Fretting fatigue Behaviour of Inconel 718 at room and high Temperature

Doctoral Thesis Mechanical Engineering Fretting fatigue behaviour of Inconel 718 at room and high temperature Author: María Moreno Rubio Advisors: Jesús Vázquez Valeo Carlos Navarro Pintado Dpto. de Ingeniería Mecánica y Fabricación Escuela Técnica Superior de Ingeniería Universidad de Sevilla Sevilla, 2025 ii Tesis doctoral: Fretting fatigue behaviour of Inconel 718 at room and high temperature Autor: María Moreno Rubio Directores: Jesús Vázquez Valeo Carlos Navarro Pintado El tribunal nombrado para juzgar el proyecto arriba indicado, compuesto por los siguientes miembros: Presidente: Vocales: Secretario: Acuerdan otorgarle la calificación de: Sevilla, 2025 El Secretario del Tribunal iv A mi familia vi Acknowledgements En la vida nos vamos encontrando obstáculos que parecen inalcanzables de superar. Pero gracias a las personas que tenemos a nuestro lado, esas personas que te ayudan, te apoyan y te van guiando para que consigas superarlos, todo parece más fácil. Al final te das cuenta de que todo se puede lograr, y que ese camino que has seguido no es más que un camino lleno de aprendizaje. Aprendizaje no solo en lo profesional, sino también en lo personal. Por ese motivo me gustaría dar las gracias a las personas que he tenido ahí para mí y que me han ayudado siempre que lo he necesitado; a esas personas que me han dado el empujón que necesitaba en el momento justo. Dicen que no hay cosa más valiosa que el tiempo, tiempo que habéis dedicado hacia mí en los últimos años. Por todo ello, solo puedo decir: gracias. Aunque hay muchas personas a las que me gustaría nombrar, en especial quiero dar las gracias a mis directores Jesús y Carlos, a mi familia y, sobre todo, a mi pareja, Fran. María Moreno Rubio Sevilla, 2025 viii Resumen En esta tesis se analiza el comportamiento del Inconel 718, una superaleación de base Níquel-Cromo, conocida por sus excelentes propiedades mecánicas y su alta resistencia a la corrosión y a la oxidación incluso a temperaturas elevadas. El estudio combina ensayos experimentales y simulaciones numéricas con el fin de caracterizar de manera precisa la respuesta del material frente al daño por fretting. Para ello se empleó una configuración de contacto cilíndrico mediante el uso de una probeta del tipo “dog-bone” con sección rectangular, presionada contra un pad cilíndrico. El conjunto se montó en un dispositivo comúnmente denominado “puente de fretting”. Se realizaron ensayos tanto de fatiga simple como de fatiga por fretting, aplicando diferentes niveles de carga axial y normal, y considerando dos condiciones de temperatura: ambiente y 650 °C. Asimismo, se llevaron a cabo ensayos interrumpidos para estudiar la evolución del daño, y se midieron los coeficientes de fricción en distintas condiciones. Los resultados experimentales obtenidos se analizaron en profundidad, identificando la influencia de la temperatura y las cargas aplicadas sobre la vida a fatiga del material. Posteriormente, se llevó a cabo un análisis fractográfico de las probetas mediante microscopía electrónica de barrido, permitiendo identificar las distintas zonas de la superficie de fractura, así como establecer correlaciones entre el tipo de daño observado y las condiciones de ensayo. Además, se estimó la tasa de propagación de grietas bajo distintas condiciones, y se compararon los resultados con datos existentes en la literatura. Para complementar el estudio experimental, se desarrolló un modelo numérico mediante el método de elementos finitos, con el objetivo de simular el campo de tensiones y deformaciones generado en la zona de contacto. Este modelo permitió analizar el efecto de las condiciones de carga y temperatura sobre el comportamiento del material. Finalmente, se aplicó un modelo de predicción de vida a fatiga por fretting, que considera conjuntamente las fases de iniciación y propagación de grietas, sin necesidad de definir explícitamente el punto de transición entre ambas. Las estimaciones obtenidas fueron comparadas con los resultados experimentales, mostrando una buena correlación y permitiendo validar la utilidad del modelo para la predicción de vida en componentes sometidos a fretting. En conjunto, el trabajo desarrollado proporciona una caracterización completa del comportamiento del Inconel 718 frente a la fatiga por fretting, ofreciendo información valiosa para el diseño de componentes críticos en condiciones severas de servicio. xvi Figure 4-1. Dimensions of plain fatigue test specimen at RT (a) and HT (b) in mm. .............. 34 Figure 4-2. HT testing equipment. ...................................................................................................... 34 Figure 4-3. Dimensions of fretting fatigue test specimen at RT (a) and HT (b) in mm. .......... 35 Figure 4-4. Scheme of fretting fatigue testing (in this case with a test specimen for testing at HT). ............................................................................................................................................................ 36 Figure 4-5. Dimension of fretting bridge in mm. ............................................................................. 37 Figure 4-6. Fretting bridge calibration scheme. ............................................................................... 38 Figure 4-7. Accesory for alignment. .................................................................................................... 38 Figure 4-8. Illustration of HT test, (a) plain fatigue test and (b) fretting fatigue test. .............. 39 Figure 4-9. Interrupted fretting fatigue test scheme of the first type - FFI-1. ............................ 41 Figure 4-10. Interrupted fretting fatigue test scheme of the second type - FFI-2. .................... 41 Figure 4-11. Interrupted fretting fatigue test scheme of the third type - FFI-3. ........................ 41 Figure 4-12. Friction coefficient test scheme. .................................................................................... 43 Figure 4-13. Plain fatigue lives of this work versus Kawagoishi’s results at RT. ..................... 45 Figure 4-14. Plain fatigue lives at RT respect to parameter SWT . ............................................... 45 Figure 4-15. Plain fatigue and fretting fatigue test at RT and HT (650 °C). ............................... 49 Figure 4-16. Fretting fatigue test at RT and HT, with N of 10 kN and 7 kN.............................. 51 Figure 4-17. Fracture surface of interrupted fretting fatigue test at HT, type 1 and 2............. 52 Figure 4-18. Fracture surface of marked fretting fatigue test at HT, type 3. .............................. 55 Figure 4-19. Fatigue life values of normal and marked fretting fatigue test at HT.................. 56 Figure 4-20. Crack evolution of fretting fatigue tests at 750 MPa of maximum axial load and 10 kN of normal load at 650 °C. ........................................................................................................... 57 Figure 5-1. Fretting fatigue fracture surface of specimens tested at σmax = 650 MPa at a) room temperature FF-RT(1)-4 and b) 650 °C FF-HT(1)10. c) and d) profile of the specimen tested at RT and HT [80]. .................................................................................................................................. 60 Figure 5-2. Fracture surface near the surface in fretting fatigue samples tested at RT and HT with 650 MPa. .......................................................................................................................................... 60 Figure 5-3. Details of fracture surface near the surface in fretting fatigue samples tested at RT and HT with 650 MPa. .................................................................................................................... 61 Figure 5-4. Fracture surface for a sample tested at RT and 650 MPa at a distance of a) 0.841 mm, b) 1.707 mm, c) 1.902 mm and d) 2.925 mm. ........................................................................... 62 Figure 5-5. Oxygen concentration (% by weight) on the fracture surface (image taken at SEM). ......................................................................................................................................................... 63 Figure 5-6. Photographs of the fracture surface of the analysed specimens, showing the characteristic tonalities in specimens at RT (grey) and at HT (blue and gold). ........................ 64 Figure 5-7. Oxide layer in FCZ. ............................................................................................................66 Figure 5-8. Microstructure at the edge of the specimen's surface after high temperature testing. Image (a) depicts the region subjected to contact (CZ), and image (b) corresponds the area unaffected by contact (FCZ). .................................................................................................67 Figure 5-9. Chemical composition of the CZ and NCZ of fretting fatigue test specimens tested at room temperature and 650 °C, under σmax of 650 and 400 MPa. ..................................68 Figure 5-10. Cross-sectional maps showing the distribution of the elements for the Inconel 718 after fretting fatigue test with σmax of 650 MPa at RT (FF-RT(1)-4) and HT (FF-HT(1)-10): test at a) RT in CZ, b) RT in FCZ, c) HT in CZ, and d) HT in FZC. ..............................................69 Figure 5-11. Crack morphology of four plain fatigue test: a) RT sample tested at σmax = 850 MPa with R = -1, b) RT sample tested at σmax = 550 MPa with R = -1, c) HT sample tested at σmax = 850 MPa with R = 0.1, and d) HT sample tested at σmax = 850 MPa with R = 0.1. ............71 Figure 5-12. Crack morphology of 4 fretting fatigue test: a) RT sample tested at σma x = 650 MPa, b) RT sample tested at σmax = 400 MPa, c) HT sample tested at σmax = 650 MPa, and d) HT sample tested at σmax = 400 MPa.....................................................................................................72 Figure 5-13. Crack scar for two samples, one tested at RT and another at HT, FF-RT(1)-8 and FF-HT(1)-11 respectively. ......................................................................................................................74 Figure 6-1. Simplification of modelled geometry.............................................................................75 Figure 6-2. Design phase in ANSYS. ...................................................................................................76 Figure 6-3. Mesh (elements). .................................................................................................................77 Figure 6-4. Master node. ........................................................................................................................78 Figure 6-5. Boundary conditions. .........................................................................................................79 Figure 6-6. Step in FEM. .........................................................................................................................80 Figure 6-7. Axial stress. ..........................................................................................................................80 Figure 6-8. Surface normal stress: analytical vs. FEM. ....................................................................81 Figure 6-9. Surface axial stress: analytical vs. FEM. .........................................................................81 Figure 6-10. Surface shear stress: analytical vs. FEM. .....................................................................82 Figure 6-11. FEM superficial axial stress. ...........................................................................................83 Figure 6-12. Scheme of eccentricity and stick zone. .........................................................................83 Figure 6-13. FEM superficial shear stress. ..........................................................................................84 Figure 6-14. Schematic of SWT parameter at the surface. ..............................................................85 Figure 6-15. Path inside the test specimen for calculation of stresses and strain in the interior of the test specimen. ................................................................................................................................86 Figure 6-16. Axial stress inside the specimen by FEM. ...................................................................87 Figure 6-17. Axial stress inside the specimen by FEM near the surface. ....................................87 xviii Figure 7-1. Stages for determining the constants of the Paris crack growth law. .................... 89 Figure 7-2. Measured point of specimen PF-RT-4, and analysis for obtaining FCGR. ........... 91 Figure 7-3. Measured point of specimen PF-HT-1, and analysis for obtaining FCGR. .......... 91 Figure 7-4.Crack growth rate for plain fatigue test at RT and HT. .............................................. 92 Figure 7-5. Scheme of corner-crack and surface-crack configurations with elliptical crack front [86]. ................................................................................................................................................... 93 Figure 7-6. Evolution of the aspect ratio a/c as a function of a, for a crack initiated at a corner. ..................................................................................................................................................................... 96 Figure 7-7. Evolution of the aspect ratio a/c as a function of a, for a crack initiated at the surface. ...................................................................................................................................................... 96 Figure 7-8. Crack growth rate (da/dN) vs. ∆𝐾𝐼 (MPa m1/2) of plain fatigue test at RT and HT. ..................................................................................................................................................................... 97 Figure 7-9. Comparative between crack growth rate, stress intensity factor, and crack length for plain fatigue. ...................................................................................................................................... 99 Figure 7-10. Comparative with other authors of FCGR vs. ∆𝐾𝐼 of plain fatigue test at RT. 101 Figure 7-11. Comparative with other authors of FCGR vs. ∆𝐾𝐼 of plain fatigue test at HT.102 Figure 7-12. Crack growth rate for fretting fatigue tests performed at a) RT and b) HT. .... 104 Figure 7-13. Comparative fracture surface for sample tested at 650 MPa, a) at RT at a distance of 1.55 mm from the surface and b) at HT at a distance of 1.46 mm from the surface. .................................................................................................................................................... 105 Figure 7-14. CGR (da/dN) vs. ∆𝐾𝐼 (MPa m1/2) of fretting fatigue tests at RT and HT. .......... 107 Figure 7-15. Crack growth rate vs. ∆𝐾𝐼 of fretting fatigue tests at RT and HT grouped by temperature. ........................................................................................................................................... 108 Figure 7-16. Comparative between crack growth rate, stress intensity factor, and crack length for fretting fatigue. ................................................................................................................... 109 Figure 7-17. Comparative with other authors of CGR vs. ∆𝐾𝐼 of fretting fatigue test at RT. ................................................................................................................................................................... 110 Figure 7-18. Comparative with other authors of CGR vs. ∆𝐾𝐼 of fretting fatigue test at HT. ................................................................................................................................................................... 111 Figure 7-19. Comparative between plain and fretting fatigue crack growth rate at RT. ...... 112 Figure 7-20. Comparative between plain and fretting fatigue crack growth rate at HT. ..... 113 Figure 8-1. Example of the curves obtained applying the fatigue model. ............................... 116 Figure 8-2. Fatigue crack estimation and stress distribution. ..................................................... 116 Figure 8-3. Scheme for obtaining the initiation curves. ................................................................ 121 Figure 8-4. Scheme for calculating average SWT parameter. ..................................................... 121 Figure 8-5. Scheme for determining initiation life based on SWT and evaluated crack length. .................................................................................................................................................................. 121 Figure 8-6. Estimated life vs. experimental life, at both temperature, RT and HT. ............... 122 Figure 8-7. S-N curve for estimated type 1 and experimental lifetimes. .................................. 125 Figure 8-8. S-N curve for estimated type 2 and experimental lifetimes. .................................. 126 Figure 8-9. Initiation length vs. estimated life and initiation phase % vs. estimated life for Type 1 at RT (left) and HT (right). .................................................................................................... 127 Figure 8-10. Initiation length vs. estimated life and initiation phase % vs. estimated life for Type 2 at RT (left) and HT (right). .................................................................................................... 127 Figure 8-11. Curve of the number of elapsed cycles. .................................................................... 128 Figure 8-12. Crack evolution estimation for maximum axial stress of 900 MPa. ................... 129 Figure 8-13. Estimated crack length for type 1 vs. real crack length. ........................................ 130 xx LIST OF TABLES Table 3-1. Chemical compositions of Inconel 718 alloy (wt%). .....................................................29 Table 3-2. Mechanical properties. ........................................................................................................31 Table 3-3. Fatigue strength and ductility parameters. ....................................................................31 Table 3-4. Grain size of microstructure of Inconel 718 alloy after the heat treatment. ............32 Table 4-1. Results of friction coefficient test.......................................................................................42 Table 4-2. Results of plain fatigue test at RT......................................................................................44 Table 4-3. Results of plain fatigue test at HT. ....................................................................................46 Table 4-4. Results of fretting fatigue test at RT (R = 0.1, f = 8Hz). .................................................47 Table 4-5.Result of fretting fatigue test at HT. ..................................................................................48 Table 4-6. Crack length of interrupted test. Type 1 and 2. .............................................................53 Table 4-7. Results of the FFI-3 test........................................................................................................54 Table 4-8. Results of total crack length and total fatigue life of FF-HT(1) test. ..........................56 Table 5-1. Summary of test conditions and cycles for fracture surface analysis. ......................63 Table 5-2. % Oxygen on the surface of the specimen. .....................................................................65 Table 5-3. Oxide layer thickness in CZ and FCZ. .............................................................................66 Table 5-4. Comparison of experimental and Hertzian contact semi-widths in fractured and non-fractured zones. ...............................................................................................................................73 Table 7-1. Coefficient of Paris’ law, C and m, in simple fatigue test. ........................................ 100 Table 7-2. Coefficient of Paris’ law, C and m, in fretting fatigue test. ....................................... 110 Table 8-1. GME and GSDE parameter to estimated life at RT. .................................................. 124 Table 8-2. GME and GSDE parameter to estimated life at HT. .................................................. 124 1 1 INTRODUCTION his chapter will provide an introduction to fatigue and fretting fatigue, covering the fundamental concepts and distinctions between the two phenomena. Additionally, it will include a historical overview of fretting fatigue, with a particular focus on its relevance to Inconel. 1.1 Introduction to fatigue Fatigue can be defined as a phenomenon that occurs when a metallic component is subjected to variable loads over time. The repetition of these loads over a certain period can induce the formation of a cracks, which may grow until the failure of the component. This type of damage is very common in mechanical elements, accounting for approximately ninety percent of the failures [1], such as railway wheels and axles, gas or steam turbine disk-blades joints, connecting rods and crankshafts, connections, etc. For this reason, it has been subject of study for engineers. Nowadays, it is not known exactly who first used the term fatigue; it is said that the credit is shared by the Frenchman Poncelet in 1854 and the Englishman Braithwaite in 1854 [2]. The study of this concept began a great interest at the beginning of the Industrial Revolution, driven mainly by the two major industries of the time: mining and railways. During this period, it was observed that various mechanisms began to fail continuously, which sparked interest in its study. It could be said that the first paper was written in 1837 by Albert [3], a German who sought to understand why the chains of mining carts were failing. At this time, there were no standard regulations explaining how to conduct experimental tests, making it very challenging for him. He analysed not only the materials but also the components in an effort to understand the cause of failure. In particular, the railway sector was a key factor in advancing knowledge about fatigue. However, it was not until Wöhler's work that fatigue began to be significantly understood. Wöhler was a German railway engineer who began investigating the cause of numerous railway accidents of the time, discovering in 1858 that fatigue damage depended on the amplitude of the cyclic stress to which components were subjected. Following this discovery, Wöhler’s Law was formulated in 1870 [4]. Wöhler’s Law states that the repeated T The measure of intelligence is the ability to change. Albert Einstein 2 Fretting fatigue behaviour of Inconel 718 at room and high temperature application of stresses can lead to the failure of a material. Therefore, it can be said that he was the first to recognize the significance of tensile stress [2]. Wöhler's experimental results were not published until 1924 by Gough. It should be noted that Wöhler presented his results in table form, and it was not until 1910 that these results were graphically represented, thanks to the Basquin [5]. Basquin plotted Wöhler's results by relating the applied loads to the number of cycles until material failure, which it is known today as the S-N curve, an example of S-N curve for aluminium 2024-T351 obtained in air at 20 kHz can be seen in Figure 1-1 [6]. Figure 1-1. Fatigue test results for aluminium 2024-T351 obtained in air at 20 kHz. Due to the numerous railway accidents at that time (see Figure 1-2, illustrating a characteristic fatigue crack pattern in a train wheel), many researchers were trying to understand why continuous circumferential cracks appeared around a journal at the corner of a keyway in an axle. Rankine was one of the first to highlight the premature failure of axles due to notches and recommended reducing sharp corners on them [7]. However, Rankine did not specifically focus on this area of study, unlike Neuber, who developed a method before World War II to calculate strain and stresses at sharp notches, known as the Neuber approach [8]. Later, the researcher Peterson published a comprehensive handbook on stress concentration factors in different geometries [9]. It was not until 1903 that Ewing and Humfrey conducted the first metallurgical study on cyclic fatigue damage in metals thanks to use an optical microscope [10]. During the following years, research on fatigue continued, and it was between 1939 and 1960 that the study of fatigue experienced a resurgence. This was primarily due to the technological focus on the aerospace sector, driven initially by military necessity and later by commercial applications. All of these developments encouraged scientists to continue their research and progress, leading to the knowledge we possess today. Introduction 3 Figure 1-2. Close-up view of fatigue cracking on the exterior wheel rim of a German high speed train. 1.2 Introduction to fretting Fretting is characterised by small-amplitude oscillatory relative displacements that occur between two components in contact under pressure [11]. The first observation of fretting fatigue was made in 1911, when Eden et al. [12] noticed oxide residues on the steel clamps of their testing machine while conducting fatigue test with steel specimens. However, it was not until 1927 that Tomlinson [13] conducted the first formal study on fretting. For this, he designed two machines to produce small rotational movements between two bodies in contact. As a result of these experiments, the term “fretting corrosion” emerged, as small oxide particles were observed on the steel specimens, distinguishing this phenomenon from common corrosion caused by a chemical reaction in the material. Tomlinson recognised that the corrosion could be attributed to the displacements occurring between the two bodies, on the order of a tenth of a micron. In 1941, Warlow-Devies [14] conducted a series of tests consisting of two stages. In the first, he induced fretting damage and subsequently subjected the material to fatigue testing. In doing so, he was able to relate the influence of fretting damage on fatigue life, finding that the fatigue life was reduced by between 13% and 17% compared to pure fatigue. Therefore, we can say that fretting damage is divided into three types:  Fretting corrosion: damage caused when the surfaces in contact generate oxide and surface cracks.  Fretting wear: damage that occurs when wear and crack formation on the surface are of greater importance. This type of damage is often used synonymously with fretting corrosion since the wear that occurs on the contact surface generally leads to the formation of oxide. 11 2. FRETTING FATIGUE retting fatigue is a specific case of fretting where two components are in contact, and one of them is also subjected to a variable global stress. Figure 2-1 schematically illustrates the differences between plain fatigue, fretting, and fretting fatigue. In plain fatigue (Figure 2-1a), the component is subjected to a cyclic global stress, denoted as 𝜎𝑔𝑙𝑜𝑏𝑎𝑙, resulting from an externally applied stress σ. In contrast, fretting (Figure 2-1b) involves one body against another by a normal force N, maintaining contact between them. Simultaneously, a tangential force Q induces a relative slip at the contact interface, leading to a localised stress distribution within the materials, represented as 𝜎𝑙𝑜𝑐𝑎𝑙. Finally, Figure 2-1c illustrates fretting fatigue, which combines aspects of both plain fatigue and fretting. In this phenomenon, one of the bodies is pressed against another by a normal force. The second body is then subjected to axial loading, while friction at the contact interface gives rise to a tangential force. As a result, both local and global stress distributions develop. In general, both the contact forces and the global stress may vary over time. Figure 2-1. Scheme of plain fatigue, fretting and fretting fatigue. Depending on the values of tangential force Q and the normal force N, two types of a) Plain fatigue b) Fretting c) Fretting fatigue F Power is not determined by your size, but the size of your heart and dreams! Monkey D. Luffy 12 Fretting fatigue behaviour of Inconel 718 at room and high temperature contact conditions may arise between the surfaces. It is possible for the friction coefficient μ to be equal to the ratio Q/N = μ, or for it to be greater, such that Q/N ≤ μ. In the first case, global slip occurs, meaning there is full relative motion across the entire contact interface between the two surfaces. In the second case, partial slip takes place. In contrast to global slip, partial slip is characterised by a contact region in which part of the surface remains adhered, while the rest undergoes relative sliding. This phenomenon is illustrated in Figure 2-2. Figure 2-2. Displacement of the adhesion zone. In the case of partial slip under an applied tensile load, the adhered region becomes displaced by a certain amount e, known as eccentricity, relative to the configuration without global stress, see Figure 2-3. This displacement arises because the deformations in each body within the contact zone are not zero; instead, the deformation corresponds to that induced by the global tensile stress [56]. Figure 2-3. Scheme of eccentricity in partial condition. Numerous factors influence the behaviour of fretting fatigue [57], including the geometry of the contact regions, normal load, tangential load, bulk stress, coefficient of friction, surface finish, residual stresses, fatigue behaviour, and loading frequency, among others. These factors can generally be grouped into two main categories: those related to the material properties and behaviour, and those related to the stress and strain fields [58]. The first group focuses on determining material properties through various experimental tests, which assess parameters such as crack growth rate, wear, and fracture toughness. The second group is concerned with the stresses generated at the contact surface. GLOBAL SLIP PARTIAL SLIP Slip zone Slip zone Adhesionzone Fretting fatigue 13 2.1. Contact Mechanics Experimental studies can be carried out using both real and simplified geometries. The advantage of using simplified geometries is that they can be analysed analytically, assuming plane strain conditions—this is valid when the contact area is small relative to the specimen thickness. However, if the contact area is large compared to the specimen thickness, the analytical approach under plane strain is no longer valid, and the finite element method (FEM) is required. In simplified geometries, mechanical contacts can generally be divided into two main types: complete contact and incomplete contact. The distinction is based on whether the contact area depends on the normal load. In incomplete contact, the contact area varies with the normal load, whereas in complete contact, it remains constant regardless of the applied load. The following are examples of these simplified geometries, along with relevant details. All equations used in this section are based on reference [59]. In that reference, the stress distributions corresponding to each of the cases discussed below can also be found. 2.1.1. Cylindrical contact This case involves a cylinder of radius R coming into contact with a plane, corresponding to an incomplete contact, as illustrated in Figure 2-4. This type of contact is currently one of the most widely used, primarily because it does not exhibit stress singularities and its alignment is easy. Furthermore, it is a simple geometry for which an analytical solution for the contact-induced stresses is available. This geometry will be used throughout the entire study, which is why section 2.2 places particular emphasis on the development of the expressions employed. In the figure, the contact zone exhibits a width of 2a, a dimension that varies with the applied normal load N, which usually in fretting test is considered constant. A variable tangential load Q, and, usually a variable axial load σ, are also present, being both typically applied in phase. The analysis begins with the derivation of the normal pressure expression induced by the load N, which is expressed as a load per unit thickness. 𝑝(𝑥)=−𝑝0 √1−(𝑥𝑎)2 (2-1) Here, p0 represents the maximum normal pressure, and a is the half-width of the contact zone. The expressions for both parameters are given as follows: 𝑝0 =2 𝑁 𝜋 𝑎 (2-2) 𝑎=√8 𝑁 𝑅 (1− 𝜈2) 𝜋 𝐸 (2-3) 14 Fretting fatigue behaviour of Inconel 718 at room and high temperature Figure 2-4. Scheme of cylindrical contact with rounded edges. In fretting, in addition to the normal load, a tangential load is applied, leading to the formation of two well-defined regions: - Slip zone: 𝑐≤|𝑥|<𝑎 - Stick or adhesion zone: |𝑥|<𝑐 The relationship between the half-width of the contact zone and the stick zone is defined by the following expression: 𝑐𝑎=√1− 𝑄 𝜇 𝑁 (2-4) It can be observed that the half-size of the stick zone, c, depends on the ratio 𝑄 𝜇 𝑁. As the value of 𝑄 𝜇 𝑁 increases, the stick zone decreases. The limiting case occurs when 𝑄=𝜇 𝑁, in which no portion of the interface remains adhered and full sliding occurs, this corresponds to global slip. In fretting fatigue, in addition to normal and tangential loads, an axial load P is also applied. This axial load induces an eccentricity e, displacing the stick region. The presence of eccentricity causes the stress distribution at the surface to vary depending on the position along the contact interface. The analytical solution for this problem was developed by Hills Fretting fatigue 15 and Nowell [59], as a perturbation of Mindlin’s original formulation [60]. The magnitude of the eccentricity e assuming plain strain is given by the following expression: 𝑒= 𝑎 𝜎 4 𝜇 𝑝0 (2-5) where σ is the axial stress in the specimen 2.1.2. Spherical contact Figure 2-5. Scheme of spherical contact. This case involves the contact between a sphere of radius R and a flat plane. As in the previous case, this is considered an incomplete contact, as illustrated in Figure 2-5. It is similar to the cylindrical contact configuration, with the main difference being that the contacting body is a sphere. As before, the system is subjected to a constant normal load N, a cyclic axial stress σ, and an oscillatory tangential load Q, and 𝜎𝑄, which represents the stress induced by Q. The normal pressure exerted at the contact is given by: 𝑝(𝑥)=−𝑝0 √1−(𝑟𝑎)2 (2-6) 16 Fretting fatigue behaviour of Inconel 718 at room and high temperature where 𝑟=√𝑥2+𝑦2, 𝑝0 is the maximum normal pressure, and a is the radius of the contact zone. The contact generated by pressing the sphere against the plane forms a circular area with radius a. The stick zone has a radius c, and due to the global tensile stress, it is displaced by an amount e in the direction of the axial load. The radius of the contact zone is determined by: 𝑎=√3 𝑁 𝑅 (1− 𝜈2) 2 𝐸 3 (2-7) The relationship between the slip zone and the stick zone is given by the following expression: 𝑐=𝑎 √1− 𝑄 𝜇 𝑁 3 (2-8) As in the case of cylindrical contact, the stick zone is displaced by an eccentricity, considering plane strain, which is defined by: 2.1.3. Flat contact This is the case of a flat rigid punch on a plane. The main advantage of this geometry is its ease of fabrication and use in fretting setups. However, a major drawback is the occurrence of singularities at the edges of the contact zone, theoretically causing infinite stresses, though in reality, plastic deformation mitigates this. To address this, two approaches can be taken: consider an elastoplastic stress field (which increases complexity), or assume a linear elastic field while accepting error near the contact edges depending on the size of the plastic zone. In this geometry, see Figure 2-6, the punch base is 2a (a being the half-width of the contact zone), and it is pressed against an elastic plane with a normal load N per unit thickness. Because the contact width is fixed and does not change with N, this constitutes a complete contact. In that figure, it can also observe Q, which corresponds to the tangential load that is in phase with the global stress σ. Additionally, it appear 𝜎𝑄, which represents the stress induced by Q and acts in the opposite direction to σ. In this way, if the global stress applies tension to the plane, the tangential stress will have a negative value. Conversely, if the global stress applies compression to the plane, the tangential stress will have a positive value. The normal stress distribution (neglecting friction) is given by: 𝑒= 8 𝑎3 𝜎 3 𝜇 𝑁 (1−𝜈) (4−3𝜈) (2-9) Fretting fatigue 17 𝑝(𝑥)=𝑁 𝜋 √𝑎2−𝑥2 (2-10) As seen, stress tends to infinity as x approaches ± a. As discussed in previous sections, the displacement of the adhesion zone within the contact can be calculated analytically. However, in the case of a flat contact, there is no analytical solution for this displacement caused by the global stress, and numerical techniques must be used instead. Moreover, it is important to note that in this case, the size of the contact and adhesion zones does not depend on the tangential load, but rather on the coefficient of friction and Poisson's ratio. Another disadvantage of the flat rigid punch contact is that due to the frictional force occurring at the contact surface, a moment is generated in the contact element causing a rotation around the y-axis. 2.1.4. Flat contact with rounded edges This configuration involves a flat rigid punch with rounded edges. It is a modification of the flat contact case, aimed at avoiding singularities and errors due to plastic deformation. Here, the corners have a radius R, with a flat central portion of width b, see Figure 2-7. The contact zone half-width a varies with the applied normal force, making this an incomplete contact. Figure 2-6. Scheme of flat contact. 18 Fretting fatigue behaviour of Inconel 718 at room and high temperature This geometry has been studied analytically by [61]. In partial slip, the relationship between the contact zone and the adhered region is defined by: 𝑁=− 𝑎2 𝐸 4(1−𝜈2)𝑅[𝜋 2−sin−1𝑏𝑎−𝑏𝑎 √1−(𝑏𝑎)2] (2-11) where N is the applied normal load, a is the half-width of the contact zone, E is the Young's modulus, ν is Poisson’s ratio, R is the radius of curvature of the corners, b is the half-width of the flat part of the punch. In the case of partial slip, the relationship between the size of the contact zone and the adhered zone can be obtained by means of the following equation: 𝑄 𝜇 𝑁=1−(𝑐/𝑎)2 [𝜋 2−sin−1𝑏𝑎−𝑏𝑎 √1−(𝑏𝑎)2] [𝜋 2−sin−1𝑏𝑎−𝑏𝑎 √1−(𝑏𝑎)2] (2-12) where Q is the tangential force, μ is the frictional force and c is the half-width of the adhesion zone. As in the previous types of contact, the effect of the global stress, 𝜎, causes a displacement of the adhesion zone. This displacement can be determined using the technique described in Section 2.1.1. Figure 2-7. Scheme of flat contact with rounded edges. Fretting fatigue 19 2.2. Cylindrical contact This section focuses on analysing the expressions related to cylindrical contact, previously introduced in Section 2.1.1. The equations used throughout this analysis are based on the formulations presented in reference [60]. The input data are divided into two categories: applied loads and material properties. The applied loads are as follows:  Normal load, 𝑁=1000 N/mm  Tangential load, 𝑄=250 N/mm  Axial stress, 𝜎=400 MPa The directions of the loads applied in this section are defined as: the normal load acts in the negative y-direction, the tangential load in the negative x-direction, and the axial stress in the positive x-direction. These directions are illustrated in Figure 2-8: The material used is Inconel 718, which will be described in detail in Chapter 3. However, for the present evaluation, three key properties must be known: Young’s modulus and Poisson’s ratio are provided in the following chapter, Table 3-2; while the coefficient of friction was obtained through experimental tests, as indicated in Section 4.4, Table 4-1. The values of these parameters are:  Coefficient of friction, 𝜇=0.43  Poisson’s ratio, 𝜈=0.29  Young’s modulus, 𝐸=200 GPa The analysis begins by considering a cylindrical contact subjected initially to a normal load and the tangential load. The maximum normal pressure, the contact zone, and the stick region are defined by expressions (2-2) through (2-4), from which the following values are obtained: 𝑝0=1180.40 MPa, 𝑎=0.54 mm, 𝑐=0.35 mm. Figure 2-8. Direction of applied loads in the theoretical study of surface stress. N Q σ Component 1 Component 2 y x 27 3. MATERIAL: INCONEL 718 There is nothing more noble than a humble material. Tadao Ando nconel is a trademark registered by the company Special Metals, focusing specifically on Nickel-chromium-based superalloys. In this work, Inconel 718, which is a frequently used Nickel-based superalloy, as before said, was used. This type of superalloy is widely used in high temperature applications due to its excellent resistance at elevated temperatures, with a melting point ranging from approximately 1260 to 1336 °C. Additionally, it exhibits remarkable resistance to both corrosion and oxidation. It is primarily used in the aerospace industry, where it can be found in components such as turbine coils, discs, and shafts, as well as compressor discs and blades. However, its use is not limited to aerospace; it also has a wide range of other applications, including liquid-fuel rockets, terrestrial gas turbines, cryogenic tanks, and nuclear power plants, among others [62] and [63]. [64] [65] [66] [67] The microstructure of Inconel 718 alloy consists of an austenitic matrix ( phase) present in a face-centered cubic (FCC) structure. Due to the heat treatment, different phases can be produced, such as gamma prime (’) present in a simple cubic (SC) structure (Ni3(Al,Ti)), gamma double prime (’’) present in a tetragonal body-centered (TBC) structure (Ni3Nb), delta (δ) phases with a simple orthorhombic structure (Ni3Nb), and others [64] - [67]. Inconel 718 alloy is primarily strengthened through the precipitation of ’ and ’’ phases, while the δ phase is a non-hardening and incoherent precipitate that typically forms as needles at grain boundaries and is identified by the chemical formula Ni3Nb. The degradation of strength observed in this alloy is primarily attributable to the depletion of the ’’ phase. Nonetheless, the δ phase has been found to enhance the material's resistance to creep rupture and sliding at grain boundaries [64] - [67]. The researchers Brooks et al. [68] investigated the phase transformations occurring in Inconel as a function of time and temperature, leading to the development of the TTT (Time-Temperature-Transformation) diagram. Furthermore, through their studies, they were able to determine the most appropriate heat treatment for this material. Brooks et al. concluded that a solution annealing process is necessary to achieve a homogeneous integration of aluminium, titanium, and niobium, elements which, along with nickel, are responsible for imparting hardness to the material in the ’ and ’’ phases. On the other hand, researcher Wang et al. [69] focused on identifying the most suitable heat treatment method to optimize the precipitation of the ’ and ’’ phases. He established that, I 28 Fretting fatigue behaviour of Inconel 718 at room and high temperature in order to achieve the desired mechanical properties, a double aging process at 720 °C and 620 °C, each for a duration of 8 hours, was required. The heat treatment proposed by Wang was taken into consideration in the development of the AMS 5663 standard, which is based on his study. The aim of this treatment is to achieve the desired microstructure, relieve residual stresses, limit hardness to prevent stress corrosion cracking, and avoid hydrogen embrittlement. 3.1. Heat treatment This material is usually commercialised in annealed condition, which allows a better machining process. Once the parts have been machined, frequently it is necessary to apply a heat treatment to enhance the mechanical properties. SD Metals recommends two main heat treatments for this type of alloy, with the most effective being a solution annealing between 925 °C and 1010 °C, followed by rapid cooling—usually in water—and then a precipitation hardening cycle. This consists of aging at 718 °C for 8 hours, followed by furnace cooling to 621 °C, maintaining this temperature for a total aging time of 18 hours, and finishing with air cooling. This process complies with AMS 5596 and AMS 5663 standards. The material studied was heat treated by holding at 980 °C for 1 hour, followed by rapid air cooling. Subsequently, precipitation hardening was performed at 718 °C for 8 hours, then furnace cooling to 621 °C, with a hold at this temperature to complete an aging period of 18 hours, followed by nitrogen cooling. This process is illustrated in Figure 3-1. Figure 3-1. Heat treatment performed. 0 5 10 15 20 25 0 200 400 600 800 1000 8 h. 8 h. Temperature (°C) Time (hours) Anneal Aging treatment 1 h. Material: inconel 718 29 3.2. Chemical composition The Special Metals company provides a certified materials test report for the batch of supplied material, where the chemical composition and the main mechanical properties at RT and at 650 °C for the annealing and aging treatment condition appear. This chemical composition is presented in Table 3-1. Table 3-1. Chemical compositions of Inconel 718 alloy (wt%). C Mn Fe S Si Cu Ni Cr Al Ti 0.03 0.11 18.02 0.0004 0.7 0.13 53.52 18.23 0.47 1.03 Co Mo Nb Ta B Bi P Pb Se 0.13 3.01 5.18 0.004 0.002 0.000001 0.008 0.00003 <0.000001 In addition, three specimens were analysed using a scanning electron microscope (SEM) equipped with energy-dispersive X-ray spectroscopy (EDS) in order to verify their chemical composition against the nominal composition of the base material. The analysis was conducted both at the grain boundaries and within the grain matrix. The average mass percentage results from this analysis are presented graphically in Figure 3-2. As shown, the concentrations of the main elements—nickel, iron, chromium, and niobium—within the grain matrix are consistent with the values provided in the technical data sheet. Another noteworthy observation is the presence of niobium at the grain boundaries, which is in agreement with previously discussed findings. Figure 3-2. Chemical composition of Inconel 718. OAl Si Nb Mo Ti Cr Mn Fe Co Ni Cu 0 10 20 30 40 50 60 % Weight Elements Grain boundary Matrix 30 Fretting fatigue behaviour of Inconel 718 at room and high temperature Additionally, two SEM images of the internal microstructure of the specimen are included to complement the chemical analysis. In these images, Figure 3-3, the grain boundaries are clearly visible, and as previously discussed, a significant niobium precipitation can be observed along these boundaries. Figure 3-3. SEM images of two test specimen. 3.3. Mechanical properties To validate the certified data, a tensile test was conducted at RT using the hydraulic universal testing machine MTS 810, the relationship between engineering stress and strain is illustrated in Figure 3-4. The ultimate tensile strength, yield strength and Young’s modulus were obtained from this test and compared with the values provided in the technical sheet, where it can be seen that the difference is very small. These mechanical properties after the heat treatment are presented in Table 3-2. The material hardness was measured using a Micro Vickers Hardness Tester Phase II in our laboratory. Figure 3-4. Tensile test at room temperature showing engineering stress-strain behaviour. 0 2 4 6 8 10 12 14 0 200 400 600 800 1000 1200 1400 1600 Stress (MPa) Deformation (%) Material: inconel 718 31 Table 3-2. Mechanical properties. Material properties Room temperature (Experimentally) Room temperature (Technical sheet) 650 °C (Technical sheet) Tensile strength 𝜎𝑢 1435 MPa 1476 MPa 1172 MPa Yield strength (RP0.2%) 𝜎𝑦 1310 MPa 1326 MPa 1080 MPa Young’s modulus 𝐸 217 GPa 200 GPa 163 GPa Poisson’s ratio 𝜈 - 0.294 0.283 Vickers Hardness 𝐻𝑣 454 435 - Other mechanical properties that will be discussed throughout this work will be the fatigue strength and ductility coefficients and exponents for both room temperature and 650 °C. These parameters will be obtained by means of the fatigue curves that will be discussed in Chapter 4. Also, the parameters defining the Ramberg-Osgood stress-strain equation will be obtained. Table 3-3. Fatigue strength and ductility parameters. Material properties Room temperature High Temperature Fatigue strength coefficient 𝜎𝑓′ 4332 MPa 1169 MPa Fatigue ductility coefficient 𝜀𝑓′ -0.000674 49.03 Fatigue strength exponent 𝑏 -0.1537 -0.05282 Fatigue ductility exponent 𝑐 -4.8 -10.01 Ramberg-Osgood cyclic hardening coefficient 𝐾′ 2007 [70] 1406 [71] Ramberg-Osgood cyclic hardening exponent 𝑛′ 0.098 [70] 0.10527 [71] 3.4. Grain Size Figure 3-5 shows the microstructure of Inconel 718 alloy after the heat treatment. These images were obtained using an optical microscope after being etched with a solution of 15 cc HCl, 10 cc HNO3 and 10 cc glacial acetic acid. Image a) corresponds to the microstructure 32 Fretting fatigue behaviour of Inconel 718 at room and high temperature of a sample tested at room temperature (RT) and the image b) corresponds to a sample tested at high temperature (HT) after testing. The grain size was measured with the Snyder Graff methods, for each microstructure (RT and HT) longitudinally and transversally in three different zones. The average of the measurements and their standard deviation are shown in Table 3-4. Note that for both cases, RT and HT, the measured grain sizes are nearly the same. For the RT case, the measured longitudinal grain size of 41 µm is similar to the values obtained by other researchers; for instance, Yiting Zhang et al. [72] observed that, for the similar heat treatment, the grain size was 36 µm. Figure 3-5. Microstructure of Inconel 718 alloy after heat treatment, a) test specimen at RT and b) test specimen at HT. Table 3-4. Grain size of microstructure of Inconel 718 alloy after the heat treatment. Test specimen Measurement direction Grain size (µm) Standard deviation (µm) RT Longitudinal 41 3.0 Transversal 27 3.1 HT Longitudinal 42 2.3 Transversal 27 3.9 33 4. EXPERIMENTAL TEST One of the virtues of the engineer is efficiency. Guang Tse xperimental tests are a critical source of information in the development of any fatigue study, as they provide empirical data on the issue under investigation. These tests replicate the behaviour of real components under laboratory-scale simplifications allowing details insights into the damage produced under specific conditions. In this work, two types of experimental tests were carried out: simple fatigue tests and fretting fatigue tests. Furthermore, temperature variability was introduced, meaning that both types of fatigue tests were conducted at room temperature (RT) and high temperature (HT), the latter being at 650 °C, as previously mentioned. In addition to these, interrupted tests and friction coefficient tests were also performed. This chapter is divided into five sections. The first four sections focus on the types of experimental tests performed. They give details on the specimens used, the necessary equipment, and the experimental procedures followed. The final section presents the results obtained. 4.1. Simple fatigue test The simple fatigue test specimens are flat with a continuous radius between their ends. The radius measures 60 mm, with a central width of 7 mm and a thickness of 5 mm, resulting in a cross-sectional area of 7x5 mm. High temperature tests were performed using a commercial oven. Due to the limited interior volume of this oven, it was necessary to manufacture some mechanical fixtures in order to introduce the test specimens inside the oven. The fixture is a kind of extension piece that is fixed to the test specimen by 2 bolts per fixture, called “extensor cord”. For this reason, there is a difference, in terms of geometry, between the RT and HT specimen. This difference lies solely in the grips, where two holes are introduced to accommodate the bolts that secure the specimen to the extensor cord. This modification does not affect the central area, which retains a radius of 60 mm and a cross-sectional area of 7x5 mm at its midpoint. The geometry of both test specimens is shown in the Figure 4-1: E 34 Fretting fatigue behaviour of Inconel 718 at room and high temperature Figure 4-1. Dimensions of plain fatigue test specimen at RT (a) and HT (b) in mm. Figure 4-2. HT testing equipment. a) b) Experimental test 35 In plain fatigue tests a cyclic axial load, σ, was applied to the specimen by means of a servo hydraulic machine MTS with a capacity of 100 kN. Plain fatigue tests at RT were performed with a R = -1 and R = 0.1 stress ratios and in both cases the load frequency was 10 Hz. On the other hand, plain fatigue test at HT were performed solely with a stress ratio of R = 0.1, due to the usage of the extension pieces, which increase the slenderness of the assembly, but the frequency was also 10 Hz. During the HT test, the temperature was measured using a thermocouple inserted into the oven, exactly in its middle length and close to the centre of the test specimen. The temperature was always kept constant within a ± 1.5 °C variation during the test at high temperature. Besides, the test specimens were kept at the test temperature (650 °C) for 1 hour before starting the tests, thus guaranteeing the temperature of the specimen. Additionally, to preserve the oven’s thermal integrity and ensure uniform heat retention, insulation was added to all its openings, including the side, top, and bottom joints. Additionally, for the high temperature tests, refrigerated clamps were required to prevent damage to the grips and the machine's load cell. To operate them, a chiller unit was installed to provide the necessary refrigeration water flow to these clamps. The chosen chiller unit was a commercial TAEevo TECH Mini, which maintained the water temperature between 20 and 24 °C. A schematic of all the elements necessary for the proper execution of these HT tests is shown in Figure 4-2. 4.2. Fretting fatigue test Figure 4-3. Dimensions of fretting fatigue test specimen at RT (a) and HT (b) in mm. a) b) 42 Fretting fatigue behaviour of Inconel 718 at room and high temperature 4.4. Friction coefficient test A fundamental parameter in fretting fatigue problems is the friction coefficient between the contact surfaces, as variations in this parameter affect the contact stresses and, consequently, the fretting fatigue life. The method used to determine the friction coefficient involves conducting a fretting fatigue test under partial slip conditions (the same conditions as in a normal fretting fatigue experimental test). In this test, again an ad hoc test specimen cut in half is used, an essential step to ensure that all the axial stress is transmitted through the fretting bridge, thereby matching the tangential load generated in the fretting bridge, Q, to half of the applied axial load, σ (i.e., Q = P/2). The schematic representation of the load distribution under this configuration is the same as that used to calibrate the fretting bridges and is shown in Figure 4-6. Here, it is possible to see how the total normal load of 10 kN is divided into the two pads of the fretting bridge, Npad = N/2. This test consists of progressively increasing the cyclic load amplitude, resulting in a continuous increase in the amplitude of the tangential load throughout the test. It is known that the friction coefficient increases in the slip zone during, approximately, the first thousand cycles and then stabilizes. It is this final value the one needed in the simulations. Therefore, the increasing loading amplitude must be applied with care not to produce global sliding, damaging and displacing the contact zone, before the stabilised friction coefficient has been reached in the entire contact zone. Initially, the test operates in the partial slip regime, where only a portion of the contact area experiences slip. As Q increase, the slip contact zone also grows. As long as the displacement amplitude increases progressively, the system remains in the partial slip regime, meaning that Q < μNpad. During this progressive cyclic increase, there will be a point at which the slip zone covers all the contact zone and the slip suddenly increases, indicating the transition from partial slip to gross slip. The moment just before this transition marks the maximum tangential load in the test, following the relation μ = Qmax/Npad. As this process unfolds, the normal load remains constant, just as in the experimental fretting fatigue test. Under the described procedure, four tests were conducted, one at RT, three at HT as shown in the Table 4-1. For subsequent studies, the average of the three HT values was used, which was 0.33. Table 4-1. Results of friction coefficient test. RT HT 0.43 0.37 - 0.29 - 0.33 A schematic of this process can be seen in Figure 4-12: Experimental test 43 Figure 4-12. Friction coefficient test scheme. 4.5. Results of experimental tests In this section the results obtained by the different test types carried out will be shown: plain fatigue, fretting fatigue, interrupted fretting fatigue and friction coefficient test. 4.5.1. Plain and fretting fatigue results First of all, the plain fatigue and fretting fatigue test results at RT and 650 °C will be analysed. Table 4-2 shows the data and results of the RT plain fatigue tests. The data included in the table are: test number; stress ratio, R; test frequency, f; maximum stress applied, σmax; equivalent maximum stress for R = 0.1, σmax,eq; the SWT parameter; and fatigue life obtained. The SWT parameter, defined as 𝑆𝑊𝑇=∆𝜀 2𝜎𝑚𝑎𝑥, was obtained by assuming elastic behavior of material, where ∆𝜀=∆𝜎/𝐸, since the tests did not exceed the elastic limit. The equivalent maximum stress, σmax,eq represents the value of the maximum stress for R = 0.1 obtained according to Goodman, that is expected to produce the same fatigue life as that applied with R = -1. The tests were carried out in two different batches, the first corresponds to tests performed at a stress ratio, R = -1. For this purpose, 6 different axial load levels were used. These load levels produced on the test specimens a maximum bulk axial stress, σmax, equal to 1000 MPa, 850 MPa, 700 MPa, 600 MPa, 550 MPa and 500 MPa. Tests carried out with the first 3 highest stress levels were repeated 2 times, tests with maximum stresses of 600 and 550 MPa were repeated 3 times, and finally the tests with a maximum stress value of 500 MPa were repeated 4 times, where in one of these tests, 5 million cycles were reached without any damage, considering this value as runout. The total number of plain fatigue tests performed at RT with R = -1 was 16. The second batch of this test type corresponds to those executed with an R = 0.1. Only 3 tests were carried out at different σmax levels, 1094 MPa, 974 MPa, and 831 MPa. The purpose of these tests was to verify if the equivalent alternating stress obtained with Goodman’s diagram, which transforms the applied stresses with R = -1 to an equivalent with R = 0.1, works well, and thus be able to compare plain fatigue tests with simple fatigue and fretting fatigue test performed at R = 0.1. Q(t) gross slip Tangential load Displacement Time Qmax (t) partial slip 44 Fretting fatigue behaviour of Inconel 718 at room and high temperature Table 4-2. Results of plain fatigue test at RT. Nº Test R f (Hz) σmax (MPa) σmax eq (MPa) (R = 0.1) SWT Nf (Cycles) PF-RT-1 -1 10 1000 1200 4.608 7001 PF-RT-2 -1 10 1000 1200 4.608 5815 PF-RT-3 -1 10 850 1096 3.329 42720 PF-RT-4 -1 10 850 1096 3.329 40663 PF-RT-5 -1 10 700 975 2.258 128562 PF-RT-6 -1 10 700 975 2.258 110410 PF-RT-7 -1 10 600 882 1.659 322755 PF-RT-8 -1 10 600 882 1.659 142585 PF-RT-9 -1 10 600 882 1.659 156340 PF-RT-10 -1 10 550 832 1.394 370509 PF-RT-11 -1 10 550 832 1.394 418916 PF-RT-12 -1 10 550 832 1.394 278023 PF-RT-13 -1 10 500 779 1.152 5.00E6 PF-RT-14 -1 10 500 779 1.152 228945 PF-RT-15 -1 10 500 779 1.152 275483 PF-RT-16 -1 10 500 779 1.152 275460 PF-RT-17 0.1 10 1094 - 2.481 44213 PF-RT-18 0.1 10 974 - 1.969 54616 PF-RT-19 0.1 10 831 - 1.432 184767 The plain fatigue results at RT were compared with the result obtained by Kawagoishi et al. [77] who tested Inconel 718 under the same heat treatment with R = -1, this comparison can be seen in Figure 4-13. The blue symbols represent the tests developed in this work, where the solid blue circles show the test carried out at R = 0.1 transformed into their equivalent at R = -1 by means of Goodman’s relation, the hollow blue circles show the test executed at R = -1, and the pink circles indicate the Kawagoishi's results. In addition to obtaining very similar life results, it can be seen that in both results the fatigue limit is around 500 MPa of maximum axial load. The difference in life between the two results may be due to the fatigue test specimen geometry used to perform them, Kagawoishi uses circular cross- Experimental test 45 section specimens while in this work rectangular cross-section specimens are used. The rectangular cross-section specimens have sharp edges that imply the appearance of weaknesses due to the higher ease of slip of the not confined grains at the sharp edges, fact that causes a decrease in fatigue life compared to the circular cross-section specimens according to standard ASTM E 466-07 [32]. Besides, in the same way as in the previous graphic, it is possible to see how Goodman’s correlation agrees well. Figure 4-13. Plain fatigue lives of this work versus Kawagoishi’s results at RT. 103104105106107 500 600 700 800 900 1000 Fatigue life Kawagoishi et el. Maximun Axial Stress - RT (MPa) Number of cycles to failure R=-1 Figure 4-14. Plain fatigue lives at RT respect to parameter SWT . 103104105106107 1 2 3 4 5 6 Plain fatigue (R=-1) Plain fatigue (R=0.1) SWT Number of cycles to failure 46 Fretting fatigue behaviour of Inconel 718 at room and high temperature In order to evaluate the effect due to different mean stress in the applied bulk load with different stress ratio, the fatigue parameter Smith-Watson-Toper (SWT) [78] was used, as it provides a way to account for mean stress effects. The results using this parameter are displayed in Figure 4-14, where the solid blue point represent the life for plain fatigue with R = -1 and hollow blue point for R = 0.1, and it can be seen that the lives obtained follow the generated trend line perfectly. The number of HT simple fatigue tests performed is smaller than that at RT, but it is sufficient to observe the influence of temperature on the fatigue behaviour of the material. The results of these tests can be seen in Table 4-3. They were conducted with R = 0.1 and a frequency of 10 Hz. The data included in the table are the following: test number, R, f, σmax and the obtained life. A total of 7 tests were performed with 5 different axial load levels: 1094, 1050, 1000, 900, and 831 MPa. Each axial load level was tested once, except for a σmax of 1000 MPa, which was repeated 3 times. Table 4-3. Results of plain fatigue test at HT. Nº Test R f (Hz) σmax (MPa) Nf (Cycles) PF-HT-1 0.1 10 1094 6800 PF-HT-2 0.1 10 1050 6242 PF-HT-3 0.1 10 1000 54469 PF-HT-4 0.1 10 1000 11278 PF-HT-5 0.1 10 1000 9857 PF-HT-6 0.1 10 900 155763 PF-HT-7 0.1 10 831 271480 Table 4-4 shows the data and results of the RT fretting fatigue tests. All these tests were carried out with R = 0.1 and frequency of 8 Hz. In this work, two different normal loads, N, were applied to the bridge, 10 and 7 kN, that leads to a normal load per unit length of 1000 N/mm and 700 N/mm for each contact pad. For both normal loads, it can be measured the actual normal load applied, and the maximum and minimum tangential load by means of a HBM (QuantumX) acquisition system. All of these data and the life obtained are shown in the table. First of all, to investigate the behaviour at different regimes for N = 10 kN, 10 different axial load levels were carried out. One level with short life and one with long life were repeated, corresponding to a σmax of 650 MPa and 320 MPa, respectively. So, a total of 12 tests were carried out. The runout was found at a load level of 240 MPa. Then, to investigate regimes for N = 7 kN, 4 axial load levels were performed: 900, 700, 400 and 275 MPa, repeating the axial load of 700 MPa. In the case of the fretting fatigue test at HT, the procedure was similar to the fretting fatigue tests done at RT. The tests were performed with R = 0.1 and a frequency of 8 Hz. Additionally, the same two normal loads were applied: 10 kN and 7 kN. The main difference was that, in this case, the normal and tangential loads could not be measured Experimental test 47 during the test, therefore the normal load was applied and measured at RT before introducing the setup into the oven. Table 4-4. Results of fretting fatigue test at RT (R = 0.1, f = 8Hz). Nº Test σmax (MPa) N (kN) theoretical N (kN) measured Qmin (kN) measured Qmax (kN) measured Nf (Cycles) FF-RT(1)-1 1100 10 10.06 -1.99 2.07 17136 FF-RT(1)-2 900 10 9.78 -1.79 1.70 33225 FF-RT(1)-3 650 10 9.76 -1.21 1.24 117619 FF-RT(1)-4 650 10 9.45 -0.87 1.56 108770 FF-RT(1)-5 575 10 10.03 -0.82 1.33 196902 FF-RT(1)-6 450 10 10.03 -0.83 0.89 431624 FF-RT(1)-7 400 10 9.91 -0.56 0.99 648348 FF-RT(1)-8 350 10 10.03 -0.13 1.22 900000 FF-RT(1)-9 320 10 9.92 -0.13 1.02 1.05E+06 FF-RT(1)-10 320 10 9.80 -0.24 0.95 1.21E+06 FF-RT(1)-11 275 10 10.05 0.08 1.20 2.83E+06 FF-RT(1)-12 240 10 10.18 -0.10 0.84 5.00E+06 FF-RT(2)-1 900 7 6.93 -1.60 1.65 28048 FF-RT(2)-2 700 7 6.88 -1.32 1.17 69158 FF-RT(2)-3 700 7 6.98 -1.36 1.17 74149 FF-RT(2)-4 400 7 7.09 -0.62 0.84 702049 FF-RT(2)-5 275 7 7.00 -0.09 1.00 1.88E+06 Table 4-5 presents σmax, N, the heating time inside the oven before start-up (HTO), and the lifetime. For N = 10 kN, 9 different axial loads were tested. Some of them were repeated to assess whether a longer preheating time before testing influenced the specimen's lifespan. Specimens under high loads, which had shorter test duration, were expected to be the most sensitive to HTO variations, as the preheating time represented a larger proportion of their total thermal exposure compared to low-load, long-duration test. 48 Fretting fatigue behaviour of Inconel 718 at room and high temperature Table 4-5.Result of fretting fatigue test at HT. Nº Test σmax (MPa) N (kN) theoretical HTO (Hours) Nf (Cycles) FF-HT(1)-1 1100 10 1 2960 FF-HT(1)-2 1100 10 2.5 2796 FF-HT(1)-3 900 10 1 8502 FF-HT(1)-4 900 10 2.5 7525 FF-HT(1)-5 750 10 1 17321 FF-HT(1)-6 750 10 2.5 14766 FF-HT(1)-7 750 10 16.5 14812 FF-HT(1)-8 750 10 2 12678 FF-HT(1)-9 750 10 2 13138 FF-HT(1)-10 650 10 1 26173 FF-HT(1)-11 575 10 1 32564 FF-HT(1)-12 500 10 1 53556 FF-HT(1)-13 450 10 1 110680 FF-HT(1)-14 400 10 1 166148 FF-HT(1)-15 350 10 1 4.30E+06 FF-HT(1)-16 350 10 1 5.00E+06 FF-HT(2)-1 700 7 1 18276 FF-HT(2)-2 575 7 1 29767 FF-HT(2)-3 450 7 1 106095 FF-HT(2)-4 400 7 1 83766 FF-HT(2)-5 400 7 1 5.00E+06 FF-HT(2)-6 400 7 1 811930 For this reason, the tests at 1100, 900, and 750 MPa were repeated, increasing the initial preheating time from 1 hour to 2.5 hours. This first modification revealed that, although the lifespan decreased, the reduction was not substantial except in the case of the 750 MPa axial load. Due to this observation, three additional tests were conducted at 750 MPa with Experimental test 49 varying preheating times of 16.5 hours and 2 hours, to determine whether the observed differences were due to the preheating duration or the natural variability of experimental tests. The results were conclusive in determining that a preheating up to 16.5 hours did not negatively affect fatigue life. For instance, the 750 MPa tests with 2 hours of preheating showed a shorter lifespan than those conducted with 2.5 and 16.5 hours of preheating. A total of 16 tests were conducted for N = 10 kN, identifying a runout at 350 MPa. At this axial load level, two tests were performed: the first had to be stopped at 4.3 million cycles, which was below the predefined 5 million-cycle runout threshold, due to machine maintenance. Consequently, the test was repeated, confirming the achievement of 5 million cycles and, therefore, the fatigue limit. For the tests conducted with a normal load of 7 kN, a total of 6 tests were performed with 4 different axial load levels, with the lowest load level of 400 MPa being repeated. Among the three tests conducted at this load level, one achieved infinite life, establishing this value as the fatigue limit. Figure 4-15 depicts the S-N curve with the results shown in the tables 1 to 4. In this figure, PF (plain fatigue) tests are represented by circles, while FF (fretting fatigue) tests are represented by triangles. The color represents the test temperature: blue shades correspond to RT and red shades to HT. The symbology for PF tests at RT has already been discussed earlier in this section; however, for further clarification, it is reiterated here. Solid blue circles represent tests actually conducted with R = 0.1, whereas hollow blue circles represent tests carried out at R = -1 and converted to their equivalent at R = 0.1 using Goodman's relation. Figure 4-15. Plain fatigue and fretting fatigue test at RT and HT (650 °C). 103104105106107 200 400 600 800 1000 1200 1400 PF - RT PF - HT FF - RT(1) FF - HT(1) FF - RT(2) FF - HT(2) Maximun Axial Stress (MPa) Number of cycles to failure R = 0.1 (1) N=10 kN (2) N=7 kN 50 Fretting fatigue behaviour of Inconel 718 at room and high temperature In fretting fatigue tests, the contact between the fretting bridge pads and the test specimen acts as a stress concentrator [79]. This causes a decrease in fatigue life compared to plain fatigue, a phenomenon that is appreciable both at room temperature (RT) and high temperature (HT). This type of behavior is typical in the vast majority of materials [53] [52]. In our case it is noticeable that, at both temperatures the fretting fatigue life (NfFF) is shorter than plain fatigue life (NfPF). Moreover, for any fatigue life, the ratio NfFF/NfPF decreases rapidly as the applied stress decreases. This implies that at the high cycle fatigue regime, fretting fatigue damage dominates over plain fatigue damage. Additionally, it can be observed that for high axial loads, around 1100 MPa, the fatigue life under both, plain fatigue and fretting fatigue conditions, is very similar. This suggests that at such load levels, the stress concentration from the contact between elements does not significantly affect fatigue. Furthermore, it is important to analyse the effect of temperature on each type of test. It is noteworthy that the plain fatigue lives at both temperatures are quite similar at certain stress levels. The effect of temperature appears to be significant only for fatigue lives shorter than approximately 10⁵ cycles. A similar trend is reported by Kawagoishi et al. [77], where a comparison of plain fatigue tests at different temperatures is shown. For plain fatigue at room temperature the fatigue limit is around 240 MPa. In contrast, for fretting fatigue tests, the effect of temperature is noticeable across all fatigue regimes. Both curves exhibit approximately the same slope but with different offsets, with fatigue life at HT being four times shorter than at RT, except for 1100 MPa, where it is six time shorter. This exception can be explained by the fact that at high axial load levels, stress concentration from the contact does not play a significant role. For fretting fatigue, the fatigue limit is approximately 240 MPa at RT and 350 MPa at HT, as discussed above. The reason why fretting fatigue life is shorter at high temperature compared to room temperature, while in simple fatigue the life remains similar in both conditions, is likely related to crack growth behaviour. In simple fatigue, a significant portion of the total fatigue life is often spent in the crack initiation stage compared with fretting fatigue, where the propagation phase generally plays a more dominant role due to the typically faster initiation process. Therefore, this hypothesis suggests that if fatigue lives are similar in simple fatigue—where initiation is the dominant phase—then crack initiation is not significantly affected by temperature. On the other hand, in fretting fatigue, where initiation occurs much more rapidly and crack propagation plays a more important role in the total life, a clear difference is observed between room and high temperatures. This indicates that temperature has a significant effect on crack propagation. This hypothesis will be further discussed in Chapters 7 and 8, where crack growth rates and life prediction are analysed. Another aspect to analyse is the influence of normal load on fretting fatigue tests, which can be observed in Figure 4-16. In this figure, tests conducted with a normal load of N = 10 kN are represented by upright triangles (marked as (1)), while those performed with N = 7 kN are represented by inverted triangles (marked as (2)). Additionally, to facilitate quick identification, tests with this lower normal load have been circled. It is noticeable that, for both RT and HT conditions, the results for both normal loads are Experimental test 51 quite similar, with no clear differences between them. However, a slight difference was observed in the fatigue limit at HT, where the lower normal load resulted in a slightly higher fatigue limit. The fatigue limit was not analysed at RT. Some researchers, including Mall et al. [51] and Kwon et al. [52], suggest that the variation of mechanical properties (such as tensile strength, yield stress, and Young’s modulus) with temperature affects the experimentally obtained results. As mentioned in the introduction, in the case of Mall et al. [51], since the mechanical properties initially increased with temperature, the obtained lives at high temperature also increased, something similar was mentioned by Kwon et al. [53], [52]. In this case, since the properties were similar, the obtained lives were similar too. Nevertheless, in both cases temperatures were lower than in our tests, 600 °C in the case of Mall et al. and 350 °C in the work of Know et al. In our case, temperature has a significant influence on mechanical properties, and this phenomenon can also be observed in the fretting fatigue life. A decrease in fretting life with the increase in temperature test has also been observed by other researchers such as [50] and [54]. Figure 4-16. Fretting fatigue test at RT and HT, with N of 10 kN and 7 kN. Several studies have examined the formation of a glaze oxide layer in nickel-based superalloys when exposed to high temperatures, which leads to a decrease in the friction coefficient and, consequently, influences fretting fatigue behaviour. It is important to note that the oxide layer also forms during heat treatment. Some researchers remove this layer before conducting experimental tests by polishing the surface, allowing the oxide to regenerate during high temperature testing. In this project, however, the oxide layer produced during the heat treatment was preserved. Based on this approach, the authors hypothesize that this oxide layer results in similar surface conditions in both RT and HT tests, and that the reduction in fatigue life is primarily due to variations in mechanical 103104105106107 200 400 600 800 1000 1200 1400 (2) (1) (2) FF - RT(1) FF - HT(1) FF - RT(2) FF - HT(2) Maximun Axial Stress (MPa) Number of cycles to failure R = 0.1 (1) N=10 kN (2) N=7 kN (1) 59 5. FRACTURE SURFACE MORPHOLOGIES Fracture is not just the end of a material, but the beginning of a story written on its surface, a narrative that tells us about the forces that brought it to its limit. George R. Irwin he fracture surface resulting of specimens tested under fretting fatigue will be analysed in this section. Figure 5-1a and Figure 5-1b show two different fracture surface morphologies of fretting fatigue test specimens at room temperature (RT) and at 650 °C (HT), respectively. In both cases, they were subjected to a maximum axial stress of 650 MPa. In these images, four distinct regions can be identified: region I, where fretting cracks initiate from the contact surface; region II, corresponding to stable crack growth in the y-axis direction; region III, where unstable crack growth occurs; and finally region IV, which shows a secondary crack initiation at the opposite contact area, though this crack does not contribute to the final fracture of the specimens [80]. These secondary cracks are not always present and tend to appear more frequently at room temperature than at high temperature. In view of Figure 5-1a and Figure 5-1b, it is evident that the surface colour changes with temperature. At high temperatures, the surface exhibits various shades of blue depending on the specific region, with the most intense blue appearing in region III, the unstable fracture zone. This coloration indicates the presence of surface oxidation. On the other hand, Figure 5-1c and Figure 5-1d depict the crack pattern of the test specimens at room temperature (RT) and high temperature (HT), respectively, viewed from the z-direction of the specimen. The viewpoint for these profiles is indicated in the same Figure 5-1a and Figure 5-1b. The four regions are also visible in Figure 5-1c and Figure 5-1d. Additionally, crack growth in region I, II and IV appears to be approximately perpendicular to the contact surface, while region III corresponds to the unstable fracture of the primary crack originating from regions I and II. In addition, region I was examined using a Scanning Electron Microscope (SEM) for both cases mentioned above, specifically at RT and HT at both tests with a stress of 650 MPa. Some images from the SEM analysis are presented in Figure 5-2 and Figure 5-3, showing areas of crack initiation close to the fretted surface at higher magnification. Upon reviewing these figures, no noticeable differences are found between the fracture surfaces at RT and HT. T 60 Fretting fatigue behaviour of Inconel 718 at room and high temperature Figure 5-1. Fretting fatigue fracture surface of specimens tested at σmax = 650 MPa at a) room temperature FF-RT(1)-4 and b) 650 °C FF-HT(1)10. c) and d) profile of the specimen tested at RT and HT [80]. I) Crack initiation II) Crack growth III) Unstable crack growth IV) Second crack initiation 7 mm 5 mm x yz x yzz yxzy x a) b) c) d) Pro?le A Pro?le B Pro? le A Pro? le B Profile A Profile A Profile B Profile B FF-RT(1)-4 FF-HT(1)-10 Figure 5-2. Fracture surface near the surface in fretting fatigue samples tested at RT and HT with 650 MPa. Fracture surface morphologies 61 The region II, the stable crack growth region, was also observed in SEM, where it clearly shows the typical striations caused by fatigue damage. Figure 5-4 shows a series of photographs in which it is possible to appreciate the advance and evolution of the striations with the crack length. In the area closer to the surface, a fine striation, small distance between peaks and valley, appears. When advancing along the fracture surface in the direction of crack growth, the distance between peaks and valley of the striations increases in size up to a certain point where dimples begin to appear, leading to the appearance of ductile fracture and unstable crack growth, due to crack stress intensity factor close to the fracture toughness. Figure 5-4 further demonstrates that for the same number of cycles (marked by the black line within the micrographs), the striation spacing widens progressively as the crack propagates deeper into the specimen. Figure 5-3. Details of fracture surface near the surface in fretting fatigue samples tested at RT and HT with 650 MPa. 62 Fretting fatigue behaviour of Inconel 718 at room and high temperature Figure 5-4. Fracture surface for a sample tested at RT and 650 MPa at a distance of a) 0.841 mm, b) 1.707 mm, c) 1.902 mm and d) 2.925 mm. 5.1. Composition analysis: oxide layer It was previously mentioned that the fracture surface colour in high temperature tests differs from that in room temperature tests, displaying a range of blue tones as shown in Figure 5-1b, however, these tones are not always blue. Observations indicate that the surface colour can vary depending on how much time the specimen has remained in the oven after fracture. A blue colour appears when the oven was turned off immediately after the test concluded, while a golden colour is observed when the specimen remained in the high temperature oven for more than six hours after the test has finished, i.e., post-fracture. For that reason, the fracture surface was analysed using Energy Dispersive X-ray Spectroscopy (EDS) in the SEM to determine the cause of the colour variations. This analysis was conducted along a straight line at different depths, starting from the contact area and up to the final fracture point. The primary aim of this analysis is not to assess the chemical composition of the surface but rather to examine the oxidation that has occurred and how it varies in each case. Therefore, only the weight percentage of oxygen on the crack surface is indicated. Based on various literature sources, it could initially be suggested that a greater exposure to high temperatures would result in increased oxide formation. A total of five fracture surfaces were analysed: two surfaces from specimens tested at room temperature (RT) and three at high temperature (HT), from the tests labelled as FF-RT(1)-1, Fracture surface morphologies 63 FF-RT(1)-7, FF-HT(1)-1, FF-HT(1)-14, and FF-HT(2)-3. The following table summarizes the characteristics of each test and the resulting life (these data were previously presented in Chapter 4). Observing Figure 5-5, it can clearly be seen that independently of the test temperature in all cases there are a high percentage of oxygen at the edge of the surface. This oxygen is associated with the oxide layer generated by the aging of the thermal treatment applied, which was not subsequently removed, as previously mentioned. Table 5-1. Summary of test conditions and cycles for fracture surface analysis. Nº test Temperature N (kN) σmax (MPa) Nf (Cycles) Surface tonality FF-RT(1)-1 RT 10 1100 17136 Grey FF-RT(1)-7 RT 10 400 648384 Grey FF-HT(1)-1 650 °C 10 1100 2960 Blue FF-HT(1)-14 650 °C 10 400 166148 Blue FF-HT(2)-3 650 °C 7 450 106095 Golden* * More than 7 hours inside of the oven at 650 °C after the failure Figure 5-5. Oxygen concentration (% by weight) on the fracture surface (image taken at SEM). It is noticeable that in the tests at room temperature FF-RT(1)-1 and FF-RT(1)-7, oxidation tends to approach zero as we move away from the contact area. In contrast, in the high temperature tests, there is a significant amount of oxygen throughout the entire fracture surface. In cases where the oven is turned off immediately after the specimen fractures, FFHT(1)-1 and FF-HT(1)-14, the oxygen percentage decreases slightly along the fracture, 7 6 5 4 3 2 1 0 0 2 4 6 8 10 12 14 16 18 20 Oxygen ( % Weight) FF-RT(1)-1 - 1100 MPa FF-RT(1)-7 - 400 MPa FF-HT(1)-1 - 1100 MPa FF-HT(1)-14 - 400 MPa FF-HT(2)-3 - 450 MPa Length (mm) FF-HT(1)-14 - 400 MPa 64 Fretting fatigue behaviour of Inconel 718 at room and high temperature indicating that the area exposed for a shorter duration is the least affected, which correlates with the crack growth rate. Conversely, when the specimen is exposed to an environment of 650 °C for an extended period after fracture, there is a substantial amount of oxygen that fluctuates within a nearly fixed range. When comparing the FF-HT(1)-14 specimen with the FF-HT(2)-3 specimen, it can be noted that although the former has a longer number of cycles, the latter, which was more than 7 hours in the oven after failure, exhibits a higher amount of oxygen. This relationship suggests that the exposure time of the piece to high temperature correlates with the oxide generated and, consequently, the coloration of the surface. The fracture surface of the test specimens can be seen in Figure 5-6. Figure 5-6. Photographs of the fracture surface of the analysed specimens, showing the characteristic tonalities in specimens at RT (grey) and at HT (blue and gold). As can be observed in Figure 5-5, the contact surfaces of all specimens show a significant amount of oxygen, regardless of whether they were tested at room temperature or at 650 °C, indicating the apparent presence of an oxide layer in both cases. In the literature, the formation of an oxide layer is typically reported exclusively when specimens are subjected to high temperatures. Therefore, the key question at this point is to understand the source of oxygen in the low-temperature specimens in this study. The main hypothesis, which was later supported by the analyses conducted, was that this oxygen originated from the thermal treatment applied, as it was maintained in this study, contrary to the majority of researchers who removed it after the treatment. To confirm the proposed hypothesis, an analysis was performed using EDS to quantify the percentage of Oxygen on the surface of six specimens: three tested at RT and three at HT. The analysis was initially carried out in areas far from the contact zone (FCZ) to avoid the influence of fretting fatigue. In both temperatures, three test specimens subjected to different load levels (1100, 650, and 400 MPa) were evaluated to cover different life ranges and, consequently, varying exposure times to high temperature in the HT tests. For each specimen, at least two measurements were taken, except for specimen FF-HT(1)-1, where only one measurement was possible. Table 5-2 presents the average values obtained for each specimen, as well as a global average based on the test temperature. According to the data presented in Table 5-2, it is observed that the oxygen values are practically the same at both low and high temperatures, being the percentage around 1215%, with the exception of the specimens tested at 1100 MPa, where a value of 15.01% at RT and 9.55% at HT was obtained. The latter value differs from the other two high temperature specimens measured and could not be corroborated with additional measurements to Fracture surface morphologies 65 confirm its validity. Based on these results, it can be concluded that a pre-existing oxide layer is present on all specimens and similar among them with an average at RT of 14.09% and at HT of 12.03% before the test, so this oxide layer is consequence of the heat treatment performed. Subsequently, to evaluate the influence of fretting contact on the oxidation of the specimen, the same analyses were performed in the slip zone (SLCZ) and the stick zone (STCZ), comparing the results with the unfretted zone (FCZ). The data obtained, presented in Table 5-2 across all zones, both at room temperature (RT) and high temperature (HT). However, a slight difference is observed in the SLCZ compared to the STCZ at HT, with the latter being slightly lower. Table 5-2. % Oxygen on the surface of the specimen. RT HT Nº Test σmax (MPa) %O FCZ %O SLCZ %O STCZ Nº Test σmax (MPa) %O FCZ %O SLCZ %O STCZ FF-RT(1)-1 1100 15.01 10.4 13.85 FF-HT(1)-1 1100 9.55 14.98 6.72 FF-RT(1)-4 650 12.53 13.75 12.61 FF-HT(1)- 10 650 13.33 13.19 12.13 FF-RT(1)-7 400 14.73 14.05 14.83 FF-HT(1)- 14 400 14.10 13.6 13.12 Average 14.09 12.73 13.75 Average 12.03 13.92 10.66 Another significant issue at this point is the depth of the oxide layer and whether it varies depending on the test temperature and the zone under consideration, FCZ or CZ (contact zone). To address this question, the thickness of the oxide layer in four specimens was measured: FF-RT(1)-4, FF-RT(1)-7, FF-HT(1)-10, and FF-HT(1)-14. Each specimen was measured at different points to obtain an adequate number of data points, with approximately six measurements per zone and specimen. Cross-sectional SEM images of Inconel 718 test specimens oxidised for the heat treatment and tested at RT and HT is given in Figure 5-7, which shows images of the aforementioned oxide layer in areas FCZ. By analysing these images in areas FCZ and, therefore, unaffected by the contact, it would be possible to determine whether the thickness of the oxide layer increases or decreases solely due to exposure to elevated temperatures when there is already a pre-existing layer. Table 5-3 reflects the average thickness of the measured oxide layer at RT and HT in both zone, CZ and FCZ, where all of them apparently exhibit a similar oxide layer thickness, being this around 0.66 µm. In the contact zone, a reduction in the oxide layer thickness would be expected due to friction between the surfaces, particularly at RT. However, this not observed in these measurements. This discrepancy may be attributed to the variability in oxide growth across the surface. Previous studies indicate that after 6 hours at 900 °C, the oxide layer thickness in Inconel 718 reaches 0.63 µm, increasing to 0.9 µm after 15 hours [81], which is consistent with the values obtained in this study due to the heat treatment. 66 Fretting fatigue behaviour of Inconel 718 at room and high temperature Figure 5-7. Oxide layer in FCZ. Table 5-3. Oxide layer thickness in CZ and FCZ. Test temperature Contact zone (CZ) No contact zone (FCZ) RT 702 nm 632 nm HT 626 nm 695 nm In conclusion, the presence of an oxide layer on the surface of the specimens is confirmed, both in RT and HT. Furthermore, it is observed that the thickness of this layer is similar in all cases. These results confirm the authors' hypothesis: the thermal treatment generates an oxide layer on the surface, and the reduced fatigue life observed in high temperature tests is attributed to changes in the mechanical properties of the material rather than the presence of the oxide layer formed at high temperature [51] - [53]. In order to understand the possible effect produced by contact loading on the material microstructure, two regions of different test specimens were analysed using EDS in the SEM: a zone just beneath contact zone and a zone far from the contact zone, the same as before. Firstly, several areas of each zone (CZ and FCZ) were analysed at different depths from the surface, ranging from 4 µm to 350 µm, with the objective of obtaining an average chemical composition, in this case only for a test specimen at HT with σmax of 650 MPa, specifically FF-HT(1)-10.The results obtained showed that, regardless of the area analysed, CZ or NCZ, the chemical composition was similar, with the following elements predominating: 52% Ni, 19% Cr, 18% Fe, 4.26% Nb, 1.24% Ti y 0.5% Al (% weight). Furthermore, since the zone most susceptible to microstructural changes due to contact loading occurs near the surface, a more exhaustive analysis was developed below the 1 mµ 1 mµ 1 mµ 1 mµ FF-RT(1)-4 - 650 MPa FF-RT(1)-10 - 650 MPa FF-RT(1)-7 - 400 MPa FF-HT(1)-14 - 400 MPa 0.491 µm 0.504 µm 0.619 µm 0.355 µm Fracture surface morphologies 67 surface of the fretting fatigue sample. The same test specimens used to analyse the thickness of the oxide layer were also examined in this section: FF-RT(1)-4, FF-RT(1)-7, FF-HT(1)-10, and FF-HT(1)-14. These test specimens where tested with σmax of 650 and 400 MPa at both temperatures. Figure 5-8 a) and b) show the surface of the specimen FF-HT(1)-10 for CZ and NCZ respectively. In these images 3 zones can be visually distinguished, corresponding the first of them to the oxide layer. For each layer and zone (CZ and NCZ), measurements were done at least two different points with the EDS, and the average of these points was taken. These same procedure was executed for the rest of the samples and it is shown in Figure 5-9. Subsequently, as previously discussed throughout this chapter, the analysed samples exhibit an initial layer with a high oxygen content, further confirming the formation of a well-defined oxide layer. Previous studies, such as those by Waterhouse [45], Delaunay [81], and Al-hatab [82], have demonstrated that the predominant oxide layer in Inconel 718 is Cr₂O₃, which serves as a protective barrier against oxidation. In this particular case, a significant increase in the concentration of titanium (Ti) and aluminium (Al) can be noted in the outer layer compared to the original composition of the material matrix. Specifically, the mass content of Ti increases from 1% to 10-14%, while Al increases from 0.47% to 5-7%. This redistribution indicates that the first layer is enriched Figure 5-8. Microstructure at the edge of the specimen's surface after high temperature testing. Image (a) depicts the region subjected to contact (CZ), and image (b) corresponds the area unaffected by contact (FCZ). 74 Fretting fatigue behaviour of Inconel 718 at room and high temperature the theoretical predictions. Figure 5-13. Crack scar for two samples, one tested at RT and another at HT, FF-RT(1)-8 and FF-HT(1)-11 respectively. e = 4.83 mm e? e = 4.83 mm e? A = 1.938 mm 12 A = mm3.949 22 A = mm5.315 2 a) b) c) d) FF-RT(1)-8 upper fre ing bridge pad Fractured zone FF-RT(1)-8 lower fre ing bridge pad Non-fractured zone FF-HT(1)-11 upper fre ing bridge pad Non-fractured zone FF-HT(1)-11 lower fre ing bridge pad Fractured zone e = 4.60 mm e? e = 4.60 mm e? A = mm5.543 2 A = 3.773 mm 12 A = mm1.950 22 A = A + A = 5.723 mm exp 2 1 2 A = A + A = 5.887 mm exp 2 1 2 75 6. FINITE ELEMENT MODEL There is nothing to fear in life, only things to understand. Marie Curie he finite element analysis will be conducted on a simplified geometry, taking advantage of the symmetry conditions present in the experimental setup. As a result, only a quarter of the fretting bridge and specimen assembly will be modelled, meaning the study will analyse one of the four contact elements. Additionally, the model was designed as a two-dimensional (2D) model under plane strain assumption. This approach was chosen because studies have shown that the results from three-dimensional (3D) and two-dimensional models are comparable, thus significantly reducing computational time [85]. Figure 6-1. Simplification of modelled geometry. 6.1. Modeling The finite element model (FEM) program used to simulate this type of fatigue, fretting fatigue, is ANSYS Mechanical APDL. This simulation allows to predict the stresses/strains to which the fretting fatigue test specimen will be subjected during experimental tests. This program consists of three clearly defined phases: pre-processing, resolution, and postT 76 Fretting fatigue behaviour of Inconel 718 at room and high temperature processing. The components of each phase are illustrated in Figure 6-2. Figure 6-2. Design phase in ANSYS. The first phase, pre-processing, involves designing the model to be used. This begins with defining points (keypoints), from which lines and areas of the geometry are established. Subsequently, a mesh is generated. The coding for this initial phase was programmed in the mathematical software MAPLE, generating a data file that was later imported into ANSYS. The structural element type used in the model is PLANE182, which is suitable for modelling solid structures in 2D. This type of element is defined by four nodes, each with two degrees of freedom, allowing for translations in the x and y nodal directions. Linear 4node elements are preferred over 8-node quadratic elements because, although the latter provide greater accuracy in linear analyses and curved geometries, in nonlinear analyses they cause problems in the interpretation of contact nodal forces, since reaction forces vary significantly between corner and mid-side nodes. The elements used to simulate the contact pair between the fretting bridge and the specimen are TARGE169 and CONTA171, commonly employed to represent 2D surface contact between two nodes. The contact algorithm selected is the Augmented Lagrange method, which detects contact points on the nodal point-normal from the contact surface. This method incorporates an automated adjustment that reduces penetration iteratively, based on the current mean stress of the underlying elements. Regarding the normal and tangential stiffness values, they are set to default in ANSYS, where the software automatically computes these values based on the material properties and element size, ensuring appropriate contact behaviour. The normal stiffness (FKN) and tangential stiffness (FKT) values are crucial in defining the response of the contact pair. These values are automatically adjusted during the simulation to maintain proper contact conditions and prevent unrealistic penetrations or sliding Preprocessor Resolution Postprocessor Keypoints Line Areas Mesh(Elements) Applicationof boundaryconditions Applicationof loads Obtaining results Finite element model 77 behaviour. The default settings in ANSYS are typically sufficient for standard analyses, but they can be fine-tuned for specific cases where more precise control over the contact interaction is required. 3 Figure 6-3. Mesh (elements). To acquire a good stress gradient estimation, it is necessary more precision in the contact 78 Fretting fatigue behaviour of Inconel 718 at room and high temperature zone, the most critical area. For this reason, the mesh size along this zone is smaller than the rest of the geometry with an approximate dimension of 6 µm width and 3.7 µm length. An image of the mesh ca be seen in Figure 6-3. In order to properly apply the normal load to this geometry, the so-called "master node" is used, as shown in Figure 6-4 and Figure 6-5. This node is responsible for receiving the entire applied normal load and, in turn, for distributing it appropriately among all the nodes that are actually affected. The connection between the master node and the bridge nodes (slave nodes) is shown in pink in Figure 6-4 and Figure 6-5. Figure 6-4. Master node. M A S T E R N O D E MASTER NODE SLAVE NODES MASTER NODE Finite element model 79 Subsequently, the boundary conditions imposed on the model due to symmetry are introduced. On the one hand, there is symmetry with respect to the x-axis, which requires constraining displacements along the y-axis at the bottom of the specimen. On the other hand, there is symmetry with respect to the y-axis on the right side of the assembly, and therefore, displacements along the x-axis are also constrained in that region. The master node also includes displacement constraints along the x-axis, since the normal load is applied along the y-axis. All these constraints are shown in blue in Figure 6-5. It is worth noting that an elastic-plastic model was considered for the development of these simulations. To this end, the cyclic stress-strain curves of Inconel 718 at both RT and 650 °C were introduced. These curves were obtained from the literature: those corresponding to RT were provided by Changhao et al. [70], while those for high temperature (HT) were reported by Jingyu et al. [71], and it can be seen in Table 3-3. Figure 6-5. Boundary conditions. In the solution phase, the system loads are applied, specifically, the normal load N and the axial load σ, which is applied at the left end of the specimen. The first consideration when applying these loads is the orientation of the coordinate axes: the x-axis is defined as positive to the right, and the y-axis as positive upwards. The normal load is applied in the opposite direction of the y-axis and remains constant, whereas the axial load varies cyclically to simulate an experimental test. A stress ratio of 0.1 is used, i.e., 𝑅=𝜎𝑚𝑖𝑛/𝜎𝑚𝑎𝑥=0.1. This simulation is carried out through a series of steps that simulate the progressive application of loads, similar to an experimental procedure, as illustrated in Figure 6-6. The most relevant steps are as follows:  Step 10: the normal load N is applied progressively until the target value is reached. 80 Fretting fatigue behaviour of Inconel 718 at room and high temperature  Step 20: the axial load is gradually applied from zero to 𝜎𝑚𝑖𝑛.  Step 30: the axial load is progressively increased from 𝜎𝑚𝑖𝑛 to 𝜎𝑚𝑎𝑥.  Step 40: the axial load is progressively decreased from 𝜎𝑚𝑎𝑥 to 𝜎𝑚𝑖𝑛. While the axial loading cycle could be repeated further, the focus here is limited to the first 40 loading steps. Figure 6-6. Step in FEM. The third and last phase of the ANSYS program is to visualize and obtain the desired results of this simulation. Some of the results we can obtain are: von Mises stress, stresses in the different axes, displacements, etc. Figure 6-7 shows that the maximum axial stresses occur in the contact zone between the bridge and the specimen, with higher values on the left edge. It is also noticeable that the axial stress at the specimen ends tends to match the applied axial stress of 350 MPa. Figure 6-7. Axial stress. 6.2. Comparative between analytical and FEM model This section will compare the generated finite element model with the surface stresses obtained from the analytical model (discussed previously in Chapter 2) with the results from ANSYS. 010 20 30 40 50 60 max Loads Step min -N -N  Finite element model 81 For this comparative, as in the previous section, a normal force of 1000 N/mm and a maximum axial stress of 350 MPa will be applied, considering that the axial load is introduced at the left end of the specimen in the direction opposite to the x-axis. The comparative results for normal stresses, axial stresses, and shear stresses on the surface are presented in Figure 6-8 to Figure 6-10, respectively. A slight discrepancy between the analytical and finite element models can be observed in the figures. This difference arises because the analytical model is simplified and does not account for the bridge geometry or its potential minor rotations—it only considers the indenter's (bridge) curvature radius. The analysis of the results leads to the conclusion that, in the context of this type of testing, the analytical model does not constitute a valid tool for accurately evaluating the stresses in the specimen, due to the simplifications it entails. Figure 6-8. Surface normal stress: analytical vs. FEM. Figure 6-9. Surface axial stress: analytical vs. FEM. -1,0 -0,5 0,0 0,5 1,0 -1400 -1200 -1000 -800 -600 -400 -200 0 Analytical model FEM yy (MPa) x (mm) -1,0 -0,5 0,0 0,5 1,0 -1000 -500 0 500 1000 1500 Analytical model FEM xx (MPa) x (mm) 82 Fretting fatigue behaviour of Inconel 718 at room and high temperature Figure 6-10. Surface shear stress: analytical vs. FEM. 6.3. FEM Analysis The finite element model is simulated for various levels of axial load, while maintaining the normal load constant at 10 kN. This results in a linear load of 1000 N/mm per contact pad, calculated by distributing the total load equally between the two pads of the fretting bridge (5 kN per pad) and dividing by their thickness, 5 mm. The objective of these simulations is to analyze the stresses and strains both on the surface and within the interior of the specimen, and to observe how they vary with the applied axial stress. These stress and strain results will subsequently be used to estimate the fatigue life of the test specimen. 6.3.1. Surface stresses from FEM The first set of results to be analysed in this section corresponds to the axial stresses at the surface, calculated in step 40, as shown in Figure 6-11. It can be observed that the region of highest axial stress, independently of the axial load applied, occurs at the left edge of the contact zone, consistently located at the same x-axis position across all loading levels. Furthermore, a decrease in stress is evident as the applied axial load decreases. To better understand the upcoming shear stress results, a brief recap of key concepts from previous sections is helpful. Under partial slip conditions, which apply in this case, the contact area is divided into a slip zone and a stick zone, as described in Figure 6-12, and bounded by the parameter c, defined in Equation (2-4). Additionally, the stick zone may be displaced by a certain length e, known as eccentricity, which results from the axial load applied to the specimen. As shown in Equation (2-5), the value of e is directly influenced by -1,0 -0,5 0,0 0,5 1,0 0 100 200 300 400 500 Analytical model FEM xy (MPa) x (mm) Finite element model 83 the axial load. The influence of partial shear stress and eccentricity at the surface is illustrated in Figure 2-10, Figure 2-11, and Figure 2-12. A schematic representation of the above considerations is shown in Figure 6-12. Figure 6-11. FEM superficial axial stress. Figure 6-12. Scheme of eccentricity and stick zone. After revisiting some key concepts regarding surface stresses in fretting fatigue, the analysis -1,0 -0,5 0,0 0,5 1,0 -1000 -500 0 500 1000 1500 2000 Surface axial stress (MPa) x (mm) 1100 MPa 900 MPa 650 MPa 575 MPa 450 MPa 400 MPa 350 MPa S t i c k z o n e -a a -c+e c+e c c Stick zone 90 Fretting fatigue behaviour of Inconel 718 at room and high temperature 7.1. Plain fatigue crack growth rate The FCGR was analysed in four plain fatigue test specimens, two for RT and two for HT:  PF-RT-4: σmax = 850 MPa at R = -1, equivalent to 1094 MPa at R = 0.1.  PF-RT-12: σmax = 550 MPa at R = -1, equivalent to 832 MPa at R = 0.1.  PF-HT-1: σmax = 1094 MPa at R = 0.1.  PF-HT-7: σmax = 832 MPa at R = 0.1. To begin the first stage, it is necessary to understand the experimentally observed crack growth pattern. This initial visualization process was performed in Chapter 5, where it was clearly identified that in the RT condition, 83% of the fracture surfaces, exhibited crack initiation at the corner and grew following a quarter-elliptical shape. In contrast, in the HT condition, the majority of the tests (57%) showed crack initiation at the surface, and thus growing according to a semi-elliptical shape. The tests specimens listed above correspond to crack initiation at the corner for RT and at the surface for HT, the cracks shapes are shown in Figure 5-11. Once the crack growth pattern has been established, the next step involves analysing the surfaces using SEM. For this purpose, different photographs were taken along the fracture surface (as shown in Figure 5-4), starting in a range of approximately 0.10 - 0.50 mm from the surface crack initiation point and extending to the unstable growth zone. In each SEM photograph, a certain number of striations were counted. Since each of these striations corresponds to one fatigue cycle, and with the scale provided by the images, it is possible to determine the total length of the counted striations. The ratio between this distance and the number of counted cycles (striations) provides the fatigue crack growth rate, da/dN, at that crack length. The crack length is known because in each observed zone it is possible to obtain, via the SEM device, the precise location of the analysed point X and Y position. Figure 7-2 and Figure 7-3 illustrate the acquisition process of the crack growth rate. Initially, a point cloud can be observed, representing the location of all the point analysed on the specimen’s crack surface for two specific cases, PF-RT-4 and PF-HT-1, respectively. Additionally, an image captured for one of these points is shown, showing how the surface striations allow the calculation of the fatigue crack growth rate. In the case of PF-RT-4, five striations can be counted using the arrow indicated in the SEM image, corresponding to five cycles. Additionally, the length of the arrow shows that the distance of the five striations is 3.056 µm, resulting in a crack growth rate of 0.611 µm/cycle. On the other hand, for the specimen PF-HT-1, three striations were counted, corresponding to three cycles, with a total length of 4.867 µm, yielding a crack growth rate of 1.622 µm/cycle. To determine the total crack length at each measured point, it is necessary to consider both the crack initiation point and the X and Y coordinates of the analysed location. The distance taken corresponds to the total length between these two points, which is obtained using the SEM device. Fatigue crack growth 91 Figure 7-2. Measured point of specimen PF-RT-4, and analysis for obtaining FCGR. Figure 7-3. Measured point of specimen PF-HT-1, and analysis for obtaining FCGR. A total of 705 images were taken: 176 images for specimen PF-RT-4, 283 images for PF-RT12, 116 images for PF-HT-1, and 130 images for PF-HT-7. With all these data, a dense cloud of points indicating the crack growth rate and its variation with crack length was obtained, as shown in Figure 7-4.This figure represents the crack growth rate in m/cycle as a function of crack length, a, in metres (considering its origin at the crack initiation point). The figure reflects the typical trend observed in fatigue, where the FCGR increases as the crack length increases. Furthermore, for the same crack length value, it is possible to observe how the FCGR increases with higher applied axial loads. Moreover, the same figure clearly shows two well-differentiated regions, marked by a distinct change in slope for each of the measured specimens. In all cases, this transition occurs at approximately one millimetre of crack length, except for the specimen tested at 1094 MPa under HT conditions, where the change occurs at around 0.30 millimetres. This behaviour may be attributed to the presence 3 2 1 00 1 2 3 Y (mm) Measured points X (mm) 5 cycles - 3.056 m 0.611 m/cycle 5 m PF-RT-4 3 2 1 00 1 2 3 PF-HT-1 Measured points Y (mm) X (mm) 3 cycles - 4.867 m 1.622 m/cycle 5 m 92 Fretting fatigue behaviour of Inconel 718 at room and high temperature of small cracks, leading to an almost constant crack growth rate over a period in which the crack is not yet considered a long crack. Figure 7-4.Crack growth rate for plain fatigue test at RT and HT. The third step of this study involves analysing the SIF corresponding to each of the obtained points in order to represent the fatigue crack growth rate in a more practical manner, i.e. da/dN vs. SIF range. For this purpose, the crack pattern established at the beginning of this process, quarter or semi-elliptical becomes important again, as the model used to analyse the SIF will depend on the established crack pattern. As said before, two patterns for the cracks as they growth are found: corner-initiated cracks growing in the shape of a quarter-ellipse and surface-initiated cracks growing in a semielliptical shape. Both cases were analysed by Newman in [86], who provided a series of equations for the mode I SIF:  Quarter-Elliptical Corner Crack: In this case, the SIF calculation will be obtained using the following expression: 𝐾=𝜎 𝐹√𝜋𝑎 𝑄 (7-1) where σ is the stress produced by the applied axial load (using the positive part of the cycle), F is the boundary correction factor that adjusts the stress intensity factor by accounting for the influence of various boundaries, depending on parameters such as crack depth, crack length, plate thickness, plate width, and the parametric angle of the elliptical crack front. Additionally, a represents the crack length, and Q 10-4 10-3 10-2 10-8 10-7 10-6 10-5 PF-RT-4 (850 MPa - R=-1) PF-RT-12 (550 MPa - R=-1) PF-HT-1 (1094 MPa - R=0,1) PF-HT-7 (832 MPa - R=0,1) Crack growth rate, da/dN (m/cycles) Crack length, a (m) Fatigue crack growth 93 is a value that depends on the crack aspect ratio, as given by Equation (7-2): 𝑄=1+1.464(𝑎𝑐)1.65 (7-2) In the aspect ratio, the parameter a corresponds to the length of the semi-axis of the ellipse in the x-direction (thickness of the specimen), and the parameter c corresponds to the length of the semi-axis in the y-direction (width of the specimen), as shown in Figure 5-11. A schematic representation of this can be seen in Figure 7-5. Figure 7-5. Scheme of corner-crack and surface-crack configurations with elliptical crack front [86]. The parameter F also depends on the aspect ratio, as well as the crack depth and the angle at which the analysed point is located, with a value of 0° when the point is at c and 90° when it is at a. 𝐹=[𝑀1+𝑀2(𝑎𝑡)2+𝑀3(𝑎𝑡)4]𝑔1𝑔2𝑓1𝑓2 (7-3) Each of the functions appearing in Equation (7-4) through (7-11) are described below: 𝑀1=1.08−0.03𝑎𝑐 (7-4) 𝑀2=−0.44+1.06 0.3+𝑎𝑐 (7-5) 𝑀3=−0.5+0.25𝑎𝑐+14.8(1−𝑎𝑐)15 (7-6) 𝑔1=1+[0.08+0.4(𝑎𝑡)2](1−sin𝜃)3 (7-7) 94 Fretting fatigue behaviour of Inconel 718 at room and high temperature 𝑔2=1+[0.08+0.15(𝑎𝑡)2](1−cos𝜃)3 (7-8) 𝑓1=[(𝑎𝑐)2cos2𝜃+sin2𝜃]1/4 (7-9) 𝑓2=1−0.2𝜆+9.4𝜆2−19.4𝜆3+27.1𝜆4 (7-10) 𝜆=𝑐𝑏√𝑎𝑡 (7-11) where 2b corresponds to the width of the specimen and t to its thickness. It is important to note that Equation 7.11 is restricted to c / b < 0.5.  Semi-Elliptical Surface Crack: The SIF calculation will be obtained using the expression (7-1), where the value of Q will again be given by Equation (7-2), and the value of boundary correction factor will be expressed by Equation (7-12): 𝐹=[𝑀1+𝑀2(𝑎𝑡)2+𝑀3(𝑎𝑡)4]𝑔𝑓1𝑓2 (7-12) Each of the functions appearing in Equation (7-13) through (7-18) are described below: 𝑀1=1.13−0.09𝑎𝑐 (7-13) 𝑀2=−0.54+0.89 0.2+𝑎𝑐 (7-14) 𝑀3=0.5− 1 0.65+𝑎𝑐+14(1−𝑎𝑐)24 (7-15) 𝑔=1+[0.1+0.35(𝑎𝑡)2](1−sin𝜃)2 (7-16) 𝑓1=[(𝑎𝑐)2cos2𝜃+sin2𝜃]1/4 (7-17) 𝑓2=[sec(𝜋𝑐 2𝑏√𝑎𝑡)]1/2 (7-18) where 2b corresponds to the thickness of the specimen and t to its width. It is important to note that Equation (7-18) is restricted to c/b < 0.5. In this case, the parameter a corresponds to the length of the semi-axis of the ellipse Fatigue crack growth 95 in the y-direction (width of the specimen), and the parameter c corresponds to the length of the semi-axis in the x-direction (thickness of the specimen), a schematic representation of this can be seen in Figure 7-5. As observed in both cases, obtaining a value for the SIF requires knowledge of the crack aspect ratio at each point of interest. Given the infinite possible crack fronts that may pass through a certain point determined by its X-Y coordinates, it is essential to estimate the aspect ratio throughout the crack growth process. For this purpose, it is essential to recall the Paris’ law, which describes the crack growth rate as a function of the SIF range: 𝑎 𝑁= (∆𝐾𝐼,𝑎)𝑚 , 𝑐 𝑁= (∆𝐾𝐼,𝑐)𝑚 (7-19) where a and c represent the crack lengths in two perpendicular directions along the crack front. ∆𝐾𝐼,𝑎 corresponds to the SIF range in Mode I at the crack front at along a (located at 90°), while ∆𝐾𝐼,𝑐 represent the SIF at the crack front along c (located at 0°). By dividing both equations, the growth rate ratio between these two directions is obtained: 𝑎 𝑐=(∆𝐾𝐼,𝑎 ∆𝐾𝐼,𝑐)𝑚 (7-20) This equation governs the evolution of the crack aspect ratio during propagation. If ∆𝐾𝐼,𝑎> ∆𝐾𝐼,𝑐, the crack will grow more rapidly in the a-direction. Conversely, if ∆𝐾𝐼,𝑎<∆𝐾𝐼,𝑐, growth in the c-direction will dominate. It is important to note that, as seen in Equation (7-20), calculating the aspect ratio requires prior knowledge of the Paris’ law exponent, m, which is initially unknown. For this reason, an iterative process is necessary until convergence is achieved (i.e., successive iterations produce identical values of m). The final value adopted for m is provided in this section in Table 7-1. To integrate this equation and obtain the evolution of the aspect ratio, it is necessary to define an initial crack aspect ratio as a starting point. In this regard, the evolution of the crack aspect ratio has been evaluated for initial values of 0.25, 0.5, 0.75, and 1, assuming an initial length of 1 µm. The results obtained show that the initial aspect ratio loses influence roughly for a crack length, a, above 50 µm, a similar conclusion have been obtained in previous studies [85]. In the case of cracks initiated at the surface, approximately the influence disappears for a crack length beyond 10 microns. Figure 7-6 shows the results for a corner crack, and Figure 7-7 presents those for a surface crack. In both graphs the y-axes represent de aspect ratio and the x-axes de crack length a. As the crack grows, the aspect ratio, a/c, tends to stabilize, reaching approximately 0.99 for corner cracks and 0.90 for surface cracks. 96 Fretting fatigue behaviour of Inconel 718 at room and high temperature Figure 7-6. Evolution of the aspect ratio a/c as a function of a, for a crack initiated at a corner. Figure 7-7. Evolution of the aspect ratio a/c as a function of a, for a crack initiated at the surface. Additionally, Figure 5-11 shows the comparison between the estimated aspect ratio (dotted lines) and the actual fracture surface of the four specimens used to measure crack growth rate above mentioned. The agreement between the estimation and experimentally observed is remarkably high, validating the methodology used. The exact measurements of the estimated crack front are indicated with dimensions in the image. Once the crack evolution is obtained, expressed by Equation (7-21), the relationship at each point can be predicted based on its X and Y coordinates, considering the centre of the ellipse as the origin of coordinates. On the other hand, the ellipse equation is defined by Equation (7-22). 𝑎𝑐=𝑓(𝑎) (7-21) 110 100 0,25 0,50 0,75 1,00 a/c a (m) a/cinitial=0.25 a/cinitial=0.50 a/cinitial=0.75 a/cinitial=1.00 110 100 0,25 0,50 0,75 1,00 a/c a (m) a/cinitial=0.25 a/cinitial=0.50 a/cinitial=0.75 a/cinitial=1.00 Fatigue crack growth 97 (𝑎 𝑥)2+(𝑐𝑦)2=1 (7-22) The system of equations formed by Equation (7-21) and (7-22) allows determining the most probable values of a and c, which characterize the shape of the crack front when passing through a certain x-y location. Once the values of a and c are determined and also its angle θ (see Figure 7-5), the SIF range, ∆𝐾𝐼, is calculated for that point. Figure 7-8. Crack growth rate (da/dN) vs. ∆𝐾𝐼 (MPa m1/2) of plain fatigue test at RT and HT. 10 100 10-7 10-6 10-5 PF-RT-4 (850 MPa - R=-1) PF-RT-12 (550 MPa - R=-1) PF-HT-1 (1094 MPa - R=0.1) PF-HT-7 (832 MPa - R=0.1) K (MPa m1/2) Crack growth rate, da/dN (m/cycles) a) 10 100 10-7 10-6 10-5 discarded HT points b) Set of points unified according to temperature PF-RT PF-HT Crack growth rate, da/dN (m/cycles) K (MPa m1/2) 98 Fretting fatigue behaviour of Inconel 718 at room and high temperature Repeating this process for all points where the FCGR was obtained, it is possible to define a relationship between da/dN (m/cycles) and ∆𝐾𝐼 (MPa m1/2). This relationship is represented in Figure 7-8, where the figure a) shows the data points categorised by the test specimen examined, while the figure b) categorizes the data points by test temperature: RT points correspond to quarter elliptical corner crack analyses, while HT points correspond with semi-elliptical surface crack analyses. In this way, blue points represent the collective dataset for all measurements at RT, while red points correspond to those at HT, allowing for a clearer comparison between both test conditions. In the previous figure, Figure 7-8, it can be seen how the values obtained for different load levels are very similar as a function of temperature. A clear difference between low and high temperatures is observed, which increases as the stress intensity factor increases, being very similar around 20 MPa m1/2 in all cases. That is, for low levels of stress intensity factor, the crack growth rate is very similar. This could explain why, in the experimental simple fatigue tests under low axial stresses, the fatigue lives tend to show little variation. Besides, it is worth noting that in the HT results of this study, a small region is observed where the crack growth rate remains almost constant despite an increase in SIF, specifically for SIF values below 20 MPa m1/2. A possible explanation for this behaviour is that the crack can be considered physically short [87] [88]- [89] [90]. Physically short cracks are typically those with a length below 1–2 mm, which roughly corresponds to the observed range. Specifically, Connolley et al. [90] conducted a study comparing long and short cracks, showing that short cracks exhibit an extended SIF range (~4 - 20 MPa m1/2) where the growth rate remains nearly constant. Therefore, for the RT condition, all data points will be considered in the evaluation of the Paris’ law parameters, whereas for the HT case, only the data points corresponding to stress intensity factor values greater than 20 MPa m1/2 will be used. Figure 7-9 presents a comparative analysis of crack growth rate, stress intensity factor, and crack length under simple fatigue conditions within a single graph. Figure 7-9 (a) displays the data for specimens tested at room temperature, while Figure 7-9 (b) shows the corresponding values at 650 °C. In both cases, the x-axis represents the stress intensity factor (SIF), the left y-axis corresponds to the fatigue crack growth rate (FCGR), and the right yaxis indicates crack length. The relationships are illustrated as follows: SIF vs. FCGR using circles, and SIF vs. crack length using squares. The colour of the symbols denotes the testing temperature. In Figure 7-9, for example, for a SIF = 40 MPa m1/2, the following results were obtained:  PF-RT-4: crack growth rate of 5.33·10-7 m/cycle, with a crack length of 1.30 mm.  PF-RT-12: crack growth rate of 4.48·10-7 m/cycle, with a crack length of 2.1 mm.  PF-HT-1: crack growth rate of 1.20·10-6 m/cycle, with a crack length of 1.74 mm.  PF-HT-7: crack growth rate of 1.44·10-6 m/cycle, with a crack length of 1.11 mm. In general, except for specimen PF-HT-1, the results show an inverse relationship between crack growth rate and crack length. That is, for the same SIF value, shorter cracks tend to grow at a higher rate than longer crack. [91] [92] [77] [93] [94] [95] [96] Fatigue crack growth 99 Figure 7-9. Comparative between crack growth rate, stress intensity factor, and crack length for plain fatigue. Other authors have suggested that this difference could be due to the plastic zone size ahead of the crack being similar to the subgrain size or dislocation cells [88]. However, if the change in slope were caused by this similarity between the plastic zone and subgrains or dislocation cells, it should be reflected in previous studies on the fatigue crack growth behaviour of this alloy [77], [91] - [96]. Unfortunately, this change in crack growth rate slope is not observed in any of them, reinforcing the idea that the change in crack growth behaviour is due to the physical shortness of the cracks. 10 100 10-7 10-6 10-5 K vs CGR K vs a K (MPa m1/2) Crack growth rate, da/dN (m/cycles) a) 10-4 10-3 10-2 PF-RT-4 (850 MPa - R=-1) PF-RT-12 (550 MPa - R=-1) Crack length, a (m) 10 100 10-7 10-6 10-5 K vs CGR K vs a PF-HT-1 (1094 MPa - R=0.1) PF-HT-7 (832 MPa - R=0.1) K (MPa m1/2) Crack growth rate, da/dN (m/cycles) 10-4 10-3 10-2 b) Crack length, a (m) 106 Fretting fatigue behaviour of Inconel 718 at room and high temperature To calculate this, considering the crack pattern and its nearly perpendicular orientation to the surface, it is assumed that crack driving force is the Mode I SIF. In the present work the Mode I SIF was calculated using the weight function method, specifically the approach proposed by Bueckner for a surface through-thickness crack [98]. According to the weight function method, the Mode I SIF is calculated as follows: 𝐾𝐼=√2 𝜋 ∫𝑤(𝑠) 𝜎𝑥(𝑠) 𝑎 0 𝑠 (7-24) where w(s) is the weight function, and σx(s) is the opening stress perpendicular to the prospective crack plane but obtained considering the uncracked geometry. This stress distribution was determined using the methodology explained in Chapter 6, where a 2D EFM model was applied. According to Bueckner’s work, the weight function is defined as: 𝑤(𝑡)=1 √𝑠(1+𝑚1𝑠𝑎+𝑚2(𝑠𝑎)2) (7-25) where m1 and m2 are functions depending on a/W; which is the ratio between the crack length, a, and the specimen width, W. The data points representing the relationship between da/dN (m/cycle) and ΔK (MPa m1/2) are plotted in Figure 7-14 for all the cases analysed above, except for the specimen tested at 1100 MPa at HT, as the FEM did not converge. In other words, the FEM was unable to provide a solution under these load and friction coefficient conditions. This lack of convergence could be due to the absence of adhesion between the fretting bridge and the sample, mainly caused by the low friction coefficient at HT. Experimentally, it is not possible to determine with certainty whether the test was conducted under gross slip or partial slip conditions, as the observation of the fretting scars was inconclusive. However, the available evidence suggests that the test was likely performed under gross slip. This hypothesis is supported by the fact that, at RT, under the same axial load of 1100 MPa, the condition was already close to the gross slip limit, with a value of Q / (μN) = 0.95. Considering that the coefficient of friction at low temperature is typically higher than at HT, it is probable that under HT conditions, the gross slip threshold was exceeded or, at the very least, the system was operating right at the limit. Since the Mode I SIF has been used as the crack driving force, the values obtained at room temperature for different load levels are quite similar, especially in the second region of the graph, where the dispersion of data points is much lower, making this similarity even more evident. For a more comprehensive comparison, all the data points have been grouped by test temperature in Figure 7-15, which clearly shows the difference between RT and HT conditions. The FCGR at high temperature is significantly higher than at room temperature, confirming that the crack propagation is accelerated by temperature, a trends that is also evident in the fretting fatigue life results presented in Figure 4-15. In both Figure 7-14 and Figure 7-15, clearly three distinct trends in the crack growth rate can be observed. A first transition occurs around a SIF value of 40 MPa m1/2 and the second Fatigue crack growth 107 around 90 MPa m1/2, leading to a change in slopes, which suggests a variation in the crack growth process. The behaviour of the first transition was also observed in HT plain fatigue tests, although to a lesser extent. As in that case, the most reasonable explanation is the influence of the short crack phase. The second transition may have occurred due to entering the region of unstable fracture on the surface. In fretting fatigue test, the unusual behaviour of physically small cracks seems to be related to the lack of early contact between the crack faces behind the crack tip during propagation. This suggests that the crack closure effect is ineffective in small cracks, allowing full opening even at minimum load. Similar results reported in [89] indicate that this effect is more pronounced for negative R-ratio, Kmin/Kmax, as in the fretting fatigue case. Figure 7-14. CGR (da/dN) vs. ∆𝐾𝐼 (MPa m1/2) of fretting fatigue tests at RT and HT. 10 100 10-8 10-7 10-6 10-5 FF-RT(1)-1 (1100 MPa) FF-RT(1)-3 (650 MPa) FF-RT(1)-7 (400 MPa) FF-RT(1)-10 (320 MPa) Crack Growth Rate, da/dN (m/cycle) K (MPa m1/2) a) 10 100 10-7 10-6 10-5 FF-HT(1)-10 (650 MPa) FF-HT(1)-14 (400 MPa) Crack growth rate, da/dN (m/cycle) K (MPa m1/2) b) 108 Fretting fatigue behaviour of Inconel 718 at room and high temperature Figure 7-15. Crack growth rate vs. ∆𝐾𝐼 of fretting fatigue tests at RT and HT grouped by temperature. Figure 7-16 presents a comparison between the crack growth rate (left y-axis), stress intensity factor (x-axis), and crack length (right y-axis) for fretting fatigue tests, at room temperature (a) and 650 °C (b), respectively. In these figures, the relationship between the stress intensity factor and crack length is represented by hollow triangles, while the relationship between the stress intensity factor and crack growth rate is shown using solid triangles. It also allows for the identification of the different regions previously discussed. In the case of the RT tests (left graph), the first region corresponds to crack lengths ranging from approximately 0.30 to 2 mm, which are considered small cracks. On the other hand, for the HT tests, the first region includes cracks smaller than approximately 1.00–1.60 mm, also categorized as small cracks, while Region III begins at crack lengths greater than 2 mm. The Paris’ law parameters were calculated fitting these data to a regression line in log-log scale. The resulting parameters, C and m, obtained from the fitting can be seen in the Table 7-2. For this purpose, the data points were segmented based on the predominant slope observed in each case. At RT, two distinct regions can be identified, with a transition at approximately 40 MPa m1/2 In contrast, at HT, three regions are observed, with transitions occurring at around 40 and 90 MPa m1/2. The evaluation of C and m was performed exclusively using the data points within each of these defined regions. The values of C and m were obtained using the CGR expressed in m/cycles and the SIF in MPa m1/2. 10 100 10-8 10-7 10-6 10-5 FF-RT FF-HT Crack growth rate, da/dN (m/cycle) K (MPa m1/2) Fatigue crack growth 109 Figure 7-16. Comparative between crack growth rate, stress intensity factor, and crack length for fretting fatigue. 10 100 10-8 10-7 10-6 10-5 FF-RT(1)-3 (650 MPa) FF-RT(1)-7 (400 MPa) FF-RT(1)-10 (320 MPa) K (MPa m1/2) Crack growth rate, da/dN (m/cycle) K vs CGR K vs a a) 10-5 10-4 10-3 10-2 Crack length, a (m) 10 100 10-8 10-7 10-6 10-5 b) K vs CGR K vs a FF-HT(1)-10 (650 MPa) FF-HT(1)-14 (400 MPa) K (MPa m1/2) Crack growth rate, da/dN (m/cycle) 10-5 10-4 10-3 10-2 Crack length, a (m) 110 Fretting fatigue behaviour of Inconel 718 at room and high temperature Table 7-2. Coefficient of Paris’ law, C and m, in fretting fatigue test. Test σmax (MPa) 1º Zone 2º Zone 3º Zone C m C m C m FF-RT(1)-1 1100 6.27E-06 -1 1.35E-11 2.56 - - FF-RT(1)-3 650 1.68E-08 0.42 2.28E-11 2.29 - - FF-RT(1)-7 400 1.12E-08 0.73 4.20E-12 2.73 - - FF-RT(1)-10 320 1.84E-09 1 1.91E-10 1.78 - - FF-HT(1)-10 650 - - 1.49E-11 2.69 3.72E-8 0.93 FF-HT(1)-14 400 2.338 0.17 7.34E-10 1.81 1.01E-7 0.71 FF-RT(1) Unified 6.27E-6 -1.02 2.28E-11 2.29 - - FF-HT(1) Unified 2.338 0.17 1.01E-10 2.25 4.89E-7 0.87 Figure 7-17. Comparative with other authors of CGR vs. ∆𝐾𝐼 of fretting fatigue test at RT. 10 100 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 NASGRO (R=0.1) CHEN ET AL. (R=-1) LEE A.J. ET EL. (R=0.05) MMPDS - 14 (R=0.05) CLAVEL ET AL. (R=0.1) MERCER ET AL. (R=0.1) SUMIN KIM ET AL (R=0.1) FF-RT (R=0.1) Crack growth rate, da/dN (m/cycle) K (MPa m1/2) Fatigue crack growth 111 Figure 7-18. Comparative with other authors of CGR vs. ∆𝐾𝐼 of fretting fatigue test at HT. In Chapter 5, where the morphology of the crack was analysed, an image of the FF-HT(1)- 14 specimen tested under fretting fatigue (Figure 5-12) was presented. This image clearly shows that the crack initiated at the contact surface and grew with a semi-elliptical front until its c semi-axis reached the full width of the specimen. Due to this, there was reason to question whether the assumption of considering the crack as through-thickness from the beginning was correct. To validate this assumption, the relationship between FCGR and SIF through-thickness crack, as previously described. For this analysis, the NASGRO software [94] was used, providing a new initial SIF value under this revised assumption. NASGRO evaluates multiaxial systems using Glinka’s equation and incorporates a weight function to account for the specimen’s width. In this particular case, only the points measured near the crack front along a semi-axis were analysed, meaning those perpendicular to the contact surface, taking the crack initiation point as the reference. After performing this analysis, it was observed that the influence of the semi-ellipse was minimal and only affected a small portion of the data analysed in this study. This influence was found in the first zone of the data obtained in this work, where the crack growth rate remains nearly constant within that SIF range. Based on these results, it is concluded that the initial assumption of considering the crack as through-thickness for the determination of the Paris’ law parameters is valid. Finally, the fatigue crack growth rates obtained in this work were compared with data from the literature, following the same approach as in plain fatigue tests. Figure 7-17 presents the comparison at RT, while Figure 7-18 does so at HT. 10 100 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 NASGRO (R=0.1 - 621°C) CHEN ET AL. (R=-1 - 600°C) LEE ET EL. (R=0.05 - 649°C) MMPDS - 14 (R=0.05 - 649°C) CLAVEL ET AL. (R=0.1 - 550°C) SUMIN KIM ET AL (R=0.1 - 650°C) CONNOLLEY ET AL (R=0.1 - 600°C) FF-HT (R=0.1 - 650°C) Crack growth rate, da/dN (m/cycle) K (MPa m1/2) 112 Fretting fatigue behaviour of Inconel 718 at room and high temperature In both cases, a non-negligible difference is observed between the literature data and the results obtained in this study. As discussed in the previous section, these differences may not be due to the measurement methodology but rather to the stress ratio used in each test. In fretting fatigue, a multiaxial problem occurs, in which the stress ratio varies throughout the test. Another important factor to consider is that the SIF calculated in this study only accounts for Mode I fracture. 7.3. Comparative of plain and fretting fatigue crack growth rate In this section, a comparative analysis between the two types of tests will be presented, as the previous sections analysed them separately. Therefore, the following figures present a comparison between specimens subjected to simple fatigue (represented by circles) and those tested under fretting fatigue (represented by triangles). For these comparisons, the data has been unified according to the type of test and temperature. Temperature is indicated by the color of the curves, where specimens tested at low temperature are represented in blue, and those tested at high temperature are shown in red. These comparisons are presented in Figure 7-19 and Figure 7-20. It is noticeable that, under the same methodology for obtaining the crack propagation law, both cases yield very similar slopes but with an offset between them. This indicates that for the same crack propagation rate, the stress intensity factor is greater in fretting fatigue, which is expected due to the conditions of the test. This was also evidenced in the experimental results, where it was observed that fretting fatigue tests exhibited a shorter fatigue life compared to plain fatigue tests. Figure 7-19. Comparative between plain and fretting fatigue crack growth rate at RT. 10 100 10-8 10-7 10-6 10-5 Plain fatigue at RT Fretting fatigue at RT Crack Growth Rate, da/dN (m/cycles) K (MPa m1/2) Fatigue crack growth 113 Figure 7-20. Comparative between plain and fretting fatigue crack growth rate at HT. Furthermore, when comparing both simple fatigue and fretting fatigue crack growth rate in terms of temperature, it is evident that the crack propagation rate is higher in the high temperature case. This is attributed to the modification of the material's mechanical properties when subjected to elevated temperatures, as previously observed in Chapter 4. The correlations derived from these graphs will be used in the subsequent chapter on life estimation. Specifically, the parameters of the Paris’ law, considering the unified data points, will be used for this purpose. 10 100 10-8 10-7 10-6 10-5 Plain fatigue at HT Fretting fatigue at HT Crack Growth Rate, da/dN (m/cycles) K (MPa m1/2) 115 8. FRETTING FATIGUE LIFE PREDICTION o estimate the fatigue life of a component, it is essential to understand the process by which failure occurs, which is usually divided into two phases. The first phase, known as the initiation phase, is when a crack nucleates or initiates. The second phase, the propagation phase, involves the growth of an already initiated crack until the component ultimately fails. There are many fatigue life prediction models. Some focus exclusively on the initiation phase, assuming that crack propagation is negligible in comparison to initiation. In this context, various multiaxial fatigue criteria have been proposed to estimate fatigue life, such as the models of Fatemi and Socie [99], Smith-Watson-Topper (SWT) [78], and Crossland [100]. These multiaxial criteria have been applied in the study of fretting fatigue; however, they neglect the influence of the propagation phase, which also plays a significant role. Other approaches only consider the propagation phase, based on the assumption that initiation is short in comparison to propagation. However, under high cycle conditions, the propagation phase is negligible relative to the initiation phase. When stress gradients are high and load levels are moderate, crack initiation may occur rapidly compared to the propagation phase. In extreme cases, where stress concentration and load levels are high, the initiation phase becomes irrelevant. In addition to these local approaches, non-local fatigue models have also been developed, particularly for applications with steep stress gradients, such as fretting fatigue. These models estimate fatigue life based on a damage parameter computed not at a single point, but over a region near the surface, usually at a critical distance. This allows them to capture the influence of the stress field around the potential crack initiation site. One notable contribution is from Araújo et al. [101], who proposed non-local formulations based on classical multiaxial fatigue parameters like SWT, showing improved prediction accuracy in fretting fatigue scenarios. [102] [103] [104] [105] [106]. The life prediction model applied in this work is the phase combination model, where total life combines separately the initiation and propagation phases. This method eliminates the need to explicitly define when the initiation phase ends and the propagation phase begins, i.e. the crack initiation length as the boundary (in terms of the crack length) between these phases is identified during the process. This model has been previously applied in earlier studies, yielding highly satisfactory results [102] - [106]. In the initiation phase analysis, the number of cycles required to produce a crack of a certain length, ai, is determined. In this phase, and using the S-N or ε-N curve, it is possible to obtain a curve (ai-Ni), see Figure 8-1, which represents the relationship between the number of initiation cycles, Ni, and different discrete values of ai. Furthermore, the propagation phase is considered as the number of cycles required to propagate a crack of an initial length ai until the failure of the component say Np. The curve T 122 Fretting fatigue behaviour of Inconel 718 at room and high temperature 8.3. Results This section presents the estimated fatigue life for Type 1 and Type 2 under both room temperature and high temperature conditions. Figure 8-6 provides a comparative analysis between estimated and experimental fatigue life for all cases. In all graphs, the experimental life is represented on the X-axis, while the estimated life is on the Y-axis. Where, if the estimated and experimental lives are equal, data points will align with the central line. If the estimated life exceeds the experimental life, the points will lie above the central line; conversely, if the estimated life is lower, the points will fall below it. The "x2" and "x3" lines indicate where the estimated life is double or triple the experimental value, while the two symmetric lines represent estimated lives that are 1/2 and 1/3 of the experimental life. These reference lines help evaluate the accuracy of the fatigue life predictions. The symbols used in the graphs indicate the methodology applied, with circles representing Type 1 and triangles representing Type 2. The colour coding corresponds to the crack Figure 8-6. Estimated life vs. experimental life, at both temperature, RT and HT. 104105106107 104 105 106 107 104105106107 104 105 106 107 104105106107 104 105 106 107 104105106107 104 105 106 107 x3 x2 HT Kth 10 MPa m1/2 Kth 13.45 MPa m1/2 Kth 15.2 MPa m1/2 Estimated Life (Cycles) RT x3 x2 x3 x2 x3 x2 Type 2 Type 1 Kth 6 MPa m1/2 Kth 7 MPa m1/2 Kth 8 MPa m1/2 Kth 11 MPa m1/2 Kth 10 MPa m1/2 Kth 13.45 MPa m1/2 Kth 15.2 MPa m1/2 Estimated Life (Cycles) Experimental life (Cycles) Kth 6 MPa m1/2 Kth 7 MPa m1/2 Kth 8 MPa m1/2 Kth 11 MPa m1/2 Experimental life (Cycles) Fretting fatigue life prediction 123 growth threshold (∆𝐾𝑡ℎ) used in the analysis. An important aspect to evaluate in these graphs is the selected ∆𝐾𝑡ℎ value, as previous studies have reported significant variability in the threshold values recommended for crack growth analysis. For room temperature, three ∆𝐾𝑡ℎ values are analysed: 10, 13.45 (comparable to 13.25 from Clavel), and 15.2 MPa m1/2. For high temperature, four ∆𝐾𝑡ℎ values are considered: 6, 7, 8, and 11 MPa m1/2. At first glance, most results show good agreement with experimental data, as most of the points fall within the x3 scatter band. However, the life estimation model used here has a key limitation: it becomes unreliable in high cycle fatigue regimes, especially when stress levels approach the crack growth threshold, as predicted lives tend to increase excessively, where predicted lives tend toward infinity. Additionally, the variation in the ∆Kth introduces significant uncertainty, making life prediction in this regime particularly challenging. While graphical representations provide an overview of the results, a more detailed analysis requires evaluating the geometric mean error (GME) and geometric standard deviation error (GSDE). The GME quantifies estimation accuracy, representing the average deviation of the data from the x1 line, while the GSDE measures the dispersion of the analysed points. For this purpose, the ratio between the estimated and experimental life is calculated, and the logarithm of each ratio is obtained: 𝛼=log 𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒 𝑙𝑖𝑓𝑒 𝐸𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝑙𝑖𝑓𝑒 (8-10) The mean value, 𝛼, and the standard deviation, 𝜎𝛼, are then calculated as: 𝛼=1 𝑛∑𝛼 (8-11) 𝜎𝛼=1 𝑛−1∑(𝛼−𝛼)2 (8-12) Finally, the antilogarithms of these values yield the parameter GME and GSDE: 𝐺𝑀𝐸=10𝛼  (8-13) 𝐺𝑆𝐷𝐸=10𝜎𝛼 (8-14) If GME = 1, the estimated and experimental lives are, on average, identical, aligning with the x1 line. If GME > 1, the model overestimates fatigue life, whereas GME < 1 indicates underestimation. For GSDE, a value of 1 signifies perfect dispersion, while GSDE > 1 reflects data variability, with larger values indicating greater dispersion. The results for these parameters are summarised in Table 8-1 for RT conditions and Table 8-2 for HT conditions. It is important to note that infinite life estimates have been excluded from these calculations. 124 Fretting fatigue behaviour of Inconel 718 at room and high temperature Table 8-1. GME and GSDE parameter to estimated life at RT. ∆𝐾𝑡ℎ Type 1 Type 2 GME GSDE GME GSDE 10 0.43 2.01 2.32 1.11 13.45 0.47 1.06 2.82 1.01 15.2 0.73 1.17 3.50 1.03 Table 8-2. GME and GSDE parameter to estimated life at HT. ∆𝐾𝑡ℎ Type 1 Type 2 GME GSDE GME GSDE 6 0.54 1.08 1.26 1.21 7 0.90 1.12 1.97 1.04 8 1.21 1.45 2.82 1.18 11 2.96 4.3 6.44 2.56  Analysis of room temperature data: For Type 1, all three models exhibit an underestimation bias, with GME values ranging from 0.43 to 0.73, where the underestimation increases as the ∆𝐾𝑡ℎvalue decreases. In terms of dispersion, variability ranges are between 1.06 and 2.01. Among the analysed values, ∆𝐾𝑡ℎ = 13.45 MPa m1/2 provides the most reasonable balance of bias and variability, whereas ∆𝐾𝑡ℎ = 10 MPa m1/2 results in excessive dispersion. For Type 2, a different trend is observed. In this case, the model tends to overestimate fatigue life, with lower ∆𝐾𝑡ℎvalues resulting in lower overestimation. Regarding dispersion, the trend is similar to Type 1, ∆𝐾𝑡ℎ = 10 MPa m1/2 provides the most unfavourable results, whereas ∆𝐾𝑡ℎ = 13.45 MPa m1/2 is the most suitable, very similar to ∆𝐾𝑡ℎ = 15.2 MPa m1/2. Overall, for both Type 1 and Type 2, threshold values of 13.45 MPa m1/2 and 15.2 MPa m1/2 yield consistent and reasonable results, demonstrating good agreement with experimental data. Comparing the two types, the fatigue life estimates based on Type 1 data appear more reliable, as they are derived from Paris' law parameters obtained from single-fatigue characterization.  Analysis of high temperature data: Fretting fatigue life prediction 125 For Type 1, all ∆𝐾𝑡ℎvalues provide reasonable accuracy, except for ∆𝐾𝑡ℎ = 11 MPa m1/2. The ∆𝐾𝑡ℎvalues of 6 and 7 MPa m1/2 tend to underestimate fatigue life, whereas ∆𝐾𝑡ℎvalues of 8 and 11 MPa m1/2 result in overestimation. In terms of dispersion, ∆𝐾𝑡ℎ = 11 MPa m1/2 performs poorly, leading to its exclusion. The ∆𝐾𝑡ℎ values of 6, 7 and 8 MPa m1/2 show acceptable dispersion. Overall, ∆𝐾𝑡ℎ = 7 MPa m1/2 offers the best results, closely followed by ∆𝐾𝑡ℎ = 8 MPa m1/2. For Type 2, a similar pattern is observed. The ∆𝐾𝑡ℎvalues of 6, 7, and 8 MPa m1/2 provide acceptable results, while ∆𝐾𝑡ℎ = 11 MPa m1/2 is inadequate. Across all cases, the model tends to overestimate fatigue life, with ∆𝐾𝑡ℎ = 6 MPa m1/2 showing the least overestimation, followed by ∆𝐾𝑡ℎ = 7 MPa m1/2, which also performs well. In summary, for HT conditions, ∆𝐾𝑡ℎvalues of 7 and 8 MPa m1/2 yield comparable results. Similar to RT conditions, Type 1 data provide more reliable fatigue life estimates. These results highlight the importance of selecting an appropriate crack growth threshold (∆𝐾𝑡ℎ) in fatigue life estimation. Although most predictions fall within the x3 scatter band, significant differences exist in the estimated life values. A precise experimental determination of ∆𝐾𝑡ℎ, or better understanding of how it varies with factors such as the load ratio (R) or crack length, is essential for achieving accurate fatigue life predictions. For the two ∆𝐾𝑡ℎ values that yielded the best results according to parameters GME and GSDE – 13.45 and 15.2 MPa m1/2 for RT, and 7 and 8 MPa m1/2 for HT – S-N curves were generated and compared with the experimental results, as shown in Figure 8-7 for Type 1 and Figure 8-8 for Type 2. In both figures, the fatigue life estimations are represented by lines, while the experimental results are shown as triangles. Results at RT are displayed in blue tones, and those at HT in red tones. Figure 8-7. S-N curve for estimated type 1 and experimental lifetimes. 103104105106107 200 400 600 800 1000 1200 N=10 kN R=0.1 FR-1 FH-1 Maximum Axial Stress (MPa) Number of cycles Estimated Life for Type 1: RT  Kth=13.45 MPa m1/2 RT  Kth=15.2 MPa m1/2 Kth=7 MPa m1/2 HT  Kth=8 MPa m1/2 126 Fretting fatigue behaviour of Inconel 718 at room and high temperature Figure 8-8. S-N curve for estimated type 2 and experimental lifetimes. In the Type 1 graph (Figure 8-7), it can be observed that the life estimations at RT tend to underestimate the experimental data up to an axial load of approximately 575 MPa. Beyond this value, the model predicts infinite life regardless of the threshold used. This suggests that, for RT, the life prediction model is not suitable for the high cycle fatigue regime and indicates the need for further investigation to understand the behaviour beyond this region. In contrast, for HT conditions, the estimated lives closely match the experimental results, regardless of the threshold applied. In the Type 2 graph (Figure 8-8), it is evident that for both temperatures, the estimations tend to overestimated fatigue life, although the slopes are very similar to those observed in the experimental tests. Furthermore, this model reflects the behaviour observed in the fretting fatigue tests where the S-N curve shifts to the left parallel to itself for HT while in plain fatigue the difference between RT and HT is minimum. This supports the idea that the HT affects mainly the propagation of cracks and not so much the initiation. As in Type 1, the RT estimations do not accurately capture the behaviour in the high cycle fatigue regime. Finally, in both figures, it is clear that the predicted fatigue lives for HT are consistently lower than those for RT, which is in agreement with the experimental observations. To complete the fatigue life prediction analysis, the initiation length and the percentage of initiation life are evaluated as a function of the estimated life, for both temperatures for Type 1 in Figure 8-9 and for Type 2 in Figure 8-10. First, when analysing the initiation length, a nonlinear behaviour is observed with respect to the estimated life. Initially, as the number of load cycles increases, the initiation length decreases. However, after reaching a certain point, this trend reverses, and the initiation length begins to increase as the cycles continue to grow. Regarding the percentage of initiation life, it is evident that for lives shorter than 10000 cycles in RT and 50000 cycles in HT, this percentage is nearly zero. As fatigue life increases, the initiation percentage also rises, becoming predominant as it 103104105106107 200 400 600 800 1000 1200 N=10 kN R=0.1 FR-1 FH-1 Maximum Axial Stress (MPa) Number of cycles Estimated Life for Type 2: RT  Kth=13.45 MPa m1/2 RT  Kth=15.2 MPa m1/2 HT  Kth=7 MPa m1/2 Kth=8 MPa m1/2 Fretting fatigue life prediction 127 approaches one million cycles. This confirms the hypothesis proposed in Chapter 4, where a clear difference between simple fatigue and fretting fatigue was observed, suggesting that this difference may be due to a greater proportion of the propagation phase in the case of fretting fatigue and a faster crack growth rate in fretting fatigue compared to plain fatigue. Figure 8-9. Initiation length vs. estimated life and initiation phase % vs. estimated life for Type 1 at RT (left) and HT (right). Figure 8-10. Initiation length vs. estimated life and initiation phase % vs. estimated life for Type 2 at RT (left) and HT (right). 103104105106107108 1 10 100 Estimated life (Cycles) Initial crack length (m) RT HT 0 20 40 60 80 100 Kth 6 MPa m1/2 Kth 7 MPa m1/2 Kth 8 MPa m1/2 Kth 11 MPa m1/2 % Initiation 103104105106107108 1 10 100 Initiation length vs. estimated life Initiation phase % vs. estimated life Kth 10 MPa m1/2 Kth 13.45 MPa m1/2 Kth 15.2 MPa m1/2 Estimated life (Cycles) Initial crack length (m) 0 20 40 60 80 100 % Initiatition 104105106 1 10 100 1000 Kth 15.2 MPa m1/2 Estimated life (Cycles) Initial crack length (m) Kth 6 MPa m1/2 Kth 7 MPa m1/2 Kth 8 MPa m1/2 Kth 11 MPa m1/2 Kth 10 MPa m1/2 Kth 13.45 MPa m1/2 0 20 40 60 80 100 % Initiation 104105106107 1 10 100 1000 Estimated life (Cycles) Initial crack length (m) 0 20 40 60 80 100 Initiation length vs. estimated life Initiation phase % vs. estimated life % Initiatition RT HT 128 Fretting fatigue behaviour of Inconel 718 at room and high temperature The first phase, where the initiation length decreases with increasing cycles, coincides with the phase in which the initiation percentage is zero. This indicates that during this stage, the values depend exclusively on the propagation phase. This trend is consistent with findings reported in the literature by Vázquez et al. [85]. 8.4. Crack evolution estimation at 650 °C By applying the combined-phase model proposed in this work, it is possible to determine the evolution of crack length as a function of the number of cycles. The number of cycles required to reach a crack of a certain length a is given by the following expression: 𝑎<𝑎𝑖 𝑁𝑡(𝑎)=𝑁𝑖(𝑎) (8-15) 𝑎>𝑎𝑖 𝑁𝑡(𝑎)=𝑁𝑖(𝑎𝑖)+𝑁𝑝|𝑎 𝑎𝑖=𝑁𝑡∗−𝑁𝑝(𝑎) For crack lengths smaller than ai, the number of elapsed cycles Nt(a) corresponds to the number of cycles required to initiate the crack, Ni(a). For crack lengths greater than ai, the number of elapsed cycles is obtained by summing the number of cycles required to initiate a crack of length ai and the number of cycles necessary to propagate it from that length to a. Alternatively, this is equivalent to subtracting from the total fatigue life (i.e., the number of cycles until failure) the number of cycles required to propagate a crack from length a to final fracture, Np(a), as illustrated in Figure 8-11. Figure 8-11. Curve of the number of elapsed cycles. In this study, the estimation of crack growth evolution was carried out for the estimation of Type 1, under a normal load of 10 kN and maximum axial load of 900 MPa and for ∆𝐾𝑡ℎ values of 13.45 and 15.2 MPa m1/2 for RT, and 7 and 8 MPa m1/2 for HT. These estimates can be seen in Figure 8-12, where it is noticeable how the crack for HT grows faster that in the RT specimen and also that the higher the crack growth threshold, the longer it takes to propagate. 𝑁𝑝(𝑎) 𝑎𝑖𝑎𝑓 𝑎 𝑁𝑡∗ (𝑇 𝑡𝑎𝑙 𝑙𝑖𝑓𝑒) Cycles Crack length Fretting fatigue life prediction 129 Figure 8-12. Crack evolution estimation for maximum axial stress of 900 MPa. Besides, the crack evolution estimation was compared with the experimental result for high temperature fretting fatigue test with a maximum axial stress of 750 MPa, under a normal load of 10 kN These loading conditions were selected because they correspond to the largest number of fretting fatigue tests performed in this work, including both conventional fretting fatigue test to the failure and fretting fatigue interrupted test. Additionally, crack growth thresholds of 7 and 8 MPa m1/2 were considered. These threshold values were selected due to their similar performance in the analysis of the GME and GSDE parameters. The resulting estimations are compared with all the interrupted tests conducted, as well as with the final fracture surfaces of the specimens, as previously discussed in Chapter 4. Figure 8-13 displays the experimentally measured crack lengths on the fracture surfaces using symbols, while the dashed lines represent the model estimations: the red line corresponds to a ∆𝐾𝑡ℎof 7 MPa m1/2, and the dark red dashed line to a ∆𝐾𝑡ℎof 8 MPa m1/2. From this figure, it can be observed that the ∆𝐾𝑡ℎof 7 MPa m1/2 tends to underestimate the total life of the specimen, and that the ∆𝐾𝑡ℎof 8 MPa m1/2 matches perfectly with the experimental data. For short life both curves producing very similar values, but increasingly diverging results as the crack approaches final failure. Finally, it can be concluded that the results indicate a satisfactory agreement between the crack evolution estimated and experimental observations at HT. 101102103104 0 500 1000 1500 2000 2500 3000 max. axial= 900 MPa Crack length (m) Cycles Estimated crack length at RT Kth=13.45 MPa m1/2 Estimated crack length at RT Kth=15.5 MPa m1/2 Estimated crack length at HT Kth=7 MPa m1/2 Estimated crack length at HT Kth=8 MPa m1/2 130 Fretting fatigue behaviour of Inconel 718 at room and high temperature Figure 8-13. Estimated crack length for type 1 vs. real crack length. 103104 0 500 1000 1500 2000 2500 3000 FFI-1 - 1º Crack length FFI-1 - 2º Crack length FFI-2 - 1º Crack length FFI-2 - 2º Crack length FFI-31º Crack length FFI-3 - 2º Crack length FFI-3 - Total crack length (1 step) FFI-3 - Total crack length (2 step) FF-HT(1) - Total crack length Estimated crack length - Kth=7 MPa m1/2 Estimated crack length - Kth=8 MPa m1/2 Crack length (m) Cycles 131 9. CONCLUSIONS AND FUTURE WORK n the present work, the influence of fretting fatigue at both room temperature and 650 °C has been analysed for the Inconel 718 alloy with a conventional heat treatment. To this end, plain fatigue and fretting fatigue tests were carried out under different axial load levels, with the experimental procedure having been previously validated in the laboratory. In total, 19 plain fatigue tests were conducted at room temperature, 7 at high temperature, 17 fretting fatigue tests at room temperature (12 with 10 kN and 5 with 7 kN of normal load), and 22 fretting fatigue tests at 650 °C (16 with 10 kN and 6 with 7 kN of normal load). In all cases, a decrease in fatigue life was observed as the applied axial load increased. At both temperatures, fretting fatigue resulted in shorter lives compared to plain fatigue, with this difference being more pronounced at lower axial loads. However, around 1100 MPa, the fatigue lives for plain and fretting conditions were similar, suggesting that at such stress levels, the stress concentration due to contact does not have a significant effect. Regarding the temperature effect, plain fatigue tests yielded very similar results, although a slight reduction in life was observed at high temperature for lives below 10⁵ cycles. In contrast, fretting fatigue tests at 650 °C exhibited fatigue lives approximately four times shorter than those at room temperature, indicating a clear influence of elevated temperature. Furthermore, the analysis of the normal load effect revealed that, for the tested conditions (7 kN and 10 kN), no significant variation in fatigue life was observed in the experimental tests. Interrupted tests allowed for the study of crack morphology, revealing the initiation of multiple semi-elliptical cracks at the contact edge, which rapidly evolved into throughthickness cracks. The number of interruptions during a test appeared to result in a slight reduction in fatigue life, although not significantly. Moreover, the number of interruptions did not seem to affect the final crack length. Scanning electron microscopy (SEM) and energy-dispersive X-ray spectroscopy (EDS) analysis revealed a considerable presence of oxides on the specimen surface, with similar thicknesses at both room and high temperatures, attributed to the heat treatment. Additionally, the amount of oxygen on the fracture surface increased with exposure time at high temperature. In terms of crack morphology, corner cracks were predominantly observed under plain fatigue at room temperature, while semi-elliptical surface cracks appeared under high temperature conditions. In fretting fatigue tests, multiple semielliptical cracks formed along the contact edge, which merged to become through-thickness cracks. Furthermore, crack growth rate analysis was performed on 11 specimens (4 under plain fatigue and 7 under fretting fatigue), revealing higher crack growth rates at 650 °C compared to room temperature. An increase in crack growth rate with higher axial stress was also observed. When plotting the crack growth rate against the stress intensity I