# Experimental verification of the crack analogue model

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## Basic contact mechanic

Usual experimental contact configurations

Industrial components and structures usually show complex geometry which can hardly be reproduced in laboratory. That’s why different simplifications of the contacts have been developed to allow easier experiments (Figure 2-1). Three reference contacts are generally defined:
– Spherical fretting pads loaded against a plane specimen. This contact geometry is unidimensional and so does not require alignment (Kuno et al., 1989). Therefore it is easier to implement experimentally. However, the computation for this representation is more complicated because it is a 3 dimensional problem.
– Cylinder fretting pads loaded against a plane specimen. This two-dimensional contact is regularly used even if it requires a cautious alignment of the pad. Moreover, it is described by the Hertz theory (Hertz, 1981) and analytical solution already exists (Hills and Nowell, 1994).
– Flat fretting pads loaded against a plane specimen. This the three-dimensional configuration is the closest from the industrial configurations. However, it is scarcely used because of the difficulty encountered for the alignment. Moreover, this geometry is characterised by high discontinuities for the pressure and shearing’s profiles at the bordering of the contact, nevertheless analytical solutions have been developed (Alexandrov et al. (2001), Ciavarella et al. (2003b)).
The industrial contact involved for the COGNAC project is complex to model. However, the main objectives of this study are to improve the today knowledge about the gradient effect and the size effect. Hence, a simplified geometry can be considered. The one selected for this study is the cylinder/plane contact because its analytical solution is easy to implement and different gradient of stress can be encountered by modifying the radius of the pad. Thus, the methodology to obtain the subsurface stress field will be developed for this configuration.

The determination of the internal fields is divided into two stages. First, the contact pressure distribution has to be determined by solving the contact problem. Then, the internal stress, strain and displacement field is found using the expressions of the surface tractions deduced previously.
Plane monotonically problem: Figure 2-2 shows a cylindrical pad loaded by a normal force P against a flat specimen. A tangential force, Q, is applied to the pad. The Hertz’s theory (Hertz, 1881) can be used as it is a non-conformal contact. Then, by applying the hypothesis of plane strain, the stress state may be extracted with an Airy function solution of the bi-harmonic equation (Timoshenko and Goodier, 1951):
Returning to the determination of the shear traction, Figure 2-4 depicts the variation of the tangential load among the time. When the load increases monotonically from 0 to Qmax, the maximum amplitude of the tangential force, the shear traction is defined as described by the equation (2.21) and (2.22)
Now consider that the load has been infinitesimally reduced from its maximum value, point A, to point B. Hence, it will have a change of sign in the rate of the tangential displacement ⁄ , so the equation (2.3) is no longer valid and instantaneous stick must occur over the entire contact.
If the fretting load is further reduced until the point C, a relative slip will appear at the contact edges. In these new slip zones ( < | | ≤ ), the shear traction will have changed from ( ) to − ( ). The corrective traction necessary to prevent this slip is: Effect of the bulk load: In the present work, a fretting-fatigue loading is considered and so the flat specimen is submitted to a bulk stress which modifies the classical Mindlin solution for the shear traction. This will be necessary as the bulk load will cause a strain in the specimen that is not present in the pads. This mismatch in strain caused an additional term in the tangential equation (2.8), and the resultant shear tractions will differ from those arising in pure fretting configuration. Nowell’s experimental configuration is used to illustrate this situation (Figure 2-6).
In this case, the partial derivative of the relative tangential displacement is no longer zero in the stick zones. The left hand side of the equation (2.8) can then be written in form of stress yields.
The effect of the bulk stress, , is to offset the stick zone, which was centrally positioned in the absence of bulk stress. The domain of the perturbation in the full solution now becomes | − | < rather than the former symmetrical | | < , where e is the offset of the centre stick zone from the centre of the contact. Application of appropriate boundary conditions in and outside of the stick zone together with the integration of the equation (2.7) fix the value e.
Thus the perturbation term for the shear traction becomes: − (2.32) ( )=1− |−|<
The solution developed above is satisfactory for moderate values of the bulk tension. If larger values of tension are applied, one edge of the stick zone will approach the edge of the contact. The current solution is therefore only valid for + ≤ .

Once the surface traction has been determined, it is possible to evaluate the stresses in the specimen by superposing the effects of the normal pressure, the shear traction and the specimen bulk load using Muskhelishvili’s potential theory (Muskhelishvili, 1953). For most of the stress components, it is possible to use a half-plane assumption in the computation of stresses due to the contact.
If the material remains elastic linear, the evaluation of the xx component of stress due to the normal load can be obtained with the superposition of the results for the elliptical tractions. Although, the shifted origins of the perturbations terms, q’(x) and q’’(x), will have to be taken into account. It is particularly worthy to note that four different combinations of superposition will be necessary to express the stress field at the maximum and minimum load, and during unloading and reloading. For instance, the normalised xx components of stress at each stages will be: Where c is the half width of the stick zone, e is the offset of the centre of the stick zone from the centre of the contact, , , are the xx components of the stress due to the normal, tangential and bulk loads respectively, and is the stress correction for the finite thickness effect.
Similar formulations can be derived for the yy and xy components of stress at the same loading stages, and the zz components may be obtained from the other two direct stresses (plane strain condition). The function in brackets may be evaluated using Muskhelishvili’s potentials (Muskhelishvili, 1953, Hills and Nowell, 1993).

Determination of the friction coefficient: Oxford’s method

Previous study (Hills and Nowell, 1994) has shown that, due to the complexity of the loading for the fretting fatigue, the coefficient of friction is not constant under the contact surface. However, for this study a constant coefficient of friction will be assumed. One methodology used to determine the coefficient of friction is the method developed by Hills and Nowell (1994).
At the beginning of the experiment, the coefficient of friction, fO, is uniform under the contact (Figure 2-7). It is the combination of stick and slip zone, which results from the fretting fatigue loading, which will raised a local coefficient of friction, fs, within the slip zones (Figure 2-8).
Figure 2-7: Coefficient of friction under the contact at Figure 2-8: Coefficient of friction under the contact the beginning of the experiment.
A graph of this function, with different value of Q/P is shown on the Figure 2-9. This graph may allow to determine the coefficient of friction of the material. However, it may be observed that for low value of Q/P, there is only a few difference of the value of the mean coefficient of friction, fm, between two points. Then, it is better to use a coefficient Q/P > 0,2 to determine correctly fs.
Figure 2-9: Prediction of the coefficient of friction within the slip zones measured from average values (Araújo, 2000).
For the sake of simplicity, a simplified contact geometry will be considered during this study: the cylinder-plane contact. For this kind of contact, an analytical formulation of the surface tractions exists and the stress field under the contact may be determined by using the Muskhelishvili potential’s theory. This 2D analytical formulation of the contact problem will be coded using a Python script, after the determination of the coefficient of friction corresponding to our contact. This will allow a rapid computation of the stress field corresponding to the different experiments which will be realised.

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Multiaxial fatigue

The fracture of materials under service conditions depends on many factors which will interact on each other’s. Some of these parameters are listed below:
– Material properties and microstructure;
– Stress concentration;
– Stress multiaxiality;
– Scale of structural components;
– Environment;
– Temperature;
– Material defects;
– …
Two regimes of fatigue can be observed: the Low Cycle Fatigue and the High Cycle Fatigue. In the first case, the structure experiences high levels of loads. Plastic deformations may be observed, due to that high level of forces, and the life is relatively short, under 105 cycles. In the other hand, for HCF, the level of solicitation stay low and elastic deformations are firstly observed. In that case, concepts of infinite life and fatigue limit stress may be defined. Usually, it will be considered that after 106 or 107 cycles the material will never break.
In order to deal with all the issues listed above, different criteria have been formulated.
One of these criteria will be presented below, a criteria based on the critical plane.
The critical plane approach (Findley, 1959, Brown and Miller, 1973, Socie, 1987, Fatemi and Socie, 1988, and McDiarmid, 1991), to identify the fatigue strength estimated of a component, is really attractive from a mechanical point of view. This is because it will not only give the fatigue strength of the component, but it will also provide information about the location and the direction estimated for the crack initiation.
Cracks initiate in preferential material planes, usually associated with high shear stresses. However, the normal mean stress plays also a role on the mechanism. It will keep the crack faces open and makes easier their growth. The idea behind the critical plane method is therefore to find the plane which will experiencing the highest combination of some equivalent normal and shear stress. There are several method to compute the equivalents shear stress and normal stress. Dang Van (1973) and later Papadopoulos (1994) proposed that the equivalent shear stress is given by the radius of the minimum circle circumscribing the stress path. However, such a minimum circle does not exist for a non-convex polygonal stress path. Improvement of this method were proposed later, Li et al. (2000) considered the minimum ellipsoid enclosing the deviatoric stress path, but it may fails for some paths such as the rectangular ones, and Mamiya et al. (2002) proposed a new definition of the minimum ellipsoid based on its Frobenius norm and they observed that for elliptical paths this norm was easily computed from the axes of rectangular Hull. Based on this observation, Mamiya et al. (2009) proposed a new method: The Rectangular Hull.

#### Rectangular Hull Method

The Rectangular Hull Method, as proposed by Mamiya et al. (2009) in the deviatoric stress space and later by Araújo et al. (2011) on the material plane, is a method to compute the equivalent shear stress, , characterizing the fatigue damage under multiaxial loading. In the first case, while working in the deviatoric space, some loading paths may need a rotation of the Hull around up to five axes, which will require an optimizing algorithm. This is no longer necessary if the Rectangular Hull Method is applied on the material planes. Indeed the rotation of the hull is done on the material plane and so a simple rotation in two dimensions is sufficient to calculate (Figure 2-10).
Figure 2-10: Stress path ψ, its convex hull and the identification of the characteristic lengths used to compute the shear stress amplitude in a material plane.
Thus, the first step to find the equivalent shear stress amplitude on a material plane Δ is to compute the halves size of the sides of a rectangular hull ( ) and ( ): ( ) = 1 max ( , ) − min ( , ) = 1,2 (2.43)
Finally, the equivalent shear stress which will be used is the one which maximises the equation (2.43): = max ( ) + ( ) (2.45)

The Modified Wöhler Curve Method

This method developed by Susmel and Taylor (2003) is founded on the hypothesis that metallic materials have a linear, elastic, homogeneous isotropic behaviour. Then, damages due to the fatigue can be estimated with the modelling of the initiation and the propagation of the micro, or miso, cracks with the continuum mechanic.

Chapter 1. Introduction
1.1. Context of the study
1.2. Review of the state of the art
1.3. Thesis scope
Chapter 2. Theory
2.1. Introduction
2.2. Basic contact mechanic
2.3. Multiaxial fatigue
2.4. Critical distance
2.5. Crack analogue
Chapter 3. Material and methods
3.1. Introduction
3.2. Titanium alloy
3.3. Group Tests
Chapter 4. Experimental verification of the crack analogue model
4.1. Introduction
4.2. Group I Tests: Determination of tests configuration
4.3. Analysis of the gradient effect
4.4. Influence of the maximum normal stress on the crack initiation process
4.5. Discussion of the analysis
Chapter 5. Study of the size effect on fretting fatigue
5.1. Introduction
5.2. Group II-a Tests: Study of the influence of the width of the specimens on fretting fatigue
5.3. Group II-b Tests: Influence of the damaged area on fretting fatigue
5.4. Post-failure investigation
5.5. Determination of the volume’s influence using the Weibull’s theory
5.6. Discussion of the results
Chapter 6. Conclusions
6.1. Overview
6.2. Main conclusions
6.3. Suggestions for future work
References.

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