Theoretical and experimental modelling of multiple site ... · Theoretical and Experimental...
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Theoretical and Experimental
Modelling of Multiple Site Damage in
Plate Components
By
Donghoon Chang
B. Sc., M. Eng.
A thesis submitted for the degree of Doctor of Philosophy at the
School of Mechanical Engineering
The University of Adelaide
Australia
Submitted: 3 October 2013
Accepted: 7 November 2013
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Abstract
Fracture and fatigue assessment of structures weakened by multiple site
damage (MSD), such as two or more interacting cracks, currently represents a
challenging problem. The lifetime prediction of structural components with MSD is
still largely based on 2D single crack solutions available in various handbooks or
derived from the simplified finite element analysis. Such simplifications could
often result in non-conservative predictions, overestimating the actual fatigue life
of the structural components. Therefore, there is a strong motivation for the
development of more advanced modelling approaches, which could incorporate the
effects of the interaction between multiple cracks, 3D and other nonlinear
phenomena.
The primary objective of this study is to develop analytical and numerical
models for the evaluation of the residual strength and fatigue crack growth of two
through-the-thickness cracks in a plate of finite thickness subjected to monotonic
and cyclic loading. The selected problem represents the simplest type of MSD,
however the obtained results can serve as benchmark solutions for modelling and
assessment of more complicated practical MSD problems.
The nonlinear interactions between the cracks as well as the 3D effects, such
as the effect of the plate thickness, are investigated with the help of the classical
strip yield model, plasticity induced crack closure concept and fundamental 3D
solution for an edge dislocation in an infinite plate. The computational procedure is
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based on the Distributed Dislocation Technique and Gauss-Chebyshev quadrature
method, which provide an effective way for obtaining highly accurate solutions to
fracture mechanics problems. An experimental study was conducted to evaluate the
effect of the plate thickness and crack interaction on the residual strength levels and
fatigue crack growth rates of two closely spaced through-the-thickness cracks in
aluminium plate specimens. The outcomes of the experimental study were also
utilised to validate the theoretical approach and estimate the accuracy of the
analytical and numerical predictions.
The major outcomes of the thesis can be formulated as follows:
� An original analytical 3D model for the evaluation of residual strength of
two collinear cracks of equal length was developed and compared with the
existing 2D models and outcomes of the experimental program conducted
by the candidate;
� Analytical and numerical models for the assessment of the fatigue crack
growth of two collinear through-the-thickness cracks subjected to a constant
amplitude cyclic loading were developed;
� The effects of the nonlinear interactions between two cracks, plate
thickness and plasticity induced crack closure on fatigue crack growth rates
were identified and analysed using the developed theoretical and
experimental techniques;
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� Further recommendations for analytical and numerical modelling of MSD
were provided.
iv
Declaration
This work contains no material which has been accepted for the award of any
other degree or diploma in any university or other tertiary institution and, to the
best of my knowledge and belief, contains no material previously published or
written by another person, except where due reference has been made in the text.
I give consent to this copy of my thesis when deposited in the University
Library, being made available for loan and photocopying, subject to the provisions
of the Copyright Act 1968. The author acknowledges that copyright of published
works contained within this thesis (as listed on the following pages) resides with
the copyright holder(s) of those works.
I also give permission for the digital version of my thesis to be made available
on the web, via the University’s digital research repository, the Library catalogue
and also through web search engines, unless permission has been granted by the
University to restrict access for a period of time.
Donghoon Chang Date
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Acknowledgments
I wish to express my deepest gratitude to my supervisors Associate Professor
Andrei Kotousov and Dr. John Codrington for their invaluable guidance throughout
my PhD study. Their abundant scientific knowledge and immense practical help
are sincerely acknowledged.
Many thanks go to Assoc Prof Reza Ghomashchi, Dr Erwin Gamboa, Dr
Antoni Blazewicz, Assoc Prof Eric Hu, Dr Zonghan Xie, Assoc Prof Anthony
Zander and Mr Ian Brown for providing me encouragement and support in many
aspects.
I feel grateful to my university friends, Aditya Khanna, Luiz Bortolan Neto,
Munawwar Ahmad Mohabuth, Roslina Mohammad, Ladan Sahafi, Houman
Alipooramirabad, Pouria Aryan, Michael Bolzon, Farzin Ghanadi, Di Lu and
Maung Myo, for their insightful discussion as well as their friendly greetings,
which helped me to unwind after a tiring day at university.
Lastly, special thanks go to my family. Without their love, patience and
understanding, this long journey would have been impossible.
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List of Publications
Journal publications
1. J. Codrington, A. Kotousov and D. Chang (2012), “Effect of a Variation in
Material Properties on the Crack Tip Opening Displacement”, Fatigue and
Fracture of Engineering Materials and Structures, v 35, pp 943-952.
2. D. Chang and A. Kotousov (2012), “A strip yield model for two collinear
cracks”, Engineering Fracture Mechanics, v 90, pp 121-128.
3. D. Chang and A. Kotousov (2012), “A strip yield model for two collinear cracks
in plates of arbitrary thickness”, International Journal of Fracture, v 176, pp 39-
47.
4. D. Chang (2013) “Assessment of the interaction between two collinear cracks in
plates of arbitrary thickness using a plasticity-induced crack closure model”,
Fatigue and Fracture of Engineering Materials and Structures, v 36, pp 1113-
1122.
Conference publications
1. D. Chang and S. Harding (2010), “A compact solution for the interface corner
stress intensity factor of a cylindrical butt joint”, 6th
Australian Congress on
Applied Mechanics, Perth, Australia.
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2. A. Kotousov, D. Chang and A. Blazewicz (2010), Scale effects at failure of bi-
material joints and structures. 37th Solid Mechanics Conference, Warsaw,
Poland.
3. D. Chang and A. Kotousov (2012), “Plasticity-induced crack closure model for
two collinear cracks in plates of arbitrary thickness”, 7th
Australian Congress on
Applied Mechanics, Adelaide, Australia.
4. D. Chang and A. Kotousov (2013), “A computational and experimental analysis
of interaction between neighbouring collinear cracks in a plate of arbitrary
thickness”, 8th
International Conference on Structural Integrity and Fracture,
Melbourne, Australia.
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Nomenclatures
2a Crack length
2b Length between outer crack tips of two collinear cracks
2c Length between inner crack tips of two collinear cracks
2d Centre-to-centre distance of cracks
g�x� Crack opening displacement
2h Plate thickness
s, t Transformed coordinates
u� y-displacement
w Tensile plastic zone size
w� Compressive plastic zone size
x, y Cartesian coordinates
∆b��ξ� Infinitesimal Burgers vector
B��ξ� Edge dislocation density function in y-direction
E Young’s modulus
G�����x, ξ� Two-dimensional Cauchy kernel for y-direction
G�����x, ξ� Three-dimensional kernel for y-direction
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K� Stress intensity factor in mode I
∆K Stress intensity factor range
∆K��� Effective stress intensity range
K��·�, K��·� Modified Bessel functions of the second kind
Q Contact-free length ratio
R Load ratio �σ��� /σ�"# �
W% Weight function
β Crack contact zone size
σ�� Remotely applied stress in y-direction
���� n
th minimum cyclic stress
σ�"#��� n
th maximum cyclic stress
σ'(��� n
th opening cyclic stress
σ� Flow stress
σ) Yield strength
ε) Yield strain
σ+ Ultimate strength
δ- Residual plastic stretch
./0s%2 Non-singular function
x
µ Shear modulus
κ Kolosov’s constant
ν Poisson’s ratio
Subscripts
i Inner crack tip
o Outer crack tip
max Maximum load
min Minimum load
op Crack opening load
Abbreviations
CA Constant amplitude
CTOD Crack tip opening displacement
DBEM Dual boundary element method
DDT Distributed dislocation technique
DTD Damage tolerant design
EPFM Elastic plastic fracture mechanics
FEA Finite element analysis
LEFM Linear elastic fracture mechanics
LT Longitudinal transverse
MSD Multiple site damage
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PICC Plasticity induced crack closure
RHS Right-hand-side
VA Variable amplitude
WFD Widespread fatigue damage
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Table of Contents
Abstract ....................................................................................................................... i
Declaration................................................................................................................ iv
Acknowledgments ..................................................................................................... v
List of Publications ................................................................................................... vi
Nomenclatures ........................................................................................................ viii
Table of Contents .................................................................................................... xii
1 Introduction ........................................................................................................ 1
1.1 Significance of Multiple Site Damage ........................................................ 2
1.2 Current Models for Assessment of MSD .................................................... 4
1.3 Plate Thickness Effect ................................................................................ 7
1.4 Objectives of the Research.......................................................................... 8
1.5 Overview of the Research Outcomes .......................................................... 9
2 Background and Literature Review.................................................................. 15
2.1 Introduction ............................................................................................... 15
2.2 Fundamental Fatigue Cracking Mechanisms and Models ........................ 16
2.2.1 Paris law and its limitations ............................................................... 17
2.2.2 Crack Tip Plasticity ........................................................................... 19
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2.2.3 Plasticity induced crack closure ........................................................ 22
2.2.4 Plasticity induced crack closure models ........................................... 28
2.3 Structural Integrity of Plates with MSD Cracks ....................................... 36
2.3.1 Prediction models on residual strength of plates with MSD ............. 37
2.3.2 Prediction models of fatigue lifetime of components subjected to
MSD...… .......................................................................................................... 41
2.4 Research Gaps .......................................................................................... 44
3 A Strip Yield Model for Two Collinear Cracks .............................................. 47
3.1 Introduction .............................................................................................. 47
3.2 Problem Formulation ................................................................................ 49
3.3 Analytical Solution: Inversion of Föppl Integral ..................................... 55
3.4 Numerical Solution: Gauss–Chebyshev Quadrature Method .................. 57
3.5 Results and Discussion ............................................................................. 60
3.5.1 Plastic zone size ................................................................................ 60
3.5.2 Crack tip opening displacement ........................................................ 63
3.6 Conclusions .............................................................................................. 65
4 A Strip Yield Model for Two Collinear Cracks in a Plate of Arbitrary
Thickness ................................................................................................................ 69
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4.1 Introduction ............................................................................................... 69
4.2 Problem Formulation and Distributed Dislocation Approach .................. 71
4.2.1 Plane stress case ................................................................................ 75
4.2.2 Plane strain case................................................................................. 76
4.2.3 Finite thickness case .......................................................................... 77
4.3 Gauss-Chebyshev Quadrature Method ..................................................... 78
4.4 Results and Discussion ............................................................................. 82
4.4.1 Local plastic collapse of two collinear cracks in a plate of finite
thickness ........................................................................................................... 82
4.4.2 Variation of plastic zone size and crack tip opening displacement of
two collinear cracks in a plate of finite thickness ............................................ 84
4.5 Conclusions ............................................................................................... 91
5 A Plasticity Induced Crack Closure Model for Two Collinear Cracks in a Plate
of Arbitrary Thickness ............................................................................................. 95
5.1 Introduction ............................................................................................... 95
5.2 Problem Formulation for the Governing Integral Equation ...................... 97
5.2.1 Plane stress condition ........................................................................ 99
5.2.2 Plane strain condition ...................................................................... 100
5.2.3 Finite thickness plate ....................................................................... 100
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5.3 Discrete Form of the Governing Integral Equation ................................ 102
5.4 Boundary Conditions and Criteria for Solution Process ........................ 103
5.4.1 Maximum load ................................................................................ 105
5.4.2 Minimum load ................................................................................. 106
5.4.3 Opening load ................................................................................... 109
5.5 Validation of the Theoretical Model: Crack Closure of a Single Crack at
Minimum Load .................................................................................................. 109
5.6 Results and Discussion ........................................................................... 111
5.6.1 Crack closure at minimum load ...................................................... 111
5.6.2 Crack opening load ......................................................................... 115
5.7 Conclusions ............................................................................................ 117
6 A Fatigue Crack Growth Model for Two Collinear Cracks in a Plate of
Arbitrary Thickness .............................................................................................. 119
6.1 Introduction ............................................................................................ 119
6.2 Transient Crack Growth Model .............................................................. 121
6.3 Validation Study: Single (Non-interacting) Crack ................................. 132
6.4 Effect of Crack Interaction and Plate Thickness on Fatigue Behaviour 145
6.4.1 Fatigue crack growth prediction for two collinear cracks ............... 145
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6.4.2 The influence of plate thickness on fatigue crack growth of two
collinear cracks ............................................................................................... 155
6.5 Conclusions ............................................................................................. 159
7 Experimental Study of Plastic Collapse of the Ligament between Two
Collinear Cracks .................................................................................................... 163
7.1 Introduction ............................................................................................. 163
7.2 Experimental Approach .......................................................................... 164
7.2.1 Material property test and specimen preparation ............................ 164
7.2.2 Plastic collapse testing ..................................................................... 167
7.3 Results and Discussion ........................................................................... 169
7.4 Conclusions ............................................................................................. 173
8 Experimental Study of Fatigue Crack Growth of Two Interacting Cracks .... 177
8.1 Introduction ............................................................................................. 177
8.2 Experimental Study ................................................................................. 178
8.3 Experimental Results and Discussion ..................................................... 182
8.4 Fatigue Crack Growth Modelling and Discussion .................................. 184
8.5 Conclusions ............................................................................................. 190
9 Conclusions and Future Work ........................................................................ 193
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9.1 Introduction ............................................................................................ 193
9.2 Analytical and Numerical Approach (Chapters 3-6) .............................. 194
9.3 Experimental Approach (Chapters 7-8) ................................................. 201
9.4 Conclusions ............................................................................................ 203
9.5 Future Work ........................................................................................... 206
References ............................................................................................................. 207
1
Chapter 1
1 Introduction
The concept of damage tolerance, which was originally introduced in the
aircraft industry in 1978 (Pitt & Jones 1997), is now extensively employed in the
design procedures and standards for various types of engineering structures, such
as ships, bridges, pipelines and pressure vessels (Pidapartia, Palakalb & Rahmana
2000). This concept assumes that every structure has initial flaws or defects that
can grow under service loading. Therefore, in order to ensure the safe operation of
(damaged) structures, it is essential to predict their service life accurately through
the rigorous implementation of analytical, numerical and experimental approaches.
The life of an in-service structure is determined by predicting the residual
strength and/or fatigue crack growth from initial structural defects or detected
damage. Unfortunately, however, there are many factors which make this task
extremely challenging. Among these factors are the effects of the load spectrum
and environmental conditions (which are often unknown), distribution and
dimensions of initial defects, presence of residual stresses, effects of multiple site
damage and scatter of material properties (Anderson et al. 2004; Calì, Citarella &
Perrella 2003; Codrington 2008; Lee 2009). The focus of the present study is on the
investigation of multiple site damage and three-dimensional geometry effects on
the residual strength and fatigue lifetime of plate components.
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1.1 Significance of Multiple Site Damage
Multiple site damage (MSD) is a type of structural damage characterised by
the presence of mutually-interacting multiple cracks. Typical MSD is often found
in an aging aircraft fuselage where fatigue cracks grow from rivet holes, and some
of them coalesce into a major crack (Koolloos et al. 2001). The importance of
MSD was first recognised as early as 1978 when an increasing number of aircraft
were forced to operate beyond their original design life (Collins & Cartwright
1996). However, this special type of structural damage did not receive much
attention until the catastrophic in-flight failure of the fuselage of an Aloha Airlines
Boeing 737 in 1988 (Hendricks 1991). The subsequent investigation into this
accident revealed that a sudden coalescence of small cracks emanating from the
collinear rivet holes undermined the damage tolerance capability of the fuselage,
leading to the catastrophic structural failure (Hendricks 1991). This accident
revealed the lack of understanding of MSD and heavy potential consequences of
neglecting the crack interaction in load bearing structures.
It is now well-recognised that the presence of MSD can pose a serious threat
to the overall structural integrity. The residual strength of a panel with a major
crack flanked by small cracks is lower than that of a panel with a single crack of
the same length (Koolloos et al. 2001). Even extremely small multiple cracks,
which are in-service undetectable, have the potential for a substantial reduction of
the residual strength of the structure (Swift 1994). More importantly, the possible
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link-up or coalescence of small and tolerable cracks (if considered as isolated) can
be very dangerous. This is because of (1) the possibility of a sudden coalescence of
the cracks resulting into a continuous crack with the length greater than the critical
one and (2) the link-up of multiple cracks can significantly accelerate the crack
growth and shorten the fatigue life (Kamaya 2008; Moukawsher, Grandt & Neussl
1996; Shkarayev & Krashanitsa 2005).
MSD is a highly complex phenomenon. The nonlinear crack interaction,
among others, is the key factor which complicates the development of accurate
mathematical models. The presence of interaction between closely located cracks
can have a significant impact on the plastic zone formation, fracture controlling
parameters and the stress/strain fields around the crack tip (Moukawsher, Grandt &
Neussl 1996). As a result, fracture and fatigue behaviour of this type of damage can
be very different from non-interactive or isolated cracks. The intensity of the crack
interaction can change substantially during fatigue crack growth depending on
various factors, such as the relative location, the relative size and the shape of the
cracks (Kamaya 2008). A quantitative analysis of the crack interaction is therefore
essential to provide a reliable estimate of the structural integrity of plate and shell
components containing MSD.
The application of conventional failure assessment methods for MSD often
has many limitations and often results in non-conservative predictions (Carpinteri,
Brighenti & Vantadori 2004; Collins & Cartwright 1996). This is because many of
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the conventional prediction tools are based on solutions obtained from the analysis
of non-interactive crack problems. Hence, incorrect conclusions can be made if the
crack interaction effects are ignored. As an example, Kuang and Chen (1998)
conducted a case study related to MSD, which clearly demonstrated that a failure
evaluation method that did not consider the crack interaction over predicts the
residual strength as compared with the corresponding experimental results. They
also indicated that the residual strength of a panel with MSD may be overestimated
up to 40% if the crack interaction is not incorporated into the predictive model.
1.2 Current Models for Assessment of MSD
It is hardly surprising that there has been a lot of research to address the MSD
problem since the Aloha Airlines accident (Hendricks 1991). These vast research
efforts mainly focused on the development of analytical and numerical tools which
are capable of providing accurate assessments on the integrity and lifetime of aging
structures containing MSD by taking into account the crack interaction effects.
This common research trend towards the development of more accurate and
reliable predictive tools is well justified because the optimization and selection of
maintenance or inspection intervals in many industries has a large impact on the
cost and efficiency of safe operation (Wang, Brust & Atluri 1997).
5
Residual strength models for MSD based on the strip yield model
Advanced analytical approaches for analysis of multiple crack problems are
often based on the strip yield model, which was originally introduced by Dugdale
(1960) and Barenblatt (1962). The popularity of this model is due to a relative
simplicity of the mathematical formulation, which enables closed-form analytical
solutions. In the strip yield model, it is assumed that plastic deformation ahead of a
crack tip is confined to an infinitesimally thin strip along the crack line. Therefore,
the calculation of the fracture controlling parameters, such as the crack tip opening
displacement, can be greatly simplified. The analytical solutions can often be
obtained by using the superposition principle and utilising well-known results for
crack problems obtained within the linear theory of elasticity (Anderson 2005).
The strip yield concept was implemented to multiple cracks and a number
of analytical solutions are currently available in the literature. Various criteria for
the assessment of the residual strength of plate and shell components with MSD
have been proposed by many researchers. The most popular among them is the
plastic zone coalescence criterion (Collins & Cartwright 2001; Kuang & Chen
1998; Nishimura 2002; Theocaris 1983; Tong, Greif & Chen 1994; Wu & Xu 2011;
Xu, Wu & Wang 2011), also known as ligament yield criterion, because of its
simplicity and conservative predictions in many practical situations. According to
the plastic zone coalescence criterion, the ligament between cracks is assumed to
fail if the plastic zones at the crack tips contact with each other. Other well-known
6
criteria for the assessment of the residual strength of MSD structures include the
classical elastic-plastic fracture mechanics parameters utilised for a single crack,
such as crack tip opening angle or displacement (Galatolo & Nilsson 2001), crack
opening displacement (Mukhtar Ali & Ali 2000; Xu, Wu & Wang 2011) and
energy-based parameters (Duong, Chen & Yu 2001; Labeas & Diamantakos 2005;
Labeas, Diamantakos & Kermanidis 2005; Wang, Brust & Atluri 1997). However,
the implementation of these criteria is generally associated with very extensive
numerical simulations, which makes them less attractive for practical assessment of
MSD.
Fatigue prediction models for MSD
Considerable efforts have been made to predict the fatigue behaviour of
mutually interactive multiple cracks. The vast majority of these theoretical research
efforts have been directed to the implementation of numerical techniques, such as
the finite element method (Kamaya 2008; Shkarayev & Krashanitsa 2005; Silva et
al. 2000) and dual boundary element method (Calì, Citarella & Perrella 2003;
Citarella 2009; Wessel et al. 2001). However, these techniques require a lot of care
and an extensive validation. The numerical techniques often produce inconstant
results due to unavoidable variations between different studies, specifically, in the
mesh density, crack advance scheme, identification of crack closure stress and
contact stresses (Solanki, Kiran, Daniewicz, S. R. & Newman Jr, J. C. 2004).
7
1.3 Plate Thickness Effect
The specimen thickness can significantly affect the stress distribution near
crack tips and formation of the crack tip plastic zone (Dougherty, Srivatsan &
Padovan 1997). Many experimental studies confirmed that the fatigue crack growth
can be considerably impacted by the three-dimensional effects such as plate
thickness. Based on theoretical and experimental data, generally from non-
interactive cracks, it has been shown by many researchers that the fatigue crack
growth rates increase significantly with an increase in the specimen thickness
(Bhuyan & Vosikovsky 1989; Codrington & Kotousov 2009a; Costa & Ferreira
1998; de Matos & Nowell 2009; Guo, Wang & Rose 1999; Newman Jr 1998; Park
& Lee 2000). This highlights the importance of accounting for the thickness effect
in assessing structural integrity of mechanical components. Although the thickness
effect has been largely investigated based on the analysis of isolated cracks, it is
likely that it has the same or even greater influence on the behaviour of mutually
interactive cracks, or in the case of MSD. Moukawsher, Grandt and Neussl (1996)
discussed the importance of including the effect of three-dimensional thickness
among other factors in an analysis of MSD problems. However, there are no
systematic theoretical or experimental studies on the effect of plate thickness or
other 3D effects on the residual strength and fatigue life of plate components
subjected to MSD.
8
1.4 Objectives of the Research
The overall objective of this study is to investigate the residual strength and
fatigue crack growth behaviour of two through-the-thickness collinear cracks of
equal length in a plate of finite thickness using theoretical and experimental
approaches. The selected crack geometry may represent one of the simplest types
of MSD and has a rather limited practical application. However, the investigated
mechanisms of crack coalescence (local plastic collapse) and the various nonlinear
effects can provide vital insight into more practical problems involving multiple
cracks. Moreover, the developed methods for the assessment of fatigue crack
growth can be readily generalised for more complicated geometries of MSD.
In accordance with the overall objective of this study, theoretical models for
the assessment of residual strength and fatigue crack growth in plates subjected to
MSD were developed. These models are based on the classical strip yield model,
plasticity induced crack closure concept and fundamental three-dimensional
solution for an edge dislocation in an infinite plate (Kotousov & Wang 2002). The
numerical procedure for obtaining two-dimensional and three-dimensional
solutions utilises the distributed dislocation technique (DDT) and Gauss-
Chebyshev quadrature method, which can provide an effective way for obtaining
highly accurate solutions to various types of fracture mechanics problems (Bilby,
Gardner & Smith 1958; Codrington & Kotousov 2007a; Hills et al. 1996). The
methods and solutions were verified against analytical, numerical and experimental
9
studies conducted in the past. Further, the developed models are applied to
investigate the nonlinear interaction effect between the cracks as well as the three-
dimensional plate thickness effect.
To support the theoretical findings, an experimental program incorporating
the investigation of the residual strength and fatigue crack growth were developed
and conducted on aluminium plate specimens containing two collinear cracks. The
residual strength and the conditions leading to the local plastic collapse of two
collinear cracks of equal length were investigated by an original experimental
method, while the fatigue crack studies followed a quite standard approach utilising
optical measurements of the fatigue crack propagation. The outcomes of the
experimental study were used to (1) evaluate and confirm experimentally the effect
of the plate thickness and crack interaction on the residual strength and fatigue
crack growth rates of two interacting cracks and (2) validate the outcomes of the
theoretical models developed in the current study.
1.5 Overview of the Research Outcomes
The major outcomes of the conducted research presented in Chapters 3-8 are
briefly summarised below:
10
Chapter 3 A Strip Yield Model for Two Collinear Cracks
� An analytical two-dimensional model for two collinear cracks of equal
length has been formulated based on the strip yield model and the
distributed dislocation technique;
� Two distinctive approaches to solving the two-dimensional model have
been developed: inversion of Föppl integral (analytical approach) and
Gauss-Chebyshev quadrature (numerical approach);
� Using the theoretical model, the crack interaction effect was investigated in
terms of the crack tip opening displacement (CTOD) and plastic zone size.
Chapter 4 A Strip Yield Model for Two Collinear Cracks in a Plate of Arbitrary
Thickness
� An analytical and numerical three-dimensional model for the evaluation of
residual strength of two collinear cracks of equal length has been developed
based on the strip yield model, the distributed dislocation technique and the
fundamental three-dimensional solution for an edge dislocation;
� The results for the residual strength of collinear cracks under plane stress
have been validated against previous studies;
� The effect of the crack interaction in conjunction with the plate thickness on
the residual strength of two collinear cracks was investigated using the
developed three-dimensional model.
11
Chapter 5 A Plasticity Induced Crack Closure Model for Two Collinear Cracks in a
Plate of Arbitrary Thickness
� A theoretical three-dimensional crack closure model for the evaluation of
crack closure/opening of two collinear cracks of equal length has been
developed using the linearly increasing plastic wake hypothesis under the
assumption of steady-state crack growth;
� This model has been validated by comparing previous crack closure data
from a single isolated crack under plane stress with corresponding results
from the model;
� The variation of the crack closure/opening as a function of the crack
interaction as well as the plate thickness was studied based on the three-
dimensional crack closure model;
Chapter 6 A Fatigue Crack Growth Model for Two Collinear Cracks in a Plate of
Arbitrary Thickness
� The steady-state three-dimensional crack closure model in Chapter 5 has
developed into a transient fatigue crack growth model for the analysis of
interacting two collinear cracks under constant amplitude cyclic loading.
� After being validated against past fatigue data from single crack cases, the
transient growth model was used to assess the combined effects of the crack
12
interaction with the plate thickness on the fatigue behaviour of closely
spaced cracks in a plate of arbitrary thickness.
Chapter 7 Experimental Study of Plastic Collapse of the Ligament between Two
Collinear Cracks
� A novel experimental technique, based on the plastic zone coalescence
criterion, has been developed to measure the residual strength of the
ligament between collinear cracks;
� The influence of the crack interaction and specimen thickness on the
residual strength of aluminium specimens was experimentally investigated,
and the results were used to validate the theoretical model for residual
strength prediction.
Chapter 8 Experimental Study of Fatigue Crack Growth of Two Interacting Cracks
� The influence of the crack interaction and specimen thickness on the growth
of closely spaced cracks has been experimentally identified and analysed
using specimens fabricated from aluminium plates of different thicknesses;
� The experimental results were also used to calculate the growth rate versus
the effective stress intensity range needed for the theoretical model
developed in Chapters 5 and 6.
13
� Using the growth rate data, predictions were made on the fatigue behaviour
of the test specimens by the current growth model, and they were analysed
and compared with the test results.
The overall conclusions and suggestions for further work are provided in Chapter 9
of this thesis.
14
15
Chapter 2
2 Background and Literature Review
2.1 Introduction
The damage tolerance philosophy, which is now widely used in the design
process of various types of engineering structures, was introduced in the US
Federal Aviation Regulations Part 25 (the Airworthiness standards) / Section 571
(damage tolerance and fatigue evaluation of structure), or FAR 25.571, in
December 1978 (FAA Advisory Circular 25.571-1 1978; MIL-HDBK-1530 1996).
FAR 25.571 includes two key requirements related with the structural integrity of
aircraft: (1) The residual strength evaluation must show that the remaining structure
is able to withstand static ultimate loads corresponding to the several conditions
specified in the regulation (residual strength requirement) and (2) Inspections must
be established to prevent catastrophic failure. In other words, a detected or
undetected fatigue crack must not grow to the size associated with the static
ultimate load levels by next inspection (fatigue lifetime requirement). The
regulation also requires that special consideration for widespread fatigue damage
(WFD), sometimes termed multiple site damage (MSD), must be included where
the design is such that this type of damage could occur. These requirements
demand reliable engineering tools which are capable of quantitatively evaluating
16
crack growth and residual life of cracked structures, taking into account the
interaction effect between neighbouring cracks.
Accordingly, the purpose of this chapter is to provide an overview into
previous efforts made to predict the fatigue crack growth and lifetime expectancy
of plates with MSD cracks. In the beginning of this chapter the fundamental fatigue
crack growth law, based on the concept of the stress intensity factor, will be briefly
reviewed. This will be followed by the outline of key elastic-plastic fracture/fatigue
mechanisms frequently utilised by researchers to explain various crack growth
phenomena. The most popular crack growth prediction models will also be
summarised. After the brief introduction into fatigue growth modelling, the
importance of MSD in failure analysis will be discussed, and various residual
strength and lifetime prediction models for plate and shell components with MSD
will be outlined. Finally, the current research gaps drawn from the provided
literature review will be presented and justified.
2.2 Fundamental Fatigue Cracking Mechanisms and Models
Enormous effort to study fatigue crack growth mechanisms was made based
on the use of linear elastic fracture mechanics (LEFM) (Donahue et al. 1972;
Klesnil & Lukas 1972; Kujawski 2001; Paris & Erdogan 1963) and elastic plastic
fracture mechanics (EPFM) (Budiansky & Hutchinson 1978; Elber 1970; Newman
1981; Schijve et al. 2004; Wheeler 1972) theories. Due to simplicity and relatively
17
low cost, a considerable portion of this research effort was devoted to the
investigation of fatigue growth of through-the-thickness cracks in flat plate
specimens subjected to tensile loading. These test results led to the development of
various fatigue life prediction techniques, which were utilised for various industrial
applications (Chaudonneret & Robert 1996; Newman 1995; Newman, Phillips &
Swain 1999).
2.2.1 Paris law and its limitations
It was pointed out by Paris and Erdogan (1963) that the rate of fatigue crack
growth per cyclic loading, da/dN , could be well characterized by the stress
intensity factor range, ∆K�9 K�"# : K���� . Here, ∆K is the amplitude range
between the maximum and the minimum stress intensities at the crack tip and is a
function of applied load and the crack length as well. Their hypothesis was based
on the following consideration: within the small scale yielding (SSY) assumption
(plastic or process zone size is much smaller than other characteristic dimensions
of the problem) the stress intensity factor (SIF) is the single parameter
characterising the crack tip conditions (Paris & Erdogan 1963). Accordingly, they
proposed a power law relationship, known as the Paris equation, to describe the
linear region (in log-log domain) of fatigue crack growth:
dadN 9 C�∆K��, (2.1)
18
where C and m are supposed to be material constants which are determined through
experimental tests on combinations of various crack geometries and loading
conditions. This is basically an empirical equation obtained by combining the
theoretical concept of SIF with experimental observations. The concept of using
∆K in correlating crack growth data is the basic principle of many fatigue crack
growth models. This concept can also be regarded as the starting point of damage
tolerant design because the total number of loading cycles which will lead to a
certain amount of increment in crack length can be determined through the
integration of equation (2.1) as long as the crack growth stage continues to follow
the Paris regime. The Paris equation is effective in correlating growth rates of
relatively long fatigue cracks subject to a constant amplitude (CA) (in terms of ∆K)
cyclic loading (Anderson 2005). However, this equation is unable to account for
the effects of the load history on fatigue growth rates. If an applied cyclic loading
is not constant, i.e. variable amplitude (VA) loading, use of the Paris equation will
lead to large error in the predictions (Lin & Smith 1999). Furthermore, fatigue data
from different tests is also largely affected by many parameters, such as specimen
thickness (Guo, Wang & Rose 1999; Newman Jr 1998; Yu & Guo 2012) and
applied load ratio (Newman Jr & Ruschau 2007; Yu & Guo 2012). This can result
in large scatter in the fatigue crack growth estimates based on the Paris equation
(Newman Jr 1998).
19
2.2.2 Crack Tip Plasticity
Within the LEFM theory, the stress at the tip of a crack approaches infinite
value due to the idealisations of a sharp crack and the linear elastic material model.
This stress behaviour at the crack tip is called a stress singularity. However, the
material can not sustain infinite stresses, and the stresses at the crack tip must be
bounded (Hills et al. 1996). As a result of a very high stress concentration, plastic
deformation occurs in the region close to the crack tip where the stress level
exceeds the yield strength of the material, giving rise to a redistribution of the
elastic stress field around the crack tip. The plastically deformed region is called a
crack tip plastic zone. The use of LEFM theory is justified by the notable argument
that the true crack tip stress and strain fields approach the LEFM solution outside
of the plastic yield zone provided that the plastic zone is very small (Anderson
2005). In other words, classic LEFM theory is valid only when the nonlinear plastic
deformation is confined to a very small region compared to the crack size or any
other characteristic size of the problem. This situation is known as the small scale
yielding condition.
The size of the plastic zone near a crack tip under a remote tensile loading
can be approximated using an analytical model, for example, the strip yield model
(Dugdale 1960). It can be considered as the first self-consistent model of crack tip
plasticity (Nowell 1998). Dugdale (1960) assumed that all plastic deformation near
the crack tip is confined to a narrow strip ahead of the crack along its direction, and
20
superimposed two elastic solutions, one for a through crack under remote tension
and the other one for a through crack with closure stresses at both tips of the crack.
The latter approximates the effect of the plastic stresses near the crack tips. From
these representations, he was able to estimate the elastic-plastic behaviour of the
material near the crack tips, for example the plastic zone size and crack tip opening
displacement.
Crack tip plasticity was used to explain the effect of an overload on fatigue
crack growth (Broek 1986; Schijve 1962; Wheeler 1972). It is now well known that
the application of a tensile overload during CA cyclic loading will introduce a
significant retardation in the crack growth rate over some period of the following
CA loading. The crack growth curve will eventually recover the rate expected by
the corresponding pure CA cyclic loading. Wheeler (1972) proposed a theoretical
model for the overload-retardation behaviour by utilising the crack tip plasticity
concept. The retardation model was based on the assumption that the magnitude of
retardation can be characterised by the ratio of the current plastic zone size to the
enlarged plastic zone size caused by the overload. In Wheeler’s model, the
retardation exists only while the current crack tip plasticity is surrounded by
plasticity zone caused by the overload as shown schematically in Figure 2.1.
21
Figure 2.1 Schematic diagram of crack tip plasticity after an overload.
In accordance with Schijve (1962) and Broek (1986), a large tensile plastic
zone is developed around a crack tip after the application of an overload. During
the unloading process, elastic contraction of the material surrounding the plastically
deformed zone applies compression on it. This can lead to the development of
compressive residual stresses ahead of the crack tip as illustrated in Figure 2.2. The
compressive residual stress essentially reduces the opening of the crack or the
effective stress intensity factor, during subsequent load cycles resulting in a
decrease of fatigue crack growth rates. Due to formation of residual compressive
stress, the crack growth after an overload cycle will occur only if the following
load cycles have enough stress intensity magnitude to fully open the crack. A
comprehensive literature review on various residual stress models was recently
provided by Machniewicz (2012).
current plastic zone
overload plastic
zone
crack growth
after overload
22
Figure 2.2 Development of compressive residual stress at the crack tip after
overload.
The Wheeler approach and the residual compressive stress approach can be
rebutted by the phenomenon of delayed retardation after an overload. Typical
experimental observations show that immediately after the application of an
overload, the crack growth rate goes up sharply for a short time before it starts to
decrease(Anderson 2005). This delayed retardation cannot be properly explained
based on these two popular approaches.
2.2.3 Plasticity induced crack closure
Elber (1970) was the first who discovered that fatigue cracks under cyclic
loading can be closed even when the applied load is not compressive, but still
compressive yield
zone
overload plastic
zone
σ
< x
:
�x, y 9 0�
y,
23
tensile during unloading. Elber asserted that the permanent tensile plastic
deformation left on the crack faces induces the premature crack closure, i.e.
plasticity induced crack closure (PICC). The extent of crack closure was
determined by measuring the change in the compliance of the specimen (Elber
1970). Elber also postulated that the premature crack closure during unloading has
an effect, which leads to reduction of the driving force during fatigue crack growth
because a crack under cyclic loading would grow only when it is fully open. From
this understanding, the crack growth driving force is governed by not only the
stress/strain fields ahead of the crack tip but also by the surface conditions and
plastic stretch behind the crack tip. To quantify the driving force of the crack
growth, Elber (1970) introduced the use of the effective stress intensity range,
∆K��� �9 K�"# : K'(�. Here, K�"# and K'( represent the maximum applied stress
intensity factor and the opening applied stress intensity factor, above which the
crack faces are completely open, respectively.
After Elber’s discovery of fatigue crack closure, a variety of other closure
mechanisms, including oxide induced closure, roughness induced closure,
transformation induced closure, viscous fluid induced closure, crack bridging and
crack deflection, have been suggested by many researchers (Suresh 1998). Among
various crack closure sources, plasticity induced crack closure (PICC) is known as
the most influential crack closing mechanism for various circumstances (Bichler &
24
Pippan 2007; Dougherty, Srivatsan & Padovan 1997; Solanki, Kiran, Daniewicz, S.
R. & Newman Jr, J. C. 2004).
Underlying mechanism of PICC
The underlying mechanism of PICC is based on the crack tip plasticity
phenomenon. If a crack grows under cyclic loading, a tensile plastic zone is formed
ahead of the crack tip (Figure 2.3 (a)). As the crack continues to propagate through
the plastic zone, the plastically stretched material is then left on the crack faces,
resulting in the formation of a plastic wake on them (Figure 2.3 (b)). The additional
material layers on the crack surfaces can now contact each other before the crack is
unloaded (Figure 2.3 (c)). This is the essence of the PICC mechanism. Therefore,
the application of this concept may be limited to certain materials and load
conditions where this particular closure mechanism prevails (Codrington 2008).
25
Figure 2.3 Schematic diagram of plasticity induced crack closure: (a) crack tip
plasticity at the start of fatigue loading, (b) development of plastic wake on crack
faces after load cycles (c) crack closure during unloading.
Use of PICC in explaining various fatigue phenomena
In modern modelling approaches to fatigue crack growth prediction, PICC
is the key mechanism, which controls fatigue crack growth behaviour of metals.
The effects of various fatigue factors, for example the applied load level, load ratio
and plate thickness, as well as the overload retardation can be explained based on
the crack closure concept.
The PICC concept has been a promising theoretical concept, which
provided a way for accounting for load interaction effects (i.e. previously applied
plastic zones (a)
(b)
(c)
plastic wake
crack face
contact
26
loads affect current crack growth behaviour in the case of VA loading), such as by
fully explaining the overload-retardation behaviour. An overload in CA load cycles
introduces an enlarged plastic zone ahead of a crack tip. This may lead to the
formation of a hump of additional material in the plastic wake on the crack faces as
illustrated in Figure 2.4. The additional thickness can in turn cause earlier crack
closure during unloading portion of the subsequent CA load cycles, reducing ∆K��� and hence the crack growth rate. Through the crack blunting mechanism, PICC is
also able to explain the delayed retardation (Anderson 2005). Immediately after an
overload is applied, crack blunting prevents the crack faces from contacting each
other. The crack growth rate is temporarily increased due to the lack of crack
closure. However, shortly after that, it drops quickly far below the previous steady
state level as the crack penetrates into the overload affected plastic zone, and the
zone begins to cause a change in the formation of the residual plastic wake on the
crack faces. This change introduces enhanced crack closure, leading to a beneficial
slowdown in crack propagation (Nowell 1998).
27
Figure 2.4 Formation of additional thickness material after an overload (Nowell
1998).
The concept of PICC has been used to correlate crack growth data from
tests where different parameters affect the fatigue crack closure. These parameters
include not only those describing the applied loading conditions, such as the load
ratio and maximum applied load levels, but also the three-dimensional geometry
factor (i.e. the specimen thickness) as well as thermo-mechanical effects
(Codrington, Kotousov & Chang 2012). Numerous previous attempts to correlate
crack growth results, through use of PICC and thus the effective stress intensity
factor range, have turned out to be very successful. For example, Schijve et al.
(2004) and Newman Jr and Ruschau (2007) successfully predicted the effect of the
load ratio and the maximum applied load level on the fatigue crack growth rate
under CA loading. They also demonstrated that the use of the effective stress
intensity range could significantly reduce the scatter in the experimental results
obtained on various test samples and under various conditions. The advantages of
crack yield zone
Additional thickness material after overload
28
using PICC to correlate crack growth data from different specimen thicknesses has
been investigated by other researchers including Costa and Ferreira (1998),
Codrington and Kotousov (2009a) and Yu and Guo (2012), to name a few.
Particularly, Codrington and Kotousov (2009a) developed a theoretical crack
growth model to explain the plate thickness effect on the fatigue crack growth
rates. It is notable that the model used the first-order plate theory in order to
account for the three-dimensional plate thickness effect without relying on the
semi-empirical out-of-plane constraint factor (Newman et al. 1995), which must be
determined by finite element analysis and/or experimental approach, or selected
based on the best fit. Using a crack closure model, they showed that the application
of the effective stress intensity range can lead to a significant reduction in the
scatter in crack growth results (plotted as the crack growth rate versus the effective
stress intensity factor) obtained on different specimens of various thicknesses. All
the previous studies mentioned above were, however, limited to the analyses of a
single crack.
2.2.4 Plasticity induced crack closure models
A number of fatigue crack growth prediction models have been developed
based on PICC (Budiansky & Hutchinson 1978; Chermahini & Blom 1991;
Cochran, Dodds & Hjelmstad 2011; Codrington & Kotousov 2007a; de Koning
1981; Dougherty, Padovan & Srivatsan 1997; Newman 1981; Newman et al. 1995;
Nowell 1998; Rodrigues & Antunes 2009; Roychowdhury & Dodds 2003;
29
Shercliff & Fleck 1990; Skinner & Daniewicz 2002; Solanki, Kiran, Daniewicz, S.
R. & Newman Jr, J. C. 2004). Calculation of the crack opening stress is the
essential part of these prediction models because the effective stress intensity range,
∆K��� , which is the crack growth driving force in accordance with Elber’s
hypothesis, is a function of the crack opening stress. Various analytical and
numerical approaches were applied to determine the crack opening stress, which
will be briefly discussed next.
Analytical models
Based on the complex potentials theory Budiansky and Hutchinson (1978)
were one of the first to develop an analytical model of PICC using the Dugdale
strip yield model. In this model, the plastic wake residual stretch and the crack
opening loads are analytically calculated as functions of applied load ranges. This
is achieved by extending the classical Dugdale model to the case of a growing
crack under CA loading. The effect of cyclic hardening and softening on fatigue
crack closure was also investigated with this model. This PICC model gave
theoretical support to the experimentally observed PICC phenomenon and provided
grounds for the use of the effective stress intensity range for characterising the
crack growth retardation effects described earlier in this chapter.
Newman (1981) developed a fatigue crack growth model under variable
amplitude loading utilising the PICC concept and Dugdale strip yield hypothesis.
Newman developed a semi-analytical crack closure model using simplified bar
30
elements with an effective flow stress, ασ' , to represent the plastic zone and
residual plastic deformation left on the crack faces for the two extreme cases of
plane stress (α 9 1) and plane strain (α 9 3) with α being an empirical constraint
factor, as shown in Figure 2.5. In this model, the remotely applied opening stress
intensity, K'( , is determined from the residual stress distribution in the plastic
wake on the crack faces. Then K'( is used to obtain the effective stress intensity
range, ∆K��� , for further calculation of fatigue crack growth per cyclic loading,
da/dN, under variable spectrum loading. Newman et al. (1995) extended a
previous two-dimensional model to simulate three-dimensional effects caused by
finite plate thickness. In the new model, various values of constraint factor, α, for
each different plate thickness were employed to take into account the out-of-plane
stress conditions. Experimental tests conducted under single overload as well as
finite element methods were used to determine the values (Newman, Phillips &
Swain 1999). In-between constraint factor values are basically determined by
interpolating data points, which were obtained by using the finite element method
or experimental studies. However, because this model is reliant on the use of finite
element analysis or test results to account for such effects as the specimen
geometry, extensive calculations and laborious tasks are inevitably included into
the crack growth modelling procedure. Furthermore, significant errors can be
introduced in the determination of the constraint factor. This is due to the
ambiguous nature of the interpolation in the case of the experimental approach, and
31
due to the nonlinearity associated with plasticity and crack contact and the use of
different crack closure conditions and crack advance schemes in the case of the
FEA approach (Pitt & Jones 1997; Solanki, K., Daniewicz, S. R. & Newman Jr, J.
C. 2004).
Figure 2.5 Newman closure model and stress distribution along the crack line
(Newman 1981).
S�"# S���
x
y y
x
x
ασ'
σ x
:ασ'
σ
32
A crack closure model for prediction of fatigue crack growth under VA
loading was presented by de Koning (1981). This simple model accounts for load
interaction effects as well as the transition from plane strain to plane stress when
the specimen plate thickness is decreased. In the model, the shape of plastic wake
is assumed to be covered with humps, which are associated with crack tip plastic
zones generated by previous spike loads. The individual hump opening stress is
then approximated using a delay switch. The delay switch is set on after application
of a spike load and set off if the crack has grown through the spike load plastic
zone. This leads to the opening stress change from zero to a positive value S'(� ,
which is a function of a maximum load S�"#� and a minimum load S���� of nth
spike
load. The crack is next assumed to be closed as long as one or more of the humps
are in contact with its counterpart on the opposite crack surface. The illustration of
this concept, when the hump that loses contact determines the crack opening stress
σ'(, is shown in Figure 2.6.
33
Figure 2.6 Opening behaviour of a crack tip in the case of three significant humps
on the crack surface (de Koning 1981)
Nowell (1998) developed a plane stress boundary element model for PICC
based on the strip yield model. The model has a similar physical foundation as
those of Budiansky and Hutchinson (1978) and Newman (1981). In this model,
however, the displacement discontinuity boundary elements are used to represent
the crack and yield zone. Quadratic programming techniques are utilised to
establish correct boundary conditions automatically. Nowell’s model was very
effective and convenient to use in terms of computational simplicity, although only
the plane stress case was taken into account in it.
Recent work by Codrington and Kotousov (Codrington & Kotousov 2007a,
2007b, 2009a) has seen the development of a new semi-analytical approach, which
is based on PICC concept and distributed dislocation technique (DDT). Their
model takes into account the effects of finite plate thickness without any empirical
1 2
3
Time
Crack Tip
S'(� S'(� S'(�
σ'( 9 S'(�
The crack is
effectively open
Hump 2 breaks contact
Hump 1 breaks contact
Hump 3 breaks contact
2’ 3’
1’
34
parameters or extensive finite element calculations. This is accomplished by
utilising the solution for an edge dislocation in an infinite plate of finite thickness,
which is based on first order plate theory (Kotousov & Wang 2002). This model
provides a powerful and efficient tool for analysing the fatigue behaviour of an
isolated crack under cyclic loading with a single overload.
The PICC concept has also led to the development of several commercial
fatigue lifetime prediction codes including NASA’s FASTRAN (Newman 1981)
and NASGRO(de Koning & Liefting 1988). However, these models are reliant on
the use of finite element data or empirical parameters to account for such effects as
the plate thickness. A lot of ambiguity is therefore inevitably included due to the
use of interpolation or trial-and-error methods in determining those empirical
parameters.
Numerical models
Numerical approaches are known to be very versatile in dealing with
various factors related to the crack opening, including load ratio, strain hardening
specimen thickness and material constants (Chang, Li & Hou 2005). A vast number
of numerical studies on PICC have been carried out using finite element analysis
(FEA), which can provide solutions for stress/strain/displacement fields at any
point of the model under consideration. The crack opening/closing due to plasticity,
which is one of deterministic characteristics related to the crack growth driving
force, can thus be determined through use of the FEA. A number of FEA of fatigue
35
crack closure have been carried out using two and three-dimensional models. The
majority of research efforts have been undertaken to assess two-dimensional PICC
of through-the-thickness straight front cracks under plane stress or plane strain
conditions (Cochran, Dodds & Hjelmstad 2011; Dougherty, Padovan & Srivatsan
1997; Rodrigues & Antunes 2009; Shercliff & Fleck 1990; Solanki, K., Daniewicz,
S. R. & Newman Jr, J. C. 2004). On the other hand, three-dimensional PICC
models have attracted some of attention among researchers (Chermahini & Blom
1991; Roychowdhury & Dodds 2003; Skinner & Daniewicz 2002). These three-
dimensional models were mainly used to investigate the effect of plate thickness on
local stress distribution around the crack tip and thus the crack opening behaviour.
The capability of crack growth prediction techniques was enhanced by the three-
dimensional consideration because real plate and shell specimens always
experience some sort of the three-dimensional stress state regardless of the plate
thickness and other parameters or conditions.
The two- or three-dimensional finite element models provided a better
understanding of the PICC mechanism. However, in order to accurately capture the
stress fields it is often required to generate a vast number of elements, especially
around crack tips. This task itself can be very time-consuming and lead to a large
amount of CPU time to solve the problem. Furthermore, this type of analysis can
face significant numerical issues to deal with including such problems as mesh
refinement, crack advance scheme, crack face contact and changes in crack front
36
shape or crack location (Pitt & Jones 1997; Solanki, Kiran, Daniewicz, S. R. &
Newman Jr, J. C. 2004). These numerical issues become more complicated for
three-dimensional analysis and can limit the application of FEA in fatigue crack
analysis. As a consequence of these numerical issues, the results of independent
numerical studies on crack closure and fatigue crack growth can differ significantly
(Kelly & Nowell 2000; Solanki, Kiran, Daniewicz, S. R. & Newman Jr, J. C. 2004).
2.3 Structural Integrity of Plates with MSD Cracks
As mentioned in the Introduction, since the Aloha accident (Hendricks 1991),
MSD has been of great concern in industry, and the importance of including MSD
conditions in failure analysis has been recognized by many researchers. However,
the MSD phenomenon is very difficult to handle with conventional methods. The
presence of interaction between closely located cracks, which is a key issue in
MSD problems, can have a significant impact on the plastic zone formation and the
stress distribution near the crack tips. Because of these complications, conventional
methods, which are mainly based on solutions obtained from the analysis of
isolated single crack problems, have limitations in assessing MSD damages
(Carpinteri, Brighenti & Vantadori 2004; Collins & Cartwright 1996).
37
2.3.1 Prediction models on residual strength of plates with MSD
A significant amount of research effort into the problem of MSD has
concentrated on the development of theoretical approaches for predicting the
structural integrity of flat plate components with interactive cracks. The strip yield
model has been widely employed in the development of procedures for prediction
of the residual strength, or crack link-up conditions. The strip yield hypothesis
allows incorporating important parameters into the mathematical modelling
procedure. The popularity of this simplified model was mainly because of its
capability to provide a reasonable balance between computational effort and
accuracy of the prediction (Codrington & Kotousov 2009b).
Plastic zone coalescence criterion (plastic zone touch or ligament failure criterion)
Many previous investigators utilised the plastic zone coalescence criterion
to determine the residual strength of structural components with multiple cracks. In
accordance with this criterion, the ligament between cracks is assumed to fail if the
plastic zones at the crack tips become in contact with each other. This is an
adequate assumption because the contact of plastic zones indicates the occurrence
of complete plastic collapse of the ligament, which in turn will lead to very large
plastic deformation of the material between crack tips (Collins & Cartwright 1996).
Theocaris (1983) extended the strip yield model to consider the plastic
zones development in two collinear and unequal cracks. In the extended model, the
applied tensile stress levels for the complete plastic collapse of the ligament
38
between the two neighbouring cracks were calculated. The model was also used to
estimate the variation of inner and outer plastic zones associated with two collinear
cracks as a function of applied loading and crack length to separation gap ratios.
This work did not explicitly refer to the plastic zone coalescence criterion.
However, it formed a theoretical foundation for many MSD studies, especially
those which focused on the ligament failure of MSD.
Tong, Greif and Chen (1994) developed a method for analysis of multiple
cracks based on the hybrid finite element technique in conjunction with the
complex variable theory of elasticity. Their results included construction and
interpretation of residual strength diagrams for stiffened panels with multiple
cracks. Crack tip plasticity played an important role in the residual strength
calculations. Net section yielding stresses in the presence of multiple cracks were
calculated based on the asymptotic formula for stresses near a crack tip, which is
based on the Irwin and Dugdale formulas. Kuang and Chen (1998) suggested an
alternating iteration method for modelling the interaction between crack tip plastic
zones. In this approach, alternating iterations are implemented to solve the problem.
Their study indicated that the residual strength of plates with MSD can be
significantly overestimated (by 40%) if the interaction of plastic zones between
neighbouring cracks is not taken into account. Collins and Cartwright (2001)
constructed a strip yield model for two equal-length collinear cracks using a
complex stress function approach. The complex stress functions for this problem
39
were determined by applying the condition that the stress intensity factor vanishes
at the crack tip. In their research, the change of the plastic zone sizes at the inner
and outer crack tips with varying applied stress was presented. Nishimura (2002)
suggested an alternative approach based on the Fredholm integral equation method
(Yi-Zhou 1984) in conjunction with the strip yield model to establish the numerical
solutions of key parameters, such as plastic zone sizes and crack tip opening
displacements, for two collinear cracks. Wu and his co-workers (Wu & Xu 2011;
Xu, Wu & Wang 2011) extended the use of a weight function approach to the MSD
analysis. In their research, the strip yield model was formulated for a coalesced
centre crack in a finite width panel. Then, weight functions were used to obtain the
stress intensity factors and displacements, which were needed for enforcing a no
stress singularity condition at the crack tips and a zero crack opening condition
along the coalesced region. Through this, they predicted the plastic zone link-up
strength for panels with a lead and several small collinear cracks.
All these studies showed strong influence of crack interaction on the
residual strength of plates containing multiple collinear cracks. However, they
were limited to two-dimensional analysis, and the out-of-plane constraint effect due
to the plate thickness was not taken into account in them.
Energy based criterion: strain energy, work of fracture and T* integral
Alternative criteria for the prediction of the residual strength of MSD
structures include the energy-based approaches and T* integral. Labeas and co-
40
workers (Labeas & Diamantakos 2005; Labeas, Diamantakos & Kermanidis 2005)
developed a crack link-up criterion based on the strain energy difference before and
after the ligament failure. In this approach, the increase of the ‘specific’ total strain
energy, which is the total strain energy divided by the ligament area, due to the
ligament failure in the absence of plastic deformation in the ligament is considered
a critical value for the ligament fracture.
A similar criterion was proposed by Duong, Chen and Yu (2001) based on
the concept of total work of fracture. Their approach utilised the assumption that
the specific work required to cause ligament failure is a linear function of the
normal extent of the plastic region.
T* integral criterion was employed by Wang, Brust and Atluri (1997) and
Gruber, Wikins and Worden (1997) for the assessment of the residual strength of
MSD structures. T* integral is the energy flux per unit crack growth into a contour
enclosing the crack tip. T* integral is now a well known fracture parameter, and it
has been successfully used in predicting stable crack growth in elastic-plastic
material and characterising creep crack growth (Anderson et al. 2004). In this
approach, the crack is assumed to grow when the T* integral value attains a critical
value, i.e. T* integral resistance, which is calculated from tests. This approach is
also capable of dealing with three-dimensional crack problems. However, it
demands extensive computational power supported by experimental studies.
Crack tip opening angle and crack tip opening displacement criteria
41
Another well-known criterion for the residual strength is based on
parameters related to crack tip opening behaviour, such as crack tip opening angle
(CTOA) and crack tip opening displacement (CTOD). Chen, Wawrzynek and
Ingraffea (1999) suggested the use of CTOA in assessing the residual strength in
the case of multiple cracks representing the MSD. In their model, it was assumed
that the angle maintains a constant value during stable crack growth, and the
residual strength was directly obtained from the crack growth data. Galatolo and
Nilsson (2001) developed a residual strength model making use of these parameters,
CTOA and CTOD. In their model, the onset of the crack growth is assumed to
occur when CTOD reaches a critical value while the stable crack propagation was
modelled to be driven by CTOA only. Furthermore, for the initiation phase of crack
growth, they proposed a more refined model where the crack growth is assumed to
occur at a constant CTOA until the crack propagates by a certain length.
2.3.2 Prediction models of fatigue lifetime of components subjected to
MSD
The prediction of crack growth for multiple cracks, especially when they
are closely located to interact with each other, represents a challenging research
topic. Such predictions have to take into account the crack interaction effect. There
are many previous studies focusing on calculating the lifetime in the presence of
MSD. This research effort employed simple LEFM approaches as well as advanced
numerical approaches, such as FEA and the dual boundary element method.
42
LEFM models
Pártl and Schijve (1993) and Collins and Cartwright (1996) developed
crack growth prediction models of MSD cracks through application of LEFM
theory. In these simple models, no plastic yielding in the uncracked ligament was
allowed, and the incremental crack growth scheme was mainly implemented
through utilization of the stress intensity factor range. The stress intensity factor
range was determined by analytical methods, such as the compound method (Pártl
& Schijve 1993) and the stress function method (Collins & Cartwright 1996).
Because of the reliance on the analytical approaches, the applications of these
models were limited to plane stress and plane strain conditions.
FEA models
FEA was widely employed to investigate the crack interaction effect on the
fatigue behaviour of MSD cracks by Silva et al. (2000), Shkarayev and Krashanitsa
(2005) and Kamaya (2008), just to name a few. These studies used FEA to perform
the stress analysis of structures with MSD before crack initiation and during crack
growth, and then implemented any appropriate crack growth law without taking
into account the crack closure. Due to the versatility of FEA, it can deal with
problems involving various configurations of multiple cracks and three-
dimensional cracks; however, it also has intrinsic problems such as vast amount of
computation time, meshing and re-meshing issues and ambiguities related with
crack advance scheme as discussed previously.
43
DBEM models
The dual boundary element method (DBEM) has also been used as a
powerful tool to handle MSD problems. This method incorporates two independent
boundary integral equations: one for the displacement at a collocation point on one
surface of the crack and the other for the traction at a corresponding collocation
point on the opposite surface. Because this numerical technique does not normally
necessitate integration over the entire problem domain, the workload for domain
discretisation can be reduced considerably. In addition to this benefit, the
modelling of crack propagation can be automated without too much difficulty.
Wessel et al. (2001), Calì, Citarella and Perrella (2003) and Citarella (2009)
investigated the problem of three-dimensional multiple crack growth using the
DBEM in conjunction with automatic crack propagation modelling. In order to
determine the local out-of-plane direction of crack growth, they employed the
minimum strain energy criterion. By this criterion, the direction of crack
propagation is assumed to follow the direction which has the minimum strain
energy density value. This method has been successful in circumventing some
drawbacks of FEA, but the solution time can be very long because conventional
DBEM matrices are non-symmetrical and dense. Furthermore, it can be difficult to
incorporate elastic-plastic considerations with the DBEM because the problem
domain that is probable to yield must be discretised (Armentani & Citarella 2006;
Wessel et al. 2001).
44
2.4 Research Gaps
The strip yield model has been one of the most widely used theoretical
concepts for modelling crack tip plasticity effects. Along with various classical
elastic-plastic fracture parameters, the strip yield model has also formed the
theoretical backbone for the development of residual strength prediction models for
plate and shell components experiencing MSD. These developments provided a
better understanding of failure of the ligament between interacting cracks and
showed a strong crack interaction effect on the structural integrity of these
components. However, most of the analytical models are two-dimensional, which
largely disregard the effect of the plate thickness on fatigue crack growth. Some of
these models investigated three-dimensional cracks, although they relied on elastic-
plastic parameters which are often determined through extensive experimental
work and/or three-dimensional FEA.
From the conducted literature review, PICC has been proved to be very
promising in the development of fatigue crack growth prediction models. However,
the majority of the previous investigations on PICC have been implemented
without taking into account the interaction between closely spaced cracks. No
PICC crack growth models which are readily available and practical for the case of
interactive cracks (or MSD) were found.
45
The literature review showed that several MSD crack growth models have
been developed. Those models were however largely based on advanced numerical
approaches, such as the FEA and the DBEM without considering the crack closure.
Such approaches have many issues, and independent studies often lead to different
or non-validated, non-reproducible results as mentioned earlier in this chapter.
Among specific problem, it can be stated that the plate thickness effect in
MSD has not been properly investigated so far. The out-of-plane constraint due to
the plate thickness is now well known to affect the local stress field near the crack
tip and thus crack plasticity, fatigue crack closure and fatigue crack growth. Even
though this plate thickness effect has been shown based on the studies of single
crack cases, it is highly likely it also has a significant impact on the residual
strength and fatigue behaviour in the case of MSD.
The primary objective of this study is to develop analytical and numerical
models for the evaluation of the residual strength and fatigue crack growth of two
through-the-thickness cracks in a plate of finite thickness subjected to monotonic
and cyclic loading. An experimental study was also conducted with the aim to
investigate the crack interaction effect and to validate the theoretical prediction
models. It is believed that the conducted theoretical and experimental studies make
an important contribution to the understanding of MSD and to the improvement of
the damage tolerance design of engineering structures, which can experience MSD
during their service life.
46
47
Chapter 3
3 A Strip Yield Model for Two Collinear Cracks
3.1 Introduction
Multiple cracks under fatigue loading often grow initially as isolated or non-
interacting defects. With an increase in the crack length and a decrease in the
distance between cracks, the deleterious interaction between the cracks increases
rapidly. This makes theories and experimental data obtained from a single crack
analysis inappropriate to evaluate the fatigue life in the case of MSD. Therefore, it
is not surprising that new evaluation methods which can take into account
interactions between multiple cracks are very important and currently in strong
need. Vast research efforts have been devoted to the solution of various crack
problems incorporating two or more interacting cracks (Collins & Cartwright 2001;
Duong, Chen & Yu 2001; Labeas, Diamantakos & Kermanidis 2005; Wu & Xu
2011). Wu and Xu (2011) applied the weight function approach to the MSD
analysis. In their study, the Dugdale strip yield model was used to determine the
conditions of coalescence of two collinear cracks in a finite width plate. In this
work, the weight function approach was applied to obtain the stress intensity
factors and crack tip opening displacements, which were needed to enforce the
finite stress conditions at the tips of the crack as well as zero crack opening along
the coalesced region. Finally, theoretical results for the plastic zone link-up
48
strength and local collapse loads for panels with a lead crack and several small
collinear cracks were provided.
An analytical approach for the evaluation of the coalescence load for two
collinear equal length cracks was suggested by Collins and Cartwright (2001).
These researchers constructed an analytical solution using a complex stress
function approach (Rice 1968). The complex stress functions for this problem were
determined by enforcing the additional condition, which removes the stress
intensity factor (stress singularity) at the crack tip. The plastic zone sizes at the
inner and outer crack tips as functions of the remotely applied stress were then
calculated.
Other researchers employed an energy based approach to address MSD
problems (Duong, Chen & Yu 2001; Labeas, Diamantakos & Kermanidis 2005). In
this approach the strip yield model was implemented to reduce a MSD problem to a
problem of a centre cracked plate subjected to remote stresses and a crack
subjected to surface tractions. After solving these two problems for the normal
displacements (opening) along the crack, the total work required to cause ligament
failure using the obtained displacement field was determined. The developed
approach allows the prediction of the ligament link-up (also called as local plastic
collapse) loads for plates containing major and adjoining minor cracks.
In this chapter, a new computational method for an analysis of two equal-
length collinear cracks, which are the simplest form of MSD, is developed. The
49
general procedure may also be extended to various types of MSD without
significant modifications. The cracks and yielding strips are represented by an
unknown distributed dislocation density function, and two alternative approaches
are developed to find the solutions to the problem. In the first approach, the
solution is obtained analytically by solving Föppl integral equation; in the second
approach a numerical procedure based on the Gauss-Chebyshev quadrature method
(Erdogan & Gupta 1972) is implemented. The latter approach can be easily
extended to the analysis of three-dimensional problems. The developed approaches
produce practically identical results when applied to solve a test problem, and both
approaches are also validated against previously published studies demonstrating a
very good agreement. In addition, the new results for the crack tip opening
displacement in the case of two equal-length collinear cracks subjected to remote
tensile stress on infinity are presented.
3.2 Problem Formulation
The geometry of the problem is shown in Figure 3.1, where two collinear
cracks, of equal physical length 2a, are subjected to remotely applied tensile stress,
σ�� . The plasticity zones are represented by thin strips of plastic yielding ahead of
the tips as first suggested in the classical Dugdale strip yield model (Dugdale 1960).
The lengths of the inner and outer plastic zones are denoted as w� and w'
respectively, and are normally different from each other because the inner and
50
outer crack tips are affected differently by the neighbouring crack. The distance
between the inner crack tips is 2c and the distance between the outer tips of crack is
2b (see Figure 3.1).
Figure 3.1 Problem geometry and coordinate system.
The distribution of the y -stress, σ�� , along the x -axis for the above
formulated problem, in the case of frictionless crack contact (Johnson 1985), can
be found from the solution of the following boundary value problem (Chang, Dh &
Kotousov, A. 2012):
u��x, 0� 9 0 , |x| E c, (3.1a)
u��x, 0� 9 0 , |x| G b, (3.1b)
a a c c
d
x
y
σ��
σ��
a
d
a w� w� w' w'
b b
51
σ���x, 0� 9 σ�, c H |x| E c < w�, (3.1c)
�x, 0� 9 0 , c < w� H |x| E b : w', (3.1d)
σ���x, 0� 9 σ� , b : w' H |x| E b, (3.1e)
where u� represents the y-displacement, and σ� is the flow stress of the material.
Usually, the flow stress is taken as the average of yield strength and ultimate
strength and can include the effect of plastic hardening (Koolloos et al. 2001). The
sizes of the inner and outer plastic zones are not known a priori and have to be
found from the solution procedure. The above boundary-value problem represents a
standard Riemann-Hilbert problem, and the solution can be obtained by the
Cauchy type integral method as demonstrated, for example, by Collins and
Cartwright (2001). This solution method does however require rather tedious
calculations. Moreover, it is normally unsuitable for complicated geometries and
loading conditions. From the solution obtained by Collins and Cartwright, it is also
quite difficult to calculate some important fracture controlling parameters, such as
the crack tip opening displacement (CTOD) or opening angle. In this chapter, two
alternative approaches based on the distributed dislocation technique will be
developed, which overcome the above mentioned difficulties and limitations.
To solve the boundary-value problem of equations (3.1a), the distributed
dislocation technique is applied (Kotousov 2007; Kotousov & Codrington 2010).
This approach involves representing the crack and plastic zone line by an unknown
52
distribution of dislocation to simulate strain nuclei, i.e. the edge dislocation in this
study. The superposition principle is then used to find stress field solutions to this
problem. From this principle, the stresses that would be present in an uncracked
body subject to the same external forces are superimposed on the stresses produced
by the distribution of strain nuclei (Hills et al. 1996). Accordingly, let us introduce
a function, B��ξ� , which represents the edge y -dislocation density in the x -
direction. It is associated with the crack opening as (Hills et al. 1996; Kotousov
2007),
B��ξ� 9 : dδ�ξ�dξ , (3.2)
where the function δ�ξ� represents the crack opening displacement along the
physical crack region and the plastic stretch along plasticity regions. Utilising the
symmetry of the problem (Codrington & Kotousov 2007b) the following singular
integral equation can be written:
2π K ξB��ξ�x� : ξ� dξLM
9 F�x� , (3.3)
where the function on the right side of (3.3) is:
53
F�x�9
OPQPR κ < 12µ 0σ� : σ�� 2 for c H |x| E c < w� and b : w' H |x| E b,
: κ < 12µ σ�� for c < w� H |x| E b : w'W . (3.4)
In (3.4) µ is the shear modulus of the material, and κ is Kolosov’s constant, which
is equal to (3–ν)/(1+ν) in plane stress and 3–4ν in plane strain with ν being the
Poisson’s ratio. Singular equation (3.3) represents a standard Föppl integral
equation (Maiti 1980), and B��ξ� is the unknown function to be solved for.
In the case of the local plastic collapse, wherein the inner plastic zones fully
occupy the centreline between two cracks ( c 9 0, see Figure 3.2), the size of the
outer plastic zone can be found analytically by the superposition of several well-
known analytical solutions for a single equivalent crack of total length, 2b 92w� < 4a < 2w' (Collins & Cartwright 2001). More specifically, in the beginning
the stress intensity factor solution at the tip of a single equivalent crack of length 2b
under remotely applied tensile stress, σ�� , is invoked. Next, the stress intensity
factor due to the compressive surface tractions of the magnitude equal to flow
stress, σ�, over the inner and outer plastic zones is constructed by using the well-
established stress intensity factor solutions. Then all these solutions are
superimposed to derive the stress intensity factors at the inner tips of the cracks.
Setting these stress intensity factor to be zero to meet the bounded stress condition
at the crack tip produces the relationship between the geometry, flow stress and the
54
applied stress, which leads to local plastic collapse. This relationship can be written
as (full details can be found in Collins and Cartwright (2001)):
b 9 Zw�� < [�2a < w�� sec ]π2 σ�� σ� ^ : w� tan ]π2 σ�� σ� ^`� . (3.5)
Figure 3.2 Local plastic collapse.
From equation (3.5), by setting w� a 0, the plastic zone size solution for a single
crack of length 4a can be obtained. This expression corresponds to Rice’s
analytical solution (Rice 1966) for a single isolated crack in an infinite plate, which
is:
w� w� a a
d
x
y
σ��
σ��
a
d
a w' w'
b b
55
b 9 2a sec bπ2 σ�� σ� c . (3.6)
These two equations (3.5) and (3.6) can serve for validating a more general
solution to be obtained for two interacting cracks in the next section.
3.3 Analytical Solution: Inversion of Föppl Integral
The solution to the Föppl integral equation (3.3) can be written in two
alternative forms (Maiti 1980). The first form is:
B��x� 9 : 2π bb� : x�x� : c�c� �⁄ K bξ� : c�b� : ξ�c� �⁄ F�ξ�ξx� : ξ� dξ LM
< P�x� : c��� �⁄ �b� : x��� �⁄ , (3.7)
where the constant P is given by:
P 9 bK�k� 2π K K bb� : x�x� : c�c� �⁄ bξ� : c�b� : ξ�c� �⁄ F�ξ�ξx� : ξ� dξdxLM
LM
, (3.8)
and K�k� is a complete elliptic integral of first kind with the k parameter described
as:
k 9 bb� : c�b� c� �⁄ . (3.9)
The second form of the solution to the Föppl integral equation (3.3) is:
56
B��x� 9 : 2π bx� : c�b� : x�c� �⁄ K bb� : ξ�ξ� : c�c� �⁄ F�ξ�ξx� : ξ� dξLM
< Q�x� : c��� �⁄ �b� : x��� �⁄ , (3.10)
where the constant Q is given by:
Q 9 bK�k� 2π K K bx� : c�b� : x�c� �⁄ bb� : ξ�ξ� : c�c� �⁄ ξF�ξ�x� : ξ� dξdx .LM
LM
(3.11)
The absence of the stress singularity condition at the crack tips x 9 c and x 9 b is
applied to equation (3.7) and (3.10) respectively, resulting in the equations:
2π K F�ξ�ξgb� : ξ�gξ� : c� dξ < Pb� : c� LM
9 0 , (3.12)
2π K F�ξ�ξgb� : ξ�gξ� : c� dξ : Qb� : c�L
M9 0. (3.13)
The solution to the system of integral equations (3.12) and (3.13) can be
obtained by using a simple iterative procedure as follows. At a fixed inner plastic
zone size, w�, and a guess value of the outer plastic zone size, w', the required
applied tensile stress σ�� is first found from equation (3.12) exactly. Then, all three
values (w�, w' and σ�� ) are substituted into equation (3.13). If equation (3.13) is
satisfied with the desired accuracy, then these values are taken as a solution to the
problem at the specified w� value. If the desired accuracy is not achieved, then a
57
new and corrected value of w' and the new calculated value of σ�� from equation
(3.12) are substituted again into equation (3.13). The procedure is repeated until the
desired accuracy or a specified convergence condition is achieved. The results of
the calculation procedure can be represented as functions of one of the problem
governing parameters, w�, w' or σ�� (see Fig. 3.1).
3.4 Numerical Solution: Gauss–Chebyshev Quadrature Method
To implement the Gauss–Chebyshev quadrature method, which can also
provide a solution to more complicated and more general problems, a scale
transformation of coordinates is carried out first by introducing the new variables t (:1 E t E 1) and s (:1 E s E 1) such that:
x 9 b < c2 < b : c2 t , (3.14a)
ξ 9 b < c2 < b : c2 s . (3.14b)
The integral in equation (3.3) is then transformed to an integral over the
range -1 to 1 based on equations (3.14a) and (3.14b):
1π K B��s�G�t, s�ds�h�
9 F�t� , (3.15)
where G�t, s� is a kernel of the integral equation, which can be expressed as:
58
G�t, s� 9 1t : s : 1t < s < 2 b < cb : c , (3.16)
and F�t� is given by equations (3.4) and (3.14a). In addition, because there should
be no net dislocation if we integrate the dislocation density from one end of the
crack to the other, the dislocation density, B/��s�, satisfies the following condition,
K B/��s�ds�h�
9 0 . (3.17)
Solution to the singular integral equation (3.15) with (3.17) can be obtained
by the standard Gauss–Chebyshev quadrature. For this purpose, an unknown
regular function ./�s� is first introduced such that:
B/��s� 9 ./�s�√1 : s� . (3.18)
This converts equation (3.15) into the following system of N algebraic equations to
N unknowns, .�s��:
b : c2N j ./k�l�
�s��G�tm, s�� 9 F�tm� k 9 1,2 n N : 1 , (3.19)
πN j ./k�l�
�s�� 9 0 , (3.20)
where the discrete integration and collocation points are given by:
59
s� 9 cos ]π 2i : 12N ^ , i 9 1, 2, … N, (3.21)
tm 9 cos ]π kN^ , k 9 1, 2, … N : 1, (3.22)
respectively.
This system of linear equations (3.19) and (3.20) can be easily solved through a
standard computer-based procedure for the solution of a N by N system of linear
algebraic equations.
Through an asymptotic analysis, the stress intensity factors at the tips of the
crack (x 9 c and x 9 b) can be respectively found as (Lonwengrub & Srivastav
1970):
KM 9 : 2µκ < 1 qπ2 �b : c�./�:1� , (3.23)
KL 9 2µκ < 1 qπ2 �b : c�./�1� . (3.24)
To ensure a bounded stress field condition, the dislocation density must be zero at
the tips of the plastic zones:
./�r1� 9 0 . (3.25)
The solution to the system of integral equations (3.19) and (3.20) with an additional
condition (3.25) can be obtained by using a similar iterative procedure used in
Section 3.3, but equation (3.25) is now employed to ensure the specified accuracy
to be met. In the case of two collinear equal length cracks the results converges at
60
the number of integration points, N~ 100. In the next Section some selected results
of calculations will be presented and discussed.
3.5 Results and Discussion
3.5.1 Plastic zone size
The results of the analysis of two equal-length collinear cracks are shown in
Figure 3.3 to Figure 3.6. The results from the Gauss–Chebyshev quadrature
approach, equations (3.19) - (3.24), are practically identical to the corresponding
results obtained by the inversion of the Föppl integral equations (3.12) and (3.13).
Figure 3.3 illustrates the variation of the normalised inner plastic zone size,
w�/w, as a function of the normalised applied stress, σ�� /σ� , for four different
separations between the cracks. To represent results in a dimensionless form, the
inner plastic zone size is divided by the plastic zone size, w, of a single crack of the
same length, which can be determined from Rice’s classical analytical solution
(Rice 1966). As shown in Figure 3.3, a significant interaction between the two
cracks is predicted if they are closely located with small c/b values while minimal
interaction is seen when the cracks are substantially separated from each other with
large c/b values. For example, as the c/b ratio changes from 0.4 to 0.05 (the cracks
are approaching each other) at some intermediate ratio of the applied stress to flow
stress, σ�� /σ� 9 0.3; the normalised inner plastic zone size, w�/w, considerably
61
increases from 1.05 to 1.75. In general, the interaction is larger at lower applied
stress levels and this tendency becomes stronger as the two cracks are located
closer. In Figure 3.3, analytical results by Collins and Cartwright (2001) are also
presented. They show a very good agreement with the present results, although the
former have a tendency to indicate slightly higher interaction and the difference
moderately increases as the interacting cracks are more closely positioned. The
difference between the present results and Collins and Cartwright data is within 4%
for the considered range of crack geometries, 0.05 E c/b E 0.4.
Figure 3.3 Normalised inner plastic zone size as a function of normalised applied
stress for four different c/b values (0.05, 0.1, 0.2 and 0.4).
1.0
1.2
1.4
1.6
1.8
2.0
2.2
0 0.5 1
c/b increasing
Present results
Collins & Cartwright
w�w
σ�� /σ�
62
Figure 3.4 shows the change in the inner to outer plastic zone size ratio,
w�/w', as a function of the value of the remotely applied stress ratio, σ�� /σ�. The
results are compared with those obtained by Collins and Cartwright (2001). As
expected, both studies are in a good agreement showing less than 1.5% difference
when c/b range is between 0.05 - 0.4. The overall trend in the variation of the
plastic zone size ratio is very similar to that of the inner plastic zone size.
Figure 3.4 Ratio of the inner to outer plastic zone size ratio as a function of
normalised applied stress for four different c/b values (0.05, 0.1, 0.2 and 0.4).
1.0
1.2
1.4
1.6
1.8
0 0.5 1
c/b increasing
Present results
Collins & Cartwright
w�w'
σ�� /σ�
63
3.5.2 Crack tip opening displacement
In this section, the effect of crack interaction on the crack tip opening
displacement (CTOD), which is equal to the plastic stretch at the physical crack tip,
is considered. CTOD is calculated for a wide range of geometries and levels of the
applied stress. The variations of the dimensionless inner CTOD, δ�/δ, and the ratio
of inner to outer CTOD, δ�/δ', are given in Figure 3.5 and Figure 3.6 respectively.
In Figure 3.5, the inner CTOD is normalised by the CTOD for a single isolated
crack, δ, which is given by Rice’s analytical solution for the CTOD (Rice 1966).
Both figures show that the interaction between two cracks is stronger at smaller
separations, and that this interaction is more significant at relatively lower stress
levels (at fixed separation or c b⁄ ratio). However, if compared to the variation of
the plastic zone size (see Figures 3.3 and 3.4), CTOD dependences as functions of
the ratio of the applied stress to the flow stress (see Figures 3.5 and 3.6) converge
to the solution for a single crack (corresponding to unit value on the y-axis) more
rapidly.
64
Figure 3.5 Normalised inner CTOD as a function of normalised applied stress for
four different c/b values (0.05, 0.1, 0.2 and 0.4).
1.0
1.2
1.4
1.6
1.8
2.0
2.2
0 0.5 1
c/b increasingδ�δ
σ�� /σ�
65
Figure 3.6 Inner to outer CTOD ratio as a function of normalised applied stress for
four different c/b values (0.05, 0.1, 0.2 and 0.4).
3.6 Conclusions
This chapter describes the development of two alternative approaches for
analysis of two equal-length collinear cracks. The considered problem is very
simple but provides insight and valid tendencies for many other problems and
applications. Both approaches are based on a representation of the cracks by a
distribution of edge dislocation, and the Dugdale strip yield model is adopted to
represent the elastic-plastic behaviour of the material at crack tips. The first is an
analytical approach and founded on the direct inversion of Föppl integral equation.
1.0
1.2
1.4
1.6
1.8
0 0.5 1
c/b increasingδ�δ'
σ�� /σ�
66
The second is a semi-analytical approach and utilises the Gauss-Chebyshev
quadrature method to calculate the dislocation density function and fracture
controlling parameters.
The two developed approaches have been compared with each other showing
practically identical results. Moreover, these approaches have been validated
against an earlier developed procedure, which utilises the complex variable and
Cauchy type integral methods (Collins & Cartwright 2001). A slight difference
(less than 4 % for all range of the considered crack geometries) has been observed,
which is likely, due to numerical errors in calculation of elliptic integrals in the
analytical solution of Collins and Cartwright (2001).
New results on interaction of the crack tip opening displacement of two equal
length collinear cracks have also been presented. Throughout the conducted study,
it has been demonstrated that the crack interaction is significantly affected by both
the crack spacing and the ratio of the applied stress to the flow stress. As an
example, the obtained results indicate that the inner plastic zone of two interactive
cracks can be up to 75% larger than that of a single crack at the same applied stress
when the spacing ratio c/b = 0.05, and the applied stress level is 0.3 of the
material’s flow stress.
The semi-analytical approach, which is based on Gauss-Chebyshev
quadrature method can be easily generalized to other crack geometries and loading
conditions and can be coupled with a suitable crack advance scheme to analyse
67
fatigue crack growth and fatigue crack interaction. Moreover, by replacing the
dislocation influence function (3.16) in the integral equation (3.15) (Rice 1966)
with a corresponding one for a finite thickness plate, the approach can be extended
to analyse the effect of plate thickness for static fracture and fatigue behaviour.
This will be demonstrated in the following chapters of this thesis. It will be also
demonstrated that the plate thickness, through the change of the out-of-plane
constraint altering the yield conditions, will significantly affect the fracture and
fatigue controlling parameters as well as the conditions of the local plastic collapse.
In these further investigations, the results obtained in this chapter will serve as
benchmark solutions for limiting cases of very thick (plane strain) and very thin
(plane stress) plates. These results will be used to validate a more general theory,
which will be developed in the next chapter for finite thickness plates.
68
69
Chapter 4
4 A Strip Yield Model for Two Collinear Cracks in a Plate of
Arbitrary Thickness
4.1 Introduction
Various methods and techniques have been proposed by different researchers
to solve MSD problems. Kaminskii, Gutsul and Galatenko (1987) suggested the
use of a distribution of displacements method combined with the complex potential
approach to investigate the interaction between two collinear cracks. After that,
Nilsson and Hutchinson (1994) introduced a modified strip yield model in
conjunction with the concept of damage-reduced fracture toughness. In this work
the weakening of the plate due to the extension of the plastic zone of the macro-
crack into the micro cracked area was modelled by changing the material’s flow
stress. Kuang and Chen (1998), Ali and Ali (2000), Collins and Cartwright (2001),
Bhargava and Hasan (2011), and Wu and Xu (2011), to name just a few, carried
out further studies on MSD based on the strip yield model. These previous research
activities were, however, limited to plane stress and/or plane strain problems, and
three-dimensional nature of the crack tip stress fields and various three-
dimensional effects were not taken into account.
The out-of-plane constraint due to the plate thickness is now well known to
affect the local stress field near the crack tip. The thickness effect on the plastic
70
zone size and fracture controlling parameters have been investigated exhaustively
for isolated cracks (Codrington & Kotousov 2009a; Costa & Ferreira 1998; de
Matos & Nowell 2009; Guo, Wang & Rose 1999; Newman Jr 1998). It is expected
that the plate thickness has also a significant impact on the residual strength and
fatigue behaviour of MSD plates.
This chapter aims to propose a new theoretical method for the analysis of
stationary MSD cracks. As an example, two collinear cracks of equal length in a
plate of arbitrary thickness will be considered in detail. The method is based on the
classical strip yield model (Dugdale 1960) and the distributed dislocation technique
(Hills et al. 1996). The analytical modelling of the nonlinear plate thickness effect
is implemented through use of the three-dimensional fundamental solution for an
edge dislocation derived by Kotousov and Wang (2002) in the frame of the
generalized plane strain theory. This three-dimensional model will be used to
investigate the residual strength of plates containing two collinear cracks. The
remotely applied tensile stress levels required for the complete plastic collapse of
the ligament will be predicted with respect to the change in the spacing of cracks as
well as the change in the plate thickness. In addition, the obtained results will be
compared with previously published solutions for special cases, such as the
solution for a single crack in a finite thickness plate (Kotousov 2004) and the
analytical solution obtained within plane stress/plane strain assumption for two
cracks subjected to remote loading (Collins & Cartwright 2001). The comparison
71
will demonstrate if the new method correctly recovers the previously obtained
results for these special cases. Further, the obtained solution will be validated using
the appropriate experimental results from an experimental program to be described
later in this thesis.
4.2 Problem Formulation and Distributed Dislocation Approach
Figure 4.1 shows the problem geometry of two collinear cracks of identical
length, 2a, with centre-to-centre crack distance, 2d, located in an infinite plate of
finite thickness, 2h. The plate is subject to a remotely applied tensile stress, σ�� .
When a cracked plate is loaded, plastic zones are formed at the crack tips. In this
work the plastic zones are modelled with the help of the classical Dugdale strip
yield model. Here, we also adopt the rigid perfectly plastic material model and
assume the uniform length of the plastic strip across the crack front (Codrington &
Kotousov 2009). These assumptions are quite common in analytical studies and
allow the analytical treatment of crack problems. It was also demonstrated in the
literature that, in many cases, the analytical solutions and results obtained by other
methods, experimentally or numerically, correlate rather well (Kotousov &
Codrington 2010). The analytical solutions however have many advantages
including versatility and better ability to reproduce and analyse.
72
The inner and outer plastic zone sizes are denoted by w� and w', respectively.
The effective crack length is defined as the sum of sizes of an actual crack and
surrounding plastic zones. The distance between the inner effective crack tips of
two cracks is denoted as 2c, whereas the distance between the outer effective crack
tips is 2b (Figure 4.1). The origin of the coordinate system is set at the middle point
of the two collinear cracks as shown in Figure 4.1.
Figure 4.1 Problem geometry and coordinate system.
The distributed dislocation technique can be effectively applied to
investigate the problem of mutually interacting cracks (Kotousov 2007; Kotousov
w� a a w' w� w' a a
2h
u b d u d
b
z
x
y
σ��
σ��
73
& Codrington 2010). This approach involves representing the crack and plastic
zone line by an unknown distribution of dislocation to simulate strain nuclei (see
Chapter 3 for details). As a result, a set of governing integral equations for x, y and
z stress fields along the positive x-axis can be obtained through use of the
distributed dislocation approach. By taking advantage of the symmetry of this
problem, the equations are:
σ##�x� 9 1π K B��ξ� G##�x, ξ�dLM
ξ , (4.1a)
σ���x� 9 1π K B��ξ� G���x, ξ�dLM
ξ < σ�� , (4.1b)
σww�x� 9 1π K B��ξ� Gww�x, ξ�dLM
ξ , (4.1c)
where B��ξ� is an edge dislocation density function which corresponds to a
location ξ between c and b on the x-axis and causes Burgers vectors in the y-
direction, and G##�x, ξ� , G���x, ξ� and Gww�x, ξ� are kernels in the x, y and z
directions, respectively. The kernels can be considered as induced stresses in the
direction of the subscript at an arbitrary point x due to an unit Burgers vector in the
y-direction located at a point ξ, and they become singular at the point where x 9 ξ.
The dislocation density function, B��ξ�, can be determined by enforcing boundary
conditions such as traction-free on crack faces and material yielding in plastic
74
zones. If the Tresca yield criterion is employed, assuming that σ�� x σ## x σww,
stresses in the plastic zone must satisfy the following equation (or yielding
criterion):
yσ�� : σwwy 9 σ� (4.2)
where σ� is the material’s flow stress. The crack opening displacement, g�x�, is
associated with the distributed dislocation density, B��ξ�, through the notion that
the sum of negative infinitesimal Burgers vectors, :B��ξ� dξ, from the point c to
any arbitrary point x positioned between c and b leads to the amount of crack
opening displacement at that point x, such that:
g�x� 9 : K B��ξ�d#M
ξ , (4.3a)
or,
B��ξ� 9 : dg�ξ�dξ . (4.3b)
Therefore, the physical meaning of the dislocation density function can be regarded
as the negative gradient of the crack opening displacement at a point between two
crack tips.
Through the use of the traction free condition on crack faces and the Tresca
yield criterion in plastic zones, the governing integral equations for problems of
75
two collinear cracks in a plate with plane stress, plane strain or finite thickness can
now be reduced to the following equation:
σ�x� 9 1π K B��ξ� G�x, ξ�dLM
ξ < σ�� , (4.4)
where σ�x� and G�x, ξ� are a representative stress and a representative kernel,
respectively, and are provided for each stress condition of the cracked plate in the
following subsections.
4.2.1 Plane stress case
The first limiting case is that of plane stress. The representative stress and
the kernel in equation (4.4) are given as:
WG�x, ξ� 9 G�����x, ξ� : G�����x, :ξ�σ�x� 9 σ���x� 9 0 z for c < w� E |x| H { : w' , (4.5a)
WG�x, ξ� 9 G�����x, ξ� : G�����x, :ξ�σ�x� 9 σ���x� 9 σ� z for c E |x| H u < w� or b : w'E |x| H { ,
(4.5b)
where G�����x, ξ� is the two-dimensional Cauchy kernel for the case of a single
isolated crack in plane stress or plane strain and is written as (Hills et al. 1996):
G�����x, ξ� 9 2µκ < 1 ] 1x : ξ^ (4.6)
76
In the equation above, µ is the shear modulus and κ is the Kolosov’s constant equal
to �3 : ν� �1 < ν�⁄ in plane stress with ν being the Poisson’s ratio. To obtain the
kernel of the system in equations (4.5a), the Cauchy kernel has been modified
based on the notion that a dislocation of equal magnitude and opposite sign is
placed at x 9 :ξ for every dislocation placed at x 9 ξ when two identical cracks
are lying on the x-axis with the origin of the coordinate system located in the
middle of them.
4.2.2 Plane strain case
If a plate with two collinear cracks is regarded to be in a plane strain
condition, the representative stress and the kernel in equation (4.4) can be given as:
WG�x, ξ� 9 G�����x, ξ� : G�����x, :ξ�σ�x� 9 σ���x� 9 0 z for c < w� E |x| H { : w' , (4.7a)
WG�x, ξ� 9 �1 : 2ν�|G�����x, ξ� : G�����x, :ξ�}σ�x� 9 σ���x� : σww�x� 9 σ� z for c E |x| H u < w� or b : w' E |x| H { ,
(4.7b)
where G�����x, ξ� represents the same Cauchy kernel expressed in equation (4.6)
except that κ is 3 : 4ν in plane strain. To derive the representative term G�x, ξ� in
equation (4.7a), the z-direction kernel in plane strain, Gww���x, ξ�, is first determined,
by making use of σww 9 ν�σ## < σ���, equations (4.1a) - (4.1c) with σ�� 9 0 and
G##���x, ξ� 9 G�����x, ξ� in plane stress or plane strain condition, as:
77
Gww���x, ξ� 9 2νG�����x, ξ� (4.8)
Lastly, the Tresca yield criterion is applied, leading to the kernel of the system as
shown in equation (4.7b).
4.2.3 Finite thickness case
For two collinear cracks in an infinite plate with finite thickness, the
representative stress and the kernel in equation (4.4) can be expressed using the
kernels by Kotousov and Wang (2002) for an edge dislocation in three-dimensional
stress state. This gives:
WG�x, ξ� 9 G�����x, ξ� : G�����x, :ξ�σ�x� 9 σ���x� 9 0 z for c < w� E |x| H { : w' , (4.9a)
WG�x, ξ� 9 G�����x, ξ� : G�����x, :ξ� : Gww���x, ξ� < Gww���x, :ξ�σ�x� 9 σ���x� : σww�x� 9 σ� z for c E |x| H u < w� or b : w' E |x| H { ,
(4.9b)
where the three-dimensional kernels in the y-direction, G�����x, ξ�, and in the z-
direction, Gww���x, ξ�, are:
G�����x, ξ� 9 : E4�1 : ν�� 1ρ � 4ν��λρ�� : �1 : ν�� : 2ν�K��λ|ρ|�: 2�2 < λ�ρ��ν�K��λ|ρ|�λ|ρ| � ,
(4.10)
78
Gww���x, ξ� 9 E2�1 : ν�� λνK��λ|ρ|� ρ|ρ| , (4.11)
where ρ 9 x : ξ, E is the Young’s modulus, λ is a parameter, which can be written
as :
λ 9 1h Z 61 : ν , (4.12)
and K0�·� and K1�·� are the modified Bessel functions of the second kind, which are
the solutions to the modified Bessel differential equation, and represent the zero-th
and the first order solutions, respectively.
4.3 Gauss-Chebyshev Quadrature Method
The Gauss–Chebyshev quadrature method will be applied to obtain the
solution to the formulated problem. A scale transformation of coordinates is first
carried out by introducing new parameters t and s such that:
x 9 b < c2 < b : c2 t , (4.13a)
ξ 9 b < c2 < b : c2 s . (4.13b)
The integral equation (4.4) is then transformed to change the integrals to the range
from -1 to 1:
79
σ/�t� 9 1π K B/�s�G/�t, s� b : c2 ds�h�
< σ�� (4.14)
where the terms with a bar notate that they have been transformed through the
equations (4.13a) and (4.13b). The values for σ/�t� and G/�t, s� in equation (4.14)
can be determined by applying the coordinate transformation equations (4.13a) to
equations (4.5a), (4.7a) or (4.9a), depending on the stress condition. In addition, it
is notable that the transformed dislocation density function, B/�s� , satisfies the
following condition, because there should be no net dislocation from one end of the
crack to the other:
K B/�s�ds�h�
9 0 . (4.15)
The solution of equation (4.14) can now be obtained by introducing an unknown
regular function ./�s� such that:
B/�s� 9 ./�s�√1 : s� . (4.16)
This converts integral equation (4.14) into N : 1 algebraic equations with N
unknowns of ./�s�� for the N number of discrete points by using discrete
integration points, s�, and collocation points, tm, as below:
σ/�tm� : σ�� 9 b : c2 j W� ./k�l�
�s�� G/�tm, s�� , k 9 1,2 n N : 1 , (4.17)
80
where:
W� 9 1N , (4.18)
s� 9 cos ]π 2i : 12N ^ , i 9 1, 2, … N , (4.19)
tm 9 cos ]π kN^ , k 9 1, 2, … N : 1 . (4.20)
The Nth equation comes from equation (4.15) to give:
πN j ./k�l�
�s�� 9 0 (4.21)
Through standard computer-based procedures, a system of N linear algebraic
equations with N unknowns, which is expressed in equations (4.17) and (4.21), can
be readily solved.
Employing an asymptotic analysis (Lonwengrub & Srivastav 1970), the
stress intensity factors at the inner (x 9 c) and the outer (x 9 b) tips of cracks can
be respectively found as follows:
KM 9 : 2µκ < 1 qπ2 �b : c�./�:1� , (4.22a)
KL 9 2µκ < 1 qπ2 �b : c�./�1� . (4.22b)
in the case of plane stress or plane strain, and
81
KM 9 : E4�1 : ν�� qπ2 �b : c�./�:1� , (4.23a)
KL 9 E4�1 : ν�� qπ2 �b : c�./�1� , (4.23b)
in the case of finite thickness plates. For a finite thickness plate, the plane strain
conditions dominate in the vicinity of the crack tip and, consequently, equations
(4.23a) have to be utilised. From the above equations, the dislocation density must
be zero at the tips of cracks to ensure the bounded stress field condition:
./�r1� 9 0 . (4.24)
To determine the inner and the outer plastic zone sizes in this problem, initial
values for these zones are first assumed, and corresponding ./�r1� values are
calculated. The next step is to employ an iterative procedure to alter the initially
guessed values until equation (4.24) is met with a specified accuracy. The crack tip
opening displacement can be calculated based on equation (4.3a) once the function
B/�s� is determined. Gauss–Chebyshev quadrature can be applied to convert the
integral into a sum of B/�s�� at integration points, similar to equation (4.17)
(Kotousov 2004).
82
4.4 Results and Discussion
4.4.1 Local plastic collapse of two collinear cracks in a plate of finite
thickness
The remotely applied tensile stress levels required for the particular case of
local plastic collapse where the inner plastic zones fully extend to the centre line
and coalesce into one (c = 0 in Figure 4.1) have been calculated based on the
developed model. The variation of the plastic collapse stress normalised by the
flow stress, σ(M /σ�, as a function of the ratio of a crack length, 2a, to the distance
between centres of cracks, 2d, is illustrated in Figure 4.2. Results from the analysis
of plane stress and plane strain are shown along with data from two different finite
thickness cases. In the graph the plate thickness, 2h, is normalised by the crack
length, 2a, which is the characteristic dimension of this infinite plate problem.
Additionally, for validation purposes, previously published analytical results by
Collins and Cartwright (2001) for plane stress are plotted in the figure.
According to Figure 4.2, the applied stress level for the complete plastic
collapse of the ligament between cracks is highly dependent on a/d . A lower
applied stress is required for a higher value of a/d, i.e. closer cracks, to cause the
ligament failure. The figure also shows that the plastic collapse stress is
considerably influenced by the plate thickness with thicker plates having higher
plastic collapse stress at a fixed a/d value. This is due to an increase in the out-of-
83
plane constraint around the crack tip with increasing plate thickness. The increased
constraint leads to smaller tensile plastic zone and hence larger applied stress level
for the complete plastic deformation of the ligament. As a/d changes from 0
(infinite spacing) to 0.5 (moderate spacing), the plastic collapse stress drops by 44%
for plane stress and 24% for plane strain. This theoretically demonstrates the
presence of strong crack interaction as well as the plate thickness effect in terms of
ligament failure. It is also interesting to point out that the solutions for finite
thickness plates recover the plane stress solution as a/d a 0 while they recover the
plane strain solution as a/d a 1. For validation purposes, analytical results by
Collins and Cartwright (2001) for plane stress are also plotted in the figure. The
present results for the case of plane stress are in very good agreement with the
previously published analytical results. To the best of the author’s knowledge no
previously published data for plane strain or finite thickness are currently available
in the literature.
84
Figure 4.2 Variation of calculated plastic collapse stress levels with the ratio of
crack length to centre-to-centre distance of cracks for different plate thicknesses
(c 9 0, h/a = plane stress, 0.3, 1.0, plane strain).
4.4.2 Variation of plastic zone size and crack tip opening displacement
of two collinear cracks in a plate of finite thickness
Figure 4.3 to 4.5 illustrate the variation of the plastic zone sizes and the
crack tip opening displacement (CTOD) at both the inner and outer crack tips of
two collinear cracks in a plate which is in plane stress, plane strain or two different
finite thickness conditions.
0
0.5
1
0 0.5 1
h/a= increasing
symbol: analytical results for plane stress
(Collins and Cartwright 2001)
line: present results
σ(M σ�
a/d
85
The c/b ratio, which is a parameter determining the geometry of two
effective cracks, is fixed at a reasonably low value (0.1) to ensure that substantial
effect of interaction between the inner plastic zones is observed through a wide
range of the applied stress, and results for this fixed separation of two effective
cracks are displayed in Figure 4.3.
In this figure, the distribution of the inner plastic zone to the corresponding
half crack length ratio, w�/a, with respect to the normalised applied stress, σ�� /σ�, for various plate thicknesses is displayed by solid lines, presenting the dependence
of the inner plastic zone size on the thickness of a plate. Throughout the normalised
applied stress span from 0.1 to 0.9, the plane stress curve shows the greater inner
plastic zone formation at a certain stress level while the smallest inner plastic zone
is observed in the curve for plane strain. Each curve for two finite thickness plates
(h/a = 0.3, 1.0) is arranged between the two plane stress and plane strain curves
with curves for higher plate thickness being closer to the plane strain one.
Furthermore, as is shown in this figure, the two curves for finite thickness plates
tend toward the plane strain curve at lower applied stress levels, but these curves
approach the plane stress solution as the applied stress is near the yield stress.
Figure 4.3 also presents the distribution of outer plastic zone size ratio,
w'/a , which is represented by the dotted lines. As is shown, due to the less
interaction at the outer crack tip by the neighbouring crack, the outer plastic zone
size is always smaller than the corresponding inner plastic zone through the whole
86
applied stress values. In addition, the variation of interaction between two collinear
cracks throughout the applied stress can be deduced based on the calculated result
that pairing curves for the inner and the outer plastic zones converge as the applied
stress level goes toward the yield stress. As the applied stress is increased with the
fixed c and b values, the plastic zones at both inner and outer crack tips grow,
resulting in a decrease in the actual crack size, 2a. The reduction of the actual crack
size without substantial change in the distance between the centres of cracks, 2d,
has the effect of isolating the cracks and finally diminishing the interaction
between them.
87
Figure 4.3 Variation of inner and outer plastic zone size against applied stress ratio, σ�� /σ� (c/b 9 0.1, h/a = plane stress, 0.3, 1.0, plane strain).
Results from the analysis of plastic zone sizes and crack tip opening
displacement for a fixed applied stress level (σ�� /σ� 9 0.5) are presented in Figure
4.4 and Figure 4.5, respectively. Curves for various plate thickness to crack length
ratios along with the plane stress and plane strain cases are plotted against the crack
size to centre-to-centre crack distance ratio, a/d, which is a parameter directly
determining the interaction between two cracks.
Figure 4.4 illustrates the variation of inner and outer plastic zone sizes due
to the interaction between two collinear cracks, and the dependence of the
-3
-2
-1
0
1
0 0.5 1
h/a increasing
σ�� /σf
w�/a w'/a
log�� �w� a � or
log�� �w' a �
88
interaction on the plate thickness. As shown in the figure, if two cracks are spaced
closer with a larger a/d ratio, the inner plastic zone grows significantly while the
corresponding outer plastic zone shows a gradual increase. With a thicker plate, the
sharp rise in an inner plastic zone size occurs at more closely spaced cracks,
corresponding to a larger value of a/d. The right end of each curve represents the
point where the two neighbouring inner plastic zones merge with each other. If two
cracks are separated apart (a/d a 0), the interaction effect disappears and each
solution for the inner and the outer plastic zones converges on that for an isolated
single crack problem. Table 4.1 compares the present solutions for widely spaced
two cracks with the analytical solutions for a single crack by Kotousov (2004).
From the table, it can be seen that there is a good correlation between the present
results for the case of a large distance between two collinear cracks and those for a
single crack (Kotousov 2004). As a practical guide: the two collinear equal length
cracks can be considered as isolated if the ratio a/d H 0.6 for relatively thick plates
and a/d H 0.4 for relatively thin plates. Other crack configurations can be
considered in a similar way using the developed method.
89
Figure 4.4 Variation of inner and outer plastic zone size against crack size to
separation ratio, a/d ( σ�� /σ� 9 0.5, h/a = plane stress, 0.125, 0.5, plane strain).
Table 4.1 Comparison of present solutions for widely spaced two cracks (a/d 90.05) and solutions for a single crack (Kotousov 2004).
h/a
σ��∞ /σ� 9 0.3 σ��∞ /σ� 9 0.5
Present
results a/�a < w��
Single crack
solution a/�a < w�
Present
results a/�a < w��
Single crack
solution a/�a < w�
Plane
stress 0.89 0.89 0.71 0.71
0.125 0.94 0.94 0.76 0.77
0.5 0.97 0.96 0.85 0.85
Plane
strain 0.98 0.98 0.91 0.93
0
0.2
0.4
0.6
0.8
0 0.5 1
h/a increasing
a/d
w�/a
or w'/a
w�/a w'/a
90
Figure 4.5 displays the change of inner CTOD, which is a measure of the
plastic stretch at the inner crack tip, to a half crack length ratio, δ�/a, together with
the variation of outer CTOD to a half crack length ratio, δ'/a, as a function of the
crack size to centre-to-centre crack distance ratio, a/d. In general, the inner and the
outer CTOD curves follow the trend of the matching plastic zone size variations.
The inner CTOD curves, however, shows a more gradual change with a/d ratio,
and the pairing inner and outer CTOD curves do not display such a strong
convergence tendency as the plastic zone size curves at low a/d ratios.
Figure 4.5 Variation of inner and outer CTOD ratios against crack size to
separation ratio, a/d (σ�� /σ) 9 0.5, h/a = plane stress, 0.125, 0.5, plane strain).
0
0.2
0.4
0.6
0 0.5 1
h/a increasing
a/d
δ'/a
δ�/a
or
δ�/a δ'/a
91
4.5 Conclusions
This chapter was aimed to investigate the thickness effect on the plastic
collapse conditions and crack tip opening displacement for two equal length
collinear cracks in an elastic-plastic plate of arbitrary thickness. The strip yield
model, the distributed dislocation approach and three-dimensional fundamental
solution for an edge dislocation in an infinite plate are used to investigate this
problem. The obtained results demonstrate, as in the previous studies, a significant
interaction between two closely located cracks and, in addition, show a substantial
dependency of the interaction on the plate thickness. The present results are in
good agreement with the previously published analytical solutions for the plane
stress condition. In addition, the present solutions converge to the case of a single
crack in a finite thickness plate (Kotousov 2004) as the gap between the two cracks
increases. The developed approach can be applied to investigate other multiple
crack geometries as well as more complicated boundary conditions.
The obtained results can also be used to assess the plastic collapse conditions
for plate of finite thickness. It is expected that the account of the thickness effect
will result in much better agreement with experiments. Unfortunately, previous
experimental studies on MSD fully relied on the two-dimensional framework,
which does not allow the investigation of the plate thickness effect. Therefore, an
experimental program was developed to specifically address this issue. A
92
comparison of the theoretical predictions with the experimental results will be
provided later in this thesis.
In addition to the plastic collapse calculations, the present results can be
applied to investigate fatigue crack growth under MSD conditions. Again, as it was
demonstrated in many studies (see for example Kotousov and Codrington (2010)),
the constraint conditions influencing the plasticity at the crack tip and CTOD are
significantly affected by the plate thickness. In this chapter, it is demonstrated that
the thickness effect in the case of MSD is much stronger than for a single crack,
and evaluations based on two-dimensional consideration might not be very
accurate as the constraint and plastic conditions can change dramatically while the
cracks approach each other due to the crack growth.
Finally, the importance of an analytical modelling should be highlighted.
Despite that such modelling usually relies on significant assumptions and
simplifications, the analytical solutions provide much better insight into the
investigation of fracture and fatigue phenomena. Analytical approaches can avoid
many difficulties associated with numerical nonlinear modelling of fracture
problems. It is important to note that analytical results can be reproduced and
verified. This is very difficult to achieve with numerical modelling of nonlinear
fatigue crack growth because many computational parameters, which are not
directly connected to the problem formulation, can significantly influence the
numerical results. These computational parameters have been previously discussed
93
in this thesis, and the inconsistences in numerical modelling were demonstrated in
many papers as well (Pitt & Jones 1997; Solanki, Kiran, Daniewicz, S. R. &
Newman Jr, J. C. 2004).
94
95
Chapter 5
5 A Plasticity Induced Crack Closure Model for Two Collinear
Cracks in a Plate of Arbitrary Thickness
5.1 Introduction
The objective of this chapter is to develop a crack closure model for the
analysis of two collinear cracks of equal length in a plate of finite thickness
subjected to constant amplitude (CA) cyclic loading. Based on the developed
model, the effects of crack interaction on the crack closure behaviour will be
investigated. Another issue to be studied with the present model is the plate
thickness effect. The importance of incorporating the plate thickness effects into
the crack closure is now well recognised (Dougherty, Padovan & Srivatsan 1997).
However, these effects have not previously been considered in the case of mutually
interactive multiple cracks.
The present model is based on the plasticity induced crack closure (PICC)
concept (see Chapter 2). The most commonly employed theoretical approach to
modelling PICC problem is the strip yield model (Dugdale 1960). Budiansky and
Hutchinson (1978) developed a theoretical model to analyse PICC phenomenon in
the case of a semi-infinite single crack under steady-state loading conditions based
on a plane stress strip yield model. The assumption of uniform plastic wake
thickness was adopted in their model. Newman (1981) extended Budiansky and
96
Hutchinson (1978)’s two-dimensional model by introducing a constraint factor to
account for the three-dimensional effects caused by the plate thickness. In the
extended model, various values of constraint factor for each different plate
thickness were determined through laborious experimental tests and extensive
finite element studies. Rose and Wang (2001) extended Budiansky and Hutchinson
(1978)’s work by employing the assumption of a linearly increasing plastic wake
along the crack faces instead of that of a uniform plastic wake. The linearly
increasing plastic wake model can be regarded to be more appropriate for a centre
crack of finite length because the wake thickness is believed to increase linearly
according to so called self-similar growth as the crack propagates under cyclic
loading. Rose and Wang (2001) investigated the variation of the length of a
contact-free zone as a function of the load ratio. However, their analytical model
was limited to a plane stress assumption and disregarded the effect of the plate
thickness.
The current PICC model for two collinear cracks to be presented in this
chapter can be used to analyse three-dimensional plate thickness effect without the
need for empirical constraint factors. This has become possible by incorporating
the fundamental three-dimensional solution for an edge dislocation (Kotousov &
Wang 2002) into the current PICC model. Following Rose and Wang (2001)’s
approach, this model uses the assumption of linearly increasing plastic wake
thickness to represent the variation of the plastic wake on the crack faces. For
97
validation purposes, predictions of the stress level corresponding to the crack
closure are made and compared with the corresponding values for a single crack
analytical solution obtained under plane stress conditions (Rose & Wang 2001).
This solution represents one of the limiting cases of the problem under
consideration when the gap between two cracks is sufficiently wide and the length
of these cracks is much larger than the plate thickness. After the validation, detailed
results on the effects of crack interaction and plate thickness on crack closure of
two closely spaced cracks are presented.
5.2 Problem Formulation for the Governing Integral Equation
The crack geometry of the problem is described in Figure 4.1. This figure
shows an idealised geometry of two through-the-thickness collinear cracks of
identical length, 2a, with centre-to-centre distance of cracks, 2d, in an infinite plate
of finite thickness, 2h. The plate is subject to remotely applied tensile load (mode
I), σ�� . The inner and outer tensile plastic zone sizes at maximum loading are
denoted by w� and w', respectively. The lengths of these tensile plastic zones can
be different due to the difference in interaction between the cracks.
The mathematical treatment undertaken for the modelling of collinear fatigue
cracks under a self-similar growth condition (when the plastic wake can be
represented by a linear function) is outlined in this section. A number of
98
simplifications, which were used previously to analyse single crack problems in
many studies, are adopted here. For example, the plastic zones at crack tips will be
evaluated based on the strip yield model. This significantly reduces the complexity
of the elastic-plastic analysis of crack problems, and it was used extensively in the
past with a great success to elucidate various fracture phenomena as well as fatigue
crack growth behaviour (Newman 1981). Another important simplification adopted
in the model is that the cracks and yielding strips are assumed to be represented by
an unknown distributed dislocation density function to simulate strain nuclei. The
governing equation to the problem of two collinear cracks can then be derived
(Kotousov 2007; Kotousov & Codrington 2010) by using the principle of
superposition. The resultant stress components along the crack line can now be
written as:
σ�x� 9 1π K B��ξ� G�x, ξ�dξ��"����h"h��
< σ�� , (5.1)
where B��ξ� is an unknown edge dislocation density function, defined in the
interval from d : a : w� to d < a < w' along the x-axis; σ�x� and G�x, ξ� are the
stress component and kernel corresponding to the edge dislocation solution,
respectively.
The corresponding crack opening displacement, g�x� , is related to the
distributed dislocation density, B��ξ�, as:
99
B��ξ� 9 : dg�ξ�dξ , (5.2a)
leading to:
g�x� 9 : K B��ξ�#�h"h��
dξ . (5.2b)
Therefore, the physical meaning of the dislocation density function is the negative
gradient of the crack opening displacement at a point between two crack tips.
Below we will consider two limiting cases of plane stress and plane strain along
with a case of a finite thickness plate. In all these cases, tensile plastic deformation
under maximum applied load will be assumed to occur according to the Tresca
yield criterion (Codrington & Kotousov 2007a).
5.2.1 Plane stress condition
The stresses and kernel in (5.1) at the maximum applied load (σ�� 9 σ�"# ),
which are obtained by using the superposition principle, corresponding to plane
stress condition, are given by the following equations:
WG�x, ξ� 9 G�����x, ξ� : G�����x, :ξ�σ�x� 9 σ���x� 9 0 z crack zone, (5.3a)
WG�x, ξ� 9 G�����x, ξ� : G�����x, :ξ�σ�x� 9 σ���x� 9 σ� z tensile plastic zones, (5.3b)
100
where σ� is a flow stress which is normally taken as an average of yield strength
and tensile strength of the material, and G�����x, ξ� is the two-dimensional Cauchy
kernel for the case of a single isolated crack, which is written as (Hills et al. 1996):
G�����x, ξ� 9 2µκ < 1 ] 1x : ξ^. (5.4)
In the equation above, µ is the shear modulus and κ is the Kolosov’s constant equal
to �3 : ν� �1 < ν�⁄ for plane stress condition with ν being the Poisson’s ratio.
5.2.2 Plane strain condition
If a plate with two collinear cracks is regarded to be under plane strain
condition, the stresses and kernel at the maximum applied load can be given as
below:
WG�x, ξ� 9 G�����x, ξ� : G�����x, :ξ�σ�x� 9 σ���x� 9 0 z crack zone, (5.5a)
WG�x, ξ� 9 �1 : 2ν�|G�����x, ξ� : G�����x, :ξ�}σ�x� 9 σ���x� : σww�x� 9 σ� z tensile plastic zones, (5.5b)
where G�����x, ξ� represents the same Cauchy kernel expressed in (5.4) except that
the parameter κ is equal to 3 : 4ν for plane strain condition.
5.2.3 Finite thickness plate
For two collinear cracks in an infinite plate of finite thickness, the
corresponding stress component and kernel in the integral equation at the
101
maximum applied load can be obtained using a similar method to the cases of plane
stress and plane strain conditions. However the kernel is replaced by Kotousov and
Wang (2002)’s three-dimensional solution for an edge dislocation:
WG�x, ξ� 9 G�����x, ξ� : G�����x, :ξ�σ�x� 9 σ���x� 9 0 z crack zone , (5.6a)
WG�x, ξ� 9 G�����x, ξ� : G�����x, :ξ� : Gww���x, ξ� < Gww���x, :ξ�σ�x� 9 σ���x� : σww�x� 9 σ� z tensile plasticzones , (5.6b)
where the three-dimensional kernels in the y-direction, G�����x, ξ�, and in the z-
direction, Gww���x, ξ�, are:
G�����x, ξ� 9 : E4�1 : ν�� 1ρ � 4ν��λρ�� : �1 : ν�� : 2ν�K��λ|ρ|�: 2�2 < λ�ρ��ν�K��λ|ρ|�λ|ρ| � ,
(5.7)
Gww���x, ξ� 9 E2�1 : ν�� λνK��λ|ρ|� ρ|ρ| , (5.8)
where ρ 9 x : ξ, E is the Young’s modulus, and λ is a term including one half of
the plate thickness, h, such that:
λ 9 1h Z 61 : ν , (5.9)
and K��·� and K��·� are the modified Bessel functions of the second kind, which
are the solutions to the modified Bessel differential equation.
102
5.3 Discrete Form of the Governing Integral Equation
The analytically derived governing integral equation (5.1) needs to be
rewritten in a discrete form in order to facilitate the numerical analysis. The
integration domain in (5.1) is first divided into three regions: the first region
spreads over the inner plastic zone (d : a : w� E x H � : �), the second region
over the actual crack zone (d : a E x H � < �) and the third region over the outer
plastic zone (d < a E x H � < � < w'). Such division of the integration domain is
necessary to ensure a computational flow, which automatically adjust the size of
the plastic zones. Therefore, (5.1) can now be rewritten as below:
σ�x� : σ�� 9 1π K B��ξ�� G�x, ξ��dξ� <�h"�h"h��
1π K B��ξ�� G�x, ξ�� dξ���"
�h"< 1π K B��ξ�� G�x, ξ��dξ� ,��"���
��"
(5.10)
where subscript numbers 1, 2 and 3 refer to the first, second and third regions,
respectively. After that, the Gauss–Chebyshev quadrature method for the second
integral term and the direct placement of edge dislocations for the other integral
terms in the right hand side of (5.10) are applied to obtain a discrete form, which
can be given by the following equation:
103
σ�x� : σ�� 9 1π j ∆b�0ξ�,%2 G0x, ξ�,%2k�
%l� < j W% ./k�
%l� 0s%2 G/0x, s%2
< 1π j ∆b�0ξ�,%2 G0x, ξ�,%2k�
%l� , (5.11)
where ∆b��ξ� can be treated as an infinitesimal Burgers vector, W% is a weight
function, ./0s%2 is a non-singular function, and s% is a non-dimensional coordinate
of the position along the crack. More details on the derivation of (5.11) can be
found in the previous publication by the author (Chang, D. & Kotousov, A. 2012).
In addition, ∆b�0ξ�,%2, ./0s%2 and ∆b�0ξ�,%2 in (5.11), are unknown functions to be
found from a solution process. These functions have to satisfy proper boundary
conditions, which will be briefly described in the next section.
5.4 Boundary Conditions and Criteria for Solution Process
The problem formulation was provided in the previous section; it is now
necessary to find solutions to the problem for the minimum and maximum values
of the applied cyclic load. This is because the solution procedure for crack opening
load, which is the key information for the prediction of fatigue crack growth rates,
relies on the solutions for values of the minimum and maximum loads. The
solution approach here utilises an iterative procedure, in which arbitrary initial
values for unknown dimensional parameters, including w� and w' at the maximum
104
value of applied cyclic load and w��, w�', β� and β' at the minimum value, are first
used in an initial step of the solving procedure. Figure 5.1 illustrates the profile of
crack opening displacement and the dimensional parameters at the maximum and
minimum loads. In this figure, w, w� and β represent the tensile plastic zone size at
maximum applied load, compressive plastic zone size and crack contact zone size
at minimum applied load, respectively. The subscripts i and o denote the inner and
outer part of the right-hand-side crack of two interacting collinear cracks.
The next step is to apply appropriate boundary conditions for the maximum
and minimum load cases along the crack and plastic zones to solve (5.11) for the
unknown functions, ∆b�0ξ�,%2, ./0s%2 and ∆b�0ξ�,%2. The obtained solution is then
evaluated and corrected for the next iteration until the required accuracy with the
initially estimated dimensional parameters is achieved. In the following sections,
this computational procedure for two cases of maximum and minimum loading is
briefly outlined.
105
(a) At maximum load, σ�"#
(b) At minimum load, �
Figure 5.1 Configuration of right-hand-side crack of two interacting collinear
cracks at (a) maximum and (b) minimum load.
5.4.1 Maximum load
The boundary conditions at the maximum value of cyclic loading were
provided in Section 5.2 (see equations (5.3), (5.5) and (5.6)). Next, it is necessary
to determine the tensile plastic zone sizes at the inner and outer crack tips under
maximum load. For this, use is made of the requirement that the stresses at the
crack tips of the plastic zones should be bounded, resulting in zero value of the
stress intensity factor, or, K� 9 0. From an asymptotic analysis (Lonwengrub &
w�� a a w�'
d
x
y
β� β'
0 δ-,� δ-,'
w� w'
g�x����2
w� a a w'
d
x
y
0
g�x��"#2
plastic wake
106
Srivastav 1970), the stress intensity factors at the inner (x 9 d : a : w�) and outer
(x 9 d < a < w') tips of plastic zones for plane stress and plane strain conditions
can be found as follows:
K�,�� 9 :√2π 2µκ < 1 ∆b�0ξ�,k�2gξ�,k� : �d : a : w�� , (5.12)
K�,'+� 9 √2π 2µκ < 1 ∆b�0ξ�,�2gd < a < w' : ξ�,� , (5.13)
respectively.
In the case of a plate of finite thickness, the plane strain conditions at the
crack tips would prevail (Codrington & Kotousov 2009a) and the corresponding
value for Kolosov’s constant κ 9 4 : 3ν is used. From (5.12) and (5.13), it
follows that the infinitesimal Burgers vectors must be zero at the tips of the plastic
zones to ensure the bounded stress condition, and this constitutes the criterion for
the determination of the inner and outer plastic zone sizes.
5.4.2 Minimum load
For the minimum load in the cyclic loading, the solution procedure is
similar to that of the maximum load. However, instead of the Tresca yield criterion,
use is made of the notion that compressive yielding takes place when σ�� 9 :σ� assuming that the out-of-plane constraint is negligible during compressive yielding
(Newman 1981). This compressive yielding leads to the generation of compressive
plastic zones, w�� and w�' , located within the former tensile plastic zones. The
107
other region of the former tensile plastic zones is here named as “no-change zone”
because there is no change in crack opening displacement in this zone at minimum
load. This is due to the adoption of an elastic-perfectly-plastic material model in
plastic zones (Dugdale 1960). The application of minimum load also divides the
whole crack region into two zones: crack contact zones and crack contact-free
zones. The possible formation of contact zones even when the minimum loading is
still tensile is associated with the development of plastic wake on the crack faces.
The specific boundary condition for each zone at minimum load can be expressed
as below:
WG�x, ξ� 9 G���x, ξ� : G���x, :ξ�σ�x� 9 σ���x� 9 :σ� z compressive plastic zones , (5.14a)
g�x�W9 g�x��"# � no : change zones , (5.14b)
WG�x, ξ� 9 G���x, ξ� : G���x, :ξ�σ�x� 9 σ���x� 9 0 z crack contact : free zones , (5.14c)
Wdg�x�dx 9 δ-,'a or : δ-,�a � crack contact zones . (5.14d)
In the above equations, G���·,·� can be either G�����·,·� for plane stress and plane
strain conditions or G�����·,·� for a finite thickness plate condition, and δ-,� and δ-,'
are inner and outer residual plastic stretches at minimum load, respectively.
Furthermore, in (5.14d), linearly increasing plastic wake thickness model is used to
108
represent the variation of the plastic wake in the contact zones (Rose & Wang
2001).
The solution procedure for minimum load is similar to that for maximum
load. Initially four dimensional parameters ( w�� , w�' , β� , β' ) are roughly
estimated, and then they are determined by enforcing the following criteria. The
first criterion corresponding to the compressive plastic zones is expressed by (5.15a)
and the next one is enforced in the no-change zone by (5.15b), where y-stresses are
greater than the negative flow stress. In addition, (5.15c) and (5.15d) describes
criterion to be satisfied in the contact-free zones and contact zones, respectively.
Wg�x���� H g�x��"# � compressive plastic zones , (5.15a)
Wσ���x� G :σ� � no : change zones , (5.15b)
W: δ-,�a H dg�x����dx H δ-,'a � crack contact : free zones , (5.15c)
W�x� H 0 � crack contact zones . (5.15d)
Using the criteria expressed in equations (5.15), the residual plastic stretches, δ-,� and δ-,', at minimum load can also be determined through an iteration procedure.
In the iterative procedure, δ-,� and δ-,' values are first initially guessed, and then
new values are calculated based on equations (5.15). This procedure is repeated
until the required level of convergence is achieved.
109
5.4.3 Opening load
The value of the applied load at which the crack completely opens is
defined as the crack opening load or opening stress, σ'( . After the crack opening
displacement at minimum load, g�x����, being calculated, the crack opening stress
can now be found by applying the boundary conditions described below:
WG�x, ξ� 9 G���x, ξ� : G���x, :ξ�σ�x� 9 σ���x� 9 0 z crack zone, (5.16a)
g�x�W9 g�x���� � tensile plastic zones , (5.17b)
To determine the opening stress, tensile loading is increased until the crack
tip opens again. The slope of g�x� is compared to the slope of g�x���� at the crack
tip to decide the crack opening. This process is separately implemented at the inner
and outer crack tips.
5.5 Validation of the Theoretical Model: Crack Closure of a Single
Crack at Minimum Load
As a validation of the developed method, a single crack problem is first
considered. Figure 5.2 shows how the load ratio and the applied stress level
significantly affect the crack closure behaviour of a single isolated crack in plane
stress at minimum load. The computational results for the contact-free length ratio,
Q �9 1 : �β� < β'�/2a� as a function of the load ratio, R �9 σ��� /σ�"# �, for two
110
different values of σ�"# /σ� are given in the figure. The value for a/d is 0.1, which
corresponds to a situation where the two cracks are widely spaced so that the
interaction between them is small enough to be neglected. According to the graph,
the contact-free length ratio declines with a decrease in the load ratio, and the ratio
curve from the higher applied stress level shows a more gradual drop, resulting in a
complete crack closure (Q 9 0) at a lower load ratio. Figure 5.2 also presents Rose
and Wang (2001)’s analytical results for an isolated single crack in plane stress. As
it can be seen here, the present calculations in overall agree well with the
previously published analytical results throughout the whole R region except for
the region where the crack is completely closed with small Q. The present results
predict earlier complete crack closure than the Rose and Wang’s results as R
sweeps from 1 to -1.
111
Figure 5.2 Comparison of contact-free length ratio variation for two widely spaced
collinear cracks from the present model against that for a single isolated crack from
Rose and Wang (2001)’s analytical model.
5.6 Results and Discussion
5.6.1 Crack closure at minimum load
The crack closure behaviour of two collinear cracks subjected to constant
amplitude cyclic loading has been investigated based on the developed model. The
focus of this study is to investigate the effect of the interaction between two
collinear cracks of equal length on plasticity induced crack closure. Accordingly, a
theoretical analysis has been conducted on the variation of the contact-free length
ratio of two collinear cracks, using various separation gap and plate thickness
values. Figure 5.3 (a) and (b) show the variation of the contact-free length ratio as a
0
0.2
0.4
0.6
0.8
1
-1 -0.5 0 0.5 1
σ�"# σ� 9 0.2, 0.7
plane stress
�a/d 9 0.1�
line : present results
symbol : Rose and Wang’s
results
R
Q
112
function of the load ratio for two separation gaps and four plate thicknesses,
respectively. Figure 5.3 (a) illustrates how the separation gap between two collinear
cracks affects fatigue crack closure as the load ratio changes at fixed maximum
applied stress level ( σ�"# /σ� 9 0.3 ) under plane stress. The centre-to-centre
distance of cracks is here a parameter which affects the interaction between two
cracks, with a smaller crack distance leading to a stronger interaction effect. It is
seen that a significant interaction effect exists between the two collinear cracks in
terms of crack closure. More specifically, a higher interaction between them (i.e.
higher a/d ratio) has the effect of increasing the contact free length ratio. This
interaction effect appears to be more substantial at smaller load ratio values.
However, even though the interaction seems to be negligible at high load ratios (i.e.
R G 0.1), it still has strong effects of decreasing the inner contact free length, β�, and increasing the outer contact free length, β' (not shown in the plot). Because the
increase in β' cancels out the decrease in β� at the high load ratios, the overall
contact free length remains nearly unchanged as a/d increases from 0.1 to 0.77. It
is also noteworthy that the load ratio for a complete crack closure ( Q 9 0 )
decreases with an increase in interaction between the cracks. This can lead to a
delayed complete crack closure as R sweeps from 1 to -1. Therefore, it is evident
that the influence of the separation gap between two collinear cracks, or the
interaction between them, has a significant impact on the fatigue crack closure
throughout the whole region of load ratios considered. Shown in Figure 5.3 (b) is
113
the effect of the plate thickness on the variation of contact free length in highly
interacting two collinear cracks (a/d 9 0.77� at σ�"# /σ� 9 0.3. In the plot, two
limiting cases of plane stress and plane strain are considered along with two finite
thickness plate cases. The finite thickness results are shown to be well bounded by
those from the two limiting cases. It is interesting to note that there is a pivot point
(R � 0.03 in Figure 5.3 (b)) around which the effect of the plate thickness vanishes,
and the contact free length ratio is the same regardless of the plate thickness.
However, the ratio of non-contact decreases with the decreasing plate thickness
when R is greater than the pivot point while the opposite is observed when R is
smaller than it. The pivot point is expected to move depending on both the
maximum applied stress level, σ�"# /σ�, and the ratio of crack length to centre-to-
centre distance of cracks, a/d, though this is beyond the scope of this research. It
can also be seen in the plot that a decrease of the plate thickness in closely located
collinear cracks has a similar effect of increasing the maximum applied stress level
in a single isolated crack case. This leads to a more gradual change in the contact
free length ratio and lower load ratios for complete crack closure.
114
(a)
(b)
Figure 5.3 Variation of contact-free length ratio of two collinear cracks for (a)
different separation gaps and (b) different plate thicknesses.
0
0.2
0.4
0.6
0.8
1
-0.5 0 0.5 1
0
0.2
0.4
0.6
0.8
1
-0.5 0 0.5 1
Q
R
Q σ�"# σ� 9 0.3
ν 9 0.3 ad 9 0.77
ha 9 plane strain, 0.125, 0.05, plane stress
ad 9 0.1, 0.77
σ�"# σ� 9 0.3
plane stress
R
115
5.6.2 Crack opening load
Theoretical results for the crack opening load are presented in this section.
Based on the current semi-analytical model, the variation of the crack opening
stress in the case of two collinear cracks is investigated. The results for the applied
opening stress to flow stress ratio, σ'( /σ�, at the inner and outer crack tips as a
function of the ratio of crack length to centre-to-centre distance of cracks, a/d, are
shown in Figure 5.4. The results are presented for four different thicknesses
including the plane stress and plane strain conditions. The maximum load ratio,
σ�"# /σ�, is set at 0.5, and the load ratio, R, at 0.
According to the obtained results, in general, as the crack spacing decreases
(a/d increases), the opening stresses for the inner and outer crack tips decrease
gradually. The overall decrease in the opening stress due to the reduction in the
crack spacing confirms the fact that interaction between multiple cracks can have a
substantial effect on fatigue crack growth. It is worthwhile to point out that the
opening stress at the inner crack tip is greater than that at the outer tip when the
interaction effect is large with high a/d values, though this phenomenon is not
observed in the case of plane strain. This higher crack opening stress at the inner
crack tip is thought to stem from the larger tensile plastic zone formed at the inner
tip of the crack at maximum applied load in comparison with the plastic zone at the
outer crack tip. The larger plastic zone can cause the formation of thicker plastic
wake at the inner part of the crack face, and this can in turn induce the delayed
116
crack opening during unloading. However, as the crack spacing increases (or when
a/d becomes smaller than approximately 0.2), the difference between the crack
opening stresses for the inner and outer crack tips becomes negligible. The results
shown in Figure 5.4 also demonstrate that the plate thickness has a significant
effect on the crack opening behaviour. The crack opening stresses of both the inner
and outer crack tips decrease considerably with an increase in the plate thickness.
These trends are in agreement with previously published theoretical and
experimental results (Codrington & Kotousov 2009a; Costa & Ferreira 1998).
Figure 5.4 Variation of σ'( /σ� with regard to a/d for different plate thicknesses.
0.32
0.4
0.48
0 0.5
ha 9 plane stress, 0.1, 0.3, plane strain R 9 0 ν 9 0.3
inner crack tip
outer crack tip
a/d
σ'( σ� σ�"# /σ� 9 0.5
117
5.7 Conclusions
The primary objective of this chapter was to investigate the effect of the
interaction between two collinear cracks of equal length on the plasticity induced
crack closure behaviour. Accordingly, a theoretical model of fatigue crack closure
has been developed based on the strip yield model, plasticity induced crack closure
concept and distributed dislocation technique. The considered problem of two
cracks represents the simplest case of multiple site damage, but this simple model
can provide new insight into how the interaction between closely located cracks
can affect one another and how closely they should be positioned to cause
substantial interaction between them. An important feature of the current model is
that it can take into account the effect of the plate thickness by using the three-
dimensional fundamental solutions for an edge dislocation.
Based on the developed model, a theoretical analysis of the crack closure and
opening behaviour has been carried out for the problem of two collinear cracks.
The present study especially focuses on the effect of the interaction between
neighbouring cracks on crack closure phenomenon. The study also associates the
interaction effect with the plate thickness effect in terms of crack opening stress
variation. Throughout the analysis, it has been demonstrated that both the
interaction and the thickness effects have a significant impact on the crack
closure/opening of two collinear cracks over a wide range in load ratio. It has also
118
been shown that when the interaction is prevalent in neighbouring cracks, the inner
crack tip shows higher opening stress than the outer crack tip.
In conclusion, the obtained results demonstrate that the crack closure of two
closely spaced collinear cracks is highly dependent on the crack interaction as well
as the thickness of the plate. Furthermore, the study provides a new theoretical
model which is capable of predicting the crack opening behaviour of interacting
cracks, directly taking into account the plate thickness effect.
119
Chapter 6
6 A Fatigue Crack Growth Model for Two Collinear Cracks in
a Plate of Arbitrary Thickness
6.1 Introduction
Fracture and fatigue behaviour of closely spaced cracks can be significantly
affected by the crack interaction effects. An application of fatigue crack growth
data and lifetime procedures utilising solutions and methods adopted for a single
(non-interacting) crack to the case of MSD can often lead to unacceptable errors
and non-conservative predictions (Carpinteri, Brighenti & Vantadori 2004; Collins
& Cartwright 1996). Therefore, multiple crack damage has to be treated with
caution and appropriate methods to provide a reliable assessment of the durability
and integrity.
It is widely recognised that the plate thickness is another important factor,
which can significantly influence the yield conditions at the crack tip as well as the
fracture controlling parameters, such as the effective stress intensity range in the
case of fatigue loading. The influence of this factor (plate thickness) was
thoroughly investigated in many theoretical and experimental studies
predominantly for non-interactive cracks (Bhuyan & Vosikovsky 1989; Codrington
& Kotousov 2009a; Costa & Ferreira 1998; de Matos & Nowell 2009; Guo, Wang
& Rose 1999; Newman Jr 1998; Park & Lee 2000). It is expected that the influence
120
of the plate thickness will be even more pronounced for closely spaced cracks
leading to a synergistic effect.
Accordingly, the objective of this chapter is to develop an accurate and
reliable three-dimensional transient fatigue crack growth model for the analysis of
two collinear cracks in a plate of finite thickness. This particular geometry
configuration represents the simplest case of MSD; however, the developed
theoretical model and computational procedures can be relatively easily extended
to analyse more complicated problems, which can incorporate more complex
geometries as well as more complex loading conditions.
In this chapter, the steady state crack closure model developed earlier in
Chapter 5 will be extended to capture the transient nature of the plastic wake
formation as a result of fatigue loading. Based on the loading history, the crack
opening stress at the inner and outer crack tips will be calculated in order to
evaluate the crack growth driving force (or the effective stress intensity range). For
simplicity, the current analysis will be limited to mode I only and CA cyclic
loading. The outcomes of previous experimental studies, which provided accurate
fatigue crack growth data for single cracks, will be utilised to validate the current
model in the cases when the crack interaction effects are negligible and multiple
cracks can be considered as non-interactive.
There weren’t many experimental studies conducted in the past on fatigue
and fracture behaviour of MSD. Therefore, one of the objectives of the current
121
study was to generate fatigue crack growth and fracture data for two collinear
cracks. These experimental results will be described later in this thesis and be also
used to further validate the theoretical model. After these careful validation studies,
the developed model will be applied to investigate the combined effects of the
crack interaction and the plate thickness on the fatigue behaviour of two closely
spaced cracks.
6.2 Transient Crack Growth Model
The steady state crack growth model described in Chapter 5 was developed
under the assumption that the thickness of the plastic wake on the crack faces
increases linearly according to so called self-similar growth (Rose & Wang 2001).
This model has provided a benchmark solution as well as a fundamental
understanding of the role of the plastic wake left on the crack faces in the
generation of crack closure during the unloading stage of the load cycle. However,
the linear plastic wake thickness idealisation is regarded to be an effective (it
significantly reduces the complexity of calculations) yet oversimplified assumption.
In the present transient crack growth model, the driving force and incremental
crack growth are calculated at each load cycle based on the plastic wake profile
formed as a result of the previous load and fatigue crack growth history.
Consider a problem of two collinear cracks of equal length in a plate of finite
thickness subjected to CA cyclic loading (see Figure 4.1). Taking advantage of the
122
symmetry of this problem only the right-hand-side (RHS) crack will be analysed.
The fatigue crack propagation model developed in this chapter employs the cycle-
by-cycle calculations of the effective stress intensity factors, crack increments and
crack lengths to account for the transient nature of the process of the plastic wake
formation.
A typical CA loading sequence is shown in Figure 6.1 (a). Two characteristic
parameters for the nth
load cycle in this sequence are the minimum, ����, and
maximum , σ�"#���, cyclic stresses, respectively. Variables σ'(,�����
and σ'(,'+����,
represent the opening stresses for the inner and outer crack tips, respectively. These
variables may have different values, as illustrated in the Figure 6.1 (a), because the
crack interaction affects the inner and outer crack tips differently. The fatigue crack
growth occurs only in the dotted parts of the loading cycle when the corresponding
crack tip (inner or outer) is fully open and subjected to tensile loading. Figure 6.1 (b) illustrates schematically the crack configurations at the
minimum and maximum applied stresses corresponding to the nth
and n+1th
load
cycles. In this figure; d is the distance from the symmetry axis to the centre line of
the crack; w is the length of the direct plasticity region; w′ is the length of the
reverse plasticity region; β describes the length of the contact region. The subscript
“in” and “out” denote the inner and outer crack tip, respectively. This figure also
shows the variation of crack opening displacement and residual stretch, g�x�, along
123
the x-axis in various regions. Superscripts in the adopted notations represent the
corresponding load steps.
(a)
Figure 6.1 (a) Cyclic load sequence and (b) corresponding crack configurations at
each load step.
����
σ�"#���
������
σ�"#�����
σ'(,����� σ'(,�������
σ��
Time
σ'(,'+����
σ'(,'+������
Occurrence of crack growth at inner crack tip outer crack tip
σ�"#
�
124
(b)
Figure 6.1 Continued.
d
x
y
w������ a����h��
a'+���h��
w����� a�����
x
a'+���� w'+����
w�'+����
∆a����� ∆a'+����
x
y
y
w������� a�������
x
a'+������ w'+������
∆a������� ∆a'+������
w�������� a�����
a'+���� w�'+������
y
Tensile plastic zone Compressive plastic zone
Plastic wake
β����� β'+����
β������� β'+������
No-change zone
g�x�������
g�x��"#���
g�x���������
g�x��"#�����
����
σ�"#���
������
σ�"#�����
125
The transient crack growth assessment procedure begins with the
calculation of the nonlinear solution for the minimum applied stress of the nth
cycle.
The solution for the initial 1st cycle can be determined, for example, by utilising the
results from the steady state analysis (Chapter 5). The solution at ���� is
determined based on the crack opening displacement functions, g�x��"#��h�� and
g�x������h��, known from the analysis of the previous load step. The boundary
conditions can be written for various regions along the crack length as:
Wg�x������� 9 g�x��"#��h�� � no change region , (6.1a)
Wσ���x� 9 :σ� � reverse plasticity region , (6.1b)
Wdg�x�������dx 9 dg�x������h��
dx � crack contact region, (6.1c)
W�x� 9 0 � free from stress region , (6.1d)
where, σ� is the material’s flow stress. The flow stress has to be selected based on
the analysis of the dominant deformations associated with the particular problem
and actual loading conditions. In this work the flow stress is simply taken as the
average of yield strength and ultimate tensile strength, which is a good
approximation of the plastic stresses over the wide range of plastic deformations
(Koolloos et al. 2001; Swift 1994).
126
As shown in equations (6.1a), four different regions can be identified at the
minimum value of the cyclic loading, σ������. In the “no change region” the material
is plastically deformed due to the maximum load at the previous loading cycle but
does not experience reverse plastic deformation at the minimum load applied at the
current cycle (nth
). In this region, there is no change in the plastic stretch during
unloading, and the crack opening, g�x�, follows the plastic streach profile as found
at the previous load cycle (n -1)
th.
In the next region, which is “reverse plasticity region”, the material strip
does experience the reverse (or compressive) plastic deformation during unloading.
The length of this region, which is denoted by w�, is known to have a considerable
influence on the formation of the plastic wake and hence the crack closure
phenomena and fatigue crack growth (Chang, Li & Hou 2005). The other two
regions are “crack contact region” and “free from stress region”. The presence of
the contact region, of which the length is represented by variable �, can occur even
if the minimum cyclic load is still tensile. This is attributed to the plastic wake
formed on crack faces as the crack propagates through the previously plastically
stretched material.
For each cycle the opening stresses at the inner and outer crack tips, σ'(,�����
and σ'(,'+����, are determined independently based on the crack opening
displacement solution, g�x�������, and the following boundary conditions:
127
W�x� 9 0 � crack region , (6.2a)
Wg�x�'(��� 9 g�x������� � opening at the crack tip. (6.2b)
In the numerical procedure developed for determining the opening loads, the slopes
of g�x�'(��� at the crack tips are monitored during the incremental loading. The
applied stress, at which the slope just starts to move in the opening direction, is
defined as the inner or outer crack opening stress, σ'(,����� or σ'(,'+����
.
Through use of the analytical stress intensity factor solution for two
collinear cracks of identical length in an infinite plate subject to mode I loading
(Erdogan 1962; Lin & Tsai 1990; Vialaton, Brunet & Bahuaud 1980; Vialaton et al.
1976), the effective stress intensity factor range can be determined. Thus, the
effective stress intensity factor ranges at the inner crack tip, ∆K���,�����, and at the
outer tip, ∆K���,'+����, at n
th cyclic load are:
∆K���,����� 9 F���σ�"#��� : σ'(,����� �√πa (6.3a)
∆K���,'+���� 9 F'+��σ�"#��� : σ'(,'+���� �√πa , (6.3b)
respectively, where the half crack length is:
a 9 a����� < a'+����2 , (6.3c)
and the stress intensity magnification factors at the inner, F��, and outer, F'+�, crack
tips are:
128
F�� 9 d : a2a Zd : ad �]d < ad : a^� E�k�K�k� : 1� , (6.3d)
F'+� 9 d < a2a Zd < ad [1 : E�k�K�k�` , (6.3e)
respectively. In the equations above, d refers to half of the centre-to-centre distance
between cracks (see Figure 6.1(b)), and K�k� and E�k� are the complete elliptic
integrals of the first and second kind, respectively, given by the following
expressions:
K�k� 9 K �1 : k�sin�θ�h�/�dθ�/�� , (6.3f)
E�k� 9 K �1 : k�sin�θ��/�dθ�/�� , (6.3g)
with:
k 9 Z1 : ]d : ad < a^� . (6.3h)
By using a fatigue crack growth law which is a function of the effective
stress intensity range, for example the modified Paris law, da/dN 9 C�∆K�����,
the inner and outer incremental crack growth, ∆a����� and ∆a'+����
, at nth
load cycle can
be now determined as:
129
∆a����� 9 C ∆K���,����� ¡�, (6.4a)
∆a'+���� 9 C ∆K���,'+���� ¡�, (6.4b)
where C and m are Paris coefficients for the constant amplitude crack growth rate
data. These constants depend on the material. Alternatively, instead of using the
modified Paris law, a table-lookup procedure (Newman Jr & Ruschau 2007) can be
applied. The ∆K��� range, in that procedure, is divided into multiple linear segments,
and each segment may be represented by different sets of C and m material
constants.
After determining ∆a����� and ∆a'+����
, the inner and outer crack sizes are
updated: a����� 9 a����h�� < ∆a����� and a'+���� 9 a'+���h�� < ∆a'+����
, and nth
maximum load,
σ�"#���, is applied and solved based on the following boundary conditions
W�x� 9 0 � crack region , (6.5a)
Wσ���x� 9 σ� � direct plasticity region. (6.5b)
The application of σ�"#��� can provide new tensile plastic zones, w, at both ends of
the crack, changing the lengths of the reverse plasticity region.
The same procedure as for the nth
load cycle considered above is repeated in
the next cycle, leading to the next values of the inner and outer crack sizes, a�������
and a'+������. The procedure is run until the two neighbouring cracks coalesce. The
130
crack growth algorithm developed in this chapter is schematically illustrated in
Figure 6.2.
131
Figure 6.2 Algorithm for the crack growth.
Start
Application of nth
minimum cyclic load, ����.
Determination of nth
inner and outer opening
cyclic loads, σ'(,�����and σ'(,'+����
, and stress
intensity ranges, ∆K���,�����and ∆K���,'+����
.
Calculation of nth
inner and outer crack
increment, ∆a����� and ∆a'+����
. Re-meshing of the increased crack
Application of nth
maximum cyclic load, σ�"#���.
End
No
Yes
Determination of plastic wake thickness for
the nth
incremental inner and outer crack
regions
n ¢ n < 1
Coalescence
of cracks?
132
6.3 Validation Study: Single (Non-interacting) Crack
To the best of the author’s knowledge, there is no fatigue data available in the
literature for two collinear through-the-thickness crack geometry considered in this
chapter. As a validation of the developed model and numerical procedure, Newman
Jr and Ruschau (2007)’s fatigue crack growth tests on specimens with single cracks
will be utilized. In their work fatigue crack growth tests on 2024-T3 aluminium
specimens with a single through-the-thickness centre crack under CA loading
(mode I) were performed. The yield strength, σ), and ultimate strength, σ+, of the
material (aluminium alloy) are 360 and 490 MPa, respectively. Half of the initial
crack length is a 9 9.15 mm including a pre-cracked length of 1.3 mm at each end
of the crack, the width of the specimen is 76mm, and the thickness 2.3mm. In these
experiments, the specimens were cyclically loaded in the LT orientation, which
corresponds to the loading in the longitudinal direction and crack propagated in the
transverse direction (relative to the rolling direction of the specimens). Three
maximum applied stress levels (σ�"# 9 51.7, 69.0 and 138.0 MPa) and two load
ratios ( R 9 σ��� /σ�"# 9 0.05 and : 1 ) were applied. However, the case of
σ�"# 9 138 MPa with R 9 :1 were excluded from the analysis because the crack
growth rates in this case were extremely high in comparison with all other cases.
The fatigue test results have been simulated using the developed transient
crack growth model for two collinear cracks with a very large centre-to-centre
distance between cracks, 2d, compared to the crack length, 2a. The reason for using
133
a very large spacing is to suppress any interaction effect between cracks to make
the model applicable to describe the experimental results for single crack. The
initial a/d ratio selected for this simulation of the single crack fatigue tests is 0.0095.
In other words, the initial crack length is less than 1% of the spacing. The crack
interaction can hence be disregarded. In addition, the three-dimensional finite plate
thickness solution, rather than plane stress or plane strain assumptions, was
invoked in the analysis to account for the out-of-plane constraint effect due to the
finite specimen thickness.
In the beginning, the material’s relationship between the crack growth rate
and the effective stress intensity range were determined. The relationship was
derived employing the so called forced growth model, which is the modified
version of the present transient crack growth model. The role of the modified
model is to calculate the corresponding crack opening stresses and hence effective
stress intensity ranges as a crack is grown accordingly to the experimental crack
growth data. To use this model, growth rates at each cycle of loading are first
obtained from the experimental crack growth results. Then these values are
substituted into the forced transient crack growth model to finally determine the
effective stress intensity range versus the experimental crack growth rate data.
Figure 6.3 (a) and (b) illustrate the variation of crack growth rate, da/dN, as a
function of the stress intensity range, ∆K , and the effective stress intensity
range, ∆K���, respectively, for various CA loading conditions. To obtain the stress
134
intensity range and the effective stress intensity range, Tada, Paris and Irwin
(1985)’s stress intensity factor solution for a cracked plate of a finite width has
been used to account for the finite width of the specimen. The solution is given by
the following equation:
K� 9 σ√πa �sec πa2W¡�/�� [1 : 0.025 aW¡� < 0.06 aW¡¦` (6.6)
where W is the half width of the specimen. The growth rate against the stress
intensity range data shows a consistent disparity between different load ratios over
the whole stress intensity range values. In contrast, if the crack growth data is
plotted against the effective stress intensity range, it displays much less scatter
between different load ratios. It has now been demonstrated based on the present
calculations that the use of the effective stress intensity ranges significantly reduces
the scatter in the growth rate data obtained from experimental test results. Figure
6.3 (b) shows that the scatter becomes slightly higher at higher values of ∆K���. The
overall relation between the rate and the effective stress intensity range, determined
based on the figure, is also shown in Table 6.1.
135
(a)
Figure 6.3 Crack growth rate against (a) stress intensity range and (b) effective
stress intensity range.
1.0E-09
1.0E-07
1.0E-05
1 10 100∆K §MPa√m¨
51.7 0.05
51.7 -1
69.0 0.05
69.0 -1
138.0 0.05
σ�"# §MPa¨ R da/dN §m/cycle¨
136
(b)
Figure 6.3 Continued.
1.0E-09
1.0E-07
1.0E-05
1 10 100∆K��� §MPa√m¨
51.7 0.05
51.7 -1
69.0 0.05
69.0 -1
138.0 0.05
σ�"# §MPa¨ R da/dN §m/cycle¨
137
Table 6.1 Crack growth rate against stress intensity range relation for 2024-T3
alloy based on Newman Jr and Ruschau (2007)’s crack growth experiments and
present model.
∆K��� §MPa√m¨ da/dN §m/cycle¨ 6.00 2.00E-09
6.80 2.49E-08
7.08 6.50E-08
8.12 1.43E-07
9.87 2.52E-07
11.30 3.30E-07
13.04 4.80E-07
15.82 9.80E-07
19.40 2.13E-06
23.00 4.40E-06
34.00 3.90E-05
After the determination of the relationship between the crack growth rate
and effective stress intensity range, the transient crack growth model is used to
simulate Newman Jr and Ruschau (2007)’s experimental results. A table-look-up
approach, rather than a growth equation approach, was adopted in this simulation
of crack growth because it provides a better accuracy than a Paris type fitting curve.
In other words, in the growth model, an effective stress intensity range value at a
crack tip is first calculated at a given load cycle, and then the corresponding crack
growth rate is looked up in Table 6.1 to determine the incremental crack growth.
138
The variation of characteristic parameters of the crack closure model, such
as the lengths of the direct plasticity region, reverse plasticity region and crack
contact region as well as the crack opening stress, versus the crack length for four
different CA cyclic loading tests are shown in Figure 6.4. In each region 150
integration points were placed to ensure the convergence of the solution process,
which was evaluated and verified by increasing the number of the integration
points until the numerical results converge.
Figure 6.4 (a) and (b) show the lengths of the direct plastic region at the
maximum applied load and the reverse region at the minimum applied load,
respectively, as the crack grows under two maximum applied stress levels (σ�"# 951.7 MPa and 69.0 MPa) and two load ratios (R 9 :1 and 0.05). It can be seen
that the lengths of direct and reverse plasticity region increase with an increase in
the crack length due to a higher stress intensity level for a longer crack. In
particular, Figure 6.4 (b) demonstrates that the length of the reverse plastic region
is not only a function of the maximum applied load (stress) but also depends on the
load ratio as well. According to the dependences in this figure, a higher maximum
stress and a lower load ratio increase the length of the reverse plasticity region
ahead of the crack tip. This general trend is in line with the previous theoretical
finding by Codrington and Kotousov (2007a). The length of the reverse plasticity
region at the minimum applied stress may have a significant impact on crack
closure and opening behaviours (Broek 1986; Schijve 1962). This is because the
139
formation of a large zone of compressive stress ahead of the crack tips during the
unloading stage of cyclic loading has a large impact on the fracture controlling
parameters such as the effective stress intensity factor range.
Figure 6.4 (c) and (d) show the variation of the crack contact region at the
minimum stress level and the crack opening stress as a function of the crack length.
As shown in Figure 6.4 (c), the crack contact zone region is directly proportional to
the fatigue crack length. Furthermore, it can be seen in this figure that the load ratio
has a huge impact on the crack closure behaviour while the maximum applied
stress level shows a marginal effect on the crack closure at the minimum applied
stress. Especially, when R 9 :1, the difference between the curves for σ�"# 951.7 MPa and 69.0 MPa is almost not distinguishable because in these two cases
the crack is mostly closed for a significant part of the load cycle. This phenomenon
of the crack closure behaviour at relatively large negative load ratios (R = -1) was
also predicted by Rose and Wang (2001)’s analytical study conducted within plane
stress theory of elasticity.
Figure 6.4 (d) presents the variation of the crack opening stress as a
function of the crack length. This is the most important result from the conducted
computational analysis because the calculation of the crack growth rate is closely
associated with the effective stress intensity range, which is a function of the crack
opening stress. According to the figure, the crack opening stress values are
relatively high, and these values increase with an increase in the crack length.
140
Figure 6.4 (d) also shows that the maximum applied stress level and load ratio have
a similar effect on the variation of the crack opening stress. The theoretical
modelling predicts that an increase in the maximum applied stress under fixed load
ratio leads to an increase in the crack opening stress, resulting in a delay in the
crack opening during loading. This delayed crack opening, in turn, reduces the
crack growth driving force. In contrast to the case considered above, a decrease in
the load ratio under fixed maximum applied stress decreases the crack opening
stress. The conducted analysis elucidates the fact that the crack opening stress is
significantly affected by the load history, which has a large impact on the crack
growth predictions.
141
(a)
Figure 6.4 Calculated length of (a) the direct plastic region, (b) reverse plasticity
region, (c) crack contact region and (d) crack opening stress as a function of crack
length for different CA cyclic loading.
0
0.02
0.04
0.06
5 10 15 20 25 30Half crack length, a §mm¨
Direct pl
astic zone
size,w§m
m¨ σ�"# 9 51.7 MPa σ�"# 9 69.0 MPa
142
(b)
(c)
Figure 6.4 Continued.
0
0.01
0.02
5 10 15 20 25 30
0
1
2
5 10 15 20 25 30
R 9 :1
R 9 0.05
σ�"# 9 51.7 MPa σ�"# 9 69.0 MPa
Half crack length, a [mm]
Compres
sive plast
ic zone si
ze,w�§mm
¨
R 9 0.05
R 9 :1
σ�"# 9 51.7 MPa σ�"# 9 69.0 MPa
Half cont
act zone s
ize,β§mm
¨
Half crack length, a [mm]
143
(d)
Figure 6.4 Continued.
The predicted as well as measured crack lengths versus the accumulated
number of cycles are presented in Figure 6.5 for the considered load ratios and
maximum applied stress levels. The theoretical results are plotted as solid lines
while the experimental data is represented by symbols. Taking into account typical
scatter in fatigue crack growth experiments, a good agreement is observed between
the test results and the model predictions. The present model provides a
conservative evaluation of the crack growth for all CA cyclic loading cases
considered except for the case where σ�"# 9 69 MPa and R 9 :1. Overall, the
validation study outlined in this section provides a reasonable confidence in the
0
0.2
0.4
0.6
5 10 15 20 25 30
σ�"# 9 51.7 MPa σ�"# 9 69.0 MPa
R 9 0.05
R 9 :1
σ'( σ�"#
Half crack length, a [mm]
144
developed theoretical approach for the fatigue growth prediction, however more
validation results will also be presented in Chapters 7 and 8. The results will be
based on the experimental data obtained by the author, specifically for the case of
interactive cracks.
Figure 6.5 Comparison between measured(Newman Jr & Ruschau 2007) and
predicted fatigue crack growth using current model.
0
10
20
30
0.00E+00 5.00E+04 1.00E+05 1.50E+05
No. of cycles, N [cycles]
Hal
f cr
ack
len
gth
, a
[mm
]
51.7 0.05
51.7 -1
69.0 0.05
69.0 -1
138.0 0.05
σ�"# §MPa¨ R
symbol : measured by Newman et al.
symbol : predicted by present model
: predicted by Newman et al.
145
6.4 Effect of Crack Interaction and Plate Thickness on Fatigue
Behaviour
6.4.1 Fatigue crack growth prediction for two collinear cracks
In this section the developed life prediction code, which is based on the
crack closure model, is used to evaluate the fatigue crack growth of two collinear
cracks in an infinite plate subjected to remote uni-axial loading. The crack growth
calculations are run until plastic collapse of the ligament between the two
neighbouring cracks occurs. The material constants in the present study are the
same as those described in Section 6.3. The load cases are limited to four CA cyclic
load combinations: σ�"# 9 51.7 MPa, and 69.0 MPa with R 9 :1 and 0.05. The
initial configuration of the cracks is selected to produce sufficient crack interaction.
Half of the initial centre-to-centre crack distance is set to d 9 18.3 mm, which
corresponds to ratio a/d 9 0.5. In the numerical simulations, the inner and outer
crack geometry parameters as well as the inner and outer opening stresses are
calculated for each load cycle, so that the inner crack size, a��, and the outer crack
size, a'+�, are determined separately at each cyclic load step (see Figure 6.1(b)). In
order to demonstrate the effect of interactions between two collinear cracks, the
non-interactive crack growth predictions are also performed for a single isolated
centre crack of the same length as the two collinear cracks. The three-dimensional
transient model taking into account the plate finite thickness effect is utilised in
these simulations.
146
Selected results of the simulations are presented in Figure 6.6 and Figure
6.7. In Figure 6.6, the variation of the crack geometry parameters, opening stresses
and effective stress intensity ranges at the inner and outer crack tips of two
collinear cracks are given as a function of the crack length. The theoretical results
correspond to the CA loading case of σ�"# 9 51.7MPa with R 9 0.05. The crack
geometry parameters include the lengths of the direct plasticity region, reverse
plasticity region and crack contact region. This figure also presents results for a
non-interactive growing crack for the purpose of comparison, revealing the crack
interaction effect. The abscissas of the graphs represent the average of a�� and a'+� when the results for two collinear cracks are given.
According to Figure 6.6 (a)-(d), the plasticity regions, crack contact zones
and the opening stresses at the inner and outer tips of the collinear cracks show a
linear and monotonic increase, at the early stage of the fatigue growth, similar to
the single (or non-interacting) crack behaviour. This can be explained by a weak
interaction between cracks due to a relatively wide initial separation gap selected in
these simulations. However, as the cracks advance and approach each other, the
inner crack tips are significantly affected by the interaction and start to show a
sharp acceleration in the fatigue crack growth rates. This signifies the effect of the
crack interaction. A similar behaviour is expected for other types of interactive
MSD. In addition, it can be seen from the crack opening stress history in Figure 6.6
(d) that the inner crack tips have a higher opening stress value than the outer tips.
147
This interesting phenomenon can be explained on the basis of the plasticity induced
crack closure concept. The effect of crack interaction causes a larger tensile plastic
zone in the front of a growing crack, as shown in Figure 6.6 (a). The larger tensile
plastic zone, in turn, induces the development of thicker plastic wake on the crack
faces at the inner crack tip, resulting in an increased stress level needed for
complete opening of the crack. The higher opening stress at the inner tip can also
be due to the larger length of the reverse plasticity region, and hence, a greater
extent of crack contact close to the inner crack tip, which are illustrated in Figure
6.6 (b) and (c). The disparity between the inner and outer opening stress values is
small (as mentioned before) and can be neglected at the initial stages of fatigue
crack growth, but the divergence between these values increases dramatically as the
two cracks approach each other. Paradoxically, this rapid increase in the crack
opening stress at the inner crack tip, due to the strong crack interaction effect, is
expected to reduce the growth rate at the inner crack tip, which will otherwise be
extremely high.
Based on the crack opening stress values presented in Figure 6.6 (d) and the
stress intensity factor equations (6.3a), the effective stress intensity range, ∆K���, which is the crack growth driving force in the current theoretical model, can now
be determined as demonstrated in Figure 6.6 (e). Equations (6.3) show that the
effective stress intensity range for two collinear cracks of identical length in an
infinite plate is a function of the crack opening stress and the crack geometry
148
factors, such as a and d. An increase in a and a decrease in d, which is the result of
the fatigue crack growth of collinear cracks, increase ∆K��� . However, an increase
in the opening stress, which is also the result of the fatigue crack growth, reduces
∆K���. It can now be clearly seen that there is a competition between these two
mechanisms affecting ∆K��� in the opposite directions as the collinear cracks grow
toward each other. Figure 6.6 (e) shows that the effect of crack geometry factors
overpowers the influence of the opening stress on ∆K���. From the initial crack
length to the final plastic collapse, ∆K��� at the inner crack tip has a consistently
higher value than that at the outer tip, and the divergence between these values
increases as the cracks propagate and the crack interaction starts to dominate. A
comparison with the non-interactive crack results (Figure 6.6 (a)-(e)) provides
further insight into the effect of crack interaction on fatigue crack growth and
demonstrates its significance in crack growth evaluation and lifetime assessments.
149
(a)
Figure 6.6 Comparison of calculated variations of (a) length of direct plasticity
region, (b) length of reverse plasticity region, (c) length of crack contact, (d) crack
opening stress and (e) effective stress intensity range as a function of crack length
for two interacting collinear cracks of equal length and a non-interactive single
crack in an infinite plate under σ�"# 9 51.7 and R 9 0.05 (initial a/d=0.5).
0
0.2
0.4
0.6
9 12 15Half crack length, a or �a�� < a'+��/2 §mm¨
Length of
direct pl
asticity re
gion§mm
¨
inner tip (w��)
outer tip (w'+�) non-interactive crack (w)
150
(b)
(c)
Figure 6.6 Continued.
0
0.02
0.04
0.06
9 12 15
0
1
2
3
9 12 15
Half lengt
h of crack
contact§m
m¨
Half crack length, a or �a�� < a'+��/2 §mm¨
inner tip (�)
outer tip (β'+�) non-interactive crack (β)
Half crack length, a or �a�� < a'+��/2 §mm¨
Length of
reverse p
lasticity re
gion§mm
¨ inner tip (w���)
outer tip (w�'+�) non-interactive crack (w�)
151
(d)
(e)
Figure 6.6 Continued.
0
0.2
0.4
0.6
9 12 15
0
5
10
15
9 12 15
Half crack length, a or �a�� < a'+��/2 §mm¨
σ'( σ�"# inner tip
outer tip
non-interactive crack
Effective s
tress inte
nsity rang
e§MPa √m
¨
inner tip
outer tip
non-interactive crack
Half crack length, a or �a�� < a'+��/2 §mm¨
152
Figure 6.7 presents the results of the crack length against the accumulative
number of load cycles for two cases: (1) two collinear cracks of equal length, and
(2) single crack for two levels of the maximum applied stress and two values of the
stress ratio. The calculations were carried out until the inner plastic zones touch
each other, resulting in local plastic collapse or crack coalescence. As it can be
observed from this figure, for all loading cases considered, that the inner crack tip
shows a higher growth rate than the outer tip due to the interaction effect resulting
in a higher ∆K��� at the inner crack tip. A comparison with the results for a single
crack demonstrates that the strong influence of the interaction between the
neighbouring cracks can be quite dominant, in terms of crack growth rates. The
present results contradict Tan and Chen (2013)’s recent theoretical study on the
growth of two coplanar short cracks, which indicated a negligible crack interaction
before the coalescence of them. Additionally, the comparison between the
interactive and non-interactive crack growth results appears to indicate a much
higher crack interaction at a lower level of the applied loading. The latter effect can
explain the discrepancies with the theoretical study of Tan and Chen (2013).
153
(a)
Figure 6.7 Comparison of predicted crack growth curves for two interacting
collinear cracks of equal length and a non-interactive single crack in an infinite
plate under various CA loading (initial a/d=0.5).
0
10
20
0.0E+00 5.0E+04 1.0E+05
No. of cycles, N [cycles]
inner tip (a���
outer tip (a'+��
non-interactive crack (a)
σ�"# 9 51.7 MPa R 9 0.05
Hal
f cr
ack
len
gth
, a
[mm
]
154
(b)
(c)
Figure 6.7 Continued.
0
10
20
0.0E+00 5.0E+04 1.0E+05
0
10
20
0.0E+00 2.0E+04 4.0E+04
No. of cycles, N [cycles]
No. of cycles, N [cycles]
inner tip (a���
outer tip (a'+��
non-interactive crack (a)
σ�"# 9 51.7 MPa R 9 :1
inner tip (a���
outer tip (a'+��
non-interactive crack (a)
σ�"# 9 69.0 MPa R 9 0.05
Hal
f cr
ack
len
gth
, a
[mm
] H
alf
crac
k l
eng
th,
a [m
m]
155
(d)
Figure 6.7 Continued.
6.4.2 The influence of plate thickness on fatigue crack growth of two
collinear cracks
To examine the effects of plate thickness on the fatigue crack growth
behaviour of two collinear interacting cracks, numerical simulations were carried
out for different plate thicknesses at fixed CA loading case of σ�"# 9 51.7MPa
and R 9 0.05 . The specimen geometry and material are identical to those in
section 6.4.1 except that the plates have various thicknesses of 1.15, 2.3 and 4.6
mm. Figure 6.8 shows the predicted crack opening stress and fatigue crack growth
0
10
20
0.0E+00 2.0E+04 4.0E+04
No. of cycles, N [cycles]
inner tip (a���
outer tip (a'+��
non-interactive crack (a)
σ�"# 9 69.0 MPa R 9 :1
Hal
f cr
ack
len
gth
, a
[mm
]
156
results for two collinear cracks of equal length in plates of different thicknesses. As
shown in Figure 6.8 (a), the impact of plate thickness on the crack opening
behaviour is rather significant. The developed three-dimensional crack growth
model predicts a decrease in the crack opening stress with an increase in the plate
thickness. This tendency is the same for the inner and outer crack tips, and it is in
agreement with many previous theoretical and experimental findings conducted
with single or non-interactive cracks (Codrington & Kotousov 2009a; Costa &
Ferreira 1998; de Matos & Nowell 2009; Guo, Wang & Rose 1999; Newman Jr
1998). This strong plate thickness effect on the fatigue crack behaviour can be
explained based on the plasticity induced crack closure concept. An increase in
plate thickness causes an increase in the out-of-plane constraint in the vicinity of
the crack tip, resulting in a smaller size (length) of the plastic zone. This reduction
in the size of the plastic zone, in turn, decreases the thickness of plastic wake on
the crack faces, eventually leading to a reduction of the crack opening stress. In this
figure, it can be observed that there is a very rapid increase in the opening stress at
the inner crack tip close to the point of the coalescence (plastic collapse) for all the
three different plate thicknesses. A closer examination reveals that the different
plate thicknesses induce the crack interaction effect at different stages (lengths) of
fatigue crack growth. The vertical dotted lines in Figure 6.8 (a) represent the
locations where the opening stress difference between the inner and outer crack tips
is 5%. As can be seen from this figure the thinner plates develop the crack
157
interaction at earlier stages of fatigue crack growth, i.e. after a smaller amount of
crack growth. The corresponding fatigue crack growth curves are presented in
Figure 6.8 (b). In this figure the dotted line represents the inner or outer half crack
length, a�� or a'+�, and the solid line is the average of a�� and a'+�. In accordance
with these dependences, the inner crack tip exhibits consistently higher growth
rates than the outer tip, throughout its fatigue lifetime for all three plate thicknesses
considered. The fatigue crack growth rates in thicker plates are predicted to be
higher than those in thinner plates under the same CA loading. This difference can
be readily understood from the crack opening curves shown in Figure 6.8 (a). In
this figure, lower crack opening stresses and thus higher effective stress intensity
ranges correspond to thicker plates. The overall dependency of crack growth on the
plate thickness predicted by the present three-dimensional transient crack growth
model is in an excellent agreement with past experimental test results conducted
with non-interactive cracks (Bhuyan & Vosikovsky 1989; Costa & Ferreira 1998;
Park & Lee 2000). Furthermore, these general dependencies found from the
theoretical modelling will be also supported by experimental studies conducted by
the author and to be presented in Chapter 8.
158
(a)
Figure 6.8 Predicted (a) crack opening stress and (b) crack growth for two
interacting collinear cracks of equal length (initial a/d=0.5) in an infinite plate of
different thickness (thickness=1.15, 2.3 and 4.6 mm).
0.2
0.3
0.4
0.5
9 12 15
σ�"# 9 51.7 MPa R 9 0.05
Plate thickness
increasing
inner tip
outer tip
Half crack length, �a�� < a'+��/2 §mm¨
σ'( σ�"#
5% difference in
opening stress b/w
inner and outer tips
159
(b)
Figure 6.8 Continued.
6.5 Conclusions
The stationary crack closure model outlined in Chapter 5 was extended in
this chapter to develop a new three-dimensional transient crack growth model to
analyse the fatigue behaviour of interacting collinear cracks in a plate of finite
thickness. The model was validated using past experimental studies and crack
growth results obtained from isolated (non-interacting) cracks.
0
10
20
0.00E+00 5.00E+04 1.00E+05 1.50E+05
σ�"# 9 51.7 MPa R 9 0.05
Plate thickness
increasing
inner tip (a���
outer tip (a'+��
average (�a�� < a'+��/2)
No. of cycles, N [cycles]
Hal
f cr
ack
len
gth
, a
[mm
]
160
It was demonstrated based on the present model that the use of the crack
closure concept can significantly reduce the scatter in the fatigue growth rates
versus the effective stress intensity range data obtained from various load
conditions. It was observed that the fatigue crack growth predictions, which were
made based on the developed model and numerical procedure, agree well with the
selected experimental results in terms of lifetime evaluation and assessment of the
fatigue crack growth at various loading conditions.
The developed theoretical model was utilised to investigate the nonlinear
effects of the crack interaction and the plate thickness on the crack opening stress
and fatigue behaviour of closely spaced cracks. Based on this model, the presence
of significant crack interaction was predicted at the inner crack tip. It was
demonstrated that these interaction effects impact the lengths of the direct and
reverse plastic regions, crack contact region and crack opening stress as well as
crack growth rate. There were two competitive mechanisms identified that
influence the effective stress intensity factor and acceleration of fatigue crack
growth at the inner crack tip. They are: a change of the crack geometry and an
increase of plastic region length with fatigue crack growth. The former mechanism
leads to higher crack growth rates at the inner crack tip when cracks approach each
other while the latter mechanism increases the crack opening stress reducing the
effective stress intensity factor range at the same crack tip. For all the considered
161
cases, the first mechanism overpowered the second one resulting in an accelerated
crack growth rate as compared to the single crack case.
Finally, the effects of the plate thickness on the growth behaviour of two
collinear interacting cracks were studied using the present theoretical and
numerical model. The impact of the plate thickness on the crack opening, and thus
the crack growth behaviour, was notably significant. The fatigue growth rates for
thicker plates were predicted to be higher than those of thinner ones at the same
crack geometry and loading. This trend was observed for the inner as well as for
the outer crack tips. The overall dependency of crack growth on plate thickness
predicted by the present model is in an excellent agreement with past experimental
studies conducted predominantly for non-interactive cracks. Furthermore, it was
shown from the modelling results that thinner plates weakened by two collinear
cracks under fatigue loading promote the crack interaction effects at earlier stages,
i.e. after a relatively smaller amount of crack advance than the corresponding
thicker plates. This can be explained by the fact that the crack tip region in thinner
plates experiences larger plastic deformations than a crack tip region in thicker
plates.
162
163
Chapter 7
7 Experimental Study of Plastic Collapse of the Ligament
between Two Collinear Cracks
7.1 Introduction
MSD can undermine the overall strength and integrity. Due to the presence of
mutual interaction between multiple cracks, the residual strength of structures with
MSD can be significantly lower than those of structures with non-interactive cracks
(Koolloos et al. 2001; Swift 1994). In Chapter 4, local plastic collapse conditions
have been investigated theoretically based on the classical strip yield model. In
particular, it was found that a reduction in the thickness of damaged plates can
induce the local plastic collapse at a substantially lower applied stress level. The
three-dimensional modelling predicted a strong influence of the crack interaction,
material properties as well as the plate thickness on the local plastic collapse
conditions.
In this chapter, the effect of the crack interaction and the plate thickness on
plastic collapse of the ligament between two cracks will now be investigated
experimentally. The experimental results will also be utilised to verify the
theoretical approach and theoretical findings described in Chapter 4.
164
7.2 Experimental Approach
Fracture tests were performed to investigate the local plastic collapse
phenomenon of the ligament between two collinear cracks subjected to remote
loading. The assessment of the collapse conditions is based on the plastic zone
coalescence criterion, which will be described later in this chapter. The tests were
carried out in the LT orientation of specimens with respect to the rolling direction.
The following sections provide details of the fracture tests and testing methodology.
7.2.1 Material property test and specimen preparation
The material used for the current study is aluminium alloy 5005 sheet. This
is one of the most commonly used aluminium alloys, and it is prominent for its
high corrosion resistance, good machinability and reasonably good mechanical
properties. In accordance with ASTM E8M-04, material property tests were
initially conducted on standard coupons cut out from the plates in the same
orientation as to be used in the investigations of the local collapse conditions. The
measured values for Young’s modulus E, 0.2% yield strength σ), corresponding
yield strain ¬) and ultimate strength σ+ were 62 GPa, 132 MPa, 0.0041 and 152
MPa, respectively. These values were obtained using an Instron tensile machine
equipped with an extensometer. Further, six specimens in total were machined by
using the water jet cutting technology to avoid any possible formation of heat
damage. Each specimen contained two collinear slits of equal length of 10 mm and
width of 2 mm. In order to investigate the crack interaction and plate thickness
165
effects, six types of specimens were fabricated with three plate thicknesses (2h =
1.2, 2.0 and 3.0mm) and two centre-to-centre distances of collinear slits (2d = 20
and 25 mm). The specimen design is shown in Figure 7.1. To produce sharp cracks,
the slits were notched with a 0.5 mm thickness saw, and after that, pre-cracked
using a constant amplitude cyclic loading with σ�"#/σ) 9 0.3, R 9 0.05 and 5 Hz
as recommended by the ASTM standard. The measured fatigue crack lengths were
typically 0.2~1.3 mm. The specimen preparation process was accomplished by
attaching a strain gauge (FLG-1-23, gauge width 1.1mm) in the middle of the
ligament between the two collinear cracks (see Figure 7.1). The role of the strain
gauge was to measure the applied tensile strain level and identify the plastic
collapse conditions of the ligament based on the plastic zone coalescence criterion.
In other words, the local plastic collapse conditions were assumed to occur when
the strain level in the middle of the ligament reaches the yield strain of the material
measured from coupon tests. The specimen dimensions after fabrication and pre-
cracking are shown in Table 7.1.
166
Figure 7.1 Test specimen with two collinear cracks equipped with strain gauge.
Table 7.1 Specimen dimensions after saw-cutting and pre-cracking (unit: mm).
Specimen No. 2h 2a1 2a2 2d
P1 1.2 11.97 11.86 24.90
P2 2 11.6 11.86 25.16
P3 3 13.13 12.75 24.31
P4 1.2 12.78 12.88 19.69
P5 2 12.64 12.55 19.86
P6 3 11.03 12.15 19.53
Strain gauge
120 mm
220 mm
200 mm
2a1
2d
2a2
Crack1 Crack2
2h
Grip area
Grip area
167
7.2.2 Plastic collapse testing
The plastic collapse tests were carried out under displacement control
(elongation rate of 10 mm/min) using an Instron 1342 hydraulic machine (see
Figure 7.2). Each specimen was stretched under tensile quasi-static loading until
failure. The plastic collapse of the ligament is specified when the plastically
deformed regions developed at the inner crack tips expand toward the center of the
ligament and come into contact. Accordingly, strain values were monitored at the
centre of the ligament while the tensile loading increases. When the strain reached
the yield strain of the material, the corresponding remotely applied stress value
(net-section stress) was considered as the plastic collapse stress for the ligament
yielding. The development of plastically deformed regions was visible with the
naked eye. In all six specimens, no crack growth was observed. This was because
the plastic collapse stress was below the critical applied stress needed for fracture
initiation or sub-critical crack growth.
168
Figure 7.2 Instron 1342 with a test sample.
169
7.3 Results and Discussion
The results of the plastic collapse tests are shown in Figure 7.3. The results
presented in this figure were normalised by the flow stress, or �σ) < σ+�/2, of the
material, and the normalised plastic collapse stress, σ(M /σ�, was plotted against the
ratio of crack length to centre-to-centre distance of cracks, a/d , which is an
indicator to the crack interaction effect.
The variation of plastic collapse stress presented in Figure 7.3 is the result of
the combined effects of the crack interaction and plate thickness, which are
characterised by different a/d and h/a.
It is virtually impossible to fabricate test specimens with the same geometry
as the pre-cracking always produces some scatter. To investigate the effect of crack
interaction (or influence of a/d on the collapse stress) the data points which have a
similar h/a value are paired and connected with a dotted line in Figure 7.3. Due to a
small number of tests the dependences largely provide a qualitative assessment of
the crack interaction effect. The shift of the dotted lines highlighted by an arrow in
Figure 7.3 demonstrates the effect of plate thickness (or influence of h/a) on the
local plastic collapse conditions. This figure also shows the effect of crack
interaction on a plastic collapse stress. For the tested specimen geometries, on
average, a drop of 20% in the plastic collapse stress was measured with an increase
in a/d from 0.49 to 0.62 (27% increase). Overall, the plastic ligament collapse
conditions were found to be highly dependent on both the crack interaction and the
170
plate thickness. These trends have also been predicted by the three-dimensional
strip yield model for two collinear cracks, which was presented in Chapter 4.
Figure 7.3 Measured plastic collapse stresses of six specimens having two collinear
cracks.
A comparison between the test results and theoretical predictions made
using the theoretical model developed in Chapter 4 is shown in Figure 7.4. In this
figure, the theoretical predictions (lines) of the plastic collapse stress normalised by
0.5
0.6
0.7
0.8
0.4 0.5 0.6 0.7
h/a increasing
Spe. No h/a
P1 0.10
P2 0.17
P3 0.23
P4 0.09
P5 0.16
P6 0.26
a/d increasing
a/d
σ(M σ�
171
the flow stress, σ(M /σ�, as a function of the ratio of crack length to centre-to-centre
distance of cracks, a/d, are presented for four different plate thickness to half crack
length ratios. This figure also displays the plastic collapse test results (symbols).
The comparison reveals that the theoretical model leads to conservative estimates
of the plastic collapse stress of the specimens. The predicted values are
substantially lower than the corresponding test results with the relative error being
about 21%. These discrepancies are due to the use of a strip yield model, idealised
yield criterion and elastic-perfectly-plastic model of material behaviour. However,
it is highly noteworthy that the plastic collapse stress predictions show the same
trends as the experimental results. Furthermore, the differences between the
predictions and the experimental results are very consistent throughout the
measured data, as shown in Table 7.2. This can imply the usefulness of the
developed model if the disparity can be offset by employing an empirical value for
the flow stress, σ�, which is to some extent is a fitting parameter as highlighted in
the previous sections.
Accordingly, the concept of using a fitting value of the flow stress, σ�, in
normalising the measured plastic collapse stress was utilized in this chapter. The
introduced fitting flow stress in this context aims to compensate errors associated
with the yield strip idealisation, idealised yield conditions and elastic-perfectly-
plastic material behaviour. A remarkable reduction of the discrepancies between
the experimental and theoretical results can be observed with this new value of the
172
flow stress (177.5 MPA). The characteristic error is now around 1-2 % (see Table
7.2), which can be considered as an excellent agreement between theory and
experiment.
Figure 7.4 Calculated and measured plastic collapse stress levels against crack
length to separation gap ratio for different plate thicknesses.
0
0.5
1
0 0.5 1
h/a= plane stress,
0.3, 1.0, plane strain
Lines: prediction for various plate thickness
Solid symbols: test result
Line symbols: prediction corresponding to test
Spe. No h/a test pred.
P1 0.10
P2 0.17
P3 0.23
P4 0.09
P5 0.16
P6 0.26
σ(M σ�
a/d
173
Table 7.2 Predicted and measured plastic collapse stress to yield strength ratios.
Specimen No P1 P2 P3 P4 P5 P6
Prediction σ(M /σ� 0.60 0.61 0.58 0.45 0.47 0.53
Experiment
σ(M /σ� 0.75 0.77 0.73 0.56 0.60 0.65
Relative
error 20% 21% 21% 20% 22% 18%
σ(M /σ�® 0.60 0.62 0.58 0.45 0.48 0.52
Relative
error 0% 1% 1% 0% 2% -2%
7.4 Conclusions
An experimental program was developed for the investigation of the effect of
crack interaction and plate thickness on the local plastic conditions of plate
specimens having two collinear cracks subject to static tensile loading. The
measurement of residual collapse stress was conducted based on the plastic zone
coalescence criterion by utilising a strain gauge glued to the mid-point of the
ligament.
The qualitative and quantitative assessments of the crack interaction effect
were conducted, and demonstrated that the residual strength decreases dramatically
with an increase in the crack interaction. It was also proved that an increase in
three-dimensional constraint, due to an increase in plate thickness, leads to higher
174
residual strength in context of the susceptibility to plastic collapse. The test results
demonstrated the importance of taking into account the effect of interaction
between closely spaced cracks as well as the effect of three-dimensional plate
thickness in the failure and structural assessment of MSD. In particular, the
experimental results justified the need and importance of the development of three-
dimensional models and a limited applicability of numerical and analytical
calculations based on plane stress/strain simplified assumptions.
In order to verify the theoretical three-dimensional strip yield model
developed in Chapter 4, a comparison between experimental results and theoretical
prediction was conducted. The comparison revealed that the theoretical model
considerably underestimated the residual strength of the specimens containing two
collinear cracks. Even though the substantial offset between experimental results
and predictions, the model predicted exactly the same trends as observed in the
experiments. This signified its usefulness in assessing the coalescence and local
plastic collapse conditions with the aid of a fitting value of the flow stress. This
fitting value compensates the errors associated with various idealisations employed
in the theoretical modelling. A remarkable agreement between the theoretical
predictions and experimental results was observed if this fitting value of the flow
stress was used in the modelling approach. In conclusion, it can be stated that the
strip yield model combined with the best-fit flow stress is very effective in
predicting the residual strength of plates weakened by two cracks. It is expected
175
that the similar modeling approach will be also very effective in the analysis of
other types of MSD.
176
177
Chapter 8
8 Experimental Study of Fatigue Crack Growth of Two
Interacting Cracks
8.1 Introduction
The nonlinear crack interaction phenomena affect the local stress field and
fracture controlling parameters. Therefore, fatigue crack growth characteristics can
also be significantly influenced by the interacting cracks or other types of MSD.
Previous experimental studies revealed that the presence of closely spaced cracks
radically accelerates the crack growth rates, leading to a large reduction in the
fatigue life (Moukawsher, Grandt & Neussl 1996; Pártl & Schijve 1993; Silva et al.
2000). The same conclusion was also derived in theoretical studies (Collins &
Cartwright 1996; Tan & Chen 2013). Despite many investigations on the crack
interaction and fatigue behaviour of structural components weakened by MSD
being conducted in the past, the theoretical modelling and accurate prediction of
lifetime in these situations represents a challenging problem. Furthermore, the
influence of the plate thickness on the strength of the crack interaction effect has
not been previously addressed in the literature. The theoretical modelling results
presented in Chapters 5 and 6 demonstrated a strong influence of the plate
thickness and other three-dimensional effects on the crack closure as well as
fatigue growth of two collinear cracks. Therefore, a direct experimental validation
178
of the theoretical model as well as the confirmation of the predicted fatigue
behaviour attracts a strong interest.
In this chapter, the combined effect of the crack interaction and specimen
thickness on fatigue growth of closely spaced cracks will be studied experimentally
using plate specimens of various thickness and geometry. The experimental study
is limited to constant amplitude (CA) cyclic loading only. The effect of the variable
cycling loading including effect of an overload or load spike is beyond the scope of
the thesis and can be a focus of future work.
The inner and outer crack tip growth extensions/rates were measured
separately and a quantitative analysis of the crack interaction effect on fatigue
crack growth were made based on the analysis of fatigue crack growth at the inner
and outer crack tips. The outcomes of the experimental study are also utilised to
validate the three-dimensional fatigue crack growth model for two collinear cracks
of equal length developed and described in Chapter 6.
8.2 Experimental Study
Fatigue crack growth tests were conducted on flat plate specimens containing
two collinear cracks of very similar length. The material used in the current
experimental study was the same as the one used in the plastic collapse tests as
described in Chapter 7, i.e. aluminium alloy 5005. The basic material properties of
179
this material were provided in Chapter 7. The geometry of the test specimen is
shown in Figure 8.1, which was supported and verified with two-dimensional finite
element analysis (FEA). Details of FEA and verification study are omitted here.
The specimens were fabricated out of large bulk plates (all in the same LT-
orientation). After that, two collinear slits of equal length of 10 mm and width of 2
mm were cut using water jet. Then, the slits were sharpened with a 0.5 mm
thickness saw and pre-cracked prior to the fatigue growth testing. The fatigue pre-
cracking was carried out until each tip of the cracks extends by 0.2~0.5 mm under
the same loading conditions as these used for the fatigue growth testing. Two
fabricated specimens had different geometry including different thicknesses, 2h, as
well as centre-to-centre distance of cracks, 2d. The full geometry details of the
specimen geometries are presented in Table 8.1.
During the pre-cracking stage it was virtually impossible to ensure the final
length for each crack of the specimen. Therefore, Table 8.1 provides the average
length of the two final cracks after the pre-cracking stage of the fatigue test
preparation procedure. This length was used in the theoretical analysis of the
conducted fatigue tests with the developed three-dimensional fatigue crack growth
model.
180
Figure 8.1 Specimen geometry for fatigue test.
Table 8.1 Specimen dimensions for fatigue test (all dimensions are in mm).
Specimen No.
Plate
thickness
Half crack
length (average)
Half crack
spacing
Crack
interaction factor
2h a 9 a� < a�2 d a/d
F1 2 6.20 11.88 0.52
F2 3 6.32 9.84 0.64
120 mm
220 mm
200 mm
2d
Crack1 Crack2
2h
Grip area
Grip area
a�,�� a�,'+� a�,�� a�,'+� 2a� 2a�
181
The fatigue tests were conducted using an Instron 1342 hydraulic machine.
Each specimen was subjected to positive (tensile) cyclic loading with sinusoidal
waveform at 5 Hz at room temperature and humidity. The maximum applied load
and load ratio were kept constant at σ�"# /σ� 9 0.39 and R 9 0.05, respectively.
The collinear cracks were grown until their coalescence. The crack size at different
times was measured optically using handheld and desk microscopes. After a
sufficient increment in crack length, which was monitored with the aid of the
handheld microscope, the specimens were unloaded and examined with the more
precise desk microscope, which provided accuracy of crack length measurements
±0.01 mm. After these measurements, the fatigue testing was continued until
failure (local plastic collapse).
Each time when the specimen was unloaded, the sizes of the inner crack tips,
a�,�� and a�,�� , and the outer crack tips, a�,'+� and a�,'+� , were measured. The
average value of the inner crack size, a�� 9 �a�,�� < a�,���/2, and the outer crack
size, a'+� 9 �a�,'+� < a�,'+��/2, were also recorded and used to evaluate the crack
growth rates from the theoretical model. As the crack tips approached each other,
the measurement intervals were shortened to obtain a more detailed picture of the
crack growth curve specifically in the fast growth region.
182
8.3 Experimental Results and Discussion
The crack growth curves for specimens F1 and F2, which had different
thicknesses and initial centre-to-centre crack distances (see Table 8.1), are
presented in Figure 8.2. A faster crack growth was observed in specimen F2, which
was the thicker and had a smaller distance (separation) between cracks. The crack
growth at the inner crack tips in the both specimens was higher than at the outer
tips as this can be seen in the figure. As the two neighbouring cracks grew and
approached each other, the inner crack growth increases sharply while the outer
counterpart shows a monotonic increase during the fatigue testing. This is a clear
indication of the crack interaction effect on the fatigue crack growth. The crack
growth curves for each specimen and each crack tip show when and how much the
fatigue crack growth is influenced by the crack interaction.
183
Figure 8.2 Measured crack growth curves of two collinear cracks of similar length
under CA loading (σ�"# /σ� 9 0.39 and R 9 0.05).
The results presented in Figure 8.2 cannot be directly utilised to provide an
independent assessment of the plate thickness effect because the growth curves are
the result of the combined result of the plate thickness and crack interaction factors
influencing the fatigue behaviour. The assessment of these effects is possible with
the theoretical modelling of these tests, which will be presented next.
6
8
10
12
0.0E+00 2.0E+04 4.0E+04 6.0E+04 8.0E+04
No. of cycles, N [cycles]
Hal
f cr
ack
len
gth
, a �� o
r a '+� [m
m]
Spe. No. a
F1 a��
F1 a'+� F2 a��
F2 a'+�
184
8.4 Fatigue Crack Growth Modelling and Discussion
Fatigue crack growth analysis of the tests was carried out using the three-
dimensional fatigue crack growth model developed in Chapter 6. The crack growth
test results were converted into the standard crack growth data, i.e. growth rate
versus effective stress intensity range data, following the same procedure as used in
Section 6.3. To evaluate the growth data from the interacting crack, an additional
fatigue test using a centre crack specimen (2h = 2mm) was conducted. Figure 8.3
shows the fatigue crack growth rates versus the effective stress intensity range data
obtained from all fatigue tests and for each crack tip. It can be seen from this figure,
the use of the effective stress intensity range leads to a relatively narrow scatter in
the crack growth rate data presented in terms of the effective stress intensity factor
range. However, the difference in the growth rate between thicker and thinner
specimens is still noticeable. For example, specimen F2 (the thicker) displays
higher growth rates than specimen F1 (the thinner) and the single centre cracked
specimen. Based on the fatigue data obtained, da/dN versus ∆K��� baseline relation
was determined (see Table 8.2).
Figure 8.3 Crack growth data from the present tests on specimens with a single
crack or two collinear cracks (
1.0E-08
1.0E-07
1.0E-06
4
da/dN §m/cycle¨
Crack growth data from the present tests on specimens with a single
crack or two collinear cracks (σ�"# /σ� 9 0.39 and R 9 0∆K��� §MPa√m¨ 8
Spe. No. crack tip
F1 inner
F1 outer
F2 inner
F2 outer
Single cracked spe.
185
Crack growth data from the present tests on specimens with a single 0.05).
12
186
Table 8.2 Crack growth rate against stress intensity range relation for aluminium
alloy 5005 based on the present tests and model.
∆K��� §MPa√m¨ da/dN §m/cycle¨ 5.5 2.70E-08
6 3.89E-08
6.5 5.45E-08
7 7.44E-08
7.5 9.94E-08
8 1.31E-07
8.5 1.80E-07
9 2.30E-07
9.5 3.00E-07
10 4.40E-07
The calculation of the crack opening stresses is essential in predicting crack
growth rates because the effective stress intensity range, which is also the crack
growth driving force, is a function of the crack opening stress. The effective stress
intensity range is a function of the crack interaction factor, a/d , as well, in
accordance with equations (6.3). Figure 8.4 presents the histories of crack opening
stress and crack the interaction factor of specimens F1 and F2 subject to CA cyclic
loading as specified in the previous section. For both specimens, higher opening
stress values were predicted at the inner crack tip than the outer one. More
specifically, the crack opening stress curves for the outer crack tips increased only
slightly and monotonically over a wide range of the crack length increments while
the fatigue behaviour for the inner tips showed a considerable increase especially
187
when the cracks were close to coalescence. It can also be seen in the figure that
specimen F1 shows lower values in both a/d and σ'( , compared to specimen F2, at
a given crack length. According to equations (6.3), a reduction in a/d leads to a
decrease in the effective stress intensity ranges while a reduction in the opening
stress σ'( has the opposite effect. Thus, the actual crack growth behaviour is a
result of these two competitive mechanisms.
It is interesting to point out that specimen F1, as predicted from the
theoretical model, has a lower opening stresses compared to specimen F2 despite
that F1 is much thinner. This seemingly contradicts the findings of Chapter 6 and
the previous publications (Codrington & Kotousov 2009a; Costa & Ferreira 1998;
de Matos & Nowell 2009; Guo, Wang & Rose 1999; Newman Jr 1998) that thicker
plates have a lower opening stress for the same in-plane geometry and loading
conditions. However, it must be taken into account that the opening stress values
are influenced not only by the plate thickness effect but also by the crack
interaction. The latter overpowers the plate thickness effect for the considered
specimen geometry and loading conditions.
188
Figure 8.4 The calculation of the opening stress to maximum stress ratio and the
crack length to centre-to-centre crack distance ratio as a function of crack length
for specimens F1 and F2 (σ�"# /σ� 9 0.39 and R 9 0.05).
The corresponding crack growth modelling results for specimens F1 and F2
are shown in Figure 8.5 and Figure 8.6, respectively, along with the experimental
results given for comparison. The crack growth calculations were made until
plastic collapse of the ligament. In these figures, the inner crack tip growth rates
were higher than the outer tip growth rates although the higher opening stresses at
0
0.5
1
6 8 10 12Half crack length, a [mm]
a/d σ'( /σ�"# (inner tip)
σ'( /σ�"# (outer tip)
F1
F2
σ '( /σ �"#
or
a/d
the inner side were predicted. This is because the effect of the crack interaction
overpowered the effect of the opening stress and has a larger contribution to the
magnitude of ∆K��� at the inner crack tip. This predicted trend, i.e. higher crack
growth at the inner tip, is in line with the experimental finding.
As it can bee seen from the comparison, the model provided a conservative
estimate for the fatigue crack growth of specimen
overestimated the fatigue life of F2. In overall, the predicted results were in
reasonable agreement with the experimental results considering the complex nature
of fatigue crack growth phenomena and many challenges in the predictive
Figure 8.5 Measured and predicted crack growth of specimen F1.
5
10
15
0.0E+00
Hal
f cr
ack
len
gth
, a
[mm
]
the inner side were predicted. This is because the effect of the crack interaction
overpowered the effect of the opening stress and has a larger contribution to the
at the inner crack tip. This predicted trend, i.e. higher crack
growth at the inner tip, is in line with the experimental finding.
As it can bee seen from the comparison, the model provided a conservative
estimate for the fatigue crack growth of specimen F1 while it slightly
overestimated the fatigue life of F2. In overall, the predicted results were in
reasonable agreement with the experimental results considering the complex nature
of fatigue crack growth phenomena and many challenges in the predictive
Measured and predicted crack growth of specimen F1.
2.0E+04 4.0E+04 6.0E+04 8.0E+04
No. of cycles, N [cycles]
test pred. a�� a'+�
189
the inner side were predicted. This is because the effect of the crack interaction
overpowered the effect of the opening stress and has a larger contribution to the
at the inner crack tip. This predicted trend, i.e. higher crack
As it can bee seen from the comparison, the model provided a conservative
F1 while it slightly
overestimated the fatigue life of F2. In overall, the predicted results were in
reasonable agreement with the experimental results considering the complex nature
of fatigue crack growth phenomena and many challenges in the predictive models.
Measured and predicted crack growth of specimen F1.
8.0E+04
190
Figure 8.6 Measured and predicted crack growth of specimen F2.
8.5 Conclusions
An experimental study of fatigue crack growth has been conducted on
specimens of different thickness weakened by two collinear cracks.
outer crack tip growth rates
interaction and specimen thickness on the growth of closely spaced cracks has been
experimentally identified and analysed
A quantitative analysis of the interaction effect on fatigue crack growth
made based on a comparison between the inner and outer crack tip growth rates.
5
10
0.0E+00
a�� a'+�
Hal
f cr
ack
len
gth
, a
[mm
]
Measured and predicted crack growth of specimen F2.
An experimental study of fatigue crack growth has been conducted on
specimens of different thickness weakened by two collinear cracks. The inner and
outer crack tip growth rates were measured separately. The influence of the crack
thickness on the growth of closely spaced cracks has been
experimentally identified and analysed.
quantitative analysis of the interaction effect on fatigue crack growth
made based on a comparison between the inner and outer crack tip growth rates.
2.0E+04 4.0E+04
test pred.
No. of cycles, N [cycles]
Measured and predicted crack growth of specimen F2.
An experimental study of fatigue crack growth has been conducted on
The inner and
The influence of the crack
thickness on the growth of closely spaced cracks has been
quantitative analysis of the interaction effect on fatigue crack growth was
made based on a comparison between the inner and outer crack tip growth rates.
191
The comparison confirmed that the nonlinear interaction between cracks
considerably increases the fatigue crack growth rate, resulting in a reduction in
fatigue life of plates containing interactive collinear cracks. It was also found that
the interaction effect on crack growth was predominant at the inner crack tip, and
the crack growth at the outer crack tips was not as significantly affected as the
inner tips by the crack interaction. Similar conclusions can be extended to other
types of MSD.
The plate thickness effect played an important role in the fatigue growth of
two interacting cracks. It was observed experimentally that the crack growth rates
at the inner and outer crack tips of the thicker specimen were higher than the
corresponding rates of the thinner specimen. This trend was in line with past
theoretical and experimental data from non-interactive cracks (Codrington &
Kotousov 2009a; Costa & Ferreira 1998; de Matos & Nowell 2009; Guo, Wang &
Rose 1999; Newman Jr 1998). Therefore, the current results suggest that the plate
thickness effect in the analysis of fatigue crack growth of multiple interactive
cracks is an important factor, which can not be disregarded in fatigue calculations.
In the present study it was found that the plate thickness effect leads to earlier
crack opening, during loading for a thicker plate specimen. This highlights the
importance of the plasticity induced crack closure concept in explaining fatigue
behaviour. Furthermore, the strong relationship between the specimen thickness
and growth rate justifies the use of the three-dimensional crack growth model in the
192
current study. It is interesting to note that the fatigue life of a plate with two
collinear cracks decreases with increasing plate thickness under a fixed cyclic
loading conditions while the resistance of the same plate to local plastic collapse
increases with increasing plate thickness. In other words, thicker plates
experiencing MSD have lower fatigue life but higher strength against local plastic
collapse.
The fatigue tests were simulated using the developed three-dimensional
theoretical growth model for the purpose of validation. The growth rates versus the
effective stress intensity range were determined from various tests and samples.
The comparison has demonstrated that, overall, the theoretical predictions in the
case of interacting cracks have a reasonable agreement with the outcome of the
experimental study.
193
Chapter 9
9 Conclusions and Future Work
9.1 Introduction
To ensure safe and efficient operation of an engineering structure, it is
essential to be able to predict its integrity and service life accurately. There are
many factors, which can influence the integrity of the engineering structure, and
many types of structural damage, which can shorten the lifetime and undermine the
integrity. The present study is focused on the investigations of multiple site damage
(MSD) and the effect of the thickness on the strength and integrity of plate and
shell components. The interaction of MSD and the thickness effect significantly
complicates the fatigue and failure assessment of the components; however a
disregard of these factors can potentially lead to large errors in the failure
assessment and non-conservative evaluations of lifetime of the structure.
A thorough literature review confirmed that there are no crack growth models
which are readily available and practical for the analysis of interactive cracks in a
plate of finite thickness. A number of advanced crack growth models, primarily
based on PICC concept, are currently available in the literature and were reviewed
in the current work. These models are very robust and sufficiently accurate in the
evaluation of crack growth, lifetime assessment and prediction of various nonlinear
194
effects. However, these past models usually consider only non-interactive cracks
and ignore effects associated with the presence of MSD.
The primary objective of the study was to develop computationally efficient
and validated three-dimensional theoretical models for the evaluation of structural
integrity and lifetime of plates weakened by two collinear cracks. This last chapter
briefly outlines the main outcomes of the theoretical and experimental studies
undertaken in the present thesis to address this primary objective.
Recommendations for future work are also provided in the end of this chapter.
9.2 Analytical and Numerical Approach (Chapters 3-6)
The problem analysed in the current study represent a simple case of MSD,
i.e. two through-the-thickness collinear cracks of equal length. However the
general procedure and approach developed can be extended to more complicated
and more practical types of MSD. The theoretical modelling of the problem was
based on three-dimensional solutions rather than plane stress or plane strain
assumptions. The three-dimensional solutions incorporate many nonlinear effects,
such as plate thickness or Poisson’s effects. Loading conditions were restricted to
quasi-static or CA cyclic loading and opening fracture mode (or mode I). However,
more complicated types of loading can be readily incorporated into the fatigue
analysis, but this task was beyond the scope of the current study.
195
Chapter 3
Chapter 3 represents the first stage of this theoretical work. It is focused on
the two-dimensional modelling of two collinear stationary cracks under remote
quasi-static tensile loading. The purpose of this work was to develop a strip yield
model for two closely spaced cracks, which can be used for a simplified analysis of
the crack interaction effects. The governing integral equation was derived by
representing the crack opening displacement by an unknown density of edge
dislocations distributed along the crack. Two alternative approaches were applied
to find the solutions to this governing equation: inversion of Föppl integral
(analytical approach) and Gauss-Chebyshev quadrature method (numerical
approach). The application of the former approach is, however, restricted to plane
stress/strain formulations of the problems while the use of the latter approach can
be extended to the analysis of three-dimensional stress state and more complicated
types of loading, which was the subject of Chapter 4.
A rigorous analysis of the crack tip plasticity of the above formulated
problem was carried out within plane stress and plane strain assumptions. The
developed approaches produced practically identical results, which provided a
good way for validation. A significant crack interaction effect on the crack tip
plastic zone and crack tip opening displacement was observed. In particular, the
crack interaction effect led to considerably larger crack tip plasticity regions and
crack tip opening displacements primarily at the inner crack tips. The intensity of
196
the crack interaction effect was influenced not only by the geometry of the cracks
but also by the applied stress level. The smaller spacing between cracks and lower
applied stresses lead to a stronger crack interaction as predicted from the developed
model. The obtained results also showed a good agreement with previously
published studies for the same geometry and boundary conditions.
Chapter 4
In this chapter the two-dimensional strip yield model for the analysis of two
collinear stationary cracks was extended to accommodate the three-dimensional
effects. The analytical modelling of the three-dimensional problem was
accomplished by using the three-dimensional fundamental solution for an edge
dislocation in an infinite plate of finite thickness. As in the previous chapter,
Gauss-Chebyshev quadrature method was applied to obtain a numerical solution to
the governing integral equations with singular Cauchy kernel.
The three-dimensional strip yield model was utilised to investigate the
residual strength of plates containing two collinear cracks. The remotely applied
tensile stress causing the local plastic collapse of the ligament was calculated as a
function of the spacing between the cracks, yield stress as well as the plate
thickness. At the same in-plane geometry the thicker plates have a higher plastic
collapse stress. This behaviour is attributed to an increase in the out-of-plane
constraint and change of the yield conditions with an increase in the plate thickness.
A quantitative assessment of the local plastic collapse stress was presented as a
197
function of two dimensionless parameters related to the three-dimensional
geometry of the problem. The three-dimensional results were partly validated with
a past two-dimensional model for limiting cases of very thin plate (which is related
to plane stress conditions) or very thick plate (where the plane strain conditions
dominate).
The developed three-dimensional strip yield model for two collinear cracks
was also utilised to study the synergistic effects of the plate thickness and crack
interaction on the local plastic collapse. Based on the considered problem a general
conclusion can be made that thicker plates are less susceptible to the MSD than
thinner plates, and the disregard of the plate thickness can lead to large errors in the
assessment of structural integrity of plate components weakened by crack damage.
Chapter 5
The three-dimensional strip yield model developed in Chapter 4 was
combined with a crack closure model based on the PICC concept. The developed
model is capable of predicting crack closure/opening stresses, which are critical in
fatigue crack growth models, and evaluating the thickness effect on these
parameters. A steady-state self-similar crack growth was analysed in this chapter.
The development of a wake of plasticity on the crack faces was modelled according
to the linearly increasing plastic wake hypothesis, which was employed in past
studies. Another important simplification adopted in the model is that the out-of-
198
plane constraint was assumed to be negligible during the compressive stage of
cycling loading.
The reverse yielding phenomenon at the crack tips was investigated with
the model. Reverse yielding plays an important role in determining the crack
opening stress, which is defined as the applied stress at which a crack tip is fully
open. The prediction of the opening stress was evaluated, based on the developed
model, separately at the inner and outer crack tips.
Further, the effects of the crack interaction and plate thickness on the crack
closure/opening behaviour were investigated. In the beginning, as a validation of
the developed model, a single crack problem was modelled. The obtained results
showed a considerable dependency of the crack closure behaviour on the load ratio
as well as the maximum applied stress level. These obtained results were also
compared with analytical results from the literature, demonstrating good agreement.
Next, it was shown that the both effects (the crack interaction and plate thickness)
have a considerable influence on the crack opening in the case of two collinear
cracks. In particular, it was predicted that the opening stress at the inner crack tip
will be always larger than that at the outer tip. An increase in the plate thickness
leads to a significant decrease in the crack opening stresses at inner and outer crack
tips. All these trends predicted by theoretical modelling are in agreement with
previously published theoretical and experimental results obtained from non-
interactive crack cases.
199
Chapter 6
The steady state crack closure model developed in Chapter 5 was further
extended to consider the cycle-by-cycle accumulation of fatigue damage and
formation of the plastic wake in a plate weakened by two collinear cracks. In this
transient crack growth model, the plastic wake thickness for a newly increased
crack region was determined from the plastic stretch of the material around the
crack tip at the previous load cycle. Based on the history of plastic wake formation
and the load condition, the crack opening stresses at the inner and outer crack tips
were calculated to obtain the effective stress intensity ranges. Then, the crack
growth rates were determined using a lookup table containing the relationship
between the crack growth rate and the effective stress intensity range for a given
specific material.
The model was validated using past experimental studies conducted for
isolated cracks (no past studies have been found for interactive cracks). It was first
demonstrated based on the present model that the use of crack closure concept can
significantly reduce the scatter in the growth rates versus effective stress intensity
range data obtained from various geometries as well as various load conditions.
Next, the fatigue crack growth results were compared with the experimental data.
A good agreement was observed between the experiments and predictions.
Once the model was validated against selected previous studies, it was
utilised to investigate the effects of the crack interaction and plate thickness on the
200
crack opening and fatigue behaviour of closely spaced collinear cracks. The
combined effects of the crack interaction with the plate thickness were analysed.
The results showed that the presence of crack interaction significantly increases the
crack growth rate at the inner crack tips. It was reaffirmed based on this transient
growth model that the opening stress at the inner crack tip is greater than that at the
outer tip. Crack growth rates at the inner crack tip were however predicted to be
higher than those at the outer tip for different CA loading conditions. That was
attributed to an exponential rise in the stress intensity factor at the inner tip due to
the crack interaction, which overpowered the drop in the effective stress range due
to higher opening stress. The plate thickness was demonstrated to be a crucial
factor affecting the crack opening and thus crack propagation. The growth rates of
thicker plates with MSD were predicted to be higher than those of thinner plates.
The combined effect of the crack interaction and plate thickness was also analysed.
The drastic increase in the inner crack tip opening stress with respect to the crack
length, which is observed when two growing cracks approach each other, was
found to occur at much later stage for thicker plates. This phenomenon indicates
that for such plates an increase in crack opening stress has delayed influence on the
reduction of crack growth rates.
201
9.3 Experimental Approach (Chapters 7-8)
To support the theoretical findings and validate the developed models, an
experimental program was developed and carried out on aluminium plate
specimens weakened by two collinear cracks. The experimental study included
fracture and fatigue tests. The fracture condition, which caused the local plastic
collapse of the ligament between the cracks, was investigated by a new
experimental method. At the same time, the fatigue crack growth tests were
conducted following a quite standard approach.
Chapter 7
The effects of the crack interaction and plate thickness on the plastic
collapse of the ligament between two cracks were investigated using an
experimental technique developed based on the plastic zone coalescence criterion.
In this technique, the axial strain was monitored at the centre of the ligament with
an increase in tensile loading. When the strain value, measured by a strain gauge
attached in the middle of the ligament, reached the yield strain of the material, the
corresponding stress (net-section stress) was considered as the plastic collapse
stress for the ligament yielding.
In the tests, six pre-cracked specimens having various geometries were
fabricated to provide a wide range of data. The effect of the crack interaction on the
plastic collapse of the ligament was investigated first. On average, a drop of 20% in
202
the critical stress for the ligament failure was measured with an increase in the ratio
of crack length to centre-to-centre crack distance from 0.49 to 0.62 (27% increase).
The plate thickness effect was also found to be an influential factor. It was
observed that an increase in the plate thickness resulted in a higher plastic collapse
stress level. This was due to the change of the out-of-plane constraint around the
crack tip with an increase of the plate thickness as it was explained previously from
theoretical modelling. Furthermore, the experimental results were utilised to
validate the strip yield model developed in Chapter 4 for the prediction of the
plastic collapse of the ligament. A comparison between the experimental and
prediction results was conducted. The comparison revealed that the theoretical
model considerably underestimated the residual strength of the specimens
containing two collinear cracks if an average value of the plastic and ultimate
stresses is used in the model. Even though there was a substantial offset between
experimental results and predictions, the model predicted exactly the same trends
as observed in the experiments. However, when an experimental (fitting) value of
the flow stress is used in the model (to compensate a number of modelling
assumptions) a remarkable agreement between the theoretical predictions and
experimental results was observed. In conclusion, it can be stated that the
developed strip yield model is very effective in predicting the residual strength of
plates weakened by two cracks for a wide range of geometries and loading
conditions.
203
Chapter 8
Fatigue tests were conducted on flat specimens of different thicknesses with
two collinear cracks to investigate the influence of crack interaction and specimen
thickness on the growth rates of closely spaced cracks. A quantitative analysis of
the interaction effect on fatigue crack growth was made based on a comparison of
growth rates at the inner and outer crack tips. The comparison confirmed that the
interaction between cracks considerably increases the fatigue crack growth rates,
resulting in a reduced fatigue life of plates weakened by interactive collinear cracks.
This phenomenon was also predicted by the growth model developed in Chapter 6.
It was also demonstrated from the experimental study that the plate thickness plays
an important role in the fatigue growth of interacting cracks. The results indicated
that the crack growth rates at the inner and outer crack tips of the thicker specimen
were higher than the corresponding growth rates of the thinner specimen. This
trend was also in line with the theoretical predictions made in Chapter 6. Overall,
the theoretical predictions of fatigue crack growth in the case of the interacting
cracks showed a reasonable agreement (typical for such sort of studies) with the
experimental data.
9.4 Conclusions
Computationally efficient three-dimensional prediction models for the
evaluation of structural integrity and fatigue of plates containing two collinear
204
cracks were developed based on the strip yield model, the PICC concept and the
distributed dislocation technique. Furthermore, the three-dimensional plate
thickness effect has been modelled by employing the fundamental three-
dimensional solution for an edge dislocation. After careful validation study, the
predictive capabilities of the developed model were demonstrated by subsequent
experimental test results. In general, a good agreement was observed between the
test results and the predictions.
Throughout the undertaken theoretical and experimental studies, it was
shown that the crack interaction and the plate thickness have a significant impact
on the structural integrity of plates with mutually interacting cracks. It is interesting
that the presence of the crack interaction leads to a reduction in the residual
strength and fatigue lifetime; meanwhile the plate thickness effect leads to opposite
trends in the residual strength and fatigue lifetime, i.e. it leads to an increase in the
residual strength but to a reduced fatigue lifetime.
The theoretical predictions and measurement results indicated (as expected)
that the inner tips of the collinear cracks are more influenced by the crack
interaction than the outer tips. The fatigue growth at the inner tips under CA cyclic
loading was predicted and measured to be around 41% (specimen F1) and 45%
(specimen F2) larger, at the time of coalescence, than the growth at the outer tips.
Moreover, the influence of crack interaction on the size of the plasticity region and
crack opening stress was predicted to be stronger for thinner plates.
205
Despite that analytical modelling of complex problems (such as crack
problems) always relies on some radical assumptions and simplifications, such an
approach provides vital insight into the complex and nonlinear phenomena.
Analytical approaches can avoid many difficulties associated with numerical
modelling of fracture problems. More importantly, compared to the numerical
analysis, the results from an analytical approach can be independently reproduced.
This is very difficult to achieve with numerical modelling of fatigue crack growth
because many numerical parameters which are not directly connected with the
problem formulation can influence the results. These include mesh density, crack
advance scheme, contact conditions and identification of crack closure conditions.
The theoretical models developed in the research can be employed to develop
criteria for the interaction of multiple cracks. The developed models can be further
utilised to determine the effective dimensions characterising the interactive and
non-interactive crack configurations. Overall, the conducted theoretical and
experimental work may be considered as an initial study for problems with MSD. It
is believed the achieved outcomes can contribute to the understanding of the
complex nonlinear phenomena of crack interaction and plate thickness and hence to
the improvement in the damage tolerance design for engineering structures
subjected to MSD.
206
9.5 Future Work
Future investigation can be focused on the development of growth prediction
tools for more complicated MSD geometries. For example, the problem of a major
crack flanked by two small cracks can provide more practical insight into MSD
analysis. In addition, the analysis of the effect of the out-of-plane constraints can be
extended to a wider range of plate thicknesses. Another recommendation for future
research can be associated with analysis and modelling of non-steady state loading
conditions. The present three-dimensional crack growth model is restricted to
constant amplitude loading. However, real structures are more likely to be exposed
to various types of loading conditions such as overloads, underloads and spectrum
loading. Therefore, future work can also be focused on the incorporation of more
complicated load cases into the theoretical model, for example an overload cycle.
In this case, the crack opening levels at various crack growth stages should be
obtained through precise modelling of plastic wake formation after the application
of an overload cycle. The advanced models can help to theoretically investigate the
combined effects of loading conditions, geometry and crack interaction, leading to
the development of techniques capable to address the current challenges in fracture
mechanics, specifically, assessing the structural integrity of components with MSD.
Finally, the use of other fracture parameters, such as crack tip opening angle
(CTOA) or J-integral, can be considered in the analysis of large scale yielding
conditions.
207
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