Fatigue crack growth experiments and analyses- from small scale to large scale yielding atconstant and variable amplitude loading

Par Ljustell

Doctoral thesis no. 81, 2013KTH School of Engineering Sciences

Department of Solid MechanicsRoyal Institute of TechnologySE-100 44 Stockholm Sweden

TRITA HFL-0531

ISSN 1654-1472

ISRN KTH/HFL/R-12/15-SE

Akademisk avhandling som med tillstand av Kungliga Tekniska Hogskolan i Stockholmframlagges till offentlig granskning for avlaggande av teknisk doktorsexamen fredagen den1 februari kl. 10.00 i sal F3, Kungliga Tekniska Hogskolan, Lindstedtsvagen 26, Stockholm.

Failure is simply the opportunity to begin again, this time more intelligently.

Henry Ford

Abstract

This thesis is on fatigue crack growth experiments and assessments of fatigue crack growth

rates. Both constant and variable amplitude loads in two different materials are considered;

a nickel based super-alloy Inconel 718 and a stainless steel 316L. The considered load levels

extend from small scale yielding (SSY) to large scale yielding (LSY) for both materials.

The effect of different load schemes on the fatigue crack growth rates is investigated on

Inconel 718 and compact tension specimens in Paper A. It is concluded that load decreasing

schemes give a too high Paris law exponent compared to constant or increasing load amplitude

schemes. Inconel 718 is further analyzed in Paper B where growth rates at variable amplitude

loading in notched tensile specimens are assessed. The predictions are based on the fatigue

crack growth parameters obtained in Paper A. The crack closure levels are taken into con-

sideration and it is concluded that linear elastic fracture mechanics is incapable of predicting

the growth rates in notches that experience large plastic cyclic strains. Even if crack closure

free fatigue parameters are used and residual stresses due to plasticity are included. It is also

concluded that crack closure free and nominal fatigue crack growth data predict the growth

rates equally well. However, if the crack closure free parameters are used, then it is possible

to make a statement in advance on the prediction in relation to the experimental outcome.

This is not possible with nominal fatigue crack growth parameters.

The last three papers consider fatigue crack growth in stainless steel 316L. Here the load is

defined as the crack tip opening displacement parameter. Paper C constitutes an investigation

on the effect of plastic deformation on the potential drop and consequently the measured crack

length. It is concluded that the nominal calibration equation obtained in the undeformed

geometry can be used at large plastic deformations. However, two conditions must be met: the

reference potential must be taken in the deformed geometry and the reference potential needs

to be adjusted at every major change of plastic deformation. The potential drop technique

is further used in Paper D and Paper E for crack length measurements at monotonic LSY.

Constant amplitude loads are considered in Paper D and two different variable amplitude

block loads are investigated in Paper E. The crack tip opening displacement is concluded in

Paper D to be an objective parameter able to characterize the load state in two different

geometries and at the present load levels. Furthermore, if the crack tip opening displacement

is controlled in an experiment and the local load ratio set to zero, then only monotonic LSY

will appear due to extensive isotropic hardening, i.e. elastic shake-down. This is also the

reason why the linear elastic stress-intensity factor successfully could merge all growth rates,

extending from SSY to monotonic LSY along a single line in a Paris law type of diagram, even

though the generally accepted criteria for SSY is never fulfilled. For the variable amplitude

loads investigated in Paper E, the effect of plastic deformation on measured potential drop

is more pronounced. However, also here both the crack tip opening displacement parameter

and the linear elastic stress-intensity factor successfully characterized the load state.

Preface

The research presented in this thesis was conducted at the Department of Solid Mechanics,

Royal Institute of Technology (KTH). The research was financed by the Consortium for

Materials and Thermal Energy Processes (KME) and the Swedish Radiation Safety Authority

(SSM). The financial support is greatly acknowledged.

Fatigue crack growth is a very interesting and challenging area of solid mechanics. From an

industrial viewpoint, fatigue crack growth is of particular interest due to its close connection to

quality and safety, two factors which require special attention since products are increasingly

optimized in terms of material consumption, etc. With this in mind, I hope the outcome of

this thesis will help engineers in making fatigue design decisions when the current limits of

linear elastic fracture mechanics theory are exceeded.

As a PhD graduate I have had the opportunity to meet a lot of people from the Universities

and the Swedish industry. I would like to acknowledge some of them. First and foremost is my

supervisor Prof. Bo Alfredsson, whose patience, encouragement and support made it possible

for me to finalize this research project and thesis. His guidance as a scientist, mentor and

friend is greatly appreciated. Thank you Bo! My former supervisor, late Prof. Fred Nilsson

is also acknowledged. He gave me the opportunity to start my PhD studies and supported

me during the first years. Special thanks also go to Mr. Hans Oberg for contributions to

this work through his extensive experience in the experimental area. I learned a lot from the

interesting discussions we have had! The people in the mechanical workshop, Messrs Bertil

Dolk and Kurt Lindqvist, are also acknowledged for immediate support on test specimens

and fixtures. This thesis includes a few extensive numerical computations for which updated

computer hardware was required; for that I thank Mr. Per Berg. I also thank all my colleagues

and friends at the department for creating a good and inspiring atmosphere at work.

My sincere thanks also goes to Prof. Soren Sjostrom, Lic. Bjorn Sjodin (Siemens Industrial

Turbomachinery AB) and Dr. Tomas Hansson, Dr. Tomas Mansson (GKN Aerospace Sweden

AB) for the cooperation during the KME project. I really appreciate the industrial perspective

that you brought into the research!

My family is the solid foundation of my life, with whom I so gladly share my joy and

happiness. But it’s also the solid foundation that supports and keeps me up when life’s hard.

Thank you my dear Josefin for your support, encouragement and patience during these years!

Twice during my time as a PhD graduate, I have had the good fortune to become a father,

obviously the best thing that has ever happened to me. Jonatan and Elvira, you always bring

out a smile on my face and help me to focus on the important things in life, thank you!

Thanks also to my sister Camilla and my parents Mats and Mona-Britt, who have always

been there for me. Better sister and parents cannot be found!

Stockholm, November 2012

List of appended papers

Paper A: Effects of different load schemes on the fatigue crack growth rate

P. Ljustell and F. Nilsson

Journal of Testing and Evaluation, 34(4), 2006, 333 − 341

Paper B: Variable amplitude crack growth in notched specimens

P. Ljustell and F. Nilsson

Engineering Fracture Mechanics, 72(18), 2005, 2703 − 2720

Paper C: The effect of large scale plastic deformation on fatigue crack length measurement

with the potential drop method

P. Ljustell

Journal of Testing and Evaluation, 39(6), 2011, 985 − 1002

Paper D: Fatigue crack growth experiments on specimens subjected to monotonic large scale

yielding

P. Ljustell

Submitted for international publication

Paper E: Variable amplitude fatigue crack growth at monotonic large scale yielding experi-

ments on stainless steel 316L

P. Ljustell and B. Alfredsson

Submitted for international publication

Individual contributions

Paper A: Effects of different load schemes on the fatigue crack growth rate

P. Ljustell: Experiments; Numerical Analyses; Manuscript

F. Nilsson: Supervision

Paper B: Variable amplitude crack growth in notched specimens

P. Ljustell: Experiments on CT-specimens; Numerical Analyses; Manuscript

F. Nilsson: Supervision; Concept

Siemens Industrial Turbomachinery AB: Experiments on NT-specimens

Paper C: The effect of large scale plastic deformation on fatigue crack length measurement

with the potential drop method

P. Ljustell: Experiments; Numerical Analyses; Manuscript

B. Alfredsson: Supervision

Paper D: Fatigue crack growth experiments on specimens subjected to monotonic large scale

yielding

P. Ljustell: Experiments; Numerical Analyses; Manuscript

B. Alfredsson: Supervision

Paper E: Variable amplitude fatigue crack growth at monotonic large scale yielding experi-

ments on stainless steel 316L

P. Ljustell: Experiments; Numerical Analyses; Manuscript−figures

B. Alfredsson: Supervision; Manuscript−text

Contents

Introduction 11

Crack growth and fatigue of metals . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Engineering approach to the fatigue problem . . . . . . . . . . . . . . . . . . . . . 12

Project backgrounds and research motivation . . . . . . . . . . . . . . . . . . . . . 14

Crack closure 17

Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Origins to crack closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Crack closure effects on the fatigue growth parameters . . . . . . . . . . . . . . . . 19

Crack closure in growth rate assessments . . . . . . . . . . . . . . . . . . . . . . . . 21

Finite elements and crack closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

The applicability of linear elastic fracture mechanics 25

Intermediate scale yielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Large scale yielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

The potential drop method for crack length measurements 31

Characterization of fatigue crack growth outside the SSY limit 35

The ΔJ parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

The Δδ parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Conclusions 45

Summary of appended papers 49

Bibliography 53

Introduction

The subject for this thesis is fatigue crack propagation in metals. A description of the different

phases of the fatigue crack life will be given in this introductory part together with a short

overview of the two main approaches to fatigue design and a background description and

research motivation of the research project.

Crack growth and fatigue of metals

Crack growth is on the atomic level breakage and separation of the bonds linking the atoms

and/or movement and gathering of dislocations (imperfections in the atomic structure). Thus,

new surfaces are created in the solid as the crack nucleates and continues to grow. This can

be interpreted as an adaption of the material to an applied load of a critical level. Application

of a strictly increasing load to a flawed structure may not be critical at first since the load

needs to be at some critical level in order for crack growth to occur. At this point stable or

unstable crack growth occurs dependent on the characteristics of the material resistance to

continuous growth and the driving force dependence vs crack extension. However, at some

point, assuming a strictly increasing load, the component or structure will fail due to loss of

load bearing capacity or functionality.

In the case of fatigue, which is characterized by time variable loads, the amplitude of the

loads may be considerably lower than the level at which strictly increasing loading causes

fracture. This is due to local plasticity on the micro-scale, although the applied loads on a

structural level only give elastic stresses. The initiation position of a fatigue crack is often

found at some inclusion, impurity or surface flaw which acts as a local stress raiser giving

small scale plastic deformation. The inclusion may have sharp edges and very much act as

a crack right from the start or have a smooth surface giving a local stress concentration

where initiation of the crack starts. Another possibility for crack initiation is concentrated

plastic deformation on certain planes in the grains orientated such that shear deformation

occurs between them. These planes, or band of planes, are called persistent slip bands. As

the plastic deformation is locally accumulated with the number of load cycles local stress

concentrations are created and form initiations spots for fatigue cracks. At this stage the

nucleated fatigue crack size ranges from sub micrometer to a few tens of micrometers. It

11

is not certain that the fatigue crack will continue to grow. Important parameters here are

the applied load level combined with the micro-structural local conditions around the fatigue

crack. Obstacles such as grain boundaries, inclusions and unfavorable orientation of the

grains locally around the crack tip may act as shielding mechanisms for further growth. Some

materials show a distinct load limit dependence. If the load amplitude is above the limit,

then the fatigue crack continuous to grow and if it is below the limit, then the fatigue crack

stops to grow. The limit is called the endurance limit or fatigue limit, usually defined as 50 %

probability of survival at 106 load cycles. If the fatigue crack nucleates from persistent slip

bands at the surface of the component, then the first part of the fatigue life is often shear

driven with growth in the maximum shear direction. As the crack continues to grow into

the component a shift in direction generally occurs. The new direction is on the continuum

level perpendicular to the principal stress axis. If the fatigue crack nucleates at an inclusion

within the material, then the initial growth direction can be either in the maximum shear

direction or perpendicular to the principal stress axis somewhat dependent on the size, shape

and orientation of the inclusion. Further growth of the fatigue crack, of submillimeter length

and above, may be driven by plastic deformation on slip bands emanating from the crack tip.

The crack growth direction at this stage is perpendicular to the principal stress axis and the

growth rate can be charaterized from continuum theory. Finally, the fatigue crack becomes

sufficiently long for unstable crack growth to take place and the component functionality will

be lost.

Although the damage process to some extent is qualitatively known, still many questions

remain to be answered, in particular with respect to life computation. Also, a big challenge lies

within finding an explanation of how a crack is created in a material initially free from flaws.

Such local phenomena are of great complexity, both from a theoretical and computational

point of view, compared to an approach on the continuum level. Hence, the traditional

engineering treatment of the the fatigue problems with elastic theory on a continuum level.

Engineering approach to the fatigue problem

From the engineering point of view there are traditionally two ways to approach the fatigue

design problem. The first and the oldest approach is the safe-life design methodology. This

methodology is based on the assumption of no existing flaws in the component/structure and

no distinction is made between initiation and propagation of the fatigue crack. Furthermore,

the stress amplitude and mean stress level are chosen such that the fatigue life (number of

cycles to failure) is either finite or infinite based on the knowledge of the material fatigue

properties and other significant factors. In order to maintain the design safety when the

design life is used, the component is replaced with a new one, even though a considerably

large part of the functional life may remain. Another assumption is that no inspections of

12

the component/structure needs to made during the design life.

Traditionally, because of the observed large scatter in the fatigue life, safety factors are

included. These reduce the allowed load level, or instead of reducing the load level, increase

the dimensions of the component/structure. However, safety factors are hard to interpret

from reliability and a durability point of view. A better approach, allowing for an easier

interpretation of the component/structural reliability, is to use statistical methods to judge

the probability of survival. From a historical point of view the inherent scatter in the fatigue

life was observed as early as in 1860, which also is the time around which the safe-life design

methodology originates. However, the scatter was first treated on a scientific level around

1945. From the engineering point of view the use of statistical methods has still not penetrated

some branches of the mechanical industry. For a comprehensive historical review on fatigue

see for instance [1].

The second approach is called damage-tolerant design. Here it is assumed that initial

cracks exist in the component/structure and periodical inspections for the presence of these

needs to be performed in order to maintain the reliability. The size of the assumed initial

crack is connected to the largest crack size that may be missed during an inspection, which

in turn is set by the detection limit of the non-destructive technique used for the inspection.

The design life of the component/structure then becomes the number of in-service fatigue

load cycles required for the initial crack to grow to a critical size. The damage-tolerant

methodology consequently focus on predictions of the fatigue crack growth rate and the

remaining fatigue life whereas the safe-life design methodology focuses on estimating the

total life. Thus, estimations of the remaining fatigue life of a flawed component/structure

is only possible through use of the damage-tolerant approach. The strive for increasing the

accuracy and the precision of the estimated fatigue crack growth rate is driven by cutting

costs by increasing the inspection length interval since inspections are expensive (downtime

of the industrial process) while maintaining the safety requirements.

A theory suitable for the damage-tolerant design methodology is the fracture mechanics

theory. The fracture mechanics theory offers methods for estimating the driving force of the

applied load to the crack. The main part of the fracture mechanics theory was developed

during the 1950s and the 1960s. The required input information to the fracture mechanics

models for fatigue crack growth rate estimates are the in-service load, material properties and

geometry of the fatigue crack and component. Historically, the damage-tolerant approach

has been applied in areas where high demands on reliability and safety are required, often

combined with limitations on structural weight such as for the aviation and space industry.

The damage-tolerant methodology was first introduced in 1974 by the US Air Force in their

structural specifications. The cause to the change of structural specifications was that the

US Air Force did not manage to obtain sufficient structural integrity and durability of the

aircrafts [1].

13

The papers that are appended to this thesis may be categorized under the damage-tolerant

methodology. The papers address the fatigue crack growth from an experimental, numerical

and theoretical point of view based on fracture mechanics with emphasis on the first two

categories. In particular, the work focused on extending the life prediction methods to larger

crack tip loads than is currently accepted and to include some aspects of variable amplitude

loads.

Project backgrounds and research motivation

The research presented may be divided into two parts based on the research financing. The

first two appended papers, Paper A and Paper B, were conducted and reported within the

project KME-101 entitled ”Fatigue crack growth and life predictions in gas turbine materials”.

This project was a collaboration between The Department of Solid Mechanics at the Royal

Institute of Technology (KTH) and the Swedish gas turbine industry represented by Siemens

Industrial Turbomachinery AB and GKN Aerospace Sweden AB (former Volvo Aero Corpo-

ration AB). The project plan spanned three years (2001−2003) with a total budget of 4.6

MSEK, equally financed by the Swedish Energy Authority (Energimyndigheten) and the the

Swedish gas turbine industry. The project leader was the now late Prof. Fred Nilsson (KTH).

Other participants from the academia were Dr. Tomas Mansson (PhD graduate), Lic.1 Par

Ljustell (PhD graduate, who entered the project in mid-year 2002), research engineer Jan

Skantz and senior research engineer Hans Oberg. The companies’ main representatives were

Prof. Soren Sjostrom, Lic. Bjorn Sjodin (Siemens Industrial Turbomachinery AB) and Dr.

Tomas Hansson (GKN Aerospace Sweden AB).

The outcome from the project was two scientific papers, Paper A and Paper B, published

in international scientific journals, Ljustell and Nilsson [2, 3]. In addition, three other papers

were published within the project, Mansson et al. [4, 5] and Nilsson et al. [6]. The research

in the current Paper A and Paper B were conducted in parallel but mainly after the work in

papers [4, 5, 6] and naturally the papers share similar thoughts and approaches to the fatigue

crack growth problem. Furthermore, three academic exams are connected to the project; one

Doctoral degree Tomas Mansson (2002); one Licentiate of Engineering degree Par Ljustell

(2005); one Master of Science degree Peter Bodin (2004).

In order to stay competitive and increase customer value the products are optimized

and the material properties must be used as efficiently as possible. The gas turbine industry

identified the need of an improved treatment of the fatigue phenomenon which in turn required

a thorough understanding of the fatigue crack growth process. Thus, current practice needed

to be enhanced. The ultimate goal was to safeguard against fatigue failure of gas turbines

and the project goal was stated as ”To develop methods whereby complex fatigue crack

1Abbreviation for: Licentiate of Engineering degree. Swedish translation: Teknologie licentiat

14

growth situations can be analyzed and predicted using data from constant amplitude testing

of standard specimens”. The reason for developing methods based on data from constant

amplitude testing of standard specimens is that the majority of all existing fatigue propagation

data in the industry are tested accordingly. The following four topics were identified as

especially important for the gas turbine technology.

� Methodology development for testing under constant and variable amplitude loading.

� Growth of cracks initiated at stress concentrations under variable amplitude loading.

� Surface crack growth under variable amplitude loading.

� Mixed mode crack growth.

In all these areas, the goal has been to develop or verify engineering practice for treat-

ment of the fatigue crack growth phenomenon. The research presented in Paper A considers

methodology development for testing under constant amplitude loading while Paper B con-

siders predictions of growth rates of fatigue cracks initiated at stress concentrations under

variable amplitude loading.

The research in Paper C to Paper E constitutes the second part of the thesis. It was

conducted and reported within the project: ”Variable amplitude fatigue crack growth in

the low cycle life regime with reference to thermal fatigue”. This project was funded by the

Swedish Radiation Safety Authority (Stralsakerhetsmyndigheten2). The project plan spanned

three years (2004−2006) with a budget of 3.1 MSEK. The project ended in early 2007 but was

prolonged by SSM until mid 2012. In early 2009 Prof. Bo Alfredsson join the project and took

the project leadership as Prof. Fred Nilsson sadly passed away in mid 2009. Others attending

the project was senior research engineer Hans Oberg and Lic. Par Ljustell (PhD graduate).

The results from the project were published in three scientific papers currently submitted

to or published in international scientific journals; Ljustell [7, 8], Ljustell and Alfredsson [9].

Furthermore, van Eeten and Nilsson [10] established and published the constitutive model

that was later used in [7, 8, 9]. In addition, three academic exams are connected to the

project; one Doctoral dissertation Par Ljustell (this thesis); two Master of Science degrees,

Paul van Eeten (2005) and Christian Partel (2007).

The motivation for the research was that second only to stress corrosion cracking, thermal

cyclic and consequently thermal fatigue is the most common damaging mechanism in Swedish

nuclear power plants [11, 12]. The thermal fatigue mechanism is also identified as major life

limit issue worldwide. Three international projects have been finalized during the last decade,

THERFAT [13], NESC-V [14] and the Thermal Fatigue Benchmark (OECD/CSNI) [15]. The

THERFAT and NESC-V projects aimed at preparing the foundation for an European proce-

dure for handling of thermal fatigue of components in nuclear power plants. No comprehensive

2Abbreviation: SSM

15

instruction for handling the thermal fatigue problem was presented due to the complex nature

of the subject. However, the projects surveys and discusses the vast experience and interdisci-

plinary aspects on thermal fatigue, such as fluid mechanics, fracture mechanics and material

mechanics. Thus, the reports provide valuable information and guidance of handling thermal

fatigue.

The project background is that pipe systems in nuclear power plants are vulnerable to

thermal fatigue, especially at mixing points where fluids of different temperatures are mixed.

The turbulent flow in these areas act as a relatively high frequency thermal cyclic load on

the inner surface of the pipe. This is a well-known phenomenon that has been studied in the

past. The damage at the inner surface is best described as an ”elephant skin” or a ”dried up

lake bed of clay” with arresting surface cracks in a network pattern. The depth of the surface

cracks are relatively small compared to the thickness of the structure and the operational

initiation time may be less than a year. Unfortunately, several cases (about 45 in the NESC-

V report) with deep or through the thickness fatigue cracks are reported internationally for

operational components. The propagation phase is often shorter than the initiation time. The

NESC-V report discusses possible conditions for deep fatigue cracking and it is argued that

three-dimensional thermal distributions are a major factor driving the fatigue crack through

the pipe wall. The three-dimensional thermal distributions occurs at large scale instabilities

in the fluid flow, i.e. stratification of the fluid flow in combination with a turbulent flow or

turbulent penetration of a nominally stagnant fluid in a pipe. The thermal load frequencies

for these three-dimensional temperature distributions are low and consequently generating

large temperature differences throughout the wall thickness as oppose to a high frequency

turbulent flow.

Due to the difficulty in performing well-controlled crack growth experiments under thermal

loading a pure mechanical load was used for Paper C to Paper E. Therefore, some features

of thermal fatigue such as strain gradients and possible material temperature dependence

were not included here. However, in the early and preliminary experimental series on current

material (stainless steel 316L) no major difference was observed between the fatigue crack

growth rate at room temperature and at +280�C.

The research presented in Paper C establishes the experimental setup and evaluates the

important factors needed to be taken into consideration for low cycle fatigue crack propaga-

tion experiments. Paper D and Paper E reports the numerical and experimental low cycle

fatigue results with constant amplitude loading and some variable block amplitude loading,

respectively.

16

Crack closure

The fracture mechanics theory is commonly used to characterize the ”severity” of the applied

load on the fatigue crack tip. If the plastic deformation at the crack tip due to the applied

load is small relative to the characteristic dimensions of the geometry, then linear elastic

fracture mechanics (LEFM) is applicable. The parameter used for the linear description of

the load on the fatigue crack tip is called the stress-intensity factor, KI. Numerous researchers

have shown that the main factor controlling the propagation rate, in the case of fatigue crack

propagation, is the range of the stress-intensity factor, ΔKI. In 1963 Paris and Erdogan [16]

expressed the crack growth per cycle as a linear relationship with the stress-intensity factor

range in a double logarithmic diagram, cf. Fig. 1. The curve can be divided into three parts

where the linear part of the curve is described by,

da

dN= a0

(ΔKI

K0

)β

. (1)

The constant a0 is an arbitrary chosen reference growth rate and ΔKI is the entire stress-

intensity factor range. Equation 1 is a rewrite of the original definition by Paris in order to

get interpretable dimensions of the two parameters, K0 and β. The two parameters K0 and

β have shown experimentally to depend on the load ratio R = KI,min/KI,max.

Paris regime

Stage I Stage II Stage III

Stress-intensity factor range

Fatiguecrackgrow

thrate

Figure 1: Illustration of the fatigue crack growth rate vs the stress-intensity factor range.

17

Basic concepts

In the early 1970s Elber [17, 18] discovered that the fatigue crack can be partially closed even

if a global tensile load is applied. Elber observes a change of the compliance during loading

and interpreted this as a gradual change of the crack length due to closure of the crack faces

behind the crack tip. An effective stress-intensity factor range ΔKI,eff is suggested to be used

as the crack driving parameter instead of the nominal stress-intensity factor range ΔKI. The

effective stress-intensity factor range is defined as,

ΔKI,eff = KI,max −KI,closure, if KI,closure > KI,min (2)

where KI,closure is the stress-intensity factor level as the crack opens/closes. Elber finds that

the use of ΔKI,eff eliminates the R-ratio dependence in fatigue parameters K0 and β and all

fatigue data coincide to a common line in double-logarithmic diagram as long as ΔKI,eff is

correctly measured/calculated. Andersson et al. [19], Dougherty et al. [20] and many others

show the same result, eliminating the R-ratio dependence in fatigue parameters by use of an

effective stress-intensity factor range. Elber also states that the crack closure phenomenon

accounts for acceleration and retardation effects in variable amplitude crack growth. The

phenomenon has since then provided a rationale for different observed behaviours such as:

retardation of the fatigue crack growth rate due to overloads, the load ratio effect on the

growth rate, some of the observed fatigue threshold effects and small crack growth behaviour

to mention a few.

Another indirect definition of the crack closure level, with the aim of extending the appli-

cability of Eq. 1 below the load ratio where crack closure first is seen, is the one stating that

the difference between crack propagation rates at different R-ratios is due to crack closure.

This procedure requires the growth law, Eq. 1, to be first established by initial experiments

where the minimum nominal stress-intensity factor exceeds the closure level. These growth

law parameters, K0 and β, from ΔKI = ΔKI,eff are then independent of the R-ratio. The

effective stress-intensity factor range in other experiments with K0 (R) and β (R) dependent

of the R-ratio are then computed as

ΔKI,eff = K0

(ΔKI

K0 (R)

)β(R)β

. (3)

The crack closure level in R-dependent experiments is finally defined as the limit giving the

effective stress-intensity factor range from Eq. 3, i.e.

KI,limit = KI,max −ΔKI,eff, if KI,limit > KI,min. (4)

This idea is not new, cf. [21]. It eliminates the need for accurate crack closure measure-

ments during the actual test, but requires a fatigue crack growth rate experiment where no

18

crack closure is present. Also, with the two different definitions mentioned above it is possible

to assess the accuracy of the other crack closure measuring techniques, cf. Mansson et al.

[4]. This indirect definition of the crack closure level is used in Paper A and Paper B where

a nickel-based superalloy Inconel 718 is used at +400�C. Most of the existing fatigue crack

growth data in the industry today is of constant amplitude at different load ratios, since

such procedures are recommended in standards. By conducting one additional crack closure

free fatigue crack growth experiment it is possible to estimate the crack closure level in all

other constant amplitude experiments. This knowledge is of high value for the engineer when

predicting the fatigue crack propagation life of a component.

Origins to crack closure

Crack closure effects may have several different origins. The common base is some change of

length of the crack faces behind the crack tip or some obstacle between the crack faces. The

origin to this may be due to corrosive environment that causes the newly created crack faces

to oxidize/corrode. Another cause is the plastic deformation formed due to the growth of the

crack leaving a wake of deformed material behind the crack tip. This plastic stretch will vary

if the load amplitude is variable causing the closure level to change with crack length. This

type of crack closure is called plasticity induced crack closure. A third type may occur if a

Mode I growing fatigue crack is subjected to a Mode II overload. This mismatch between

the crack surfaces is called roughness induced crack closure. Due to the relative displacement

in the crack plane, retardation effects occur that may affect the crack growth rate several

millimeters ahead, Dahlin and Olsson [22, 23]. For a summary of proposed mechanisms, see

for instance Suresh [24].

Crack closure effects on the fatigue growth parameters

A study on the effect of different load schemes on the fatigue crack rate data, i.e. the exponent

β and the constant K0 in Eq. 1 is described in Paper A. It is found that ΔKI-reducing load

schemes effects the fatigue data and may result in a too high exponent β. Figure 2 shows the

evaluation of the exponent on the ordinate for three different experiments with constant load

ratios, two with load decreasing schemes and one with an load increasing scheme. Here the

stress-intensity factor range is kept constant at each load level for a crack extension sufficiently

long for attaining steady-state growth conditions before ΔKI is increased or decreased. Each

level is divided into five equally long growth increments Δa. The number on the abscissa tells

the number of increments included, counted from the last increment at constant ΔKI and

towards the point of load change, when determining the crack growth parameters.

Figure 2 shows that a shielding mechanism is present, reducing the fatigue crack growth

rate, in the two experiments with ΔKI-reducing load schemes. The shielding effect also

19

1 2 3 4 5

3

3.5

4

4.5

5

Number of crack length inc., Δa, incl. in the eval. of the fatigue exp.T

he e

xpon

ent,

β, in

Par

is la

w

Load decreasing test, R = 0.05, Δa = 0.4 mmLoad decreasing test, R = −1, Δa = 0.2 mmLoad increasing test, R = 0.5, Δa = 0.2 mm

Figure 2: The effect of different load schemes on the slope of the fatigue crack growth data. (Paper A).

diminish and finally disappears as the crack tip extends away from the point of load change,

observed as converging value of β when fewer increments Δa are included in the evaluation

of the fatigue parameters. Thus, plasticity induced crack closure is present. This transient

shielding effect is not present in the ΔKI-increasing experiment. The main recommendation

from Paper A is to use load increasing or constant ΔKI schemes at fatigue crack growth

testing. Decreasing load schemes introduce a risk of obtaining load history effected fatigue

crack growth data. This has implications not only on the fatigue crack growth parameters

but also on the fatigue crack growth threshold cf. Newman [25].

By arguments of self-similarity in LEFM the crack closure level should be proportional

to the maximum stress-intensity level. Thus, fatigue crack growth rates described by curves

with constant R-values should also be parallel in diagram such as Fig. 1, i.e. the curves

should have the same exponent β.

15 20 25 30 35

3

4

5

6

7

8x 10

−5

Crack length a / mm

Cra

ck g

row

th r

ate

/ mm

/cyc

R=

0.05

R=

0.1

R=

0.15

R=

0.2

R=

0.25

R=

0.3

R=

0.35

R=

0.4

R=

0.45

R=

0.5

R=

0.55

R=

0.05

R=

0.1

R=

0.15

R=

0.2

R=

0.25

R=

0.3

R=

0.35

R=

0.4

R=

0.45

R=

0.5

R=

0.55

Figure 3: Crack growth rate at constant ΔKI = 18 MPa√m and incremental increase of R. (Paper C).

One way of investigating the load ratio dependence of a material is to keep the ΔKI con-

stant while increasing R-value, starting the experiment with a low R-value. In the author’s

20

view, based on the research findings in Paper A and Paper C, the load ratio dependence ob-

served primarily consists of two components, crack closure and in some materials a maximum

load effect. Inconel 718 show no dependence of the maximum load level used at +400�C while

the stainless steel 316L used in Paper C may be sensitive to the maximum load level at room

temperature, Fig. 3. In the stainless steel 316L the fatigue crack growth rate increases rather

proportional to the increase of the R-ratio up to a R-ratio of approximately 0.2. Above this

limit the average crack growth rate is rather unaffected by the R-ratio until a R-ratio of 0.35

is applied. Here the crack growth rate starts to increase again indicating a dependence on

the KI,max-level.

Crack closure in growth rate assessments

Incorporating estimates of the crack closure level in predictions of the fatigue life can be

easy or very difficult dependent on the type of load history. If the component is exposed to

a constant amplitude load history a good estimate of the crack closure level would be the

experimentally obtained one, Eq. 3. For a totally random load history it is a lot harder to

estimate the crack closure level on a cycle by cycle basis. It would require complete knowledge

about the load history and the coupling between the load level and crack closure level since

plastic deformation is load history dependent. If complete knowledge is available, then it is

theoretically possible, although not practically possible to compute the crack closure level

with the finite element method (FEM). Consequently, empirical models such as the Wheeler

[26] and Willenborg et al. [27], which only account for retardation and not acceleration due

to underloads, and later Chang [28], which also includes acceleration are proposed. These

estimate the yield-zone size in which the crack growth is assumed to be affected. Empirical

crack closure models are also proposed, see for instance Bell and Schijve [29] and de Koning

[30]. Semi-analytical models such as the Dugdale or strip-yield models are also developed

and numerically implemented. These allow for numerical simulation of the crack closure

behaviour opposed to the empirical models which assume a specific behaviour. Since the

Dugdale and strip-yield models are based on plane stress a constraint parameter is introduced

in the numerical models. The constraint parameter allows in an artificial way to account for

geometrical constraints such as material thickness, crack location relative external boundaries

and type of applied load (tension/bending type of load). Furthermore, models assume elastic-

perfectly-plastic constitutive behaviour. Basically if none of the empirical or semi-analytical

models are at hand or unsuitable to the considered application, then the option for the

engineer is to make an average estimate of the crack closure level based on for instance

experimental data.

A third type of load history are those that are irregular but possible to describe with

repeatable load blocks. The typical application for such load histories may be start-stop type

21

of applications. In Paper B the crack growth rates are predicted in notched specimens for

a number of different variable amplitude load histories that are described by load blocks.

The predictions are based on constant amplitude experiments conducted in Paper A and on

a cycle weighted average of Eq. 1. For each load block the average crack growth rate are

computed as

(da

dN

)average

=n

n+m

da

dN(ΔKI,small) +

m

n+m

da

dN(ΔKI,large) , (5)

where the n and m are the number of small and large cycles in the variable amplitude block

sequence. The stress-intensity factor ranges are either chosen as nominal (the whole stress-

intensity factor range acts detrimentally to the crack tip) or as effective (the crack closure

level is taken into account, which reduces the stress-intensity factor range) ranges depending

on the evaluation method used. Here results from the effective method are presented in Fig.

4(b), i.e. Reduced data and the Closure computed with FEM for one type of load block, see

inset in Fig. 4(a). For the complete set of data see Fig. 6. In the Reduced data the crack

closure level is assumed to be constant and controlled by the largest load cycle in the load

block. This assumption is justified by the small distance traveled by the crack tip between two

large load cycles compared to the plastic zone size given by the large cycles. A steady state

crack closure level is thus expected, exclusively determined by the the large load cycle. Such

an approach are also used by Socie [31] and Mansson et al. [4, 5]. The effective stress-intensity

factor ranges are then computed according to Eq. 3.

0 1 2 3 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Crack length a / mm

Nor

mal

ized

clo

sure

val

ues

Pcl

/Pla

rge,

max

−1−0.5

00.5

1

P/P

larg

e,m

ax

(a)

0 1 2 3 4 5 610

−4

10−3

10−2

Crack length a / mm

da/d

N /

mm

/cyc

le

Closure computed with FEMReduced dataExperimental data, specimen no. 1Experimental data, specimen no. 2

(b)

Figure 4: (a) The crack closure level computed with the finite element method. Inset in the figure is also thetype of load cycle. (b) Predictions of experimental fatigue crack growth rates. (Paper B).

22

Finite elements and crack closure

Another way to obtain the crack closure level is to use the finite element method. The

development and implementation of the finite element models into computer codes in the

1970s gave further possibilities to analyze crack closure induced by plastic deformation at

the crack tip. The early models were very small and simple. But along with the immense

development of the computer capacity larger and more complex models can now be solved.

Solanki et al. [32] present an extensive review paper on simulations of plasticity induced

crack closure by the finite element method. The basic approach is to model the crack and

apply a cyclic load onto the component. Crack growth is then accomplished by releasing the

crack tip node at some stage in the load cycle. Combined with an elastic-plastic constitutive

model the plastically deformed material is built up in the wake and the crack closure level

can be studied as the first node behind the crack tip comes into contact (crack closure) or

leaves contact (crack opening) with the opposite crack surface. In Paper B, finite element

simulations of the crack closure levels are presented for two different types of load blocks. A

two-dimensional plane strain model is used and the plastic part of the constitutive model is

based on incremental, rate-independent classical plasticity with linear kinematic hardening.

Figure 4(a) shows the computed crack closure level averaged over two consecutive load blocks

vs crack length. Figure 4(b) shows the prediction of the fatigue crack growth rates (Eq. 5)

with the simulated crack closure level included (Closure computed with FEM ). One should

have in mind when interpreting such simulations that several researchers show a dependence

of the crack closure level on the finite element size, cf. McClung and Sehitoglu [33, 34],

Solanki et al. [35] and Gonzales and Zapatero [36]. Furthermore, the choice of constitutive

model may also influence the the results, cf. Jiang et al. [37] and Pommier and Bompard [38]

and Pommier [39].

23

24

The applicability of linear elastic

fracture mechanics

A prerequisite for accurate fatigue life predictions is to have reliable fatigue data at hand.

From the damage-tolerant point of view this means to have a proper description of the relation

between the applied load at the crack tip and the corresponding measured fatigue crack

growth rate. The purpose of conducting fatigue crack growth experiments is to establish

this relationship. The relationship needs only to be applicable to the specific geometry and

load from the fatigue design point of view. Thus, full-scale fatigue tests are an alternative

to establish the fatigue data and total life of the component. However, full-scale tests are

often very expensive in terms of time, technique and money and often practically impossible

to use as a stage in the iterative design process. The other choice is to rely on the concept of

similitude or transferability. The concept states that if there are enough similarities between

two situations one may be able to infer that the results would be the same or similar. This gives

the opportunity to carry out material tests on small specimens that are far cheaper than full-

scale tests and assume the same behaviour in both cases. The iterative design process would

then take place as simulations in the computer environment, with the material data from

the small test specimens, and drastically lowering the development costs. Fracture mechanics

theory provides the theoretical basis and measures for transfer of fatigue propagation data

from the small specimen in the laboratory to the full-scale field application.

The use of LEFM would be very limited if it only applies to the completely elastic ma-

terials. Fortunately LEFM can be used in more general situations under certain limitations.

These limitations are mainly connected to the size of the non-linearly deformed zone relative

the dimensions of the component. From a theoretical point of view considering the auton-

omy of the crack tip state, one can conclude that the only requirement needed for control of

the crack tip field by the stress-intensity factor is to have elastic boundaries and vanishing

contribution of higher order terms on the boundary between the linear and non-linear defor-

mation, cf. [40]. However, from a practical engineering point of view, a small scale yielding

(SSY) requirement is imposed on the non-linearly deformed zone in order for the similitude

concept to be applicable. The requirement states that the size of the plastic zone must be

significantly smaller than any dimension of the component. This limit is generally expressed

25

by the stress-intensity factor and yield strength, σY of the material,

min (crack length, uncracked ligament, thickness) > κ

(KI

σY

)2

. (6)

The constant κ is obtained by systematic experimentation or numerical computation by solv-

ing the full non-linear problem. In the linear elastic fracture toughness standard ASTM

E399 κ is conservatively set to 2.5. If Eq. 6 is fulfilled along with other requirements at

crack growth initiation, then the applied stress-intensity factor value is stated as the material

fracture toughness, KIc. This size conditions, i.e.

min (crack length, uncracked ligament, thickness) > 2.5

(KIc

σY

)2

, (7)

is often considered as a general limit of application of LEFM. However, the requirement on

the thickness is not connected to the applicability of the LEFM and the control of the crack

tip state by the stress-intensity factor. It is connected to the similitude concept since the

plastic zone size relative the component thickness determines the state of plane stress or

plane strain. The linear elastic fracture toughness, KIc depends on the component thickness

with a higher fracture toughness for plane stress states. This is not a problem as long as

the small experimental specimen is of the same thickness as the component to be used in the

application. However, the ASTM E399 standard has a requirement on the thickness implying

a plane strain state and assuring specimen stability against out of plane buckling.

In the case of fatigue crack growth the same kind of requirement as Eq. 6 is used. In the

Standard Test Method for Measurement of Fatigue Crack Growth Rates ASTM E647 κ is set

to 4/π,

min (crack length, uncracked ligament) >4

π

(KI

σY

)2

, (8)

primarily based on empirical results applicable to compact tension (CT)-specimens. In Eq.

8 the thickness requirement is relaxed in order for the standard to be applicable to thin

structures. However, one should be aware of the results may be dependent on specimen

thickness and the limitations connected to the concept of similitude.

Predictions of variable load amplitude fatigue crack growth rates in notched specimens,

Eqs 3 and 5 are presented in Fig. 6 and Paper B based on constant load amplitude parameters

from Paper A. For load block type, see inset in each subfigure, Fig. 6. The numbers on the

inset ordinates are normalized load levels based on the maximum load in each block. Two

different geometries are used, i.e. the concept similitude is tested. The fatigue parameters

obtained in Paper A fulfill Eq. 8 while the target fatigue data in Paper B do not. Figure 5

shows the maximum stress-intensity factor applied and the two measures on the right hand

side of the diagram reflects how well the conditions of LEFM are satisfied. According to ASTM

26

0 1 2 3 4 5 60

50

100

150

200

KI,m

ax /

MP

a(m

)½

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

(KI,m

ax/σ

Y)2 /B

0 1 2 3 4 5 60

2

4

6

8

10

12

14

16

18

20

(KI,m

ax/σ

Y)2 /a

Crack length a / mm

KI,max

(KI,max

/σY)2/a

(KI,max

/σY)2/B

SSY limit

Figure 5: The scale of the left axis gives the maximum stress-intensity factor. The two scales of the right axescorrespond to the two measures of the applicability of linear elastic fracture mechanics. (Paper B).

0 1 2 3 4 5 610

−4

10−3

10−2

Crack length a / mm

da/d

N /

mm

/cyc

le

Cycle type 1

0 1 2 3 4 5 610

−5

10−4

10−3

10−2

Crack length a / mm

da/d

N /

mm

/cyc

leCycle type 2

0 1 2 3 4 5 610

−4

10−3

10−2

Crack length a / mm

da/d

N /

mm

/cyc

le

Cycle type 3

0 1 2 3 4 5 610

−4

10−3

10−2

Crack length a / mm

da/d

N /

mm

/cyc

le

Cycle type 4

0 1 2 3 4 5 610

−4

10−3

10−2

Crack length a / mm

da/d

N /

mm

/cyc

le

Cycle type 5

0 1 2 3 4 5 610

−4

10−3

10−2

Crack length a / mm

da/d

N /

mm

/cyc

le

Cycle type 6

0

0.5

1

0

0.5

1

−1−0.5

00.5

1

0

0.5

1

−1−0.5

00.5

1

−1−0.5

00.5

1

Figure 6: Experimental fatigue crack growth rates (markers) from two specimens for each cycle type. Thepredictions (dashed dotted line) include consideration of crack closure according to Eqs 3 and 5, i.e.corresponding to Reduced data in Fig. 4(b). The numbers on the inset ordinates are normalizedload levels based on the maximum load in each block. (Paper B).

27

E647 they should fall below 0.785 for LEFM to be applicable in a CT-specimen. Reasonable

predictions of the fatigue crack growth rates are obtained as the fatigue crack becomes longer

than the notch affected zone, despite that the concept of SSY is never fulfilled. At crack

lengths a > 0.5 mm Fig. 5 gives κ = 1/15 ≈ 0.07, which is considerably smaller than

4/π ≈ 1.27 in Eq. 8. Note the growth rates from one experiment (cycle type 3 filled markers)

does not conform with the general trend. The specimen developed shear lips right from the

start which caused a decrease in the growth rates throughout the ligament.

The applicability of LEFM is further investigated for load levels above the commonly

accepted monotonic SSY limit given by Eq. 8, see Paper C to Paper E. A stainless steel

316L is used and the crack tip opening displacement (CTOD), denoted δ, is controlled in the

experiments. The applied load levels relative to the SSY limit given by Eq. 8 are presented

in Fig. 7(a). Each line represents a constant CTOD at the crack tip. The flow stress σFS

used in Eq. 8 is defined as the average of the yield strength and ultimate tensile strength,

since extensive hardening occurred. The maximum load levels applied in the experiments are

above the monotonic SSY limit. However, at cyclic loading following Rice’s scheme [41] by

replacing KI with ΔKI and σY with 2σY,cyc

min (crack length, uncracked ligament) >4

π

(ΔKI

2σY,cyc

)2

, (9)

the load levels fall below the cyclic SSY limit due to the extensive isotropic hardening and

elastic shake-down of the material, Fig. 7(b). Consequently, monotonic large scale yielding

(LSY) rather than cyclic LSY took place in the experiments.

15 20 25 30 35 400

0.5

1

1.5

2

2.5

3

3.5

4

Crack length a / mm

(KI,m

ax/σ

FS)2 /a

δmax

= 8 μm

12

16

20

24

28

SSY limit

Increasing δmax

(a)

15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Crack length a / mm

Δδ = 7.6 μm11.4

15.219

22.826.6

(ΔK

I/(2σ

Y,c

yc))

2 /a

SSY limit

Increasing Δδ

(b)

Figure 7: Comparison of load and crack length to the condition for LEFM stated in ASTM E647, (a) mono-tonic increase to the maximum load, and (b) the condition adapted to cyclic loading. (Papers Cto Paper E).

The experimentally obtained fatigue crack growth rates for Δδ = 7.6−26.6 μm in Paper D

are presented in Fig. 8. The stress-intensity factors are computed from measured crack

28

lengths and applied forces. Two different specimens are used; the CT-specimen and an in-

house developed edge crack tension panel, the (ECT)-specimen. Surprisingly, all data are

consolidated along the extrapolated line from the SSY crack closure free data, the blue circles

in Fig. 8(a), i.e. the stress-intensity factor is able to characterize and correlate the load state

at the crack tip throughout the entire load range from SSY to monotonic LSY. Two possible

explanations for the encouraging results may be: firstly, no crack closure is present in the

LSY fatigue data since it conforms with the SSY fatigue data obtained at crack closure free

conditions; secondly, elastic conditions is prevailing at steady state conditions due to cyclic

isotropic hardening and elastic shake-down.

10 20 30 40 50 60 708010

0

101

102

103

104

ΔK / MPa(m)1/2

da/d

N /

nm/c

ycle

Const. KI,max

, CT 10.0 mm,R

G = [0.5, 0.8]

CT 10.0 mm, RG

= 0.5

CT 10.0 mm, RL = 0.05

CT 19.5 mm, RL = 0.5

CT 19.5 mm, RL = 0.05

ECT 9.8 mm, RL = 0.5

ECT 9.8 mm, RL = 0.25

Lsq curve fit to const. KI,max

data

(a)

20 30 40 50 60 70 8010

2

103

104

ΔK / MPa(m)1/2

da/d

N /

nm/c

ycle

(b)

Figure 8: Linear evaluation of the fatigue crack growth data. (a) Summary view. (b) Enlarged view on themonotonic LSY fatigue data. (Paper D).

(a) (b)

Figure 9: Deformed specimens. (a) The CT-specimen and (b) the ECT-specimen. (Paper D).

Figure 9 show the deformed specimens after the experiments from which some of the

29

highest fatigue crack growth rates in Fig. 8 are obtained. The specimens show large rotation

and plastic deformation in the crack plane and also through the thickness.

The results in Figs 5−8 shows that the ASTM E647 criterion, Eq. 8 for applicability

of LEFM is conservative and may be relaxed in some situations. In Paper B a nickel-based

superalloy, Inconel 718, is used which shows limited hardening compared to the stainless

steel 316L used in Paper C to Paper E. The stress-intensity factor range characterized the

fatigue crack growth in both materials at loads outside the LEFM criterion limits. Thus, this

knowledge may be very valuable for the experienced engineer in the design situation. However,

for safety reasons a conservative criterion is necessary to insure the concept of similitude in

all situations.

Intermediate scale yielding

This load region is characterized by a plastic zone size at the crack tip that is about the same

as the characteristic dimensions of specimen; still the plastic zone should not interact with the

outer boundaries of the body. As stated above, the stress-intensity factor may be a possible

characterizing measure even in this region but will not always work. A better measure would

be the J -integral or the CTOD, which are able to account for the non-linear deformation.

The validity condition for elastic-plastic fracture mechanics (EPFM) is not equally well

established and developed as for LEFM. A similar criterion is suggested, cf. Hutchinson and

Paris [42] that any characteristic dimension l in the body should be larger than

l > κ2

(J

σY

), (10)

for the J -integral to be applicable.

The non-dimensional constant κ2 is usually set to 25−100 dependent on the type of

loading (pure bending or pure tension). In the ASTM E1820 standard for non-linear fracture

toughness testing κ2 is set to 25. If Eq. 10 is fulfilled, then the experimentally obtained

J-value may be interpreted as size independent fracture toughness value, JIc.

Large scale yielding

When extensive plasticity occurs and the size of the yield zone becomes large compared to

characteristic dimensions in the body the single parameter description breaks down. The

size of the J -dominated zone becomes strongly dependent on the configuration of the test

specimen, which means that the distributions of stresses and strains in the crack tip vicinity

are not unique.

The Paper B to Paper E make no distinction between ISY and LSY. Thus, data that falls

outside the monotonic SSY limit is termed LSY.

30

The potential drop method for

crack length measurements

The potential drop (PD) is one of two techniques recommended in ASTM E647 for measuring

the fatigue crack length at SSY conditions. The idea is to pass a constant electrical current

through the specimen and measure the potential difference across the crack mouth. The PD

across the crack increases with increasing crack length. Thus, the technique is well suited for

crack length measurements at SSY. Several researchers have also used the method for crack

closure measurements. The idea is that the current passes through the contact between the

crack surfaces as long as contact exists. Consequently, when the contact is released and the

crack opens, changes in the PD signal takes place until the crack is fully opened after which

the PD value remains constant. However, a number of studies indicates problems with the

PD technique for crack closure measurements, Bachmann and Munz [43], Pippan et al. [44],

while others find the technique useful, Shih and Wei [45], Spence et al. [46] and Andersson

et al. [47]. The problems are mainly connected to fatigue crack propagation in vacuum and

air in connection to the formation of an oxide layer on the crack surfaces which prevents

current flow. The PD technique was used in the present Paper C to Paper E for crack length

measurements. By keeping the local load ratio, RL = δmin/δmax above zero in all experiments

no crack closure is expected. Consequently, the PD technique is primarily used for crack

length measurements even though crack closure effects are searched for but never found.

Different factors that affect the measured PD values and consequently the calibration

equation that connects the normalized potential to the normalized crack length are inves-

tigated in Paper C. The material used is the stainless steel 316L. For SSY fatigue, several

studies have been conducted that investigate different techniques for establishing the calibra-

tion equation, cf. Clark and Knott [48], Wei and Brazill [49], Johnson [50]. However, when

using a normalized calibration equation a reference potential is needed giving the correct

crack length at an arbitrary but known calibration point. If a significantly higher load level is

applied relative to the load level at which the original reference potential was taken, a change

of the state may be expected due to geometry and resistivity changes. Thus, the original

reference potential and shape of the calibration curve may be affected by the plastic defor-

mation at the substantial load change, compared to the nominal calibration curve. Hence,

31

substantial load changes in the LSY range may give errors in measured fatigue crack length.

The error seems to be very small in the SSY region, cf. Ritchie and Bathe [51], Hicks and

Pickard [52] and Wilson [53].

The change in measured PD due to plastic deformation origins from a resistance change

which in turn consists of a geometrical part and resistivity part. Figure 10 shows the results

from a cyclic tension-tension test on a smooth cylindrical specimen where the PD is contin-

uously measured. The strain range is controlled and increased in steps up to a total strain

range of 3 % with 10−30 cycles at each strain range.

0 0.5 1 1.5 2 2.5 3−500

−400

−300

−200

−100

0

100

200

300

400

500

Plastic strain εpl

/ %

Tru

e st

ress

σ /

MP

a

(a)

0 0.5 1 1.5 2 2.5 3−1

0

1

2

3

4

5

6

7

Rel

ativ

e ch

ange

of r

esis

tanc

e ΔR

/R0 /

%

Plastic strain εpl

/ %

(b)

0 0.5 1 1.5 2 2.5 3−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Plastic strain εpl

/ %

Rel

ativ

e ch

ange

of r

esis

tivity

Δρ/

ρ 0 / %

(c)

Figure 10: (a) Cyclic stress-strain curve for Rε = 0. (b) Cyclic relative change of resistance for Rε = 0. (c)Cyclic relative change of resistivity for Rε = 0. (Paper C).

As shown in Fig. 10(a) the stainless steel 316L exhibits extensive strain hardening. The

Figs 10(b)−10(c) show the cyclic change of resistance and resistivity, respectively. The relative

change of resistance is on the same order as the plastic strain while the relative change of

resistivity is an order lower compared to the applied strain. The same relative change is

observed at a monotonic tension test, i.e. the resistivity contributed with <10 % of the total

32

resistance change. Assuming that the same relation holds in the PD measured crack length

situation, which may not be the case, the major part (>90 %) of the error in measured crack

length can be ascribed to a change in geometry while the effect of the material property

change will be relatively small (<10 %).

As stated, the plastic deformation will affect the resistance and consequently the crack

length. Figure 11(a) shows the ratio between the first and forthcoming PD measured crack

lengths for 15 different experiments on CT-specimens with two different thicknesses, 10 mm

and 19.5 mm. The maximum load level applied in each experiment corresponds to the load

level used later in the fatigue experiment. A shift of the PD measured crack length within

±10 % immediately occurred as the large load is applied. It means that an error of ±10 % in

measured crack length, or even more, can be expected if the calibration point is taken in the

nominal geometry. Expected outcome would be and an artificial increase in crack length but

surprisingly about half of experiments showed an artificial decrease in crack length at the first

load cycle. The reason for the diverging results is not known to the author. However, finite

element simulations of the geometrical effect on the PD signal all point towards an artificial

increase in crack length, see Paper C.

0 100 200 300 400 500 6000.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

a PD

/aP

D,0

Cycle number

10 mm19.5 mm

11639

(a)

100 120 140 160 1801

3

5

7

CM

OD

/ m

m

Time / s

100 120 140 160 180

0.74

0.76

0.78

V1 /

mV

Time / s

100 120 140 160 1801.341.361.381.4

1.42

V2 /

mV

Time / s

(b)

Figure 11: (a) The ratio between the first and the forthcoming PD measured crack lengths for 15 exper-iments. (b) Measures at first application of the load level used during the fatigue experimenton specimen 11639. Top: The crack mouth opening displacement, middle: the potential in thecracked specimen, below: the potential in the reference specimen. (Paper C).

Figure 11(b) shows the crack mouth opening displacement (CMOD), denoted ν, the PD

signal over the fatigue crack, V1 and the PD signal in the reference specimen, V2 from exper-

iment 11639. The first large load is applied at ∼ 110 s and initial drop in V1 is measured.

Towards the end of the load increase, V1 also started to increase and finally a too long crack

length is measured. The PD in the reference specimen is temporarily affected by the initial

large load change. Changes in V2 is not desired since it ought to be unaffected by changes in

other parts of the system. However, no long term affect from changes in geometry, material

33

hardening/softening or crack extension is observed in V2, which is most importantly.

Figure 12 shows the measured crack lengths vs cycle number. In order to verify the

PD sampled crack lengths additional visual measurements are conducted on one side of the

specimen. The main observation is the overall conformity of the two measurements techniques.

However, in some experiments with a crack growth over at least 10 mm, a weak tendency

existed for the PD method to give a too short crack length estimate compared to the visually

measured crack length. That tendency is consistent with the ideas that Wilkowski and Maxey

[54] schematically present. In spite of the weak tendency, as the results show, it seems that

the calibration curve taken from the nominal geometry and applied in the monotonic LSY

case (calibrated in the deformed state) still yields a reliable measurement.

0 1 2 3 4 5

x 104

10

15

20

25

30

35

40

Cycle number

Cra

ck le

ngth

/ m

m

aPD

aVisual

1163811631

Disturbances in the PD signal

11639

Figure 12: Crack length measurements on 10 mm thick CT-specimen. (Paper C).

34

Characterization of fatigue crack

growth outside the SSY limit

The limits given by Eqs 8−9 define in a conservative manner the applicability of LEFM.

Outside these limits LEFM may still be applicable. However, the applicability of LEFM

beyond Eqs 8−9 can only be judged based on EPFM which takes the non-linear constitutive

behaviour into consideration. The two main EPFM parameters suggested in literature for

characterization of fatigue crack growth at non-linear conditions are the ΔJ and Δδ.

The ΔJ parameter

The J -integral and ΔJ -integral are two measures used in the EPFM field. This means that

the measures are capable of characterizing the state at the crack tip even if the plastic zone

is not confined to a very small region compared to the in-plane dimensions. Still the plastic

zone may not interact with any in-plane dimension for the theory to be applicable.

The J contour integral is presented in 1968 by Rice [55]. He idealizes the elastic-plastic

non-linear constitutive behaviour to a non-linear elastic behaviour. Rice then shows that the

non-linear energy release rate J, can be written as a path independent line integral,

J =

∫Γ

(Wdy − Ti

∂ui∂x

ds

). (11)

Here are W the strain energy density, Ti are the components of the traction vector, ui are

the displacement vector components and ds is the length increment along the a contour Γ

around the crack tip.

The following equation needs to be satisfied in order for the contour integral to be strictly

path independent,

σij =∂W

∂εij. (12)

The requirement of elastic material response, linear or non-linear, may be relaxed to a least

approximately include elastic-plastic materials under the condition of monotonic loading. The

35

response of the two different constitutive models is approximately identical (apart from a small

change of the Poisson’s ratio) as long as no unloading occurs. In three dimensions this may

not be true because unloading may occur due to stress redistribution in the elastic-plastic

material. Still, the assumption of non-linear elastic behaviour may be a good assumption in

cases where a high triaxiality exists at the crack tip and no global unloading occurs.

The J -value can also be calculated as a deformation-J denoted JD. The deformation

JD is basically calculated as an integration of the load-deformation path. In an elastic-

plastic material, however, history dependence exists that is removed when the integration is

performed. The J -value from the contour integral and the load-deformation path may differ

somewhat dependent on the amount of crack growth in the elastic-plastic material.

The situation becomes a lot more complicated in the case of fatigue. When unloading

occurs in an elastic-plastic material, deformation plasticity theory no longer describes the

actual behaviour of the material. Thus, highly path dependent values result in the situation

of unloading and are therefore questionable to use in the fatigue situation. Despite the loss

of generality several researchers have tried to use the ΔJ when correlating the crack growth

rate with different degrees of success. The definition of ΔJ is first presented by Lamba [56]

in 1975 in connection to stress concentration factors.

The material in front of the crack tip experiences a cyclic stress-strain range and conse-

quently the process is characterized by Δσij and Δεij. Figure 13 shows a cyclic stress-strain

loop where the initial value is indicated with number 1 and final value with number 2. The

range of the J -integral is then defined as [56], Dowling and Begley [57] and Lambert et al.

[58]

ΔJ =

∫Γ

(ψ (Δεij) dy −ΔTi

∂Δui∂x

ds

). (13)

Here Γ is the integration path around the crack tip. Further, ΔTi and Δui are the changes

in traction and displacement between the initial point 1 and the finial point 2. In addition,

ψ corresponds to the strain energy density and is defined in an analogous way by

ψ (Δεij) =

∫ ε2ij

ε1ij

Δσijd (Δεij) =

∫ ε2ij

ε1ij

(σij − σ1ij

)dεij. (14)

Usually only the loading part of the cyclic loop is included in the integration of the strain

energy density rather than the whole loop. If the initial point 1 is located at zero stress

and strain then ΔJ = J . Equation 13 is a generalization of Eq. 11 to include the situation

when the stress and strain at point 1 is not at zero. The requirements for ΔJ to be path

independent are analogous to those for the original J mentioned above, i.e. σij = ∂W/∂εij .

As in the case of monotonic loading it is also possible to calculate the ΔJD from the cyclic

load-displacement curve. The general form is written as

36

ΔJD =η

Bb

∫ ΔV

0ΔPd (ΔV ) =

η

Bb

∫ Vmax

Vmin

(P − Pmin) dV, (15)

where η is dimensionless constant, B the specimen thickness and b the uncracked ligament.

Pmax, Pmin, Vmax and Vmin are the maximum and minimum load and displacement, respec-

tively, during the specific load cycle.

(σ1ij , ε1ij)

(σ2ij, ε2ij)

Strain

Stress

Figure 13: Stress-strain curve.

If Eq. 10 is met, then Eqs 13 and 15 should give the same result. In general, Eqs 13 and

15 do not necessarily give the same value under equal conditions. This may happen when

excessive plastic deformation occurs in the geometry (LSY). Still, the state at the crack tip

may be characterized by Eq. 15 meaning that the path independence is lost in Eq. 13.

Several researches: Tanaka et al. [59], Mowbray [60] and Rolfe and Barsom [61] show

experimentally that ΔJD is an unambiguous parameter correlating the fatigue crack growth

rate at SSY and cyclic LSY conditions. The fatigue crack growth rates fall onto a single line

plotted in a double logarithmic diagram vs ΔJD. However, the parameter is not suitable for

all situations. For instance, fatigue crack growth at high load ratios, load control and LSY

is a situation shown to be difficult for ΔJD to correlate due to ratcheting effects, Tanaka et

al. [62] and Dowling [63]. While ΔJD has shown to correlate fatigue data at both SSY and

LSY one cannot expect ΔJ calculated as a contour integral to do the same. If the body is

plastically deformed out to the boundaries, i.e. LSY then, the behaviour locally at the crack

tip becomes dependent on the overall behaviour of the geometry and thereby result in a path

dependent ΔJ . However, in the SSY region [64] and in the ISY region both parameters are

expected to give similar results.

The advantage of the ΔJD measure, compared to both ΔJ and Δδ, lays with the possibility

to evaluate the parameter from the load-displacement data. Furthermore, the theoretical

base for use of ΔJ in elastic-plastic materials is not without doubts. The definition of the

path independence of the J -integral relies on linear elastic, non-linear elastic or deformations

37

plasticity. Another issue is the crack closure mechanism which is problematic to incorporate

into the contour integral, however, possible in the ΔJD concept, cf. [57]. Furthermore, the

physical meaning of ΔJ and ΔJD is also hard to interpret.

The Δδ parameter

An easily interpreted measure is the crack tip opening displacement (CTOD), denoted δ.

Originally this parameter was developed by Wells [65] who discovers that several structural

steels could not be characterized by LEFM, i.e. KIc is not applicable. He also discovers while

examining the fracture surfaces that the crack surfaces moves apart prior to fracture. Plastic

deformation precedes the fracture and the initially sharp crack tip is blunted. The plastic

deformation increases with increasing fracture toughness and Wells proposes the opening at

the crack tip as a fracture toughness parameter.

The CTOD has no unique definition. The most common one was defined by Rice [55].

It is measured as the distance between two lines separated by 90◦ lines extending from the

crack tip and intersecting the crack surfaces behind the tip, shown in Fig. 14. Others define

the CTOD at the elastic-plastic boundary or at a distance beyond the plastic zone where the

crack face is no longer deformed. The CTOD range is used in connection to fatigue, i.e. Δδ.

90◦ δ

Figure 14: A common definition of the CTOD.

The drawback with the CTOD measure, compared with the JD, is the rather expensive

experimental equipment that is required for the measurements. However, the CTOD measure

has several advantages. It has an obvious physical interpretation and crack closure can easily

be incorporated. Furthermore, it is a local parameter defined close to the crack tip and

possible to evaluate with finite elements, even though a fine mesh is required.

The definition of δ that Paper C to Paper E use is very similar to the one in Fig. 14.

However, instead of using the intersection between the crack surface and the two straight lines

emanating from the crack tip at 90◦ a straight line is fitted to node number 2 and 3 behind

the crack tip and that line is regarded as the crack surface, see Fig. 15. By excluding the

first node behind the crack tip less scatter is achieved in the parameter. Hence, node number

38

15.49 15.492 15.494 15.496 15.498 15.50

1

2

3

4

5x 10

−3

Crack length a / mm

Def

orm

atio

n / m

m

Crack tip

45°

Lsq fit to node 2 and 3

Node number 2 and 3 behind the crack tip

½ δ

Figure 15: Definition of δ. (Paper C to Paper E).

2 and 3 are used as a good compromise between capturing the local behaviour at the crack

tip while minimizing the scatter in the computed Δδ.

The mesh size dependence in δ is investigated in Paper D. No obvious convergence is

observed. However, in order to have a properly computed Δδ-value a minimum element size

is required. The required element size is found to be strongly dependent on the definition of

the CTOD parameter and the mesh structure in the crack tip region. No published results

are found on the convergence behaviour of the CTOD parameter in connection to fatigue

and decreasing element size including possible crack closure. A comparison with convergence

studies on the required element size for computation of the crack closure level show that

no convergence is achieved with decreasing element size, cf. Gonzalez-Herrera and Zapatero

[36]. The comparison is justified by fact that both measures (δ and the crack closure level)

are evaluated in the vicinity behind the crack tip.

The fatigue crack growth experiments in Paper C to Paper E are controlled through δ.

Based on finite element computations for the two geometries in Fig. 9 the coupling between

δ, applied external load P , and crack length a, is derived as

Δδ = 2a0

(ΔP

PL (a0)

)2

f1

( a

W

)(16)

and

δmax = 2a0

(Pmax

PL (a0)

)2

f2

( a

W

). (17)

Here, a0 is a constant reference crack length set to 12.5 mm for the CT-specimen and 6 mm

for the ECT-specimen; PL (a0) is the limit load at a0; ΔP and Pmax are the load range and the

maximum load; f1 and f2 dimensionless functions obtained by non-linear least square curve

fits to finite element results, see Fig. 16 for results on the ECT-specimen. At linear conditions

δ is proportional to the square of the applied load P . Hence, the square in Eqs 16−17. The

limit load and the reference crack length a0 are added in order to have dimensionless functions

f1 and f2.

39

Figure 16 displays the result from two-dimensional plane deformation finite element com-

putations on the ECT-specimen where a cyclic load with constant maximum and minimum

load is applied. The global load ratio, RG = Pmin/Pmax was 0.5 in all computations. Each

computation continued until gross plastic deformations occurred and finally stopped due to

plastic collapse of the ligament, i.e. a crack is propagated through the ligament. Each line

shows the development of δ as a function of crack length. Filled markers represent an applied

maximum load that is larger than the limit load at the evaluated crack length and opposite,

an unfilled marker represent a load level lower than the limit load. The black unmarked lines

in Figs 16(c)−16(d) are the dimensionless functions f1 and f2.

5 10 15 20 25 30 350

2

4

6

8

10

12

Crack length a / mm

Δδ/2

/ μm

10.26 kN/mm12.82 kN/mm15.39 kN/mm17.95 kN/mm20.51 kN/mm23.08 kN/mm25.64 kN/mm

Increasing load

(a)

5 10 15 20 25 30 350

2

4

6

8

10

12

14

16

18

Crack length a / mm

δ max

/2 /

μm

10.26 kN/mm12.82 kN/mm15.39 kN/mm17.95 kN/mm20.51 kN/mm23.08 kN/mm25.64 kN/mm

Increasing load

(b)

5 10 15 20 25 30 350

1

2

3

4

5

6

7x 10

−3

Crack length a / mm

Δδ/(

2a0(Δ

P/P

g(a0))

2 )

10.26 kN/mm12.82 kN/mm15.39 kN/mm17.95 kN/mm20.51 kN/mm23.08 kN/mm25.64 kN/mmNonlinear Curve Fit

f1

(c)

5 10 15 20 25 30 350

0.5

1

1.5

2

2.5

3

3.5x 10

−3

Crack length a / mm

δ max

/(2a

0(Pm

ax/P

g(a0))

2 )

10.26 kN/mm12.82 kN/mm15.39 kN/mm17.95 kN/mm20.51 kN/mm23.08 kN/mm25.64 kN/mmNonlinear Curve Fit

f2

(d)

Figure 16: The CTOD vs crack length for different load levels in the ECT-specimen. (a) Δδ/2. (b) δmax/2.(c) Normalized Δδ/2. (d) Normalized δmax/2. (Paper D).

In Paper D constant amplitude loading is investigated in terms of δ, i.e. the force is

continuously decreased with the fatigue crack growth. The applied load level is computed on

a cycle-by-cycle routine based on Eqs 16 and 17. Given the user specified Δδ and δmax along

with measured crack length (aPD) in the previous load cycle, ΔP and Pmax are computed

40

and applied in the next load cycle. The crack lengths are simultaneously measured using

direct current PD, compliance and visually by a low magnifying long distance microscope.

The compliance based measurements are excluded from the figures due to large scatter. The

microscope allowed for manual in-plane crack length measurements on one side of the speci-

men. Since LSY occurred and in the case of the CT-specimen large accumulated rotations,

see Figs 9 and 17(b), this is taken into account by correcting for the rotation according to

aVisual = anotch +

√a2meas +

(ymeas

2

)2, (18)

where ymeas is the vertical distance at the electro discharge notch root and ameas the horizontal

distance from the notch root to the crack tip, see Fig. 17(a). Here, anotch = 10 mm.

ameas

10 aVisual

ym

eas

Initial loadline

Current loadline

(a) (b)

Figure 17: (a) Deformed specimen that illustrates the rotational compensation of visual measured cracklength, aVisual. (b) Specimen 11901, after completed experiment. (Paper E).

Figure 18(a) shows the measured crack lengths vs cycle number from the constant am-

plitude experiments on the ECT-specimen. The results show a relatively linear relationship

between the crack length and cycle number. This supports the assumption that the CTOD

is able to characterize the crack tip load state at monotonic LSY fatigue crack growth. This

rather linear relation is also seen in the results from the CT-specimen, see Paper D. The

deviation from the linear trend towards the end in some experiments are connected to large

extrapolations, i.e. the validity range for the control functions are 5 ≤ a ≤ 35 mm for the

ECT-specimen.

41

0 2 4 6 8

x 104

15

20

25

30

35

40

45

50

55

12518−4

Cycle number

Cra

ck le

ngth

/ m

m

12518−2

12519

12520

aPD

aVisual

(a)

15 20 25 30 35 40 45 50 550

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

Crack length / mm

Cra

ck g

row

th r

ate

/ μm

/cyc

le

12520

1251912518−4

12518−2

PD dataVisual dataPrediction(a

PD)

(b)

Figure 18: Fatigue crack growth data for the 9.8 mm thick ECT-specimen. (a) Crack length vs cycle number.(b) Fatigue crack growth rate vs crack length. (Paper D).

The corresponding fatigue crack growth rates vs crack lengths for the ECT-specimen are

shown in Fig. 18(b). As a consequence of the relatively linear relations in Fig. 18(a) the rates

are rather constant, except towards the end of the experiments due to extrapolations. Figure

18(b) includes predictions of the growth rates based on LEFM. These predictions correspond

to the data presented in Fig. 8.

0.1 1 1010

0

101

102

103

104

Δδ / μm

da/d

N /

nm/c

ycle

Const. KI,max

, CT 10.0R

G=[0.5,0.8]

CT 10.0, RG

= 0.5

CT 10.0, RL = 0.05

CT 19.5, RL = 0.5

CT 19.5, RL = 0.05

ECT 9.8, RL = 0.5

ECT 9.8, RL = 0.25

Lsq curve fit to constant KI,max

data

(a)

2 3 4 5 6 7 8 910 20 30 4010

2

103

104

Δδ / μm

da/d

N /

nm/c

ycle

(b)

Figure 19: The fatigue crack growth data vs Δδ. The black line is a least square curve fit to the con-stant KI,max-data. (a) Summary view. (b) Enlarged view on the monotonic LSY fatigue data.(Paper D).

Figure 19 summarizes all fatigue crack growth rates vs Δδ from both the ECT- and CT-

specimen. The blue circles are the constant KI,max-data obtained at SSY conditions. The

black line is a least squares fitted line, exclusively fitted to the constant KI,max-data. The

other markers are the fatigue crack growth data obtained at monotonic LSY conditions. All

42

fatigue crack growth rates are consolidated vs Δδ, cf. Tanaka et al. [59] and McClung and

Sehitoglu [66]. Hence, the objectivity of Δδ was checked up to monotonic LSY and δ can be

considered as a correlating parameter giving transferable results between geometries. This is

a prerequisite for predictive purposes in engineering fatigue design.

Furthermore, the fatigue crack growth at variable amplitude loading is described in Pa-

per E. Experiments similar to the ones in Paper D are conducted. However, here two different

load histories are used, see Fig. 20. The load blocks consist of one large cycle and eight small

cycles and are continuously repeated throughout the experiments. The load ratios for the

large cycles are RL = 0.05 and for the small cycles RL = 0.7 − 0.8 in sequence 1, Fig. 20(a).

In sequence 2 the load ratio is RL = 0.05 for all cycles, Fig. 20(b).

0 1 2 3 4 5 6 7 8 9

δ

Cycle number

(a)

0 1 2 3 4 5 6 7 8 9

δ

Cycle number

(b)

Figure 20: Block load: (a) sequence 1, large cycle followed by 8 small cycles at high maximum load, i.e.effect of underload, and (b) sequence 2, large cycle followed by 8 small cycles at low load, i.e.effect of overload. (Paper E).

Figures 21(a) and 21(b) display examples of measured and predicted fatigue crack growth

rates for an experiment with load sequence 1 and 2 respectively. The data are based on

three independent crack length measurement systems; the low magnifying long distance mi-

croscope (Visual data); the direct current PD technique (PD data and Prediction(aPD));

post-experiment measurement of the start and stop crack length according to the average

nine-point formula along the crack front recommended in ASTM E1820 (Prediction(aASTM)).

Note that all rates are plotted against aPD, which was selected for the abscissa because it was

used for controlling the experiments.

The predictions are based on crack closure free fatigue parameters from Paper D (the

KI,max-experiment) and on the cycle weighted average of the growth rates according to Eq.

5. No crack closure are observed in the PD data. This is also supported by the fact that

RL > 0 for all experiments and crack lengths. The respective ΔKI-values are determined

using force and crack lengths from the experiments. The predictions based on aPD are of the

43

same size but consistently lower than the measured crack growth rates. The difference are

due to a consistently shorter measured aPD relative to aVisual. The difference also increased

with crack extension due to accumulated plastic deformations and consequently increased

deflections. The reason is the effect of geometrical changes on the reference potential, which

could have been solved by taking a new reference potential. However, that is not conducted.

The aASTM-based predictions utilize the measured aPD-values but linearly interpolate the

intermediate crack lengths to coincide with the start and stop aASTM-values. The predictions

are consistently higher than those based on aPD and in most cases somewhat below the

measured rates but in some cases they over predict the crack growth rates.

27 28 29 30 31 32 33 34 35 360

0.4

0.8

1.2

1.6

2

2.4

2.8

3.2

3.6

4

4.4

Specimen: 12475

Crack length / mm

Cra

ck g

row

th r

ate

/ μm

/cyc

le

PD dataVisual dataPrediction(a

PD)

Prediction(aASTM

)

(a)

19 20 21 22 23 24 25 26 270

0.6

1.2

1.8

2.4

3

3.6

4.2

4.8

5.4

6

Specimen: 11897

Crack length / mm

Cra

ck g

row

th r

ate

/ μm

/cyc

le

PD dataVisual dataPrediction(a

PD)

Prediction(aASTM

)

(b)

Figure 21: Crack growth rates at variable amplitude experiments for (a) sequence 1 with underloads and for(b) sequence 2 with overloads. (Paper E).

The present experiments are performed on CT-specimens and a directly comparable load

measure is not available in literature. McClung and Sehitoglu [66] quantify the cyclic LSY load

on centre cracked specimens in terms of strain over the crack. For variable load amplitudes

and R = −1, the strain ranges are Δεlarge = 0.5 % and Δεsmall = 0.2 % in their experiments.

Returning to Paper D the growth rates on both the ECT- and CT-specimens are transferable

between specimen types and the deformation in one ECT-experiment corresponded to Δε =

0.1 − 0.3 % with crack growth. The load in the actual experiment is Δδ = 5 μm and

δmax = 10 μm. For the variable load amplitude series in Paper E Δδlarge ≈ 20 − 28 μm and

Δδsmall ≈ 4− 11 μm, which suggests that the present loads correspond to Δεlarge ≈ 0.4− 1.2

% and Δεsmall ≈ 0.1 − 0.3 % for a set-up following that of McClung and Sehitoglu [66].

Hence, expressed in specimen Δε the variable amplitude loads in Paper E seemed to be in the

same range as the LSY fatigue loads that McClung and Sehitoglu use. However, the present

variable load amplitude experiments are closure free with RL = 0.05 or 0.5 (RG ≈ 0.4− 0.6)

and on a material with substantial monotonic hardening whereas McClung and Sehitoglu use

RG = −1 with crack closure and reversed plastic deformation at the crack tip.

44

Conclusions

During the present work a series of conclusions were made on fatigue crack growth experi-

ments, the possibilities to use linear elastic fracture mechanics beyond its established limits

and conditions for such extensions. The results are presented in detail in the appended papers.

The main conclusions are summarized here.

� The accuracy of fatigue crack growth rate assessments and life predictions are heavily

dependent on the fatigue crack growth rate data obtained from experiments. In order

to ensure generality of the fatigue growth data and successful transferability of results

between test specimen and applications no load history effects is desired in the fatigue

crack growth experiments.

– When establishing the fatigue crack growth data, the experimental ΔK-reducing

procedures should be avoided. The reason is to minimize the risk of introducing

transient crack closure effects into the data, which is not representative for the

load history of the assessed application.

– One method of producing fatigue crack growth data with minimal transient load

history effects is to propagate the crack under constant low ΔK for a couple of

millimeters and then do the same at a high ΔK. Thereby the ligament is used for

producing fatigue data with high accuracy at two different load levels.

– The constant KI,max-method gives a statistically significant lower slope of the fa-

tigue data compared to other load schemes in the nickel-based superalloy Inconel

718 at +400�C. The difference in fatigue crack growth rates is small in the actual

measuring range but extrapolations should be avoided as the errors may become

significant.

� Even though engineering structures in general exhibit variable amplitude loading his-

tories, most of the existing fatigue crack growth data in the industry are of constant

amplitude type. Ideally, it should be possible to predict variable amplitude crack growth

rates with experimental data from constant amplitude fatigue crack growth experiments.

– Mean fatigue crack growth rates are computed from variable amplitude loading

experiments described with repeated simple load blocks. Predictions are based on

45

constant amplitude fatigue crack growth data. Although experimental data showed

rather large a scatter it was concluded that variable amplitude crack growth rates

can be predicted with reasonable agreement using constant amplitude fatigue data.

– Due to the relative levels of large and small loads and the rather limited number

of small cycles within each load block in the present experiments, the large cycle

influenced the overall crack growth rate for a block.

� It is desired from an engineering point of view to use linear elastic fracture mechanics in

fatigue life assessments due to the increased complexity of the analyses if one substitutes

linear elastic fracture mechanics with elastic-plastic fracture mechanics. Examples of

such situations are fatigue crack growth in notch affected regions and fatigue crack

growth at load levels above the commonly used monotonic small scale yielding criterion

given by American Society for Testing and Materials (ASTM).

– The crack growth rates in a notch affected region could not be predicted by linear

elastic fracture mechanics despite consideration of crack closure and residual stress

effects. However, the fatigue crack growth rates are predicted as the length is

comparable to the plastic zone size due to the notch (0.5 mm).

– The use of crack closure free fatigue crack growth data in fatigue crack growth

rate assessments enables statements in advance about the prediction accuracy in

relation to the expected outcome from the application. This is not possible to do

with nominal fatigue crack growth data.

– For the nickel-based superalloy Inconel 718, crack closure free fatigue growth from

the small scale yielding load range is able to predict growth rates where the load

exceeded the monotonic small scale yielding criterion by a factor of about 15.

– For the stainless steel 316L, crack closure free fatigue growth data at small scale

yielding is able to predict growth rates where the load exceeded the monotonic

small scale yielding criterion with approximately a factor 4. The main reasons are

concluded to be the large isotropic hardening of the material that gave small scale

cyclic yielding and the absence of crack closure.

– For the stainless steel 316L it is noted that if the crack tip opening displacement

range Δδ is kept constant and the local load ratio is close to zero, then the corre-

sponding cyclic small scale yielding criterion will still be fulfilled due to the elastic

shake-down in the material. Also, in a fatigue experiment with maximum load

levels far beyond the monotonic small scale yielding limit.

� The crack tip opening displacement range Δδ, which is a non-linear elastic-plastic frac-

ture mechanics parameter, is investigated as measure for characterizing the load state

46

at the fatigue crack tip at monotonic large scale yielding. Both constant amplitude

and variable amplitude load histories constituting of repeated load blocks are consid-

ered. The crack tip opening displacement is controlled in the experiments and ideally

an applied constant Δδ would result in a constant fatigue crack growth rate.

– The experimental results show sufficiently constant fatigue crack growth rates to

conclude that Δδ is able to characterize the load state at monotonic large scale

yielding constant and variable amplitude fatigue crack growth.

– The fatigue crack growth rates from two different geometries based on Δδ, are

consolidated to a straight line in a Paris law type of diagram. Thus, Δδ is concluded

to be an objective parameter for the current geometries and load levels.

� The direct current potential drop method is a recommended technique for crack length

measurements in standard small scale yielding fatigue crack growth rate testing. The

technique is also applicable to crack length measurements at large scale yielding when

certain conditions are fulfilled.

– The nominal calibration curve obtained from the undeformed geometry can be

used at large scale yielding if the reference potential is adjusted at every major

change of (plastic) deformation. This applies in particular, at the first fatigue load

cycle.

– The major part of the difference between the nominal and the true calibration

curve in the large scale yielding case should be attributed to geometry change and

only a minor part to the material resistivity change.

– The fatigue crack length can be measured in situ with the potential drop technique,

i.e. the experiment does not need to be stopped.

47

48

Summary of appended papers

Paper A: Effects of different load schemes on the fatigue crack growth rate

This paper presents an experimental study on a nickel-based superalloy, Inconel 718, at

an elevated temperature of +400�C. Inconel 718 is preferably used in high temperature appli-

cations and extreme environments such as in gas turbines, chemical processing applications

and pressure vessels. The basic question was how to conduct fatigue crack growth experi-

ments based on linear elastic fracture mechanics in order to obtain growth data as free from

load history effects as possible in the stage II region. This was investigated by use of four

different load sequences on the compact tension geometry. An important question was to in-

vestigate if the so called constant KI,max-method provides an upper bound of the fatigue crack

growth rate in the stage II region. The results indicated that the constant KI,max-method

gave an upper bound in fatigue crack growth rate in the current measuring range but also

gave a statistically significant lower exponent of the crack growth equation. A more reliable

method to establish the fatigue crack growth parameters was to propagate the crack under

constant stress-intensity factor range. This should be done at two different stress-intensity

factor ranges for a few millimeters at each range. Also, ΔK-reducing (constant load ratio)

procedure should not be used, when performing fatigue crack growth testing in the stage II

region, in order to minimize the risk of transient crack closure effects. Transient crack closure

effects showed up as a too steep gradient of the fatigue growth rates vs ΔK plotted in a

double logarithmic diagram, i.e. too large an exponent in Paris law.

Paper B: Variable amplitude crack growth in notched specimens

The typical conditions for an engineer, in the fatigue life design situation, are to have

constant amplitude fatigue crack growth data from fracture mechanics specimens at specific

load ratios at his disposal. However, variable load amplitude and complex geometries are

rather the rule in applications. The load history can sometimes be divided into load blocks.

Especially if the product is intended for a repeated start-stop type of use under similar

conditions as for a gas turbine. In this paper, the constant amplitude fatigue crack growth

data from Paper A was used for predictions of variable amplitude fatigue crack growth in

49

notched tensile specimens. The variable load consisted of load blocks repeatedly applied to

the specimens until final failure. The notched specimens were manufactured from the nickel-

based superalloy, Inconel 718, and the experiments were conducted at an elevated temperature

of +400�C and relative high loads. Mean crack growth rates were computed over each load

block in the notched specimens. Crack closure levels were computed, from the compact

tension specimens and applied in the prediction of the fatigue crack growth rates in the

notched specimens, by comparison of crack closure free experimental data obtained with the

so called constant KI,max-method and nominal data, i.e. fatigue data at a constant load ratio.

Also, numerical simulations of the closure level throughout the notched tensile specimens were

performed. Although the experimental data from the notched specimens showed rather large

a scatter it was concluded that variable amplitude crack growth rates can be predicted with

reasonable agreement using constant amplitude fatigue data. This conclusion is valid despite

that the load levels applied to the notched specimens were considerably above the ASTM

criterion for linear elastic fracture mechanics to be applicable. Furthermore, crack growth

rates in the inelastic notch affected zone could not be predicted by linear elastic fracture

mechanics, even if absence of crack closure was assumed.

Paper C: The effect of large scale plastic deformation on fatigue crack length measurement

with the potential drop method

One of several difficulties encountered in fatigue crack growth testing, as the constitutive

response moves from linear elastic to non-linear elastic-plastic, is to ensure proper crack

length measurements under the conditions which the measurements technique is to be used.

In this Paper A combined experimental and numerical investigation was conducted on the

effects of plastic deformation and material resistance on the relationship between the potential

difference and crack size, denoted calibration curve, or equation. The stainless steel 316L was

used at room temperature for investigating the limitations of the calibration curve. It is

concluded that the nominal calibration equation, obtained from the undeformed geometry,

can be used for fatigue crack length measurements at large plastic deformation. However, the

reference potential must, for reliable crack length measurements, be measured at the deformed

state and later adjusted at every major change of (plastic) deformation. Furthermore, the

major part (∼ 90 %) of the change in reference potential was concluded to be attributed to

the geometry change and only a minor part (∼ 10 %) to the resistivity change in the current

material. The scatter in the potential drop measured crack length, measured on a cycle by

cycle basis, was about 30 times smaller here compared to the compliance measured crack

length. Also, in situ potential drop sampling was possible, i.e., the test did not need to be

stopped for crack length measurements. Thus, the constitutive time dependent effects were

minimized.

50

Paper D: Fatigue crack growth experiments on specimens subjected to monotonic large scale

yielding

The conclusions in Paper C formed guidelines for further experiments where the fatigue

crack growth behaviour was investigated for through thickness long cracks at monotonic large

scale yielding. The stainless steel 316L was used for two different specimens; the compact ten-

sion specimen and an in-house developed edge cracked tension specimen. Constant amplitude

loading defined as the crack tip opening displacement, δ, was investigated. Finite element

computations on the coupling between the applied load, the crack length and δ, were con-

ducted. These coupling equations were used to control Δδ in situ in the experiments based on

the potential drop measured crack length. The Δδ was able to characterize and correlate the

fatigue crack growth load state for the present geometries and loads. Also the stress-intensity

factor range, ΔKI, could predict the growth rates due to large isotropic hardening at cyclic

conditions and absence of crack closure.

Paper E: Variable amplitude fatigue crack growth at monotonic large scale yielding experi-

ments on stainless steel 316L

Fatigue crack growth in stainless steel 316L was investigated for two different variable

amplitude block loads in the monotonic large scale yielding load range. The experiments

were performed on compact tension specimens. The crack loads were defined as crack tip

opening displacement, δ. The crack length was continuously monitored by potential drop

with reference potential from the deformed configuration. The force for the next load cycle

was in situ controlled using the measured crack length from the previous cycle. The control

program used numerically determined control functions that combined the measured crack

length and target Δδ to the desired force. The average growth rates over the block sequences

were determined from potential drop and visually measured crack lengths. The experimental

growth rates were also predicted, based on small scale yielding fatigue crack growth data from

closure free experiments, using a straight forward average procedure. The possibility to use

small scale yielding material data was explained the by the large isotropic material hardening

and closure free crack growth.

51

52

Bibliography

[1] W. Schutz. A history of fatigue. Engineering Fracture Mechanics, 54(2):263–300, 1996.

[2] P. Ljustell and F. Nilsson. Effects of different load schemes on the fatigue crack growth

rate. Journal of Testing and Evaluation, 34(4):333–341, 2006.

[3] P. Ljustell and F. Nilsson. Variable amplitude crack growth in notched specimens. En-

gineering Fracture Mechanics, 72(18):2703–2720, 2005.

[4] T. Mansson, H. Oberg, and F. Nilsson. Closure effects on fatigue crack growth rates

at constant and variable amplitude loading. Engineering Fracture Mechanics, 71(9-

10):1273–1288, 2004.

[5] T. Mansson, J. Skantz, and F. Nilsson. High temperature fatigue crack growth in two

metals under constant and variable amplitude loading. International Journal of Fatigue,

24(11):1159–1168, 2002.

[6] F. Nilsson, T. Hansson, and T. Mansson. Growth of surface crack under constant and

variable amplitude loading. Engineering Fracture Mechanics, 71(12):1725–1735, 2003.

[7] P. Ljustell. The effect of large scale plastic deformation in fatigue crack length measure-

ment with the potential drop method. Journal of Testing and Evaluation, 39(6):985–1002,

2011.

[8] P. Ljustell. Fatigue crack growth experiments on specimens subjected to monotonic large

scale yielding. Submitted for international publication.

[9] P. Ljustell and B. Alfredsson. Variable amplitude fatigue crack growth at monotonic

large scale yielding experiments on stainless steel 316L. Submitted for international

publication.

[10] P. van Eeten and F. Nilsson. Constant and variable amplitude cyclic plasticity in 316L

stainless steel. Journal of Testing and Evaluation, 34(4):1–14, 2006.

[11] Stralsakerhetslaget vid de svenska karnkraftsverken. Technical Report 2011:18,

ISSN:2000-0456, Swedish Radiation Safety Authority (Stralsakerhetsmyndigheten), 2011.

53

[12] Skador i svenska karnkraftanlaggningars mekaniska anordningar 1972-2000. Technical

Report SKI Rapport 02:50, ISSN 1104-1374, ISRN SKI-R–01/X–SE, Swedish Radiation

Safety Authority (Stralsakerhetsmyndigheten), 2002.

[13] K.-J. Metzner and U. Wilke. European THERFAT project−thermal fatigue evaluation

of piping system ”Tee”-connections. Nuclear Engineering and Design, 235(2):473–484,

2005.

[14] M. Dahlberg, K.-F. Nilsson, N. Taylor, C. Faidy, U. Wilke, S. Chapuliot, D. Kalkhof,

I. Bretherton, M. Church, J. Solin, and J. Catalano. Development of a european proce-

dure for assessment of high cycle thermal fatigue in light water reactors: final report of

the NESC-thermal fatigue project. Technical report, EUR 22763 EN, 2007.

[15] FAT3D−An OECD/NEA benchmark on thermal fatigue in fluid mixing areas. Tech-

nical report, Nuclear energy agency, Committee on the safety of nuclear installations,

CSNI integrity and agening group, NEA/CSNI/R(2005)2, 2005.

[16] P. Paris and F. Erdogan. Fatigue crack closure under cyclic tension. Journal of Basic

Engineering, 85:528–534, 1963.

[17] W. Elber. Fatigue crack growth under cyclic tension. Engineering Fracture Mechanics,

2(1):37–45, 1970.

[18] W. Elber. The significance of fatigue crack closure. In Damage tolerance in air craft

structures, ASTM STP 486, pages 230–242. American Society for Testing and Materials,

1971.

[19] H. Andersson, C. Persson, and T. Hansson. Crack growth in IN718 at high temperature.

International Journal of Fatigue, 23(9):817–827, 2001.

[20] J. D. Dougherty, J. Padovan, and T. S. Srivatsan. Fatigue crack propagation and closure

behaviour of modified 1070 steel: Experimental results. Engineering Fracture Mechanics,

56(2):167–187, 1997.

[21] M. Lang. Description of load interaction effects by the ΔKeffI concept. In R. C. McClung

and J. C. Jr Newman, editors, Advances in fatigue crack closure measurements and

analysis: second volume. ASTM STP 1343, pages 207–223. American Society for Testing

and Materials, 1999.

[22] P. Dahlin and M. Olsson. Reduction of mode I fatigue crack growth rate due to occasional

mode II loading. International Journal of Fatigue, 26(10):1083–1093, 2004.

[23] P. Dahlin and M. Olsson. Mode I fatigue crack growth reduction mechanisms after a

single mode II load cycle. Engineering Fracture Mechanics, 73(13):1833–1848, 2006.

54

[24] S. Suresh. Fatigue of Materials, 2nd edition. Cambridge, UK: Cambridge University

Press, 1998.

[25] J. C. Newman Jr. The merging of fatigue and fracture mechanics concepts: historical

perspective. Progress in Aerospace Science, 34(5-6):347–390, 1998.

[26] O. E. Wheeler. Spectrum loading and crack growth. Journal of Basic Engineering,

94(1):181–186, 1972.

[27] J. D. Willenborg, R. M. Engle, and H. A. Wood. A crack growth retardation model using

and effective stress concept. AFFDL-TM-71-1-FBR, Dayton, OH.

[28] J. D. Chang, R. M. Engle, and J. Stolpestad. Fatigue crack growth behaviour and

life predictions for 2219-T851 aluminum subjected to variable-amplitude loadings. In

Fracture mechanics, ASTM STP 743, pages 3–27. American Society for Testing and

Materials, 1981.

[29] J. Schijve. Prediction methods for fatigue crack growth in aircraft material. In Fracture

mechanics, ASTM STP 700, pages 3–34. American Society for Testing and Materials,

1980.

[30] A. U. de Koning. A simple crack closure model for prediction of fatigue crack growth

rates under variable-amplitude loading. In Fracture mechanics, ASTM STP 743, pages

63–85. American Society for Testing and Materials, 1981.

[31] D. F. Socie. Prediction of fatigue crack growth in notched members under variable

amplitude loading histories. Engineering Fracture Mechanics, 9(4):849–865, 1977.

[32] K. Solanki, S. R. Daniewicz, and J. C. Newman Jr. Finite element analysis of plasticity-

induced crack closure: an overview. Engineering Fracture Mechanics, 71(2):149–171,

2004.

[33] R. C. McClung and S. Sehitoglu. On the finite element analysis of fatigue crack closure−1.

basic modelling issues. Engineering Fracture Mechanics, 33(2):237–252, 1989.

[34] R. C. McClung and S. Sehitoglu. On the finite element analysis of fatigue crack closure−2.

basic modelling issues. Engineering Fracture Mechanics, 33(2):253–272, 1989.

[35] K. Solanki, S. R. Daniewicz, and J. C. Newman Jr. Finite element modeling of plasticity-

induced crack closure with emphasis on geometry and mesh refinement effects. Engineer-

ing Fracture Mechanics, 70(12):1475–1489, 2003.

[36] A. Gonzalez-Herrera and J. Zapatero. Influence of minimum element size to determine

crack closure stress by the finite element method. Engineering Fracture Mechanics,

72(3):337–355, 2005.

55

[37] T. Jiang, M. Feng, and F. Ding. A re-examination of plasticity-induced crack closure

in fatigue and crack propagation. International Journal of Plasticity, 21(9):1720–1740,

2005.

[38] S. Pommier and P. Bompard. Bauschinger effect of alloys and plasticity-induced crack

closure: a finite element analysis. Fatigue and Fracture of Engineering Materials and

Structures, 23(2):129–139, 2000.

[39] S. Pommier. A study of the relationship between variable level fatigue crack growth and

the cyclic constitutive behaviour of steel. International Journal of Fatigue, 23(SUPPL.

1):111–118, 2001.

[40] F. Nilsson. Fracture Mechanics - from Theory to Applications. KTH, Stockholm, 2001.

[41] J. R. Rice. Mechanics of crack tip deformation and extension by fatigue. In Fatigue

Crack Propagation, ASTM STP 415, pages 247–311. American Society for Testing and

Materials, 1967.

[42] J. W. Hutchinson and P.C. Paris. Stability analysis of J -controlled crack growth. In

J. D. Landes, J. A. Begley, and G. A. Clarke, editors, Elastic-Plastic Fracture, ASTM

STP 668, pages 37–64. American Society for Testing and Materials, 1979.

[43] V. Bachmann and D. Munz. Fatigue crack closure evaluation with the potential method.

Engineering Fracture Mechanics, 11(1):61–71, 1979.

[44] R. Pippan, G. Haas, and H. P. Stuwe. Comparison of two methods to measure crack

closure in ultra-high vacuum. Engineering Fracture Mechanics, 34(5-6):1075–1084, 1989.

[45] T. T. Shih and R. P. Wei. Study of crack closure in fatigue. Engineering Fracture

Mechanics, 6(1):19–32, 1974.

[46] S. H. Spence, W. J. Evans, and N. Medwell. Crack growth response of IMI 804 under

variable amplitude loading. International Journal of Fatigue, 19(1):33–41, 1997.

[47] M. Andersson, C. Persson, and S. Melin. Experimental and numerical investigation

of crack closure measurements with electrical PD technique. International Journal of

Fatigue, 28(9):1059–1068, 2006.

[48] G. Clark and J. F. Knott. Measurement of fatigue cracks in notched specimens by means

of theoretical electrical potential calibrations. Journal of the Mechanics and Physics of

Solids, 23(4):265–276, 1975.

[49] R. P. Wei and R. L. Brazill. An assessment of A-C and D-C potential systems for monitor-

ing fatigue crack growth. In Bucci Hudak Jr., editor, Fatigue crack growth measurement

56

and data analysis, ASTM STP 738, pages 103–119. American Society for Testing and

Materials, 1981.

[50] H. H. Johnson. Calibrating the electrical potential method for studying slow crack

growth. Materials Research and standards, 5(9):442–445, 1965.

[51] R. O. Ritchie and K. J. Bathe. On the calibration of the electrical potential techniques for

monitoring crack growth using finite element methods. International Journal of Fatigue,

15(1):47–55, 1979.

[52] M. A. Hicks and A. C. Pickard. A comparison of theoretical and experimental methods

of calibrating the electrical potential drop technique for crack length determination.

International Journal of Fatigue, 20(2):91–101, 1982.

[53] W. K. Wilson. On the electrical potential analysis of a cracked fracture mechanics test

specimen using the finite element method. Engineering Fracture Mechanics, 18(2):349–

358, 1983.

[54] G. M. Wilkowski and W. A. Maxey. Review and applications of the electric potential

method for measuring crack growth in specimens, flawed pipes, and pressure vessels. In

J. C. Lewis and G. Sines, editors, ASTM STP 791, pages 266–294. American Society for

Testing and Materials, 1983.

[55] J. R. Rice. A path independent integral and the approximate analysis of strain concen-

tration by notches and cracks. Journal of Applied Mechanics, 35:379–386, 1968.

[56] H. S. Lamba. The J -integral applied to cyclic loading. Engineering Fracture Mechanics,

7(4):693–703, 1975.

[57] N. E. Dowling and J. A. Begley. Fatigue crack growth during gross yielding and the

J -integral. In Mechanics of crack growth, ASTM STP 590, pages 82–103. American

Society for Testing and Materials, 1976.

[58] Y. Lambert, P. Sillard, and C. Bathias. Application of the J -concept to fatigue crack

growth in large scale yielding. In T. A. Cruse, editor, ASTM STP 969, pages 318–329.

American Society for Testing and Materials, 1988.

[59] K. Tanaka, T. Hoshide, and N. Sakai. Mechanics of fatigue crack propagation by crack-tip

plastic blunting. Engineering Fracture Mechanics, 19(5):805–825, 1984.

[60] D. F. Mowbray. Use of a compact-type strip specimen for fatigue crack growth rate

testing in the high-rate regime. In J. D. Landes, J. A. Begley, and G. A. Clarke, editors,

Elastic-Plastic Fracture, ASTM STP 668, pages 736–752. American Society for Testing

and Materials, 1979.

57

[61] S. T. Rolfe and J. M. Barsom. Fracture and Fatigue Control in Structures. Prentice-Hall

Inc., 1977.

[62] K. Tanaka, T. Hoshide, and M. Nakata. Elastic-plastic crack propagation under high

cyclic stress. In C. F. Shih and J. P. Gudas, editors, Elastic-Plastic Fracture: Sec-

ond Symposium, Volume II - Fracture Resistance Curves and Engineering Applications,

ASTM STP 803, pages 708–722. American Society for Testing and Materials, 1983.

[63] N. E. Dowling. Geometry effects and the J -integral approach to elastic-plastic fatigue

crack growth. In J.L. Swedlow and M.L. Williams, editors, Cracks and Fracture, ASTM

STP 601, pages 19–32. American Society for Testing and Materials, 1976.

[64] A. H. Rosenberger and H. Ghonem. High temperature elastic-plastic small crack growth

behaviour in a nickel-base superalloy. Fatigue and Fracture of Engineering Materials and

Structures, 17(5):509–521, 1994.

[65] A. A. Wells. The condition of fast fracture in aluminum alloys with particular reference

to comet failures. Technical report, British Welding Research Association Report, 1955.

[66] R. C. McClung and H. Sehitoglu. Closure behaviour of small cracks under high strain

fatigue histories. In J. C. Newman and Jr. W. Elber, editors, Mechanics of Fatigue Crack

Closure, ASTM STP 982, pages 279–299. American Society for Testing and Materials,

1988.

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