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 the
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 [
In [
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 [
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.
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 as
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 Figure
(a) Crack fracture modes in plane; (b) left: couple of unitary forces opening the crack; right: general traction distributions on the crack faces.
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 [
In order to give a brief overview of the physical meaning of the WFs, let us consider a simple geometry shown in Figure
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 Figure
(a) Superposition principle for the SIF in a cracked panel. (b) Panel with an oblique crack in tension.
When the geometry is not symmetrical as in the previous example, for example, when the crack is oblique as in Figure
Let us consider a homogenous, isotropic linear elastic half-plane, lying in the part
Given the normal
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 length
(a) Scheme of the geometry: from the real problem to a cracked half-plane. (b) Reference frames and other symbols.
For an oblique edge crack in a half-plane, two sources for the WFs can be found: [
Comparison of the WFs from [
In this simplified geometry, fatigue actions are described as travelling loads that move along the normal loads ( tractive loads (
We will assume that these loads simply translate along the
In dry conditions, these are the external loads. Additionally, contact actions
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 Figure
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 [
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, with
The model of the oblique surface crack has been written in a completely symbolic form in Mathcad. Equations (
Then, since the nominal normal and shear stresses are required along the crack faces for applying (
It can be noticed that positive values of
In order to reduce the computational time, it is convenient to select some discrete values of
At this point, a check of the obtained SIFs has to be done since, as already stated,
Therefore, also
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 faces
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 [
Results are usually presented in scaled form, with the dimensionless SIFs written as
At first, a simple case with only Hertzian contact along the boundary has been investigated, taken from [
Results are shown in Figure
(a) Dimensionless SIF for mode II, for two crack lengths. Comparison with data of Figure 8 in [
From a computational point of view, each curve is obtained by estimating 25 values in the predefined range
The role of motion direction has already been discussed in the literature also in relation to the presence a tractive load
In Figure
Dimensionless SIFs for mode II (a) and mode I (b), for different values of the friction coefficient. (c) Tangential load direction with
Figure
The effects of fluid pressure on the crack faces have been added, as described in (
Results are shown in Figure
Dimensionless SIFs for pressurized crack (numerical case:
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. Figure
(a)-(b) Dimensionless SIFs produced by nominal normal (
The main contribution is produced by the normal stresses due to Hertzian pressure (
This “split” analysis can be useful to achieve fast indications for other cases, such as for increased friction
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.
The general expressions of the WFs in [
As already stated, the coefficients
(a) Plots of the coefficients of the WFs in Eq. (
In the literature, positive values of
Given the polynomial expressions in (
Crack length
Hertzian contact half-width
Position of the travelling load; the origin
Coefficient of sliding friction
Weight Function for calculating
Weight Function for calculating
Weight Function for calculating the contribution of
Weight Function for calculating the contribution of
Imaginary unit
Maximum Hertzian pressure (
Pressure at the crack mouth
Complex variable
Airy’s function
Dimensionless SIFs
Stress intensity factor for the first mode (opening)
Stress intensity factor for the second mode (sliding)
Contribution of pressure
Normal actions along the half-plane border (>0 when directed as the
Rotation matrix between
Fixed reference frame,
Fixed reference frame,
Reference frame travelling with the load, axes parallel to
Tangential actions along the half-plane boundary (>0 when directed as the
Stress tensor (
Angle of the crack with respect to the surface
Corrective factor when the crack lips are closed, ratio between
Contact actions on the crack lips
Nominal normal stress along the crack lips in the uncracked body (>0 when tractive; 0> when compressive)
Nominal shear stress along the crack lips in the uncracked body (arbitrary positive sign)
Muskhelishvili’s potential functions.
The author declares that there are no conflicts of interest regarding the publication of this paper.