Accounting for non-linear effects in fatigue crack propagation simulations using FRANC3D

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Accounting for non-linear effects in fatigue

crack propagation simulations using

FRANC3D

Christian Busse

Division of Solid Mechanics

Master Thesis

Department of Management and Engineering

LIU-IEI-TEK-A- -14/01916- -SE

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Accounting for non-linear effects in fatigue

crack propagation simulations using

FRANC3D

Master Thesis in Mechanical Engineering Department of Management and Engineering

Division of Solid Mechanics

Linköping University

Christian Busse

LIU-IEI-TEK-A- -14/01916- -SE

Supervisor Siemens

David Gustafsson

Combustor Mechanical Integrity Siemens Industrial Turbomachinery

Supervisor LiU

Kjell Simonsson

Division of Solid Mechanics Linköping University

Examiner

Daniel Leidermark

Division of Solid Mechanics Linköping University

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Christian Busse:

Accounting for non-linear effects in fatigue crack propagation simulations using FRANC3D

Master Thesis in Mechanical Engineering Linköping University

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Abstract

In this thesis methods to account for non-linear effects in fatigue crack propagation simulations using FRANC3D are evaluated. FRANC3D is a crack growth software that supports automated crack growth in the FE mesh using the power of an external FE code.

Introductorily, a theoretical base in fracture mechanics, especially regarding crack propagation models is established. Furthermore, the functionality of FRANC3D is shown for several different applications.

As a benchmark for the investigated methods the associated results are compared to data from laboratory tests. The conditions in the test are closely modeled, but with relevant simplifications. The cyclic life-times are calculated using Paris’ law incorporating the stress intensity factors computed by FRANC3D and with material parameters derived from a different set of experiments than those simulated. When comparing the calculated cyclic life-time with the test data it can be seen that the pure linear elastic simulation, for this particular test set-up, gives nearly as good results as the investigated approaches that account for non-linear effects.

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Acknowledgements

I would like to thank my supervisor Dr. David Gustafsson for his committed help and guidance in all situations during my work at SIEMENS in Finspång. The same is for my colleagues MSc. Patrik Rasmusson and Dr. Björn Sjödin who always supported me with valuable input that made me develop my skills and gave me a better understand-ing about fracture mechanics. Furthermore, I would like to thank my supervisor at the University Professor Kjell Simonsson for all his helpful feedback.

Finally I would like to thank my girlfriend, family and friends for all their support.

Christian Busse Linköping, May 2014

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Contents

List of Figures XI

List of Tables XIII

Nomenclature XV

1 Introduction 1

1.1 Background . . . 1

1.2 Fatigue . . . 2

1.3 FRANC3D . . . 3

1.4 Aims of the thesis . . . 4

1.5 Scope of work . . . 5

2 Fundamentals 6 2.1 Stress concentrations . . . 6

2.2 Stress intensity factors . . . 7

2.3 Fatigue life analysis . . . 8

3 Testing and Material 10 3.1 Material . . . 10 3.2 Testing . . . 11 3.2.1 Conditions . . . 11 3.2.2 Test data . . . 12 4 Preparation work 13 4.1 Model . . . 13 4.2 Functionality of FRANC3D . . . 15

4.2.1 Theoretical background to FRANC3D’s automatic crack growth function . . . 16

4.2.2 Handbook solution . . . 18

4.2.3 Growth of Edge, Center and Through Cracks . . . 20

4.2.4 Influence of increment length on accuracy . . . 22

4.2.5 Determination of the shape function . . . 23 4.2.6 Automated crack growth analysis applied on the test specimen . 26

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5.1 Settings . . . 28

5.2 Accounting for nonlinear effects in FRANC3D . . . 28

5.2.1 Linear elastic analysis . . . 29

5.2.2 Case 1 . . . 30

5.2.3 Case 2 . . . 33

5.3 Cyclic life-time analysis . . . 36

6 Results 37 6.1 Results of the accounting for nonlinear effects in FRANC3D . . . 38

6.1.1 Comparison between stress and strain based loadings . . . 38

6.1.2 Different applied strains . . . 39

6.2 Cyclic life-time analysis . . . 42

7 Discussion 44 7.1 Result discussion . . . 44

7.1.1 Accounting for nonlinear effects . . . 44

7.1.2 Cyclic life time and comparison to test data . . . 46

7.1.3 Sources of error . . . 46

7.1.3.1 Material and Testing . . . 46

7.1.3.2 Computations . . . 47

7.2 Method discussion . . . 47

8 Conclusion and future work 49 8.1 Conclusion . . . 49 8.2 Future work . . . 50 References 51 A Mesh study 53 B Figures 55 C Boundary Conditions 56 D ABAQUS sub-routine 57 X

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List of Figures

1.1 Cross-section of the gas turbine SGT-700. Courtesy of Siemens . . . 1

1.2 General procedure FRANC3D . . . 4

2.1 Example for stress concentration . . . 6

2.2 The three modes of loading that can be applied to a crack . . . 7

2.3 Crack growth rate over SIF-range in log scale for a certain R value . . . 8

3.1 Notched test specimen . . . 11

3.2 Applied load and strain as a function of number of load cycles . . . 12

4.1 CAD model of the specimen . . . 14

4.2 Final mesh . . . 15

4.3 Coordinate system at crack tip . . . 16

4.4 Mesh elements around crack tip, with courtesy of FAC . . . 18

4.5 Handbook solution. Geometry of a simple plate . . . 18

4.6 Comparison of stress intensity factors . . . 19

4.7 Position of the initial cracks indicated by the predefined crack template 20 4.8 SIFs for different initial crack positions . . . 21

4.9 Evolution of the crack fronts . . . 21

4.10 Comparison of SIFs for different crack growth increment lengths . . . . 22

4.11 Shape function for the given geometry . . . 24

4.12 Comparison of SIF between test data using the shape function and FRANC3D for ε = 0, 6 % . . . 24

4.13 Comparison of SIF between test data using the shape function and FRANC3D for ε = 0, 7 % . . . 25

4.14 Comparison of SIF between test data using the shape function and FRANC3D for ε = 0, 8 % . . . 25

4.15 Comparison of SIFs of handbook solutions . . . 27

5.1 Superposition for crack face traction . . . 29

5.2 Workflow - linear-elastic computation . . . 30

5.3 Normalized loading over time for Case 1 . . . 31

5.4 Hysteresis loop for the uncracked geometry for Case 1 . . . 31

5.5 Workflow - Case1 . . . 32

5.6 Normalized loading over time for Case 2 . . . 33

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5.8 (a) crack in virgin material, (b) crack blunting and plastic zone forma-tion from applied tensile load, (c) compressive residual stresses at the

crack tip after unloading . . . 35

5.9 Workflow - Case2 . . . 35

6.1 Notch plane . . . 37

6.2 Stress controlled: Comparison of KI(a) of Case 1, Case 2 and a linear elastic computation for σ = 145MPa . . . 38

6.3 Strain controlled: Comparison of KI(a) of Case 1, Case 2 and a linear elastic computation for ε = 0, 65 % . . . 39

6.4 ε = 0.6 %: SIFs over crack length . . . 40

6.5 ε = 0.7 %: SIFs over crack length . . . 40

6.6 ε = 0.8 %: SIFs over crack length . . . 41

6.7 Difference in SIFs between Case 2 and the linear elastic case for all tests 41 6.8 ε = 0.6 %: crack length over loading cycles . . . 42

6.9 ε = 0.7 %: crack length over loading cycles . . . 43

6.10 ε = 0.8 %: crack length over loading cycles . . . 43

7.1 Plastic zone around the crack tip . . . 48

A.1 Mesh study: Maximum strain . . . 54

A.2 Mesh study: Minimum strain . . . 54

B.1 Composition of the local(left) and global(right) sub-model . . . 55

C.1 Applied boundary conditions . . . 56

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List of Tables

3.1 Composition of Inconel 718 . . . 10

3.2 Material properties of Inconel 718 . . . 10

3.3 Properties of the tests with ∆εnom and Rε prescribed . . . 12

5.1 Overview of computations . . . 28

6.1 Ratios of average nominal stress at notch plane and yield stress for the uncracked specimen . . . 38

8.1 Summary . . . 49

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Nomenclature

Roman Symbols

Kf Fatigue notch factor [-]

Kt Stress concentration factor [-]

KIC Fracture toughness [P a√m]

KIII Stress intensity factor mode III [P a√m]

KII Stress intensity factor mode II [P a√m]

KI Stress intensity factor mode I [P a√m]

Rε Strain ratio [-]

b Material property for Coffin Manson [-]

C Material property for Coffin Manson [-]

c Material property for Coffin Manson [-]

E Young’s modulus [P a]

f Shape function mode I [-]

F Force [N]

g Shape function mode II [-]

h Shape function mode III [-]

N Number of loading cycles [-]

n Exponent in Paris’ law [-]

Greek Symbols

ν Poisson ratio [-]

ρ Density [kg/m3]

σmax Maximum stress [P a]

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σnom Nominal stress [P a]

σyeng Engineering stress [P a]

σyreal Real stress [P a]

σyy Remote stress for mode I [P a]

τyx Remote stress for mode II [P a]

τyz Remote stress for mode III [P a]

θ Kink angle [ εa Strain amplitude [-] Abbreviations Al Aluminium C Carbon Cr Chrome

EPFM Elasto-Plastic Fracture Mechanics

FAC Fracture Analysis Consultants Inc.

FCC structure Face Centered Cubic structure

FE Finite Elemente

Fe Iron

FRANC3D Fracture Analysis Code 3D

HCF-test High-Cycle-Fatigue tests

LEFM Linear Elastic Fracture Mechanics

Mo Molybdenum

Nb Niobium

Ni Nickel

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SIF Stress Intensity Factor

SIT AB Siemens Industrial Turbomachinery AB

Ti Titanium

TMF-tests Thermo-Mechanical-Fatigue-tests

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1

Introduction

1.1

Background

Siemens Industrial Turbomachinery AB (SIT AB), develops and manufactures gas as well as steam turbines. Their steam turbines generate power in a range of 60 and 250 MW and the gas turbines between 15 and 60 MW. Turbines generate a rotational work that can be used in different ways, e.g. generating electric power or, as in aircraft turbines thrust. The investigations performed in this thesis are mainly relevant for gas turbines. In general the gas turbine can be divided into three parts, a compressor, a combustor and a turbine stage, cf. Figure 1.1. The service temperatures in the turbines are high enough to limit the use of steel due to creep and oxidation. Therefore materials like Nickel-base superalloys are often used in gas turbines.

Figure 1.1: Cross-section of the gas turbine SGT-700. Courtesy of Siemens The degree of efficiency of gas turbines is highly dependent on their service temper-ature. Therefore manufacturers always strive towards a design which enables as high temperatures as possible. The use of high temperatures requires a great material re-sistance against creep, oxidation and fatigue. There are two ways to ensure a safe use of gas turbine components. Firstly, it can run at lower temperatures incorporat-ing a high safety margin to the limits of its components. Secondly, the maintenance intervals can be held shorter to ensure that flaws are detected before they can cause failures. Since both options have obvious disadvantages, i.e. lower performance at lower temperature and higher cost due to more frequent maintenances, a main focus of research is the development and testing of high temperature resistant materials and

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2 1 INTRODUCTION reliable life-time prediction. Increasing the life time of the components and increasing the temperature, i.e. higher efficiency in the generation of power has also advantages regarding the aspects of environment and a sustainable society. Due to the longer life time, less parts have to be exchanged which results in less waste. The higher efficiency makes it possible to generate the same amount of power with fewer turbines and less fuel.

No questions regarding ethics, morals or gender equality are raised by this thesis.

1.2

Fatigue

Fatigue is the limiting factor in high temperature components which are exposed to cyclic loadings. In the high temperature parts of the gas turbines the predominant kinds of fatigue are Low-Cycle Fatigue (LCF) and Thermo-Mechanical Fatigue (TMF). That means that the cyclic life-time of the components is relatively short with some thousand cycles in comparison to High-Cycle Fatigue (HCF) where several hundred thousand loading cycles can be endured. This thesis investigates LCF, while TMF and HCF are not considered.

In complex geometries, there are frequently certain regions where stress concentrations appear, e.g. notches. At these stress concentrations, plastic flow in the material can occur at loading levels where in the rest of the component only elastic deformations take place. In handbooks, stress concentrations are incorporated in fatigue

calcula-tions via a fatigue notch factor Kf. Plastic deformation is the driving force of crack

initiation. Therefore, it is more accurate to use the strain amplitude instead of the

stress amplitude for calculation of the fatigue life time. The strain amplitude εa can

be determined by a method suggested by Neuber [1]. The most common relation used to calculate the fatigue life time was proposed by Coffin and Manson [2], see below:

εa = C1Nb+ C2Nc (1.1)

In Equation 1.1 parameters C1, b, C2 and c are material parameters. The first part

of the sum may be interpreted as the elastic strain and the second part as the plastic strain. In order to predict the crack growth, certain fatigue crack growth models have been developed. A rather simple but widely used method is the crack growth model called Paris’ law proposed by P. C. Paris [3].

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1.3 FRANC3D 3

da

dN = C(∆K)

n (1.2)

It states a power law relationship between the Stress Intensity Factor-range (SIF-range) and the crack growth rate, see Equation 1.2, where C and n are material constants, N is the number of loading cycles, and a is the crack length. This model is mainly used in this thesis, and is discussed in more detail in Sub-section 2.3. It may be noted that there are crack growth models that account for the mean stress effects like the Walker [4] or the Forman [5] model.

Modern approaches to analyze fatigue crack growth and the fatigue life of components are based on computer programs like e.g. NASGRO [6]. This thesis deals with the accounting for non-linear effects in fatigue crack propagation simulations by using the program FRANC3D [7], see below.

1.3

FRANC3D

FRANC3D is a software that supports automated crack growth in the FE mesh using the power of an external FE code. In FRANC3D the user can extract a sub-model from the complete model and insert cracks from prepared templates, where only crack size and shape have to be input. FRANC3D is used for meshing the corresponding part and executing the external analysis program, which in this work is ABAQUS [8]. After calculating the fracture parameters, FRANC3D is used for expanding the crack. Finally, FRANC3D is used to compute the SIF history and the fatigue life [9]. The general procedure can be seen in Figure 1.2. As illustrated, ABAQUS is used to build the global model and to define the sub-model. The uncracked geometry is exported to FRANC3D, where the user inserts a crack from a template. In FRANC3D the cracked geometry is remeshed and sent to ABAQUS for a stress analysis. Afterwards, the results are sent back to FRANC3D for computing the SIF and extending the crack. Then the crack is extended again and the model is remeshed and sent back to ABAQUS for a stress analysis. This loop continues until the user-specified number of crack growth steps is reached. Finally, the user can determine the cyclic life time using Paris’ law in a post-processing step as will be explained later.

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4 1 INTRODUCTION

Figure 1.2: General procedure FRANC3D

1.4

Aims of the thesis

Geometry induced stress concentrations can cause complex stress allocations, implying non-linear effects such as plastic flow. Since these can considerably affect the fatigue life time of a component they have to be taken into account in the fatigue analysis. Accounting for plasticity in FRANC3D is the main goal of this thesis. Furthermore, the influence of the incorporated effects on the crack propagation and the fatigue life time is to be assessed. In order to validate the results, a comparison to a series of material tests is performed and also handbook solutions were used as a complement where possible. The considered test specimens were made of the material Inconel 718, which is a Nickel-base superalloy. Linear Elastic Fracture Mechanics (LEFM) does not account for global non-linear phenomena and therefore can limit the accuracy of the fatigue life time prediction. Since FRANC3D does not support the involvement of non-linear material effects, i.e. Elasto-Plastic Fracture Mechanics (EPFM), there is a need to develop methods to account for those in a more realistic way. Conclusively, results closer to reality will hopefully lead to better prediction of the cyclic life time. This thesis proposes methods to account for the stress concentrations, i.e. the plastic flow at local regions, and to incorporate that effect into the automatic crack growth analysis of FRANC3D. Different approaches are evaluated and compared with linear-elastic computations. Finally their influence on the predicted cyclic life time is dis-cussed.

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1.5 Scope of work 5

1.5

Scope of work

In this section the framework of this thesis is explained and the necessary steps in order to achieve the final conclusions are presented. In order to improve the understanding of the thesis there is a first section where some needed mechanical fundamentals are described. After that the testing procedure and the material of the test specimens are presented. Before starting the main part of the thesis, the preparation work is shown, including the model setup and several examples of the functionality of FRANC3D. The main part starts with a methodology section describing the adopted methods. Following, the results are presented and discussed. Finally, the methods are discussed and final conclusions are drawn. The thesis ends with an outlook on future work.

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2

Fundamentals

In order to understand some theory that appear in this thesis, a certain basic knowledge concerning Solid Mechanics and in particular Fracture Mechanics is required. Some of the important topics are covered briefly in the following sub-sections.

2.1

Stress concentrations

Stress concentrations will appear where the geometry of the component shows an irregular shape that contains stress raisers, e.g. notches or cracks. All geometric features that make a component experience a localized increase in the stress field are considered as stress concentrators. Generally, fatigue cracks initiate at these regions.

In order to quantify the stress raisers, stress concentration factors Kt are calculated

at those regions in the following way:

Kt = σmax/σnom (2.1)

where σmax is the maximum stress at the stress concentration and σnom is the nominal

stress at the stress concentration. The notch factor for the specimen used in this

work is approximately Kt = 3.7.An example of a stress concentration can be seen in

Figure 2.1. Due to the decrease of cross section at the hole in the center of the plate the stresses at the edges are higher. This is visualized by the lines marked in red. For further informations and examples, see [10].

Figure 2.1: Example for stress concentration[11]

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2.2 Stress intensity factors 7

2.2

Stress intensity factors

The SIF gives a measure of the magnitude of the stress singularity at the crack tip. SIFs

are expressed for three modes: KI, KII and KIII, which correspond to the different

modes of loading a crack can be exposed to, cf. Figure 2.2. Mode I corresponds to opening of the crack, Mode II to in-plane shearing and Mode III to out-of-plane shearing. Mathematically they are expressed by:

KI = σyy√πaf (2.2)

KII = τyx√πag (2.3)

KIII = τyz √

πah (2.4)

where a is the crack length, and where f, g, and h are functions of the geometry (shape functions) and the type of loading [1]. All mentioned stresses are remote stresses. In this thesis the predominant load case is Mode I. Mode II and III are neglected due to their vanishingly small influence on the results. The definition of the SIFs is only relevant in LEFM.

Figure 2.2: The three modes of loading that can be applied to a crack

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8 2 FUNDAMENTALS

2.3

Fatigue life analysis

In order to calculate the cyclic life-time when knowing K as a function of the crack length there are several different approaches. A rather simple but widely used method is the so called Paris’ law proposed by P. C. Paris, see above Equation 1.2.

Figure 2.3: Crack growth rate over SIF-range in log scale for a certain R value The crack growth rate may generally be divided into three regions, cf. Figure 2.3, where the regions I and III are ignored in Equation 1.2. Although neglecting regions I and III, Paris’ law is widely used since in most applications most time is spent in region II. It is only valid for the R-values used in the test by which Paris’ law has been calibrated. As shown in Figure 2.3, the curve shifts left with increasing R-values. R is defined as the ratio of the minimal over the maximal loading.

The SIF-range ∆K according to [1] is defined as follows:

Kmin ≥ 0 ⇒ ∆K = Kmax− Kmin (2.5)

Kmin < 0 ⇒ ∆K = Kmax (2.6)

Kmax < 0 ⇒ ∆K = 0 (2.7)

The actual effective SIF ∆Kef f can be defined as: ∆Kef f = Kmax−Kopen, where Kopen

is the SIF at the time when the crack opens, which is not necessarily zero, but can Accounting for non-linear effects in fatigue crack propagation simulations using

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2.3 Fatigue life analysis 9

be greater than zero. Using this Kopen, the effective SIF can decrease in comparison

to the ∆K from Equation 2.5, which can result in a lower crack growth rate da/dN. However, this issue has not been considered in this work.

By integrating the Equations 1.2 and 2.2 the number of loading cycles it takes for a crack to grow to a certain length can be determined, see Equation 2.8.

N = 1 C(∆σyy√π)n ˆ af inal ainitial da {√af (a)}n (2.8)

This relation is used in the later sections of the thesis in order to calculate the cyclic

life-time. It is commonly assumed that a part fails when Kmax reaches the critical

value of Kc, the fracture toughness, which is a material property.

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3

Testing and Material

In this section the material under study, the testing procedure and the evaluation of the test results are explained. At first Nickel-base superalloys in general and in particular the considered material Inconel 718 are presented.

3.1

Material

Superalloys generally have good mechanical properties such as long-time strength and toughness at high temperatures and this is the reason why they are often used in gas turbines. Modern Nickel-base superalloys exhibit a complex alloy composition as well as an intricate phase chemistry and structure [12]. They consist of various phases such

as gamma γ, gamma prime γ′

, gamma double prime γ′′

, delta δ and several other carbides and borides, cf [13]. An additional reason for the good temperature behavior of Nickel is the FCC structure that makes it both ductile and tough. The specimen investigated in this report consists of Inconel 718 which has an alloy composition as shown in Table 3.1 [14].

Table 3.1: Composition of Inconel 718

Element Ni Cr Mo Nb Al Ti Fe C

Weight% balance 19.0 3.0 5.1 0.5 0.9 18.5 0.04

Inconel 718 has several good mechanical properties like a high yield limit, cf. Table 3.2. The data is taken from an internal SIT report [15] except for the Paris’ law parameters

C and n which are taken from the work conducted by Månsson et al. [16]. Due to the

compromise of the good mechanical properties and its relatively low cost, it is the most frequently used Nickel-base superalloy [12].

Table 3.2: Material properties of Inconel 718

Property ρ ν E σyeng σyreal KIC C n

Unit [kg/m3

] [/] [P a] [P a] [P a] [P a√m] [m/(P a√m)n] [/]

T=400 °C 8220 0.3 179E+9 1055E+6 1139E+6 125E+6 3.34E-29 2.911

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3.2 Testing 11

3.2

Testing

A notched specimen with an additional radius at the opposite side according to Fig-ure 3.1 is used in the low-cycle fatigue tests. The additional radius counteracts a bending of the specimen that can occur during tensile loading. The specimen is loaded a limited amount of cycles under strain control. The strain is in all cases evaluated by an extensometer which has a length of 12,5 mm and is placed centered over the notch. The specimen is fixed at one side and pulled from the other. The prescribed

strain ratio Rε = εmin/εmax is zero in all tests, i.e. the minimum strain is considered

zero. Before the first cycle, the specimen is exposed to a 24 h enduring, so called dwell time, where the specimen is kept loaded under maximum strain for 24 h. This is done according to standard practice in order to reduce the scatter in the results between different specimens, since this makes the hysteresis loop stabilize faster [17].

Figure 3.1: Notched test specimen

3.2.1 Conditions

The test specimens are loaded with a strain rate of 6 %/min where the displacement is applied according to a triangular cycle shape. The tests are performed in a furnace to keep a constant uniform temperature of 400 °C in an air environment.

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12 3 TESTING AND MATERIAL

3.2.2 Test data

The data gained from the performed LCF-tests with prescribed maximum and mini-mum strain is the crack length as a function of cycles. In addition the test machine also registers the required force in order to achieve the displacement. Crack initiation and propagation evaluation is performed based on images obtained by a camera that is directed to the side of the specimen and captures an image every 1, 2, 5 or 10 cycles, cf. [18].

For the LCF-test three sets of data are considered. An overview of the test parameters is given in Table 3.3.

Table 3.3: Properties of the tests with ∆εnom and Rε prescribed

LC11301 LC11303 LC11305

∆εnom 0,8 0,6 0,7

σmax [MPa] 1049 841 915

σmin [MPa] -187 -95 -157

Rε 0 0 0

Since the crack is extending with an increasing number of loading cycles the required force, or stress, decreases while the strain range stays constant. An example of the results for one of the laboratory tests is illustrated in Figure 3.2. The applied strain is constant for all load cycles.

Note: The strains depicted in Figure 3.2a have been shifted downwards due to

cali-bration issues of the specimen. The corresponding Rε is equal to zero, which implies

that the actual strains are εmax = 0, 8 and εmin = 0.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Number of loading cycles [−]

Prescribed strain at extensometer [

]

Max strain Min strian

(a) Applied strain in %

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 40 60 80 100 120 140 160 Loading cycles N [−]

Stress at measured by testing machine [MPa]

(b) Applied load

Figure 3.2: Applied load and strain as a function of number of load cycles

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4

Preparation work

In this section the work carried out as a preparation of the actual investigation is explained. In the first section the model and boundary conditions using ABAQUS are described. In order to ensure that the results are mesh independent a convergence study has been performed. Finally, the functionality of FRANC3D is discussed in Sub-section 4.2 where only a linear elastic material behavior is considered.

Note: For further sections, a basic knowledge of FRANC3D is required. For additional information the reader is referred to [19].

4.1

Model

In this section the used model will be shown and the boundary conditions explained. The specimen is modeled according to Figure 3.1 and shown in Figure 4.1. The handles where the specimens are fixed and pulled, respectively, are not modeled since those regions are not of interest. The boundary conditions have been chosen in order to make the simulation represent, as close as possible, the laboratory testing. As explained in Sub-section 3.2 the specimen is fixed on one edge and is loaded on the other. This results in a fixed support at the lower surface on the model and a boundary condition that restraints the nodes to only move in vertical direction on the the upper surface. At the same upper surface a controlled displacement is applied in order to induce the same strain as measured in the test, cf. Figure C.1. The displacement can be positive and negative in order to induce tension and compression. The applied displacement is controlled by a written ABAQUS User-subroutine, cf. Appendix D.

Since the temperature is assumed to be constant in the whole specimen, it can be neglected in the model. The uniform expansion of the material would not induce any further stresses.

In the following Sub-section 4.2 boundary conditions are set corresponding to a tensile test with force control. This is done in order to achieve closer results to the handbook solutions the FRANC3D simulations are compared with.

A mesh independency study has been performed to ensure that the mesh gives the best possible accuracy with as little nodes and elements as necessary. The final mesh is shown in Figure 4.2, and consists only of second order tetrahedron elements of the type C3D10. Other elements and mesh densities have been evaluated, but this configuration

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14 4 PREPARATION WORK

Figure 4.1: CAD model of the specimen

showed the best compromise between computational cost and accuracy. The whole study is shown in Appendix A. It can be seen that the area around the notch is refined multiple times in order to catch the stress concentration induced by the notch. The remaining areas have a lower mesh resolution since there are no complex effects that have to be captured.

For all simulations the whole model has to be used since the only possible symmetry plane is the notch plane, see Figure 6.1. In the notch plane the crack is inserted and the whole sub-model around it is remeshed with each crack growth step. This coincides with the use of a symmetry condition.

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4.2 Functionality of FRANC3D 15

Figure 4.2: Final mesh

4.2

Functionality of FRANC3D

In this section the functionality of the automated crack growth feature of the program FRANC3D is described, beginning with a comparison to a handbook solution. Using the test specimen geometry, the propagation of different types of crack is shown and examined. Moreover, two handbook solutions are used to validate the application of the automated crack growth to the test specimen geometry. Introducing the following study, a short theoretical background about the automatic crack growth function in FRANC3D is given.

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16 4 PREPARATION WORK

4.2.1 Theoretical background to FRANC3D’s automatic crack growth

function

FRANC3D features a function that automatically expands a crack from the initial size according to the parameters the user sets. The computed SIFs along the existing crack are used to calculate the direction and new local length increment by which the crack will extend. The direction is given by the kink angle Θ, cf. Figure 4.3 [9]. FRANC3D offers five different algorithms to do so. The default algorithm is the calculation of the maximum tensile stress, which assumes that the crack will propagate in the direction of the maximum hoop stress. The overall crack increment length can be set prior to the computation. In this case FRANC3D tries to fit the growth steps by calculating the extension of the crack front points by a power law relationship:

∆ai = ∆amedian(Ki/Kmedian)n (4.1)

Figure 4.3: Coordinate system at crack tip[20]

Here ∆amedianis the specified crack extension for the crack front. Kmedian is the median

K-value for all points along the crack and Ki is the SIF at point i. The exponent n is

a user specified value, which is 2 by default. It has been shown in practice that values between 2 and 3 are to be chosen. A study of this parameter has been performed and has shown that the results for value of n = 2 show similar SIFs as in a handbook solution and a reasonable tunneling behavior of the crack, i.e. it grows from initial

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4.2 Functionality of FRANC3D 17

penny crack into a through crack with increasing crack length. For more information, see [7]. In order to determine the SIFs the user may choose between two different approaches: The use of the M-integral, which is default and used in this work and the displacement correlation method, see Equation 4.3 and 4.2, respectively.

M(1,2) = ˆ ! σij(1)∂u (2) i ∂x1 + σij(2)∂u (1) i ∂x1 − W (1,2)δ 1j " ∂q ∂xj dV i = 1, 2, 3 j = 1, 2 (4.2) ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ u v w ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ = 2(1 + ν) E (r 2π ⎛ ⎜ ⎜ ⎝ KI ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ cos(θ 2)[1 − 2ν + sin2( θ 2)] sin(θ 2)[2 − 2ν − cos2(θ2)] 0 ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ + KII ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ sin(θ 2)[2 − 2ν + cos2( θ 2)] cos(θ 2)[1 + 2ν + sin2(θ2)] 0 ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ + KIII ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 0 0 2 sin(θ 2) ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ ⎞ ⎟ ⎟ ⎠ (4.3)

Generally it can be said that the M-integral gives more accurate results but that the displacement correlation method is more robust. Due to its robustness it provides a good possibility to double-check the results. In the displacement correlation method, the SIFs are calculated by the local displacement behind the crack tip, where θ corre-sponds to the kink angle and r is the distance to the crack tip. The M-integral is an energy method that calculates the SIFs by a volume integral, see Equation 4.2. For fur-ther information about the mathematical background, the reader is referred to [21]. In order to get accurate results the automatic mesher of FRANC3D uses different element types at the crack tip to ensure a smoother transition to the remaining tetrahedron elements. Since the integral gives the best results when it is evaluated along a consis-tent distance from the crack tip, the elements closest to the tip are wedges. Following the wedges there are several rings of brick elements. Finally between the bricks and the tetrahedron elements there are pyramids to make the transition compatible [22]. This set-up is illustrated in Figure 4.4 [23].

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18 4 PREPARATION WORK

Figure 4.4: Mesh elements around crack tip, with courtesy of FAC

4.2.2 Handbook solution

In order to establish a comparison to a handbook solution, the geometry of a simple plate is chosen, c.f. Figure 4.5. It is used to validate FRANC3D through comparing the stress intensity factors at the crack tip under a tensile load for different crack lengths. This is done by three different approaches. The handbook solution of a 2D-plate, the automated crack growth function of FRANC3D and the calculation of the J-integral using ABAQUS are compared.

Figure 4.5: Handbook solution. Geometry of a simple plate

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4.2 Functionality of FRANC3D 19

The procedure using ABAQUS corresponds to calculating the J-integrals around the crack tip for the different crack lengths and then determines the stress intensity factors by:

KI =

JE (4.4)

The whole procedure can be reviewed in [24]. The handbook solution is taken from [25]. The comparison of the stress intensity factors as a function of the crack length is shown in the following Figure 4.6 below. It can be seen that the FRANC3D result agrees with the handbook solution even better than the ABAQUS result. The remaining deviation can be explained by the difference between the 2-D assumption in the hand-book solution and the 3-D analysis. Furthermore, it is not assured that the handhand-book solution is perfectly accurate. Generally some percents of error to reality are to be expected. 1 1.5 2 2.5 3 x 10−3 90 100 110 120 130 140 150 160 170 180 Crack length a [m] SIF K I [MPa sqrt(m)] Handbook solution ABAQUS FRANC3D

Figure 4.6: Comparison of stress intensity factors

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20 4 PREPARATION WORK

4.2.3 Growth of Edge, Center and Through Cracks

In this section the propagation of a crack through the test specimen is shown. The crack initiates at different positions and with different shapes. Considered are an edge and a centered penny crack as well as a through crack. The locations of the cracks are visualized in Figure 4.7. The depicted model represents the local sub-model of the specimen. The composition of the local and global model is shown in Figure B.1. The same model is used for all the following computations.

(a) Through crack (b) Centered penny crack (c) Corner penny crack

Figure 4.7: Position of the initial cracks indicated by the predefined crack template For each of the three cases an automated crack growth simulation has been performed. The magnitude of the SIFs in relation to the crack length is shown in Figure 4.8. The initial crack length was 0,43 mm for all cases. As shown, the SIFs differ only in the early stages of the simulation. As soon as the crack grows to a length of around 2,5 mm the curves yield the same SIF for all crack lengths. For larger crack lengths this behavior stays the same. This was to be expected since the corner and the center penny crack will grow through the whole thickness finally obtaining the same crack front shape as the through crack. The evolution of the crack growth for the different cases is schematically shown in Figure 4.9. This simulation shows that the position of the nucleation and the shape of the crack has some influence of the SIFs and thus also on the crack growth rate in the early stages. Since the influence is only visible for short cracks and it is not possible to predict where a crack nucleates this phenomena is neglected in the following investigation. Therefore, for all further analyses a through crack is used.

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4.2 Functionality of FRANC3D 21 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10−3 50 55 60 65 70 75 80 85 90 95 100 Crack length a [m] SIF K I [MPa sqrt(m)] Through Crack Center Penny Edge Penny

Figure 4.8: SIFs for different initial crack positions

(a) Through crack (b) Center penny

crack

(c) Edge penny

crack Figure 4.9: Evolution of the crack fronts

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22 4 PREPARATION WORK

4.2.4 Influence of increment length on accuracy

Since the computation time is highly dependent on the number of crack increments, i.e. on the increment length, in the crack growth analysis it is important to investigate its influence on the results. In order to do this a study containing four different increment lengths is performed and the associated results compared. The specimen is loaded with a 60 MPa far field stress in all cases. The considered increment lengths are 0.2 mm, 0.5 mm, 1 mm and 2 mm, respectively. The comparison of the SIFs for the different crack lengths is shown in Figure 4.10. It can be seen that to a certain extent the choice of the increment length is robust and does not affect the solution. Only the 2 mm increment length does not produce good results. It has to be said that this length was chosen to see the limits of the program and is not a reasonable choice for this kind of geometry since it corresponds to approximately 15 % of the specimen’s thickness. Only in case a quick evaluation of the magnitude of the SIFs is desired it can be useful since the computational time is considerably lower. As a conclusion, it can be said that FRANC3D is robust in terms of result accuracy for crack growth increment lengths up to a certain limit. If the increments are reasonably chosen suitable to the problem geometry the program can be trusted in delivering good results.

0 1 2 3 4 5 6 7 8 9 x 10−3 0 10 20 30 40 0 10 20 Crack length a [m] SIF K I [MPa sqrt(m)] 1mm increment 2mm increment 0,5mm increment 0,2mm increment

Figure 4.10: Comparison of SIFs for different crack growth increment lengths

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4.2 Functionality of FRANC3D 23

4.2.5 Determination of the shape function

As explained in Section 2.2 the SIFs are determined by KI = σ√aπf = F√aπ ˜f where

˜

f, in this application, is assumed to be only dependent on the crack length a, i.e.

˜

f = ˜f (a). For a fast evaluation of KI for the considered specimen based only on test

data, which provides the force F as a function of crack length, the shape function ˜f (a)

is to be considered so that:

KI(a) = F (a)√aπ ˜f (a) (4.5)

In order to do so a FRANC3D simulation incorporating a linear elastic material model with a certain displacement load is performed and the corresponding force F for each

considered crack length a and SIF KI are reported. With this, a series of data points

according to Equation 4.6 can be obtained. Using this data points in a curve fitting establishes the shape function.

˜

f (a) = KI(a)

F (a)√aπ (4.6)

The resulting curve fitting graph as well as the used data points are shown in Fig-ure 4.11. Its function is a sixth order polynomial, see Equation 4.7.

˜ f (a) = p1 · a6− p2· a5+ p3 · a4− p4· a3+ p5· a2− p6 · a + p7 (4.7) p1 = 1, 31E18 p2 = 4, 072E16 p3 = 4, 072E16 p4 = 3, 593E12 p5 = 1, 413E10 p6 = 3, 135E7 p7 = 6, 016E4

In order to validate the shape function the SIF is calculated using the applied forces F(a) with its corresponding crack lengths taken from the test data as shown in

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24 4 PREPARATION WORK 1 2 3 4 5 6 7 8 x 10−3 2.8 3 3.2 3.4 3.6 3.8 4x 10 4 Crack length a [m] Shape function f [1/ m 2]

Figure 4.11: Shape function for the given geometry

tion 4.6. This is done for all three tests, i.e. all three applied strains of 0,6 %, 0,7 % and 0,8 %. The comparison between the SIF calculated with the test data in Equation 4.5 and by FRANC3D is shown in the Figures 4.12 - 4.14. It can be seen that the determination of the SIFs using the above derived shape function gives good approx-imations for the tests with the applied strain of 0,6 % and 0,7 %, cf. Figure 4.12 and 4.13. For a higher load, i.e. ε = 0, 8 % the difference between the SIFs calculated by FRANC3D and the SIFs determined by use of the shape function is significantly higher. This shows that the approach to determine the SIFs using the shape functions gives increasing accuracy with lower loads. It shall only be used for rough estimations and has to be validated by other means, e.g. FRANC3D.

0 1 2 3 4 5 6 7 8 x 10−3 35 40 45 50 55 60 65 70 75 Crack length a [m] SIF K I [MPa sqrt(m)] Test data FRANC3D

Figure 4.12: Comparison of SIF between test data using the shape function and FRANC3D for ε = 0, 6 %

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4.2 Functionality of FRANC3D 25 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10−3 30 40 50 60 70 80 Crack length a [m] SIF K I [MPa sqrt(m)] Test data FRANC3D

Figure 4.13: Comparison of SIF between test data using the shape function and FRANC3D for ε = 0, 7 % 0 1 2 3 4 5 6 7 x 10−3 65 70 75 80 85 90 95 100 Crack length a [m] SIF K I [MPa sqrt(m)] Test data FRANC3D

Figure 4.14: Comparison of SIF between test data using the shape function and FRANC3D for ε = 0, 8 %

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26 4 PREPARATION WORK

4.2.6 Automated crack growth analysis applied on the test specimen

In this section the automated crack growth analysis of the previous Sub-section 4.2.3 is validated using the two handbook solutions. This is necessary since the specimen has a rather complex geometry that cannot be approximated by only one handbook case. It is separated into one part that represents the case of a crack close to the notch and the other area where the influence of the stress concentration has vanished. The chosen handbook solution for a crack near the notch is depicted in [25]. The second handbook solution for the complementary area which is not affected by the stress concentration at the notch, corresponds to the 2D plate used as an elemental case above in Sub-section 4.2.2. In order to transform the problem on the notched specimen the crack length has to be considered as the actual crack length plus the notch depth of 3 mm. The two handbook solutions and the calculations by the automated crack growth function by FRANC3D are compared in Figure 4.15. The SIFs are compared for a crack length between 0,5 mm and 4,5 mm. It can be seen that in the early stages of the crack growth the FRANC3D calculations agree better with the handbook solution of the notched specimen. It can be concluded that after around 1,5 mm the influence of the stress concentration fades and the crack growth behaves as for the cracked plate. After the crack has propagated a certain length the graphs start to deviate again due to the stress concentration of the radius at the opposite end of the specimen. It has to be noted that there are some limitations to the handbook solutions, which cause deviations:

Notched plate

• It assumes an infinite plate • It is 2-dimensional

Simple plate

• It is 2-dimensional

• Does not take into account radius

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4.2 Functionality of FRANC3D 27 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 10−3 30 40 50 60 70 80 90 100 110 120 Crack length a [m] SIF K I [MPa sqrt(m)]

Handbook solution notch

Handbook solution plate (without radius) FRANC3D

Figure 4.15: Comparison of SIFs of handbook solutions

Despite of the found deviations it can be concluded that FRANC3D calculates the SIFs accurately while automatically growing the crack.

In the succeeding section below, methods to implement nonlinear effects into FRANC3D are investigated.

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5

Methodology

5.1

Settings

In the succeeding part the following settings are used: • small scale yielding at the crack tip is assumed

• constant and uniform temperature throughout whole model is assumed • all crack growth analyses in FRANC3D are linear elastic

• in ABAQUS all FE simulations, to calculate the residual stresses, a perfectly plastic material model is used

• the strains are corresponding to the displacements at the gauge, i.e. 12,5 mm from the notch

• through crack is used for all simulations

To summarize, the following cases are investigated, where the different cases are defined below and where ε = 0, 65% corresponds to an applied load of 145MPa at the uncracked specimen:

Table 5.1: Overview of computations

Strain controlled Stress controlled

Linear elastic ε = 0, 65 %; 0, 6 %; 0, 7 %; 0, 8 % σ = 145MPa

Case 1 ε = 0, 65 %; 0, 6 %; 0, 7 %; 0, 8 % σ = 145MPa

Case 2 ε = 0, 65 %; 0, 6 %; 0, 7 %; 0, 8 % σ = 145MPa

5.2

Accounting for nonlinear effects in FRANC3D

This section contains the approaches to account for plasticity effects in FRANC3D. Due to stress concentrations in the notch areas of the component, plastic deformations appear, which influence the crack growth behavior of the material. The investigated approaches make use of the associated residual stress field. In the following the ap-proaches will be referred to as Linear elastic Case, Case 1 and Case 2, respectively.

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5.2 Accounting for nonlinear effects in FRANC3D 29

All cases are investigated with stress and strain controlled loading according to the stresses and strains obtained in the test data.

In Case 1 and Case 2 residual stresses are first obtained from the uncracked model, cf. (b) in Figure 5.1. At the imaginary crack, the crack face tractions that cause the closure are identified. In (c) crack face tractions with the same magnitude but opposite direction are applied to the cracked model. A superposition of (b) and (c) gives (a), which corresponds to the desired model with the crack. This superposition technique makes it possible to handle difficult loads and boundary conditions.

Note: Since the applied stresses or strains are constant for all simulations for all computational steps, the computations using a constant stress will deviate considerably from the test data, which represents a case of constant strain.

Figure 5.1: Superposition for crack face traction

5.2.1 Linear elastic analysis

In order to establish a first set of results, the simplest analysis case of a linear elastic material behavior is carried out. The model subjected to a constant strain according the loads in the tests. No residual stress fields are involved. The workflow of the general procedure is schematically illustrated in Figure 5.2.

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30 5 METHODOLOGY

Apply forces/displacements to model

Perform automated crack growth analysis

Calculate cyclic life-time using Paris’s law A B A Q U S F R A N C 3 D Linear elastic

Figure 5.2: Workflow - linear-elastic computation

5.2.2 Case 1

This approach incorporates the residual stress field found by an elasto-plastic analysis after one and a half loading cycles. This means that the specimen is loaded, then unloaded and loaded again with the same load as in the previous cycle, cf. Figure 5.3. The corresponding stress-strain response is illustrated in Figure 5.4.

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5.2 Accounting for nonlinear effects in FRANC3D 31 0 5 10 15 20 25 0 0.25 0.5 0.75 1 Time t [s] Normalized Strain

Figure 5.3: Normalized loading over time for Case 1

0 1 2 3 4 5 6 7 8 x 10−3 −1,500 −1,000 −500 0 500 1,000 1,500 Strain [−] Stress [MPa]

Figure 5.4: Hysteresis loop for the uncracked geometry for Case 1

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32 5 METHODOLOGY

The strains in this loop correspond to the testing explained above. Up to this point all simulations are done in ABAQUS only. The stress field at the end of the analysis, corresponding to the maximum stress/strain, marked with a red ring in Figure 5.4, is exported into FRANC3D and applied to the model as a so called crack face traction (residual stress field). This approach makes it possible to incorporate the effect of plastic rearrangements of stresses and strains at the notch in the analyses. The suc-ceeding analysis in FRANC3D uses a linear-elastic material behavior. Using the stress field from the elasto-plastic computation at the last stage under maximum tension, this load incorporates the effect of the initial plastic flow at the notch as well as the linear elastic stress response outside the stress concentration. As mentioned before, using this approach the applied stress field act as a constant stress on the model for all computational steps. The workflow is illustrated in Figure 5.5

Apply forces/displacements to model

Run simulation according to load plot

At final increment export stress field to FRANC3D

F R A N C 3 D A B A Q U S Elasto-plastic Linear elastic

Apply stress field to F3D model as crack face traction

Perform automatic crack growth analysis

Calculate cyclic life-time using Paris’ law

Figure 5.5: Workflow - Case1

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5.2 Accounting for nonlinear effects in FRANC3D 33

5.2.3 Case 2

This approach is somewhat similar to Case 1 since it also uses the stress field of a preceding elasto-plastic analysis. Compared to Case 1, here only the residual stress after one load cycle is used. This means that the specimen is loaded to maximum stress or strain, respectively, and than released, cf. Figure 5.6. The stress-strain response corresponding to this is shown in Figure 5.7, where the final increment from which the residual stress field is exported is marked with a red ring.

0 2 4 6 8 10 12 14 0 0.25 0.5 0.75 1 Time t [s] Normalized Strain

Figure 5.6: Normalized loading over time for Case 2

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34 5 METHODOLOGY 0 1 2 3 4 5 6 7 8 x 10−3 −1,500 −1,000 −500 0 500 1,000 1,500 Strain [−] Stress [MPa]

Figure 5.7: Hysteresis loop Case 2

This causes plastic flow in the area of the stress concentration which results in a compressive stress state at the stress concentration. This phenomenon is illustrated in Figure 5.8. The notch and the influence of the plastic flow after unloading is shown. The residual stress field is obtained and used in the FRANC3D model as a crack face traction as previously mentioned. In addition to this stress field the specimen is loaded as in the testing but with a linear elastic response. This is done once for a stress and once for a strain controlled loading. For the strain controlled loadings an ABAQUS sub-routine has been written that ensures that the strain over the extensometer length is constant throughout the analysis even though the crack length increases. The sub-routine is attached in Appendix D. The workflow for Case 2 is schematically illustrated in Figure 5.9.

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5.2 Accounting for nonlinear effects in FRANC3D 35

Figure 5.8: (a) crack in virgin material, (b) crack blunting and plastic zone forma-tion from applied tensile load, (c) compressive residual stresses at the crack tip after unloading

Apply forces/displacements to model

Run simulation according to load plot

At final increment export stress field to FRANC3D

F R A N C 3 D A B A Q U S Elasto-plastic Linear elastic

Apply stress field to F3D model as crack face traction + a constant load

Perform automatic crack growth analysis

Calculate cyclic life-time using Paris’ law

Figure 5.9: Workflow - Case2

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36 5 METHODOLOGY

5.3

Cyclic life-time analysis

Using the calculated SIFs as functions of crack lengths, FRANC3D calculates the cyclic life-time by integrating Paris’ law as explained previously. A constant amplitude

analysis according to R = 0 is used since Kopen is assumed to be Kopen = Kmin = 0.

The obtained simulation results are then compared to the test data for validation.

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6

Results

In this section the results of the above explained approaches are shown. A summary of the ratios of the average nominal stress in the notch plane and the yield stress σnomnotch/σy for the different cases is shown in the following Table 6.1 for the uncracked

specimen. Furthermore the local stress σlocal at the notch is shown which incorporates

the notch factor Kt = 2, 21, i.e. σlocal = σnomnotch · Kt. Its purpose is to give an idea

about the degree of plasticity occurring for each case. The notch plane is shown in Figure 6.1.

Figure 6.1: Notch plane

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38 6 RESULTS Table 6.1: Ratios of average nominal stress at notch plane and yield stress for the uncracked specimen

Case σnomnotch [MPa] σy [MPa]

σnomnotch σy [%] σlocal [MPa] σ = 145 MP a 895 1139 78,6 1997,95 ε = 0, 65 % 895 1139 78,6 1997,95 ε = 0, 6 % 872 1139 76,6 1927,21 ε = 0, 7 % 912 1139 80,1 2015,52 ε = 0, 8 % 1049 1139 92 2318,29

6.1

Results of the accounting for nonlinear effects in FRANC3D

6.1.1 Comparison between stress and strain based loadings

Automated crack growth analysis has been performed incorporating the linear elastic case, Case 1 and Case 2 with stress controlled boundary conditions. The resulting

KI(a) graphs are shown in Figure 6.2. It can be seen that Case 1 and Case 2 agree

good for all crack length. The linear elastic computation deviates only up to a crack length of two millimeters.

0 1 2 3 4 5 6 7 8 x 10−3 40 60 80 100 120 140 160 180 200 Crack length a [m] SIF K I [MPa sqrt(m)] Case1 Linear elastic Case2

Figure 6.2: Stress controlled: Comparison of KI(a) of Case 1, Case 2 and a linear

elastic computation for σ = 145MPa

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6.1 Results of the accounting for nonlinear effects in FRANC3D 39

With strain controlled tension the KI(a) graphs look as depicted in Figure 6.3. Case

2 and the linear elastic computation behave similarly. The SIFs calculated by Case 1 deviate a lot at all crack length.

0 1 2 3 4 5 6 7 8 x 10−3 50 100 150 200 crack length a [m] SIF K I [MPa sqrt(m)] Case 1 Case 2 Linear−elastic

Figure 6.3: Strain controlled: Comparison of KI(a) of Case 1, Case 2 and a linear

elastic computation for ε = 0, 65 %

6.1.2 Different applied strains

From this point on only Case2 and the linear elastic case are considered, since Case 1 is obviously performing less well under strain controlled conditions. The results of the the automated crack growth analyses for the different applied strains of ε=0.6 %, 0.7 % and 0.8 % are displayed in Figures 6.4 - 6.6. In Figure 6.7 the difference of the SIFs between Case 2 and the linear elastic case for the three tests is illustrated.

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40 6 RESULTS 0 1 2 3 4 5 6 7 8 x 10−3 30 35 40 45 50 55 60 65 70 75 80 Crack length a [m] SIF K I [MPa sqrt(m)] Linear elastic Case 2

Figure 6.4: ε = 0.6 %: SIFs over crack length

0 1 2 3 4 5 6 x 10−3 40 45 50 55 60 65 70 75 80 85 90 Crack length a [m] SIF K I [MPa sqrt(m)] Case 2 Linear elastic

Figure 6.5: ε = 0.7 %: SIFs over crack length

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6.1 Results of the accounting for nonlinear effects in FRANC3D 41 0 1 2 3 4 5 6 x 10−3 70 75 80 85 90 95 100 Crack length a [m] SIF K I [MPa sqrt(m)] Case 2 Linear elastic

Figure 6.6: ε = 0.8 %: SIFs over crack length

0 1 2 3 4 5 6 x 10−3 0 2 4 6 8 10 12 14 16 Crack length a [m] Difference in K I

between Case 2 and

the linear elastic Case in %

0.8 % Strain 0.6 % Strain 0.7 % Strain

Figure 6.7: Difference in SIFs between Case 2 and the linear elastic case for all tests

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42 6 RESULTS

6.2

Cyclic life-time analysis

In this section the cyclic life-time analysis for the different applied strains is performed. Since it is discovered that Case 1 is not suitable for a comparison to the test data, it is not included in the following results. The reasons for the rejection are explained in Section 7.1.1. The following graphs result from the comparison between the linear-elastic case, Case 2 and the test data:

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 1 2 3 4 5 6 7 8x 10 −3 Loading cycles N [−] Crack length a [m] Case 2 Test data Linear elastic

Figure 6.8: ε = 0.6 %: crack length over loading cycles

Accounting for non-linear effects in fatigue crack propagation simulations using FRANC3D

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6.2 Cyclic life-time analysis 43 0 500 1000 1500 2000 2500 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x 10 −3 Loading cycles N [−] Crack length a [m] Case 2 Test data Linear elastic

Figure 6.9: ε = 0.7 %: crack length over loading cycles

0 500 1000 1500 2000 0 1 2 3 4 5 6 7 8x 10 −3 Loading cycles N [−] Crack length a [m] Linear elastic Test Data Case 2

Figure 6.10: ε = 0.8 %: crack length over loading cycles

Accounting for non-linear effects in fatigue crack propagation simulations using FRANC3D

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7

Discussion

In the following the results and the used methods to achieve the results are scrutinized. Furthermore possible sources of error and their influence on the results are discussed.

7.1

Result discussion

In this section the results of the foregoing analyses are discussed. It has to be men-tioned that only the strain controlled approach is consistent with the testing of the specimen since it is also strain controlled with a constant strain over all loading cases. Due to the extension of the crack, the stress decreases with an increasing number of loading cycles.

7.1.1 Accounting for nonlinear effects

The methods to account for nonlinear effects, i.e. Case 1 and Case 2 can be divided into two parts with different load cases; in a part with stress controlled loading and a part with strain controlled loading.

First the stress controlled analysis is discussed. As shown in Figure 6.2 the results for Case 1 and Case 2 agree very well over the whole crack length interval. The SIFs of the linear elastic computation differs only up to a crack length of around two millimeters. This can be explained with the stress concentration at the notch which influence Case 1 and Case 2 in that region. To conclude, the material rearrangements due to the plastic flow are accounted for using applied residual stress fields. Due to the compressive stresses caused by the plastic flow close to the notch the local stress does not rise as much as in the linear elastic computation. Accounting for this effects, the cyclic life-time of the components can probably be evaluated more accurately using this relatively simple method.

In the strain controlled loading approach the three cases differ considerably more. It can be observed that a similar behavior to the stress controlled approach of the linear elastic computations and Case 2 occurs, i.e. they converge after a certain crack length. The calculated SIFs by Case 1 show a very different behavior, cf. Figure 6.3. This can be explained by the fact that the used method is not compatible with a displacement controlled loading. As mentioned earlier the investigated test is strain controlled, which means that the maximum strain is constant over all test cycles but the corresponding far field stress is not, c.f Figure 3.2. Since Case 1 uses the stress

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7.1 Result discussion 45 field of the maximum strain state of the uncracked specimen, basically a constant load is applied to the specimen for all crack increments, i.e. simulation steps. This results in stronger increase of the SIFs compared to Case 2 and the linear elastic computation. Since Case 2 only incorporates the compressive material rearrangements close to the notch and then applies the constant displacement loads for each crack increment, a different behavior of the SIF curve can be obtained. This reflects the characteristics of the strain controlled laboratory test much more accurately. Moreover the same effect of the plastic flow close to the stress concentration is found as explained before in the stress controlled approach.

It can be mentioned that the computations using Case 1 are more convenient from a computational cost perspective since only one ABAQUS simulation to obtain the stress field is necessary. No further loadings have to be considered. Case 2 needs one further loading that has to be applied. The difference in workload is not enormous for the rather small model that is considered. For more complex models this might induce a considerable difference in computational cost which is always an important factor. Considering the magnitude of the differences in SIFs between the Case 2 and the linear-elastic simulation for the different applied strains ε = 0.6 %, 0.7 % and 0.8 %, cf. Figures 6.4 - 6.6, it can be seen that the deviation between the simulations using Case 2 and the linear-elastic case differ more with increasing applied strain range. This is expected since a larger applied strain leads to a higher stress at the stress concentration and thus more plastic flow. However, still in the simulations with an applied strain of ε = 0.8 % the deviations are only notable for short crack lengths. For longer cracks the influence is considerably smaller. In the simulations with an applied strain of ε = 0.6 %, 0.7 % the deviation of the SIF is rather small over the whole crack length range. The influence of the applied strain on the difference of the SIF between Case 2 and the linear elastic Case for all tests is illustrated in percentage in Figure 6.7. It can be observed that the higher the applied strain the greater the deviations between Case 2 and the linear elastic Case. However, except for the small crack lengths in the 0,8 % strain test all deviations of the SIF are in a range lower than 5 %. Since all analyzed tests incorporate a high degree of plasticity, cf. Table 6.1, it can be concluded that the linear elastic analyses give a good approximation of the results of Case 2.

In the following section it is shown how the methods for accounting of the nonlinear effect influences the cyclic life- time.

Accounting for non-linear effects in fatigue crack propagation simulations using FRANC3D

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