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

i

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

ii

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;

iii

� 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

v

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.

vi

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.

vii

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.

viii

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

ix

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

xi

PICC Plasticity induced crack closure

RHS Right-hand-side

VA Variable amplitude

WFD Widespread fatigue damage

xii

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

xiii

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

xiv

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

xv

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

xvi

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

xvii

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.

2

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

3

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

4

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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