accounting for non-linear eﬀects in fatigue crack propagation simulations using...

Accounting for non-linear effects in fatiguecrack propagation simulations using

FRANC3D

Christian Busse

Division of Solid Mechanics

Master Thesis

Department of Management and Engineering

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

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 GustafssonCombustor Mechanical Integrity

Siemens Industrial Turbomachinery

Supervisor LiU

Kjell SimonssonDivision of Solid Mechanics


Examiner

Daniel LeidermarkDivision of Solid Mechanics


Christian Busse:

Accounting for non-linear effects in fatigue crack propagation

simulations using FRANC3D

Master Thesis in Mechanical Engineering


Handling period: 20. January 2014 - 11. June 2014

Upphovsrätt

Detta dokument hålls tillgängligt på Internet — eller dess framtida ersättare — un-

der 25 år från publiceringsdatum under förutsättning att inga extraordinära om-

ständigheter uppstår. Tillgång till dokumentet innebär tillstånd för var och en att läsa,

ladda ner, skriva ut enstaka kopior för enskilt bruk och att använda det oförändrat

för icke- kommersiell forskning och för undervisning. Överföring av upphovsrätten vid

en senare tidpunkt kan inte upphäva detta tillstånd. All annan användning av doku-

mentet kräver upphovsmannens medgivande. För att garantera äktheten, säkerheten

och tillgängligheten finns det lösningar av teknisk och administrativ art. Upphovs-

mannens ideella rätt innefattar rätt att bli nämnd som upphovsman i den omfattning

som god sed kräver vid användning av dokumentet på ovan be- skrivna sätt samt

skydd mot att dokumentet ändras eller presenteras i sådan form eller i sådant sam-

manhang som är kränkande för upphovsmannens litterära eller konstnärliga anseende

eller egenart. För ytterligare information om Linköping University Electronic Press se

förla- gets hemsida http://www.ep.liu.se/

Copyright

The publishers will keep this document online on the Internet — or its possible re-

placement — for a period of 25 years from the date of publication barring exceptional

circumstances. The online availability of the document implies a permanent permis-

sion for anyone to read, to download, to print out single copies for his/her own use

and to use it unchanged for any non-commercial research and educational purpose.

Subsequent transfers of copyright cannot revoke this permission. All other uses of

the document are conditional on the consent of the copyright owner. The publisher

has taken technical and administrative measures to assure authenticity, security and

accessibility. According to intellectual property law the author has the right to be men-

tioned when his/her work is accessed as described above and to be protected against

infringement. For additional information about the Linköping University Electronic

Press and its procedures for publication and for assurance of document integrity, please

refer to its www home page: http://www.ep.liu.se/

©Christian Busse

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.

V

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

VII

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

5 Methodology 28

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

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


FRANC3D for ε = 0, 7% . . . . . . . . . . . . . . . . . . . . . . . . . . 25


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

5.7 Hysteresis loop Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

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



6.7 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



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

XII

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

A.1 Mesh study: Comparison of meshes . . . . . . . . . . . . . . . . . . . . 53

Nomenclature

Roman Symbols

Kf Fatigue notch factor [-]

Kt Stress concentration factor [-]

KIC Fracture toughness [Pa√m]

KIII Stress intensity factor mode III [Pa√m]

KII Stress intensity factor mode II [Pa√m]

KI Stress intensity factor mode I [Pa√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 [Pa]

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 [Pa]

XV

σnom Nominal stress [Pa]

σyeng Engineering stress [Pa]

σyreal Real stress [Pa]

σyy Remote stress for mode I [Pa]

τyx Remote stress for mode II [Pa]

τyz Remote stress for mode III [Pa]

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

XVI

SIF Stress Intensity Factor

SIT AB Siemens Industrial Turbomachinery AB

Ti Titanium

TMF-tests Thermo-Mechanical-Fatigue-tests

XVII

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

1

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 + C2N

c (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].

Accounting for non-linear effects in fatigue crack propagation simulations usingFRANC3D

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.


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.


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.


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]

6

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


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 ∆Keff can be defined as: ∆Keff = Kmax−Kopen, where Kopen

is the SIF at the time when the crack opens, which is not necessarily zero, but can


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 afinal

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.


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 CWeight% 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 σyrealKIC C n

Unit [kg/m3] [/] [Pa] [Pa] [Pa] [Pa√m] [m/(Pa

√m)n] [/]

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

10

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.


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 0,7σmax [MPa] 1049 841 915σmin [MPa] -187 -95 -157Rε 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 [−]

Pres

crib

ed s

train

at e

xten

som

eter

[−]

Max strainMin strian

(a) Applied strain in %

0 1000 2000 3000 4000 5000 6000 7000 8000 900040

60

80

100

120

140

160

Loading cycles N [−]

Stre

ss a

t mea

sure

d by

test

ing

mac

hine

[MPa

]

(b) Applied load

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


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

13

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.


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.



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 ∆amedian is 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



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

!

σ(1)ij

∂u(2)i

∂x1+ σ(2)

ij

∂u(1)i

∂x1−W (1,2)δ1j

"

∂q

∂xj

dV i = 1, 2, 3 j = 1, 2 (4.2)

⎧

⎪

⎪

⎨

⎪

⎪

⎩

u

v

w

⎫

⎪

⎪

⎬

⎪

⎪

⎭

=2(1 + ν)

E

√

r

2π

⎛

⎜

⎜

⎝

KI

⎧

⎪

⎪

⎨

⎪

⎪

⎩

cos( θ2)[1 − 2ν + sin2( θ

2)]

sin( θ2)[2 − 2ν − cos2( θ

2)]

0

⎫

⎪

⎪

⎬

⎪

⎪

⎭

+ KII

⎧

⎪

⎪

⎨

⎪

⎪

⎩

sin( θ2)[2 − 2ν + cos2( θ

2)]

cos( θ2)[1 + 2ν + sin2( θ

2)]

0

⎫

⎪

⎪

⎬

⎪

⎪

⎭

+ KIII

⎧

⎪

⎪

⎨

⎪

⎪

⎩

0

0

2 sin( θ2)

⎫

⎪

⎪

⎬

⎪

⎪

⎭

⎞

⎟

⎟

⎠

(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].



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



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 handbook

solution is perfectly accurate. Generally some percents of error to reality are to be

expected.

1 1.5 2 2.5 3x 10−3

90

100

110

120

130

140

150

160

170

180

Crack length a [m]

SIF

K I [MPa

sqr

t(m)]

Handbook solutionABAQUSFRANC3D

Figure 4.6: Comparison of stress intensity factors



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.



0 0.5 1 1.5 2 2.5 3 3.5 4x 10−3

50

55

60

65

70

75

80

85

90

95

100

Crack length a [m]

SIF

K I [MPa

sqr

t(m)]

Through CrackCenter PennyEdge Penny

Figure 4.8: SIFs for different initial crack positions

(a) Through crack (b) Center pennycrack

(c) Edge pennycrack

Figure 4.9: Evolution of the crack fronts



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 9x 10−3

0

10

20

30

40

0

10

20

Crack length a [m]

SIF

K I [MPa

sqr

t(m)]

1mm increment2mm increment0,5mm increment0,2mm increment

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



4.2.5 Determination of the shape function

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

√aπf̃ where

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

f̃ = f̃(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 f̃(a)

is to be considered so that:

KI(a) = F (a)√aπf̃(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.

f̃(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.

f̃(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 Equa-



1 2 3 4 5 6 7 8x 10−3

2.8

3

3.2

3.4

3.6

3.8

4x 104

Crack length a [m]

Shap

e fu

nctio

n 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 8x 10−3

35

40

45

50

55

60

65

70

75

Crack length a [m]

SIF

K I [MPa

sqr

t(m)]

Test dataFRANC3D

Figure 4.12: Comparison of SIF between test data using the shape function andFRANC3D for ε = 0, 6%



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x 10−3

30

40

50

60

70

80

Crack length a [m]

SIF

K I [MPa

sqr

t(m)]

Test dataFRANC3D


0 1 2 3 4 5 6 7x 10−3

65

70

75

80

85

90

95

100

Crack length a [m]

SIF

K I [MPa

sqr

t(m)]

Test dataFRANC3D




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 3mm.

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



0.5 1 1.5 2 2.5 3 3.5 4 4.5x 10−3

30

40

50

60

70

80

90

100

110

120

Crack length a [m]

SIF

K I [MPa

sqr

t(m)]

Handbook solution notchHandbook 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.


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

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

28

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.


30 5 METHODOLOGY

Apply forces/displacements to model

Perform automated crack growth analysis

Calculate cyclic life-time using Paris’s law

ABAQUS

FRANC3D

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.



0 5 10 15 20 250

0.25

0.5

0.75

1

Time t [s]

Nor

mal

ized

Stra

in

Figure 5.3: Normalized loading over time for Case 1

0 1 2 3 4 5 6 7 8x 10−3

−1,500

−1,000

−500

0

500

1,000

1,500

Strain [−]

Stre

ss [M

Pa]

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


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


Run simulation according to load plot

At final increment export stress field to FRANC3D

FRANC3D

ABAQUS

Elasto-plastic

Linear elastic

Apply stress field to F3D modelas crack face traction

Perform automatic crack growth analysis

Calculate cyclic life-timeusing Paris’ law

Figure 5.5: Workflow - Case1



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 12 140

0.25

0.5

0.75

1

Time t [s]

Nor

mal

ized

Stra

in

Figure 5.6: Normalized loading over time for Case 2


34 5 METHODOLOGY

0 1 2 3 4 5 6 7 8x 10−3

−1,500

−1,000

−500

0

500

1,000

1,500

Strain [−]

Stre

ss [M

Pa]

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.



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 afterunloading


Run simulation according to load plot

At final increment export stress field to FRANC3D

FRANC3D

ABAQUS

Elasto-plastic

Linear elastic

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

Perform automatic crack growth analysis

Calculate cyclic life-timeusing Paris’ law

Figure 5.9: Workflow - Case2


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.


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

37

38 6 RESULTS

Table 6.1: Ratios of average nominal stress at notch plane and yield stress for theuncracked specimen

Case σnomnotch[MPa] σy [MPa] σnomnotch

σy[%] σlocal [MPa]

σ = 145MPa 895 1139 78,6 1997,95ε = 0, 65% 895 1139 78,6 1997,95ε = 0, 6% 872 1139 76,6 1927,21ε = 0, 7% 912 1139 80,1 2015,52ε = 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 8x 10−3

40

60

80

100

120

140

160

180

200

Crack length a [m]

SIF

K I [MPa

sqr

t(m)]

Case1Linear elasticCase2

Figure 6.2: Stress controlled: Comparison of KI(a) of Case 1, Case 2 and a linearelastic computation for σ = 145MPa


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 8x 10−3

50

100

150

200

crack length a [m]

SIF

K I [MPa

sqr

t(m)]

Case 1Case 2Linear−elastic

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


40 6 RESULTS

0 1 2 3 4 5 6 7 8x 10−3

30

35

40

45

50

55

60

65

70

75

80

Crack length a [m]

SIF

K I [MPa

sqr

t(m)]

Linear elasticCase 2

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

0 1 2 3 4 5 6x 10−3

40

45

50

55

60

65

70

75

80

85

90

Crack length a [m]

SIF

K I [MPa

sqr

t(m)]

Case 2Linear elastic



6.1 Results of the accounting for nonlinear effects in FRANC3D 41

0 1 2 3 4 5 6x 10−3

70

75

80

85

90

95

100

Crack length a [m]

SIF

K I [MPa

sqr

t(m)]

Case 2Linear elastic


0 1 2 3 4 5 6x 10−3

0

2

4

6

8

10

12

14

16

Crack length a [m]

Diff

eren

ce in

KI b

etw

een

Cas

e 2

and

the

linea

r ela

stic

Cas

e in

%

0.8 % Strain0.6 % Strain0.7 % Strain

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


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 50000

1

2

3

4

5

6

7

8x 10−3


Cra

ck le

ngth

a [m

]

Case 2Test dataLinear elastic

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


6.2 Cyclic life-time analysis 43

0 500 1000 1500 2000 25000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10−3


Cra

ck le

ngth

a [m

]

Case 2Test dataLinear elastic


0 500 1000 1500 20000

1

2

3

4

5

6

7

8x 10−3


Cra

ck le

ngth

a [m

]

Linear elasticTest DataCase 2



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

44

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.


46 7 DISCUSSION

7.1.2 Cyclic life time and comparison to test data

From the knowledge of the limitation of the three different approaches, Case 1 is

rejected for the following cyclic life-time analysis since it will not give reasonable

results, see Figure 6.3. The results for the different applied strain magnitudes of

ε=0,6 %, 0,7 % and 0,8 % show only slightly different characteristics when comparing

the linear-elastic case and Case 2, cf. Figures 6.8 - 6.10. Case 2 gives slightly less

conservative results than the linear elastic case. Except for the analysis of the applied

strain of ε = 0, 6%, cf. Figure 6.8, all simulations give conservative results after 500

loading cycles in comparison to the test data. In general a good agreement between

the test data and the simulations can be observed. Furthermore, the slope of the

curves, i.e. the crack growth rates da/dN agree very well with the test data after

a certain amount of loading cycles. One reasons for the delayed agreement could be

that complex phenomena occurring at small cracks have not been accounted for, see

Sub-section 7.2.

This strengthens the conclusions made in Sub-section 7.1.1 that the linear elastic

approach gives good results very close to Case 2. Since the workload in order to

prepare the linear elastic case as well as the computational cost is lower in comparison

with Case 2, the linear elastic approach is the favored method in this work. Since tests

with a high grade of plasticity are considered, cf. Table 6.1, this should be even more

valid for tests with less plasticity.

7.1.3 Sources of error

Material tests and FE-computations can never be perfectly accurate. In the following

possible sources of error and their influence on the results are discussed.

7.1.3.1 Material and Testing

While performing material testings it can never be assured that all specimen are sub-

jected to the same initial conditions. There are always impurities or minor defects

in the material structure. Even though the initial dwell time of 24 h is supposed to

counteract this, creating similar conditions, all specimens will never show the same

properties in their microstructure.

Furthermore, the procedure to determine the crack length is relatively uncertain due

to following sources of error:


7.2 Method discussion 47

• Camera registers only one side of the specimen

• Limited camera resolution

• Human errors

Concerning the fact that the camera only registers one side of the specimen it has

to be mentioned that the actual crack length can differ from the measured length.

Even though a semi-elliptical crack shape is expected, the crack is further progressed

in the middle of the specimen which would be the corresponding length measured in

the FE-computations. However, it cannot be assured that the crack always initiates

in the middle as a centre crack and grows like a semi-elliptical crack. Its shape can be

somewhat arbitrary so that the length registered by the camera does not represent the

actual crack length. This can cause uncertainties concerning the crack growth data,

for small crack lengths. A possible solution to this could be the use of a potential drop

testing procedure.

7.1.3.2 Computations

Finite-element computations depend greatly on the quality of the mesh. In order to

ensure the sufficiency of the mesh, an independency study is performed as described

earlier. Although the study showed convergency, see Appendix A, an even finer mesh

could have given more accurate results. However, the computational cost would rise

immensely.

Furthermore, the initial dwell time of the specimen prior to the LCF-testing is ne-

glected. It is not accounted for since at the relatively low temperature of 400 °C stress

relaxation is not considered an issue.

Additionally, the used perfectly plastic material model is not entirely corresponding

to reality, although it is deemed sufficiently good.

7.2 Method discussion

In this section the above used method is discussed and alternatives are briefly pre-

sented. The used method incorporates only the plastic flow that occurs in the un-

cracked model. This gives a good approximation since the crack propagation is only

influenced by it in the early stages when it is close to the stress concentration, i.e.

notch. For the succeeding states a linear elastic response is used and thus the plastic


48 7 DISCUSSION

zone around the crack tip is neglected, see Figure 7.1. In an analysis that accounts for

elasto-plastic material behavior, elasto-plastic fracture parameters would have to be

calculated which is generally much more difficult because one has to keep track of the

deformed shape, stress and strain state at each step of the growth and map the state

forward as the crack grows. Since the elements at the crack tip blunt significantly

the remeshing is a considerable issue. Furthermore, it would make the use of SIFs

impossible which is a convenient measure of the state of the crack. All these issues

and the rather small influence of the results are the reasons for the choice of the above

used method.

In the above used method the complex behavior of small cracks is neglected since there

is no constitutive model to describe it. Inventing a model would go beyond the scope

of this thesis.

Figure 7.1: Plastic zone around the crack tip [26]


8 Conclusion and future work

8.1 Conclusion

To conclude, it can be stated that the two methods, Case 1 and Case 2 make it

possible to account nonlinear effects into the automated crack growth analyses by

FRANC3D. Both approaches are suitable for a stress controlled load case. If a model

requires a strain controlled load only Case 2 can be used due to above stated use of

the crack face traction which would result in a stress controlled simulation for Case

1. This implies that Case 2 can be used for both load cases, whereas Case 1 is only

suitable for stress controlled situations. This makes Case 2 a more universal approach.

However, as mentioned above Case 1 is the simpler choice in terms of setting up the

simulation and computational cost. Additionally to those two approaches the linear

elastic case, which does not account for non-linear effects, has been investigated. It

was found that for this particular load case and geometry and the considered strain

ranges and R-value, i.e. Rε = 0 it gives results very close to the ones of Case 2.

Thus, it is concluded that the linear elastic approach describes the crack propagation

and the cyclic life-time behavior reasonably well under the given conditions. This was

validated by the comparison to the test data which gave results close to the ones of

the linear elastic approach. Finally, it is concluded that the choice of method, i.e.

Case 1, Case 2 and the linear elastic case has to be chosen depending on the given

problem. In order to generalize the conclusions above, further geometries, R-values,

load ranges and materials have to be investigated. The three considered tests do not

provide sufficient information regarding the applicability on arbitrary situations.

Table 8.1: Summary

Computational time set-up comparison to testLinear elastic short easy good

Case1 medium medium not suitableCase2 medium more demanding good

49

50 8 CONCLUSION AND FUTURE WORK

8.2 Future work

Succeeding this work there are more issues to investigate. The influence of the load-

ing ratio R, the geometry, different load ranges and materials on the results is to be

assessed in order to verify that the, in this thesis, presented approaches are suitable

for arbitrary set-ups. Furthermore, the investigations of automatic crack growth sim-

ulations of anisotropic materials such as single crystals are desirable. For these kinds

of materials the crack growth models are not sufficient yet to achieve good results.

Improved models have to be established. Finally, the above shown methods shall be

utilized on more ductile materials where very high degrees of plasticity occur and the

results evaluated. In short:

• Different R-ratios, geometries, load ranges and materials

• Anisotropic materials

• More ductile materials


References

[1] Tore Dahlberg. Failure Fracture Fatigue. Studentliteratur, 2002.

[2] L.F. Coffin S.S. Manson. Coffin-manson relation. 1953.

[3] P.C. Paris M.P. Gomez and W.E. Anderson. A rational analytic theory of fatigue.

The trend in engineering, 13, 1961.

[4] K. Walker. The effect of stress ratio during crack propagation and fatigue for

2024-t3 and 7075-t6 aluminum.

[5] R.G. Foreman V.E. Keary and R.M. Engle. Numerical analysis of crack propa-

gation in cyclic loaded structures. Journal of Basic engineering, 89, 1967.

[6] Southwest research institute. www.nasgro.swri.org.

[7] Fracture Analysis Consultants Inc. Franc3D Reference Manual, November 2011.

[8] Dassault systèmes. Abaqus. ABAQUS Documentation.

[9] Fracture Analysis Consultants Inc. www.fracanalysis.com, 2014.

[10] T.L. Anderson. Fracture mechanics - Fundamentals and applications. Taylor and

Francis Group, 2005.

[11] Kaidor. Stress concentrations. [CC-BY-SA-3.0], via Wikimedia Commons.

[12] David Gustafsson. High temperature fatigue crack propagation behaviour of In-

conel 718. PhD thesis, Linköpings Universitet, 2012.

[13] F.V. Anthunes. High temperature fatigue crack growth of Inconel 718. 2000.

[14] R.C. Reed. The superalloys - Fundamentals and applications. Cambridge univer-

sity press, 2006.

[15] SIT. Material 638103. Technical Report K-190-638103, 2012.

[16] Tomas Månsson Jan Skanty and Fred Nilsson. High temperature fatigue crack

growth in two metals under constant and variable amplitude loading. Interna-

tional Journal of Fatigue, 24, 2002.

[17] Johan Moverare. Private communication, 2014.

51

52 REFERENCES

[18] Johan Moverare. Material technology base development - notch lcf testing of

blade superalloys at 500 ◦c. Technical Report 1CS59166, SIT, 2006.

[19] Fracture Analysis Consultants Inc. Franc3D ABAQUS Tutorial, November 2011.

[20] Bbanerje. Coordinate system at a crack tip. [CC-BY-SA-3.0], via Wikimedia

Commons.

[21] P.A. Wawrzynek and B.J. Carter. The m-integral for computing stress intensity

factors in generally anisotropic materials. Nasa, 2005.

[22] Valeriy Kruity. Sub-modelling of thermal mechanical fatigue crack propagation.

Master’s thesis, Rensselaer Polytechnic Institue, Hartford Connecticut, 2009.

[23] Fracture Analysis Consultants Inc. Advances in simulation of arbitrary 3d crack

growth using franc3d/ng. Presentation at conference.

[24] J. Loureiro C. Busse and R. Winberg. Stress intensity factors using abaqus. 2014.

[25] H. Tada G.R. Irwin and P.C. Paris. The stress analysis of cracks handbook, p.

303. ASME Press, 2000.

[26] Bbanerje. Plastic zone around the crack tip. [CC-BY-SA-3.0], via Wikimedia

Commons, May 2008.


Appendix

A Mesh study

As mentioned previously, the mesh independency study is an important tool in order

to ensure a sufficient quality of the mesh. Five different mesh configurations according

to Table A.1 have been investigated. For the study the stress at the notch has been

evaluated. Mainly the mesh density around the notch has been modified. Considered

are the critical load cases of the where the model is loaded with maximum and mini-

mum strain. The maximum strain is evaluated at the first and the second load cycle.

The minimum strain is considered only in the first load cycle. The results are shown

in Figure A.1 and A.2. As concluded above the mesh can be said to be independent

with mesh 3 at around 220.000 nodes. Further refining does not change the result

in a matter that would justify the increasing computational cost. The computational

power that was available for this simulations was:

• 8 FEA compute nodes, each with 2x hex-core Intel Xeon X5675 3.06 GHz (West-

mere), 96 GB RAM, 6x 300 GB HDD

• 6 FEA compute nodes, each with 2x octo-core Intel Xeon E5-2670 2.6 GHZ

(Sandy Bridge), 128 GB RAM, 8x 600 GB HDD

, where 16 cores have been used.

Table A.1: Mesh study: Comparison of meshes

Meshes 1 2 3 4 5No. of nodes 24033 74651 217527 376165 475819

Computational time [min] 11 18 31 300 1500

53

54 A MESH STUDY

0 50,000 100,000 150,000 200,000 250,0001.24

1.26

1.28

1.3

1.32

1.34

1.36

1.38

1.4

1.42

1.44x 109

Number of nodes

Stre

ss a

t not

ch [P

a]

Max stress 1Max stress 2

Figure A.1: Mesh study: Maximum strain

0 100,000 200,000 300,000 400,000 500,000−1.46

−1.44

−1.42

−1.4

−1.38

−1.36

−1.34

−1.32

−1.3

−1.28

−1.26x 109

Number of nodes

Stre

ss a

t the

not

ch [P

a]

Figure A.2: Mesh study: Minimum strain


B Figures

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

55

C Boundary Conditions

Figure C.1: Applied boundary conditions

56

D ABAQUS sub-routine

1 SUBROUTINE UAMP(2 * ampName, time, ampValueOld, dt, nProps, props, nSvars, svars,3 * lFlagsInfo,4 * nSensor, sensorValues, sensorNames, jSensorLookUpTable,5 * AmpValueNew,6 * lFlagsDefine,7 * AmpDerivative, AmpSecDerivative, AmpIncIntegral,8 * AmpDoubleIntegral)9 !

10 INCLUDE ’ABA_PARAM.INC’1112 ! time indices13 parameter (iStepTime = 1,14 * iTotalTime = 2,15 * nTime = 2)16 ! flags passed in for information17 parameter (iInitialization = 1,18 * iRegularInc = 2,19 * iCuts = 3,20 * ikStep = 4,21 * nFlagsInfo = 4)22 ! optional flags to be defined23 parameter (iComputeDeriv = 1,24 * iComputeSecDeriv = 2,25 * iComputeInteg = 3,26 * iComputeDoubleInteg = 4,27 * iStopAnalysis = 5,28 * iConcludeStep = 6,29 * nFlagsDefine = 6)30 dimension time(nTime), lFlagsInfo(nFlagsInfo),31 * lFlagsDefine(nFlagsDefine)32 dimension jSensorLookUpTable(*)33 dimension sensorValues(nSensor), svars(nSvars), props(nProps)3435 character*80 sensorNames(nSensor)36 character*80 ampName37 !38 PARAMETER (DispTot=0.3400E-04,StepTime=1.0)39 PARAMETER (zero=0.0d0, one=1.0d0, two=2.0d0, four=4.0d0)4041 DOUBLE PRECISION MonitorDisp1,MonitorDisp2,AmpValueOld42 DOUBLE PRECISION MonitorDisp3,Strain,init4344 lFlagsDefine(iComputeDeriv) = 045 lFlagsDefine(iComputeSecDeriv) = 046 lFlagsDefine(iComputeInteg) = 047 lFlagsDefine(iComputeDoubleInteg) = 048 !49 MonitorDisp1 = GetSensorValue(’SENSOR1’,50 * jSensorLookUpTable,51 * sensorValues)52 MonitorDisp2 = GetSensorValue(’SENSOR2’,53 * jSensorLookUpTable,54 * sensorValues)55 MonitorDisp3 = GetSensorValue(’SENSOR3’,56 * jSensorLookUpTable,57 * sensorValues)5859 tim = time(iStepTime)60 AmpValueOld = svars(1)61626364 init=0.001465 Strain=0.0086667 ! Diff Calculates the procentual differences of the current strain to target strain68 ! AmpValueNew corrects the old Amplitude via Diff69 IF (MonitorDisp1.GT.zero) THEN

57

58 D ABAQUS SUB-ROUTINE

70 Diff=(((MonitorDisp1-MonitorDisp2)/0.012371)-Strain)/Strain71 AmpValueNew=AmpValueOld*(1-Diff)7273 ! Init gives an initial value7475 ELSE76 AmpValueNew=init77 ENDIF7879 svars(1)=AmpValueNew8081 RETURN82 END


accounting for non-linear eﬀects in fatigue crack propagation simulations using...

Documents