## Abstract

In this paper, the crack initiation at contact surface of solids is investigated on the basis of the concept of potential energy release rate. The expressions for path-independent integral vector *J*_{i} (*i*=1, 2) are derived and applied to the consideration of the process of crack initiation. The relationship is then established between the value of the path-independent integral vector *J*_{i} and the potential energy release rate for crack initiation in an arbitrary orientation. This allows the prediction of crack initiation angle on the basis of the maximum energy release rate criterion. The surface crack initiation angle in fretting fatigue is determined analytically as a function of the friction coefficient of the edge contact. This theoretical result is compared with the existing experimental results reported in the literature and a good agreement is found. The formulation provides a novel basis for numerical modelling of the complex process of fretting fatigue.

## 1. Introduction

### (a) Problem background

Surface cracking in solids due to stress concentration at the edges of frictional contacts between components of an assembly can be encountered in many mechanical apparatuses. Fretting fatigue can significantly reduce the durability of a variety of engineering devices, from bolted mechanical joints to dovetail connections in aircraft jet engines.

The phenomenon of fretting fatigue consists of two distinct stages. Stage I refers to the process of crack initiation, driven by the stress concentration at the edge of a frictional contact. This stress state arises due to the combination of compressive and shear tractions at the edge of the contact region. The crack initiation is therefore associated with shear and is observed to occur at a sharp angle to the surface, as illustrated in figure 1 (Antoniou & Radtke 1997). The initial inclination of the stage I fretting fatigue crack is frequently found to be approximately 45° (Waterhouse 1981). Stage II of the fretting fatigue processes is associated with the presence of alternating tensile stress within the component. As the crack initiated by the contact tractions becomes longer, the contact-induced stresses become less significant, while the presence of an underlying tensile stress in the component induces the crack to re-orientate to a path perpendicular to the direction of action of this principal stress component. This re-orientation can be clearly seen in figure 1. It is the re-orientation and propagation of the crack in stage II that eventually leads to component failure.

The prediction of the surface crack initiation under fretting fatigue conditions is thus of great practical importance. Of the two stages described, the growth in stage II can be addressed with conventional crack growth methods based on fracture mechanics.

The situation in stage I appears to be somewhat more complex, particularly in the case of sharp-edged (complete) contact. Traditional fracture theories cannot be employed directly and many factors are involved (e.g. local material microstructure, temperature, etc.). A fundamental analysis of the mechanics of this problem has, however, to rely on the analysis of the stress environment in the neighbourhood of the contact edge, in a way similar to the conventional treatment of crack propagation conditions.

The aim of the present study is to establish a consistent approach and to formulate criteria for the determination of surface crack initiation and its orientation.

A great deal of effort has been devoted to the study of the surface cracking mechanism due to the concentration of tractions at the edge of contact. The developments in this area follow fairly closely the historical development of the traditional fracture mechanics concepts (e.g. Giannakopoulos *et al*. 1998; Yang & Mall 2001; Xie & Hills 2003). Giannakopoulos *et al*. (1998) proposed the crack analogue approach to the analysis of the fretting contact and identified some important aspects of the equivalence between contact mechanics and fracture mechanics. They validated the approach under the conditions of small-scale yielding. This approach facilitates the analysis of crack initiation in fretting fatigue in cases of high stress concentration. In their analysis of fretting fatigue within the framework of linear elastic fracture mechanics (LEFM), Yang & Mall (2001) considered adopting either the maximum circumferential stress criterion or the maximum shear stress criterion, to predict crack initiation in the opening mode and the shear mode, respectively. Recently, using the minimum strain energy density criterion (*S*-theory), Xie & Hills (2003) investigated the crack initiation angle.

The use of criteria based on one stress component for the analysis of crack initiation must be questioned, since fracture initiation may not in fact be governed by a single one of the six independent stress components, but rather a combination may play the role of the critical parameter. The use of strain energy density (*S*-theory) requires validation, as well as clarification of the physical basis. In contrast, the use of the potential energy release rate in the analysis of crack initiation is a logical application of the central principle of fracture mechanics. In the literature, the application of the maximum energy release rate (MERR) approach in fracture mechanics has been widely discussed, including such developments as the extension to include cases of material anisotropy (Beom & Atluri 1996).

### (b) Framework for analysis

#### (i) Rounded edge contact and the small scale yielding (SSY) concept

It is very important to appreciate that, although the analysis presented in this paper is based on the consideration of square root singular behaviour at the edge of contact between a flat rigid punch and a semi-infinite elastic solid, it is not directly aimed at understanding the conditions of crack initiation at the corners of sharp-edged contacts. The reason for this is twofold. Firstly, it becomes apparent after brief consideration that, just like there does not exist a perfectly rigid punch, there does not exist a perfectly sharp material edge. At any rate, even if such a corner could be prepared by cleavage, it would not survive for any length of time the combined application of normal and shear loads to the punch. This situation is similar to that encountered in engineering fracture mechanics: although atomically sharp cleavage cracks may exist in extremely brittle crystalline solids, in practice engineering structural material always possess sufficient ductility to induce crack tip blunting. There are therefore two effects responsible for deviation from the brittle elastic theory: geometric rounding of the tip and plastic deformation resulting in stress–strain nonlinearity.

Despite the disagreement in the detail of local crack tip geometry and stress response between practice and theory, LEFM has been used with considerable success to describe the strength of most practical materials. The reasons for this situation have been elucidated by Rice (1968*a*) through the introduction of the concept of small scale yielding (SSY). The SSY approach relies on the fact that, despite the fact that local deviations from linear elastic behaviour occur within the small region immediately surrounding the crack tip (the damage and plastic zones), it can be postulated that the processes occurring there are entirely determined and controlled by the linear elastic solution within an annular intermediate zone immediately surrounding the plastic and process zones. Further away from the crack tip the solution becomes sensitive to the detailed geometry of the object, the particular conditions of load application, etc. We wish to borrow the concept of small-scale yielding from this classical treatment to consider the deformation state at the edge of contact between a semi-infinite plane elastic solid and a rigid punch that is flat, but slightly rounded at the edge. If the rounding radius is significantly smaller than the punch width, then in the first approximation its effect on the global (long range) traction distribution can be ignored and the contact described by the square root singular distribution of normal traction. This gives rise to the so-called unbounded asymptotic (Giannakopoulos *et al*. 1998; Sackfield *et al*. 2003). However, in strong similarity with small-scale yielding in fracture, two effects contribute to moderate the traction magnitudes at the contact edge. Firstly, it is the rounding of the punch edge. Secondly, it is the plastic deformation induced in the substrate within the highly stressed region.

We consider the geometric nonlinearity first. Unlike the crack tip blunting problem which does not readily allow rigorous analytical elastic solution, the problem of the flat and rounded punch can be readily solved (Ciavarella *et al*. 1998). Moreover, a transition can be made to the case of semi-infinite rigid punch with a rounded edge (Sackfield *et al*. 2003), that corresponds to the scale of consideration being brought from global to intermediate and local level: the effects of the other end of the punch and substrate boundaries are ignored, the traction profile remotely from the edge follows the decaying inverse square root asymptote. Locally, on the other hand, the traction remains finite and drops to zero at the point of separation between punch and substrate, as required in incomplete contact. The situation is illustrated in figure 2, where the hyperbolic dashed curve indicates the outer, unbounded square root singular asymptote. Alongside that the inner, bounded asymptote is indicated by the parabolic curve. The concept of bounded asymptotes in the context of fretting contact problems was first introduced Sackfield *et al*. (2003) and used with some success to predict crack initiation in fatigue (Dini & Hills 2004). The authors observed that the generalized stress intensity factor for the bounded asymptote was fully determined by the rounding radius, friction coefficient and the stress intensity factor of the outer, unbounded asymptote.

We now turn our attention to the nonlinearity of local material behaviour within the highly stressed region at the edge of contact. Schematically we indicate this region of plastic deformation by the grey shaded semi-circle corresponding to the location of normal tractions (figure 2). In practice, the shape of that region is not semi-circular and its extent and location depend on the multi-axial stress state. However, for the purposes of the present discussion this schematic serves adequately well. Outside the grey shaded region, we indicate by another semi-circular contour, the annular region within which the outer unbounded asymptotic dominates. It thus provides an adequate description of the deformation and stress state and imposes the boundary conditions for the inner grey region.

The main point that we would like to make now is that, in similarity with the SSY hypothesis in fracture mechanics, in the present case, we can postulate that the conditions for crack initiation (that would occur due to plastic deformation within the grey shaded process zone) are completely determined by the intermediate square root singular unbounded asymptote or more precisely, by the combination of stress intensity factors *K*_{I} and *K*_{II} describing the normal and shear traction distributions, respectively.

#### (ii) Contact traction and bulk stress

One further assumption must be made before we can proceed with the analysis. In practice, fretting fatigue occurs due to a combination of two principal factors. The first factor is surface damage due to relative displacement of contacting surfaces and tractions induced by the contact. It may be thought that the corresponding stresses are particularly important in controlling crack initiation. Secondly, a cyclically varying underlying bulk stress acting in the substrate in the direction parallel to the surface provides the driving force for crack propagation that becomes ever more important as the crack grows longer. In the present study, we focus our attention on crack initiation and ignore the effect of bulk stress on this process. It will become apparent from the analysis that, while contact tractions produce *stress intensification* even in the absence of an incipient crack, bulk stress does not contribute to this effect.

#### (iii) Grain size and contact process zone size

Finally, we wish to address the issue of material inhomogeneity and grain structure and its possible effect on crack nucleation. In preparing many structural materials, particular measures are often used to control the grain size so as to keep it below 20–30 μm and often less. On the other hand, in many engineering applications chamfering or de-burring of the edges results in rounding radii in the order of fractions of millimetres. It is these radii of the edges of contacting surfaces that control the dimensions of the process zones for crack nucleation. In other words, an order of magnitude difference usually exists between the process zone size and grain size. If this observation is taken into account together with the three-dimensional nature of real contacts (which increases manyfold the number of grains within the process zone), then it is logical to assume that even under the conditions considered in the present paper, the material can be adequately represented by the continuum model with average properties.

In this paper, we use MERR to shed some light on the problem of surface crack initiation under the action of concentrated stresses arising at the edge of frictional contacts. The paper is constructed as follows. Firstly, the stress and displacement fields around the edge of a rectangular contact is reviewed in §2. In §3, the MERR analysis of crack initiation in terms of the vector *J*_{i} integral are derived, from which the critical parameter for surface crack initiation and its orientation are given. In §4, to demonstrate the new theory, a set of the typical experiments described in the literature and the analytical solutions obtained by the currently proposed theory are compared. Good agreement between the predictions and experimental observations of crack initiation from fretting contacts is found. Conclusions are drawn and a discussion is presented in §5.

## 2. Elastic field of an edge of contact with a rectangular punch

Neglecting the coupled effect between normal and tangential deformations due to mode I and mode II loads (Giannakopoulos *et al*. 1998; Yang & Mall 2001; Xie & Hills 2003), the singular stress field at the sharp edge of the sliding contact of the rectangular rigid punch pressing into the linear elastic substrate was obtained by Nadai (1963) and can be expressed in the asymptotic form in polar coordinates in figure 3 as follows:(2.1)The associated displacements are(2.2)where (*κ*=(3−4*ν*) for plane strain and *κ*=(3−*ν*)/(1+*ν*) for generalized plane stress. It has been pointed out by Giannakopoulos *et al*. (1998) that the stress and displacement field of the left contact edge in figure 3 is in the same asymptotic form as the crack tip under remote load obtained by Williams (1957). The parameters *K*_{I} and *K*_{II} have the conventional fracture mechanics meaning of mode I and mode II stress intensity factors, respectively. For example, consider a rectangular rigid punch of width 2*a* pressed into the surface of an elastic half-plane with a normal force *P* per unit width and a shear force *Q* per unit width. Then *K*_{I}=−*P*/(*πa*)^{1/2}, *K*_{II}=*Q*/(*πa*)^{1/2}. It should be emphasized that for crack problems *K*_{I}≥0 (if *K*_{I}≤0, that means that the singularity of the open-crack tip will vanish), while for contact problems *K*_{I}≤0. This clearly implies the negative (compressive) normal traction at the contact surface.

It should be explained that in the analytical treatment presented in this paper, the sliding condition only needs to be fulfilled locally at the edge of the contact, within the zone in which the unbounded asymptotic approximation persists as expressions (2.1) and (2.2). Thus, gross slip or partial slip conditions (with the above caveat) are required for the treatment to be valid. It is clear that, e.g. under the conditions of gross slip, damage (i.e. the effects of local stress concentration and surface rubbing) is distributed over a region of the substrate that depends on the fretting amplitude. Note that we do not seek, at this point, to incorporate this very important effect into our analysis, but merely wish to identify which material parameters ought to be included in the criterion and whether the direction of crack initiation under these conditions can be predicted. Expressions (2.1) and (2.2) form the basis of mathematical manipulations presented in the following sections.

## 3. The path-independent integral vector *J*_{i} and the potential energy release rate for crack initiation

The original concept of path-independent integral was developed by Eshelby (1951) to characterize the generalized forces on singularities and/or inhomogeneities in elastic fields. In application to the analysis of the conditions for crack propagation, the concept of *J*-integral was introduced by Rice (1968*a*) as follows:(3.1)where *Γ* denotes a curve surrounding the crack tip. The curve is traversed in the anticlockwise sense. Variable *s* indicates the arc length and **T**_{j}=*σ*_{ji}**m**_{i} denotes the traction vector on *Γ*, defined with respect to the outward unit vector ** m** normal to the curve.

The *J*-integral defined in equation (3.1) has the physical interpretation of the rate of change of potential energy with respect to the incremental change of crack length along the crack line, i.e. the *x*_{1}-axis (Rice 1968*a*,*b*). Cherepanov (1979) considered the same problem from the point of view of energy flux and sought to apply for *J*-integral (or *Γ*-integral in his notation) to the study of the curved crack problem. The concept of path-independent *J*-integral has found widespread application to various aspects of fracture mechanics (Sanders 1960; Knowles & Stenberg 1972; Budiansky & Rice 1973; Cherepanov 1979 and others).

In this section, we present the derivation of the path-independent vector *J*-integral formulation to establish the relationship between the vector *J*-integral and the potential energy release rate of crack initiation. For this purpose, alongside the component *J*_{1} introduced in equation (3.1), we shall require another component, *J*_{2}, to be introduced. Our derivation is aimed at the analysis of crack initiation conditions for a surface crack.

Consider a half-plane edge contact illustrated by figure 4. The body has perimeter enclosing an area *a*. Traction ** T** acts on the part of the boundary, denoted by

*K*

_{I}and

*K*

_{II}, while on the remaining part of the boundary, , displacement boundary conditions are prescribed in the form of the displacement vector

**u**_{0}. The coordinate system can always be chosen so that the origin lies at the infinitesimal initiation crack tip O, even when the crack is advancing. We suppose that the traction and displacement boundary conditions on

*Γ*

_{0}are fixed and that O

^{+}and O

^{−}denote the points reached by approaching point O from its left and right side, respectively. The potential energy

*Π*of the body is given by(3.2)where the strain energy density and

*α*is the crack initiation angle. Now consider the origin O undergoing a virtual displacement by an infinitesimal distance d

*l*orientated at an arbitrary angle

*α*which produces an infinitesimal crack as shown in figure 4. The energy release rate due to the formation of new crack surfaces can be written as(3.3)Since the coordinates are always attached on the crack tip and the perimeter

*Γ*

_{0}is fixed, it is established that(3.4)where

*n*

_{1}=cos

*α*and

*n*

_{2}=sin

*α*. Equation (3.3) can be re-written as(3.5)Noting(3.6)and the principle of virtual work(3.7)we obtain the following equation from equation (3.5)(3.8)where(3.9)

*J*

_{i}was introduced by Knowles & Stenberg (1972) and Budiansky & Rice (1973). We note that the above

*J*-integral corresponds to a component of a vector.

Irwin (1957) introduced symbol *G* to denote the rate of change of potential energy associated with the crack advance, *G*=−d*Π*/d*l*. Equation (3.8), therefore, states that(3.10)or(3.11)We conclude that the potential energy release rate *G*, that can also be interpreted as the crack initiation driving force, is determined not only by *J*_{1} (i.e. *J*), but also by *J*_{2}. The relationship thus established is different from the well-known formulation *G*=*J* for brittle fracture mechanics when the crack tip advances along the *x*-axis which is chosen to parallel the original crack surfaces in the literature (Rice 1968*b*). If once *J*_{1} and *J*_{2} are evaluated, the crack initiation driving force *G* can be considered as a function of the virtual cracking angle *α*. A schematic illustration of the relationship between *G* and *α* given by equation (3.11) is plotted as the solid arc in figure 5.

The MERR criterion, i.e. maximum crack initiation driving force, on the basis of equation (3.11) can be expressed in the form(3.12)where 0≤*α*≤*π* and *k* is an integer number.

Equation (3.12) provides a method for evaluating both the maximum crack initiation driving force and also the possible boundary crack initiation angle. Once *G*_{max} reaches a critical value *G*_{c} (toughness, a material parameter) represented by the unity circle in figure 5, the boundary crack will initiate.

Let us now turn our attention back to equation (3.9). Since there is no singularity in the area closed by the contour in figure 4, it follows:(3.13)From equation (3.13), we can get(3.14)From equations (3.9) and (3.14), the expression for the path-independent vector *J*_{i}-integral is obtained in the form(3.15)where in figure 4. Clearly, we can select an arbitrary contour *Γ* which starts from O^{−}, surrounds the initiating crack and arrives at O^{+}. For ease of calculation, we choose the contour to be a half circle with radius *r*. Substituting equations (2.1) and (2.2) into the first and last integrals in the above expression and letting *r*→0, it is found that(3.16)Then equation (3.15) can be re-written more simply as(3.17)Substitution equations (2.1) and (2.2) into equation (3.17) leads to(3.18)Inserting into equation (3.12) shows(3.19)(3.20)The Coulomb friction law for this case can be introduced in terms of the stress intensity factors from equation (2.1) as follows:(3.21)From equations (3.19)–(3.21), we obtain(3.22)and(3.23)The variation of the normalized potential energy release rate for surface crack initiation as a function of the friction coefficient *f* (*f*≥0) is shown in figure 6. One noteworthy feature of this figure is the rapid rise with increasing *f*. This indicates, evidently, that the surface crack initiation driving force *G*_{max} is significantly reduced if small values of the friction coefficient are maintained.

## 4. Analysis and discussion

### (a) Analysis

The deformation mechanics of a complete contact subject to fretting action is complex. The analysis presented in this paper is asymptotic in the sense that it aims to capture the salient features of the stress field by considering an idealized geometry. Although such analyses do not cover all possible geometries, they provide a framework for considering key geometric features of contacts, e.g. nominally sharp edged, but slightly rounded punch shapes. Using this basis several criteria from the LEFM were adapted for the analysis of fretting (Giannakopoulos *et al*. 1998; Yang & Mall 2001; Xie & Hills 2003 and others). Here, the mechanics of contact crack initiation is analysed on the basis of the concept of potential energy release rate. Equations for the driving force for boundary crack initiation of contacting surface were developed and expressed explicitly in terms of the remote contact loads measured by *K*_{I} and *K*_{II}.

The critical energy release rate *G*_{c} is the central parameter used in the analysis. The objective is to propose that values of *G*_{c} determined from conventional fracture mechanics experiments could be used for the assessment of contact crack initiation. This statement requires validation through experimentation and analysis.

A further important result of the present study concerns the prediction of crack initiation angle. The maximum crack initiation angle is 45° (figure 7). This is attained only if *f*=1 from equation (3.23). For an illustration of the validity of this result, we refer to some experimental observations (figure 8) available in the literature (Pape & Neu 1999, 2001). In these studies, fretting contact was established between a substrate and a pad with sharp rectangular corners. The friction coefficient *f* was in the range (0.8–0.9). Figure 8 illustrates the fact that contact crack initiation angles are close to the prediction of 45°.

### (b) Discussion

The theoretical results obtained above must be applied with caution, with full account being taken of the analysis framework presented in the introductory section. Practical characterization of the process of fretting fatigue crack initiation is complicated by a number of problems, e.g. it may be difficult to draw a clear distinction between initiation and propagation; the effect of underlying bulk stress may manifest itself in crack turning at early stages, making the observation of the initiation angle difficult; materials with particularly large grain size may show increased scatter in crack initiation statistics, etc.

It is nevertheless the view of the authors that the analysis proposed in the present study will be useful for the purpose of identifying underlying correlations between important problem parameters (e.g. details of the punch edge geometry) and crack initiation conditions.

## 5. Conclusions

In the present paper, we used the analysis of energy to establish the relationship between the potential energy release rate and the path-independent vector *J*_{i}-integral for surface crack initiation at previously un-cracked boundaries.

The first important result shows that the energy release rate *G* depends not only on the first component of the vector integral, *J*_{1}-integral, conventionally used as a measure of the forward driving force on the crack tip. The second component of the vector integral, the *J*_{2}-integral, must also be introduced and its influence on crack initiation taken into account.

Secondly, using the MERR theory, we obtained analytical expressions for the maximum potential energy release rate *G*_{max} available for the possible initiation angle. The surface crack initiation angle predicted is in good agreement with the experimental results reported in the literature. Furthermore, the basis for assessing surface crack initiation has thus been developed using material toughness *G*_{c}, the key material parameter of LEFM.

The results presented are likely to improve the understanding of the complex processes of contact crack initiation in fretting fatigue, including such problems as the prediction of growth trajectories from frictional contacts and conditions for crack initiation at contacts with rounded corners.

## Acknowledgments

This research is partially supported by the National Natural Science Foundation of China (Contract No. 10502040). The authors would also like to acknowledge the support of Rolls-Royce plc through University Technology Centre in Solid Mechanics, Department of Engineering Science, University of Oxford.

## Footnotes

- Received April 24, 2005.
- Accepted December 21, 2005.

- © 2006 The Royal Society