aτnxc,ddxc∫0ahI,σxc,aσnxc,ddxc.Therefore, alsoKII must be recalculated as(22)KIId=∫0a1+λhII,σxc,aσnxc,d+hII,τxc,aτnxc,ddx.Due to these contact actions, also frictional loads on the crack faces can arise, whose contribution to the SIFs is frequently negligible and thus not considered in this study.For wet conditions, fluid pressure on the crack facesp(xc,d) must be taken into account adding a pressurization contribute to the SIFs Kp; that is,(23)KIpd=∫0ahI,σxc,apxc,ddxc,KIIpd=∫0ahII,σxc,apxc,ddxc.As already mentioned, p(xc,d) may be constant o linearly decreasing to zero at the crack tip; that is,(24)pxc,d=pmdorpxc,d=pmd1-xca,where pm is the pressure at the crack mouth, from (13)(25)pmd=0ifd≤-aHord≥aHp01-daH2otherwise.Thus, Kp can be easily calculated as(26)KMpd=pmd∫0ahM,σxc,adxcorKMpd=pmd∫0ahM,σxc,a1-xcadxcwith M=I or II, where the integrals can be estimated in closed form, producing a scalar quantity depending on a and α (see Appendix).


### 3.5. Numerical Example

In order to validate the proposed model, some comparisons with results from the literature are discussed in Results. Unless stated otherwise, the following numerical data taken from a case used by Datsyshyn and Marchenko [8] has been considered: α = 25°, p0=840 MPa, aH=0.22 mm, and a/aH=0.5. For this value of α only the WF in [18] can be used.


## 3.1. Geometrical Assumptions

Rolling contact fatigue affects many mechanical components, from gears to roller bearings. Each case is characterized by its own geometry and working conditions. However, with some simplifying assumptions, the presented approach can be applied to most of them.As far as the geometry is concerned, we will assume that the crack lengtha is much smaller than the main dimension of the problem (e.g., radius of curvature r≫a) so that the crack region can be described as a half-plane with a surface crack, as shown in Figure 3(a). A reference frame S on the half-plane is assumed, with origin at the crack mouth (Figure 3(b)), x-axis along the boundary, and the body extending over y≤0, as for (10)–(12).A surface crack usually initiates from the free surface and grows through mode I and II mechanisms at an angle α with respect to the surface. Typical values of α are in the range 20°–30° with the tip of the crack shifted towards the rolling direction with respect to its mouth.Figure 3
(a) Scheme of the geometry: from the real problem to a cracked half-plane. (b) Reference frames and other symbols.
(a) (b)


## 3.2. Weight Functions

For an oblique edge crack in a half-plane, two sources for the WFs can be found: [18, 19]. In the first study, a solution is provided for 20°≤α≤90° through 80 coefficients of power series expansions, while in the second case 8 coefficients are given, but only for some discrete values of α (15°, 30°, 45°, 60°, and 90°) (see Appendix for complete details). Such functions are compared in Figure 4 for two angles of interest, that is, α = 15° and α = 30°, for a unitary crack length. It can be observed that the four functions are very similar apart from the mixed term hI,τx,a in the range 0≤x/a≤0.6. Actually, they are also proposed assuming a different positive direction for the tangential actions τn (and thus for KII), as detailed in Appendix, so care should be taken when choosing the WFs.Figure 4
Comparison of the WFs from [18, 19] for an oblique edge crack in a half-plane for α = 15° and α = 30°, a=1. (solid line [18], dashed line [19]).


## 3.3. Loading Conditions

In this simplified geometry, fatigue actions are described as travelling loads that move along thex-axis. In rolling contact fatigue, travelling loads are usually distinguished in(a)
normal loads (N-type): contact pressures, typically approximated with Hertzian contact formulas,(b)
tractive loads (T-type): tangential actions, representing frictional forces, generally expressed as f|N|, with f coefficient of friction. A sign is attributed to f to simulate driving (i.e., in the motion direction, f>0) or driven (i.e., opposite to the motion direction, f<0) conditions.We will assume that these loads simply translate along thex-axis maintaining their characteristics. Thus, a second reference frame St{Ω,xt,yt} is introduced where they can be easily defined and remain constant. Accordingly, the normal load due to Hertzian pressure can be expressed as(13)Nxt=-p01-xtaH2for-aH≤xt≤aH,where p0 is the maximum pressure and aH the contact half-width. The minus sign indicates a compressive action. Therefore, the tangential load is(14)Txt=fp01-xtaH2.In dry conditions, these are the external loads. Additionally, contact actionsσc between the crack lips can arise to avoid their overlapping as a consequence of compressive forces. Actually, the crack can be totally or partially closed, depending also on the crack size and on the load position (in particular on the ratio a/aH). A correct formulation of the problem can be found, for example, in [8, 11], and introduces important complexities to the solution passing through the estimation of the displacement field (not only at the crack tip). Thus, most frequently general cases are studied with finite element analyses, as in [11]. In this paper, a simplification proposed in [9, 16] is adopted, which can be useful for dealing with short cracks, where the crack is mainly open or totally closed. The simplification assumes a proportionality between those contact actions σc on the crack lips and the nominal normal tractions σn. The scale factor between σn and σc is calculated imposing KI=0, as described in Section 3.4 (see (20)-(21)). Contact actions can imply further frictional tangential tractions between the crack lips that can be in stick or slip condition [7, 24].In the so-called wet conditions, hydraulic loads due to the presence of fluid (lubricant) within the crack must also be considered. They can affect crack propagation in a double way, as shown in Figure5: direct “hydraulic pressure mechanism” and entrapment of the fluid inside the crack as stated by Bower in [7].Figure 5
Actions on the crack faces due to the presence of fluid.The description of the first effect requires an assumption on the value of fluid pressure within the crack. Generally, it is considered uniform and equal to the pressure at the crack mouth (as in [7]), or varying linearly from the mouth to the tip where it becomes zero (as in [25]). Both cases can be easily considered with the WFs.On the other side, fluid entrapment is quite difficult to be evaluated even if it represents a very critical condition for defects propagation. It takes into account the fact that as the contact passes over the crack mouth, a local closure of the faces may be observed in rather long cracks, that is, withaH<a. In this case, for some time, the lubricant may remain trapped inside the crack though the lips are closed at the mouth. A volume conservation for the lubricant is usually considered in order to evaluate the actions at the crack tip. As already stated, dealing with contact between the crack lips results in important complications, particularly when the conservation of volume must be guaranteed. However, the condition of fluid entrapment can be considered less dangerous than the crack pressurization as it is assumed that pressure in the trapped fluid is lower than the external one.


## 3.4. Model Implementation

The model of the oblique surface crack has been written in a completely symbolic form in Mathcad. Equations (10)–(14) have been used to estimate the nominal stress field that, for a given load, will be written as a function of xt and yt. Then, the stress tensor components inS are simply obtained replacing xt→x-d and yt=y, obtaining the symbolic functions(15)σyy=σyyx,y,d,σxx=σxxx,y,d,σxy=σxyx,y,d.Then, since the nominal normal and shear stresses are required along the crack faces for applying (4), the stress tensor T must be expressed in a rotated reference frame Sc{O,xc,yc} with O, at the crack mouth and xc along the crack (Figure 3(b)). That can be done as(16)TSc=RTSRT,where, for positive α,(17)R=cos⁡α-sin⁡αsin⁡αcos⁡α.Then, the nominal normal and shear stresses on the crack lip (σn and τn) having normal yc can be written as a function of xc and d, replacing x=xccos⁡αy=-xcsin⁡α, obtaining(18)σnx,y,d=σxxx,y,dsin2⁡α+σyyx,y,dcos2⁡α+2σxyx,y,dcos⁡αsin⁡ασnxc,d⟷σnxccos⁡α,-xcsin⁡α,dτnx,y,d=σxxx,y,d-σyyx,y,dcos⁡αsin⁡α+σxyx,y,dcos2⁡α-sin2⁡ατnxc,d⟷τnxccos⁡α,-xcsin⁡α,d.It can be noticed that positive values ofσn and τn mean stresses in agreement with the direction of yc and xc, respectively. At this point, the SIFs can be calculated by means of (4), changing the sign of τn if the WFs from [19] are chosen, that is, as anticipated in Section 3.2 and detailed in the Appendix.In order to reduce the computational time, it is convenient to select some discrete values ofd, for example, for -3aH≤d≤3aH, and of xc, for 0≤xc<a. Consequently, the integrands in (4) can be numerically estimated, fitted with a spline in xc and integrated for every value of d.At this point, a check of the obtained SIFs has to be done since, as already stated,KI must be nonnegative; that is,(19)KId=∫0ahI,σxc,aσnxc,d+hI,τxc,aτnxc,ddxc≥0.When (19) is not satisfied, it means that the crack is closed and contact actions σc on the crack lips must be introduced. According to the simplification introduced in the previous section, such contact actions are estimated as(20)σcxc,d=λdσnxc,d,where λ(d) is calculated imposing KI=0; that is,(21)∫0ahI,σxc,aσnxc,d+hI,τxc,aτnxc,ddxc+λ∫0ahI,σxc,aσnxc,ddxc=0⟶λ=-∫0ahI,σxc,aσnxc,d+hI,τxc,aτnxc,ddxc∫0ahI,σxc,aσnxc,ddxc.Therefore, alsoKII must be recalculated as(22)KIId=∫0a1+λhII,σxc,aσnxc,d+hII,τxc,aτnxc,ddx.Due to these contact actions, also frictional loads on the crack faces can arise, whose contribution to the SIFs is frequently negligible and thus not considered in this study.For wet conditions, fluid pressure on the crack facesp(xc,d) must be taken into account adding a pressurization contribute to the SIFs Kp; that is,(23)KIpd=∫0ahI,σxc,apxc,ddxc,KIIpd=∫0ahII,σxc,apxc,ddxc.As already mentioned, p(xc,d) may be constant o linearly decreasing to zero at the crack tip; that is,(24)pxc,d=pmdorpxc,d=pmd1-xca,where pm is the pressure at the crack mouth, from (13)(25)pmd=0ifd≤-aHord≥aHp01-daH2otherwise.Thus, Kp can be easily calculated as(26)KMpd=pmd∫0ahM,σxc,adxcorKMpd=pmd∫0ahM,σxc,a1-xcadxcwith M=I or II, where the integrals can be estimated in closed form, producing a scalar quantity depending on a and α (see Appendix).


## 3.5. Numerical Example

In order to validate the proposed model, some comparisons with results from the literature are discussed in Results. Unless stated otherwise, the following numerical data taken from a case used by Datsyshyn and Marchenko [8] has been considered: α = 25°, p0=840 MPa, aH=0.22 mm, and a/aH=0.5. For this value of α only the WF in [18] can be used.


## 4. Results

Results are usually presented in scaled form, with the dimensionless SIFs written as(27)FI/II=KI/II1p0πaHplotted versus a dimensionless distance d/aH. However, other scaling factors are used in the literature; thus comparisons can require some adaptation.


### 4.1. Short and Long Crack under Hertzian Contact

At first, a simple case with only Hertzian contact along the boundary has been investigated, taken from [8]. No tractive force T and no friction between the crack lips are considered but the model is applied to compare short and long cracks, that is, a/aH equal to 0.552 and 1.374, respectively.Results are shown in Figure6. Only the dimensionless FII is reported in Figure 6(a) as FI is zero. For both crack lengths, the obtained results are in rather good agreement with the literature. Higher errors are observed in peak values, overestimated by the model for about 8% for the long crack. In Figure 6(b), the values of λ estimated according to (21) are depicted. It can be observed that, in the central phase of the contact, that is, when the load passes over the crack mouth (-1<d/aH<1), λ is nearly -1. This means that FII is “produced” almost completely by nominal tangential actions, as supposed in [16].Figure 6
(a) Dimensionless SIF for mode II, for two crack lengths. Comparison with data of Figure 8 in [8]. (b) Proportionality coefficient for estimating lip contact actions as in (21).
(a) (b)From a computational point of view, each curve is obtained by estimating 25 values in the predefined range-1.5<d/aH<2.5 requiring about 15 minutes for all the curves.


### 4.2. Tractive Loads

The role of motion direction has already been discussed in the literature also in relation to the presence a tractive loadT due to nonnegligible friction effect. Results obtained with the proposed approach are compared to those obtained by finite element simulations described in [11].In Figure7(a), a plot of FII is shown where a tractive force T is added to the normal one N considering positive and negative values of the friction coefficient. In order to simplify the comparison, that is, to reproduce the given figure, the sign of FII is modified; thus with respect to Figure 6 curves are turned upside-down. It can be observed that the model can reproduce satisfactorily the trend of the literature.Figure 7
Dimensionless SIFs for mode II (a) and mode I (b), for different values of the friction coefficient. (c) Tangential load direction withf. (d) Conditions for positive KI (numerical case: a/aH=0.5; α=25°).
(a) (b) (c) (d)Figure7(b) shows the time history of KI as the load passes over the crack, with different values of the friction coefficient. Tangential actions cause positive values of the first SIF when the load is approaching the crack mouth for driven contact or when the load is leaving for braking contacts, in agreement with the literature (Figures 7(c) and 7(d)).


### 4.3. Hydraulic Pressure

The effects of fluid pressure on the crack faces have been added, as described in (24)–(26) and (A.3) in Appendix. A constant distribution has been considered, in order to compare the present study with others in the literature.Results are shown in Figure8, where the obtained dimensionless SIFs are matched with curves from [11], obtained by FE analyses which in turn are almost overlapped to those by Bower [7] and Beghini and Santus [9]. The effect is obviously limited to the range -1<d/aH<1 when the Hertzian pressure at the crack mouth is not zero. As expected, the pressurization of the crack increases markedly FI (see Figure 6 for comparison with curve for f=-0.05) and is considered one of the main causes of crack propagation.Figure 8
Dimensionless SIFs for pressurized crack (numerical case:a/aH=0.5; α=25°, f=-0.05, constant fluid pressure on the crack faces).


### 4.4. Separate Contributions to the SIFs

As final result, it is interesting to observe an important advantage of this model that enables splitting the single contributions of external load to the SIFs. Figure8 shows the separated curves of FI and FII due to the travelling loads N and T and to the fluid pressure, whose sum gives plots in Figure 9. For N and T, the separate contribution of nominal normal (σ) and shear stresses (τ) is also detailed.Figure 9
(a)-(b) Dimensionless SIFs produced by nominal normal (σ) and shear stresses (τ), produced by external loads N and T and by pressure on the crack lips (p) (numerical case: a/aH=0.5; α=25°, f=-0.05, constant fluid pressure on the crack faces).
(a) (b)The main contribution is produced by the normal stresses due to Hertzian pressure (σN) and the fluid pressurization, as expected. The two curves have similar magnitude, opposite sign, and a small shift that causes their sum to be positive and negative for FI and FII, respectively. Other contributions are one order smaller. It is possible to appreciate the positive contribution to FI of both stress components due to T (σT and τT) being f=-0.05.This “split” analysis can be useful to achieve fast indications for other cases, such as for increased frictionf=-0.1, since the superposition principle allows simply doubling the values of the curves due to σT and τT.


## 4.1. Short and Long Crack under Hertzian Contact

At first, a simple case with only Hertzian contact along the boundary has been investigated, taken from [8]. No tractive force T and no friction between the crack lips are considered but the model is applied to compare short and long cracks, that is, a/aH equal to 0.552 and 1.374, respectively.Results are shown in Figure6. Only the dimensionless FII is reported in Figure 6(a) as FI is zero. For both crack lengths, the obtained results are in rather good agreement with the literature. Higher errors are observed in peak values, overestimated by the model for about 8% for the long crack. In Figure 6(b), the values of λ estimated according to (21) are depicted. It can be observed that, in the central phase of the contact, that is, when the load passes over the crack mouth (-1<d/aH<1), λ is nearly -1. This means that FII is “produced” almost completely by nominal tangential actions, as supposed in [16].Figure 6
(a) Dimensionless SIF for mode II, for two crack lengths. Comparison with data of Figure 8 in [8]. (b) Proportionality coefficient for estimating lip contact actions as in (21).
(a) (b)From a computational point of view, each curve is obtained by estimating 25 values in the predefined range-1.5<d/aH<2.5 requiring about 15 minutes for all the curves.


## 4.2. Tractive Loads

The role of motion direction has already been discussed in the literature also in relation to the presence a tractive loadT due to nonnegligible friction effect. Results obtained with the proposed approach are compared to those obtained by finite element simulations described in [11].In Figure7(a), a plot of FII is shown where a tractive force T is added to the normal one N considering positive and negative values of the friction coefficient. In order to simplify the comparison, that is, to reproduce the given figure, the sign of FII is modified; thus with respect to Figure 6 curves are turned upside-down. It can be observed that the model can reproduce satisfactorily the trend of the literature.Figure 7
Dimensionless SIFs for mode II (a) and mode I (b), for different values of the friction coefficient. (c) Tangential load direction withf. (d) Conditions for positive KI (numerical case: a/aH=0.5; α=25°).
(a) (b) (c) (d)Figure7(b) shows the time history of KI as the load passes over the crack, with different values of the friction coefficient. Tangential actions cause positive values of the first SIF when the load is approaching the crack mouth for driven contact or when the load is leaving for braking contacts, in agreement with the literature (Figures 7(c) and 7(d)).


## 4.3. Hydraulic Pressure

The effects of fluid pressure on the crack faces have been added, as described in (24)–(26) and (A.3) in Appendix. A constant distribution has been considered, in order to compare the present study with others in the literature.Results are shown in Figure8, where the obtained dimensionless SIFs are matched with curves from [11], obtained by FE analyses which in turn are almost overlapped to those by Bower [7] and Beghini and Santus [9]. The effect is obviously limited to the range -1<d/aH<1 when the Hertzian pressure at the crack mouth is not zero. As expected, the pressurization of the crack increases markedly FI (see Figure 6 for comparison with curve for f=-0.05) and is considered one of the main causes of crack propagation.Figure 8
Dimensionless SIFs for pressurized crack (numerical case:a/aH=0.5; α=25°, f=-0.05, constant fluid pressure on the crack faces).


## 4.4. Separate Contributions to the SIFs

As final result, it is interesting to observe an important advantage of this model that enables splitting the single contributions of external load to the SIFs. Figure8 shows the separated curves of FI and FII due to the travelling loads N and T and to the fluid pressure, whose sum gives plots in Figure 9. For N and T, the separate contribution of nominal normal (σ) and shear stresses (τ) is also detailed.Figure 9
(a)-(b) Dimensionless SIFs produced by nominal normal (σ) and shear stresses (τ), produced by external loads N and T and by pressure on the crack lips (p) (numerical case: a/aH=0.5; α=25°, f=-0.05, constant fluid pressure on the crack faces).
(a) (b)The main contribution is produced by the normal stresses due to Hertzian pressure (σN) and the fluid pressurization, as expected. The two curves have similar magnitude, opposite sign, and a small shift that causes their sum to be positive and negative for FI and FII, respectively. Other contributions are one order smaller. It is possible to appreciate the positive contribution to FI of both stress components due to T (σT and τT) being f=-0.05.This “split” analysis can be useful to achieve fast indications for other cases, such as for increased frictionf=-0.1, since the superposition principle allows simply doubling the values of the curves due to σT and τT.


## 5. Conclusions

An analytical model of a surface crack in a half-plane has been described in detail as well as results of its application compared to other studies in the literature. The model is based on the application of Linear Elastic Fracture Mechanics, in particular taking advantage of the Weight Function Method for evaluating the stress intensity factors which characterize the behaviour and propagation of the crack. The theoretical foundations are also reported in the background.The model is rather simple to be implemented and its results are in satisfactory agreement with the literature. It can be conveniently applied to investigate many problems of rolling contact fatigue and hopefully it could help to clarify some aspects of this complex phenomenon that are still debated.



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*Source: 6948741-2018-03-27.xml*