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## Macrozones of the Ti-6Al-4V alloy

Le Biavant (2000) observed the presence of millimetric ghost structure in the Ti-6Al-4V alloy. Those macrozones, also named microtextures by other authors (Bantounas et al. 2010), can be observed with a chemical reaction or with simple mechanical tests (Fig.2.25). In these macrozones, the p phase has a preferred orientation and neighbouring macros zones may display very dierent major p grain orientation. The ex–grains, from the rst forging, is responsible for this macro structure. Indeed, despite the second forging in the + , the p grains and s colonies inherited from a prior tend to keep the same orientation or a Burgers variant.

Le Biavant et al. (2002) showed a high heterogeneity of crack initiation between dierent macrozones (Fig.2.25(b)). Indeed, in macrozones with orientation maximizing the shear stress on slip system, a high number of tiny cracks nucleate and coalescence is favored by the similar orientation of p grain. For uniaxial tests, it results in a heterogeneity of the crack density at the scale of the macrozones and apparition of paralleled cracks inside each macrozone.

### Microstructure and material properties observed in literature

In most cases, the Ti-6Al-4V alloy used in the industry has a bimodal microstructure consisting of equiaxed p and transformed -grains. However, depending on the parameters (e.g. temperature,

time, cooling rate) of the thermomechanical treatment or heat treatment, the microstructure and mechanical properties can vary in a wide range (Oberwinkler et al. 2010; Stoschka et al. 2009; Wu

et al. 2013). Indeed, the fatigue strength highly depends on the microstructural parameters such as

the p content and size in the bimodal microstructure.

Tab.2.3 summarizes information about microstructure parameters and high cyclic fatigue (HCF) strength found in the literature from 1984 to present. All fatigue tests were performed at room temperature in air under axial loading conditions with a sine wave on unnotched specimens. Note that fatigue limits are not dened as the stress amplitude, a, but as the maximum stress, max = m+a, leading to innite life. Fig.2.26 summarizes in a graph the maximum strength in a tensile test of those literature data against the life limit considered. Despite the resemblance, note that this gure is not S-N curve. Each point corresponds of the fatigue limit after dierent test campaigns. Studies results show a dispersion contained in 20% the average fatigue strength. Basic uniaxial tests were not part of this work, and basic fatigue properties had to be taken from the literature. Hence, careful appreciation of some prediction results presented in this thesis will be necessary.

#### Uniaxial tests and in uence of the loading ratio.

In general, fully reversed push-pull fatigue tests are used to generate a S-N curve. The classical results of Goodman (1899) and Haigh (1917) have shown that the addition of an mean tensile stress leads to a decrease of the fatigue limit, whereas superimposition of a mean compressive stress signicantly increases the fatigue limit. The Haigh diagram is a common method of representing the fatigue limit or endurance limit stress of a material in terms of alternating stress (the vibratory stress amplitude) and the mean stress (Nicholas 2006). Dierent equations were proposed to represent the eect of the mean stress: Goodman equation (or Modied Goodman equation): a = 1 1 m u.

**Gradient eect – Local vs Non local approach**

Until now, plain specimens with uniform loading were implicitly considered, but structural components generally present stress concentrations promoting crack initiation. From the beginning, notched specimens were considered to study the eect of stress concentration. The eect of notches having a nite tip radius on the linear-elastic stress eld distribution is usually quantied through the stress concentration factor, Kt. The S-N curve of Fig.2.34, adapted from the results of MacGregor and Grossman (1952), presents the fatigue test data for an unnotched bar (Kt = 1) and for a notched specimen with Kt = 3:1. The blue plain line represents estimated data for the corresponding notched geometry. Notice that the experimental results are above this prediction.

**Method of distribution of dislocations.**

The stress intensity factor at the crack tip might also be obtained by the method of distribution of dislocations, which is described in detail by Hills and Nowell (1993). A rapid description of the method will be presented here.

The technique is based on the principle developed by Bueckner (1958), which is basically a superposition principle. Let suppose a body with a crack subjected to a contact loads as shown in Fig.2.40. Equivalence of the original problem can be obtained by superposition of (a) and (b) where (a) represents the body without the crack subjected to contact loads and (b) represents a cracked body in the absence of external loads but stress on the crack faces has inversely equal to the stress obtained on the line of (a), so that when adding the two problems, the crack surfaces are free of stresses.

**Table of contents :**

Contents

**1 Introduction. **

1.1 Context of the study

1.2 The contact problem

1.3 Stress gradient and scale eect

1.4 Thesis scope

**2 Literature Review. **

2.1 Fretting

2.1.1 Introduction

2.1.2 Tribological behaviour of contact

2.1.3 Subsurface stress eld

2.1.4 Numerical modelling of fretting fatigue

2.2 Titanium Ti-6Al-4V

2.2.1 Macrozones of the Ti-6Al-4V alloy

2.2.2 Microstructure and material properties observed in literature

2.3 Fatigue of materials

2.3.1 Basic concepts

2.3.2 Uniaxial tests and inuence of the loading ratio

2.3.3 Criteria based on stress invariants

2.3.4 Critical plane approach

2.3.5 Gradient eect – Local vs Non local approach

2.3.6 Scale eect – Probabilistic approach

2.4 Fracture mechanics

2.4.1 Introduction

2.4.2 Fracture modes

2.4.3 Asymptotic solution

2.4.4 J-integral

2.4.5 Method of distribution of dislocations

2.4.6 Paris Law

2.4.7 The Kitagawa{Takahashi (K{T) diagram

2.4.8 Crack closure or eect of the loading ratio

2.4.9 Gradient eect – Critical distance

2.4.10 T-stress

2.5 Summary of the literature review

**3 Test design and experimental protocol **

3.1 Introduction

3.2 Apparatus conguration and preliminary test

3.2.1 Pads and Specimen

3.2.2 Experimental set-up used for cylinder-plane fretting fatigue research .

3.2.3 Digital Image Correlation applied to fretting fatigue

3.2.4 Experimental set-up for spherical-plane fretting bi-axial fatigue research

3.2.5 Coecient of friction

3.3 Finite element modelling of fretting fatigue tests

3.3.1 Basic 2D model

3.3.2 Comparison with 3D model

3.3.3 Crack modeling

**4 Experimental results **

4.1 Results of cylinder/plane fretting fatigue test

4.1.1 First set of tests

4.1.2 Comparison among dierent gradients

4.2 Results of spherical/plane fretting fatigue test

4.2.1 Objective

4.2.2 Experimental test realized

4.3 Conclusion of the experimental tests

**5 Prediction of fretting fatigue life **

5.1 Introduction

5.1.1 Material properties and coecient of friction of Ti-6Al-4V alloy

5.1.2 On the use of the analytical formulation of the contact problem

5.1.3 Location of the critical site for crack initiation

5.2 Multiaxial stress fatigue model

5.2.1 Fretting map and comparison with experimental results

5.2.2 Discussion on the model

5.2.3 Proposition of criterion parameter

5.3 Short crack arrest approach

5.3.1 Introduction

5.3.2 Comparison with experimental results

5.3.3 Eect of the loading ratio

5.3.4 Conclusion

5.4 A new methodology – Using the T-stress

5.4.1 Assumptions

5.4.2 Expression of the criterion

5.4.3 Identication of the parameters

5.4.4 Representation of the criterion

5.4.5 Application to fretting

**6 Conclusions, recommendations and future work **

6.1 Overview

6.2 Methodology used in this study

6.2.1 Experimental set-up

6.2.2 Analytical and numerical modeling of the problem

6.2.3 Prediction of fretting fatigue lifetime

6.3 Main conclusions

6.3.1 Fretting fatigue experiments

6.3.2 Prediction of the fretting fatigue lifetime

6.4 The scale eect

6.5 Notch analogy