Modelling of TMF Crack Growth in Polycrystalline Gas Turbine Alloys : Accounting for Crack Closure Effects

Modelling of TMF Crack Growth in

Polycrystalline Gas Turbine Alloys

Accounting for Crack Closure Effects

Jordi Loureiro-Homs

Link¨

oping Studies in Science and Technology

Licentiate Thesis No. 1885

Modelling of TMF Crack Growth

in Polycrystalline Gas Turbine

Alloys

Accounting for Crack Closure Effects

Jordi Loureiro-Homs

Solid Mechanics

Link¨oping University

SE–581 83 Link¨oping, Sweden

Cover:

A single edge notch specimen at the latest stage of a TMF crack growth test

Printed by:

LiU-Tryck, Link¨oping, Sweden, 2020

ISBN: 978-91-7929-792-3 ISSN: 0280-7971

Distributed by:

Link¨oping University

Solid Mechanics

SE–581 83 Link¨oping, Sweden

© 2020 Jordi Loureiro-Homs

This document was prepared with LA_{TEX, September 25, 2020}

No part of this publication may be reproduced, stored in a retrieval system, or be transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the author.

Preface

The work presented in this Licentiate of Engineering thesis has been carried out at

Link¨oping University. This research has been funded by the Swedish Energy Agency

and Siemens Energy through ”Turbines for Future Energy Systems” (Turbiner f¨or

framtidens energisystem), Grant No.44100-1, the support of which is gratefully acknowledged.

A big thanks to all the team members from Link¨oping University and Siemens,

whose support has been invaluable.

To my office mate, Thomas Lindst¨om, I am sure your dream car is right around

the corner: the mighty Niva Lada will be yours. Eventually.

Abstract

The main objective of the work presented in this Licentiate of Engineering thesis is to investigate and model the fatigue crack propagation behaviour of the nickel-based superalloy Inconel 792, with special attention to the industrial lifing of high-temperature components. In-phase (IP) crack propagation tests have been performed at different temperatures and loading regimes, including extended hold times. The observations from these tests have been the basis for establishing several hypotheses to describe the crack growth behaviour, which progressively have been verified experimentally and numerically. Most prominently, it has been observed that crack closure has a substantial impact on crack growth and can explain, to a large degree, the crack growth behaviour for this material under the conditions studied. This phenomenon has been observed experimentally and modelled numerically to extend further the precision of the methodology.

Sammanfattning

Huvudsyftet med arbetet som presenteras i denna licentiat avhandling ¨ar att

unders¨oka och modellera utmattnings sprickv¨axtbeteendet hos den nickelbaserade

superlegeringen Inconel 792, med s¨arskild uppm¨arksamhet riktad mot liusl¨

angs-dmodellering av h¨ogtemperaturkomponenter i en industriell kontext. I-fas (IP)

sprickv¨axtprov har utf¨orts vid olika temperaturer och belastningsregimer, inklusive

h˚alltider. Observationerna fr˚an dessa tester har legat till grund f¨or hypoteser f¨or

att f¨orklara spricktillv¨axtbeteende, vilka successivt har verifierats experimentellt

och numeriskt. Mest framtr¨adande har det observerats att sprickslutning har en

v¨asentlig inverkan p˚asprickv¨axten, och kan i stor utstr¨ackning f¨orklara sprickv¨axten

f¨or detta material under studerade f¨orh˚allanden. Detta fenomen har observerats

experimentellt och modellerats numeriskt f¨or att f¨orb¨attra metodens precision.

List of papers

In this thesis, the following papers have been included:

I. Jordi Loureiro-Homs, D. Gustafsson, P. Almroth, K. Simonsson, R. Eriksson, D. Leidermark, (2020). Accounting for initial plastic deformation for fatigue crack growth predictions under TMF loading condition, International Journal of Fatigue, Volume 136, 2020, 105569

II. Jordi Loureiro-Homs, P. Almroth, F. Palmert, D. Gustafsson, K. Simonsson, R. Eriksson, D. Leidermark, (2020). Accounting for Crack Closure Effects in TMF Crack Growth Tests With Extended Hold Times in Gas Turbine Blade Alloys, International Journal of Fatigue, Volume 153, 2021, 105917

Note

The appended papers have been reformatted to fit the layout of the thesis. Some of the figures, originally in color, may have been reproduced in black and white and the associated text adjusted accordingly.

The published appended papers have been reprinted with the permission of the respective copyright holder and all appended papers have been reformatted to fit the layout of the dissertation

The author’s contribution

The research presented in the appended papers as well as the writing has been carried out primarily by the author.

Contents

Part I – Background and Theory

1

1 Introduction 3

1.1 The role of gas turbine engines for power generation on a sustainable

transition . . . 3

1.2 Aim of the work . . . 4

2 Turbine engine conditions 5 2.1 Engine loading conditions . . . 5

2.2 Engine materials . . . 6

2.2.1 Nickel-based superalloy Inconel 792 . . . 6

2.3 Relevance of fatigue crack growth assessment in turbine components 8 3 Fatigue crack propagation 9 3.1 Fatigue crack propagation . . . 10

3.2 The crack closure phenomenon . . . 10

3.3 Relevance of crack closure and residual stress in turbine blades applications . . . 13

4 Experimental and computational methods 15 4.1 Experimental procedures . . . 16

4.1.1 Experimental measurement of crack length . . . 17

4.1.2 Experimental measurement of crack closure . . . 19

4.2 Numerical methods for crack closure . . . 22

4.2.1 Mesh size . . . 22

4.2.2 Time discretisation . . . 23 xi

4.2.3 Numerical measurement of crack closure . . . 23

5 Outlook 25

6 Review of Appended Papers 27

References 29

Part II – Appended Papers

35

Paper II: Accounting for Crack Closure Effects in TMF Crack Growth

Tests With Extended Hold Times in Gas Turbine Blade Alloys . . . 49

Part I

Introduction

1

1.1

The role of gas turbine engines for power generation on

a sustainable transition

It is a fact that burning fossil fuels has led to the deployment of a great amount of the greenhouse gas CO2, considered to be the main contributor to climate change. It is also a fact that most of the gas turbines are contributing to this issue to a lesser or greater extent. Many efforts are being made from the industry to lower emissions and extend efficiency in these engines and, over the years, we have seen a remarkable improvement in this regard. The issue of greenhouse gas emissions has been internationally acknowledged and several actions have been set on place to limit the increase in the average global temperature. This goal could be achieved by reducing the CO2 emissions, reducing energy consumption and developing alternative green energy sources. A lot of effort has been focused on the replacement of fossil fuel energy sources such as wind and solar power. However, these renewable energy sources are inherently intermittent resulting in the need for both energy storage solutions and supplementary energy production. For the short run, as other technologies mature, gas turbines offer the flexibility needed to supplement demand when natural resources are not available [1, 2]. Additionally, gas turbines can be run in a variety of green fuels, including hydrogen and syngas which when sourced from the so-called power-to-fuel concepts, creates an overall CO2-neutral circular economy [3].

With all that, gas turbines will most likely be part of the solution in both the short and long term future, despite being part of the problem today. There are many engineering challenges ahead for this to happen, as gas turbines need to be able to cope with demands never required before: better cyclic capability, fast start-ups, increased efficiency and reliability among others. All these efforts are also driving the development of new technologies which require fundamental research. Prominently, the complex geometries found in gas turbine components are pushing the development of additively manufactured parts for high temperature components [4–6]. This new technology requires bespoke modelling techniques and is contributing to the development of fundamental science [7].

CHAPTER 1. INTRODUCTION

1.2

Aim of the work

This PhD project is funded by the Swedish Energy Agency and Siemens Energy

through the research program Turbines for Future Energy Systems (Turbiner f¨or

framtidens energisystem), with the aim to contribute to strengthen the capacity of turbine engines and other high-temperature components to complement renewable sources and to adapt to the new requirements with all that entitles. Efficiency, reliability and availability are key factors for a gas turbine engine suited to provide power on intermittent demands. To support the design process and maintenance operations it is important to include crack propagation. This work provides some of the required foundations to gain industrially applicable simulation methods to account for fatigue crack propagation in high-temperature turbine components.

Turbine engine conditions

2

2.1

Engine loading conditions

Gas turbines are engineering wonders that are still stretching the ingenuity of designers and engineers alike. Every part of an aero turbine is designed to minimize weight with great constraints to fulfill aerodynamic aspects, not to mention the requirements and difficulties of operating under constantly changing conditions. On the other hand, stationary gas turbines are often driven at constant speeds under, more or less, constant conditions. These turbines, which are the main focus of this work, have other requirements that make the design and operation a comparable challenge, even more so as these machines need to adapt to the new requirements. There are many factors influencing the cyclic life in a gas turbine engine; under normal conditions, the cyclic behaviour is defined by the ramping up and down during start and stop operations. Generally, after the ramp up, the engine will be subjected to a prolonged operation under its nominal working conditions or under part load if so required. Ideally, the engine will be shut down following a particular sequence that prevents abrupt changes in the turbine. If there is any fault during the operation, the control system triggers an emergency stop that will trip the turbine and shut down the fuel immediately, exposing the turbine components to abrupt temperature changes. The cyclic life of a turbine is defined as the amount of starts and stops. If the turbine experiences a trip, this usually translates to a larger reduction of its cyclic life. The maintenance intervals are defined by a threshold of either equivalent operation hours or equivalent cycles.

For the case of turbine blades, an important loading arises from the temper-ature cycling between room and operation tempertemper-ature. This effect is especially pronounced at points that are hotter or colder than the surrounding material. The existence of these points, with large temperature gradients, is often difficult to avoid and they may present a transient or stationary behaviour. The combination of loading conditions and the properties of the materials used for this application results in a cycle that stabilises after the first few cycles, behaving in essence, elastically after the first few cycles. This material characteristics enables, to some extend, the use of linear-elastic fracture mechanics.

CHAPTER 2. TURBINE ENGINE CONDITIONS

Table 1: Inconel 792 main alloyants

Ni Fe Cr Mo W Co Nb Al C Other

61.0 3.5 12.4 1.9 3.8 9.0 - 3.5 0.04 0.3 V

2.2

Engine materials

The demanding environment inside a gas turbine requires materials that can cope with high temperature, high stress and hot corrosive gases for an extended time. There are just a handful of materials capable of withstanding the conditions at the

hottest sections of a turbine, where temperatures may reach 1300 °C. Nowadays,

the use of nickel-based superalloys is generally required as the temperature at the turbine inlet has been increasing over time, driven by the need for higher efficiencies and more power. The use of these materials, in combination with ceramic coatings and advance cooling techniques, have been, and still are, key factors that allowed the possibility to obtain higher power and efficiency improvements. Nickel-based superalloys have a combination of good oxidation/corrosion resistance at high temperature, high strength, long fatigue life and, most importantly, outstanding properties in creep and stress rupture at high temperatures. These alloys contain roughly 50% by weight in nickel and many other alloying elements, each of which serve a specific role. These different alloys find places in the gas turbine engine depending on the individual requirements. The manufacturing processes involved are also tailored to meet these requirements; a high-pressure turbine blades may require to be cast in a single-crystal whereas a combustion chamber lining may be obtained from a rolled sheet. For the present work, the aim is focused on the coarse grained cast turbine blade alloy commercially known as Inconel 792.

2.2.1

Nickel-based superalloy Inconel 792

Nickel-based superalloys are typically classified in three main classes; γ’ strengthened, solid solution strengthened and oxide dispersion strengthened. The alloy on focus in this work, Inconel 792, belongs to the gamma-prime strengthened class. Its structure consists of two phases: a nickel matrix, γ, and coherent intermetallic compound γ’. This phase makes up approximately 50% in volume and it is a hard strengthening phase in a ductile matrix. The outstanding high temperature properties of the nickel matrix are attributed to the high tolerance for solutes and for the tendency to form oxides. A further reduction in oxidation and diffusion of intermetallic elements is achieved with the addition of Cr and Al. The main alloyants of Inconel 792 are included in Table 1.

The texture of this alloy is shown in Figure 1, with grain sizes ranging from 1 to 3 mm, depending on the manufacturing process of the material. A typical turbine blade casted with this material is shown in Figure 2 where it is possible to see the variation of grain sizes depending on the position (smaller grains at the leading edge and coarser in the fir tree bulk) .

2.2. ENGINE MATERIALS

0 mm 10

Figure 1: Detail of an etched cross section of an Inconel 792 bar used to manufacture test specimens.

Figure 2: Typical turbine blade geometry and a detail of the material texture on different locations of the turbine.

CHAPTER 2. TURBINE ENGINE CONDITIONS

2.3

Relevance of fatigue crack growth assessment in turbine

components

As discussed in Section 1.1, the strive to extend the turbine performance increases the already demanding loading of turbine components; higher turbine inlet temperatures, larger thermal gradients and more corrosive fuels among others. This also needs to be done without compromising the reliability and safety of a gas turbine engine. The designers involved in the process are stretching the boundaries of what the materials are capable to cope with. They can do so not only thanks to the improved control and monitoring systems, but also due to a better understanding of the material behaviour. Obviously, cracks are never desired on a turbine component; they are avoided. Nevertheless, despite all the efforts during the design and testing, sometimes they are unpreventable. Under such circumstances, it is important to be ahead of the problems and predict what the crack may do and how much time it is ahead of the component for a safe operation. This information is crucial to assist in managing and planning of maintenance operations without compromising availability and ultimately, taking sensible decisions regarding the end of life of a component.

Fatigue crack propagation

3

CHAPTER 3. FATIGUE CRACK PROPAGATION

3.1

Fatigue crack propagation

A crack present in a turbine component will tend to grow under cyclic loading. The existence of this crack may be derived from manufacturing processes or it may be initiated from cyclic fatigue. Under the right conditions, this crack can potentially grow to a critical length resulting in component failure. To avoid this situation, it is important to estimate the remaining life in a component after a crack has initiated. The Paris model for crack propagation [8, 9] (c.f. Eq. 1) is arguably the most used relation to predict crack growth rate,

da

dN = C∆K

n ₍₁₎

where C and m are experimentally obtained material constants and where ∆K is the operational stress intensity factor. This parameter is typically defined as

∆K = Kmax− Kmin. Even though ∆K is the driving force, this will be true for a

certain range of R (at low load ratio, Kmax becomes the relevant force and at high

load ratios ∆K is). For this reason, it is often assumed that negative values of the stress intensity factor have no impact on crack growth and the operational stress intensity factor is defined by [10] as,

∆KAST M= max(Kmax, 0) − max(Kmin, 0) (2)

It has been shown by many authors that experiments under negative load ratios show higher crack growth rates than predicted [11]. This may be attributed to the fact that a crack may be open even under negative loading, and in some instances, even throughout its entire cycle. This phenomenon may be attributed to residual strains, plasticity-induced crack closure or a combination of both. The plasticity-induced crack closure can be used to explain the effects of load ratio, R, on the crack growth rate behaviour by correcting the range ∆K for which the crack is actually open. The result of this correction is commonly known as the effective

stress intensity range, ∆Kef f (c.f. Eq. 3) and, in essence, it represents an intrinsic

material behaviour.

∆Kef f= Kmax− Kopen (3)

3.2

The crack closure phenomenon

Crack closure is a term coined by Elber [12, 13] in the early seventies when the tests he was studying showed an unexpected elastic compliance that he attributed to the crack faces remaining in contact at low tensile loads; in other words, the crack remained closed for a portion of the tensile loading. Assuming that a closed crack does not contribute to crack growth, the stress intensity range can be expressed 10

3.2. THE CRACK CLOSURE PHENOMENON

as the amplitude in which the crack is open. This is schematically represented in Figure 3. This phenomenon is useful to account for a wider range of fatigue crack propagation data and understanding the effects and influence of different stress ratios. Since then, many researchers have focused on the description of this phenomenon and additional mechanisms to explain premature crack-face contact were identified, although the general consensus is that plasticity-induced crack closure has the largest impact in fatigue crack growth for the materials and applications considered here [14]. time L oad time K
Ke  Kmax Kmin Kopen
K 0 0

Figure 3: Schematic representation of ∆Kef f

Plasticity-induced crack closure can be intuitively explained under plane stress conditions: the deformed region in the crack tip elongates plastically in the load direction. This strain is balanced by the reduction in thickness of the cross section, c.f. Figure 5-b. These elongated elements act as a wedge behind the crack tip (c.f. Figure 5-c) thereby reducing the fatigue crack growth rate. The localisation at the notch root, exaggerated in Figure 5, can be observed by naked eye on some tests, c.f. Figure 4. Dugdale [15], modelled this plastic wedge and based on that, a more formal description of the model was elaborated by Budiansky and Hutchinson [16]. Other strip yield models are still relevant to describe different load ratios [17] under plane stress conditions and in [18], for plane strain conditions. Another form of plasticity-induced crack closure is due to crack-tip blunting; during a prolonged hold time, creep strains reshape the crack and the tip becomes blunted. This blunting 11

CHAPTER 3. FATIGUE CRACK PROPAGATION

Figure 4: Example of a test specimen showing indications of in-plane localisation at the notch root.

requires the re-nucleation of a new crack tip thereby de-accelerating the crack growth during hold time. When the load is removed, the change in geometry due to inelastic strains at the crack tip will induce compressive or tensile stresses steaming from the elastic material surrounding the deformed material affecting the related stress intensity range ∆Kef f. This phenomenon has been studied in detail in [19, 20] and it is schematicaly illustrated in Figure 6.

Even though the overall understanding of crack closure is rather intuitive, it is difficult to measure experimentally. These measurements of crack closure often rely on variables that are an indirect consequence of the phenomenon itself and therefore require a set of assumptions to characterise crack closure in isolation. Besides, measurements at the small region around the crack tip (in which crack closure is relevant) are very difficult to obtain and require specialised experimental techniques, such as digital image correlation [21, 22]. Although providing close-up details of the evolution of strains at the crack tip, these sophisticated techniques cannot characterise crack closure through the thickness of the specimens and are, therefore, approximations [23]. A detailed discussion on the adopted experimental techniques to measure crack closure is included in Section 4.1.

With all the difficulties that an experimental measurement of crack closure entitles, researches have resourced to numerical simulations employing finite element methods to complement and facilitate the understanding of crack closure. Despite the greater control of a simulation over an experiment, the calculations involved are complex and with many parameters that may change the outcome of the problem. This renders difficult a meaningful comparison with experimental values. The necessary discretisation of the problem into elements makes it unfeasible to simulate the phenomenon explicitly, as in reality, the crack will propagate continuously 12

3.3. RELEVANCE OF CRACK CLOSURE AND RESIDUAL STRESS IN TURBINE BLADES APPLICATIONS

a) b) c)

Figure 5: Schematic representation of the material ”wedging” the crack, illustrating the concept of plasticity-induced crack closure.

(rather than in increments), over a large number of cycles. Despite all the difficulties, many researchers started early focusing on the problem of fatigue crack closure employing numerical simulations. The work from Newman, [17, 24] , McClung and Sehitoglu [25, 26], are among the pioneer researchers that lay the foundation of the important aspects of crack closure in a simulation, which are highly relevant to the present day. Some of these aspects will be discussed in Section 4.2.3.

3.3

Relevance of crack closure and residual stress in turbine

blades applications

Crack closure and residual stress effects in crack growth are tightly related as they share the underlying mechanism. Residual stresses are commonly understood as the consequence of initial plastic deformations in the uncracked body; plasticity-induced crack closure is commonly attributed to the plastic deformations occurring in front of the crack as it propagates through the material. In both cases, it is ultimately the inelastic deformations at the crack tip that induces a level of stress which prevents or facilitates the opening of the crack tip. Most of the components in a turbine engine will present some degree of residual stress due to manufacturing processes or loading history. Many features of a turbine blade act as stress raisers. In locations in which the loading is severe enough to risk fatigue crack initiation, it will undergo stress and strain redistribution during the first few starting cycles. In a turbine blade, the majority of the inelastic strains will occur before a crack is initiated. A growing crack will need to plough its way through this region with the related increase (or decrease) of the associated stress intensity factors. Turbine blade materials are not capable of undergoing cyclic inelasticity under thermomechanical fatigue and they must be designed to present, except for localized regions, a closed 13

CHAPTER 3. FATIGUE CRACK PROPAGATION

Incubation Crack growth

Time

ti

a0

Crack len

gth

Initial sharp crack, t=0

Blunting of the crack tip, t<ti

Nucleation of a new crack tip, t=ti Crack growth, t>ti

Figure 6: Schematic representation of crack tip blunting process during extended hold times.

hysteresis loop behaving elastically after the first start. On such locations, if the residual stresses are taken into account, it is possible, in principle, to use linear elastic fracture mechanics. If, on the other hand, the loading conditions are such that the residual stress changes over time, the whole loading history should be taken into consideration which in essence is equivalent to analyse plasticity-induced crack closure.

Experimental and computational methods

4

CHAPTER 4. EXPERIMENTAL AND COMPUTATIONAL METHODS

4.1

Experimental procedures

For the present work, the test campaign has been focused on a set of three relevant

temperatures: Room temperature, 750°C and 850°C tests under in-phase (IP)

loading (Tables 2, 3 and 4, respectively). The tests at elevated temperatures have been run both with and without hold times. The tests were performed at different load ratios and under stress and strain control, respectively.

Table 2: LCF crack propagation tests at room temperature.

Test ∆σ [MPa] R Temp. [°C]

LCF3921 295 0 RT LCF3941 735 -1.5 RT LCF3942 727 -1.5 RT Test ∆ R Temp. [°C] LCF3827 0.6% 0 RT LCF3830 0.6% 0.33 RT LCF3828 0.6% −∞ RT LCF3829 0.6% 3 RT LCF3910 0.6% 0 RT LCF3911 0.6% 0 RT

Table 3: TMF and LCF crack propagation tests at 750°C.

Test ∆ R Temp. [°C] Hold

TMF0202 0.6% 0 100-750, IP 5 min.

TMF0204 0.5% 0 100-750, IP 5 min.

Test ∆σ [MPa] R Temp. [°C] Hold

TMF3908 493 0 100-750, IP 5 min.

LCF4213 500 -1 750 no hold

All the tests have been performed for single-edge-notch specimens, and two different geometries have been used (c.f. Figure 8), which were machined from cast

round bars. The material was subjected to hot isostatic pressing at 1195°C and

150 MPa for 3 h. A two-step aging process is performed, first by a solution heat

treatment at 1121°C for 2 h and second, by aging at 845 °C for 24 h. For all the

experiments, the pre-cracking was achieved by isothermal-cycling at the maximum temperature until a Mode I crack, of at least 0.8mm measured from the notch root, was created. The testing was performed using a test setup identical to the one described in [27] using a 100kN servo-hydraulic machine, c.f Figure 7. The heating of the specimen was done by induction-heating and cooling was performed using compressed air from two nozzles. The heating and cooling were set to a

constant rate of 2°C/s. This temperature rate was controlled with a spot-welded

thermocouple, located in the centre of the specimen, on the opposite side of the 16

4.1. EXPERIMENTAL PROCEDURES

Table 4: TMF and LCF crack propagation tests at 850°C

Test ∆σ [MPa] R Temp. [°C] Hold

LCF00177 420 -1 850 0

TMF00151 360 -1 100-850 1 h

TMF3912 500 -1 100-850 1 h

TMF00182 440 -1 100-850 1 h

TMF4208(*) 430 -1 100-850 6 h

(*) - Test performed for the specimen geometry SEN-2

Figure 7: Testing rig

notch. The displacements were measured with an extensometer located on the same side as the notch, with a length of 12mm.

4.1.1

Experimental measurement of crack length

A challenging aspect of crack propagation testing is the accurate measurement of the crack length. Some of the most common methods are optical, potential drop and compliance-based. Perhaps the most basic form of crack length measurement is the optical method, in which a travelling microscope is used and the crack lengths are measured manually as the test progresses. The main disadvantage of this method is that the crack is measured only on the visible surfaces and therefore crack tunnelling cannot be accounted for and it is difficult to automate the process. Potential drop methods are capable of measuring crack lengths in real-time and are therefore very attractive for time-dependent crack growth tests. Unfortunately, this method

CHAPTER 4. EXPERIMENTAL AND COMPUTATIONAL METHODS 12 Diam. Ø15 (2x) R25 ± 0.25 (4x) 72 ± 0. 1 64.5 ± 0.1 R25 ± 0. 25 (4x) 144 ± 0.1 0.25 ± 0.02 15 ± 0.05 1 ± 0.05 B B 6 ± 0.05 11 ± 0.05 A A B A A a) b)

Figure 8: Tests specimens used in the test campaign: a) SEN-1 and b) SEN-2

requires special rigs that are difficult to use with induction-heating techniques and regardless, under TMF conditions, the necessary parameters are difficult to calibrate to obtain accurate measurements.

In the tests performed in this work a combination of optical and compliance-based methods were used to measure crack lengths. This method has been thoroughly described in e.g. [28, 29]. The method was validated by comparing with optical measurements and heat tints, as shown in Figure 9.

In a nutshell, the compliance-based method to estimate crack lengths uses the stiffness drop due to a growing crack. This is characterised beforehand with the aid of a 3D crack propagation software [30] under linear-elastic conditions. With the normalized characterisation of the stiffness drop as a function of crack length, it is possible to post-process the experiments and thereby obtain a crack length measurement based on that calibrated curve. The experimental estimations of stiffness are done during the unloading ramp, where linear-elastic conditions and a fully-open crack are found. These measurements are taken always at the same points with respect to the cycle (at the same temperature). With this, the changes of young modulus do not require to be accounted for, nor does any other material parameter; only the specimens initial uncracked stiffness is required. It is worth mentioning that this method accounts for an average crack length measurement, c.f. Figure 9. A major drawback of this method is the difficulty to measure the crack

4.1. EXPERIMENTAL PROCEDURES

Figure 9: Illustration of the verification of the compliance method by optical measurement of the heat tints and side scrives.

growth as a function of time; since the crack is estimated during the unloading, and therefore the measurement is as a function of cycles.

4.1.2

Experimental measurement of crack closure

There are several possibilities to measure crack closure experimentally. The two most common techniques involve either the measurement of global displacements or the measurement of strains near the crack tip. In the former methodology, the crack closure measurement is taken based on the relative change of compliance of the specimen; in the latter, it is inferred from the strains at the crack tip employing laser interferometry or digital image correlation. A review and comparison of several experimental methods to estimate crack closure is included in [31, 32], conclud-ing that compliance-based methods, albeit providconclud-ing less accuracy than surface measurements, are generally the recommended choice for most situations. The compliance-based methods are favoured due to its relative simplicity: the methodol-ogy can be applied with the post-process of the gathered displacements during the test without the need for further equipment. These displacement measurements,

CHAPTER 4. EXPERIMENTAL AND COMPUTATIONAL METHODS

which may be taken at the crack mouth, at the strain clip-gauge, or on the back of the specimen, consider a global crack closure throughout the entire thickness of the specimen. Even though these measurements, relatively far from the crack-tip, may appear less sensitive than surface methods, the former presents the advantage of providing an averaged value of crack closure through the thickness, whilst it is a pure surface measurement in the latter. The work carried out by Palmer et al, [27] provides an experimental methodology based on relative changes of compliance specifically tailored for the conditions in a TMF test. In that work, the authors also show the importance of accounting for crack closure to explain crack growth for a similar alloy as the one discussed here. Their experimental methodology has been compared to estimations of crack closure using digital image correlation techniques in [21]. For this work, the same methodology has been adopted to evaluate crack closure experimentally, which will be summarised in the following paragraphs. As in all the compliance-based techniques, the crack opening load is estimated by the relative changes in the stiffness of the specimen. The opening load is found at the inflexion point on a stress-strain plot as shown in Figure 10. This method takes into account the variation in elastic modulus as a function of temperature. Linear-elastic assumptions are only necessary for the part of the loading ramp where the crack is closed, as well as the start of the unloading when the crack is fully open. The stiffness of the specimen is calculated using the displacement measured by the extensometer pins located 6 mm away from the notch root, accounting only for the global apparent crack opening rather than near the crack-tip. With this in

consideration, the estimated opening stress, σopenobtained experimentally needs to

be understood as a thickness-averaged value.

The degree of crack opening, Di, is obtained by the ratio of the relative stiffness

drop of the specimen during the loading cycle i, (βi,load), with the relative stiffness

drop for the fully open crack during the unloading cycle i − 1, (βi−1,unload) as

described by Equation 4 . Di(τ ) = βi,load(τ ) βi−1,unload = 1 − Mi,load(τ ) M0,load(Ti(τ )) 1 −Mi−1,unload M0,unload (4)

where τ is a dimensionless time variable defined as,

τ =t − ti

tramp (5)

The denominator in Eq. 4 can be interpreted as the relative stiffness drop in cycle i, evaluated for a specific point on the unloading ramp. The numerator is the relative stiffness drop in cycle i, evaluated continuously throughout the loading ramp (from minimum to maximum load). The experimental evaluation does not require any material input other than the characterisation of the specimen’s uncracked stiffness (M0,unload) and the uncracked stiffness as a function of the 20

4.1. EXPERIMENTAL PROCEDURES 0.006 0.008 0.01 0.012 0.014 0.016 0.018 Mechanical Strain [mm/mm] -250 -200 -150 -100 -50 0 50 100 150 200 250 Str ess [MP a]

Points for secand evaluation of Mload

Data Crack opening (D=0.9) Munload -250 -200 -150 -100 -50 0 50 100 150 200 Stress [MPa] 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Degree of opening evolution Opening point Degree of opening threshold

Degr

ee of crac

k openin

g

Figure 10: Illustration of the crack opening identification by means of relative compliance changes

temperature, M0,load(Ti(τ )). Note that the specimen’s uncracked stiffness, M0,unload

and the unloading stiffness,Mi−1,unload, are evaluated at the same point through the cycle and therefore the variation in elastic modulus has no influence on the evaluation as these two points are at the same temperature. An illustration of the

typical evaluation of Mload and Munloadis included in Figure 10.

The value of D ranges from 0 to 1, corresponding to a fully closed and a fully open crack, respectively. D is evaluated for the entire loading ramp although it is during the lower part of the loading and up until the crack is fully open that it is considered valid, based on the assumption that the specimen has an elastic response up until that point. After that, the inelastic response may change the meaning of the degree of opening and it may no longer be a useful measure for that purpose. The evolution of the crack opening parameter D is illustrated in 10.

Due to the measurement uncertainties and the inherent noise in the signals, it is often necessary to relax the set value of D when the crack is considered fully open. As mentioned before, this point should be set to D = 1 for a fully opened crack, but in the experimental evaluation this criterion is conveniently set to D = 0.9, meaning that the true effective stress intensity range is slightly overestimated for the experimental evaluation. The selection of an appropriate value of D for the crack opening criterion is a trade-off between reducing scatter in the determined crack opening force and minimizing the deviation from the force corresponding to a complete crack opening. An important aspect of this trade-off is the crack length limit above which crack opening evaluation is possible: Since the scatter of the determined crack opening force decreases with increasing crack length, it is possible to evaluate tests with large cracks using a crack opening criterion higher than 0.9. On the other hand, this would narrow the outcome of the testing.

CHAPTER 4. EXPERIMENTAL AND COMPUTATIONAL METHODS

4.2

Numerical methods for crack closure

Many authors have resourced to finite element methods to study crack closure effects in fatigue crack growth, [33–40]. These techniques have been proven very useful for researchers to examine different parameters and for analysing their influence on crack closure and crack growth independently. A crack growth finite element analysis including crack closure is challenging and many aspects may affect the outcome. McClung et. al. provided an early systematic study of the important parameters to consider when resourcing to numerical simulations [25, 26]. There are two key issues to crack propagation including crack closure: time and geometry discretisation. The former is required due to the large amount of cycles involved in a real crack propagation test; accounting for all the cycles is computationally prohibitive and some assumptions are necessary for this reason. The geometry discretisation relates to the element size, particularly at the region of interest. Again, responding to computational limitations, the number of elements is a trade-off between accuracy and computational cost. In addition, the selection of the element size generally dictates the minimum crack growth increment. Early finite element models featured plane strain formulation with springs as means of contact and elastic perfecly-plastic constitutive material models. Nowadays, the improvements in computational performance have allowed the addition of more sophisticated constitutive models, refined meshes and, in some instances, simulations in 3D. The essence of the calculations and their inherent challenges are, nevertheless, much the same.

There are still many unknown influencing parameters, and for most of the cases, the requirements of each individual problem need to be assessed in a case-by-case manner. In the following sections some of the key parameters are summarised.

4.2.1

Mesh size

Crack closure is based on the plasticity around the crack tip and flanks. It is therefore important to capture the plastic zone around the crack tip as accurately as possible. McClung recomend to set the minimum element size as a function of the Dugdale’s plastic zone [15] defined by,

rpD= π 8  KI σi 2 (6) The recommendation from McClung in terms of element size was set to accom-modate 10-20 elements within the plastic zone with the aim to achieve an accurate representation of the reversed plastic zone [26]. This is a general recommendation that is being followed to this date, despite the advancements in computational power. In [41, 42], representing more modern publications devoted to the study of the effects and requirements of mesh size with modern processing,the author suggest smaller elements by orders of magnitude than the ones suggested by McClung. Without overlooking the importance of mesh size, the results from the simulations gathered up until the publication of this work suggest that the recommendations 22

4.2. NUMERICAL METHODS FOR CRACK CLOSURE

Figure 11: Detail of a mesh suitable for estimations of crack closure in FE-simulations. The mesh is progressively refined keeping quadrilateral elements.

from McClung suffice to accurately depict, qualitatively and quantitatively, crack closure for this material and test conditions, based on the experimental back-to-back comparisons. A typical mesh size detail is included in Figure 11 for 1st order, plane

stress elements. The refinement around the area of interest is of 50µm, whereas in

the bulk of the region is about 100 times larger than that.

4.2.2

Time discretisation

As mentioned before, it is not feasible to reproduce explicitly a fatigue crack growth experiment in a simulation; it is necessary to carry out a predetermined number of cycles after a crack increment has been imposed. This might have a larger impact depending on the material constitutive model, and the general consensus is to cycle until a stabilized hysteresis loop is achieved. Another important parameter is the time-resolution in the course of the loading ramp, during which, crack closure is to be monitored. Some authors have used interpolation schemes to find the load at which the crack opens. In this work, a fine time-resolution has been used instead. However, in order to optimise the computational cost, only certain increments of crack growth have been monitored. This enables a fast computation without compromising time resolution.

4.2.3

Numerical measurement of crack closure

An additional aspect of numerical assessment of crack closure simulations is to determine the instant at which the crack is considered to be open. For the case of experimental estimation, there are just a few practical ways to do so, namely compliance-based and surface-based. In numerical simulations, there are a few more additional options that some researchers have chosen responding to different reasons. Perhaps the most widespread criteria is based on displacements. In this, at the instant that the first node behind the crack tip has a relative distance from the reference surface greater than 0, the crack is considered open [25]. Another criteria commonly used is stress-based. In this, the crack is considered open at the instant

CHAPTER 4. EXPERIMENTAL AND COMPUTATIONAL METHODS

that the stress at the crack tip element changes direction (from compressive to tensile) [43, 44]. Different variations of these two are used where the parameters are taken further away from the crack tip. It is important to notice that different methods are more mesh-sensitive than others. For a relatively coarse mesh, it is expected to obtain different opening loads when different opening criteria are used and as the mesh is refined, the different criterion should present marginal differences between each other. In this work, a compliance-based and a tip-displacement criterion have been used. The former method was meant to verify the experimental results and the latter to refine the crack closure assessment by looking closer at the crack tip.

Outlook

5

The main overall target of this project is to be able to predict crack growth in turbine components. This components are subjected to high temperature and high loads for long periods of time. It is therefore important to be able to account for any time dependencies that can influence crack propagation. The work performed so far has shown that crack closure can explain, to some extent, crack growth behaviour including time-dependence in crack growth under prolonged hold times. Nevertheless, the results for tests with extensive creep also show that there is still some questions to address. The next steps for this project will be devoted to inquire creep crack growth. A new test campaign is already set on place with the aim to assist this research and others to provide the right experimental results to aid the research process.

Under cases in which creep ductile behaviour is observed, the physical basis for correlating crack-driving forces based on linear-elastic fracture mechanics, namely

∆K, ∆Kef f, might be questionable. In such cases, an appropriate crack driving

force needs to be established. C∗ is a suitable candidate to characterise crack

growth under steady state creep. This parameter accounts for long term creep conditions, with slow growing cracks to enable the development of global steady state creep. Other parameters based on that have been developed over the years to

account for short-term effects, namely C(t) and Ct. The use of these parameters in

practical applications is not straight forward and requires a careful attention and sophisticated simulations. For this reasons, linear-elastic fracture mechanics has been preferred for most of the industrial applications, even for cases where that assumption would be questionable. It would be interesting to address this issue and investigate the feasible limits of stress intensity approaches as crack driving force

compared to C∗_{, hopefully providing a comprehensive methodology to assess lifing}

of components, including time dependencies.

Review of Appended Papers

6

Paper I

Accounting for initial plastic deformation for fatigue crack growth

predictions under TMF loading condition

In this paper, the authors investigate the importance of accounting for initial plastic deformation in order to explain crack growth behaviour. In general, crack closure has a negligible impact on cases with a loading ratio above 0.7. In such cases, the impact of residual stress, arising from initial plastic deformations, can explain crack growth behaviour. The paper includes a description of a computer-aided methodology to be able to apply this to industrial components under thermo-mechanical fatigue conditions. The methodology is also compared to tests at different temperatures and different loading ratios. Taking into account residual stresses is of particular importance under strain controlled test, in which the derivation of the stress intensity factors based on the external loading alone is not completely meaningful to characterise crack growth.

Paper II

Accounting for crack closure effects in TMF crack growth tests with

extended hold times in gas turbine blade alloys

Following the research line in this project, the next paper is the logical continuation of the residual stress topic. In this paper, the effects of crack closure are investigated, numerically and experimentally, under TMF conditions at high temperature and extended hold times. The experiments are used to validate a numerical model, which in turn was used to extend the precision of crack closure estimations from the experiments. The results show how crack closure can explain fatigue crack propagation even under creep conditions.

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Part II

Papers

The papers associated with this thesis have been removed for

copyright reasons. For more details about these see:

FACULTY OF SCIENCE AND ENGINEERING

Linköping Studies in Science and Technology, Licentiate Thesis No. 1885, 2020

Solid Mechanics, Department of Management and Engineering Linköping University

SE-581 83 Linköping, Sweden